numerical methods for analyzing nonstationary dynamic...

Numerical methods for analyzing nonstationary dynamic economic models and their applications

Inna Tsener

http://www.ua.es/

http://www.eltallerdigital.com/

Departamento de Fundamentos del Analisis Economico

Facultad de Ciencias Economicas y Empresariales

Numerical methods for analyzingnonstationary dynamic economic models

and their applications

Inna Tseneraaaaaa

Memoria presentada para aspirar al grado deDOCTORA POR LA UNIVERSIDAD DE ALICANTE

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Mencion Doctora InternacionalDoctorado en Economıa Cuantitativa

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Dirigida por: Prof. Lilia Maliar, Prof. Serguei Maliar

To my parents

Acknowledgements

I would like to express my deep gratitude to my thesis supervisors Lilia Maliar andSerguei Maliar for the support and guidance during the development of this thesis. Theirconstructive suggestions and useful critiques improved this research work significantly.

I would like to thank the members of the Department of Economics and the partici-pants of its seminars for creating an optimal research environment. My grateful thanksare intended to former and present directors of the department Juan Mora and LolaCollado. I would also like to thank Jose Agullo, Pedro Albarran, Vadym Lepetyuk,Adam Sanjurjo and Francesco Turino for the advices and help provided during the lastyear of my doctoral studies.

I owe my deepest thanks to colleagues Nathan Carroll and Rafael Valero and friendsAnna Grigoriyeva, Elena Ognivtseva, Olesya Zarubegnova, Ekaterina Tkachenko, Dmit-riy Solodkiy and Andrey Trutnev who have always given me great help during theseyears.

I am deeply grateful to Fernando Garcia for encouragement and moral support attimes when it was needed. Finally, I am infinitely indebted to my parents and thankthem for being so patient.

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It is in our nature to explore,to reach out into the unknown.The only true failure would benot to explore at all.

“Shackleton’s Journey”

Contents

Resumen iii

Introduction xiii

1 A Tractable Framework for Analyzing a Class of Nonstationary MarkovModels 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 A class of nonstationary Markov economies . . . . . . . . . . . . . . . . 6

1.2.1 The stochastic environment . . . . . . . . . . . . . . . . . . . . 71.2.2 A nonstationary optimization problem . . . . . . . . . . . . . . 71.2.3 Assumptions about exogenous variable . . . . . . . . . . . . . . 81.2.4 Assumptions about the utility and production functions . . . . . 101.2.5 Optimal program . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Extended function path framework . . . . . . . . . . . . . . . . . . . . 121.3.1 Introducing extended function path framework . . . . . . . . . . 121.3.2 Theoretical foundations of EFP framework . . . . . . . . . . . . 16

1.4 Relation of EFP to the literature . . . . . . . . . . . . . . . . . . . . . 191.4.1 Early literature on stochastic growth models . . . . . . . . . . . 191.4.2 Methods constructing Markov decision functions . . . . . . . . . 191.4.3 Methods constructing a path for variables . . . . . . . . . . . . 21

1.5 Assessing EFP accuracy in a test model with balanced growth . . . . . 231.5.1 Implementation details of EFP . . . . . . . . . . . . . . . . . . 231.5.2 A comparison of four solution methods . . . . . . . . . . . . . . 24

1.6 Numerical analysis of nonstationary and unbalanced growth applications 301.6.1 Application 1: An unbalanced growth model with a CES produc-

tion function and capital-augmenting technological progress . . 301.6.2 Application 2: A nonstationary model with a parameter shift . . 331.6.3 Application 3: A nonstationary model with a parameter drift . . 371.6.4 Application 4: Calibrating a growth model with a parameter drift

to unbalanced U.S. data . . . . . . . . . . . . . . . . . . . . . . 401.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

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ii CONTENTS

2 Capital-Skill Complementarity Revisited: Can an Unbalanced GrowthModel Account for Unbalanced Growth Data 692.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 722.3 A macroeconomic model of capital-skill complementarity . . . . . . . . 762.4 Quantitative analysis of the production function . . . . . . . . . . . . . 79

2.4.1 Estimation of the parameters of the production functions . . . . 792.4.2 Implications for the skill premium . . . . . . . . . . . . . . . . . 81

2.5 General equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.5.1 Calibration and solution procedure . . . . . . . . . . . . . . . . 832.5.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 852.7 Appendix A: Data construction . . . . . . . . . . . . . . . . . . . . . . 882.8 Appendix B: Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 932.9 Appendix C: Decentralization . . . . . . . . . . . . . . . . . . . . . . . 94

3 Geometric Programming: Approaches for Solving Economic Models 993.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.2 Geometric programming . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.2.1 Geometric program . . . . . . . . . . . . . . . . . . . . . . . . . 1013.2.2 Complexity of GP and solution methods . . . . . . . . . . . . . 103

3.3 Applications in economics . . . . . . . . . . . . . . . . . . . . . . . . . 1043.3.1 Deterministic case . . . . . . . . . . . . . . . . . . . . . . . . . 1053.3.2 Stochastic case . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.3.3 Intratemporal conditions . . . . . . . . . . . . . . . . . . . . . . 110

3.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.4.1 Forward and backward shooting . . . . . . . . . . . . . . . . . . 1133.4.2 Fair and Taylor (1983) method . . . . . . . . . . . . . . . . . . 1143.4.3 Nonlinear model predictive control: finding a solution directly

from a maximization problem . . . . . . . . . . . . . . . . . . . 1163.4.4 Parametric path method . . . . . . . . . . . . . . . . . . . . . . 117

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Resumen

La mayorıa de los modelos utilizados en macroeconomıa suponen que los parametros yfunciones que rigen el comportamiento individual son constantes en el tiempo. Sin em-bargo, el mundo en que vivimos cambia continuamente y la velocidad de estos cambios,impulsados principalmente por el progreso tecnologico y la integracion global, esta cre-ciendo. Observamos que el capital, el consumo, la produccion de bienes y otras variablesmacroeconomicas muestran patrones de crecimiento no estacionarios, la volatilidad dela produccion tiene cambios estructurales y los ajustes estacionales juegan un papelimportante para el analisis del ciclo economico. Para modelar adecuadamente este en-torno economico es necesario contar con un marco que aborde con exito los problemasde la no estacionariedad .

El objetivo de esta tesis doctoral es triple: examinar metodos numericos existentespara el analisis de modelos economicos no estacionarios, introducir un nuevo metodocuantitativo, llamado el trayecto extendido para funciones (TEF), para calibracion,resolucion, simulacion y estimacion de estos modelos no estacionarios y, por ultimo,aplicar esta nueva herramienta a una coleccion de fenomenos economicos interesantesque se caracterizan como no estacionarios y de crecimiento desequilibrado. La tesisconsta de tres capıtulos, cada uno de interes independiente, que versan sobre los metodosnumericos para el analisis de modelos economicos dinamicos no estacionarios y susaplicaciones.

El primer capıtulo es un trabajo conjunto con Lilia Maliar, Serguei Maliar y JohnTaylor. Se estudia una clase de modelos estocasticos no estacionarios y de crecimientodesequilibrado en los que las preferencias, la tecnologıa y las leyes del movimiento delas variables exogenas cambian con el tiempo segun el proceso de Markov con proba-bilidades de transicion no estacionarios. En particular, las probabilidades de transicionde este proceso de Markov no estacionario pueden cambiar de un perıodo de tiempo aotro. Se demuestra que si los parametros estructurales del modelo siguen un procesode Markov con probabilidades de transicion posiblemente no estacionarios, entonces lasfunciones de decision optimas dependen tanto del estado como del tiempo. Por el con-trario, en las economıas de Markov estacionarias las funciones de decision dependen delestado y son independientes del tiempo. Esta propiedad util de los modelos estacionar-ios de Markov los hace analıticamente y numericamente tratable y, al mismo tiempo,restringe el estudio de algunas aplicaciones relevantes de crecimiento desequilibrado yno estacionario de la economıa.

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iv RESUMEN

Para estudiar los modelos economicos dinamicos no estacionarios introducimos unnuevo metodo cuantitativo, llamado el trayecto extendido para funciones (TEF). Eltrayecto extendido para funciones es una generalizacion del metodo el trayecto exten-dido (TE) de Fair y Taylor (1983) para las economıas de Markov no estacionarios. TEFaproxima con precision una solucion a una economıa con horizonte de tiempo infinitodurante los primeros τ perıodos. Se supone que en algun perıodo T τ la economıaes estacionaria y se procede en dos pasos: en primer lugar, se construyen las funcionesde decision convencionales Markov estacionarias para la economıa estacionaria en cadaperıodo de tiempo t; y luego se encuentra un trayecto para funciones de decision enperiodos t = 0, ..., T − 1 que llevan a la dada condicion terminal en perıodo T y quehace que estas funciones son de una sola vez coherentes entre sı.

Los metodos del TE y TEF son similares en que ambos extienden el trayecto, es de-cir, construyen una solucion aproximada para el horizonte mas largo T cuando realmentese necesita la solucion para el horizonte τ . Los dos metodos difieren en el objeto queconstruyen, especıficamente, el metodo de Fair y Taylor (1983) construye un trayectopara las variables bajo el supuesto del equivalente de certeza (es decir, sustituyendo laexpectativa de funciones entre los estados por sus valores esperados en esos estados),mientras TEF construye un trayecto para funciones de decision mediante la aproxi-macion de funciones de expectativas de manera precisa usando cuadratura o metodosde integracion monomios.

El TEF tambien esta relacionado con diversos metodos numericos que construyen untrayecto de variables en economıas determinısticas, en particular, metodos introducidosa economıa por Lipton et al. (1980); ver Atolia y Buffie (2009 a,b) para una discusiondetallada de los metodos de disparo. La literatura relacionada tambien incluye metodosde solucion para los modelos de tiempo continuo estudiados en Chen (1999); un metodopara la caracterizacion de equilibrio en los modelos de ciclo de vida con una rutaagregada determinista e incertidumbre idiosincrasico, propuesto por Conesa y Krueger(1999); un metodo de trayecto parametrico de Judd (2002); y un metodo de controlpredictivo, desarrollado en Grune et al. (2013). El principal inconveniente de este tipode metodos es que no proporcionan un marco coherente para el analisis de las economıascon incertidumbre.

Nosotros evaluamos la exactitud de aproximacion producida por TEF en el ambi-ente no estacionario. En particular, consideramos una version del modelo neoclasico decrecimiento con cantidad de trabajo aumentada con el cambio tecnologico. Esta versiondel modelo es consistente con la senda de crecimiento equilibrado y se puede convertiren estacionario y, por lo tanto, resuelto por metodos de solucion convencionales queson fiables y muy precisos (los residuos maximos en las ecuaciones son de orden 10−6

en simulacian estocastica; ver Judd et al (2014) para evaluaciones de precision de estosmetodos). En consecuencia, el modelo tambien se resuelve por los TE y TEF metodosbajo diferentes parametrizaciones. En todos los experimentos la diferencia entre solu-ciones “exacta” y TEF es menor que 10−6 ≈ 0, 0001% durante los primeros 50 perıodosde simulacion en los modelos resueltos con T = 200. En contraste, las mismas es-tadısticas. para el metodo Fair y Taylor (1983) son significativamente menos precisas.

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Para los perıodos t > 50 la exactitud de la aproximacion del TEF se deteriora y lasolucion se hace especialmente inexacta al final.

La exactitud de la solucion TEF depende de la duracion del horizonte temporalT y de la condicion terminal. Esta dependencia es el resultado del supuesto de queen el ultimo perıodo T una economıa modelada es estacionaria. Teniendo en cuenta,que el marco de TEF se puede utilizar para encontrar soluciones en las economıasno estacionarias con horizonte finito (como, por ejemplo, un modelo de generacionessolapadas de Hasanhodzic y Kotlikoff (2013)), ası como con horizonte infinito. Para esteultimo caso, probamos un teorema de la autopista que muestra que la aproximacionTEF converge asintoticamente a la verdadera solucion de modelo no estacionario deMarkov. El teorema de la autopista se basa en la idea de que la autopista es a menudola ruta mas rapida entre dos puntos que pueden estar muy separados, incluso si no esuna ruta directa. Desde el punto de vista de los modelos de Markov no estacionarios,el programa optimo de la economıa estacionaria en perıodo T (que es producida porTEF) sigue durante mucho tiempo el programa optimo de la economıa no estacionaria(autopista) y se aparta de ella solo en el ultimo momento para cumplir la condicionfinal. Este hecho explica la inexactitud en la cola de las soluciones producidas por TEFen nuestros experimentos.

La contribucion principal de la parte teorica de este trabajo es que distinguimosy caracterizamos una clase manejable de modelos no estacionarios de Markov y pro-ponemos un marco de TEF para la construccion de soluciones de este tipo de modelos.Ademas, este analisis contiene nuevos resultados formales. En primer lugar, el Teorema1 establece una estructura de Markov no estacionaria para una economıa general conhorizonte finito en la que la condicion terminal esta dada por una funcion de Markovcontingente de estado, mientras que la literatura anterior establece una estructura deMarkov no estacionario para un caso especial de una condicion terminal constante(vease Mitra y Nyarko (1991)). En segundo lugar, nuestro teorema de la autopista semuestra para una condicion terminal que se genera por una solucion de Markov para unmodelo estacionario en perıodo T , mientras que los teoremas de la autopista existentesen la literatura son construidas suponiendo una condicion terminal especıfica de cero;ver Majumdar y Zilch (1987), Mitra y Nyarko (1991) y Joshi (1997).

Aplicamos TEF para analizar una coleccion de aplicaciones desafiantes no esta-cionarios y con crecimiento desequilibrado que no admiten los equilibrios de Markovestacionarios convencionales, incluyendo los modelos de crecimiento con cambios y drift-ing en parametros previstos, con los avances technologicos que aumentan el capital, concambios de regimen anticipados, con la volatilidad variable con una tendencia deter-minista y con ajustes estacionales.

El progreso tecnologico que aumenta el capital. El progreso tecnologico aumenta lacapacidad de la produccion de bienes y aumenta la variedad de productos. Acemoglu(2003) incorpora el capital aumentado con el progreso tecnologico en un modelo deter-ministico de cambio tecnologico endogeno. Sin embargo, solo el supuesto del progresotechnologico que aumenta la cantidad de trabajo es consistente con la senda de crec-imiento equilibrado; ver King et al. (1988). En nuestra primera aplicacion usamos TEF

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para resolver un modelo de crecimiento no estacionario con un capital aumentado porel progreso tecnologico que no admite un equilibrio estacionario de Markov.

En el modelo los bienes se producen de acuerdo con la funcion de producion conla elasticidad de sustitucion constante (ESC) que incorpora el capital y la cantidadde trabajo aumentados por el progreso technologico. La solucion del modelo con lacantidad de trabajo aumentada por el progreso technologico es tıpica para una sendade crecimiento equilibrado: hay un camino de crecimiento determinista exponencial conuna tasa de crecimiento constante y las fluctuaciones cıclicas en torno a este camino.A su vez, la solucion del modelo con el capital aumentado por el progreso technologicotiene una tasa de crecimiento variable en el tiempo, es decir, la tasa de rendimientodel capital disminuye a medida que la economıa crece. Las propiedades cıclicas deambos modelos tienen un aspecto similar, que es un resultado novedoso ya que laspropiedades de los modelos estocasticos de crecimiento con capital aumentado por elprogreso tecnologico aun no habıan sido estudiadas en la literatura.

Cambios de regımenes anticipados y imprevistos. La literatura sobre los regımenesdistingue dos tipos de cambios: previstos e imprevistos. Los ejemplos de los cambios deregımenes anticipados pueden ser las elecciones presidenciales con resultados predeci-bles, anuncios de polıticas, cambios legislativos previstos. La idea de que los cambiosde regımenes previstos son importantes para las fluctuaciones economicas se remonta aPiguo (1927) y es defendida en Cochrane (1994) y Beaudry y Portier (2006). En nuestrasegunda aplicacion, usamos TEF para construir solucion no estacionaria de Markov deun modelo de crecimiento que experimenta una combinacion de cambios tecnologicosprevistos y no previstos.

Consideramos una version del modelo de crecimiento neoclasico donde el nivel dela tecnologıa puede tomar dos valores: alto y bajo. El enfoque se hace en el casode que los cambios de regimen estan tanto no recurrentes y anticipados por el agentedesde el principio. Por ejemplo, consideramos un escenario en el que la economıacomienza con un bajo nivel de tecnologıa en t = 0, cambia a un nivel de alta tecnologıaen t = 250 y luego cambia de nuevo a un nivel bajo en t = 550. Solucionamos elmodelo con este perfil de nivel de tecnologıa para el caso en el que se esperan estoscambios y para el caso en el que son inesperados y comparamos las soluciones. Enla solucion con los cambios de regımenes esperados, se observa un fuerte efecto deanticipacion: alrededor de 50 perıodos antes del cambio del nivel de tecnologıa de bajoa alto (de mayor a menor) el agente comienza a aumentar gradualmente (disminuir) suconsumo y disminuir (aumentar) el capital con el fin de traer (aplazar) una parte delos beneficios (perdidas) de crecimiento (disminucion) de la tecnologıa futura hasta elpresente. Cuando el cambio de tecnologıa realmente ocurre, tiene solo un efecto menorsobre el consumo. En contraste, los efectos anticipados de moderacion del consumoestan ausentes en la solucion con cambios de parametros tecnologicos inesperado.

Los cambios estacionales. El caso especial de los cambios de regımenes previstosson los ajustes estacionales. Barsky y Miron (1989) documenta la evidencia sobre laimportancia de los cambios de temporada para el ciclo economico y Saijo (2013 ) sostieneque el tratamiento inadecuado de los cambios estacionales puede dar lugar a un sesgo

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significativo en las estimaciones de los parametros.Para modelar los cambios estacionales suponemos que en el modelo de crecimiento

neoclasico cada cuarto periodo el parametro tecnologico toma un valor alto, mientrasque el resto de los perıodos se mantiene en un nivel bajo. Ademas de los cambiosestacionales, un agente se enfrenta a los shocks de productividad convencionales, demodo que la trayectoria resultante del nivel de productividad esta dada por una com-posicion de los cambios estacionales esperados y los cambios estocasticos inesperadosen la productividad.

Nos encontramos con que en esta configuracion el tamano de las fluctuaciones esta-cionales de consumo y de capital es muy pequeno comparado con la amplitud de lasfluctuaciones estacionales de productividad. El agente ajusta su capital y su consumopara aprovechar el crecimiento de la productividad estacional en promedio, como lahipotesis de consumo permanente suguiere.

Drifiting de parametros. Otro posible ambiente no estacionario que pueda surgiren la economıa es el modelo con un drifting de parametros. En comparacion con loscambios de regimen, el drifting de parametros representa la idea de cambios gradualesen los parametros del modelo . Entre otros Clarida et al. (2000) y Canova (2009)documentan la evidencia a favor del drifting de parametros en economıa. Ademas, Galı(2006) sostiene que los cambios de regimen no recurrentes con variaciones graduales enpolıticas se encuentran entre los mas empıricamente relevantes.

Drifting de parametros es, en general, incompatible con un equilibrio de Markovporque las funciones de decision cambian gradualmente con el tiempo. En un modelode crecimiento neoclasico consideramos un escenario en el que el nivel de la tecnologıacambiara gradualmente de menor a mayor nivel y viceversa. Para ser mas especıficos, elnivel de tecnologıa es bajo para los 200 primeros perıodos, luego aumenta linealmentedurante los proximos 100 perıodos y permanece en un nivel alto para los 300 perıodossiguientes; despues disminuye linealmente hacia atras y se mantiene en un nivel bajo200 perıodos mas. Similar a las aplicaciones con los cambios de regımenes y ajusteestacional, en el caso del drifting del nivel de productividad observamos el suavizo delconsumo bien pronunciado a costa de ajustes anticipados de capital.

Volatilidad estocastica frente a la volatilidad deterministica. McConnel y Perez-Quiros (2000) documentan un declive estructural monotona en la volatilidad del crec-imiento del PIB real de la economıa estadounidense. La literatura normalmente asumeque la desviacion estandar de los shocks exogenos o bien sigue un proceso de Markov oexperimenta cambios de regimen de Markov recurrentes; ver Bloom (2009), Fernandez- Villaverde, Guerron - Quintana y Rubio - Ramırez (2010).

En nuestro experimento, relajamos el supuesto de que la desviacion estandar deshocks a la productividad exogena es constante y en vez, incluimos en un modelo decrecimiento neoclasico estandar estocastico un shock de productividad que presentadisminucion en la volatilidad. Una solucion no estacionaria de Markov producida porTEF muestra que las fluctuaciones en el capital y el consumo disminuyen su amplituden respuesta a la disminucion de la volatilidad de productividad.

Calibracion y estimacion de parametros en los modelos no estacionarios y de crec-

viii RESUMEN

imiento desequilibrado. En la ultima aplicacion usamos el marco TEF para calibrar yestimar parametros en el modelo de crecimiento desequilibrado mediante la construcciony simulacion de solucion no estacionaria de Markov. Consideramos un modelo en elque la tasa de depreciacion de capital tiene tanto tendencia temporal determinista comocomponente cıclico estocastico.

Simulamos una solucion de series temporales utilizando los valores de los parametrosajustados, y obtenemos los patrones de crecimiento desequilibrados que se parecen alos que observamos en los datos sobre la economıa de Estados Unidos, en particular,las tendencias en el tiempo y las propiedades de los ciclos economicos de la produccion,del consumo, del capital y de la inversion.

En el segundo capıtulo de la tesis doctoral construyo un modelo de equilibrio gen-eral que replica bien los patrones de crecimiento desbalanceado observadas en los datosde la economıa estadounidense. En este capıtulo se contribuye a la literatura de tresmaneras: en primer lugar, se construye un conjunto de datos actualizado que contienelas variables macroeconomicas claves que caracterizan el crecimiento economico de laeconomıa estadounidense en el 1964 - 2012. El conjunto de datos incluye los variablesdel mercado de trabajo tales como la poblacion de trabajadores cualificados y no cual-ificados, las horas anuales trabajadas y el salario; ademas, incluye variables agregadostales como el consumo, capital (edificios), capital (maquinas), la inversion y los preciosrelativos. El conjunto de datos puede ser visto como una version actualizada de datosde la Krusell et al. (2000) que abarca el perıodo 1963 - 1992. En segundo lugar, seanaliza como los hallazgos de Krusell et al. (2000) son robustos a modificaciones enforma especıfica de la construccion de las variables economicas y el tamano de la mues-tra utilizada para la estimacion. En tercer lugar, se construye un modelo de equilibriogeneral que cuenta con la complementariedad entre capital y cantidad de trabajo cuali-ficado para evaluar si el modelo puede producir endogenamente patrones de crecimientodesequilibrados de consumo, capital (edificios y maquinas) similares a los observadosen los datos economicos de Estados Unidos.

El conjunto de datos se construye para la estimacion y la calibracion de los paramet-ros del modelo, ası como para evaluar el ajuste del modelo a los datos. Construyo dosgrupos de variables de la economıa de Estados Unidos: en el primer grupo se incluyenlas variables del mercado de trabajo y se construye a partir de datos microeconomicosy el segundo grupo incluye variables como la produccion, el capital y los precios y seconstruye a partir de datos macroeconomicos.

En la construccion de las variables del mercado de trabajo se distinguen dos gru-pos de trabajadores: cualificados y no cualificados. Los individuos que tienen tıtulouniversitario de la educacion o superior se consideran cualificados; los que terminanla escuela secundaria pertenecen a la categorıa de los trabajadores no cualificados; y,finalmente, el resto se dividen en dos entre estos grupos. Se define “Cantidad total detrabajo cualificado” como una suma de productos de semanas anuales trabajadas y lashoras trabajadas la semana pasada para diferentes grupos de educacion, el genero, deraza, de edad y de los perfiles de experiencia. “Las horas anuales trabajadas” es unratio de la cantidad total de trabajo de un grupo de trabajadores entre el numero de

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trabajadores en ese grupo.Se observa que en el perıodo 1963 - 2012 el numero de trabajadores cualificados

y no cualificados aumenta con el tiempo, sin embargo, el porcentaje de aumento demano de obra cualificada es mucho mayor que el aumento del trabajo no cualificado.En particular, la poblacion de trabajadores no cualificados aumenta de 50,9 millones a70,8 millones (equivale al 39,1% de aumento), mientras que la poblacion de trabajadorescualificados aumenta en el mismo perıodo de 11,1 a 70,4 millones (un aumento de casi534,2%). Las horas trabajadas por ambos tipos de trabajadores se mantienen constantesy, por lo tanto, las tendencias en las poblaciones definen las tendencias en el total delas variables de trabajo.

Una observacion interesante es que la oferta relativa de trabajadores cualificados ytanto su salario aumentan con el tiempo. Este fenomeno recibe el nombre de “puzzlede rendimiento de la educacion” y ha sido estudiado por muchos economistas; verKrusell et al. (2000), Acemoglu y Autor (2012), entre otros. El rendimiento de laeducacion se define como la relacion entre los salarios de los trabajadores cualificadosy no cualificados y en nuestro conjunto de datos aumenta a una tasa promedio de 1%por ano.

Tambien se observa que las existencias reales de capital (edificios y maquinas) au-mentan con el tiempo. Mas especıfico, la tasa de crecimiento de capital invertido enlas maquinas se acelera a partir de 1995, que refleja la introduccion y extension de lasnuevas tecnologıas, como internet, ordenadores, etc. Nos encontramos con que la cuotade trabajo en el ingreso no cambia de forma sistemıtica en el tiempo. Sin embargo,el precio relativo de los equipos y el precio relativo de maquinas ajustado por calidaddisminuyen.

En suma, los patrones de crecimiento de las principales variables macroeconomicasde la economıa estadounidense resultantes son muy desequilibrados.

Una gran parte de la literatura estudia el papel que juegan los avances tecnologicosen el proceso de produccion de bienes e investiga sus efectos sobre el capital y trabajo;para un estudio exhaustivo de la literatura ver Goldin y Katz (2008) y Acemoglu yAutor (2012). Bound y Johnson (1992) concluyen que un aumento en el rendimientode la educacion se puede atribuir a los cambios tecnologicos, mientras Krusell et al.(2000) sostienen que la dinamica del rendimiento de la educacion solo puede explicarseen terminos de factores observables. El mensaje principal de la literatura es que todaslas variables que determinan la dinamica del rendimiento de la educacion pueden serobservados directamente de los datos y por lo tanto, los economistas y los polıticosdeben concentrarse en polıticas que afectan estas variables en una manera particular(mientras que los factores no observados y exogenos no pueden ser afectados).

Se utilizan variables macroeconomicas como el capital y el trabajo de nuestro con-junto de datos para estimar los parametros de la funcion de produccion de bienes. Lafuncion de produccion que se utiliza en nuestro analisis tiene elasticidad de sustitucionconstante entre los factores de produccion que son las cantidades de trabajo de tra-bajadores cualificados y no cualificados y dos tipos de capital. Krusell et al. (2000)tambien estiman los parametros de esta funcion de produccion, sin embargo, no pode-

x RESUMEN

mos basar nuestro analisis en sus resultados debido a la diferencia en la definicion dela variable la cantidad de trabajo total. En su analisis la cantidad de trabajo total sedefine en terminos per capita, mientras que nosotros usamos medidas globales totales.

Para estimar los parametros que rigen las elasticidades de sustitucion usamos un pro-cedimiento de simulacion de pseudo maxima verosimilitud (SPMV). Este procedimientode estimacion esta en lınea con Laroque y Salanie (1993) y Krusell et al. (2000) y tienecomo objetivo encontrar las estimaciones de los parametros de un modelo de espaciolatente no lineal que consta de las ecuaciones de modelo y ecuaciones de estado. Lasecuaciones de modelo incluyen tres ecuaciones obtenidas a partir de las condiciones deprimer orden del problema de optimizacion de la empresa que maximiza los beneficios.Estas ecuaciones relacionan las variables observadas en los datos con sus homologosobtenidos a partir de la funcion de produccion ESC. Estos son (1) ecuacion que definala cuota de participacion de trabajo, (2) la razon entre salarios de trabajadores cual-ificados y no cualificados, y (3) una condicion que equivale tarifas de rendimiento noobservables de dos tipos de capital. Las ecuaciones de estado se forman por unas leyeslineales (en logaritmos) de movimiento para las eficiencias no observadas de trabajo dedos tipos de trabajadores.

Se consideran dos versiones del modelo de espacio latente no lineal descrito an-teriormente: con y sin progreso technologico que aumenta la cantidad de trabajo detrabajadores cualificados y no cualificados. En los experimentos, la elasticidad de susti-tucion entre el capital invertido en maquinas y la cantidad de trabajo no cualificadooscila desde 1,66 hasta 2,22. La elasticidad de sustitucion entre el capital invertidoen maquinas y la cantidad de trabajo cualificado varıa entre 0,5 y 0,67. Las primerosestimaciones estan en lınea con los resultados reportados en Krusell et al. (2000), mien-tras que las ultimas estimaciones estan aproximadamente dos veces por debajo. Masimportante, estos resultados confirman la idea principal del analisis de Krusell et al.(2000): en el contexto de la funcion de produccion ESC, se puede explicar la evoluciondel rendimiento de la educacion utilizando variables observables solo sin anadir ningunprogreso exogeno no observable. Sin embargo, la literatura convencional esta de acuerdoen que el avance tecnologico juga un papel importante en la explicacion de los patronesde crecimiento de otras variables macroeconomicas y, por lo tanto, decidimos manten-erlo para nuestro analisis de equilibrio general.

Se soluciona el problema de un planificador social que maximiza la utilidad sujetoa la restriccion de recursos agregada de la economıa. Para que nuestro modelo sea masrazonable incluimos varias fuentes de crecimiento exogeno: la poblacion de trabajadorescualificados y no cualificados crece con tasas diferentes, dos tipos de avances tecnologicosaumentan la productividad de la cantidad de trabajo cualificado y no cualificado, elcambio tecnologico especıfico para equipos y el cambio tecnologico neutral de Hicksayudan para que el modelo produzca los patrones que coincidan con los observados enlos datos.

Por construccion, el modelo es no estacionario y no puede ser estudiado con metodosnumericos convencionales: en la presencia de crecimiento desequilibrado, las funcionesde decision cambiar de un perıodo a otro y no tienen puntos fijos en cualquiera de

xi

las ecuaciones de Bellman o de Euler. Para hacer frente al crecimiento la literaturade ciclos economicos reales impone las restricciones sobre las tasas de crecimiento queen su lugar hace que el modelo sea coherente con la senda de crecimiento equilibradodonde las variables crecen a unas tasas (posiblemente diferentes) constantes. En talcaso, el modelo resultante se convierte en estacionario (mediante la eliminacion de lastendencias) y puede resolverse con metodos numericos convencionales; ver King et al.(1988) para una discusion cuidadosa de supuestos necesarios para la transformacion.Dado que se permiten diferentes avances tecnologicos, este procedimiento no se puedeutilizar en nuestro caso. Maliar y Maliar (2011) proponen una manera de convertir cier-tos modelos no estacionarios y no balanceados en estacionarios mediante la imposicionde restricciones en las tasas de crecimiento de algunas variables. Sin embargo, estasrestricciones no son necesariamente consistentes con los datos y su modelo no tuvo exitoen reproducion de los patrones de crecimiento observados en los datos.

Aplico el marco TEF, que se describe en el capıtulo uno de la presente tesis doctoralpara estudiar las implicaciones cuantitativas del modelo de crecimiento no estacionarioy encuentro con que el modelo puede explicar bien los patrones de crecimiento no bal-anceados claves de los datos de Estados Unidos. En particular, la dinamica de lasexistencias de capital, las cantidades totales de trabajo de los trabajadores cualificadosy no cualificados y la producion de bienes reproducen los datos. Sin embargo, el modelono es del todo exitoso para explicar el comportamiento del rendimiento de la educacion.Es decir, se predice que el rendimiento de la educacion es mas o menos constante, mien-tras que en los datos crece con el tiempo. Hay tres canales que son responsables dela dinamica creciente del rendimiento de la educacion: la cantidad relativa de traba-jadores, la eficiencia relativa de trabajadores y la complementariedad entre capital ytrabajo cualificado. Cantidad relativa de efecto laboral disminuye el rendimiento de laeducacion, mientras que los efectos de eficiencia relativa y el complementariedad entrecapital y trabajo cualificado aumentan. En virtud de los valores de los parametrosestimados los tres efectos se compensan entre sı.

En el tercer capıtulo de la tesis doctoral exploro las aplicaciones de los metodosnumericos de programacion geometrica (PG) en la economıa. Esta tecnica es adecuadapara la solucion de los ambos tipos de modelos economicos estacionarios y no esta-cionarios. Programacion geometrica fue introducida por primera vez en la ingenierıaen Zener (1961) y mas tarde se desarallo en Duffin et al. (1967). El objeto principalde la programacion geometrica es un programa geometrico que satisface las siguientescondiciones: (1) funcion objetivo y las restricciones de desigualdad son posinomios; (2)las restricciones de igualdad son monomios. Los metodos de programacion geometricase utilizan ampliamente en la ingenierıa, la teorıa de la informacion, ciencias de la ad-ministracion y las finanzas computacionales; ver Boyd (2007) para la revision de dichasaplicaciones.

El analisis de los modelos economicos dinamicos a menudo requiere encontrar solu-ciones a los problemas de optimizacion no lineal o sistemas de ecuaciones no lineales.Los metodos existente habitualmente se basan en solvers numericos disponibles (queson en muchos casos las variantes del procedimiento de Newton-Raphson), en aprox-

xii RESUMEN

imaciones lineales o en metodos iterativos como el punto fijo. El primer grupo demetodos requiere una buena suposicion inicial que puede ser difıcil de encontrar enalgunos casos. El segundo grupo de metodos, mientras que es muy fiable y sencillodesde el punto de vista cuantitativo, puede llevar a malas aproximaciones en algunoscasos. Finalmente, el ultimo grupo de metodos puede funcionar mal en aplicacionescuando la dimension aumenta. Por el contrario, los metodos numericos desarrolladospara la programacion geometrica se basan en algoritmos de puntos interiores primarios-duales que no requieren suposicion inicial de parametros y son aplicables a problemascon muchas variables de estado. Nuevos metodos numericos desarrollados en el campode la programacion geometrica pueden resolver programas geometricos grandes en unacantidad de tiempo razonable.

En este capıtulo se resuelven modelos de crecimiento neoclasicos deterministos y es-tocasticos y un conjunto de problemas de optimizacion intratemporales que, a menudo,surgen en economıa. Encontramos que las soluciones obtenidas por metodos de progra-macion geometrica a los problemas deterministas e intratemporales son exactas. Sinembargo, la solucion obtenida a un modelo de crecimiento neoclasico estocastico es in-exacta y el grado de inexactitud aumenta con el grado de no linealidad del modelo. Esteresultado es una consecuencia natural de la suposicion de equivalencia certeza de quehacemos para que el modelo estocastico sea manejable para ambito de PG. El principiode equivalencia certeza se consiste en sustituir el futuro nivel de productividad en laecuacion de Euler con su valor esperado.

En cuanto a la exactitud y la velocidad de los metodos de programacion geometrica,los experimentos sugieren lo siguiente: en primer lugar, los metodos numericos deprogramacion geometrica son menos rapidos que los metodos numericos usadas eneconomıa habitualmente, tales como, por ejemplo, el algoritmo generalizado de sim-ulacion estocastica (AGSE). En segundo lugar, la exactitud de la solucion depende delos parametros del modelo y se hace menor a medida que aumenta el parametro deaversion al riesgo, γ, o la volatilidad del shock, σ.

Se compara programacion geometrica con otros metodos numericos utilizados paracalcular las soluciones a los modelos economicos, como los metodos de disparo, el metodoFair y Taylor (1983), marco de control predictivo de modelo no lineal y el metodoparametrico de ruta. Encontramos que la conducta de estos metodos puede ser com-parada a los metodos de programacion geometrica. Ademas, la exactitud de la soluciondel modelo de crecimiento neoclasico estocastico obtenida por estos metodos es similara la solucioon obtenida por PG. Estos resultados vienen, naturalmente, teniendo encuenta que los cinco metodos se basan en la suposicion de equivalencia certeza.

Introduction

Most of the models used in macroeconomics assume that parameters and functionsgoverning individual behaviour are constant across time. However, the world we livein changes continually and the speed of these changes, driven mostly by technologicalprogress and global integration, is growing. We observe that capital, consumption, out-put and other macroeconomic variables display nonstationary growth patterns, volatil-ity of output has structural changes and seasonal adjustments play an important role forthe analysis of the business cycle. To adequately model this economic environment it isnecessary to have a framework that sucessfully addresses the issues of nonstationarity.

The aim of this thesis is threefold: to examine existing numerical methods for an-alyzing nonstationary economic models, to introduce a novel quantitative framework,called the extended function path (EFP), for calibrating, solving, simulating and esti-mating such nonstationary models and, finally, to apply this novel tool to a collectionof interesting nonstationary and unbalanced growth economic applications. The thesisconsists of three chapters, each of independent interest, that cover numerical methodsfor analyzing nonstationary dynamic economic models and their applications.

The first chapter is a joint work with Lilia Maliar, Serguei Maliar and John Tay-lor. It studies a class of stochastic nonstationary and unbalanced growth models inwhich preferences, technology and laws of motion of exogenous variables change overtime according to Markov processes with nonstationary transition probabilities. Inparticular, the transition probabilities of these nonstationary Markov processes canchange from one time period to another. We show that if structural parameters ofthe model follow a Markov process with possibly nonstationary transition probabilities,then the optimal decision functions are both state-contingent and time-dependent. Incontrast, in stationary Markov economies the decision functions are state-contingentand time-independent. This useful property of Markov stationary models makes themanalytically and numerically tractable and at the same time restricts the study of somerelevant nonstationary and unbalanced growth applications in economics.

To study nonstationary dynamic economic models we introduce a novel quantitativeframework, called the extended function path (EFP). The extended function path isa generalization of the extended path (EP) framework of Fair and Taylor (1983) tononstationary Markov economies. EFP accurately approximates a solution to an infinitehorizon economy during the first τ periods. It assumes that at some period T τ the economy becomes stationary and it proceeds in two steps: first, it constructs

xiii

xiv INTRODUCTION

conventional Markov stationary decision functions for the stationary economy in eachperiod of time t; and then it finds a path for decision functions in periods t = 0, ..., T−1(that are time-consistent with each other) such that it matches the given T -periodterminal condition.

The EFP framework is related to various numerical methods that construct a pathof variables in deterministic economies, in particular, shooting methods introducedto economics by Lipton et al. (1980); see Atolia and Buffie (2009 a,b) for carefuldiscussion of shooting methods. The related literature also includes solution methodsfor continuous time models studied in Chen (1999); a framework for characterizingequilibrium in life-cycle models with a deterministic aggregate path and idiosyncraticuncertainty, proposed by Conesa and Krueger (1999); a parametric path method of Judd(2002); and a predictive control method, developed in Grune et al. (2013). The mainshortcoming of this class of methods is that they do not provide a coherent frameworkfor the analysis of economies with uncertainty.

The main contributions in the theoretical part of this chapter are that we distin-guish and characterize a tractable class of nonstationary Markov models and that wepropose an EFP framework for constructing solutions to such models. In addition,this analysis contains new formal results. First, Theorem 1 establishes a nonstationaryMarkov structure for a general, finite horizon economy in which a terminal condition isgiven by a state-contingent Markov function while the previous literature establishes anonstationary Markov structure for a special case of a constant terminal condition (seeMitra and Nyarko (1991)). Second, our turnpike theorem is shown for a terminal con-dition that is generated by a Markov solution to a T -period stationary model, whereasthe turnpike theorems existing in the literature have been constructed by assuming aspecific zero terminal condition; see Majumdar and Zilch (1987), Mitra and Nyarko(1991) and Joshi (1997).

We apply EFP to analyze a collection of challenging nonstationary and unbalancedgrowth applications that do not admit conventional stationary Markov equilibria, in-cluding growth models with anticipated parameter shifts and drifts, capital augmentingtechnological progress, anticipated regime switches, time-varying volatility with a de-terministic trend and seasonal adjustments.

In the second chapter of the thesis I construct a general equilibrium model thatreplicates well the unbalanced growth patterns observed in the data for the U.S. econ-omy. This chapter contributes to the literature in three ways: firstly, I construct anup-to-date dataset that contains the key macroeconomics variables that characterizethe economic growth in the U.S. economy over the 1964 - 2012 period. The dataset in-cludes labor-market variables such as population of skilled and unskilled workers, theirannual hours worked and their wages; furthermore, it includes such aggregate variablesas consumption, capital structures, capital equipment, investment and relative prices.The dataset can be viewed as an updated version of the Krusell et al. (2000) datasetthat covers the period 1963 - 1992. Secondly, I analyze whether the findings of Krusell etal. (2000) are robust to modifications to the method of constructing economic variablesand the sample size used for the estimation. Thirdly, I construct a general equilibrium

xv

model that features capital-skill complementarity to evaluate whether the model canendogenously produce unbalanced growth patterns of consumption, capital equipmentand capital structures similar to those observed in the U.S. economic data.

The dataset is used for estimation and calibration of the parameters of the modelas well as for assessing the fit of the model to the data. Two groups of variables onthe U.S. economy are constructed: the first group includes labor market variables andis obtained from microeconomic data and the second group includes such variables asoutput, capital, and prices and is obtained from macroeconomic data.

I observe that over the period 1963 - 2012 the number of both skilled and unskilledworkers increases over time, however, the percentage increase in skilled labor is muchlarger than the increase in unskilled labor. In particular, the population of unskilledworkers increases from 50.9 millions to 70.8 millions (which corresponds to a 39.1%increase) whereas the population of skilled workers increases over the same period from11.1 to 70.4 millions (an increase of almost 534.2%). Hours worked by both typesof workers remained constant and, therefore, the trends in the populations define thetrends in the total labor input variables. Real stocks of capital structures and capitalequipment increase over time as well. To be more specific, the growth rate of equipmentaccelerates starting from 1995 which reflects the introduction and extension of moderntechnologies such as the internet, computers, etc. I find that labor’s share of income didnot systematically change over time. However, the relative price of equipment and thequality adjusted price of equipment decrease over time. In sum, the resulting growthpatterns of the key macroeconomic variables of the U.S. economy are highly unbalanced.

To estimate the parameters that govern the elasticities of substitution a simulatedpseudo maximum likelihood (SPMLE) procedure is used. This estimation procedureis in line with Laroque and Salanie (1993) and Krusell et al. (2000) and aims to findthe estimates of the parameters from a non-linear latent space model consisting ofmodel equations and state equations. We consider two versions of the non-linear latentspace model: with and without trend components in the labor augmenting technolog-ical changes for skilled and unskilled labor. In our estimation results, the elasticity ofsubstitution between equipment and unskilled labor ranges from 1.66 to 2.22. The elas-ticity of substitution between equipment and skilled labor varies from 0.5 to 0.67. Theformer estimates are in line with the results reported in Krusell et al. (2000), while thelatter estimates are about two times lower. Most importantly, these findings confirm themain insight of the Krusell et al. (2000) analysis: In the context of the CES productionfunction, we can account for the evolution of the skill premium using just observablevariables without adding any exogenously unobservable progresses. Nevertheless, themainstream literature agrees on different types of technological progress playing an im-portant role in accounting for the growth patterns of other macroeconomic variablesand, therefore, they are kept for our general equilibrium analysis.

By construction our model is nonstationary and cannot be studied with conventionalnumerical methods: in the presence of unbalanced growth, decision functions changefrom one period to another and they do not have fixed points on either Bellman or Eulerequations. To deal with growth the real business cycle literature imposes restrictions

xvi INTRODUCTION

on the growth rates of the exogenous parameters to make a model consistent with thebalanced growth path where the variables grow at a constant (possibly different) rate.In such a case, the resulting model is made stationary (by removing the trends) and canbe solved using conventional numerical methods; see King et al. (1988) for a carefuldiscussion of the assumptions necessary for the transformation. Since we allow fordifferent technological progresses, the approach used in the literature cannot be usedin our case.

We apply the EFP framework, described in chapter one of the present thesis, tostudy the quantitative implications or our nonstationary growth model and find thatthe model can account well for the key unbalanced growth patterns of the U.S. data.In particular, the dynamics of the stock of capital equipment, total labor inputs ofskilled and unskilled workers and output match the data well. However, the modelis not entirely successful in explaining the behavior of the skill premium. Namely,it predicts that the skill premium is roughly constant while in the data itgrows overtime. There are three effects that are responsible for the increasing dynamics of theskill premium: Relative quantity of labor, relative efficiency of labor effect and capital-skill complementarity effects. The relative quantity of labor effect decreases the skillpremium, while the relative efficiency and capital-skill complementarity effects increaseit. Under the estimated parameter values the three effects offset each other.

In the third chapter of the thesis I explore the applications of geometric program-ming (GP) numerical methods in economics. This technique is suitable for solving bothstationary and nonstationary economic models. Geometric programming was first in-troduced in engineering in Zener (1961) and later developed in Duffin et al. (1967). Theobjective of geometric programming is to solve a geometric program that satisfies thefollowing conditions: (1) objective function and inequality constraints are posynomialfunctions; (2) equality constraints are monomial functions. Geometric programmingmethods are extensively used in engineering, information theory, management scienceand computational finance; see Boyd (2007) for a review of applications.

The analysis of dynamic economic models often requires finding solutions to nonlin-ear optimization problems or systems of nonlinear equations. Existing methods usuallyrely on available numerical solvers (which are in many cases variants of the Newton-Raphson procedure), on linear approximations or on iterative methods such as fixedpoint. The first group of methods require a good initial guess which can be hard tofind in some instances. The second group of methods, while being very reliable andnumerically cheap, may deliver bad approximations. Finally, the last group of methodscan perform poorly in applications as the dimensionality of the state space increases.In contrast, numerical methods developed for geometric programming are based onprimal-dual interior-point algorithms that do not require an initial guess and are appli-cable to problems with many state variables. New numerical methods developed in thefield of geometric programming can solve large scale geometric programs in a reasonableamount of time.

In this chapter deterministic and stochastic neoclassical growth models and a setof intratemporal optimization problems that often arise in economics are solved. I find

xvii

that solutions delivered by geometric programming methods to deterministic and in-tratemporal problems are accurate. However, the solution delivered for a stochasticneoclassical growth model is inaccurate and the degree of inaccuracy increases with thedegree of nonlinearity of the model. This result is a natural consequence of the assump-tion of certainty equivalence that is imposed to make the stochastic model tractablefor the GP environment. The performance of geometric programming can be comparedto other numerical methods used to compute solutions to economic models such asshooting methods, the Fair and Taylor (1983) method, the nonlinear model predictivecontrol framework and the parametric path method.

xviii INTRODUCTION

Chapter 1

A Tractable Framework forAnalyzing a Class of NonstationaryMarkov Models

1.1 Introduction

We study a class of infinite-horizon nonlinear dynamic economic models in which pref-erences, technology and laws of motion for exogenous variables can change over timeeither deterministically or stochastically, according to a Markov process, or both. A dis-tinctive feature of our analysis is that we allow for Markov processes with time-varyingtransition probabilities. Unbalanced stochastic growth models fit into that class, butso do many other models and applications such as the entry into a monetary union, anonrecurrent policy regime switch or deterministic seasonals. The studied models arenonstationary in the sense that the decision and value functions are time-dependentand cannot be generally solved with conventional numerical methods that constructstationary Markov equilibria.

Two clarifying comments are in order: First, some nonstationary models from thestudied class can be converted into stationary, for example, a nonstationary model witha balanced growth path can be converted into stationary by using an appropriate changeof variables. We do not focus on those special cases but on a generic nonstationary modelfor which a stationary representation is unknown. Second, Markov processes can benonstationary even if all the parameters and transition probabilities are time-invariant,for example, unit root and explosive processes are nonstationary. The latter kind ofnonstationary processes in not explicitly studied in the present paper.

1

2 CHAPTER 1. EXTENDED FUNCTION PATH FRAMEWORK

We introduce a quantitative framework, called extended function path (EFP), for cal-ibrating, solving, simulating and estimating the studied class of nonstationary Markovmodels. EFP is aimed to accurately approximate time-varying decision functions in anonstationary economy during a given number of periods τ . It assumes that in someremote period T τ , the economy becomes stationary, and it proceeds in two steps:First, it constructs conventional stationary Markov decision functions for the station-ary economy for t ≥ T ; and then, it finds a path of time-varying decision functions forperiods t = 0, ....T − 1 that matches the given T -period terminal condition (i.e., thestationary Markov decision functions for t ≥ T ). If T is sufficiently large relatively toτ , the EFP approximation in the first τ periods is not sensitive to the specific terminalcondition used. Therefore, we obtain an accurate approximation for time-varying deci-sion and value functions in the first τ periods (the remaining decision functions for theperiods T − τ are discarded).

EFP resembles solution methods for finite-horizon models with a given terminalcondition (e.g., life-cycle models), however, for the studied class of infinite-horizonproblems, there is no terminal condition any finite period. Hence, EFP constructs afinite-horizon approximation to an infinite-horizon nonstationary problem by using anappropriate truncation.

We develop theoretical foundations for the EFP framework in the context of theconstructed class of nonstationary Markov models. First, we provide a set of assump-tions under which the optimal decision and value functions in the T -period stationaryeconomy are state-contingent, i.e., memoryless concerning a specific history that leadsto the current state. In our case, time-dependency takes a particular tractable formfor the model’s endogenous variables, namely, the optimal decision and value functionsfollow a Markov process with possibly time-varying transition probabilities. Second, weprove a turnpike theorem that shows that a solution to the T -period stationary modelconverges to the true solution of the nonstationary models as T increases. This resultimplies that EFP is capable of approximating a solution to a nonstationary infinite-horizon problems with an arbitrary degree of precision.

Another method in the literature that can solve nonstationary infinite-horizon Mar-kov models is the extended path (EP) framework of Fair and Taylor (1983). To dealwith uncertainty, Fair and Taylor (1983) propose to use the certainty-equivalence ap-proach, namely, they replace expectation of a function across states by a value of thefunction in the expected state. The EP and EFP methods are similar in that theyboth extend the path, i.e., they both construct an approximate solution for larger timehorizon T than time horizon τ for which the solution is actually needed (to mitigatethe effect of an arbitrary terminal condition on the approximation during the initial τperiods). However, the two methods differ critically in the object they construct andin the way they approximate expectation functions, namely, Fair and Taylor’s (1983)method constructs a path for variables by using the certainty equivalence approach,while EFP constructs a path for decision functions by using more accurate integra-tion techniques such as Monte Carlo, quadrature and monomial ones. As a result,EP can accurately solve linear models with uncertainty (in such models, the certainty

1.1. INTRODUCTION 3

equivalence approach leads to exact approximation of integrals), as well as nonlinearmodels without uncertainty, whereas EFP can also accurately solve nonlinear models inwhich the certainty equivalence approach is either not applicable or leads to inaccuratesolutions.

Although numerical examples in the paper are limited to models with two or threestate variables, we design EFP in the way that makes it tractable in large-scale applica-tions. A specific combination of computational techniques that we use includes Smolyaksparse grids (see, e.g., Krueger and Kubler (2004) and Judd, Maliar, Maliar and Valero(2014)), nonproduct monomial integration methods (such methods are inexpensive andproduce more accurate approximations to integrals than Monte Carlo methods, seeJudd, Maliar and Maliar (2011) for comparison results) and derivative-free solvers (weuse Gauss-Jacobi iteration in line with Fair and Taylor (1983)); see Maliar and Maliar(2014) for surveys of these and other computational techniques that are tractable inproblems with high dimensionality (up to 100 state variables). Examples of MATLABcode are provided in webpages of the authors.

Our numerical analysis includes two parts. First, we assess the performance of EFPin a nonstationary test model with a balanced growth path for which a high-qualityapproximation is available, and we find EFP to be both accurate and reliable. Then,we apply EFP to analyze a collection of challenging nonstationary and unbalancedgrowth applications that do not admit conventional stationary Markov equilibria. Theseapplications are discussed below.

Capital augmenting technological progress. Acemoglu (2002) argues that technicalchange may be directed toward different factors of production; and Acemoglu (2003)explicitly incorporates capital augmenting technological progress into a deterministicmodel of endogenous technological change. However, the assumption of capital aug-menting technological progress is inconsistent with a balanced growth path in the stan-dard stochastic growth model but only is the assumption of labor augmenting tech-nological progress; see King, Plosser and Rebello (1988). In our first application, weuse EFP to solve a unbalanced growth model with capital augmenting technologicalprogress that does not admit a stationary Markov equilibrium. Our numerical resultsshow that business cycle fluctuations are similar in the models with capital and laboraugmenting technological progresses, however, in the former model, the growth rate ofcapital declines over time while in the latter model, it is constant.

Anticipated versus unanticipated regime switches. The literature on regime switchesfocuses on unanticipated recurring regime switches (parameter shifts); see Sims and Zha(2006), Davig and Leeper (2007, 2009), Farmer, Waggoner, and Zha (2011), Foerster,Rubio-Ramırez, Waggoner and Zha (2013) and Zhong (2015), among others. However,there are regime switches that are anticipated nonrecurrent, e.g., presidential electionswith predictable outcomes, policy announcements, anticipated legislative changes. Theidea that anticipated shocks are important for economic fluctuations is dated back toPigou (1927) and is advocated in, e.g., Cochrane (1994), Beaudry and Portier (2006),Schmitt-Grohe and Uribe (2012). An announcement of accession of new members tothe European Union produced important anticipatory effects; see Garmel, Maliar and


Maliar (2008). In our second application, we use EFP to construct a nonstationaryMarkov solution of a growth model that experiences a combination of anticipated andunanticipated technological changes. Our analysis reveals important anticipatory effectsrelatively to naive solutions in which anticipated switches in regimes are ignored.

Seasonal changes. Seasonal adjustments are a special case of anticipated regimeswitches; see Barsky and Miron (1989) for well documented evidence on the impor-tance of seasonal changes for the business cycle. Saijo (2013) argues that inadequatetreatment of seasonal changes may lead to a significant bias in the parameter estimates.Two approaches are available in the literature to study models with seasonal changes:first, Hansen and Sargent (1993, 2013) use spectral density of variables to constructperiodic optimal decision rules; and second, Christiano and Todd (2002) linearize themodel around a seasonally-varying steady state growth path and solve for four distinctdecision rules. EFP provides an alternative simple and general framework for analyzingseasonal variations. As an example, we construct a nonstationary Markov solution toa growth model with periodic anticipated seasonal changes, and we find a dramaticsmoothing effect of seasonal changes on the model’s endogenous variables.

Parameter drifting. There is ample evidence in favor of parameter drifting in eco-nomic models; see, e.g., Clarida, Galı and Gertler (2000), Lubick and Schorfheide(2004), Cogley and Sargent (2005), Goodfriend and King (2009), and Canova (2009).Furthermore, Galı (2006) argues that nonrecurrent regime changes with gradual policyvariations are empirically relevant. However, parameter drifting is generally inconsis-tent with Markov equilibrium because decision functions gradually change (drift) overtime. In our third application, we use EFP to construct a nonstationary Markov solu-tion of a stochastic growth model with parameter drifting. We again observe importantanticipatory effects relative to naive solutions in which anticipated parameter driftingis ignored.

Stochastic volatility versus deterministic trend in volatility. A large body of recentliterature documents the importance of degrees of uncertainty for the business cyclebehavior; see, e.g., Bloom (2009), Fernandez-Villaverde and Rubio-Ramırez (2010),Fernandez-Villaverde, Guerron-Quintana and Rubio-Ramırez (2010). The literaturenormally assumes that the standard deviation of exogenous shocks either follows aMarkov process or experiences recurring Markov regime switches. However, there isempirical evidence that volatility of output has a well pronounced time trend, for ex-ample, see Mc Connel and Perez-Quiros (2000), Blanchard and Simon (2001) and Stockand Watson (2003). In our experiment, we construct a nonstationary Markov solutionof a growth model in which volatility of shocks gradually decreases over time, as sug-gested by the analysis of Mc Connel and Perez-Quiros (2000). As expected, volatilityof endogenous variables in our model gradually decreases over time in response to de-creasing volatility of shocks.

Calibration and estimation of parameters in nonstationary and unbalanced growthmodels. There are econometric methods that estimate and calibrate parameters in eco-nomic models by constructing and simulating numerical solutions, including simulatedmethod of moments (e.g., Canova (2007)); Bayesian analysis (e.g., Smets and Wouters

1.1. INTRODUCTION 5

(2003), and Del Negro, Schorfheide, Smets and Wouters (2007)); and maximum like-lihood method (e.g., Fernandez-Villaverde and Rubio-Ramırez (2007)). In the fourthapplication, we illustrate how EFP can be used to calibrate and estimate parametersin an unbalanced growth model by constructing and simulating nonstationary Markovsolutions. We specifically consider a model in which the depreciation rate of capitalhas both a deterministic time trend and a stochastic cyclical component. Shocks tothe depreciation rate are introduced in, e.g., Liu, Waggoner and Zha (2011), Gourio(2012) and Zhong (2015); also, see Karabarbounis and Neiman (2014) for evidence onthe evolution of the depreciation rate over time. We simulate a time series solution us-ing the fitted parameter values, and we obtain unbalanced growth patterns that closelyresemble those observed in the data on the U.S. economy.

The proposed EFP framework is related to several streams of economic literature.First, the theoretical foundations of EFP build on early theoretical literature that stud-ies stochastic growth models with deterministically time-varying utility and productionfunctions; see Majumdar and Zilcha (1987), Mitra and Nyarko (1991) and Joshi (1997).1

Our main contributions to that literature is that we propose a practical approach to con-structing numerical solutions of an empirically-relevant class of nonstationary Markovmodels, whereas the previous literature was limited to purely theoretical existence re-sults.

Second, EFP is related to conventional numerical techniques for constructing sta-tionary Markov decision functions. Although such techniques cannot be used for solvingthe studied class of nonstationary models, they can be used as ingredients of EFP, in-cluding projection techniques (see, e.g., Judd (1992), Christiano and Fisher (2000),Maliar and Maliar (2015)); perturbation techniques (see, e.g., Judd and Guu (1993),Collard and Juillard (2001), Schmitt-Grohe and Uribe (2004)); stochastic simulationtechniques (see, e.g., Den Haan and Marcet (1990), Judd, Maliar and Maliar (2011));and numerical dynamic programming techniques, in particular, those designed to dealwith large scale applications (see, e.g., Smith (1993), Rust (1996), Carroll (2005), Maliarand Maliar (2013)).2 There are two classes of methods that are particularly close toEFP. First, these are the methods for analyzing life-cycle models, developed in Kruegerand Kubler (2004, 2006) and Hasanhodzic and Kotlikoff (2013). However, in a life-cycleeconomy, the terminal condition is either given or is a choice variable (as in a life-cyclemodel with bequests); see Rıos-Rull (1999) and Nishiyama and Smetters (2014) for re-views of the literature on life-cycle economies. In turn, terminal condition is unknown

1In turn, this literature on nonstationary models builds on mathematical tools developed for sta-tionary models in Brock and Gale (1969), Brock (1971), Brock and Mirman (1972, 1973), McKenzie(1976), Mirman and Zilcha (1977), Brock and Majumdar (1978), Mitra and Zilcha (1981), amongothers.

2Surveys of numerical solution methods for stationary Markov models include Taylor and Uhlig(1990), Gaspar and Judd (1997), Judd (1998), Marimon and Scott (1999), Santos (1999), Miranda andFackler (2002), Adda and Cooper (2003), Aruoba, Fernandez-Villaverde and Rubio-Ramırez, (2006),Kendrik, Ruben-Mercado and Amman (2006), Canova (2007), Heer and Maußner (2010), Lim andMcNelis (2008), Stachurski (2009), Kollmann, Maliar, Malin and Pichler (2011) and Maliar and Maliar(2014).


in our infinite horizon nonstationary economy and must be constructed in a way thatensures asymptotic convergence of the EFP approximation to the true infinite horizonsolution. Second, EFP is related to a perturbation-based method of Schmitt-Groheand Uribe (2012) that can solve models with anticipated shocks of a fixed horizon (e.g.,shocks that happen each fourth or eight periods), however, unlike this method, EFPcan handle anticipated shocks of any periodicity and duration.

Finally, EFP is related to numerical methods that construct a path for variables,in particular, shooting methods for deterministic economies introduced to economicsin Lipton, Poterba, Sachs and Summers (1980) and an extended path method foreconomies with uncertainty proposed by Fair and Taylor (1983).3 The related literaturealso includes solution methods for continuous time models studied in Chen (1999); aframework for characterizing equilibrium in life-cycle models with a deterministic ag-gregate path and idiosyncratic uncertainty, proposed by Conesa and Krueger (1999); aparametric path method of Judd (2002); and a predictive control method, developed inGrune, Semmler and Stieler (2013); see also Atolia and Buffie (2009 a,b) for a carefuldiscussion of shooting methods. The main shortcoming of this class of methods is thatthe assumption of certainty equivalence does not always provide sufficiently accurateapproximations of expectation functions in nonlinear models. Adjemian and Juillard(2013) propose a stochastic extended path method that approximates expectation func-tions more accurately by constructing and averaging multiple paths for variables underdifferent sequences of exogenous shocks. EFP differs from this literature in the way itdeals with uncertainty, specifically, EFP constructs time-varying state-contingent de-cision functions that include stochastic shocks as additional arguments, whereas theabove literature constructs one or several paths for endogenous variables.

The rest of the paper is as follows: In Section 1.2, we construct a class of nonsta-tionary Markov models. In Section 1.3, we introduce EFP and provide its theoreticalfoundations. In Section 1.4, we describe the relation of EFP to the literature. In Section1.5, we assess the performance of EFP in a nonstationary test model with a balancedgrowth path. In Section 1.6, we solve a collection of nonstationary applications; andfinally in Section 1.7, we conclude.

1.2 A class of nonstationary Markov economies

We study a class of infinite-horizon nonlinear dynamic economic models in which pref-erences, technology and laws of motion for exogenous variables can change over timeeither deterministically or stochastically, according to a Markov process with possiblytime-varying transition probabilities, or both. The constructed class of models is non-stationary because the optimal decision and value functions are time dependent. The

3Earlier literature was aware that methods solving for a path of variables can be used in the contextof nonstationary problems. In particular, Lipton, Poterba, Sachs and Summers (1980, p.2) say ”...we allow for a possibility that F [model’s equations] may be time dependent (i.e., non-autonomous)”.Also, Fair and Taylor (1983) explicitly assume that the model’s equations are time dependent.

1.2. A CLASS OF NONSTATIONARY MARKOV ECONOMIES 7

existence theorems for stochastic growth models with time-varying fundamentals areestablished in Majumdar and Zilcha (1987), Mitra and Nyarko (1991) and Joshi (1997),among others.

1.2.1 The stochastic environment

Our exposition relies on standard measure theory notation; see, e.g., Stokey and Lucaswith Prescott (1989), Santos (1999) and Stachurski (2009). Time is discrete and infinite,t = 0, 1, .... Let (Ω,F , P ) be a probability space:

a) Ω = Π∞t=0Ωt is a space of sequences ε ≡ (ε0, ε1...) such that εt ∈ Ωt for all t, whereΩt is a compact metric space endowed with the Borel σ–field Et. Here, Ωt is theset of all possible states of the environment at t and εt ∈ Ωt is the state of theenvironment at t.

b) F is the σ–algebra on Ω generated by cylinder sets of the form Π∞τ=0Aτ , whereAτ ∈ Eτ for all τ and Aτ = Ωτ for all but finitely many τ .

c) P is the probability measure on (Ω,F).

We denote by Ft a filtration on Ω, where Ft is a sub σ–field of F induced bya partial history up of environment ht = (ε0, ..., εt) ∈ Πt

τ=0Ωτ up to period t, i.e., Ftis generated by cylinder sets of the form Πt

τ=0Aτ , where Aτ ∈ Eτ for all τ ≤ t andAτ = Ωτ for τ > t. In particular, we have that F0 is the course σ–field 0,Ω, and thatF∞ = F . Furthermore, if Ω consists of either finite or countable states, ε is called adiscrete state process or chain; otherwise, it is called a continuous state process. Ouranalysis focuses on continuous state processes, however, can be generalized to chainswith minor modifications.

1.2.2 A nonstationary optimization problem

As an example, we consider a nonstationary stochastic growth model in which prefer-ences, technology and laws of motion for exogenous variables change over time:

maxct,kt+1∞t=0

E0

[∞∑t=0

βtut (ct)

](1.1)

s.t. ct + kt+1 = (1− δ) kt + ft (kt, zt) , (1.2)

zt+1 = ϕt (zt, εt+1) , (1.3)

where ct ≥ 0 and kt ≥ 0 denote consumption and capital, respectively; initial condition(k0, z0) is given; ut : R+ → R and ft : R2

+ → R+ and ϕt : R2 → R are possiblytime-dependent utility function, production functions and law of motion for exogenousvariable zt, respectively; the sequence of ut, ft and ϕt for t ≥ 0 is known to the agent inperiod t = 0; εt+1 is i.i.d; β ∈ (0, 1) is the discount factor; δ ∈ [0, 1] is the depreciationrate; and Et [·] is an operator of expectation, conditional on a t-period information set.


Stationary models. A well-known special case of the general setup (3.25)–(3.28) isa stationary Markov models in which ut ≡ u, ft ≡ f and ϕt ≡ ϕ. Such a model has astationary Markov solution in which the value function V (kt, zt) and decision functionskt+1 = K (kt, zt) and ct = C (kt, zt) are both state-contingent and time-invariant; see,e.g., Stokey and Lucas with Prescott (1989, p. 391).

Nonstationary models. In a general case, a solution to the model (3.25)–(3.28)is nonstationary. The decision functions of endogenous variables ct and kt can betime-dependent for two reasons: first, because ut and ft change over time; and second,because ϕt and consequently, the transition probabilities of exogenous variable zt changeover time.

Remark 1. For presentational convenience, we assume that only zt is a random variablefollowing a Markov process with possibly time-varying transition probabilities, while theother model’s parameters evolve in a deterministic manner, i.e., the sequence of ut, ftand ϕt for all t ≥ 0 is deterministic. However, the quantitative framework we developin the paper can be used to solve models in which β, δ, as well as the parameters ofut, ft and ϕt, are all random variables, following a Markov process with time-varyingtransition probabilities. In particular, in Section 6, we consider a version of the modelin which δ follows a Markov process with time-varying transition probabilities.

1.2.3 Assumptions about exogenous variable

We provide some definitions that will be useful for explaining the assumption (3.28)about the Markov process for exogenous variable zt; these definitions are standard andclosely follow Stokey and Lucas with Prescott (1989, Ch. 8.2).

Definition 1. (Stochastic process). A stochastic process on (Ω,F , P ) is an increasingsequence of σ–algebras F1 ⊆ F2 ⊆ ... ⊆ F ; a measurable space (Z,Z); and a sequenceof functions zt : Ω→ Z for t ≥ 0 such that each zt is Ft measurable.

Stationarity is commonly used assumption in economic literature.

Definition 2. (Stationary process). A stochastic process z on (Ω,F , P ) is called sta-tionary if the unconditional probability measure, given by

Pt+1,...,t+n (C) = P (ε ∈ Ω : [zt+1 (ε) , ..., zt+n (ε)] ∈ C) , (1.4)

is independent of t for all C ∈ Zn, t ≥ 0 and n ≥ 1.

A related notion is stationary (time-invariant) transition probabilities. Let us denoteby Pt+1,...,t+n (C|zt = zt, ..., z0 = z0) the probability of the event

ε ∈ Ω : [zt+1 (ε) , ..., zt+n (ε)] ∈ C ,


given that the event ε ∈ Ω : zt = zt (ε) , ..., z0 = z0 (ε) occurs.

Definition 3. (Stationary transition probabilities). A stochastic process z on (Ω,F , P )is said to have stationary transition probabilities if the conditional probabilities

Pt+1,...,t+n (C|zt = zt, ..., z0 = z0) (1.5)

are independent of t for all C ∈ Zn, ε ∈ Ω, t ≥ 0 and n ≥ 1.

The assumption of stationary transition probabilities (1.5) implies the property of sta-tionarity (1.4) provided that the corresponding unconditional probability measures ex-ist. However, a process can be nonstationary even if transition probabilities are sta-tionary, for example, a unit root process or explosive process is nonstationary; seeStokey and Lucas with Prescott (1989, Ch 8.2) for a related discussion. This kind ofnonstationary processes is not studied explicitly in the present paper, i.e., we focus onnonstationarity that arises because transition probabilities change from one period toanother.

In general, Pt+1,...,t+n (C) and Pt+1,...,t+n (C|·) depend on the entire history of theevents up to t (i.e., the stochastic process zt is measurable with respect to the subσ–field Ft). However, history-dependent processes are difficult to analyze in a generalcase. It is of interest to distinguish special cases in which the dependence on history hasrelatively simple and tractable form. A well-known case is a class of Markov processes.

Definition 4. (Markov process). A stochastic process z on (Ω,F , P ) is (first-order)Markov if

Pt+1,...,t+n (C|zt = zt, ..., z0 = z0) = Pt+1,...,t+n (C|zt = zt) , (1.6)

for all C ∈ Zn, t ≥ 0 and n ≥ 1.

The key property of a Markov process is that it is memoryless, namely, all past history(zt, ..., z0) is irrelevant for determining the future realizations except of the most recentpast zt.

The literature typically assumed exogenous variables that are both stationary andMarkov. As follows from (3.28), we maintain the assumption of Markov process (1.6),however, we relax the restriction of stationarity (1.4), namely, we allow for the casewhen transition probabilities (1.5) of zt change over time. Below, we show an exam-ple that illustrates the type of stochastic processes that will be used in this paper formodeling exogenous variables.

Example 1. Consider a first-order autoregressive process

zt+1 = ρtzt + σtεt+1, (1.7)

where the sequences (ρ0, ρ1, ...) and (σ0, σ1, ...) are deterministically given at t = 0 andεt+1 ∼ N (0, 1). The conditional probability distribution zt+1 ∼ N (ρtzt, σt) depends


only on the most recent past zt = zt and is independent of history (zt, ..., z0) as requiredby (1.6) and hence, the process is Markov. However, if ρt and σt change over time, thenthe distribution N (ρtzt, σ

2t ) depends not only on state zt = zt but also on a specific

period t, so that transitions (1.5) are not stationary, and as a result, the process isnonstationary since it does not have time-invariant unconditional probability measure(1.4). If ρt = ρ and σt = σ for all t, then the conditional probability distributionN (ρzt, σ

2) depends only on state zt = zt but not on time, and the transitions arestationary. If, in addition, ρ < 1, then the process is stationary in the sense ((1.4)).Finally, ρt = 1 for all t corresponds to a unit root process, which is nonstationary evenif σt = σ for all t; and |ρt| > 1 for all t leads to an explosive process. As we said earlier,unit root and explosive nonstationary processes are not explicitly studied in the presentpaper.

Remark 2. Mitra and Nyarko (1991) refer to a class of Markov processes with non-stationary transition probabilities as semi-Markov processes because of their certainsimilarity to Levy’s (1954) generalization of the Markov renewal process for the caseof random arrival times; see Jansen and Manca (2006) for a review of applications ofsemi-Markov processes in statistics, operation research and other fields.

1.2.4 Assumptions about the utility and production functions

We make standard assumptions about the utility and production functions that ensureexistence, uniqueness and interiority of a solution. Concerning the utility function ut,we impose the following assumptions for each t ≥ 0:

Assumption 1. ut is twice continuously differentiable on R+.Assumption 2. u′t > 0, i.e., ut is strictly increasing on R+, where u′t ≡ ∂ut

∂c.

Assumption 3. u′′t < 0, i.e., ut is strictly concave on R+, where u′′t ≡ ∂2ut∂c2

.Assumption 4. ut satisfies the Inada conditions lim

c→0u′t (c) = +∞ and lim

c→∞u′t (c) = 0.

Concerning the production function ft, we make the following assumptions for eacht ≥ 0:

Assumption 5. ft is twice continuously differentiable on R2+.

Assumption 6. f ′t (k, z) > 0 for all k ∈ R+ and z ∈ R+, where f ′t ≡∂ft∂k

.

Assumption 7. f ′′t (k, z) ≤ 0 for all k ∈ R+ and z ∈ R+, where f ′′t ≡∂2ft∂k2

.Assumption 8. ft satisfies the Inada conditions lim

k→0f ′t (k, z) = +∞ and lim

k→∞f ′t (k, z) =

0 for all z ∈ R+.

Let us define a pure capital accumulation process kmaxt ∞t=0 by assuming ct = 0 for all

t in (3.27) which for each history ht = (z0, ..., zt), leads to

kmaxt+1 = ft (kmax

t , zt) , (1.8)


where kmax0 ≡ k0. We impose an additional joint boundedness restriction on preferences

and technology by using the constructed process (1.8):

Assumption 9. (Bounded objective function). E0 [∑∞

t=0 βtut (kmax

t )] <∞.

This assumption insures that the objective function (3.25) is bounded so that its max-imum exists. In particular, Assumption 9 holds either (i) when ut is bounded fromabove for all t, i.e., ut (c) < ∞ for any c ≥ 0 or (ii) when ft is bounded from abovefor all t, i.e., ft (k, zt) < ∞ for any k ≥ 0 and zt ∈ Zt. However, it also holds foreconomies with nonvanishing growth and unbounded utility and production functionsas long as ut (kmax

t ) does not grow too fast so that the product βtut (kmaxt ) still declines

at a sufficiently high rate and the objective function (3.25) converges to a finite limit.

1.2.5 Optimal program

Definition 1 (Feasible program). A feasible program for the economy (3.25)–(3.28)is a pair of adapted (i.e., Ft measurable for all t) processes ct, kt∞t=0 such that giveninitial condition k0 and history h∞ = (ε0, ε1...), they satisfy ct ≥ 0, kt ≥ 0 and (3.27)for all t.

We denote by =∞ a set of all feasible programs from given initial capital k0 and givenhistory h∞ = (ε0, ε1...). We next introduce the concept of solution of the studied model.

Definition 5 (Optimal program). A feasible program c∞t , k∞t ∞t=0 ∈ =∞ is called

optimal if

E0

[∞∑t=0

βt ut (c∞t )− ut (ct)

]≥ 0 (1.9)

for every feasible process ct, kt∞t=0 ∈ =∞.

Stochastic models with time-varying fundamentals are studied in Majumdar and Zilcha(1987), Mitra and Nyarko (1991) and Joshi (1997), among others. The existence resultsfor this class of models have been established in the literature for a general measurablestochastic environment, i.e., in the absence of restriction of Markov process (3.28). Inparticular, this literature shows that, under assumptions Assumptions 1-9, there existsan optimal program c∞t , k∞t

∞t=0 ∈ =∞ in the economy (3.25), (3.27), and it is both

interior and unique; see Theorem 4.1 in Mitra and Nyarko (1991) and see Theorem 7in Majumdar and Zilcha (1987). The results of this literature apply to us as well.

Remark 3. The existence of the optimal program in the economy (3.25)–(3.28) canbe shown under weaker assumptions. For example, Mitra and Nyarko (1991) use ajoint boundedness restriction on preferences and technology (the so-called Condition


E ) that is less restrictive than our Assumption 9; Joshi (1997) characterizes the optimalprograms in nonconvex economies by relaxing our Assumptions 7 and 8, etc.

While the previous literature establishes the existence and uniqueness results for theconstructed class of nonstationary model for a general non-Markov stochastic environ-ment, it does not offer a practical approach for constructing time-dependent solutionsin applications. In contrast, we will see that our additional Markov restriction (3.28)on zt leads to a tractable class of nonstationary Markov models for which the solutionscan be characterized both analytically and numerically.

1.3 Extended function path framework

We introduce a quantitative framework, which we call extended function path (EFP)framework, for approximating an optimal program in the nonstationary Markov econ-omy (3.25)–(3.28). In Section 1.3.1, we present the EFP framework, and in Section1.3.2, we develop its theoretical foundations.

1.3.1 Introducing extended function path framework

To approximate the optimal program in the nonstationary economy (3.25)–(3.28), weintroduce a supplementary economy that becomes stationary in some remote period T .

Definition 5 (T -period stationary economy). A T -period stationary economy is theversion of the economy (3.25)–(3.28) in which the utility and production functions andthe laws of motions for exogenous variables are time invariant for t ≥ T , i.e., ut = u,ft = f and ϕt = ϕ for all t ≥ T .

The key idea of our EFP framework is to approximate an optimal program in thenonstationary Markov economy (3.25)–(3.28) during the initial τ periods using a sup-

1.3. EXTENDED FUNCTION PATH FRAMEWORK 13

plementary T -period stationary economy.

Algorithm 1: Extended function path.

Step 0. Initialization. Choose some T τ and construct T -period stationaryeconomy such that ut = u, ft = f and ϕt = ϕ for all t ≥ T .

Step 1. Construct a stationary Markov solution to T -period stationaryeconomy, i.e., find a stationary capital function K satisfying:u′(c) = βE [u′(c′)(1− δ + f ′ (k′, ϕ (z, ε′)))]c = (1− δ) k + f (k, z)− k′c′ = (1− δ) k′ + f (k′, ϕ (z, ε′))− k′′k′ = K (k, z) and k′′ = K (k′, ϕ (z, ε′)).

Step 2. Construct a path for capital policy functions (K0, ..., KT ) that matchesthe terminal condition KT ≡ K and satisfies for t = 0, ...T − 1:u′t(ct) = βEt

[u′t+1(ct+1)(1− δ + f ′t+1 (kt+1, ϕt (zt, εt+1)))

]ct = (1− δ) kt + ft (kt, zt)− kt+1

ct+1 = (1− δ) kt+1 + ft+1 (kt+1, ϕt (zt, εt+1))− kt+2

kt+1 = Kt (kt, zt) and kt+2 = Kt+1 (kt+1, ϕt (zt, εt+1)) .

The first τ functions (K0, ..., Kτ ) constitute an approximate solution andthe remaining T − τ functions (Kτ+1, ..., KT ) are discarded.

A useful property of T -period stationary economy is that its optimal program is easyto characterize. First, since the economy (3.25)–(3.28) becomes stationary at T , theoptimal program is stationary Markov for t ≥ T , and Step 1 of EFP can be implementedby using conventional solution methods. Second, given the terminal condition producedby the T -period stationary economy, the sequence of T − 1 Euler equations identifiesuniquely a path for decision functions for t = 0, ..., T − 1. To construct such a path, wecan use backward induction, namely, given the capital function KT , we use the Eulerequation to compute the capital function KT−1 at T − 1; given KT−1, we use it tocompute KT−2; and so on until the entire path (KT , ..., K0) is constructed.

The term extended path indicates that EFP constructs a path of functions for largertime horizon T than the number of periods τ for which an approximate solution isactually needed, i.e., EFP extends the path from τ to T . In this respect, EFP is similarto extended path (EP) framework of Fair and Taylor (1983). By choosing sufficientlylarge T , both EFP and EP mitigate the effect of specific terminal condition on theapproximation during the initial τ periods. In turn, the term path versus function pathhighlights the key difference between the EP and EFP methods: the former methodconstructs a path for variables, whereas the latter method constructs a path for decisionfunctions. To approximate expectation functions, Fair and Taylor (1983) method relieson the assumption of certainty equivalence while the EFP method uses more accurate


integration methods such as Monte Carlo, Gauss-Hermite quadrature and monomialsmethods. As a result, EP can accurately solve linear models with uncertainty (insuch models, the certainty equivalence assumption leads to an exact approximationfor integrals), as well as nonlinear models without uncertainty, whereas EFP can alsoaccurately solve nonlinear models in which the certainty equivalence approach is eithernot applicable or leads to inaccurate solutions; in Section 1.4.3, we discuss the relationbetween EP and EFP in more details.

We implement EFP by using a combination of three techniques. First, to approxi-mate decision functions, we use Smolyak (sparse) grids. Second, to approximate expec-tation functions, we use a nonproduct monomial integration rule. Finally, to solve forcoefficients of the policy functions, we use a Gauss-Jacobi method, which is a derivative-free fixed-point-iteration method in line with Fair and Taylor (1983). The implemen-tation details are described in Section 1.5.1 and Appendix B.

In Figure 1, we illustrate a sequence of functions (function path) produced by EFPfor a version of the model (3.25)–(3.28) with exogenous growth due to labor augmentingtechnological progress (the model’s parameterization and implementation details aredescribed in Section 1.5).

Figure 1.1: Function path, produced by EFP, for a growth model with technologicalprogress

We plot the capital functions for periods 1, 20 and 40, (i.e., k2 = K1 (k1, z1), k21 =K20 (k20, z20) and k41 = K40 (k40, z40)) which we approximate using Smolyak (sparse)grids. Here, in Step 1, we construct the capital function K40 by assuming that the


economy becomes stationary in period T = 40; and in Step 2, we construct a path ofthe capital functions that (K1, ...K39) that matches the corresponding terminal functionK40. The Smolyak grids are shown by stars in the horizontal kt× zt plane. The domainfor capital (on which Smolyak grids are constructed) and the range of the constructedcapital function grow at the rate of labor augmenting technological progress.

Remark 4. In the paper, we analyze just one specific combination implementation ofEFP but there are many ways in which EFP can be implemented: First, to constructdecision functions, we can use a variety of grid techniques, integration rules, approxima-tion methods, iteration schemes, etc. that are used by conventional solution methods.Second, to construct a function path, we can use any method that solves a systemof nonlinear equations, including Newton-style solvers, Gauss-Siedel iteration used byshooting methods, Gauss-Jacobi iteration used by Fair and Taylor’s (1983) method, etc.Since EFP constructs not just one but many decision functions (i.e., a separate decisionfunction in each time period), we prefer techniques that have relatively low computa-tional expense. Furthermore, to make EFP tractable in large-scale applications, weopt for techniques whose cost does not rapidly increase with the dimensionality of theproblem (number of state variables). Maliar and Maliar (2014) survey techniques thatare designed to deal with large-scale problems, including nonproduct sparse grids, sim-ulated grids, cluster grids and epsilon-distinguishable-set grids; nonproduct monomialand simulation based integration methods, and derivative-free solvers.

Remark 5. The property of the T -period stationary economy that is essential forour analysis is that decision functions are stationary Markov at T . In our baselineimplementation of EFP, we attain this property by assuming that the preferences,technology and laws of motion for exogenous variables do not change starting fromt = T , i.e., ut = uT , ft = fT and ϕt = ϕT for all t ≥ T . Instead, we can use otherassumptions that lead to Markov decision functions at T , for example, we can assumethat at T , the economy switches to a balanced growth path. Furthermore, we canassume that the economy arrives at a zero capital stock at T with the correspondingtrivial Markov solution kt = 0 for all t ≥ T (this case allows for standard interpretationof a finite horizon economy). Finally, we can use some T -period Markov terminalcondition K (k, z) without specifying explicitly an economic model that generates thisterminal condition.

Remark 6. We have described a variant of EFP that constructs time-dependent capitalfunctions (K0, ..., Kτ ). Similarly, we can formulate a variant of EFP that constructstime-dependent value functions (V0, ..., Vτ ). Such a value-iterative EFP first solves forVT = VT+1 = V for the T -period stationary economy and then it solves for a path(VT−1, ..., V0) that satisfies the sequence of the Bellman equations for t = 0, ..., T andthat meets the terminal condition VT of the T -period stationary economy.


1.3.2 Theoretical foundations of EFP framework

We now develop theoretical foundations of the EFP framework. We prove two theorems:Theorem 1 shows that the optimal program in the T -period stationary economy is givenby a Markov process with possibly time-varying transition probabilities; and Theorem2 shows that the optimal program of the T -period stationary economy converges tothe optimal program of the original nonstationary Markov economy (3.25)–(3.28) as Tincreases.

Theorem 1 (Optimal program of the T -period stationary economy). In the T -periodstationary economy (3.25)–(3.28), the optimal program is given by a Markov processwith possibly time-varying transition probabilities.

Proof. Under Assumptions 1-9, first-order conditions (FOCs) are necessary for opti-mality. We will show that FOCs are also sufficient both to identify the optimal programand to establish its Markov structure. Our proof is constructive: it relies on backwardinduction and includes two steps that correspond to Steps 1 and 2 of EFP, respectively.

Step 1. At T , the economy becomes stationary and remains stationary forever, i.e.,ut ≡ u, ft ≡ f and ϕt ≡ ϕ for all t ≥ T . Thus, the model’s equations and decisionfunctions are time invariant for t ≥ T . It is well known that under Assumptions 1-9,there is a unique stationary Markov capital function K that satisfies the optimalityconditions that are listed in Step 1 of Algorithm 1; see, e.g., Stokey and Lucas withPrescott (1989, p. 391).

Step 2. Given the constructed T -period capital function KT ≡ K, we define thecapital functions KT−1, ..., K0 in previous periods by using backward induction. As afirst step, we write the Euler equation for period T − 1,

u′T−1(cT−1) = βET−1 [u′T (cT )(1− δ + f ′T (kT , zT ))] , (1.10)

where cT−1 and cT are related to kT and kT+1 in periods T and T − 1 by

cT−1 = (1− δ) kT−1 + fT−1 (kT−1, zT−1)− kT , (1.11)

cT = (1− δ) kT + fT (kT , zT )− kT+1. (1.12)

By assumption (3.28), zT follows a Markov process, i.e., zT = ϕT (zT−1, ε). Furthermore,by construction of the decision function K in Step 1, we have that kT+1 = KT (kT , zT )is a Markov decision function. By substituting these two results into (1.10)–(3.19),we obtain a functional equation that defines kT for each possible state (kT−1, zT−1).Therefore, the capital decisions at period T −1 are given by a state-contingent functionkT = KT−1 (kT−1, zT−1), i.e., capital decisions today are independent of history thatleads to the current state. However, the constructed decision functions depend on theparameters of the utility and production functions and the law of motions for shocksin periods T − 1 and T , and it is not generally true that KT−1 6= KT . By proceedingiteratively backward, we construct a sequence of state-contingent and possibly time-dependent capital functions KT−1 (kT−1, zT−1) , ..., K0 (k0, z0) that satisfies (1.10)–(3.19)


for t = 0, ..., T − 1 and that matches terminal function KT (kT , zT ). Definitions 3and 4 imply that kt+1 follows a Markov process with possibly time-varying transitionprobabilities.

We now show that the optimal program of the T -period stationary economy approx-imates arbitrary well the optimal program of the nonstationary economy (3.25)–(3.28)as T increases. Our analysis is related to the literature that shows asymptotic con-vergence of the optimal program of the finite horizon economy to that of the infinitehorizon economy; see, e.g., McKenzie (1976) and Joshi (1997). This kind of conver-gence results is referred to as turnpike theorems. Our T -period stationary economy canbe interpreted as a finite horizon economy characterized by a specific nonzero terminalcondition; we therefore also refer to our convergence result as a turnpike theorem.

Let us fix history h∞ = (ε0, ε1...) and initial condition (k0, z0) and construct theproductivity levels ztTt=0 using (3.28). We then use the constructed sequence of capitalfunctions K0 (k0, z0) , ..., KT (kT , zT ) to generate the optimal program

cTt , k

Tt

∞t=0

forthe T -period stationary economy such that

kTt+1 = Kt

(kTt , zt

), (1.13)

where kT0 = k0 and cTt satisfies the budget constraint (3.27) for all t ≥ 0. Then, we havethe following result.

Theorem 2 (Turnpike theorem): For any real number ε > 0 and any natural numberτ , there exists a threshold terminal date T (ε, τ) such that for any T ≥ T (ε, τ), we have∣∣k∞t − kTt ∣∣ < ε, for all t ≤ τ , (1.14)

where c∞t , k∞t ∞t=0 ∈ =∞ is the optimal program in the nonstationary economy (3.25)–

(3.28), andcTt , k

Tt

Tt=0

is the optimal program (1.13) in the T -period stationary econ-omy.

Proof. See Appendix A.

The convergence is uniform: Our turnpike theorem states that for all T ≥ T (ε, τ), theconstructed nonstationary Markov approximation

kTt

is guaranteed to be within agiven ε-accuracy range from the true solution k∞t during the initial τ periods (forperiods t > τ , our approximation may become insufficiently accurate and exit the ε-accuracy range). The name turnpike theorem emphasizes the idea that turnpike isoften the fastest route between two points which are far apart even if it is not a directroute. In terms of the studied model, this means that the optimal program of theT -period stationary economy

kTt

follows for a long time the optimal program of thenonstationary economy k∞t (turnpike) and it deviates from turnpike only at the endto meet a given terminal condition (i.e. the final destination off turnpike).

In Figure 2, we illustrate the convergence of the optimal program of the T -periodstationary economy to that of the original nonstationary economy. We again considera version of the model with long-run growth due to labor augmenting technological


progress (the parameterization of the model and the implementation details are de-scribed in Section 1.5). We fix the same initial condition and realization of shocks inall experiments. Here, k∞t denotes the true solution to the infinite-horizon nonstation-ary model (3.25)–(3.28) and kL, kT , k′ and k′′ denote the corresponding solutions tofinite-horizon economies characterized by different terminal conditions.

0

5

10

15

20

25

30

time

k t

k0

T

kT

kτ

τ

k’

kT∞

k’’

Figure 1.2: Convergence of the optimal program of T -period stationary economy

We observe the convergence of the simulated path of T -period stationary economyto that of the nonstationary economy under all terminal conditions considered. How-ever, the convergence is faster under terminal conditions κ′ and κ′′, that are locatedrelatively close to the true T -period capital k∞T of the nonstationary economy, thenunder zero terminal condition that is located farther away from the true solution. Itis clear that a zero-capital terminal condition is not an efficient choice for constructingan approximation to the infinite horizon nonstationary economy, namely, in the infinitehorizon economy, capital grows all the time, whereas in the finite horizon economy, cap-ital needs to turn down at some point to meet a zero terminal condition. Our T -periodstationary economy delivers more efficient terminal condition than the conventionalfinite-horizon approximation.

Remark 7. Our turnpike theorem shows the convergence of the optimal program ofthe T -period stationary economy to that of a nonstationary economy for a given initialcondition and given history. It is classified as an early turnpike theorem in the liter-ature; there are also medium and late turnpike theorems that prove the convergence

1.4. RELATION OF EFP TO THE LITERATURE 19

by varying initial conditions and history, respectively; see McKenzie (1976) and Joshi(1997) for discussion. We do not prove other turnpike theorems for our T -period station-ary economy because they are not directly related to the EFP quantitative frameworkintroduced in the present paper.

1.4 Relation of EFP to the literature

EFP is related to three main steams of literature: (1) early theoretical literature thatstudies properties of solutions of non-Markov stochastic growth models; (2) literature onnumerical methods for constructing solutions to stationary Markov models; (3) finally,literature on solving for a path for variables.

1.4.1 Early literature on stochastic growth models

The early literature provides theoretical foundations for stochastic growth models andcharacterizes the properties of their solutions; see Brock and Gale (1969), Brock (1971),Brock and Mirman (1972, 1973), Mirman and Zilcha (1977), Brock and Majumdar(1978), Mitra and Zilcha (1981), among others. In particular, Majumdar and Zilcha(1987), Mitra and Nyarko (1991), and Joshi (1997) study infinite horizon economieswith deterministically time-varying utility and production functions similar to ours.However, this literature is limited to purely theoretical analysis and does not offerpractical methods for constructing their nonstationary solutions in applications.

Our main contributions relative to that literature are that we distinguish a tractableMarkov class of nonstationary models and propose an EFP framework for analyzingquantitative implications of such models. In addition, we show new formal results.First, our Theorem 1 establishes Markov structure of the optimal program in the T -period stationary economy while the previous literature establishes similar results fora finite-horizon economy with a zero terminal condition; e.g., Mitra and Nyarko (1991,Theorem 4.3). Second, our Theorem 2 (turnpike theorem) focuses on terminal condi-tion that is generated by a Markov solution to a class of T -period stationary economies,whereas the turnpike theorems existing in the literature assumes a specific zero termi-nal condition, kT = 0 representing a finite-horizon economy; see, e.g., Majumdar andZilcha (1987), Mitra and Nyarko (1991) and Joshi (1997). Our two theorems providetheoretical foundations for the EFP framework.

1.4.2 Methods constructing Markov decision functions

The mainstream of economic literature relies on stationary Markov models. There isa variety of methods for constructing solutions to such models, in particular, projec-tion methods (see, e.g., Judd (1992), Christiano and Fisher (2000), Maliar and Maliar(2015)); perturbation methods (see, e.g., Judd and Guu (1993), Collard and Juillard(2001), Schmitt-Grohe and Uribe (2004)); and stochastic simulation methods (see, e.g.,


Den Haan and Marcet (1990), Judd, Maliar and Maliar (2011)); and numerical dynamicprogramming methods, in particular, those designed to deal with large scale applica-tions (see, e.g., Smith (1993), Rust (1996), Carroll (2005), Maliar and Maliar (2013)).For surveys of such methods, see Taylor and Uhlig (1990), Rust (1996), Gaspar andJudd (1997), Judd (1998), Marimon and Scott (1999), Santos (1999), Miranda andFackler (2002), Adda and Cooper (2003), Aruoba, Fernandez-Villaverde and Rubio-Ramırez (2006), Kendrik, Ruben-Mercado and Amman (2006), Canova (2007), Heerand Maußner (2010), Lim and McNelis (2008), and Stachurski (2009), Den Haan (2010).In particular, Kollmann, Maliar, Malin and Pichler (2011) and Maliar and Maliar (2014)survey numerical methods for analyzing problems with high dimensionality. The con-ventional methods for constructing stationary Markov solutions cannot generally beused for solving models with time varying parameters studied in the present paper,however, the techniques used by these conventional methods can be used as ingredientsof EFP, including a variety of grid techniques, integration methods, numerical solvers,etc.

There are three groups of Markov methods that EFP is particularly close to. Firstof all, EFP is related to numerical methods that construct decision functions in life-cycle models as in Krueger and Kubler (2004, 2006) and Hasanhodzic and Kotlikoff(2013). The decision functions in such models change from one generation to another,and the sequence of the generation-specific decision functions resembles a function pathconstructed by EFP; see Rıos-Rull (1999) and Nishiyama and Smetters (2014) for re-views of the literature on life-cycle economies. The difference is that terminal conditionis either known in the life-cycle economy or is a choice variable (as in an economywith bequests), while it is unknown in our infinite-horizon economy and must be con-structed in the way that ensures the convergence of an EFP approximation to the truenonstationary solution.

Furthermore, EFP is related to economic literature that studies Markov nonsta-tionary models with balanced growth paths; see King, Plosser and Rebello (1988) forrestrictions on preferences and technology that are consistent with a balanced growthpath. However, this class of models is limited; for example, models with labor aug-menting technological progress are generally consistent with a balanced growth pathbut not models with either capital augmenting or Hicks neutral technological progress.There are examples of constructing balanced growth path for some models that do notsatisfy the restrictions of King, Plosser and Rebello (1988) but they are also limited.4

Finally, EFP is related to the literature that incorporates certain kinds of nonsta-tionarity by augmenting the economic models to include additional state variables. Inparticular, Bloom (2009), Fernandez-Villaverde and Rubio-Ramırez (2010), Fernandez-Villaverde, Guerron-Quintana and Rubio-Ramırez (2010), among others, argue that

4Two examples are as follows: Maliar and Maliar (2004) shows the existence of a balanced growthpath in a model with endogenous growth and cycles by removing a common stochastic trend represent-ing randomly arriving technological innovations; and Maliar and Maliar (2010) constructs a balancedgrowth path in a model with capital-skill complementarity and several types of technical progress byimposing additional restrictions on growth rates of variables.

1.4. RELATION OF EFP TO THE LITERATURE 21

the behavior of real-world economies is affected by degrees of uncertainty and intro-duce models with stochastic volatility. Furthermore, Davig and Leeper (2009), Farmer,Waggoner and Zha (2011), Foerster, Rubio-Ramırez, Waggoner and Zha (2013) andZhong (2015), among others, advocate periodic unanticipated changes in regimes. Inparticular, a recent paper of Schmitt-Grohe and Uribe (2012) introduces a quantita-tive framework that allows for anticipated exogenous shocks of a fixed periodicity andlength. The key difference of our analysis from this literature in that it allows for timedependence of the model itself while the above literature expands the state space oftime-invariant models.

1.4.3 Methods constructing a path for variables

The EFP framework is related to numerical methods that construct a path for variablesin deterministic economies. To illustrate such methods, let us abstract from uncertaintyby assuming that ft depends on kt but not on zt. By substituting ct and ct+1 from (3.27)into the Euler equation of (3.25)–(3.28), we obtain a second-order difference equation,

u′t((1− δ) kt + ft (kt)− kt+1)

= β[u′t+1((1− δ) kt+1 + ft+1 (kt+1)− kt+2)(1− δ + f ′t+1 (kt+1))

]. (1.15)

Initial condition k0 is given. Let us choose a sufficiently large T and fix some kT+1

(typically, the literature assumes that the economy arrives in the steady state kT+1 =k∗). This yields a system of T nonlinear equations (1.15) with T unknowns k1, ..., kT.The turnpike theorem implies that in initial τ periods, the solution to this system isinsensitive to a specific terminal condition used if τ T .

It is possible to solve the system (1.15) numerically by using a Newton-style or othernumerical solvers, however, it could be expensive. As an alternative, the literature de-veloped numerical methods that exploit the recursive structure of the system (1.15).A well known is a class of shooting methods that solve for path (k1, ..., kT )by usingGauss-Siedel iteration. There are two type of shooting methods: a forward shootingand a backward shooting. A typical forward shooting method expresses kt+2 in terms ofkt and kt+1 and constructs a forward path (k1, ..., kT+1); it iterates on k1 until the pathreaches a given terminal condition kT+1 = k∗. In turn, a typical reverse shooting methodexpresses kt in terms of kt+1 and kt+2 and constructs a backward path kT , ..., k0; ititerates on kT until the path reaches a given initial condition k0. Shooting methodsare introduced to economics in Lipton, Poterba, Sachs and Summers (1980) who alsonoticed their potential for solving nonstationary models. A shortcoming of shootingmethods is that they tend to produce explosive paths, in particular, forward shoot-ing methods; see Atolia and Buffie (2009 a, b) for a careful discussion and possibletreatments of this problem.

Fair and Taylor (1983) introduced an extended path method that can be used tosolve economic models with uncertainty. Their method relies on a certainty-equivalenceapproximation, namely, it replaces expectation of a function across states with a value


of the function in the expected state. In terms of the economy (3.25)–(3.28), this means

Et[u′t+1 (ct+1) (1− δ + f ′(kt+1, zt+1))

]≈ u′t+1 (ct+1) (1− δ + f ′t+1(kt+1, Et [zt+1])). (1.16)

This kind of approximation is exact for linear and linearized models, and it can besufficiently accurate for models that are close to linear; see Cagnon and Taylor (1990),and Love (2010). However, it becomes highly inaccurate when either volatility and/orthe degrees of nonlinearity increase; see our accuracy evaluations in Section 1.5.

To avoid explosive behavior, Fair and Taylor’s (1983) method iterates on the econ-omy’s path at once in line with Gauss-Jacobi iteration. Namely, it guesses the economy’spath (k1, ..., kT+1), substitute the quantities for t = 1, ...T + 1 it in the right side of TEuler equations (1.15), respectively, and obtains a new path (k0, ..., kT ) in the left sideof (1.15); and it iterates on the path until the convergence is achieved. Also, Fair andTaylor (1983) propose a simple procedure for determining T that is sufficient to insurethat a specific terminal condition used does not affect the quality of approximation,namely, they suggested to increase T (i.e., extend the path) until the solution in theinitial period(s) becomes insensitive to further increases in T . In Appendix C, we de-scribe a specific implementation of Fair and Taylor’s (1983) method, which we used inSection 1.5 for comparison with EFP.

There are other methods in the literature that solve for path. In particular, Chen(1999) propose methods for finding a solution to continuous time models. Conesa andKrueger (1999) introduce a framework for characterizing equilibria in nonstationarylife-cycle models in which the aggregate economy’s path is deterministic but there isidiosyncratic uncertainty. Judd (2002) proposes a parametric path method that approx-imates a deterministic path using a family of polynomial functions. Heer and Maußner(2010) implement Fair and Taylor’s (1983) method using a Newton-style solver. Fi-nally, Grune, Semmler and Stieler (2013) develop a nonlinear model predictive controlmethod that solves for a path of variables by maximizing the objective function witha numerical solver directly, without using first-order conditions. Applications of pathmethods in economics are numerous, e.g., Chen, Imrohoroglu and Imrohoroglu (2006),Bodenstein, Erceg and Guerrieri (2009), Coibion, Gorodnichenko and Wieland (2011),Braun and Korber (2011) and Hansen and Imrohoroglu (2013).

Adjemian and Juillard (2013) propose a modification of Fair and Taylor’s (1983)method, called stochastic extended path method, that improves on accuracy of ap-proximation of conditional expectation functions. The main idea of their method is toconstruct a tree of all possible future shocks and to solve for multiple paths for variableson all branches of the tree. The expectation functions is approximated with a weightedaverage of the corresponding variables on multiple paths. The number of tree branchesand paths grows exponentially with the path length and so does the cost of this methodbut the authors propose a clever way of reducing the cost by restricting attention topaths that have highest probability of occurrence.

The EFP construction of function path is similar to the construction of variablespath in the above literature however EFP differs from this literature in the object

1.5. ASSESSING EFP ACCURACY 23

it constructs and in the way it deals with uncertainty. Namely, EFP constructs asequence of Markov state-contingent decision functions that include stochastic shocksas one of the arguments rather than solving for a path under some sequences of shocks.In this respect, EFP is similar to conventional numerical approaches that constructstate-contingent solutions to stationary Markov models.

1.5 Assessing EFP accuracy in a test model with

balanced growth

To assess the quality of approximations produced by EFP in nonstationary environ-ments, we need a test application for which a sufficiently accurate solution is avail-able. We use a version of the model (3.25)–(3.28) with labor augmenting technologicalprogress parameterized by Cobb-Douglas utility and production functions,

ut (c) =c1−γ

1− γ, and ft (k, z) = zkαA1−α

t , (1.17)

where γ > 0 and α ∈ (0, 1); At = A0gtA represents a labor augmenting technolog-

ical progress with an exogenous constant growth rate gA ≥ 1. The process for theproductivity level (3.28) is given by

ln zt+1 = ρ ln zt + σεt+1, εt+1 ∼ N (0, 1) , (1.18)

where ρ ∈ (−1, 1), σ ∈ (0,∞). This version of the model is consistent with balancedgrowth and can be converted into a stationary model; see King, Plosser and Rebelo(1988). We can first solve the stationary model very accurately using conventionalsolution methods, and we can then recover an accurate solution to the original non-stationary model (3.25)–(3.28) to be used for a comparison; see Appendix 1.7 for adescription of the stationary model.

1.5.1 Implementation details of EFP

EFP solves the original, nonstationary growth model (3.25)–(3.28) without convertingit into stationary. The path of function produced by EFP is shown in Figure 1. Below,we discuss some implementation details of EFP; further implementation details areprovided in Appendix B.

First, EFP begins by constructing a sequence of grids for t = 0, ..., T on which asequence (path) of the decision functions will be approximated. An important practi-cal question is where the grids must be centered in t = 0, ..., T . In the conventionalstationary model, we typically center a grid in the deterministic steady state. However,in a growing economy, the steady state does not exist. To address this issue, we definean analogue of steady state for non-stationary economies as a path for the model’svariables that constitutes a solution to an otherwise identical deterministic model in


which all shocks are shut down. We call such a solution a growth path and we denote itby ”∗” superscript. For example, in Figure 1, we show growth path for capital k∗1, k∗20and k∗40 for periods 1, 20 and 40, respectively; see the centers of Smolyak grids in thekt × zt plane.

In the special case of balanced growth model (3.25)–(3.28), the growth path canbe constructed analytically, by using the deterministic steady state of the correspond-ing stationary model. Namely, in the stationary model, the steady state is given by

k∗0 ≡ A0

(gγA − β + δβ

αβ

)1/(α−1)

, and in t = 1, ..., T , it evolves as k∗t = k∗0gtA. In unbal-

anced growth models, the growth path must be in general constructed using numericaltechniques and requires to specify initial and terminal conditions; see Section 1.6.1 fora discussion and examples.

Second, EFP requires us to specify a terminal condition in the form of T -perioddecision functions. (For example, in Figure 1, the terminal period is T = 40 and theterminal decision function is K40). What terminal condition do we choose? Again, fora special case of balanced growth model, it is possible to infer the ”exact” terminalcondition from the solution to the stationary model; see Appendix D for details. How-ever, in a general case, a balanced growth path and an appropriate terminal conditionis unknown. To assess the role of the terminal condition in the accuracy of solutions,we compare two different EFP solutions: in one solution, we use the ”exact” terminalcondition, which is the T -period decision functions inferred from the balanced growthmodel; and in the other solution, we use a stationary Markov solution to a T -periodstationary economy which stops growing at T .

Finally, our turnpike analysis states that we can always find a sufficiently large Tso that the approximation produced by EFP is sufficiently accurate during the first τperiods. But how do we choose T and τ in applications? A popular implementationof Fair and Taylor’s (1983) method builds on τ = 1, namely, we first construct a pathbetween given k0 and kT+1 and we take only k1 from the constructed path; we thenconstructs a path between k1 and kT+2 and takes only k2; and so on until the path ofa required length is constructed. In contrast, we implement EFP by using much largervalues of τ such as 50 or 100 by considering also larger T ’s in order to economize oncost.

1.5.2 A comparison of four solution methods

We solve the nonstationary growth model (3.25)–(3.28) using four alternative solutionmethods: (1) a conventional method that constructs a solution to the stationary modelwith a balanced growth path; (2) EFP method that solves a nonstationary model di-rectly; (3) Fair and Taylor’s (1983) method that uses a certainty equivalence assumption(1.16) to approximates expectation functions; (4) a naive method that replaces a non-stationary model with a sequence of stationary models and that solves such models oneby one. The naive method differs from EFP in that it neglects the connection betweenthe decision functions of different periods. We refer to the solutions produced by the

four methods considered as exact solution, EFP solution, Fair and Taylor’s solutionand naive solution, respectively.

The solution that we call exact is not exact but very accurate, namely, unit-freemaximum residuals in the model’s equations are of order 10−6 on a stochastic simulationof 10,000 observations; see Maliar and Maliar (2014) and Judd, Maliar, Maliar andValero (2014) for accuracy evaluations of Smolyak methods. It will suffice for us toshow that EFP can attain the same accuracy levels for nonstationary models as thestate-of-the-art conventional solution methods do for similar stationary models.

For all experiments, we fix α = 0.36, β = 0.99, δ = 0.025 and ρ = 0.95. Theremaining parameters are set in the benchmark case at γ = 5, σε = 0.03, gA = 1.01 andT = 200, and we vary these parameters across experiments. For all simulations, we usethe same initial condition and the same sequence of productivity shocks.

Our code is written in MATLAB 2013a, and we use a desktop computer with Intel(R)Core(TM) i7-2600 CPU (3.40 GHz) with RAM 12GB. The running times for EFP canbe reduced considerably if we use parallelization (our iteration, which is in line withGauss-Jacobi method, is naturally parallelizable).

Critical role of expectations in the accuracy of solutions

In the left panel of Figure 3, we represent the growing time-series solutions for the foursolution methods, as well as the (steady state) growth path for capital. In the rightpanel, we plot the time series solutions after detrending the growth path.

As is evident from the both panels, the EFP solution and the exact solution arevisibly indistinguishable except at the end of the time horizon – the last 10−15 periods,which suggests that the accuracy range τ may be large in this particular example. Fairand Taylor’s (1983) and naive methods produce the solutions that are visibly below theexact solution; and the naive solution is the least accurate of all.

We assess the accuracy of the constructed solutions numerically. We first simulateeach of the four solutions for 100 times and we then compute the mean and maximumabsolute differences in log 10 units between the accurate solution and the remainingthree solutions across 100 simulations for the intervals [0, τ ], where τ = 50, 100, 150,175 and 200. This kind of accuracy evaluation shows how the accuracy of the approx-imations depend on τ . We report the accuracy results in Tables 1 and 2, where wealso report the time needed for computing and simulating 100 solutions of length T (inseconds).

In Table 1, the difference between the exact and EFP solutions is less than 10−6 ≈0.0001% over the first 50 periods for the three experiments considered (differing in timehorizon and terminal condition). Thus, the EFP method delivers a remarkably accuratesolution for τ = 50 with time horizon T = 200.

Furthermore, the differences between the exact solution and Fair and Taylor’s (1983)solution are around 10−1.6 ≈ 2.5% in Table 1. Fair and Taylor’s (1983) method hasrelatively low accuracy because formula (1.16) used for approximating conditional ex-pectation is inaccurate. Fair and Taylor’s (1983) method is more accurate for models


0 50 100 150 2000

10

20

30

40

50

60

70

80

90

Capital

0 50 100 150 2006

7

8

9

10

11

12

13

Detrended capital

Exact solution

EFP solution

Naive solution

F&T solution

Growth path

Exact solution

EFP solution

Naive solution

F&T solution

Growth path

Figure 1.3: Comparison of the solution methods for the test model with balanced growth

with a smaller variance of shocks and /or smaller degrees of nonlinearities. For exam-ple, we assess the difference between the exact solution and Fair and Taylor’s (1983)solutions for the model with γ = 1, σε = 0.01, gA = 1.01 and T = 200, and we foundthat such a difference is around 0.1% (this experiment is not reported).

Finally, the difference between the exact and naive solutions in Table 1 can be aslarge as 10%. The poor performance of the naive may seem surprising given thatsuch a method does take into account the technology growth when constructing solu-tions. Namely, a naive method solves each t-period stationary model by assuming thatproductivities at t and t + 1 are correctly given by At = A0g

tA and At+1 = A0g

t+1A ,

respectively. Why is the naive method so inaccurate? The reason is that in each timeperiod, the naive method computes a stationary solution under the assumption thattechnology will remain at the current levels At and At+1 forever, meanwhile in the truenonstationary economy, technical change continues forever. As a result, in the formercase, the agent is ”unaware” about the future permanent productivity growth and hasexpectations that are systematically more pessimistic than those of the agent in thetrue nonstationary growing economy. It was pointed out by Cooley, Leroy and Raymon(1984) that naive-style solution methods are logically inconsistent: agents are unawareabout a possibility of parameter changes when they solve their optimization problems,however, they are later confronted with parameter changes in simulations. Our anal-ysis suggests that this effect is particularly large in growing economies. We concludethat approximating expectation functions accurately is critical for constructing accuratesolutions to nonstationary growth models.

1.5. ASSESSING EFP ACCURACY 27

Table 1.1: Comparison of four solution methods.

Fair-Taylor (1983) Naive EFP method EFP methodmethod, τ = 1 method τ = 1 τ = 200

Terminal Steady Steady - Balanced T -period Balanced T -periodcondition state state growth stationary growth stationary

T 200 400 200 200 200 400 200 200 400

Mean errors across t periods in log10 unitst ∈ [0, 50] -1.60 -1.60 -1.36 -7.30 -6.97 -7.15 -7.23 -6.75 -7.01t ∈ [0, 100] -1.42 -1.42 -1.19 -7.06 -6.81 -6.98 -7.03 -6.19 -6.81t ∈ [0, 150] -1.34 -1.35 -1.11 -6.96 -6.73 -6.91 -6.94 -5.47 -6.73t ∈ [0, 175] -1.32 -1.32 -1.09 -6.93 -6.71 -6.89 -6.91 -5.09 -6.70t ∈ [0, 200] -1.30 -1.31 -1.07 -6.91 -6.69 -6.87 -6.90 -4.70 -6.68

Maximum errors across t periods in log10 unitst ∈ [0, 50] -1.29 -1.29 -1.04 -6.83 -6.63 -6.81 -6.82 -6.01 -6.42t ∈ [0, 100] -1.18 -1.18 -0.92 -6.69 -6.42 -6.68 -6.68 -4.39 -5.99t ∈ [0, 150] -1.14 -1.14 -0.89 -6.66 -6.39 -6.67 -6.66 -2.89 -5.98t ∈ [0, 175] -1.14 -1.13 -0.89 -6.66 -6.40 -6.66 -6.66 -2.10 -5.98t ∈ [0, 200] -1.14 -1.13 -0.89 -6.66 -6.37 -6.66 -6.66 -1.45 -5.92

Running time, in secondsSolution 1.2(+4) 6.1(+4) 28.9 216.5 8.6(+3) 1.9(+4) 104.9 99.1 225.9Simulation - - 2.6 2.6 2.6 5.8 2.6 2.8 5.7Total 1.2(+4) 6.1(+4) 31.5 219.2 8.6(+3) 1.9(+4) 107.6 101.9 231.6

Notes: ”Mean errors” and ”Marimum errors” are, respectively, mean and maximum unit-free absolute

difference between the exact solution for capital and the solution delivered by a method in the column.

The difference between the solutions is computed across 100 simulations.

Terminal condition and the ”tail” of simulation

As Figure 3 shows, the exact and EFP solutions differ in the tail considerably; thisdifference is especially well seen for the detrended time series in the right panel. Thedifference in the tail is due to the difference in the terminal conditions. Namely, to con-struct the exact solution, we assume that the economy grows forever while to constructthe EFP solution, we assume that it stops growing at T . If we use the same terminalconditions in both cases, then the EFP solution would be visually indistinguishablefrom the exact solution everywhere in the figure.

In Table 1, we first consider a version of EFP that constructs the function path underτ = 1 (this is similar to the implementation of Fair and Taylor’s (1983) method usedin the literature). Namely, given the capital function KT in the T -period stationaryeconomy, we solve for decision functions for t = 0, ..., T − 1, store K0 and discard therest of the functions. Next, given KT+1, we solve for decision functions for t = 1, ..., T ,store K1 and discard the rest of the functions. We proceed forward until the whole path


(K0, ...KT ) is constructed. We consider two different lengths of time horizon T = 200and T = 400, and we consider two different terminal conditions: one comes fromthe solution of the stationary balanced growth model (and can be viewed as ”exact”terminal condition) and the other comes from the T -period stationary economy (and isfar from the ”exact” terminal condition).

EFP method with τ = 1 is very accurate in the studied example independentlyof specific terminal condition used, namely, the EFP solution differs from the exactsolution by less than 10−6 = 0.0001%. This result illustrates that the effect of specificterminal condition on the very first element of the path τ = 1 is negligible if the lengthof the path T is sufficiently large.

How to extend the path

A shortcoming of the described version of EFP with τ = 1 is its high computational ex-pense: the running time under T = 200 and T = 400 is 15 and 30 minutes, respectively.The cost is high because we need to recompute a sequence of decision functions eachtime when we extend the path by one period ahead. Effectively, we solve the model Ttimes and not just once.

Our turnpike theorem suggests a cheaper version of EFP in which we construct alonger function path but do it just once; the results for such EFP method are providedin the last three columns of Table 1. We now observe that the terminal conditionplays a critical role in the accuracy of solutions near the tail. If we use the terminalcondition from the T -period stationary economy, the errors increase as we advance intime and reach nearly 4% at the end of simulation. In contrast, if we use the terminalcondition from the stationary balanced growth model, the EFP solution is very accurateeverywhere including the tail. Finally, the most important result is shown in the lastcolumn. If we construct a function path of length T = 400, however, use only thefirst τ = 200 decision functions, the solution for the first τ = 200 periods is almost asaccurate as that produced by a sequence of functions with τ = 1. This is true eventhough we use the terminal condition from the T -period stationary economy that is faraway from the exact solution. We draw attention to the fact that constructing a longerpath is relatively inexpensive: the running time increases from about 2 minutes to 4minutes under T = 200 and T = 400, respectively.

Cost of finding solution and cost of simulation

An important advantage of EFP relatively to methods that solve for a path of variablesis its low simulation cost. Under EFP, we construct a path for decision functions justonce, and we can use the constructed functions to simulate the model as many times asneeded under different sequences of shocks. In contrast, under Fair and Taylor’s (1983)and other methods that solve for a path of variables, the solution and simulation stepsare combined: in order to produce a new simulation, we need to entirely recompute asolution to the model under a different sequence of shocks. The time that EFP needs to

compute a solution and simulate it 100 times is about 2 and 4 minutes for T = 200 andT = 400, respectively, while the respective times for Fair and Taylor’s (1983) methodare 20 and 60 minutes.

Sensitivity analysis

On the basis of the results in Table 1, we advocate the version of EFP that constructsa sufficiently long path just once by using T τ . In Table 2, we assess the accuracyof this preferred EFP version with T = 400 under different parameterizations. As aterminal guess, we use decision rules produced by the T -period stationary economy.We consider several combinations of the values of the parameters γ, σε, gA such thatγ ∈ 0.1; 1; 5; 10, σε ∈ 0.01; 0.03 and gA ∈ 1; 1.01; 1.05. Our benchmark values(see ”Model 1”) are γ;σε; gA = 5; 0.03; 1.01.

Table 1.2: Sensitivity analysis for the EFP method.

Parameters Model 1 Model 2 Model 3 Model 4 Model 5 Models 6 Model 7γ 5 5 5 5 0.1 1 10σε 0.03 0.03 0.03 0.01 0.01 0.01 0.01gA 1.01 1.00 1.05 1.01 1.01 1.01 1.01

Mean errors across t periods in log10 unitst ∈ [0, 50] -7.01 -6.67 -7.34 -7.03 -7.03 -6.61 -7.30t ∈ [0, 100] -6.82 -6.44 -7.25 -6.84 6.92 -6.48 -7.08t ∈ [0, 150] -6.73 -6.33 -7.22 -6.76 -6.89 -6.43 -6.98t ∈ [0, 175] -6.70 -6.29 -7.22 -6.74 -6.87 -6.41 -6.95t ∈ [0, 200] -6.68 -6.26 -7.21 -6.72 -6.87 -6.37 -6.93

Maximum errors across t periods in log10 unitst ∈ [0, 50] -6.42 -6.31 -7.13 -6.66 -6.08 -6.24 -6.81t ∈ [0, 100] -5.99 -6.12 -7.05 -6.54 -5.97 -6.18 -6.36t ∈ [0, 150] -5.98 -6.04 -7.05 -6.52 -5.97 -6.18 -6.35t ∈ [0, 175] -5.98 -6.01 -7.05 -6.52 -5.97 -6.13 -6.33t ∈ [0, 200] -5.92 -5.99 -7.05 -6.51 -5.96 -5.88 -6.24

Running time, in secondsSolution 225.9 150.0 193.0 216.98 836.5 300.7 245.9Simulation 5.6 5.7 5.8 5.66 5.6 5.6 5.7Total 231.6 155.7 198.8 222.64 842.1 306.3 251.6

Notes: ”Mean errors” and ”Marimum errors” are, respectively, mean and maximum unit-free absolute

difference between the exact solution for capital and the solution delivered by EFP under the param-

eterization in the column. The difference between the solutions is computed across 100 simulations.

The time horizon is T = 400 and the terminal condition is constructed by using T-period stationary

economy in all experiments.

Depending on a specific parameterization, the difference between the exact and


EFP solutions in the model’s equations for τ = 200, vary between 10−6 = 0.0001%and 10−7 = 0.00001%, which are high accuracy levels. The running time for all casesexcept for Model 5 is between 155 seconds and 306 seconds, which is reasonable. Thereis one case when the computational time increases to 842 seconds, which correspondsto to a low degree of risk aversion parameter γ = 0.1. (We find that with low degreeof risk aversion, the convergence is more fragile and we had to decrease the dampingparameter from ξ = 0.05 to ξ = 0.1). Overall, our results suggests that the EFP methodcan solve nonstationary growth models both accurately and reliably in a wide range ofthe model’s parameters at a relatively modest cost.

1.6 Numerical analysis of nonstationary and unbal-

anced growth applications

We provide a collection of numerical examples that illustrate how EFP can be used forcalibrating, solving, estimating and simulating nonstationary problems. Our examplesinclude models with an unbalanced growth path, expected and unexpected technologyshocks, seasonal adjustments and deterministically changing volatility of productivityshocks, as well as an example of calibrating and estimating parameters in an unbalancedgrowth model using the data on the U.S. economy. These applications do not allow forstationary Markov equilibria and hence, cannot be studied with conventional solutionmethods. The model’s parameterization, time horizon and terminal condition differacross applications – we describe them separately for each application considered.

1.6.1 Application 1: An unbalanced growth model with a CESproduction function and capital-augmenting technologi-cal progress

The previous section focused on a nonstationary growth model that can be convertedinto a stationary model and that can be studied with conventional solution methods. Wenow consider a nonstationary model that cannot be converted into a stationary modeland cannot be studied with conventional methods. Namely, we assume a constantelasticity of substitution (CES) production function, and we allow for both labor andcapital augmenting technological progresses,

F (kt, `t) = [α(Ak,tkt)v + (1− α)(A`,t`t)

v]1/v , (1.19)

where Ak,t = Ak,0gtAk

; A`,t = A`,0gtA`

; v ≤ 1; α ∈ (0, 1); gAk and gA` are the rates ofcapital and labour augmenting technological progresses, respectively. We assume thatlabor is supplied inelastically and normalize it to one `t = 1 for all t, and we denotethe corresponding production function by f(kt) ≡ F (kt, 1). The model with capitalaugmenting technological progress does not satisfy the assumptions in King, Plosserand Rebelo (1988) and does not admit a balanced growth path.

1.6. NUMERICAL ANALYSIS OF APPLICATIONS 31

The assumption of capital augmenting technological progress is advocated in theliterature on directed technical change. Acemoglu (2002) points out that in most cases,technical change does not apply to the same fixed factors of production all the timebut is endogenously directed to those factors of production that can give the largestimprovement in the efficiency of production.5 An implication of this argument that isrelevant for our analysis is that technical change can be directed either to capital or tolabor or other production factors depending on a particular case.

Furthermore, Acemoglu (2003) explicitly incorporates capital augmenting techno-logical progress into a deterministic model of endogenous technical change with bothlabor and capital augmenting innovations. Empirical estimates of the growth ratesof the capital augmenting technical change can be found in, e.g., Klump, Mc Adamand Willman (2007), and Leon-Ledesma Leon-Ledesma, Mc Adam and Wilman (2015).Below, we show how the model with capital augmenting technological progress can bestudied by using EFP.

A growth path for a nonstationary economy

Our first goal is to define a growth path around which the sequence of grids will becentered. For constructing the growth path, we shut down uncertainty by assumingzt = 1 for all t (similar to what we do for a model with balanced growth) and werewrite the model’s equations in the way that is convenient for identifying the path.

First, the Euler equation of period t, evaluated on the steady state path, is

1 = β

[u′(c∗t+1)

u′(c∗t )(1− δ + f ′

αAvk,t+1(k

∗t+1)

v−1 [α(Ak,t+1k∗t+1)

v + (1− α)Av`,t+1

](1−v)/v],

where c∗t and k∗t are the variables on the growth path. From the last equation, weexpress k∗t+1 as

k∗t+1 = (1− α)1/vA`,t+1

Ak,t+1

[((gu′,t+1)

−1 − β + δβ

αβ · Ak,t+1

)v/(1−v)− α

]1/v, (1.20)

where gu′,t+1 ≡u′(c∗t+1)

u′(c∗t )follows from the budget constraints (3.27) for t and t+ 1:

gu′,t+1 =u′[(1− δ) k∗t+1 +

[α(Ak,t+1k

∗t+1)

v + (1− α)Av`,t+1

]1/v − k∗t+2

]u′[(1− δ) k∗t +

[α(Ak,tk∗t )

v + (1− α)Av`,t]1/v − k∗t+1

] . (1.21)

Thus, we obtain a system of T−1 equations (1.20) with T+1 unknowns k∗0, ..., k∗T+1. This

system does not have a unique solution unless we impose two additional restrictions.

5Namely, endogenous technical change is biased toward a relatively more scarce factor when theelasticity of substitution is low (because this factor is relatively more expensive); however, it is bi-ased toward a relatively more abundant factor when the elasticity of substitution is high (becausetechnologies using such a factor have a larger market).


Identifying restrictions on initial and terminal conditions

There are many possible ways to impose identifying restrictions on the solution ofsystem (1.20), (1.21). In this specific application, we restrict the initial and terminalcapital stocks, k∗0 and k∗T+1. Namely, we restrict k∗0 by assuming that the capital growth

rate is the same in the first two periodsk∗1k∗0

=k∗2k∗1

, and we restrict k∗T+1 by assuming such

a growth rate is the same in the last two periodsk∗Tk∗T−1

=k∗T+1

k∗T. This pins down the

initial and terminal capital stocks on the growth path in terms of (k∗1, ..., k∗T ),

k∗0 =(k∗1)2

k∗2and k∗T+1 =

(k∗T )2

k∗T−1. (1.22)

The model satisfies the assumptions of King, Plosser and Rebelo (1988) if there is onlylabor augmenting technological progress, i.e., A`,t grows at a constant, exogenouslygiven rate gA` and Ak,t = Ak for all t. In this special case, the model has a balancedgrowth path on which all variables grow at a constant rate gA` and this is in particulartrue for initial and terminal periods, i.e., condition (1.22) is satisfied exactly.

In the case of capital augmenting technological progress, the growth rate of endoge-nous variables changes over time in an unbalanced manner even if we assume that Ak,tgrows at a constant, exogenously given growth rate gAk and A`,t = A` for all t. Byimposing two additional restrictions in (1.22), we solve for k∗0, ..., k

∗T+1 satisfying (1.20),

(1.21). In our applications, changes in the growth path k∗0, ..., k∗T+1 had only a minor

effect on the quality of the approximations. This is because a specific growth pathdoes not identify the solution itself but only a set of points in which the Smolyak gridsare centered. Centering a grid in a slightly different point will not significantly affectthe properties of solution in a typical application. The assumption in (1.22) can bemodified if needed.

Results of numerical experiments

For numerical experiments, we assume T = 260, γ = 1, α = 0.36, β = 0.99, δ = 0.025,ρ = 0.95, σε = 0.01, v = −0.42; the last value is taken in line with Antras (2004) whoestimated the elasticity of substitution between capital and labor to be in the range[0.641, 0.892] that corresponds to v ∈ [−0.12,−0.56]. We solve two models: the modelwith labor augmenting progress parameterized by A`,0 = 1.1130, gA` = 1.00153 andAk,0 = gAk = 1 and the model with capital augmenting progress parameterized byAk,0 = 1, gAk = 0.9867 and A`,0 = gA` = 1. (The parameters A`,0, gA` , Ak,0, gAk forboth models are chosen to approximately match the initial and terminal capital stocksfor time-series solutions of both models).

Figure 4 plots the time-series solutions for models with labour and capital augment-ing progresses, as well as their growth paths.

The solution of the model with labor augmenting technological progress is typical fora balanced growth model. There is an exponential growth path with a constant growth


0 20 40 60 80 100 120 140 1609.5

10

10.5

11

11.5

12

12.5

13

time

Capital

Capital augmenting: simulation

Labour augmenting: simulation

Capital augmenting: growth path

Labour augmenting: growth path

Figure 1.4: Technological progress in the model with the CES production

rate and cyclical fluctuations around the growth path. (In the figure, the growth pathin the model with labor augmenting technological progress is situated below the lineargrowth path shown by a solid line). In turn, the solution of the model with capitalaugmenting progress has a pronounced concave growth pattern that shows that therate of return to capital decreases as the economy grows (In the figure, the growth pathin the model with capital augmenting technological progress is situated above the lineargrowth path shown by a solid line). Finally, the cyclical properties of both models looksimilar (provided that growth is detrended). These are novel results since the propertiesof stochastic growth models with capital augmenting progress are not studied yet inthe literature (to the best of our knowledge).

1.6.2 Application 2: A nonstationary model with a parametershift

The recent literature on regime switches addresses the critiques of naive solution meth-ods of Cooley, Leroy and Raymon (1984) and provides a logically consistent way of mod-eling unanticipated regime switches. Specifically, this literature assumes that agentssolve maximization problems in which regime are possible, and thus, they can ade-quately react to regime changes in simulation as implied by their decision functions;see Sims and Zha (2006), Davig and Leeper (2007, 2009), Farmer, Waggoner, and Zha(2011), Foerster, Rubio-Ramırez, Waggoner and Zha (2013) and Zhong (2015), among


others.The above literature assumes that the regimes come at random, drawn from a

stationary probability distribution. However, there are real-world situations in whichparameter shifts are nonrecurrent and anticipated by agents in advance, e.g., seasonalchanges, presidential elections with anticipated outcome, forward-looking policy an-nouncements, anticipated technological advances, etc. A prominent example of ananticipated shock is an accession of new members to the European Union that wasannounced many years in advance and that resulted in quantitatively-important antici-patory effects; see Garmel, Maliar and Maliar (2008) for a discussion and a quantitativeassessment of such effects in a three-country general equilibrium model.

Schmitt-Grohe and Uribe (2012) propose a computational approach that allows todeal with anticipated parameter shifts of fixed time horizons in the context of stationaryMarkov models (the parameter shifts systematically occur, for example, each fourth oreach eighth periods). However, if the anticipated parameter shifts are either nonrecur-rent and do not have fixed anticipation horizons, the model does not admit stationaryMarkov solutions and cannot be studied using conventional solution methods. How-ever, EFP can solve models with such anticipated shocks. As an example, we showhow to solve a model with anticipated technology shocks, and we compare the solutionproduced by EFP to naive solutions in which shocks are unanticipated.

Anticipated technology shocks

The idea that anticipated shocks play an important role in business cycle fluctuationsgoes back to Pigou (1927). The literature that advocates the importance of anticipatedshocks for aggregate fluctuations includes, e.g., Cochrane (1994), Beaudry and Portier(2006), and Schmitt-Grohe and Uribe (2012).

We consider a version of the model (3.25), (3.27), (1.17) and (1.18) in which thetechnology level At can take two values, A = 1 (low) and A = 1.2 (high). A specialcase of this setup is a model in which A and A are unanticipated and randomly drawnfrom a given probability distribution. Such a model has a stationary Markov solutionthat can be studied using the approaches described in the literature on regime switches,e.g., Davig and Leeper (2007, 2009).

In contrast, we focus on the case when the regime switches are both nonrecurrentand anticipated by the agent from the beginning. As an example, we consider a sce-nario when the economy starts with A at t = 0, switches to A at t′ = 250 and thenswitches back to A at t′′ = 550 (instead, we could have considered any other scenariofor technology levels). We show the technology profile in the upper panel of Figure 5.The other parameters are T = 900, γ = 1, α = 0.36, β = 0.99, δ = 0.025, ρ = 0.95,σε = 0.01.

The agent solves the utility-maximization problem at t = 0 given the technologyprofile. The implementation of the EFP method for this case is similar to the one withtechnological progress studied in the previous section (as initial and terminal valuesof the growth path, k∗0 and k∗T+1, we use a steady state of the model with A). In the


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Pro

ductivity

0 100 200 300 400 500 600 700 80030

40

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Capital

time

0 100 200 300 400 500 600 700 8002.5

3

3.5

4

Consum

ption

time

EFP solution

Naive solution

EFP solution

Naive solution

Figure 1.5: Anticipated versus unanticipated technology shocks

case of the naive solution, shocks are unexpected. We construct two stationary naivesolutions under A and A. The agent follows the first solution until the first switch att′ = 250, then the agent follows the second solution until the second switch at t′′ = 550and finally, the agent goes back to the first solution for the rest of the simulation.

The two time-series solutions for capital and consumption are shown, respectively, inthe middle and lower panels of Figure 5. In simulation, we set zt = 1 for all t to makethe anticipatory effects more visible. Remarkably, in the solution with the expectedregime switches, we observe a strong anticipation effect: about 50 periods before theswitch from A and A takes place, the agent starts gradually increasing her consumptionand to decrease her capital stock in order to bring some part of the benefits from futuretechnology growth to present. When the technology switch actually occurs, it has onlya minor effect on consumption. (The tendencies reverse when there is a switch from Ato A). In contrast, consumption-smoothing anticipatory effects are absent for the naivesolution. Here, unexpected technology shocks lead to large jumps in consumption in theexact moment of technology switches. The difference in the solutions is quantitativelysignificant under our empirically plausible parameter choice. Finally, in the AppendixE, we plot the simulated solution by considering both deterministic technology switchesand stochastic productivity shocks following an AR(1) process (1.37); see Figure 10.Anticipatory effects are well pronounced in those experiments as well.


A model with seasonal changes

One empirically-relevant application that our EFP framework can deal with is a growthmodel with seasonal changes. An important role of seasonal fluctuations in the totalvariation in aggregate economic variables is well documented in the literature; see, e.g.,Barsky and Miron (1989). Ignoring seasonality when estimating dynamic stochasticgeneral equilibrium models may lead to substantial errors in the estimated parameters;see, e.g., Saijo (2013).

Two approaches have been proposed in the literature to model seasonality. Hansenand Sargent (1993, 2013) characterize seasonality in terms of the spectral density of vari-ables. They assume that seasonality comes either from seasonality in exogenous shockprocesses (with spectral peaks at seasonal frequencies) or from propagation mechanismsdetermined by preferences and technology (e.g., seasonal habit persistence) or from sea-sonal periodicity in the parameters of the preferences and technologies; in these cases,the optimal decision rules are periodic. Second, Christiano and Todd (2002) develop amodel in which exogenous shocks contain deterministic seasonal dummies and in whichinvestment process is period-specific (an investment project requires four quarters tocomplete, and current-period total investment depends on the projects started in thecurrent and three previous periods); to solve such a model, they linearize around themodel’s seasonally varying steady state growth path and solve for four distinct decisionrules.

We now show how to solve economic models with seasonal changes by using EFP. Asan example, we assume that every forth period, At takes a high value A, and the rest ofthe periods, it takes a low value A; for example, this pattern can be observed in a countryon a seacoast in which there is a high productivity season in summer. Thus, we obtainthe following sequence of technology levels: A,A,A,A,A,A,A,A, .... In additional tothe seasonal changes, the agent faces the conventional productivity shocks (1.37), sothat the resulting path for the productivity level is given by a composition of expectedseasonal changes in At and unexpected stochastic changes in productivity levels givenby a stationary autoregressive process. The parameters are the same as in the previousmodel except that we use γ = 2, β = 0.97, A = 0.98 and A = 1.06 (these parametersare fixed for expositional convenience). To construct the growth path for the EFPmethod, we set the initial and terminal conditions at 3/4k∗ + 1/4k

∗, where k∗ and k

∗

are the steady states of capital in the models with A and A, respectively. In Figure 6,we plot time series for productivity, capital and consumption (we normalize the initialvalues of all series to one). An interesting finding in Figure 6 is that the size of seasonalconsumption and capital fluctuations is very small compared to the size of seasonalproductivity fluctuations. A consumption-smoothing agent knows that the seasonalshock is temporary and that it does not pay to react much on the impact of such a shock.Instead, the agent adjusts her capital and consumption to take advantage of seasonalproductivity growth on average, as permanent consumption hypothesis suggests. Amagnitude of seasonal fluctuations in the model’s variables is far larger and comparablein size to seasonal productivity fluctuations in a naive solution in which the seasonal


0 10 20 30 40 50 60 70 80 90 1000.96

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

time

Va

ria

ble

ProductivityCapitalConsumption

Figure 1.6: Seasonality

shocks are unexpected as naive agents would fail to take into account anticipatoryeffects (we do not provide the naive solution to avoid a clutter).

1.6.3 Application 3: A nonstationary model with a parameterdrift

A class of models with parameter drifting is another interesting and empirically rele-vant type of economic applications that are characterized by nonstationary solutions.There is ample empirical evidence in favor of parameter drifting, see, e.g., Clarida,Galı and Gertler (2000), Lubick and Schorfheide (2004), Cogley and Sargent (2005),Goodfriend and King (2009), Canova (2009). The assumption of parameter driftingis advocated in Galı (2006). The previous literature focus on economic models withstationary Markov equilibria by assuming that the model’s parameters follow a station-ary autoregressive process; see, e.g., Fernandez-Villaverde and Rubio-Ramırez (2007),Fernandez-Villaverde, Guerron-Quintana and Rubio-Ramırez (2010). However, if themodel’s parameters follow a pattern with a pronounced time trend, the equilibrium de-cision rules change from one period to another and the conventional solution methodsare not applicable. Below, we show how to use EFP to solve an example of the modelwith parameter drift that includes time trends.


A growth model with a productivity drift

We consider a scenario that is similar to the one in Application 2, however, we nowassume that technology does not switch to a higher/lower level in one period but in-creases/decreases gradually. To be specific, the level of technology is low, A, for thefirst 200 periods; it increases linearly to a high level, A, for the next 100 periods; itstays constant for the following 300 periods; it decreases linearly back to a low level, A,for 200 periods and finally, it stays there for the remaining period of time. The aboveproductivity profile is shown in Figure 7. To calibrate the model, we use the same

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3.5

4

Consu

mptio

n

time

EFP solutionNaive solution

EFP solutionNaive solution

Figure 1.7: Technology drift

parameters as in Application 2.We plot the EFP time-series solution of the model with a parameter drift in the

middle and lower panel of Figure 7. For a comparison, we also provide a naive solutionin which shocks are always unanticipated. To produce the naive solution, we solve astationary model 100 times under each level of technology that occurs in the parameterdrift, and we jump from one stationary naive solution to another after each technologychange. Again, to simulate the solution, we set zt = 1 for all t for a better visibility ofanticipatory effects.

Similar to Application 2, we observe well-pronounced smoothing of consumption atthe cost of anticipatory adjustments of capital. In particular, the consumption pathwith an expected parameter drift is smoother than the one in the naive solution in thoseplaces where the parameter shift begins / ends and we observe the kink. In the AppendixE, we provide a plot of the simulated solution with both deterministic productivity shifts


and stochastic productivity shocks; see Figure 11. Again, anticipatory effects are wellpronounced in that case as well.

Example of a parameter drift: diminishing volatility

A large body of recent literature documents the importance of degree of uncertainty forthe business cycle. This literature argues that volatility changes over time. They modelvolatility (e.g., standard deviation of the productivity level) as a stochastic process oras a regime switch; see, e.g., Bloom (2009), Fernandez-Villaverde and Rubio-Ramırez(2010), Fernandez-Villaverde, Guerron-Quintana and Rubio-Ramırez (2010). The liter-ature normally assumes that the standard deviation of exogenous shocks either followsa Markov process or experiences recurring Markov regime switches. In the latter case,volatility can be treated as an additional state variable, and in the former case, theregime is an additional state variable; in both cases, it is possible to cast the modelwith changing volatility into the conventional stationary framework.

However, there is evidence that the volatility has a well pronounced time trend, forexample, Mc Connel and Perez-Quiros (2000) document a monotone structural declinein the volatility of real GDP growth in the U.S. economy. Blanchard and Simon (2001)find a nonmonotone pattern of the decline in the U.S. GDP volatility: there was asteady decline in the volatility from the 1950s to 1970, then there was a stationarypattern and finally, there was another decline in the late 1980s and the 1990s. Stockand Watson (2003) find a sharp reduction in volatility of U.S. GDP growth in thefirst quarter of 1984. This kind of evidence cannot be reconciled in a model in whichstochastic volatility follows a standard AR(1) process with time-invariant parameters.We show how to use EFP to study a model in which the volatility has both a stochasticand deterministic components.

We specifically consider the standard neoclassical stochastic growth model, modifiedto include a diminishing volatility of the productivity shock:

ln zt = ρ ln zt−1 + σtεt, σt =B

tρσ, εt ∼ N (0, 1) , (1.23)

where B is a scaling parameter, and ρσ is a parameter that governs the volatility of zt.The standard deviation of the productivity shock Bσ/tρσ decreases over time, reaching

zero in the limit, limt→∞Bσ

tρσ= 0. In our numerical example, we use T = 500, γ = 1,

α = 0.36, β = 0.99, δ = 0.025, ρ = 0.95, σε = 0.01, B = 1 and ρσ = 1.05. To solvethis model, we use EFP in which we build a grid around the deterministic steady statevalue of capital. In Figure 8, we plot a sequence of simulated productivity levels; as wesee, initially, there are large productivity fluctuations but gradually, these fluctuationsbecome smaller. As expected, fluctuations in capital and consumption also decrease inamplitude in response to diminishing volatility.


0 50 100 150 200 250

0.998

0.9985

0.999

0.9995

1

1.0005

1.001

1.0015

time

Varia

ble

ProductivityCapitalConsumption

Figure 1.8: Diminishing volatility

1.6.4 Application 4: Calibrating a growth model with a pa-rameter drift to unbalanced U.S. data

There is a large group of econometric methods that estimate and calibrate economicmodels by constructing numerical solutions explicitly, including simulated method ofmoments (e.g., Canova (2007)); Bayesian estimation method (e.g., Smets and Wouters(2003), and Del Negro, Schorfheide, Smets and Wouters (2007)); and maximum like-lihood method (e.g., Fernandez-Villaverde and Rubio-Ramırez (2007)). Normally, therelated literature imposes restrictions on the model that lead to a balanced growthpath, converts the model into stationary model and solves it for stationary Markovequilibrium by using conventional methods.

However, there are two potential problems with this approach. First, the restrictionsthat are necessary to impose for balanced growth might not be the empirically-relevantones. For example, we might want to analyze a model with nonhomothetic utilityand production functions, several kinds of technical progress and parameter shifts anddrifts. However, any deviation from the restrictions in King, Plosser and Rebelo (1988)destroys the property of balanced growth and hence, destroys the conventional Markovstationary equilibria. Second, the real world data are not always consistent with theassumption of balanced growth, in particular, different variables might grow at differentand possibly time-varying rates. In this section, we illustrate how EFP can be used tocalibrate and estimate parameters in an unbalanced growth model by using the dataon U.S. economy.


Time series to match

We took macroeconomic data on the U.S. economy from the webpages of the Bureauof Economic Analysis and the Federal Reserve Bank of St. Louis (namely, the dataon capital and investment come from the former data base, while the data on theremaining time series, as well as that on the implicit price deflator, come from thelatter data base); the sample spans over the period 1964:Q1 - 2011:Q4. Investmentis defined as nonresidential and residential private fixed investment. Consumption isdefined as a sum of nondurables and services. Capital is given by a sum of fixedassets and durables; capital series are annual (in contrast to the other series which arequarterly); we interpolate annual series of capital to get quarterly series using splineinterpolation. Output is obtained as a sum of consumption and investment. We deflatethe constructed variables with the corresponding implicit price deflator and we convertthem in per capita terms by dividing them by the series of the total population.

The model with a depreciation rate drift

While the constructed data are grossly consistent with Kaldor’s (1961) facts, we stillobserve visible differences in growth rates of across variables. We do not test whetheror not such differences in growth rates are statistically significant but formulate andestimate an unbalanced growth model in which different variables can grow at differingrates. We specifically extend the model (3.25)–(3.28) to include time-varying depreci-ation rate of capital,

maxct,kt+1t=0,...,∞

E0

∞∑t=0

βtu(ct) (1.24)

s.t. ct + kt+1 = Atztkαt + (1− dtδt) kt, (1.25)

ln δt = ρδ ln δt−1 + εδ,t, εδ,t ∼ N(0, σ2

εd

), (1.26)

ln zt = ρz ln zt−1 + εz,t, εz,t ∼ N(0, σ2

εz

), (1.27)

where dtδt stands for a time-varying depreciation rate with dt being a trend componentof depreciation, dt = d0g

td, and δt being a stochastic shock to depreciation. Our as-

sumption of a time trend in depreciation rate is consistent with the data of the Bureauof Economic Analysis. In particular, the aggregate depreciation rate changes over timebecause the compositional of aggregate capital changes over time even if depreciationrates of each type of capital are constant; see Karabarbounis and Brent (2014). In turn,shocks to depreciation rate can result from the economic obsolescence of capital andare studied in, e.g., Liu, Waggoner and Zha (2011), Gourio (2012) and Zhong (2015),in particular, this literature argues that a shock to the capital depreciation rate playsan important role in accounting for the business cycle fluctuations.


Calibration and estimation of the model’s parameters

To identify the model’s parameters, we formulate the following set of restrictions

Atzt =ytkαt, (1.28)

dtδt =itkt− kt+1 − kt

kt, (1.29)

1

β=

1

T

T∑t=1

c−γt+1

c−γt

[1− dt+1δt+1 + αAt+1zt+1k

α−1t+1

]. (1.30)

We set γ = 1 and we search for α that matches best the growth rates of variables inthe data. First, given some α, we construct Atzt using (1.28) , and we estimate theparameters ρz, σ

2εz , gA in the process for productivity zt = zρt−1 exp(εz,t) using a linear

regression method. To identify a growing and cycle components, At and zt, respectively,we assume z0 = 1. Second, we construct the data on dtδt using (1.29), and we estimatethe parameters ρδ, σ

2εδ

, gd in the process for productivity δt = δρt−1 exp(εδ,t) using alinear regression. Again, to separate growth and cycles, dt and δt, respectively, weassume δ0 = 1. Finally, we calibrate the discount factor by using the Euler equation(1.30).

Our estimation-calibration procedure gives the following values of parameters: β =0.9013, ρz = 0.9890 , σεz = 0.0054, gA = 1.002, ρδ = 0.9538 , σεδ = 0.0381 andgd = 1.002. We observe a considerable positive growth rate in the depreciation rategd = 1.002. Furthermore, we find that the best fit of our criteria for the growth rateis obtained under α = 0.7. This value for the capital share in output is larger thanis typically used in the business cycle literature, however, it is roughly in line withthe recent finding of Karabarbounis and Neiman (2014) that labor shares graduallydeclined over time; the implied gross capital shares reach 0.55.

We know that on the tail, the EFP solution will depend on a specific terminalcondition used and may be insufficiently accurate. To deal with this issue, we extrap-olate the data for 80 periods forward, using the growth rates that we estimated fromthe data on consumption, capital, output, and investment under the assumption ofexponential growth. We implement EFP to match the initial and terminal conditionsin the extrapolated data, i.e., we use T = τ + 80. To identify the growth path inour unbalanced growth model, we use assumption (1.22). We construct a sequence ofgrowing Smolyak grids. There are three state variables (kt, zt, δt) in this applicationand the corresponding second-level Smolyak grid consists of 25 multidimensional gridpoints. After we compute the EFP solution, we simulate the model using the sequenceof shocks reconstructed from the data.

Fitted time series

Figure 9 presents the simulated time-series solution for capital, output, investment andconsumption; for comparison, we also provide the corresponding time series from the

1.7. CONCLUSION 43

data. To appreciate the differences in growth rates, we scaled all four panels to havethe same percentage change in y.

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

0.1

0.15

0.2

0.25

Capital

Data

Model

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100.01

0.02

0.03

0.04

0.05

Output

Data

Model

1965 1970 1975 1980 1985 1990 1995 2000 2005 20102

4

6

8

10x 10

−3 Investment

Data

Model

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100.01

0.02

0.03

0.04

0.05

Consumption

Data

Model

Figure 1.9: Matching nonstationary macroeconomic data on the U.S. economy

First of all, we can visually appreciate the nonstationarity in the data: investmentgrows considerably faster than other variables. With the assumption of time-varyingdepreciation rate, the model (1.24)–(1.27) can closely reproduce the growth rates of allmodel’s variables.

The main goal of this application is not to advocate the role of time varying depre-ciation rate or some specific estimation and calibration technique. Rather, we wouldlike to illustrate how estimation and calibration of the parameters can be carried out inthe context of a nested fixed-point problem without assuming stationarity and balancedgrowth. Similar to the depreciation rate, we could have made all other parameters timedependent, including the discount factor β, the share of capital in production α andthe parameters of the process for the productivity level (1.27). Furthermore, our sim-ple estimation-calibration technique can be replaced by more sophisticated econometrictechniques such as maximum likelihood, simulated method of moments, etc.

1.7 Conclusion

Stationary Markov dynamic economic models are a dominant framework in recent eco-nomic literature. A shortcoming of this framework is that it generally restricts the


structural parameters of economic models to be constant, and it restricts the behav-ior patterns to be time invariant. In this paper, we construct a more flexible classof nonstationary Markov models that allows for time-varying preferences, technologyand laws of motions for exogenous variables. We propose EFP framework for solving,calibration, simulation and estimation of parameters in such models. EFP enables usto analyze economic models that do not admit stationary Markov equilibria and thatcannot be studied with conventional solution methods. Literally, EFP makes it possibleto analyze a unique historical path of real-world economies.

Our analysis can be extended in three possible directions: First, our numericalresults are produced by an Euler equation version of EFP that finds a path of decisionfunctions to satisfy a sequence of Euler equations. It is also of interest to explore theperformance of an analogous value-iterative version of EFP that first constructs a valuefunction for some remote period T and then constructs a path of time-varying valuefunctions that matches the given terminal value function.

Second, we build EFP on global approximation techniques that construct decisionfunctions on a sequence of domains covered with Smolyak grids. It is of much practicalinterest to develop also a version of EFP that builds on local perturbation techniques.The conventional EP method of Fair and Taylor (1983) is incorporated in the Dynaresoftware platform, and possibly, a perturbation-based version of the proposed EFPframework can be added there as well.6

6For more details on the Dynare software, see http://www.dynare.org and the Dynare referencemanual by Adjemian, Bastani, Juillard, Mihoubi, Perendia, Ratto, and Villemot (2011).

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54 BIBLIOGRAPHY

Appendices to ”A Tractable Framework forAnalyzing Nonstationary and Unbalanced Growth

Models”

Lilia MaliarSerguei MaliarJohn TaylorInna Tsener

Appendix A. Asymptotic convergence of T -period

stationary economy to nonstationary economy

This section elaborates the proof of Theorem 2 (turnpike theorem) formulated in Section1.3.2, specifically, it shows that the optimal program of the T -period stationary economyconverges to the optimal program of the nonstationary economy (3.25)–(3.28) as T →∞. The proof relies on three lemmas presented in Appendices A1-A3. In AppendixA1, we construct a limit program of a finite horizon economy with a terminal conditionkT = 0. In Appendix A2, we show that the optimal program of the T -period stationaryeconomy, constructed in Section 1.3.1, converges to the same limit program as doesthe finite horizon economy with a zero terminal condition kT = 0. In Appendix A3,we show that the limit program of the finite horizon economy with a zero terminalcondition kT = 0 is also an optimal program for the infinite horizon nonstationaryeconomy (3.25)–(3.28). Finally, in Appendix A4, we combine the results of AppendicesA1-A3 to establish the claim of Theorem 2. Our construction relies on mathematicaltools developed in Majumdar and Zilcha (1987), Mitra and Nyarko (1991), Joshi (1997).We use the convention that equalities and inequalities hold almost everywhere (a.e.)except for a set of measure zero.

Appendix A1. Limit program of finite horizon economy with azero terminal capital

In this section, we consider a finite horizon version of the economy (3.25)–(3.28) witha given terminal condition for capital kT . Specifically, we assume that the agent solves

maxct,kt+1Tt=0

E0

[T∑t=0

βtut (ct)

](1.31)

s.t. (3.27), (3.28), (1.32)

BIBLIOGRAPHY 55

where initial condition (k0, z0) and terminal condition kT are given. We first definefeasible programs for the finite horizon economy.

Definition A1 (Feasible programs in the finite horizon economy). A feasible programin the finite horizon economy is a pair of adapted (i.e., Ft measurable for all t) processesct, ktTt=0 such that given initial condition k0 and a partial history hT = (ε0, ..., εT ),they reach a given terminal condition kT at T , satisfy ct ≥ 0, kt ≥ 0 and (3.27), (3.28)for all t = 1, ...T .

In this section, we focus on a finite horizon economy that reaches a zero terminalcondition, kT = 0, at T . We denote by =T,0 a set of all finite horizon feasible programsfrom given initial capital k0 and given partial history hT ≡ (ε0, ..., εT ) that attain givenkT = 0 at T . We next introduce the concept of solution for the finite horizon model.

Definition A2 (Optimal program in the finite horizon model). A feasible finite horizon

programcT,0t , kT,0t

Tt=0∈ =T,0 is called optimal if

E0

[T∑t=0

βtut(c

T,0t )− ut (ct)

]≥ 0 (A1)

for every feasible process ct, ktTt=0 ∈ =T,0.

The existence result for the finite horizon version of the economy (1.31), (1.32) witha zero terminal condition is established in the literature. Namely, under Assumptions

1-9, there exists an optimal programcT,0t , kT,0t

Tt=0∈ =T,0 and it is both interior and

unique. The existence of the optimal program can be shown by using either a Bellmanequation approach (see Mitra and Nyarko (1991), Theorem 3.1) or an Euler equationapproach (see Majumdar and Zilcha (1987), Theorems 1 and 2).

We next show that under terminal condition kT,0T = kT = 0, the optimal program inthe finite horizon economy (1.31), (1.32) has a well-defined limit.

Lemma 1. A finite horizon optimal programcT,0t , kT,0t

Tt=0∈ =T,0 with a zero terminal

condition kT,0T = 0 converges to a limit programclimt , klimt

∞t=0

when T →∞, i.e.,

klimt ≡ limT→∞

kT,0t and climt ≡ limT→∞

cT,0t , for t = 0, 1, ... (A2)

Proof. The existence of the limit program follows by three arguments:i) Extending time horizon from T to T + 1 increases T -period capital of the finite

horizon optimal program, i.e., kT+1,0T > kT,0T . To see this, note that the model with time

horizon T has zero (terminal) capital kT,0T = 0 at T . When time horizon is extended from

56 BIBLIOGRAPHY

T to T + 1, the model has zero (terminal) capital kT+1,0T+1 = 0 at T + 1 but it has strictly

positive capital kT+1,0T > 0 at T ; this follows by the Inada conditions–Assumption 4.

ii) The optimal program for the finite horizon economy has the following propertyof monotonicity with respect to the terminal condition: if c′t, k′t

Tt=0 and c′′t , k′′t

Tt=0 are

two optimal programs for the finite horizon economy with terminal conditions κ′ < κ′′,then the respective optimal capital choices have the same ranking in each period, i.e.,k′t ≤ k′′t for all t = 1, ...T . This monotonicity result follows by either Bellman equationprogramming techniques (see Mitra and Nyarko (1991, Theorem 3.2 and Corollary 3.3))or Euler equation programming techniques (see Majumdar and Zilcha (1987, Theorem3)) or lattice programming techniques (see Hopenhayn and Prescott (1992)); see alsoJoshi (1997, Theorem 1) for generalizations of these results to nonconvex economies.

Hence, the stochastic processkT,0t

Tt=0

shifts up (weakly) in a pointwise manner when

T increases to T + 1, i.e., kT+1,0t ≥ kT,0t for t ≥ 0.

iii) By construction, the capital program from the optimal programcT,0t , kT,0t

Tt=0

is

bounded from above by the capital accumulation process 0, kmaxt Tt=0 defined in (1.8),

i.e., kT,0t ≤ kmaxt < ∞ for t ≥ 0. A sequence that is bounded and monotone is known

to have a well-defined limit.

Appendix A2. Limit program of the T -period stationary econ-omy

We now show that the optimal program of the T -period stationary economy, introducedin Section 1.3.1, converges to the same limit program (A2) as the optimal program ofthe finite horizon economy (1.31), (1.32) with a zero terminal condition. We denoteby =T a set of all feasible finite horizon programs that attains a terminal conditionof the T -period stationary economy. (We assume the same initial capital (k0, z0) andthe same partial history hT ≡ (ε0, ..., εT ) as those fixed for the finite horizon economy(1.31), (1.32)).

Lemma 2. The optimal program of the T -period stationary economycTt , k

Tt

Tt=0∈ =T

converges to a unique limit programclimt , klimt

∞t=0∈ =∞ defined in (A2) as T → ∞

i.e., for all t ≥ 0klimt ≡ lim

T→∞kTt and climt ≡ lim

T→∞cTt . (A3)

Proof. The proof of the lemma follows by six arguments.i). Observe that, by Assumptions 1-8, the optimal program of the T -period sta-

tionary economy has a positive capital stock kTT > 0 at T (since the terminal capital isgenerated by the capital decision function of a stationary version of the model), while

for the optimal programcT,0t , kT,0t

Tt=0∈ =T,0 of the finite horizon economy, it is zero

by definition, kT,0T = 0.

BIBLIOGRAPHY 57

ii). The property of monotonicity with respect to terminal condition implies that ifkTT > kT,0T , then kTt ≥ kT,0t for all t = 1, ..., T ; see our discussion in ii). of the proof toLemma 1.

iii). Let us fix some τ ∈ 1, ..., T. We show that up to period τ , the optimal

programcTt , k

Tt

τt=0

does not give higher expected utility thancT,0t , kT,0t

τt=0

, i.e.,

E0

[τ∑t=0

βtut(cTt)− ut(cT,0t )

]≤ 0. (A4)

Toward contradiction, assume that it does, i.e.,

E0

[τ∑t=0


]> 0. (A5)

Then, consider a new process c′t, k′tτt=0 that follows

cTt , k

Tt

Tt=0∈ =T up to period τ−1

and that drops down at τ to match kT,0τ of the finite horizon programcTt , k

Tt

Tt=0∈ =T,0,

i.e., c′t, k′tτt=0 ≡

cTt , k

Tt

τ−1t=0∪cTτ + kTτ − kT,0τ , kT,0τ

. By monotonicity ii). we have

kTτ − kT,0τ ≥ 0, so that

E0

[τ∑t=0

βtut (c′t)− ut

(cTt)]

=

= E0

[βτut(cTτ + kTτ − kT,0τ

)− ut

(cTτ)]≥ 0, (A6)

where the last inequality follows by Assumption 2 of strictly increasing ut.

iv). By construction c′t, k′tτt=0 and

cT,0t , kT,0t

τt=0

reach the same capital kT,0τ at

τ . Let us extend the program c′t, k′tτt=0 to T by assuming that it follows the process

cT,0t , kT,0t

Tt=0

from the period τ + 1 up to T , i.e., c′t, k′tTt=τ+1 ≡

cT,0t , kT,0t

Tt=τ+1

.

Then, we have

E0

[T∑t=0

βtut (c′t)− ut(c

T,0t )]

= E0

[τ∑t=0

βtut (c′t)− ut(c

T,0t )]

≥ E0

[τ∑t=0

βtut(cTt)− ut

(cT,0t

)]> 0, (A7)

where the last two inequalities follow by result (A6) and assumption (A5), respectively.Thus, we obtain a contradiction: The constructed program c′t, k′t

Tt=0 ∈ =T,0 is feasible

in the finite horizon economy with a zero terminal condition, k′T = 0, and it gives

strictly higher expected utility than the optimal programcT,0t , kT,0t

Tt=0∈ =T,0 in that

economy.

58 BIBLIOGRAPHY

v). Holding τ fixed, we compute the limit of (A4) by letting T go to infinity:

limT→∞

E0

[τ∑t=0


]=

limT→∞

E0

[τ∑t=0

βtut(cTt)]− E0

[τ∑t=0

βtut(climt)]≤ 0. (A8)

vi). The last inequality implies that for any τ ≥ 1, the limit programclimt , klimt

∞t=0∈

=∞ of the finite horizon economycT,0t , kT,0t

Tt=0∈ =T,0 with a zero terminal con-

dition kT,0T = 0 gives at least as high expected utility as the optimal limit programcTt , k

Tt

Tt=0∈ =T of the T -period stationary economy. Since Assumptions 1-8 imply

that the optimal program is unique, we conclude thatclimt , klimt

∞t=0∈ =∞ defined

in (A2) is a unique limit of the optimal programcTt , k

Tt

Tt=0∈ =T of the T -period

stationary economy.

Appendix A3. Convergence of finite horizon economy to infinitehorizon economy

We now show a connection between the optimal programs of the finite horizon andinfinite horizon economies. Namely, we show that the finite horizon economy (1.31),(1.32) with a zero terminal condition kT,0T = 0 converges to the nonstationary infinitehorizon economy (3.25)–(3.28) as T →∞.

Lemma 3. The limit programclimt , klimt

∞t=0

is a unique optimal program c∞t , k∞t ∞t=0 ∈

=∞ in the infinite horizon nonstationary economy (3.25)–(3.28).Proof. We prove this lemma by contradiction. We use the arguments that are

similar to those used in the proof of Lemma 2.i). Toward contradiction, assume that

climt , klimt

∞t=0

is not an optimal program ofthe infinite horizon economy c∞t , k∞t

∞t=0 ∈ =∞. By definition of limit, there exists a

real number ε > 0 and a subsequence of natural numbers T1, T2, ... ⊆ 0, 1, ... suchthat c∞t , k∞t

∞t=0 ∈ =∞ gives strictly higher expected utility than the limit program of

the finite horizon economyclimt , klimt

∞t=0

, i.e.,

E0

[Tn∑t=0

βtut (c∞t )− ut(climt )

]> ε for all Tn ∈ T1, T2, ... . (A9)

ii). Let us fix some Tn ∈ T1, T2, ... and consider any finite T ≥ Tn. Assumptions1-8 imply that k∞T > 0 while kT,0T = 0 by definition of the finite horizon economy witha zero terminal condition. The monotonicity of the optimal program with respect to aterminal condition implies that if k∞T > kT,0T , then k∞t ≥ kT,0t for all t = 1, ..., T ; see ourdiscussion in ii). of the proof of Lemma 1.

BIBLIOGRAPHY 59

iii). Following the arguments in iii). of the proof of Lemma 2, we can show that upto period Tn, the optimal program c∞t , k∞t

Tnt=0 does not give higher expected utility

thancT,0t , kT,0t

Tnt=0

, i.e.,

E0

[Tn∑t=0

βtut (c∞t )− ut(cT,0t )

]≤ 0 for all Tn. (A10)

iv). Holding Tn fixed, we compute the limit of (A10) by letting T go to infinity:

limT→∞

E0

[Tn∑t=0

βtut (c∞t )− ut(cT,0t )

]

= E0

[Tn∑t=0

βtut (c∞t )− βtut(climt)]≤ 0 for all Tn. (A11)

However, result (A11) contradicts to our assumption in (A9).v). We conclude that for any subsequence T1, T2, ... ⊆ 0, 1, ..., we have

E0

[Tn∑t=0

βtut (c∞t )− ut(climt )

]≤ 0 for all Tn. (A12)

However, under Assumptions 1-8, the optimal program c∞t , k∞t ∞t=0 ∈ =∞ is unique,

and hence, it must be that c∞t , k∞t ∞t=0 coincides with

climt , klimt

∞t=0

for all t ≥ 0.

Appendix A4. Proof to the turnpike theorem

We now combine the results of Lemmas 1-3 together into a turnpike-style theorem toshow the convergence of the optimal program of the T -period stationary economy tothat of the infinite horizon nonstationary economy. To be specific, Lemma 1 showsthat the optimal program of the finite horizon economy with a zero terminal conditioncT,0t , kT,0t

Tt=0∈ =T,0 converges to the limit program

climt , klimt

∞t=0

. Lemma 2 shows

that the optimal program of the T -period stationary economycTt , k

Tt

Tt=0

also con-

verges to the same limit programclimt , klimt

∞t=0

. Finally, Lemma 3 shows that the limit

program of the finite horizon economiesclimt , klimt

∞t=0

is optimal in the nonstationaryinfinite horizon economy. Then, it must be the case that the limit optimal program of

the T -period stationary economycTt , k

Tt

Tt=0

is optimal in the infinite horizon nonsta-tionary economy. This argument is formalized below.

Proof to Theorem 2 (turnpike theorem). The proof follows by definition of limit andLemmas 1-3. Let us fix a real number ε > 0 and a natural number τ such that1 ≤ τ <∞.

60 BIBLIOGRAPHY

i). Lemma 1 shows thatcT,0t , kT,0t

Tt=0∈ =T,0 converges to a limit program

climt , klimt∞t=0

as T → ∞. Then, definition of limit implies that there exists T1 > 0

such that∣∣∣kT,0t − klimt

∣∣∣ < ε3

for t = 0, ..., τ .

ii). Lemma 2 implies that the finite horizon problem of the T -period stationary

economycTt , k

Tt

Tt=0

also converges to limit programclimt , klimt

∞t=0

as T →∞. Then,

there exists T2 > 0 such that∣∣klimt − kTt ∣∣ < ε

3for t = 0, ..., τ .

iii). Lemma 3 implies the programcT,0t , kT,0t

Tt=0∈ =T,0 converges to the infinite

horizon optimal program c∞t , k∞t ∞t=0 as T →∞. Then, there exists T3 > 0 such that∣∣∣kT,0t − k∞t

∣∣∣ < ε3

for t = 0, ..., τ .

iv). Then, the triangular inequality implies

∣∣kTt − k∞t ∣∣ =∣∣∣kTt − klimt + klimt − k

T,0t + kT,0t − k∞t

∣∣∣≤∣∣kTt − klimt ∣∣+

∣∣∣klimt − kT,0t

∣∣∣+∣∣∣kT,0t − k∞t

∣∣∣ < ε

3+ε

3+ε

3= ε,

for T ≥ T (ε, τ) ≡ max T1, T2, T3.

Remark A1. Our proof of the turnpike theorem addresses a technical issue that doesnot arise in the literature that focuses on finite horizon economies with a zero terminalcondition; see, e.g., Majumdar and Zilcha (1987), Mitra and Nyarko (1991) and Joshi(1997). Their construction relies on the fact that the optimal program of the finitehorizon economy is always pointwise below the optimal program of the infinite horizoneconomy, i.e., kTt ≤ k∞t , for t = 1, ..., τ , and it gives strictly higher expected utilityup to T than does the infinite horizon optimal program (because excess capital canbe consumed at terminal period T ). This argument does not directly applies to ourT -period stationary economy: our finite horizon program can be either below or abovethe infinite horizon program depending on a specific T -period terminal condition; seethe experiments with terminal conditions κ′ and κ′′ in Figure 1, respectively. Ourproof addresses this issue by constructing in Lemma 2 a separate limit program for theT -period stationary economy.

Remark A2. We also proved a similar turnpike theorem for a more general version ofthe economy (3.25)–(3.28) (proofs are not provided). First, we relax the assumption ofMarkov structure of the stochastic process (3.28) (i.e., we consider a general stochasticenvironment that satisfies only a weak assumptions of measurability); and second, werelax the assumption that the terminal condition comes from the T -period stationaryeconomy (i.e., we consider an arbitrary terminal condition kT ). To save on space, wedo not include this more general turnpike theorem in the paper but limit ourselves tothe nonstationary Markov setup that is actually studied.

BIBLIOGRAPHY 61

Appendix B. Implementation of EFP

In this section, we describe the implementation of the EFP method used to producethe numerical results in the main text.

The EFP method is more expensive than conventional solution methods for sta-tionary models because decision functions must be constructed not just once but forT periods. We implement EFP in the way that keeps its cost relatively low: First,to approximate decision functions, we use a version of the Smolyak (sparse) grid tech-nique. Specifically, we use a version of the Smolyak method that combines a Smolyakgrid with ordinary polynomials for approximating functions off the grid. This methodis described in Maliar, Maliar and Judd (2011) who find it to be sufficiently accuratein the context of a similar growth model, namely, unit-free residuals in the model’sequations do not exceed 0.01% on a stochastic simulation of 10,000 observations). Forthis version of the Smolyak method, the polynomial coefficients are overdetermined, forexample, in a 2-dimensional case, we have 13 points in a second-level Smolyak grid, andwe have only six coefficients in second-degree ordinary polynomial. Hence, we identifythe coefficients using a least-squares regression; we use an SVD decomposition, to en-hance numerical stability; see Judd, Maliar and Maliar (2011) for a discussion of thisand other numerically stable approximation methods. We do not construct the Smolyakgrid within a hypercube normalized to [−1, 1]2, like do Smolyak methods that rely onChebyshev polynomials used in, e.g., Krueger and Kubler (2004) and Judd, Maliar,Maliar and Valero (2014). Instead, we construct a sequence of Smolyak grids aroundactual steady state and thus, the hypercube, in which the Smolyak grid is constructed,grows over time as shown in Figure 1.

Second, to approximate expectation functions, we use Gauss-Hermite quadraturerule with 10 integration nodes. However, a comparison analysis in Judd, Maliar andMaliar (2011) shows that for models with smooth decision functions like ours, thenumber of integration nodes plays only a minor role in the properties of the solution,for example, the results will be the same up to six digits of precision if instead of tenintegration nodes we use just two nodes or a simple linear monomial rule (a two-nodeGauss-Hermite quadrature rule is equivalent to a linear monomial integration rule forthe two-dimensional case). However, simulation-based Monte-Carlo-style integrationmethods produce very inaccurate approximations for integrals and are not consideredin this paper; see Judd, Maliar and Maliar (2011) for discussion.

Third, to solve for the coefficients of decision functions, we use a simple derivative-free fixed-point iteration method in line with Gauss-Jacobi iteration. Let us re-writethe Euler equation i). constructed in the initialization step of the algorithm by pre-multiplying both sides by t-period capital

k′m,t = β

J∑j=1

εj,t

[u′t(c

′m,j,t)

u′t(cm,t)

1− δ + ft+1

(k′m,tk

∗t+1, z

′m,j,tz

∗t+1

)]k′m,t. (1.33)

62 BIBLIOGRAPHY

Algorithm 1 (implementation). Extended function path.

The goal of EFP.

EFP is aimed at approximating a solution of a nonstationary model during the first τ

periods, i.e., it finds approximating functions(K0, ..., Kτ

)such that Kt ≈ Kt for t = 1, ...τ ,

where Kt and Kt are a t-period true capital function and its parametric approximation,respectively.

Step 0. Initialization.

a. Choose time horizon T τ for constructing T -period stationary economy.

b. Construct a deterministic path z∗t Tt=0 for exogenous state variable ztTt=0 satisfying

z∗t+1 = ϕt (z∗t , Et [εt+1]) for t = 0, .., T .

c. Construct a deterministic path k∗t Tt=0 for endogenous state variable ktTt=0 satisfying

u′t(c∗t ) = βu′t(c

∗t+1)(1− δ + f ′t+1

(k∗t+1, z

∗t+1

)).

c∗t + k∗t+1 = (1− δ) k∗t + ft (k∗t , z∗t ) for t = 0, .., T .

d. For t = 0, ..., T :

Construct a grid (km,t, zm,t)Mm=1 centered at (k∗t , z∗t ).

Choose integration nodes, εj,t, and weights, ωj,t for j = 1, ..., J .Construct future shocks z′m,j,t = ϕt (zm,t, εj,t).

e. Write a t-period discretized system of the optimality conditions:

i). u′t(cm,t) = βJ∑j=1

ωj,t

[u′t(c

′m,j,t)

1− δ + ft+1

(k′m,t, z

′m,j,t

)]ii). cm,t + k′m,t = (1− δ) km,t + ft (km,t, zm,t)

iii). c′m,j,t + k′′m,j,t = (1− δ) k′m,t + ft+1

(k′m,t, z

′m,j,t

)iv). k′m,t = Kt (km,t, zm,t) and k′′m,j,t = Kt+1

(k′m,t, z

′m,j,t

).

d. Assume that the model becomes stationary at T .

Step 1. Solving the T -period stationary model.

Find KT = KT+1 that approximately solves the system i).-iv). on the grid for the T -periodstationary economy fT+1 = fT , uT+1 = uT , ϕT+1 = ϕT .

Step 2. Solving for a function path for t = 0, 1, ..., T − 1.

a. Construct the function path(K0, ..., KT−1, KT

)that approximately solves the system

i).-iv) for each t = 0, ..., T and that matches the given terminal function KT constructedin Step 1.

The EFP solution:

Use(K0, ..., Kτ

)as an approximation to (K0, ...,Kτ ) and discard the remaining T − τ

functions.

We use different notation, k′m,t and k′m,t, for t-period capital in the left and right side

BIBLIOGRAPHY 63

of (1.33), respectively, in order to describe our fixed-point iteration method. Namely,we substitute k′m,t in the right side of (1.33) and in the constraints ii). and iii). in theinitialization step to compute cm,t and c′m,j,t, respectively, and we obtain a new set of

values of the capital function on the grid k′m,t in the left side. We iterate on these stepsuntil convergence.

Our approximation functions Kt are ordinary polynomial functions characterizedby a time-dependent vector of parameters bt, i.e., Kt = K (·; bt). So, operationally, the

iteration is performed not on the grid values k′m,t and k′m,t but on the coefficients of theapproximation functions. The iteration procedure differs in Steps 1 and 2.

In Step 1, we construct a solution to T -period stationary economy. For iteration i,we fix some initial vector of coefficients b, compute k′m,T+1 = K (km,T , zm,T ; b), find cm,T

and c′m,j,T to satisfy constraints ii) and iii), respectively and find k′m,T+1 from the Euler

equation i). We run a regression of k′m,T+1 on K (km,T , zm,T ; ·) in order to re-estimate the

coefficients b and we compute the coefficients for iteration i+ 1 as a weighted average,i.e., b(i+1) = (1− ξ) b(i) + ξb(i), where ξ ∈ (0, 1) is a damping parameter (typically,ξ = 0.05). We use partial updating instead of full updating ξ = 1 because fixed-pointiteration can be numerically unstable and using partial updating enhances numericalstability; see Maliar, Maliar and Judd (2011). This kind of fixed-point iterations areused by numerical methods that solve for equilibrium in conventional stationary Markoveconomies; see e.g., Judd, Maliar and Maliar (2011), Judd, Maliar, Maliar and Valero(2014).

In Step 2, we iterate on the path for the polynomial coefficients using Gauss-Jacobistyle iterations in line with Fair and Taylor (1983). Specifically, on iteration j, we take

a path for the coefficients vectorsb(j)1 , ..., b

(j)T

, compute the corresponding path for

capital quantities using k′m,t = Kt

(km,t, zm,t; b

(j)t

), and find a path for consumption

quantities cm,t and c′m,j,t from constraints ii) and iii), respectively, for t = 0, ..., T .Substitute these quantities in the right side of a sequence of Euler equations for t =0, ..., T to obtain a new path for capital quantities in the left side of the Euler equationk′m,t for t = 0, ..., T − 1. Run T − 1 regressions of k′m,t on polynomial functional

forms Kt (km,t, zm,t; bt) for t = 0, ..., T − 1 to construct a new path for the coefficientsb(j)0 , ..., b

(j)T−1

. Compute the path of the coefficients for iteration j + 1 as a weighted

average, i.e., b(j+1)t = (1− ξ) b(j)t + ξb

(j)t , t = 0, ..., T − 1, where ξ ∈ (0, 1) is a damping

parameter which we again typically set at ξ = 0.05. (Observe that this iteration

procedure changes all the coefficients on the path except of the last one b(j)T ≡ b, which

is a given terminal conditions that we computed in Step 1 from T -period stationaryeconomy).

In fact, the problem of constructing a path for function coefficients is similar to theproblem of constructing a path for variables: in both cases, we need to solve a largesystem of nonlinear equations. The difference is that under EFP, the arguments ofthis system are not variables but parameters of the approximating functions. Instead

64 BIBLIOGRAPHY

of Gauss-Jacobi style iteration on path, we can use Gauss-Siedel fixed-point iteration(shooting), Newton-style solvers or any other technique that can solve a system of non-linear equations; see Lipton, Poterba, Sachs and Summers (1980), Atolia and Buffie(2009a,b), Heer and Maußner (2010), and Grune, Semmler and Stieler (2013) for ex-amples of such techniques.

Appendix C. Fair and Taylor’s (1983) method

This appendix describes the version of Fair and Taylor’s (1983) method used to producethe results in the main text. We illustrate this method in the context of the growthmodel (3.25)–(3.28) (we assume δ = 1 and u (c) = ln (c) for expository convenience).The Euler equation and budget constraint, respectively, are:

1

ct= βEt

[1

ct+1

(1− δ + zt+1f′(kt+1))

],

ct + kt+1 = (1− δ) kt + ztf (kt) .

Combine the above two conditions to get

kt+1 = ztf (kt)−[Et

(βzt+1f

′(kt+1))

zt+1f (kt+1)− kt+2

)]−1≈

ztf (kt)−zet+1f (kt+1)− kt+2

βzet+1f′(kt+1))

, (1.34)

where the expectation function zet+1 = Et [zt+1] is approximated as implied by cer-tainty equivalence assumption (1.16). For example, for the standard AR(1) processfor productivity levels (1.18), this approximation means that for each t, we havezet+1 = E [zt+1] = zρtE [exp(εt)], where εt ∼ N(0, σ2). To solve for the path of vari-ables, Fair and Taylor (1983) use a derivative-free iteration in line with Gauss-Jacobimethod :

BIBLIOGRAPHY 65

Algorithm 2. Extended path (EP) framework by Fair and Taylor (1983).

The goal of EP framework of Fair and Taylor (1983).

EFP is aimed at approximating a path for variables satisfying the model’s equations during

the first τ periods, i.e., it finds k0, ..., kτ such that∥∥∥kt − kt∥∥∥ < ε for

t = 1, ...τ , where ε > 0 is target accuracy, ‖·‖ is an absolute value, and kt and kt are thet-period true capital stocks and their approximation, respectively.


a. Choose time horizon T τ and terminal condition kT+1.b. Construct and fix a sequence of shocks ztt=0,...,T .

c. Construct and fixzet+1

t=0,...,T

such that zet+1 = Et [zt+1] for all t.

d. Guess an equilibrium pathk(1)t

t=1,...,T ′

for iteration j = 1.

e. Write a t-period system of the optimality conditions in the form:

kt+1 = ztf (kt)−zet+1f(kt+1)−kt+2

βzet+1f′(kt+1))

.

Step 1. Solving for a path using Gauss-Jacobi method.

a. Substitute a pathk(j)t

t=1,...,T ′

into the right side of (3.53) to find

k(j+1)t+1 = ztf

(k(j)t

)−

zet+1f(k(j)t+1

)−k(j)t+2

βzet+1f′(k

(j)t+1))

, t = 1, ..., T

b. End iteration if the convergence is achieved∣∣∣k(j+1)t+1 − k(j)t+1

∣∣∣ < tolerance level.

Otherwise, increase j by 1 and repeat Step 1.

The EP solution:

Use the first τ constructed values k0, ..., kτ as an approximationto the true solution kt, ..., kτ and discard the last T − τ values.

In the original version of their EP method, Fair and Taylor (1983) use τ = 1, i.e.,

they keep only the first element k1 from the constructed path(k1, ..., kT

)and disregard

the rest of the path; then, they solve for a new path(k1, ..., kT+1

)starting from k1 and

ending in a given kT+1 and store k2, disregarding the rest of the path; and they advanceforward until the path of the given length τ is constructed. T is chosen so that furtherits extensions do not affect the solution in the initial period of the path. For instance,to find a solution k1, Fair and Taylor (1983) solve the model several times under T + 1,

T + 2, T + 3, ... and check that k1 remains the same (up to a given degree of precision).Finally, it is also possible to use Fair and Taylor’s (1983) method under larger values ofτ such as τ = 100; in this respect, Fair and Taylor’s (1983) method is similar to EFP.

As is typical for fixed-point-iteration style methods, Gauss-Jacobi iteration may failto converge. To deal with this issue, Fair and Taylor (1983) use damping, namely, they

66 BIBLIOGRAPHY

update the path over iteration only by a small amount k(j+1)t+1 = ξk

(j+1)t+1 + (1− ξ) k(j)t+1

where ξ ∈ (0, 1) is a small number close to zero (e.g., 0.01).Steps 1a and 1b of Fair and Taylor’s (1983) method are called Type I and Type II

iterations and are analogous to Step 2 of the EFP method when the sequence of thedecision functions is constructed. The extension of path is called Type III iteration andgives the name to Fair and Taylor (1983) method.

In our examples, we implement Fair and Taylor’s (1983) method using a conventionalNewton style numerical solver instead of Gauss-Jacobi iteration; a similar implementa-tion is used in Heer and Maußner (2010). The cost of Fair and Taylor’s (1983) methodcan depend considerably on a specific solver used and can be very high (as we need tosolve a system of equations with hundreds of unknowns numerically). In our simpleexamples, a Newton-style solver was sufficiently fast and reliable. In more complicatedmodels, we are typically unable to derive closed-form laws of motion for the state vari-ables, and derivative-free fixed-point iteration advocated in Fair and Taylor (1983) canbe a better alternative.

Appendix D. Solving the test model using the asso-

ciated stationary model

We first convert the nonstationary model (3.25), (3.27), (1.17), (1.18) with labor aug-menting technological progress into a stationary model using the standard change ofvariables ct = ct/At and kt = kt/At. This leads us to the following model

maxkt+1,ct

t=0,...,∞

E0

∞∑t=0

(β∗)tc1−γt

1− γ(1.35)

s.t. ct + gAkt+1 = (1− δ) kt + ztkαt , (1.36)

ln zt+1 = ρt ln zt + σtεt+1, εt+1 ∼ N (0, 1) , (1.37)

where β∗ ≡ βg1−γA . We solve this stationary model by using the same version of theSmolyak method that is used within EFP to find a solution to T -period stationaryeconomy.

After a solution to the stationary model (1.35)–(1.37) is constructed, a solution fornonstationary variables can be recovered by using an inverse transformation ct = ctAtand kt = ktAt.

For the sake of our comparison, we also need to recover the path of nonstationarydecision functions in terms of their parameters. Let us show how this can be done underpolynomial approximation of decision functions. Let us assume that a capital policyfunction of the stationary model is approximated by complete polynomial of degree L,namely, kt+1 =

∑Ll=0

∑lm=0 bm+

(l−1)(l+2)2

+1kmt z

l−mt , where bi is a polynomial coefficient,

i = 0, ..., L+ (L−1)(L+2)2

+ 1. Given that the stationary and nonstationary solutions are

BIBLIOGRAPHY 67

related by kt+1 = kt+1/(A0g

t+1A

), we have

kt+1 = A0gt+1A kt+1 = A0g

t+1A

L∑l=0

l∑m=0

bm+

(l−1)(l+2)2

+1kmt z

l−mt =

A0

L∑l=0

l∑m=0

gA1−(m−1)tb

m+(l−1)(l+2)

2+1kmt z

l−mt . (1.38)

For example, for first-degree polynomial L = 1, we construct the coefficients vector ofthe nonstationary model by premultiplying the coefficient vector b ≡ (b0, b1, b2) of the

stationary model by a vector(A0g

t+1A , A0gA, A0g

t+1A

)>, which yields

bt+1 ≡(b0A0g

t+1A , b1A0gA, b2A0g

t+1A

), t = 0, ..., T , where T is time horizon (length of

simulation in the solution procedure). Note that a similar relation will hold even if thegrowth rate gA is time variable.

Appendix E. Additional figures

In Figure 10, we plot the simulated solution to the model with both deterministic tech-nology switches and stochastic productivity shocks following an AR(1) process (1.37);

0 100 200 300 400 500 600 700 800

1

1.1

1.2

time

Pro

du

ctivity

0 100 200 300 400 500 600 700 80030

40

50

60

Ca

pita

l

time

0 100 200 300 400 500 600 700 8002.5

3

3.5

4

Co

nsu

mp

tio

n

time

EFP solution

Naive solution

EFP solution

Naive solution

Figure 1.10: Deterministic technology switches and stochastic productivity shocks

In Figure 11, we provide a plot of simulated solution with both productivity driftand stochastic productivity shocks.

68 BIBLIOGRAPHY

0 100 200 300 400 500 600 700 8000.95

1

1.05

1.1

1.15

1.2

time

Pro

ductivity

0 100 200 300 400 500 600 700 80035

40

45

50

55

Capital

time

0 100 200 300 400 500 600 700 8002.5

3

3.5

4

Consum

ption

time

EFP solution

Naive solution

EFP solution

Naive solution

Figure 1.11: Productivity shifts and stochastic productivity shocks

Chapter 2

Capital-Skill ComplementarityRevisited: Can an UnbalancedGrowth Model Account forUnbalanced Growth Data

2.1 Introduction

The economic growth determines wealth of nations: even small differences in growthrates between countries sum up into large differences in wealth after several decades.It is critical for economists and politicians to gain understanding of the driving forcesof economic growth in order to design effective government policies.

The economic growth is a complex phenomenon. The economic literature distin-guishes several complementary mechanisms of growth, such as, capital augmentingtechnological progress (machines get more efficient), labor augmenting technologicalprogress (labor becomes more skilled) and a neutral technological progress (machinesand labor are combined more efficiently in the production process). Furthermore, thegrowth rate is directly affected by consumption-saving decisions as those decisions de-termine the savings rate and hence, the speed of capital accumulation.

The resulting growth patterns for the key macroeconomic variables in the data arehighly unbalanced. In particular, for the US economy, we document the followingregularities over the 1963 - 2012 period: output and stock of structures increased by2.63 and 3.24 times, respectively; the stock of equipment increased by more than 30times; the population of skilled and unskilled workers increased by 6.3 and 1.39 times,

69

70 CHAPTER 2. CAPITAL-SKILL COMPLEMENTARITY REVISITED

respectively; the price of equipment relative to the price of consumption went downby more than 9 times; and the skill premium (defined as a ratio of wages of skilled tounskilled labor) was growing at an average rate of 1% per year.

A large body of literature study the role of technological progress in the productionprocess and investigate its effects on capital and labor inputs; for a comprehensive sur-vey of the literature see Goldin and Katz (2008) and Acemoglu and Autor (2012). Inparticular, Bound and Johnson (1992) conclude that an increase in the skill premiumcan be attributed to technological changes. Katz and Murphy (1992) document theimportance of the trend component which is interpreted in their work as a relative de-mand shifts for labor. Their analysis showed that that introducing a trend in the modelis important in accounting for an increase in the skill premium. The constant elasticityof substitution (CES) production function proves remarkable successful in accountingfor the key features of the data. An important contribution to the literature is madeby Krusell et al (2000) who argues that it is possible to account for the data withoutassuming exogenous and unobservable variables by using even more flexible and realis-tic CES production function with four production inputs, skilled labor, unskilled labor,capital equipment and capital structures: they assume that skilled labor is more com-plementary with equipment than is unskilled labor. The main message of the literatureis that all variables that determine economic growth can be directly observed from thedata and hence, the economists and politicians must concentrate on policies that affectthese variables in the way that promotes the economic growth (while unobserved andexogenous sources of growth cannot be affected).

The present paper contributes to the literature in three ways: First, we construct anup-to-date data set that contains the key macroeconomic variables that characterize theeconomic growth in the US economy over the 1964 - 2012 period. Our data set includeslabor-market variables such as the population of skilled and unskilled workers, theirannual hours worked and their wages; these variables are constructed using micro-leveldata–the Current population survey. Furthermore, our data set includes such aggregatevariables as consumption, capital structures, capital equipment, investment and relativeprices; these variables are constructed using using macro-level data–subcategories ofthe National income and product accounts. In the construction of the data, we closelyfollow the methodology of KORV (2000) and thus, our data set can be viewed asan actualized version of their data. The sample of KORV covers the period 1963 -1992. Since then, the world experienced a dramatic technological change due to thedevelopment of computers, internet, smartphones, etc. Therefore, our data allow toappreciate how growth patterns changed in the recent years. Finally, we also constructsome labor market variables that are not provided in KORV (2000), such as skilled andunskilled population of workers and their average and average annual hours worked.

Second, we analyze how the findings of KORV (2000) are robust to modifications inspecific way of constructing economic variables and the sample size used for estimation.We find that the pattern of the skill premium and implications and estimations ofthe parameters in the KORV function critically depend on specific way in which thevariables are defined. In our data the skill premium is increasing and comparable

2.1. INTRODUCTION 71

increasing pattern of skill premium is also documented by other studies, see Acemogluand Autor (2011). We estimated parameters in the production function using simulatedpseudo maximum likelihood method as in Laroque and Salanie (1993). Our estimate ofthe elasticity of substitution between equipment (skilled labor) and unskilled labor isabout 2, and that of the elasticity of substitution between equipment and skilled laborof about 0.6. The former estimate is in line with the results in Krusell et al. (2000) whoobtained 1.67, while the latter estimate is about two times lower than their estimate.Most importantly, our results confirm the main insight of KORV (2000) analysis. In thecontext of their CES production function, we can can account for the data includingthe growth pattern of skill premium using just observable variables without adding anyexogeneous unobservable progress.

Finally, the mainstream of the literature including KORV (2000) is limited to partialequilibrium analysis of the production side of the economy. Namely, they use macroe-conomic variables such as capital and labor from the data to estimate parameters inproduction functions. However, to study questions related to optimal policy choice, weneed to build a model in which growth variables are generated endogenously. Only then,we can analyze how such variables are affected by fiscal and or monetary policy andonly then, we can design economic policies that promote economic growth. Therefore,as the next step, we embody the CES production function into a general equilibriumanalysis. Specifically, we consider an economy populated by two types of consumers,skilled and unskilled, who solve the intertemporal utility maximization problems bychoosing consumption, savings and labor supply. The skilled and unskilled populationgrows at the same rates as in the data. We allow for three empirically relevant typesof technological progress, namely, capital augmenting, labor augmenting and a neutraltechnological progresses and also, we allow for technological changes that affects a rel-ative price of investment to consumption (to account for regularities observed in thedata).

We ask: ”Can a general equilibrium model with realistic degrees of capital skillcomplementarity account quantitatively for the growth patterns observed in the USdata under empirically relevant calibration?” In particular, we investigate whether ornot the model can endogeneously produce unbalanced growth patterns for consumption,capital equipment and capital structures similar to those that we observe in the USeconomy data.

The constructed model is non-stationary due to assumption of unbalanced techno-logical progress and cannot be studied with conventional numerical solution methods.To be precise, the conventional numerical methods construct stationary Markov equi-libria in which decision functions are fixed points of either Bellman or Euler equations.1

However, our model does not have a stationary Markov equilibrium: in the presence ofunbalanced growth, decision function change from one period to another and they arenot have fixed points of either Euler or Bellman equations.

1For reviews of numerical solution methods for constructing stationary markov equilibria, seeTaylor and Uhlig (1990), Judd (1998), Adda and Cooper (2003), etc.


The approach used in the real business cycle literature to deal with the growth isto impose restrictions on the model that make it consistent with a balanced growthwhen all variables grow at the same rate. The resulting model can be converted intostationary by using an appropriate change of variables and can be solved using conven-tional solution methods. However, only the assumption of labor augmenting progress isconsistent with balanced growth; see King et al. (1988) for a careful discussion. Sincewe also allow for capital and technology augmenting progresses, this approach cannotbe used in our case. Maliar and Maliar (2011) propose a way of converting certainunbalanced non-stationary models into stationary by imposing restrictions on growthrates of some variables. However, such restrictions are not necessarily consistent withthe data and their model failed to account for the growth patterns in the data.

A recent paper by Maliar, Maliar, Taylor and Tsener (2014) introduce a novel quan-titative framework that allows to calibrate, solve, estimate and simulate non-stationaryand unbalanced growth models without imposing any supplementary restrictions. Weapplied this framework to study quantitative implications of our non-stationary growthmodel, and we find that the model can account for the key unbalanced growth patternsin the data. However, the model is not entirely successful in explaining the behaviorof skill premium. Namely, it predicts that the skill premium is roughly constant whilein the data it is growing over time.There are three effects that are responsible for theincreasing dynamics of the skill premium: Relative quantity of labor, relative efficiencyof labor effect and capital-skill complementarity effects. The relative quantity of laboreffect decreases the skill premium, while the relative efficiency and capital-skill comple-mentarity effects increase it. Under the estimated parameter values the three effectsoffset each other.

The rest of the paper is organized as follows: Section 2 describes the data; Section 3describes the production function and formulates a general equilibrium model; Section4 estimates the parameters of the production function; Section 5 discusses calibrationand solution procedure; and finally, Section 6 concludes.

2.2 Data

The goal of this paper is to analyze implications of a general equilibrium macroeconomicmodel based on the CES production function. To this purpose, we first construct a dataset for estimation and calibration of the parameters of the model as well as for assessingthe fit of the model to the data.

In a seminal paper, Krusell et al. (2000) constructed a data set for estimation of apartial equilibrium model of the US economy based on the CES production function.To much extend, we follow their methodology in the construction of our dataset. Ouranalysis differs from theirs in two respects: First, we extend their data sample (theirdata sample is constructed for the years 1963-1992 while we construct the sample for1963 - 2012). Second, we construct some additional labor market variables that werenot relevant for partial equilibrium analysis of KORV, but are essential for our general

2.2. DATA 73

1965 1970 1975 1980 1985 1990 1995 2000 2005 20101400

1600

1800

2000

2200Annual hours worked

skilled

unskilled

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100

2

4

6

8x 10

7 Number of workers

skilled

unskilled

1965 1970 1975 1980 1985 1990 1995 2000 2005 20102

4

6

8

10

12

14x 10

10 Total annual labor input

skilled

unskilled

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010500

600

700

800

900

1000

1100Weekly wages

skilled

unskilled

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100.2

0.4

0.6

0.8

1

1.2Labor input ratio

Data

KORV(2000)

1965 1970 1975 1980 1985 1990 1995 2000 2005 20101.3

1.4

1.5

1.6

1.7

1.8Skill premium

Data

KORV(2000)

Figure 2.1: Microeconomic data

equilibrium analysis. Finally, we analyze how robust are some empirical regularitiesestablished in KORV to specific modeling assumptions.

We construct two groups of variables on the US economy: the first group includeslabor market variables and is constructed using microeconomic data and the secondgroup includes such variables as output, capital, and prices and is constructed usingmacroeconomic data. A detailed description of the construction of these two groups ofthe data and our methodology are provided in Appendices A.1 and A.2, respectively.

Microeconomic data To construct labor market variables, we use the current pop-ulation survey (CPS) data set. To be more specific, for years 1964-1988 we use MarchAnnual Demographic Supplement files (original Mare-Winship data) for years 1989-2012we use CPS Annual Demographic March files. The constructed variables are shown inFigure 1. We distinguished 2 representative groups by their level of skills. Many educa-tion measures show a strong secular increase in the quantity of skilled labor relative tothe quantity of unskilled labor. In our analysis skilled individuals are those who havecollege or higher degree and half of those who have some years of college education.Unskilled workers are, therefore, the rest of the sample.

We define “Total labor input” as a sum of products of annual weeks worked andhours worked last week over different education, gender, race, age and experience pro-files. In our sample all the workers are assigned to a skilled or unskilled groups ofworker. ”Annual hours worked” is a ratio of the total labor input of a particular skill


group of workers and number of workers in that group.Growth patterns observed in Figure 1 are unbalanced: the number of both skilled

and unskilled agents increases over time, however, the percentage increase in skilledlabor is much larger than increase in unskilled labor. In particular, over the period1963-2012, the population of unskilled workers increased from 50.9 millions to 70.8that corresponds to 39.1% whereas the population of skilled workers increased over thisperiod from 11.1 to 70.4 million that corresponds to almost 533.1 %. Moreover, hoursworked by skilled agents also have a pronounced upward trend while such a trend is notpresent for unskilled labor. As a result, the labor input ratio of skilled versus unskilledlabor increases over time. The weekly wages of skilled agents grow more rapidly thanthose of unskilled and therefore, skill premium defined as a ratio of wages of skilled tounskilled workers have an upward time trend.

We provide the available in KORV data variables for the sake of comparison. Annualhours worked by skilled and unskilled agents and number of workers in two skilled groupsare not reported in the KORV.

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010200

400

600

800

1000

1200

1400

1600Labor input adjusted by population growth

skilled

unskilled

skilled−KORV

unskilled−KORV

Figure 2.2: Labor Input

We observe that our data on wages are very similar, but the data on labor input aresignificantly different. The difference in labor input leads to different pattern of skillpremium. In our case, an increase in the skill premium is more pronounced and moresystematic over time than in KORV.

The main reason our labor input data differ from KORV seems to be the fact thatthe KORV variables ”total labor input” are normalized to a fixed total labor while our

2.2. DATA 75

labor input data contain trends attributable to the population growth. If we redefineour labor input variables so that to exclude the population growth, we obtain resultsthat are very similar to KORV for both labor input and skill premium. This point isillustrated in Figure 2. In this paper, we will work with original unnormalized datathat are more suitable for our analysis than normalized one. Hence, our quantitativeimplications will be to some extend different from those in KORV in particular due tothis fact.

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100

500

1000

1500

2000Capital structures

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100

2000

4000

6000

8000

10000Capital equipment

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100.6

0.65

0.7

0.75

0.8Labor share of income

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100

500

1000

1500

2000Output

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100

0.2

0.4

0.6

0.8

1Relative price of capital equipment

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100.4

0.6

0.8

1

1.2

1.4Quality adjusted price of equipment

Data

KORV

Data

KORV

Data

KORV

Data

KORV

Data

KORV

Data

CV

Figure 2.3: Macroeconomic data

Macroeconomic data In Figure 3, we report selected aggregate macroeconomicindicators for the US economy.

To construct this variable we used the data from Federal reserve Bank of St.Louisand Bureau of Economics Analysis. Capital structures and capital equipment are con-structed using capital accumulation equation for structures and equipment, respectively.We use the data on real private fixed investment of two types of capital and their pricesto recover the annual series for capital. As a measure of output we use real GDP.

The prices for equipment and consumption are the quality adjusted Tornqvistindexes constructed in line with Krusell et al. (2000) and Cummins and Violante(2002)2 using disaggregated data on different types of capital input and consumptionexpenditures.

2We refer to this paper as CV hereafter


Again, we observe unbalanced growth patterns on Figure 3. Capital structuresincrease from 363.2 to 1780.7 billions of dollars over the sample period which correspondto 390 percent while equipment increase from 265 to 8833.6 billions of dollars whichcorrespond to 3233 %. In particular, the growth rate of equipment increases startingfrom 1995 that reflects the introduction and extension of modern technologies such asinternet, computers, etc. Labor share of income did not systematically change over time.However, the relative price of equipment and the quality adjusted price of equipmentdecrease over time.

Our data on output and capital equipment are similar to KORV. Our data onstructure grows at a higher rate due to correction by a quality adjusted price index.The mean labor share of income in our sample is equal to 0.65 that differs from theone reported in KORV due to a fact the authors of the paper use a different measureof labor share. On the graph the labor share reported in KORV is normalized for thesake of comparison.

Finally, our quality adjusted price of equipment is compared not to KORV but toCV because in Cummins and Violante (2002) the relative price of equipment is reportedfor a longer time, 1947 - 2000, while in Krusell et al. (2000) the data are provided onlyup to 1992.

2.3 A macroeconomic model of capital-skill comple-

mentarity

Time is discrete and the horizon is infinite, t = 1, 2, ...,∞. The economy consists of aproduction side and a consumer side and is governed by a social planner. A decentralizedeconomy that corresponds to the planner’s problem is reported in Appendix C togetherwith the definition of a competitive equilibrium.

Producer side There are two types of agents, skilled and unskilled; their variablesare denoted by superscripts ”s” and ”u”, respectively. There are two types of capitalstocks, capital structures and capital equipment, denoted by superscripts ”b” and ”e”,respectively. The production function is of the Constant Elasticity of Substitution(CES) type:

Yt = AtG (Kbt, Ket, Lst, Lut)

= AtKαbt

[µLσut + (1− µ) (λKρ

et + (1− λ)Lρst)σρ

] 1−ασ, (2.1)

where Yt is output; At is an exogenously given level of technology (common to bothsectors); Kbt and Ket are the inputs of capital structures and capital equipment, re-spectively; α ∈ (0, 1), µ ∈ (0, 1), λ ∈ (0, 1), ρ < 1 and σ < 1 are the parametersgoverning the elasticities of substitution among structures, equipment, skilled labor

2.3. A MACROECONOMICMODEL OF CAPITAL-SKILL COMPLEMENTARITY77

and unskilled labor; and Lst and Lut are efficiency labor inputs of skilled and unskilledagents, respectively, that are related to the corresponding physical inputs by

Lst = N st h

stψ

st and Lut = Nu

t hut ψ

ut , (2.2)

where N st and Nu

t are the number of skilled and unskilled workers; hst and hut are hoursworked by each skilled and unskilled agent; and ψst and ψst represents labor-augmentingtechnological progress of skilled and unskilled labor, respectively. The representativefirm maximizes period-by-period profits by hiring capital and labor

maxKbt,Ket,Ns

t hst ,N

ut h

ut

[Yt − rbtKbt − retKet − wstN st h

st − wutNu

t hut ] , (2.3)

taking the market prices rbt, ret, wst , w

ut as given which are interest rates paid on capital

invested in structures and equipment, and wages paid to skilled and unskilled labor,respectively.

Consumer side The economy has two sectors: one sector produces consumptiongoods and capital structures and the other sector produces capital equipment. Bothsectors use the same technology, however, there is a technology factor specific to thecapital-equipment sector. We aggregate the production of two sectors by introducingan exogenous relative price between consumption (structures) and equipment, qt.

The planner solves the following intertemporal utility maximization problem

maxCst ,Cut ,hst ,hut ,

Kb,t+1,Ke,t+1∞t=0

E0

∞∑t=0

βt [θsN st U

s (Cst , 1− hst) + θuNu

t Uu (Cu

t , 1− hut )] , (2.4)

subject to the economy’s resource constraint

N st C

st +Nu

t Cut +Kb,t+1 +

Ke,t+1

qt

= AtG (Kbt, Ket, Lst, Lut) + (1− δb)Kbt + (1− δe)Ket

qt, (2.5)

where β ∈ (0, 1) is a discount factor; Et is the operator of expectation conditional oninformation set in period t; initial endowments of capital structures and equipment,Kb0 and Ke0 are given; the production function G (Kbt, Ket, Lst, Lut) is given by (2.1);N it is an exogenously given number of agents of group i ∈ s, u; the endogeneous

variables Cit , h

it, K

ibt and Ki

et are, respectively, consumption, labor, and the capitalstock of structures and equipment of an agent of group i ∈ s, u; the time endowmentis normalized to one, so the term 1−hit represents leisure; and δb ∈ (0, 1) and δe ∈ (0, 1)are the depreciation rates of capital structures and capital equipment, respectively.The period utility function U i is continuously differentiable, strictly increasing in botharguments and concave; and θs and θu are the welfare weights of skilled and unskilledagents, respectively, that are normalized to one by θs + θu = 1.


Exogenous technological progress and uncertainty To make our model consis-tent with growth patterns in the data, we introduce several sources of exogenous growthsimultaneously. First of all, we assume that skilled and unskilled population can growat deterministic differing rates, i.e.,

N st = N s

0 (γs)t and Nut = Nu

0 (γu)t , (2.6)

where γs and γu are the growth rates of the skilled and unskilled labor, respectively.Furthermore, we assume three different kinds of technological progress: two la-

bor augmenting progresses that increases efficiency of both skilled and unskilled laborgrowing at possibly different rates, ψit, i ∈ s, u; Hicks-neutral technological progressthat increases the level of technology At;, and finally, equipment-specific technologicalprogress that improves the technology of the equipment sector relative to that of theconsumption and structure sector that decreases the relative price of equipment 1

qt. All

four kinds of progress have a similar structure: they are composed of a deterministicgrowth component and a stochastic shock component, namely,

ψit = ψi0(Γi)tzit, i ∈ s, u ; (2.7)

At = A0

(ΓA)tzAt , zAt = ρAzAt−1 + σAεt; (2.8)

1

qt=

zqtq0 (Γq)t

, zqt = ρqzqt−1 + σqεt; (2.9)

where Γi, ΓA and Γq are growth rates of deterministic components, and zAt and zqtare stochastic components following AR(1) processes, i ∈ s, u, where ρA, and ρq

are autocorrelation coefficients; σA, and σq are standard deviations and εt ∼ N (0, 1);zit ∼ N (0, σ2

z) is a normally distributed shock.

Optimality conditions Optimality conditions of the problem (3.51), (2.5) with re-spect to Cs

t , Cut , hst , h

ut , kb,t+1 and ke,t+1 can be written as

θiU ic

(Cit , 1− hit

)= ηt, i ∈ s, u ; (2.10)

θsU sh (Cs

t , 1− hst) = ηtAtG3 (Kbt, Ket, Lst, Lut)ψst , (2.11)

θuUuh (Cu

t , 1− hut ) = ηtAtG4 (Kbt, Ket, Lst, Lut)ψut , (2.12)

ηt = βEt ηt+1 [1− δb + At+1G1 (Kb,t+1, Ke,t+1, Ls,t+1, Lu,t+1)] , (2.13)

ηtqt

= βEt

ηt+1

[1− δeqt+1

+ At+1G2 (Kb,t+1, Ke,t+1, Ls,t+1, Lu,t+1)

], (2.14)

where Gi is a first-order partial derivative of the function G with respect to the i-thargument, i = 1, ..., 4 given by

G1(·) = αAtKα−1bt

[µLσut + (1− µ) (λKρ


] 1−ασ, (2.15)

2.4. QUANTITATIVE ANALYSIS OF THE PRODUCTION FUNCTION 79

G2(·) = AtKαbt (1− α) (1− µ)λ (λKρ

et + (1− λ)Lρst)σρ−1Kρ−1

et ×[µLσut + (1− µ) (λKρ


] 1−ασ−1, (2.16)

G3(·) = AtKαbt (1− α) (1− µ) (1− λ) (λKρ

et + (1− λ)Lρst)σρ−1 Lρ−1st ×[

µLσut + (1− µ) (λKρet + (1− λ)Lρst)

σρ

] 1−ασ−1, (2.17)

G4(·) = AtKαbt (1− α)µLσ−1ut × [

µLσut + (1− µ) (λKρet + (1− λ)Lρst)

σρ

] 1−ασ−1. (2.18)

Optimality conditions (3.19)− (2.14) together with the resource constraint (2.5) char-acterize the equilibrium.

2.4 Quantitative analysis of the production func-

tion

Due to the difference in the definition of labor input variables we cannot rely on theparameters reported in KORV, however we can use their methodology to estimate theparameters for our dataset. In this section we briefly describe the estimation procedureand results. The details of the pseudo maximum likelihood estimation procedure usedin the analisis can be found in Appendix B.

2.4.1 Estimation of the parameters of the production func-tions

We follow the estimation strategy of KORV who base their estimation strategy onLaroque and Salanie (1993). We use the following structural equations to estimate themodel parameters:

wsthstNst + wuthutNut

Yt= lsht(ψt, Xt;φ)

wsthstNst

wuthutNut

= wbrt(ψt, Xt;φ)

(1− δs) +G1(ψt+1, Xt+1;φ) = Et

(qtqt+1

)(1− δe) + qtG2(ψt+1, Xt+1;φ)

where ψt = ψst , ψut is a vector of unobserved latent variables, Xt = Kst, Ket, Lst, Lutis a vector of endogenous variables and φ = σ, ρ, α, µ, λ;ψs0, ψ

u0 , γψs , γψu , γA,Ωω, σε, δe, δs

is a vector of parameters that will be discussed in details later.


First and second equations are biased on the wages implied by the firm’s first orderconditions for hiring skilled and unskilled labor. First equation defines total labor shareof income as a function of the parameters of the production function. Second equationsspecifies the wage bill ratio as a function of the parameters. Third equation is a proxyfor the unobserved rental rates of capital equipment and structures. It is obtained fromthe Euler equation (the derivative of Lagrangian with respect to capital equipment).

In the data we observe left-hand sides of the first two equations, in the third equa-tion we know the parameters δs, δe and relative price of equipment 1/qt. We take theestimates of the depreciation rates of capital structures and equipment from KORVand they are equal to δe = 0.125 and δs = 0.05, respectively. Our analysis assumesthat changes in the unobserved latent variables can account for fluctuations in the skillpremium. We specify the stochastic process for the unobserved latent variables ψst andψut as a trend stationary process:

log(ψt) = logψ0 + t log γψ + ωt, ωt ∼ N(0,Ωω)

The simplifying assumption that we make is to substitute the first term of the right

hand side of the third equation withqtqt+1

(1 − δe) + εt, where εt is the i.i.d. forecast

error, which is assumed to be normally distributed: ε ∼ N(0, σ2ε )

It is challenging to estimate the specified above model because of various reasons.First of all, CES production function introduces highly nonlinear patterns in the equa-tions to be estimated. Second difficulty that we encounter is the relatively high numberof parameters to be estimated given that we have few data points. At last, we have twolatent variables which are unobserved to a researcher and specified as trend stationary.

Therefore we want to estimate a non-linear latent space model of the following form:

ME: Zt = f(ψt, Xt;φ) + υt

SE: log(ψt) = logψ0 + t log γψ + ωt

where the function f(·) contains the three equations described above, which were ob-tained from our model: two labor share conditions, equality of the rates of returnconditions and the production function equation.

The dimensionality of the vector φ is 16. To simplify further our analisis wemake additional assumption. We set parameters δb = 0.05 and δe = 0.125 follow-ing KORV(2000). In addition we estimate a time series ARMA model for the relativeprice of equipment to get an estimate for the standardt deviation of ε: σε = 0.028.We have four scaling parameters µ, λ, ψs0, ψ

u0 and for identification we need to fix one

of them. We choose to fix ψs0 and we also assume that the two shock ωst and ωut areuncorrected and are distributed normally with zero mean and variance σ2

ω.We estimate our model by simulated pseudo-maximum likelihood by means of a two-

step procedure. We treat capital structures and capital equipment as predeterminedand project skilled and unskilled labor input onto exogenous variables. On the secondstep we estimate the model using fitted values of the labor inputs. The number ofsimulations is set equal to 500.

2.4. QUANTITATIVE ANALYSIS OF THE PRODUCTION FUNCTION 81

Table 2.1: Estimated parameter values

Parameters Data KORV(2000)

Efficiency of labor Latent - Latent -

ψs0 3.000 6.000 15.000ψu0 1.4001 0.0001 11.2822γψs 1.0282 1.000 1.0207 1.000γψu 1.0082 1.000 1.0103 1.000

α 0.1350 0.2981 0.0858 0.117λ 0.9176 0.6310 0.7936µ 0.7189 0.9970 0.6371σ 0.5496 0.5235 0.4126 0.401ρ -0.9773 -0.7497 -0.7336 -0.495ηw 0.090 0.2 0.100 0.2074

Remark: The number of simulations in the estimated procedureis S = 500. ¨Latent¨ and ”-” stands for the cases with a deter-ministic growth component in the unobserved efficiency variableand without growth component, respectivly. Empty cell meansthat the data was not reported.

2.4.2 Implications for the skill premium

The parameter estimates dependents on the variables used in the estimation and moreprecisely on their units of measurement. For the estimation purposes we normalizepopulation variable to be in millions of persons and the hours worked by skilled andunskilled workers are normalized to be 0.3685 and 0.2459, respectively.

We focus our attention on two version of the model specified in the section above.The first version is characterized by the growth rate in the latent labor efficiency variableγψ, and the second one assumes that unobserved labor efficiency has no growth. KORVconclude that in order to account for the variability of the skill premium in the data itis enough to have the production function that allows for capital skill complementarity.Our analisis suggest that the variability of the skill premium observed in the data can beaccounted both by the version of the model with and without growth in latent efficiencyunits of the labor.

We report the estimated coefficients for the version of the model with and withoutgrowth rate in efficiency units of labor for KORV and updated dataset in Table 1. Overall experiments, the parameter σ varies from 0.4 to roughly 0.55 which implies theelasticity of substitution between equipment and unskilled labor ranging from 1.66 to2.22. The elasticity of substitution between equipment and skilled labor differs fromthe estimates reported in KORV and varies from 0.67 to 0.5 which correspond to thevalues of parameter ρ of −0.495 and −0.9773.

The difference between estimates reported in KORV and those obtained for the up-dated dataset may result in decreasing skill premium and this feature can be translated


1965 1970 1975 1980 1985 19900.9

0.95

1

1.05

1.1

1.15

KORV(2000)

with growth in latent variables

without growth in latent variables

1965 1970 1975 1980 1985 1990 1995 2000 2005 20101.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Data

with growth in latent variables

without growth in latent variables

Figure 2.4: Estimation Results

further to our general equilibrium model. Therefore, in further analisis we use theparameters estimated for the updated dataset.

We recover the estimated shocks for the model using the labor share and wage billequations from the model. Given the estimates of the parameters and the series of thecapital stocks and labor inputs in each period of time t these two equations form asystem of nonlinear equations in two unknowns which are the shock to unobserved effi-ciencies of skilled and unskilled labor, respectively. This system has a unique solution.

We then use these estimated shocks for simulations of the skill premium for KORVand updated dataset. Result suggest that simulated skilled premium is somewhat morevolatile than the one observed in the data. Nevertheless the growth rates of the skillpremium in both cases seem to be similar. These results are illustrated on the Figure4.

On the panel b of Figure 4 we observe that the simulated skill premium for thecase with growth rate in unobserved labor efficiencies has the same pattern as the skillpremium simulated for the case when the there is no growth in ψt. Both cases havedifferent parameter values however in a simulation estimated shocks absorb possibledifference implied by alike parameters and therefore the simulated skill premium forboth cases look very similar.

2.5. GENERAL EQUILIBRIUM 83

2.5 General equilibrium

In this section we describe the calibration procedure and some aspects of the solutionprocedure. The model that we need to solve a non stationary dynamic stochastic generalequilibrium model that can be solved with a procedure describe in detail in Maliar etal. (2015).

2.5.1 Calibration and solution procedure

In calibration exercise we take some parameters from available studies and the estima-tion results from previous section and others are calibrated from the model equations.

Production function parameters α, ρ, σ, λ, µ are taken from our estimation results forthe case with latent variables and are equal to 0.135, -0.977, 0.549, 0.9176, 0.7189, re-spectively. Parameters for latent variables also come from there ψs0 = 3, ψu0 = 1.4, γψs =1.0282, γψu = 1.0082.

We take the estimates of the depreciation rates for capital structures and capitalequipment δe = 0.125, δs = 0.05 from the KORV paper.

As we have data on the relative price of equipment, qt, we can estimate the growthrate of the relative price of equipment by estimating the parameters of the equation:

log qt = β0 + β1t+ ηt, (2.19)

where q0 = eβ0 and Γq = eβ1 . Parameters ρq and σq are then estimated using fittedresiduals from equation (2.19). The same strategy was used to recover parametersA0,Γ

A, ρA, σA. We recover Solow residual from the production function equation:

At = Yt/G(Kbt, Ket, Lst, Lut). (2.20)

We assume that social planner put equal weight on two groups of agents and there-fore the values of θs and θu are equal to 0.5 both. Growth rates of the population ofskilled and unskilled workers, γs and γu, takes valued of 1.0393 and 1.0121.

Utility function U i, i ∈ s, u is of additively separable type:

U(Cit , 1− hit) = log(Ci

t) +(1− hst)1−ν

i − 1

1− νi, i ∈ s, u (2.21)

Parameters νs and νu are both set equal to 1, respectively and the discount factorβ is calibrated to 0.99.

The model we specified is a non stationary dynamic stochastic equilibrium model.We use a novel numerical solution procedure described in detail [?]. First of all, werearrange optimality conditions (3.19) − (2.14) together with the resource constraint


(2.5) to get:

kb,t+1 = βCγt χtEt

[(C−γt+1χt+1 (1− δs + At+1G1) kb,t+1

](2.22)

ke,t+1 = βqtCγt χtEt

[(C−γt+1χt+1

(1− δeqt+1

+ At+1G2

)ke,t+1

](2.23)

hst = 1−((θs)−1c−γt χtAtG3φ

st

)−1/νs(2.24)

hut = 1−((θu)−1c−γt χtAtG4φ

ut

)−1/νu(2.25)

Ct + Kb,t+1 +Ke,t+1

qt= Yt + (1− δb)Kbt + (1− δe)

Ket

qt, (2.26)

where χt = (N st (θs)1/γ + Nu

t (θu)1/γ)−γ and is obtained from conditions (3.19) forskilled and unskilled labor respectively; Ct is a total consumption of skilled and unskilledagents in period t.

In order to solve the model we approximate right hand side of the equations forkb,t+1, ks,t+1, h

st , h

ut with a second degree ordinary polynomial function. Expectation Et

is approximated by means of Gauss-Hermite quadrature nodes and weights. For eachof four policy functions we have 27 parameters to estimate. We set the simulationperiod equal to 150 to make the effect of terminal conditions negligible at the sametime maintaining computational costs on a reasonable level.

2.5.2 Numerical results

After obtaining the solution which is a set of coefficients of policy functions for kb,t+1, ks,t+1,hst , h

ut we simulate the solution using the estimated from data shocks for latent labor effi-

ciencies, Solow residual and relative price of equipment. Figure (??) plots the simulatedseries for capital equipment and structures, total labor input of skilled and unskilledagents, consumption and skill premium.

The first thing to notice on the skill premium is not increasing and rather is fluctu-ating over its mean. This result is robust to different parameter values of νs, νu, β andγ. The other shortcoming of our calibration is that the we do not match the levels of thedata. The model is calibrated in such a way that production side and consumption sideof the model are calibrated in a separate way. This may be a reason why we have sucha poor performance in this respect. It si not a trivial task to calibrate a non stationarymacroeconomic model with growth since it involves a complicated interaction betweenstocks of variables and their growth rates.

The growth rates of capital equipment is matched quite well, while the stock ofcapital structures exhibits a pronounced deviation from the trend observed in the data.The total annual labor input of skilled and unskilled workers is matched well and theconsumption is increasing.

From the equation of the skill premium its dynamics depends on the capital stockof equipment, total labor input of skilled and unskilled workers and unobserved latentvariables. The growth rates of these series generated by the model are slightly lower

2.6. CONCLUSION 85

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100

5

10

15

20

25

30

Eq

uip

me

nt

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100

1

2

3

4

Str

uctu

res

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100.5

1

1.5

2

2.5

3

Co

nsu

mp

tio

n

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100.9

1

1.1

1.2

1.3

Skill p

rem

ium

Simulation

Data

Simulation

Data

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100.5

1

1.5

2

2.5

3

3.5

Skille

d la

bo

r in

pu

t

Simulation

Data

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

0.7

0.8

0.9

1

1.1

1.2

1.3

Un

skille

d la

bo

r in

pu

t

Simulation

Data

Simulation

Data

Simulation

Data

Figure 2.5: Simulation results

than the ones observed in the data. Moreover, since the stocks are not matched thelevel of the skill premium is neither similar to the one observed in the data.

2.6 Conclusion

We document the evolution of capital and labor measures in the US economy over the1963-2012 period using the data from the Current population survey and the Nationalincome and product accounts. Overall our data is very similar to the data reported instudies for skill premium. There are some differences in data on wages and labor input.The difference in labor input leads to different pattern of skill premium. In our case, anincrease in the skill premium is more pronounced and more systematic over time thanin KORV.

We construct a dynamic general equilibrium model of capital-skill complementarityin which both the quantities of skilled and unskilled labor input and the stocks of equip-ment and structures are determined endogenously. We find that the constructed modelcan account for business cycles properties and growth patterns of the key macroeco-nomic aggregates, such as capital equipment and structures, skilled and unskilled labor,output, relative price of investment. However, the dynamics of the skill premium is notmatched with the data. In particular, in the model the skill premium is stationarywhich is a consequence of ratio of marginal disutilitites of labor of skilled and unskilledagents being roughly constant in the model.


The model needs to be modified to have the ratio of marginal disutilitites of laborincreasing. Numerical method introduced in Maliar, Maliar, Taylor and Tsener (2014)opens an avenue for solving such models. Hopefully we can incorporate new featuresand reconcile the model predictions. Policy analysis will be extremely interesting inparticular how taxes-subsidies will affect the result.

Bibliography

[1] Acemoglu, D. and Autor, D., (2011). Skills, tasks and technologies: implicationsfor employment and earnings. Volume 4 of Handbook of Labor Economics, chapter12, 1043–1171. Elsevier.

[2] Acemoglu, D. and Autor, D., (2012). What does human capital do? A review ofGoldin and Katz’s “The race between education and technology”. Working paper17820, NBER.

[3] Adda, J. and Cooper, R., (2003). Dynamic economics: quantitative methods andapplications. Volume 1 of MIT Press Books. The MIT Press.

[4] Berndt, E., Griliches, Z., Rappaport, N., (1995). Econometric estimates of priceindixes for personal computers in the 1990’s. Journal of Econometrics, 68(1),371–392.

[5] Bound, J. and Johnson, G., (1992). Changes in the structure of wages in the1980’s: an evaluation of alternative explanations. American Economic Review,82(3), 371–392.

[6] Brown, K. and Greenstein, S., (1995). How much better is bigger, faster andcheaper? Buyer benefits from innovation in mainframe computers in the 1980’s.Technical report, NBER.

[7] Cole, R., Chen, Y., Barquien-Stolleman, J., Dulberger, N., Hodge, J.,(1986).Quality-adjusted price indexes for computer processors and selected peripheralequipment. Survey of Current Business, 66(1), 41–50.

[8] Cummins, J. and Violante, G., (2002). Investment-specific technological changein the US (1947–2000): measurement and macroeconomic consequences. Reviewof Economic Dynamics, Elsevier for the Society for Economic Dynamics, 5(2),243–284.

[9] Goldin, C. and Katz, L., (2008). The race between education and technology.Belknap Press for Harvard University Press.

[10] Gordon, R., (1990). The measurement of durable goods prices. Number 1 inNBER Books. NBER, Inc.

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[12] Katz, L. and Murphy, K., (1992). Changes in relative wages, 1963–1987: supplyand demand factors. The Quarterly Journal of Economics, 107(1), 35–78.

[13] King, R., Plosser, C. and S. Rebelo, (1988). Production, growth and businesscycles, Journal of Monetary Economics 21, 195-232.

[14] Laroque, G. and Salanie, B., (1993). Simulation-based estimation of models withlagged latent variables. Journal of Applied Econometrics. John Wiley & Sons,Ltd., 8, 119–133.

[15] Maliar, L. and Maliar, S., (2001). Heterogeneity in capital and skills in a neo-classical stochastic growth model. Journal of Economic Dynamics and Control,25(9), 1367–1397.

[16] Maliar, L. and Maliar, S., (2003). The representative consumer in the neoclassicalgrowth model with idiosyncratic shocks. Review of Economic Dynamics, 6(2),368–380.

[17] Maliar, L. and Maliar, S., (2011). Capital-skill complementarity and balancedgrowth. Economics, 78(310), 240–259.

[18] Maliar, L., Maliar, S., Taylor, J., Tsener, I., (2015). A framework for nonstation-ary and unbalanced growth dynamic stochastic models. Manuscript.

[19] Taylor, J. and Uhlig, H., (1990). Solving nonlinear stochastic growth models:a comparison of alternative solution methods. Journal of Business & EconomicStatistics, 8(1), 1–17.

2.7 Appendix A: Data construction

In Appendix A.1, we first describe the construction of labor market variables usingmicroeconomic data and in Appendix A.2, we describe the construction of the remainingvariables using macroeconomic data.

Appendix A1. Microeconomic data

For each year t and each individual i in the sample we recorded age, race, sex, yearsof education, employment status, weeks worked last year, weekly hours worked, laborincome and the CPS sampling weights:

Variable ai,t contains actual age of the individual; ri divides individuals on white,black and other according to nationality reported; si splits individuals on males orfemales; ei contains information on how many years of education she has: 0 years ofeducation, elementary, high school, college 1-4, college 5, college 6+ classes attended;

2.7. APPENDIX A: DATA CONSTRUCTION 89

Variable 1964-2013 (Position) LabelAge age in years (93) ai,tRace race (179) riSex sex (183) siYears of Education highest grade (124;99) eiEmployment Status employment status (107) lfsi,tWeeks worked weeks worked last year (205) wki,t−1Weekly Hours worked hours worked last week (126) hi,tLabor Income person’s wage and salary (187) yi,t−1Weight personA´s suppl. weight µi,t

lfsi,t is used to select employed workers; wki,t−1 helps to get information on the numberof weeks worked last year, i.e. 0, 1, 2, etc.; hi,t defines number of hours worked lastweek; yi,t−1 are person’s wage and salary; µi,t person’s supplementary weight in thesample.

Individuals who had a full-time job are selected in our sample if their age is in theinterval [16, 70] and if they did not report to be self-employed and were not on a servicein armed forces. We multiply top coded values of personal wage and salary by 1.5. Wecalculate the experience of each individual if he has any education by subtracting thenumber of years spend on education and 7 from the actual age of the individual. Wefurther assume that minor individuals do not start working until age 16 regardless ofeducation status.

All individuals from this subsample of the data are further divided into groups bysex (males and females), education (no school degree, high school, some college, collegeand greater than college) and experience: 0 − 9, 10 − 19, 20 − 29, 30 − 39, 40 − 48. Inthe resulting subsample we have some individuals who have negative supplementarypersonal weight and we drop these few observations. We compute mean predictedweekly wages for a broader groups of skilled and unskilled workers as a mean weightedaverages of the relevant subgroup mean using average share of total hours worked foreach group.

For some years considered in this sample wki,t−1 is reported in intervals of weeks.For these years we assume that all individuals have the same number of weeks workedin each bean and impute for each of them the average value of weeks worked for thatbean over the years 1976− 1978. Wage in labor input data in the survey refers to oneyear earlier and therefore our sample is for the period 1963 - 2012.

Appendix A.2: Macroeconomic data

In this section we explain how to construct annual quality adjusted series for capitalequipment and structures. We have assumed the same price for consumption andstructures, pc, which is obtained as a weighted arithmetic average of the implicit pricedeflators for nondurable consumption goods and services. We used as weights the

90 BIBLIOGRAPHY

1965 1970 1975 1980 1985 1990 1995 2000 2005 20100.1

0.15

0.2

0.25

0.3

0.35

0.4

OIP

INDEQ

TRANSP

OTHER

Figure 2.6: Relative share of types equipment

shares of consumption of nondurable goods and services in the total consumption ofnondurable gods and services.

We break down equipment category into several subcategories: information process-ing equipment: computers and peripheral equipment (COMP), communication equip-ment (COMM), medical equipment and instruments, non medical instruments, photo-copy and related equipment (INST), office and accounting equipment (OFF); industrialequipment (INDEQ): fabricated metal products, engines and turbines, metalworkingmachinery, special industry machinery, n.e.c., general industrial, including materialshandling, equipment, electrical transmission, distribution, and industrial apparatus;transportation (TRANSP): trucks, buses, and truck trailers, autos, aircraft, ships andboats, railroad equipment; other equipment (OTHER): furniture and fixtures, agri-cultural machinery, construction machinery, mining and oilfield machinery, electricalequipment, n.e.c., other nonresidential equipment3.

Following KORV we aggregate these categories into four main categories: OIP,INDEQ, TRANSP, OTHER. Figure (??) shows relative nominal shares of the fourmain categories, according to NIPA. The share of office and information processingequipment increased for two times. There are downward trends in shares of industrialequipment and transportation equipment.

We construct a series of quality adjusted indexes for different groups of equipmentfor a period 1947 - 2012. For years 1947 - 1983 we rely on [?] data. For other years we

3source: BEA classification

2.7. APPENDIX A: DATA CONSTRUCTION 91

need to predict the prices of each category of equipment.Dlagg: We use the Conference board lagging economic index in the estimation. We

had the data available till 2010, after 2010 we collected the values of the index from thepress realizes reports published on the web page of the Conference board. Thereforevalues for years 2011, 2012, 2013 were recovered.

INDEQ, TRANSP, OTHER: For this three variables we can construct a quality-adjusted prices for a period after 1983 using GordonA´s data. Gordon(1990)’s datacontains quality adjusted indexes for disaggregated groups of industrial equipment.We aggregate these subgroups with a Tornquist procedure (see KORV and CV fordetails). For aggregation we need to have shares and prices. We take shares for differentsubcategories of private fixed investment from Federal Reserve of St.Louis database.For INDEQ, TRANSP and OTHER we estimate a VAR modelin which we use theconference board lagging economic index. Our final sample spans from 1959 (and notfrom 1947 as in KORV) since the economic indicator data starts from 1959.

OIP: We split the OIP category into COMP and all other than computers andperipheries, i.e. OFF, COMM and INST. For communication equipment (COMM) andinstruments (INST), we use the same forecasting technique as before, but we fit twoseparate equations this time following KORV who document that the data before 1984do not show any strong comovements. For INST we estimate an AR(1) model. Theforecasted values for INST are positive up to 2000 but then turn to be negative forthe last decade which contradicts to a definition of an index number. Therefore to goaround this fact we assume that for the last decade the change in quality adjusted pricewas equal to the change in the official BEA price. Assuming this we most probablyunderestimate the decline in the quality adjusted price index.

For the OFF category we take the official NIPA price index.For computers and peripherals (COMP) there is a large literature available. The

COMP category consists of personal computers and peripherals. Computers in itsturn are classified into personal computers (PCs) and other computers (mainframes,workstations, and midrange computers). For the years up to 1983 we rely on Gordon’sdata of quality adjusted price indexes (B1). For years 1984-1992 we make an attemptto reconstruct the quality adjusted price indexes relying on the literature available inthis field.

Keeping this classification of the categories of COMP in mind we first construct theshares and then the indexes of each subcategory.

Shares: Computers and peripherals held by consumers are not relevant for for ourmeasure of capital input; therefore we consider only the durable used in the businesssector. We compute the share of PCs used in business 4 and we assume that peripheralsare shared in the same proportion between business and private sector. We assume thatall other computers are used only in business. We compute the shares of computers andperipherals using Table 1343 and the share of personal computers and other computersusing table 1339/1273 of the Statistical Abstract of the U.S.

4Table 1340 of Statistical Abstract of the U.S.

92 BIBLIOGRAPHY

Indexes: Brown and Greenstein (1995) compute an adjusted prices for mainframesand they find that in the year 1985-1991 the prices declined by 27 percent on average,every year. We assume that this number holds for all other types of computers exceptfor PCs.

Berndt et al. (1995) compute a hedonic-adjusted price index for PCs. They reportquality adjusted price indexes for years 1989-1992, and the changes in price indexes foryears 1982-1988. We assume that the price change in 1989 is equal the average of pricechanges in 1988 and 1990. The average price change over the years 1982-1992 is equalto 28 percent per year.

Peripheral is a device that is connected to a host computer, but not part of it; itexpands the hosts capabilities but does not form part of it. According to this definitionwe are going to consider disk drives, printers and displays from Cole et al. (1986). Usingthe shares of personal computers and mainframes (and other computers, table 1339) weconstruct a weighted quality adjusted price of PC’s and other computers (using here aweighted average). Then we compute the ratio between the the change in price of thiscategory and peripherals, which is equal to 0.52. This relationship in price changes isassumed to be constant over the period 1994 - 1992 and we use this fact to recover thequality adjusted price for peripherals over this period.

Note that on this stage we use weighted average for aggregation of different subsubcategories since the data is scarce and we did not want to loose any observations inprice indexes as we would do with a Tornqvist index, since it always requires 1 moreobservation.

On the last step we use changes in quality adjusted price indexes to recover the pricefor year 1984 - 1992. We are not aware of any available quality adjusted price index foryears 1992 - 2012, therefore we use price changes of the official price index for computersand peripherals to predict the quality adjusted prices for this period. Predicted valuesare again negative and we assume that changes in the quality adjusted price is equal tothe change in the official price. The quality adjusted price tends rapidly to 0.

We aggregate all the indexes into 4 main categories using Tornqvist procedure; theweights are based on expenditure shares. When calculating shares we do not take intoaccount residential equipment, since it does not classified into categories in BEA data.Later these 4 categories are aggregates into price of equipment. Anyway this exclusionshould not make big difference since the share of residential equipment is quite low(approximately 0.1 %)

Aggregation: The relative price index is constructed by dividing the aggregateindex of equipment through pc. We construct the capital series using capital accumu-lation equation. We take the investment in equipment and structures series and deflatethem by a corresponding quality adjusted price. We then assume that the value ofstock of strutters in 1963 is equal to 363,2 and that of equipment is equal to 265 fol-lowing KORV who choose it so that to match the investment capital ratio; we also taketheir estimates for the depreciation rate of structures and equipment, δb = 0.05 andδe = 0.125, respectively. Using this information we recover the stocks of capital series

2.8. APPENDIX B: ESTIMATION 93

using the equation for the law of motion, for example for year 1964:

ke1964 = (1− de)ke1963 + ie1963, (2.27)

ks1964 = (1− ds)ks1963 + is1963. (2.28)

2.8 Appendix B: Estimation

First thing to note is that labor share constructed from data provided by KORV resultsto have an increasing trend while in the BEA data it is constant and approximatelyequal to 0.7. Wage bill ratio in the KORV data is 2.6 times higher in 1992 than in 1963which is in contrast to plots in the paper: only 2.4 (see Figure 9 in KORV).

Following the notation of KORV consider a nonlinear model:

(Model) : Zt = f(Xt, ψt;φ) + εt (2.29)

(State) : ψt = ψ0(γ)t exp(wt) (2.30)

where f(·) contains all three nonlinear observational equations; Zt is a (3×1) vector, Xt

is a set of inputs (capital structures and equipment, labor inputs of skilled and unskilledworkers), ψt is a (2×1) vector of unobserved variables; εt = [0, 0, ε3t]

′, wt = [w1t, w2t]′ are

vector of i.i.d. normally distributed random disturbances with mean zero and covariancematrix.

We allow for possible dependence between shocks and hours worked and we use a twostep SPMLE procedure developed by White(1994). We treat skilled and unskilled laborinput as endogenous and we project these variables onto a constant, current and laggedstock of capital equipment, current stock of capital structures, lagged relative priceof capital equipment, time trend and the lagged value of the Commerce Department’scomposite index of business cycle indicators. The fitted (instrumented) values of skilledand unskilled labor,Xt, are used in the next steps of the estimation.

In the next step of estimation we draw a (T × S) matrices of shocks for w1t, w2t, ε3tto construct latent variables and Zi

t :

ψit = ψ0(γ)t exp(wit) (2.31)

Zit = f(Xt, ψ

it, ε

it;φ) (2.32)

By simulating the model through these two equations we can obtain first and second(simulated moments) of Zt:

mS(Xt;φ) =1

S

S∑i=1

f(Xt, ψit, ε

it;φ)

VS(Xt;φ) =1

S − 1

S∑i=1

(Zit − f(Xt, ψ

it, ε

it;φ)

)(Zit − f(Xt, ψ

it, ε

it;φ)

)′

94 BIBLIOGRAPHY

These moments, mS(Xt;φ) and VS(Xt;φ), are constructed for each t. The simulatedpseudo maximum likelihood estimator, φ, is defined to be a minimizer of

lHT (φ) =1

2T

T∑t=1

(Zt −mS(Xt;φ)

)′VS(Xt;φ)−1

(Zt −mS(Xt;φ)

)+ log |VS(Xt;φ)| (2.33)

To find minimum of a function we first used fminsearch, but over the iterations thisfunction was going in the space of parameters that do not have an economic meaning(for example, negative values for latent variables), this in turn was causing problemswith computing inverses of the variance-covariance matrices. Therefore, we switchedto constrained optimization with fmincon.

Following KORV we define a variable qt = qt+1/qt and estimate an ARMA(1,1)model (in Gretl). The estimated standard deviation of disturbances is equal to 0.026642.

In earlier draft of the paper KORV report that they had difficulties to estimate allthe parameters together, the variance of the latent state ηw was always quickly increasedby the toolbox resulting in implausibly high variability of the skill premium. In ourresults the constraint on variance is binding as well. KORV fixed a grid of values overa realstic range for ηw and then for each of this points they estimated the rest of theparameter vector. The maximum of the pseudo likelihood function across these gridpoints delivered the final estimates.

In our experiments the parameters of the production function parameter estimatesdepended a lot on the values of ηw and ψs0.

The labor share and wage bill ratio equations used in the estimation take the fol-lowing form:

G3tLstG4tLut

=1− µµ

(1− λ)

(λ+ (1− λ)

(LstKet

)ρ)σ/ρ−1(LstKet

)ρ(LutKet

)−σ(2.34)

G3tLst +G4tLutYt

= (1− α) (2.35)

·[(1− µ)(1− λ)

(λ+ (1− λ)

(LstKet

)ρ)σ/ρ−1(LstKet

)ρ+ µ

(LutKet

)σ]

(1− µ)

(λ+ (1− λ)

(LstKet

)ρ)σ/ρ(LstKet

)ρ+ µ

(LutKet

)σ (2.36)

2.9 Appendix C: Decentralization

Let us denote by Bt a collection of all possible exogenous states in period t. Weassume that Bt follows a stationary first-order Markov process. Specifically, let < bethe Borel σ-algebra on =. Define a transition function for the distribution of skillsΠ : = × < → [0, 1] on the measurable space (=,<) such that: for each z ∈ =, Π (z, ·)is a probability measure on (=,<), and for each Z ∈ <, Π (·, Z) is a <-measurable

2.9. APPENDIX C: DECENTRALIZATION 95

function. We shall interpret the function Π (z, Z) as the probability that the nextperiod’s distribution of skills lies in the set Z given that the current distribution ofskills is z, i.e., Π (z, Z) = Pr Bt+1 ∈ Z | Bt = z. The initial state B0 ∈ = is given.We assume that there is a complete set of markets, i.e., that the agents can trade state-contingent Arrow securities. The agent’s i ∈ s, u portfolio of securities is denoted byM i

t (B)B∈< . The claim of type B ∈ < pays one unit of t + 1 consumption good inthe state B and nothing otherwise. The price of such a claim is pt (B).

Consider the utility maximization problem of a representative agent of type i ∈s, u:

maxCit ,h

it,k

ib,t+1,k

ie,t+1,M i

t+1(Z)Z∈<∞t=0

E0

∞∑t=0

βtN itU

i(Cit , 1− hit

), (2.37)

subject to

Cit + γikib,t+1 + γi

kie,t+1

qt+ γi

∫<pt (Z)M i

t+1 (Z) dZ

= withit + (1− δb + rbt) k

ibt + (1− δe)

kietqt

+ retkiet +M i

t (Bt) . (2.38)

where kib,t, kie,t are capital stock of structures and equipment of agent i holds, respec-

tively; rbt and ret are the interest rates paid on capital structures and equipment,respectively; wit are wages paid to an agent of type i and M i

t (Bt) are Arrow securities.The initial endowments of capital structures and equipment, kib,0 and kie,0, are Arrow se-curities M i

0(B0) are given. The period utility function U i is continuously differentiable,strictly increasing in both arguments and concave.

The first order conditions of this problem are:

φit = N itU

ic(C

it , 1− hit), (2.39)

witφit = N i

tUih(C

it , 1− hit), (2.40)

γiφit = βEt(φit+1(1− δb + rb,t+1)

), (2.41)

γiφitqt

= βEt

(φit+1

(1− δeqt+1

+ re,t+1

)), (2.42)

φitpt (B) γi = βφit+1 (B′) · Π Bt+1 = B′ | Bt = BB′,B∈< , (2.43)

where φit is the Lagrange multiplier associated with the resource constraint (2.38).The First Order Condition (FOC) of the problem (2.37) - (2.38) with respect to holdingsof Arrow securities is

φitpt (B) γi = βφit+1 (B′) · Π Bt+1 = B′ | Bt = BB′,B∈< . (2.44)

By taking the ratio of FOC (2.43) of a skilled agent s to that of an unskilled agent u,we obtain

φs0φu0

=φs1/γ

s

φu1/γu

= ... =φst/(γ

s)t

φut /(γu)t... ≡ φs

φu, (2.45)

96 BIBLIOGRAPHY

φs and φu are some constants. Given that φit = N itU

ic (Ci

t , 1− hit), we have that the ratioof marginal utilities of consumption of two heterogeneous consumers, adjusted to thecorresponding growth rates of population, is constant across time and states of nature

U sc (Cs

0 , 1− hs0)Uuc (Cu

0 , 1− hu0)=U sc (Cs

1 , 1− hs1)Uuc (Cu

1 , 1− hu1)= .... =

U sc (Cs

t , 1− hst)Uuc (Cu

t , 1− hut )=φs/N s

0

φu/Nu0

. (2.46)

This is a consequence of the assumption of complete markets. We rearrange the FOCswith respect to physical hours worked, capital structures and equipment of a represen-tative agent of type i, respectively, to get:

U ih

(Cit , 1− hit

)= U i

c

(Cit , 1− hit

)wit, (2.47)

U ic

(Cit , 1− nit

)= βEt

[U ic

(Cit+1, 1− nit+1

)· (1− δb + rb,t+1)

], (2.48)

U ic (Ci

t , 1− nit)qt

= βEt

[U ic

(Cit+1, 1− nit+1

)·(

1− δeqt+1

+ re,t+1

)]. (2.49)

Thus, (2.46)− (2.49) are the FOCs of the competitive equilibrium economy.Let us consider now the planner’s problem (3.51), (2.5). The FOC with respect to

consumption of the skilled and the unskilled agents, respectively, are

θsU sc (Cs

t , 1− hst) = ηt, (2.50)

θuUuc (Cu

t , 1− hut ) = ηt, (2.51)

where ηt is the Lagrange multiplier associated with the economy’s resource constraint

(2.5). Dividing (2.50) by (2.51) and setting the value of θs so thatφs/Ns

0

φu/Nu0

= θs

1−θs , we

obtain condition (2.46) of the competitive equilibrium economy. The FOC with respectto capital structures is

ηt = βEt [ηt+1 (1− δb + rb,t+1)] . (2.52)

Combining (2.50) and (2.51) with (2.52), we get condition (2.48) of the competitiveequilibrium economy. Similarly, the FOC with respect to equipment is

ηtqt

= βEt

[ηt+1

(1− δeqt+1

+ re,t+1

)]. (2.53)

After substituting conditions (2.50) and (2.51) into (2.53), we obtain condition (2.49) ofthe competitive equilibrium economy. From the firm’s problem (2.3), equilibrium wagesare given by wst = AtG3 (Kbt, Ket, Lst, Lut)ψ

st and wut = AtG4 (Kbt, Ket, Lst, Lut)ψ

ut . By

substituting these wages into a FOC with respect to physical hours worked of the plan-ner’s problem, we get (3.42). Finally, the resource constraint (2.5) is satisfied in com-petitive equilibrium which is trivial to prove by multiplying the individual constraintsby N i

t and summing them up5.

5Strictly speaking, we also need to show that the individual transversality conditions in the decen-tralized economy imply the aggregate transversality condition in the planner’s economy. This can beshown as in Maliar and Maliar (2001, 2003a).

2.9. APPENDIX C: DECENTRALIZATION 97

Competitive equilibrium: A competitive equilibrium in the economy (2.37) −(2.38) is a sequence of contingency plans for the agents’ allocationCit , h

it, k

ib,t+1, k

ie,t+1,M

it+1 (Z)

i∈s,uZ∈<,t∈T , for the firm’s allocation Kbt, Ket, Lst, Lutt∈T

and for the prices rbt, ret, wst , wut , pt (Z)Z∈<,t∈T such that given the prices:(i) the sequence of plans for the agents’ allocation solves the utility maximizationproblem (2.37), (2.38) for i ∈ s, u;(ii) the sequence of plans for the firm’s allocation solves the profit maximization problemof the firm (2.1)− (2.3);(iii) all markets clear and the economy’s resource constraint is satisfied.

Moreover, the equilibrium plans are to be such that Cit > 0 and 0 < hit < 1 for

i ∈ s, u, Kbt, Ket > 0 and pt (Z) > 0 for all Z ∈ <.

98 BIBLIOGRAPHY

Chapter 3

Geometric Programming:Approaches for Solving EconomicModels

3.1 Introduction

Geometric programming is a field of mathematical optimization that focuses on non-linear optimization problems subject to linear or nonlinear constraints. Recent devel-opments of new solution methods for geometric programming problems makes this fieldof optimization attractive for economists. New methods can solve large-scale geometricprograms efficiently and reliably in a reasonable amount of time.

The solution to dynamic economic models often requires finding solutions to non-linear optimization problems or systems of nonlinear equations. Existing methods usu-ally rely on available numerical solvers – which are in many cases variants of the NewtonRaphson procedure – or on iterative methods such as fixed point. The former grouprequires a good initial guess while the latter performs poorly in applications as thedimensionality of the state space increases. In the contrary, methods developed forgeometric programming are based on primal-dual interior-point methods that do notrequire submission of initial guess and are reliable in problems with many variables.In this paper we study extents to which geometric programming can be applied toproblems encountered in economics.

The main object of geometric programming is a geometric program that satisfiesthe following conditions: (1) objective function and inequality constraints should beformed by means of posynomial functions, (2) equality constraints should be formed

99

100 CHAPTER 3. GEOMETRIC PROGRAMMING: APPROACHES

by monomial functions. Geometric programming methods are extensively used in en-gineering, information theory, management science, computational finance and otherfields; see Boyd (2007) for a review.

In this paper we solve deterministic and stochastic neoclassical growth models and aset of intratermporal optimization problems that usually arise in economics by means ofgeometric programming. Although in the project the methods are applied to economicmodels that are stationary by nature, they could as well be used in solving nonstationaryeconomic models.

We find that geometric programming methods accurately solve deterministic growthmodel and a set of intratemporal optimization problems. However, the solution deliv-ered by geometric programming for a stochastic growth model is inaccurate and thedegree of inaccuracy increases with the degree of nonlinearity of the model. The reasonfor this inaccuracy is an assumption of certainty equivalence that we must impose tobe able to solve the model by GP methods. In particular, the main source of errors arethe errors in Euler equations which are approximated inadequately.

We compare geometric programming to other numerical methodologies used to com-pute solutions to economic models such as shooting methods, Fair and Taylor (1983)method, nonlinear model predictive control framework and parametric path method.We find that geometric programming can be compared in speed and accuracy to thesemethods. Moreover, the accuracy of the solution to stochastic neoclassical growth modeldelivered by these methods is similar to the one delivered by geometric programming.Considering that all five methods rely on the assumption of certainty equivalence theseresults come naturally.

The rest of the paper is structured as follows: In Section 2 we introduce geometricprogramming and illustrate this technique with examples. In Section 3, we solve deter-ministic and stochastic neoclassical growth models and a set of problems coming fromintratemporal first order conditions. In Section 4, we compare geometric programmingto other numerical methodologies. In Section 5, we conclude.

3.2 Geometric programming

We study a framework developed for finding solutions to nonlinear optimization prob-lems subject to linear or nonlinear constraints. Geometric programming was first in-troduced in Zener (1961) and later developed in Duffin et al. (1967). The main objectof geometric programming is a geometric program: a type of mathematical optimiza-tion problem characterized by objective and constraint functions having a special form.Most of the early research on geometric programming was focused on theoretical, algo-rithmic and application aspects; see Dambo (1976), Beightler and Philips (1976) andAvriel (1980) among others. Recent advances include Boyd et al. (2007), Boyd andVandenberghe (2004) that introduce new applications of geometric programming anddevelop efficient numerical algorithms for their solution.

3.2. GEOMETRIC PROGRAMMING 101

3.2.1 Geometric program

Geometric programming methods can solve constrained and unconstrained optimizationproblems, where the constraints on the objective functions can be presented as equalitiesor inequalities.

The main objects of geometric programming are monomials and posynomials. Con-sider a vector x = (x1, ..., xn) consisting of n real positive variables x1, ..., xn.

Definition 1. (Monomial function). A real valued function f of x is called a monomialfunction if:

f(x) = cxa11 xa22 ...x

ann ,

where c > 0 and ai ∈ R.

The constant c is called a coefficient of the monomial, and the constants a1, ..., anare referred to as the exponents of the monomial.

Definition 2. (Posynomial function). A sum of one or more monomials is calledposynomial function or posynomial:

f(x) =K∑k=1

ckxa1k1 xa2k2 ...xankn ,

where ck > 0.

Any positive constant (and any variable) is a monomial. Also, any monomial is aposynomial.1 Monomials are closed under division and multiplication. Posynomials areclosed under addition, multiplication and positive scaling.

Example 1. Consider the functions:

3x2y−0.2z, 3(x+ y)2, 3(x− y)2,

where x, y and z are positive variables. The first expression is a monomial, the secondfunction is a posynomial but not a monomial. The third function is neither a monomialnor a posynomial since a closer inspection reveals that the constant multiplying theterm xy is negative, i.e. c2 < 0.

Definition 3. (Geometric program (GP)). A geometric program is an optimization

1This follows from the Definition 2 for K = 1.


problem of the form:

min f0(x) (3.1)

subject to

fi(x) ≤ 1, i = 1, ...,m (3.2)

gi(x) = 1, i = 1, ..., n (3.3)

where fi and f0 are posynomial functions, gi are monomials, and xi are the optimizationvariables.

This formulation of the problem is called geometric program in a standard form.Not all problems are originally in the standard geometric programming form, howeverthere are techniques which can be used to transform original problems to this formor successively approximate them by other geometric programming problems. In thefollowing section we will describe briefly possible transformations needed to convert theoptimization problems considered in economics to a GP in a standard form. For a morecomplete set of examples an interested reader is referred to Boyd and Vandenberghe(2004) and Boyd et al. (2007).

In general, a primal problem (3.1)–(3.3) can be a complicated nonlinear optimizationproblem since the objective function and constraints are nonlinear functions. However,there exist a dual problem for a problem (3.1)–(3.3) for which the constraints arelinear. The relationship between the objective functions in primal and dual problemsis as follows:

K∑k=1

ck

n∏i=1

xaiki ≥K∏k=1

(ckwk

)wk, (3.4)

where wk is a weight of the value of monomial function k in the value of the objec-tive function (3.1),

∑k wk = 1. Inequality (3.4) is a the arithmetic-geometric mean

inequality and it relates the sums and products of positive numbers. This inequalityplayed a central role in early development of the mathematical foundations of geometricprogramming.

If one seeks for the minimum of the expression on the left-hand side (objective fun-tion of the primal problem) of the inequality (3.4) then she could find it by maximizingthe expression on the right-had side (objective function of the dual problem) and bydoing so drive the inequality to equality. Thus, the global minimum for the primalproblem equals the global maximum for the dual problem.2

Remark 1. In a problem (3.1)–(3.3) there is an implicit constraint that all the variablesare positive, xi > 0. These restrictions are due to the fact that the main objects ofthe GP are monomials and posynomials. Additionally, most of the solution methodsused for GP rely on arithmetic-geometric mean inequality which implies it implicitly;see Simon and Blume (1994).

2For a rigorous mathematical proof of the above statement see Beightler and Phillips (1976).

3.2. GEOMETRIC PROGRAMMING 103

3.2.2 Complexity of GP and solution methods

There are many problems in science that originally have a form of GP; see for anexample economic order quantity model (Thomopoulos and Lehman (1971), Churchmanet al.(1953)), truss design problem (Bazaraa (1993)) and others. At the same time thereare many other problems that are not expressed originally in a standard GP form, butcan be succesfully rewritten in this form or approximated by it. Formulating a practicalproblem as a GP is called GP modeling. To express a problem in a standard GP formone can make use of change of variables, functional substitutions and approximations.

Example 2. Consider the problem

min√

1 + x2 + log(1 + y/x) (3.5)

subject to

maxy, z2 ≤ 1, (3.6)

1/x+ z/y ≤ 1, (3.7)

where x, y, z ∈ R. The problem (3.5)–(3.7) is not expressed in a standard GP form,since the objective function and constraint (3.6) are not posynomials. Lets introducenew variables ξ1, ξ2, ξ3 along with inequality constraints:

1 + x2 ≤ ξ1, 1 + y/x ≤ ξ2, y ≤ ξ3, z2 ≤ ξ3. (3.8)

New variables ξ1, ξ2 and ξ3 are upper bounds on the monomials 1−x2, 1+y/x andy, z2,respectively. An equivalent GP takes form:

min ξ0.51 + (ξ2/ε)ε (3.9)

subject to

ξ3 ≤ 1, (3.10)

y ≤ ξ3, z2 ≤ ξ3 (3.11)

1/x+ z/y ≤ 1, (3.12)

where ε is an arbitrary small positive quantity.

General nonlinear optimization techniques do not impose any restrictions on theobjective function and constraints and, therefore, can be used to solve any nonlinearproblem; for review of such numerical methods see Judd (1998). In contrast, GPmodeling requires the optimization problem to be of certain structure. Example 2shows that expressing a problem in a standard GP form could be a complicated task.The advantage of using geometric programming is in the numerical methods that areavailable and allow to solve large scale GPs efficiently and reliably; see Boyd (2007) fora review.

In early works the GPs were solved mostly analytically using a dual problem; itis possible to do so only for very small problems. The first numerical methods for


geometric programming were developed in Duffin (1970), Avriel et al. (1975) andRajpogal and Bricker (1990) and were based on solving a sequence of linear programs.

Nesterov and Nemirovsky (1994), Kortanek et al. (1996) and Andersen and Ye(1998) described interior-point methods for GP. Boyd and Vandenberghe (2004) docu-ment the speed of interior-point methods: a GP problem with 1000 variables and 10000constraints can be solved in under a minute. Additionaly, interior-point methods arerobust and do not require parameter tuning, starting points or initial guesses. Thisis typically a problem of fixed-point style numerical methods used in economics suchas parameterized expectation algorithm and its varianats, see Den Haan and Marcet(1990), Christiano and Fisher (2000).

Remark 2. Interior-point methods belong to the class of methods that solve nonlinearconvex optimization problems. A GP program can be expressed as nonlinear convexoptimization problem through a change of variables. Consider a change under whichy = log x, substituting x = ey into (3.1)–(3.3) and taking logs we get:

min log f0(ey) (3.13)

subject to

log fi(ey) ≤ 0, i = 1, ...,m (3.14)

log gi(ey) = 0, i = 1, ..., n (3.15)

A problem (3.13)–(3.15) is a nonlinear convex optimization problem since the objec-tive and inequality constraints are convex functions and equality constraints are linearfunctions.

GP problems can be solved using toolboxes designed for Matlab that include primor-dual interior-point solvers, such as GGPLAB and CVX; both are available publicly andat no charge. The former is tailored for sovling GPs and generalized GPs, while thelatter allows to solve a wider range of convex optimization problems.

3.3 Applications in economics

In this section we solve a set of problems often encountered in economics by geometricprogramming methods. Firstly, we solve a deterministic neoclassical growth model. Wethen study how geometric programming methods can be applied to finding solution toa stochastic version of this model. Lastly, we apply the methods to find intratemporalchoice allocations; for multi-country model see Juillard and Villemot (2011), for laborchoice see Maliar and Maliar (2005).

3.3. APPLICATIONS IN ECONOMICS 105

3.3.1 Deterministic case

Consider as an example the following deterministic neoclassical growth model:

maxctTt=0,kt+1T−1

t=0

T∑t=0

βtu(ct) (3.16)

subject to

ct + kt+1 ≤ f(kt) + (1− δ)kt (3.17)

where initial and terminal conditions k0, kT are given; ct and kt are, respectively, con-sumption and capital; u and f are the utility and production functions, respectively;β ∈ (0, 1) is the discount factor and δ is the depreciation rate. We assume that theutility and production functions take Cobb-Douglas form and, therefore, are strictlyincreasing and concave:

u(ct) =c1−γt − 1

1− γ, f(kt) = kαt (3.18)

Without loss of generality we assume that δ = 1; this assumption is going to be relaxedlater.

In general, the problem (3.16)–(3.17) is not a GP in a standard form since (1) thesolution is a maximum and not a minimum, (2) the objective function is a posyno-mial for certain parameters of risk aversion. We describe the transformations that arenecesary to be done for the problem (3.16)–(3.17) to become a GP problem below.

Low risk aversion. For the case when the representative agent posses a low riskaversion, i.e. γ < 1, the objective function is a posynomial. However, the optimizationproblem is set for a maximum instead of minimum. We add additional variable x0 andan inequality constraint such that:

maxT∑t=0

βtu(ct) ⇐⇒ maxx0 s.t.T∑t=0

βtu(ct) ≥ x0 (3.19)

The function on the left-hand side of the constraint (3.19) is a posynomial, however,the sign of the inequality is reversed. To deal with this issue we introduce a set ofsupplementary weights φTt=0 such that:

T∏t=0

(βtu(ct)x

−10

φt

)φt≤ 1, (3.20)

where the supplementary weights are given by:

φt =βtu(ct)∑Tt=0 β

tu(ct). (3.21)


Low risk aversion – Iterative procedure

Step 0. Determine an initial feasible solution.

a. Determine steady state values of capital, k∗, and consumption, c∗.

b. Set initial conditions for capital, k0 = k∗.c. Construct a sequence of consumption: ctTt=0 = 0.7c∗.d. Construct a sequence of capital using (3.17):

kt+1 = kαt − ct.e. Check the feasibility of constructed paths for capital, ktT+1

t=0 ,and consumption, ctTt=0.

Step 1. Calculate weights on iteration j.

Substitute ctTt=0 into equation (3.21) to obtain φjt

Step 2. Solving the problem.

a. Using interior-point methods solve a GP:maxx0

s.t. ct + kt+1 ≤ kαt∏Tt=0

(βtu(ct)x

−10

φjt

)φjt≤ 1

b. Use the solution cjt , kjt+1Tt=0 to compute the weights for iteration j + 1, φj+1

t .

End iteration if the convergence is achieved ||xj0 − xj+10 || < τ , where τ is a tolerance level.

Inequality (3.21) is a monomial approximation of the constraint (3.19) that followsfrom arithmetic-geometric mean inequality: a lower bound inequality is approximatedby an upper bound inequality (3.21). An iterative procedure used to solve the problem(3.16)–(3.17) is outlined above.

Risk aversion equals to 1. When risk aversion parameter γ equals to 1 theconstant relative risk aversion (CRRA) utility function takes the form of:

u(ct) = log(ct). (3.22)

Conventional geometric programming structure does not allow for logarithmic termsto appear in either objective function or constraints. In this case we use a limitingapproximation:

limε→0

(cεtε− 1

ε

)= log(ct). (3.23)

When ε is very close to zero the expression in parenthesis approximates accurately thenatural logarithm. In our experiments ε = 0.01.


After this transformation the corresponding GP problem takes form:

maxT∑t=0

βtcεtε

(3.24)

subject to

ct + kt+1 ≤ kαt , (3.25)

High risk aversion. In the case when the risk aversion parameter is greater than1 the utility function is not a posynomial since 1− γ < 1. Hence, in solution procedurewe search for minimum of the −

∑Tt=0 β

tu(ct) instead of maximum of (3.16).

0 5 10 15 20 25 300.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

time

Con

sum

ptio

n

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

time

Cap

ital

γ = 0.01γ = 0.1γ = 1γ = 5γ = 10γ = 15

γ = 0.01γ = 0.1γ = 1γ = 5γ = 10γ = 15

Figure 3.1: Convergence to the steady state and risk aversion

In our numerical examples, we use β = 0.99, α = 0.36, δ = 1, T = 200, convergencecriterion, τ , is set to 10−6. We solve the model (3.16)–(3.17) for a set of risk aversionparameters γ = 0.01, 0.1, 1, 5, 10, 15. For each of this specifications we set the terminalcondition for capital equal to 0, kT+1 = 0. An economy departures from a point farfrom steady state, k0 = 0.1k∗. The computational times for the cases of low, unity andhigh risk aversion models are, respectively, 60.3, 3.5 and 41.0 seconds.

Figure (??) plots consumption and capital time-series solutions for different degreesof risk aversion. The lower risk aversion that agent posses the quicker she desires torestore the steady state level of consumption and capital. Therefore, the speed ofconvergence to the steady state is higher in the specification with low γ and lower in


specifications with high γ. Note, that with primal-dual interior-point methods we wereable to find solutions to models with extreme values of γ, such that 0.01 and 15. Suchextreme parametrization may cause conventional methods fail to converge.

Remark 3. Traditional models in economics assume that δ 6= 1, however in this casebudget constraint (3.17) does not satisfy definition of a posynomial function. Geometricprogramming methods can also be applied in such instances by successive approxima-tion of the constraint by monomial functions. In particular, consider a supplementaryvariable st such that:

ct + kt+1 ≤ st ≤ kαt + (1− δ)kt. (3.26)

The first inequality in (3.26) is a posynomial function, while the second one is not.Based on arithmetic-geometric mean inequality we can use an iterative procedure withweights ψt as an approximation.3

3.3.2 Stochastic case

Consider a stochastic version of the model studied in Section 3.3.1:

maxctTt=0,kt+1T−1

t=0

E0

T∑t=0

βtu(ct) (3.27)

subject to

ct + kt+1 ≤ f(kt, zt) + (1− δ)kt, (3.28)

log(zt) = ρ log(zt−1) + εt, εt ∼ N(0.σ2), (3.29)

where the initial condition (k0, z0) is given; Et is the expectation operator conditionalon the information set at time t. The difference with respect to deterministic model isin the productivity level, zt, that is determined by AR(1) process. Its autocorrelationcoefficient is ρ ∈ (−1, 1) and the variance of the productivity shock is σ > 0.

To the best of our knowledge there are few methods available to solve stochasticGP. Avriel and Wilde (1970) study the problem where the exponents of the posynomialaik are deterministic, while the coefficients ck are positive random variables. Authorsfind numerically tractable bounds on the objective function, see Dupacova for a reviewof methods and applications.

One possible way to generalize geometric programming methods to the case of uncer-tainty in the productivity level is to use a certainty equivalence assumption. ConsiderEuler equation for the problem (3.27)–(3.29):

u′(ct) = βEt (u′(ct+1) (1− δ + f ′(kt+1, zt+1)))) , (3.30)

where u′ and f ′ are the derivative of the utility function with respect to consumption andproduction function with respect to capital. Certainty equivalence principle consists in

3This procedure is similar to the one used in the case of neoclassical growth model with low riskaversion.


Table 3.1: Comparison of GP and GSSA, d = 1

GP methods GSSA

εmean εmax CPU1 CPU2 εmean εmax CPU

(γ, σ)(0.1, 0.001) -2.8111 -2.0669 - 1.27(+4) -3.2757 -2.5596 5.35(0.1, 0.01) -1.8168 -1.0972 - 1.09(+4) -2.2756 -1.5563 10.09(0.1, 0.1) -0.8076 0.0289 - 1.29(+4) -1.2744 -0.5238 37.08

(1, 0.001) -5.9554 -4.9553 29.65 64.92 -9.1490 -8.4313 3.04(1, 0.01) -5.8020 -4.8020 28.89 60.49 -9.1756 -7.4396 5.05(1, 0.1) -5.8466 -4.6568 28.84 62.59 -3.8123 -2.4622 22.50

(5, 0.001) -2.7759 -2.2185 681.2 2.55(+3) -3.1761 -2.4810 4.48(5, 0.01) -1.7515 -1.1465 683.87 2.23(+3) -2.1803 -1.4761 8.51(5, 0.1) -0.7770 -0.0512 695.53 2.61(+3) - - -

Note: εmean and εmax are the average and maximum absolute standard Euler-equation er-ros, log10; CPU1 is a computation time needed to allocate variable and constraints of theproblem in seconds; CPU2 is a time needed for GGPLAB to a solve the problem; CPU is acomputational time of the GSSA; “-” means algorithm failed to converge.

replacing the future productivity level, zt+1, with its expected value in equation (3.30):

u′(ct) ≈ βu′(ct+1) (1− δ + f ′(kt+1, Et[(zt+1])) . (3.31)

In Table 3.1 we compare the accuracy of the solution to the model (3.27)–(3.29)by geometric programming methods (GP methods) under the assumption of certaintyequivalence to a solution by GSSA (see Judd et al. (2011)). In these experiments we useT = 600, ρ = 0.95 and a set of parameters for risk aversion γ = 0.1, 1, 5 and volatilityof the productivity shock σ = 0.001, 0.01, 0.1. To compute Euler equation errorsof the future policy function kt+1 = K(kt, zt) for the solution delivered by geometricprogramming methods we run an OLS regression on the stochastic path of kt+1Tt=0.

First, geometric methods yield in speed to the state of the art numerical methods,such as GSSA. Second, the accuracy of the solution depends on the risk aversion pa-rameter (that introduces nonlinearity in the model) and the volatility of the shock. Inall experiments an increase in the volatility of the shock results in increase of the meanand maximum Euler equation errors. In general such increase would be higher for GPmethods.

Third, note that neoclassical growth model is a stationary model. Geometric pro-gramming methods can as well be applied for finding solutions to nonstationary models;for example, neoclassical growth model with labor augmenting technological change.At the same time GSSA methods can only be applied in such models if they allow for


transformation into stationary ones. In order to facilitate the comparison between thesemethods we focus on stationary models.

3.3.3 Intratemporal conditions

Labor choice. Consider a standard neoclassical growth model with a valuable leisurechoice:

maxct,kt,kt+1∞t=0

E0

∞∑t=0

βt(c1−γt − 1

1− γ+ b

(1− nt)1−σ − 1

1− σ

)(3.32)

s.t. ct + kt+1 = (1− δ)kt + ztkαt n

1−αt (3.33)

log(zt) = ρ log(zt−1) + εt, εt ∼ N(0.σ2) (3.34)

kt+1 ≥ 0, ct ≥ 0, nt ∈ [0, 1] (3.35)

where initial condition (k0, z0) is given. Here, nt is hours worked (labor); γ, σ ∈[0,∞), b > 0 are the utility function parameters. The first order conditions of thisproblem with respect to consumption, capital and labor have the following form:

c−γt = βEt[c−γt+1(1− d+ zt+1k

α−1t+1 n

1−αt+1 )], (3.36)

(1− nt)−σnαt =1

bztk

αt c−γt . (3.37)

Equation (3.36) is an Euler equation and it relates marginal utility of consumption incurrent and future time periods and, therefore, is referred to as intertemporal condition.Equation (3.37) relates consumption, capital and labor choice in current period t andis referred to as intratemporal condition. Maliar and Maliar (2005) argue that in typesof macroeconomic models with valuable leisure such as (3.32)–(3.35) it is convenient toparametrize labor function.

If instead we parametrize the consumption function ct = [δψ(β; kt, θt)]−1/γ, nt can

be restored from intratemporal condition (3.37) and kt+1, hence, will be defined by(3.33). Intratemporal condition (3.37) is a nonlinear equation that need to be solvedfor nt many timed inside of the iterative cycle of any numerical method. Such type ofnonlinear equations arise frequently in other economic models; see Maliar and Maliar(2013) for examples.

A precomputation of labor manifold outside of the main iterative cycle can beused on a par with parametrizing the labor function. One can construct a grid ofat ≡ 1

bztk

αt c−γt and solve the equation (3.37) for each grid point. Moreover, this equation

can be solved by means of geometric programming methods. Consider the equation thatneed to be solved:

(1− nt)−σnαt = at. (3.38)

The function on the left hand side of this equation f(nt) = (1 − nt)−σnαt is strictly

increasing in nt (f ′(nt) > 0) and, therefore, there is a unique value n∗t for which (3.38)holds:

(1− n∗t )−σ(n∗t )α

at= 1.


Taking into account this considerations the problem can be reformulated in a standardGP form:

max nt (3.39)

s.t.

(lt)−σ(nt)

α

at≤ 1, (3.40)

lt + nt ≤ 1. (3.41)

The above problem has two variables and two constraints. For this experiment, we fixσ = 5, α = 0.33, at = 5.25. The numerical cost of computing a solution to the problem(3.42)–(3.44) is about 0.05 seconds. Note, that we do not reduce the number of con-straints by substituting out leisure, lt, since this would result in constraint (3.43) notbeing a posynomial function.

Remark 4. Geometric programming formulation of the problem of solving theequation (3.38) can be extended to a more general case. Consider a case where thisequation need to be solved T times. Since n∗tTt=0 are independent solutions of thesame equation we can formulate a more general problem:

maxT∑t=0

nt (3.42)

s.t.

(lt)−σ(nt)

α

at≤ 1, t = 0, ..., T (3.43)

lt + nt ≤ 1, t = 0, ..., T (3.44)

that has 2T variables and 2T constraints and can be solved by geometric programmingmethods.

Multi-country models. Geometric programming methods can as well be appliedto economic problems that feature many nonlinear equations. Consider a set of firstorder conditions of the social planner’s maximization problem in the multi-country realbusiness cycle model with j = 1, ..., N countries4.

ujc(cjt , l

jt )τ

j = uj′

c (cj′, lj′)τ j

′, (3.45)

ujl (cjt , l

jt ) = −ujc(cj, lj)z

jt f

jl (kjt , l

jt ), (3.46)

N∑j=1

cjt =N∑j=1

zjt f j(kjt , ljt )− φ

2kjt

(kjt+1

kjt− 1

)2

+ kjt − kjt+1

, (3.47)

4For the set up of the economy see Maliar et al. (2011) and Juillard and Villemot (2011)


where uj and f j are utility and production functions of a country j, respectively;cjt , k

jt , z

jt , τ

j are, respectively, consumption, capital, productivity level and a welfareweight of a country j; parameter φ governs the intensity of the adjustment cost frictionon capital stock.

Conditions (3.45)–(3.47) form a system of nonlinear equations that need to be solvedfor a given triple (kjt , k

jt+1, a

jt)Nj=1 and that as well correspond to first order conditions

of the social planner’s optimization problem in period t:

maxcjt ,l

jtj=1,...,Nt=0,...,∞

N∑j

τ juj(cjt , ljt ) (3.48)

s.t.N∑j=1

cjt =N∑j=1

ajtf j(kjt , ljt )− φ

2kjt

(kjt+1

kjt− 1

)2

+ kjt − kjt+1)

. (3.49)

In our experiments we use parameters and functional forms for utility and pro-duction functions corresponding to Model A8 from Juillard and Villemot (2011). Weoutline an iterative procedure that can be used to compute the solution below:

Multi-country model


a. Discretize the space for ljt , i.e. ljt (m)Mm=1.b. Compute consumption manifolds using (3.46).

c. Guess ljtNj=1 to begin iterations.

Step 1. Solving the model on iteration i.

For a given (kjt , kjt+1, a

jt , l

jt )Nj=1 solve problem (3.48)–(3.49) by geometric programming

methods.

Step 2. Updating the guess.

a. Use precomputed manifolds for consumption to update the guess on ljtNj=1.

b. Come back to Step 2.

End iteration if the convergence is achieved ||(lit − li+1t )|| < τ , where τ is a

tolerance level and lt is a vector of labor allocations for countries j = 1, ..., N .

Numerical costs of this iterative procedure for the model with 2,4, 6 and 8 countriesare 2.12, 2.74, 2.64 and 2.85 seconds, respectively.

3.4 Comparison

In this section we compare geometric programming methods to other numerical methodsused in economics such as: shooting methods, Fair and Taylor (1983) method, nonlinear

3.4. COMPARISON 113

model predictive control and reinforcement learning; see Judd (1998), Fair and Taylor(1983), Grune et al. (2013) and others.

3.4.1 Forward and backward shooting

Shooting methods are introduced in economics in Lipton et al. (1980). These methodscan be used to find solutions to finite and infinite horizon economies. Solution to aninfinite horizon economy can be recovered from a finite horizon economy under someassumption on the terminal capital, kT+1, and long modeling horizon, T . Consideragain the problem (3.16)–(3.17). First order conditions of this problem with respect toconsumption and future capital result in a deterministic Euler equation of the form:

u′(ct) = β (u′(ct+1)(1− δ + f ′(kt+1))) , (3.50)

that together with the budget constraint (3.17) describe the evolution of the economy.For simplicity we assume that δ = 1, γ = 1 and that the economy should arrive tosteady state level of capital in T periods, kT+1 = k∗.

Forward shooting Lets express kt+2 as a function of kt and kt+1

kt+2 = f(kt+1)− β(f(kt)− kt+1)f′(kt+1) (3.51)

k0 is given and if we knew a corresponding to solution k1 we could compute k2 andin this way restore the whole sequence of capital ktT+1

t=0 . The idea of forward shootingis to find a value of k1 which will result in kT+1 = k∗. This can be done in two steps: Wefirst guess a value ki1 and compute a corresponding to it kT+1(k

i1). On step 2 we update

our guess and iterate on steps 1 and 2 until the difference between ||kT+1(k1)−k∗|| < ε.In a model considered in this section step 1 does not involve heavy computations. In

general, on step 1 we need to solve a system of nonlinear equations in many unknowns.In step 2 we can use a nonlinear solver, bisection method or Gauss - Siedel type iterationsto find the next guess, ki+1

1 .It is difficult to solve long-horizons models with shooting methods since kT+1 is very

sensitive to the choice of k1. Even a small deviation from a true k1 can lead to anexplosive path (see Figure (??)). However, this means that the reverse result holds, i.e.the terminal state k1 is relatively insensitive to kT+1. This is a key idea in backward(or reverse) shooting.

Backward shooting Consider again (3.50) and lets express kt as a function of kt+1

and kt+2:

kt = f−1((

kt+1 +f(kt+1)− kt+2

βf ′(kt+1)

))(3.52)

Given kT+1 = k∗ we want to find a value of kT that leads to k0. In many economicapplications backward shooting was found to be more stable than forward shooting,


0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1No terminal condition

Figure 3.2: Divergence patterns in forward shooting

however there is no general result and a better performance of any particular methodwill depend on the character of steady state convergence.

The literature on backward shooting has not made much progress to date in devel-oping solution algorithms for models with more than one state variables. Judd (1998)describes applications of forward shooting in multidimensional systems. Atolia andBuffie (2009a,b) develop a reverse shooting algorithm to solve models with 2-3 statevariables. Stochastic models can be solved by shooting methods under assumption ofcertainty equivalence discussed in Section 3.2.

3.4.2 Fair and Taylor (1983) method

Fair and Taylor (1983) proposed an extended path method that can solve stochasticeconomic models. Let us illustrate the idea of this method in the context of the standardgrowth model (3.27)–(3.29). Assume that the economy is stochastic and write the Eulerequation and the budget constraint by assuming a logarithmic utility function:

1

ct= βEt

(1

ct+1

(1− δ + zt+1f′(kt+1))

),

ct + kt+1 = (1− δ) kt + ztf (kt) .

Combine the two conditions to get (setting δ = 1 for expository convenience):

kt+1 = ztf (kt)−[Et

(zt+1f (kt+1)− kt+2

βzt+1f ′(kt+1))

)]−1. (3.53)

3.4. COMPARISON 115

Our objective is to compute a path satisfying (3.53) of length T , namely,(k1, k2, ..., kT

)for a given initial condition (k0, z0) is given, a fixed sequence of productivity levelsz1, ..., zT and terminal condition kT+1. This gives us T equations (3.53) with T un-knowns (k1, k2, ..., kT ).

There are several alternative ways to solve this system of equations including a non-linear solver, backward and forward shooting and a fixed-point type of iteration. TheFair and Taylor (1983) use an efficient derivative-free iteration in line with Gauss-Seidelto solves for the path as follows:

Fair and Taylor (1983).

Extended path is aimed at approximating a solution to a nonstationary deterministic

model during the first τ periods, i.e., it finds k0, ..., kτ such that∥∥∥kt − kt∥∥∥ < ε for

t = 1, ...τ , where ε > 0 is target accuracy, ‖·‖ is an absolute value, and kt and ktare the t-period true capital stocks and their approximation, respectively.


a. Choose time horizon h τ and terminal condition kh+1.b. Construct and fix z∗t t=0,...,h.

c. Guess an equilibrium pathk(1)t

t=1,...,h

for iteration 1.

d. Write a t-period system of the optimality conditions in the form:

kt+1 = z∗t f (kt)−z∗t+1f(kt+1)−kt+2

βz∗t+1f′(kt+1))

.

Step 1. Solving for a path on iteration j.

a. Substitute a pathk(j)t

t=1,...,h

into the right side of (3.53) to find

k(j+1)t+1 = z∗t f

(k(j)t

)−

z∗t+1f(k(j)t+1

)−k(j)t+2

βz∗t+1f′(k

(j)t+1))

, t = 1, ..., h

b. End iteration if the convergence is achieved∣∣∣k(j+1)t+1 − k(j)t+1

∣∣∣ < tolerance level.

Otherwise, increase j by 1 and repeat Step 1.

The EP solution:

Use the first τ constructed values k0, ..., kτ as an approximation

to the true solution k0, ..., kτ and discard the last h− τ values.

Let us make several remarks about the Fair and Taylor (1983) method. In theversion of extended path studied in Fair and Taylor (1983), they use τ = 1, i.e., they

keep only the first element k1 from the constructed pathk1, ..., kh

and disregard the

rest of the path; then, they solve for a newk1, ..., kh+1

starting from k1 and ending

at steady state at h+ 2 (i.e, extend the path 1 period ahead) and store k2 and advanceforward until the path of the given length T is constructed. The extension of path


is called Type III iteration and gives the name to the Fair and Taylor (1983) method5. If h τ , the effect of specific terminal condition on the initial path of length τ isnegligible.

To insure that specific terminal condition kT+1 has little effect on k1, we may needto choose T to be very large. This means that the cost of extended path method can bevery high. In more complicated models, we may be unable to derive the law of motionfor the state variables analytically as in our simple example and we may need to solvea system of equations with hundreds of unknowns. This can be very costly.

As is typical for fixed-point iteration style methods, they may fail to converge, inparticular, if the updating is too fast. To deal with this issue, we use damping, namely,we update the path over iteration only by a small amount k

(j+1)t+1 = ξk

(j+1)t+1 +(1− ξ) k(j)t+1

where ξ ∈ (0, 1) is a small number close to zero such as, 0.01.Adjemian and Juillard (2013), Maliar et al. (2015) assess the accuracy and compu-

tational speed of the Fair and Taylor method. The stochastic extended path (SEP) isintroduced in the former research project. The idea is to construct a tree of possiblefuture shocks and to solve for multiple paths for variables on all branches of the tree.This modification is introduced to overcome the inaccuracy resulting from the assump-tion of certainty equivalence. SEP performs better in terms of accuracy, however, thecomputational cost increases heavily.

Cost and accuracy of geometric programming methods is very similar to Fair andTaylor (1983) method. The advantage of GP methods is that they rely on efficientprimal-dual interior point algorithms and, therefore, are more numerically stable.

3.4.3 Nonlinear model predictive control: finding a solutiondirectly from a maximization problem

Heer and Maussner (p.12) describe an extended path method under a numerical solverinstead of Gauss-Seidel iteration. Grune et al (2013) evaluate the performance of thenonlinear model predictive control (NMPC) method which is similar to Fair and Taylor(1983) extended path methods. Both NMPC and Fair and Taylor use certainty equiva-lence assumption to approximate the stochastic dynamics of capital and consumption.The difference lies in how the system of first order conditions are solved: the formerattacks it with solvers, while the latter uses derivative free iteration methods.

Remark 5. Principle of certainty equivalence is used in both the Fair and Taylorand NMPC methods. The principle consists of substituting future realizations of errorterms by its expected values, i.e by 0:

Et(zt+s) = θt+1 = zρs

t . (3.54)

5In Fair and Taylor(1983) the extention of the path refers to an increase in h which is increased

until further extentions do not affect the solution. For instance, to find a solution k1 we would solvethe model several times for h + 1, h + 2, h + 3, ... periods starting form period zero. In present paperwe extend the path when we move from t to t+ 1.

3.4. COMPARISON 117

Certainty equivalence allows us to get a system of nonlinear equations that does not haveexpectation terms and can be solved by any conventional numerical method. Gruneet al (2013) substitute future realizations of the productivity levels by their expectedvalues, i.e.:

Et(zt+1) = exp(ρ log zt + σ2/2),

and therefore the variance of the shock ε is taken into account. In our experiments weassess the accuracy of the solution under both types of certainty equivalence approxi-mation and we find that there are little differences in accuracy of the solutions.

3.4.4 Parametric path method

Judd (2002) introduces parametric path method. To illustrate how this method worksconsider again the problem (3.16)–(3.17). The objective of the parametric path methodis to find a sequence of ct and kt which satisfy the equations (3.50) and (3.17) and isnon-explosive, i.e. sup

t≥0‖ct‖ < ∞ and sup

t≥0‖kt‖ < ∞. It replaces a sequence of ct and

kt with a parametrization of its components which in some way represents the beliefson how ct and kt evolve over time. For example, in perfect foresight models consumerswant to smooth their consumption and consumption path will often be very smooth.We outline the main steps of the parametric path method below:

Parametric path method


a. Choose parametrizations K(t, b, k), where b is a parameter vector.b. Form residual functions Ri(t, b) = f i (k,K(t, b, k)), where f it is formed

from first order conditions.c. Select test functions pj(t)mj=1, m is degree of polynomial function.

d. Form projections P ij(b).= 〈Ri(t, b)pj(t)〉.

Step 1. Solving for a parametric path.

a. Solve a system of equations consisting of initial conditions plus projection equations

P ij(b) = 0. Denote the solution b.

b. Compute τ = max ‖f it(k,K(t, b, k)

)‖, for t = 0, 1, ..., T .

c. Repeate Steps 0a - Step 1b until τ is sufficiently small.

Functional form K(t, b, x, z) should be flexible enough to be able to approximateany likely solution. The advantage of the parametric path method is that it reduces aproblem to a small system of nonlinear equations which can be solved by any numericalmethod available. However, as the complexity of the problem increases the computa-tional cost of solving big systems of nonlinear equations does parametric path methodless attractive.


Table 3.2: Accuracy of parametric path method

δ = 1 δ = 0.025m Emean Emax Emean Emax1 -4.79 -3.47 -3.63 -2.352 -5.33 -3.95 -3.95 -3.033 -5.94 -4.72 -4.34 -3.404 -6.48 -5.07 -4.88 -3.185 -7.06 -5.53 -4.96 -3.166 -7.21 -5.77 -4.86 -3.21

We asses the accuracy of the parametric path method for the problem (3.16)–(3.17).In our experiments we set α = 0.36, γ = 1.1, δ = 1, 0.025, β = 0.99, k0 = 0.5k∗.Following Judd (2002) we parameterize the capital policy function with a flexible form:

k(t) =

(m∑j=0

bjφj(2λt)) exp(−λt) + k∗(1− exp(−λt)

)(3.55)

where φj is the Laguerre polynomial of degree j; bjare the parameters; λ is the speedof convergence to the steady state. In Step 0d integral is approximated with Gauss-Laugerre quadrature nodes and weights; see Judd (1998) for details.

We report mean and maximum Euler equation errors for models with δ = 1 andδ = 0.025 in Table 3.2. As δ increases we need to use more and more polynomialdegrees to improve the accuracy of the approximation (see Figure ??). Parametricpath method is hardly generalizable to the case of stochastic models. In stochasticgeneral equilibrium model the productivity level zt becomes part of the state space.Therefore, the capital policy function (3.55) should include zt as an argument.

3.5 Conclusion

Geometric programming is a field of mathematical optimization that focuses on non-linear optimization problems subject to linear or nonlinear constraints. New numericalmethods can solve large-scale geometric programs efficiently and reliably in reasonableamount of time. We find that geometric programming methods accurately solve de-terministic growth model and a set of intratemporal optimization problems. However,the solution delivered by geometric programming for a stochastic growth model is in-accurate and the degree of inaccuracy increases with the degree of nonlinearity of themodel.

In this respect geometric programming can be compared in speed and accuracy toother numerical methodologies used in economics such as shooting methods, Fair and

0 200 400 600 800 1000 12000.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Capital

Time

m = 1

m = 2

m = 3

m = 4

Figure 3.3: Accuracy of the parametric path method

Taylor (1983) method, nonlinear model predictive control framework and parametricpath method. The accuracy of the solution to stochastic neoclassical growth modeldelivered by these methods is similar to the one delivered by geometric programming.Considering that all five methods rely on the assumption of certainty equivalence thisresults comes naturally.

Although in this project geometric programming methods are applied to economicmodels that are stationary by nature, the methods can as well be used in solvingnonstationary economic models.

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