chair of macroeconomics - growth and innovation · 1.1 some stylized facts however, some of these...

$: Chair of Macroeconomics - Growth and Innovation · 1.1 Some Stylized Facts However, some of these stylized facts could be challenged: I See some \distortions" in long-term trend since$
Growth and Innovation

(Module MW21.4)

PD Dr. M. Pasche

Friedrich Schiller University Jena

Creative Commons by 3.0 license – 2018 (except for included graphics from other sources)

Work in progress! Bug Report to: [email protected]

p.1

Outline:

1. The Empirical Picture of Growth

1.1 Some Stylized Facts1.2 Distribution and Convergence1.3 Environmental Impact

2. Preliminaries of Growth Theory

2.1 The Basic Solow Model2.2 Basic Model with Exogenous Technological Change2.3 Basic Model With Human Capital (Mankiw-Romer-Weil)2.4 Basic Model With Intertemporal Optimization

(Cass-Koopmans-Ramsey)2.5 Empirical Evidence

p.2

3. Innovation and Endogeneous Growth

3.1 Overview: Sources of Growth3.2 A Model with Knowledge Spillovers3.3 Models with Human Capital Accumulation3.4 R&D based Growth with Increasing Product Variety3.5 R&D based Growth with Increasing Product Quality3.6 Notes on Technology Diffusion and North-South Models3.7 Empirical Evidence and Critique

4. Growth and the Environment – Limits to Growth

4.1 Economic-ecological interaction4.2 Some empirics4.3 Green Growth? Innovation, Structural Change, and the

Environment4.4 Sustainability and De-Growth

p.3

Basic Literature:

* Barro, R.J., Sala-i-Martin, X. (2004), Economic Growth.MIT-Press (2nd ed.).

I Aghion, P., Howitt, P. (2009), The Economics of Growth.MIT Press.

I Acemoglu, D. (2008), Introduction to Modern EconomicGrowth. Princeton University Press.

I Sims, E., Garin, J., Lester, R. (2017). IntermediateMacroeconomics.https://www3.nd.edu/~esims1/gls_int_macro.pdf

References to more specific literature can be found in the slide collection.

p.4

https://www3.nd.edu/~esims1/gls_int_macro.pdf

Preliminary schedule (winter 2018/2019):

Week Tuesday Wednesday

42 ch.1 –43 ch.2 –44 ch.2 *)45 ch.2 ex46 ch.2 ex47 ch.2 ex48 ch.3 ex49 ch.3 ex50 ch.3 –51 ch.3 ex

02 ch.3 ex03 ch.3 ex04 ch.4 ex05 ch.4 ex06 ch.4 ex

*) public holiday

ex = exercise class p.5

Type of module:

I Compulsory in the specialization area “Innovation and Change”.

I Elective for the specialization “Quantitative Macroeconomics”.

I Additional (elective) for all other.

Examination:

I Written exam at the end of the term (60 minutes).

Role of mathematics:

I Necessary in order to fully understand dynamics, transitionprocesses, stability, and empirical implications of the models – andalso their limitations.

I Math is useful for training the intuition behind the models.

I For the exam not all math stuff is relevant! Explaining the idea andintuition behind the theories is always important!

p.6

No-screen-policy:

I Nobody can be forced to attend the course in person. But ifyou attend (which is highly recommendable!) a key successfactor is attention.

I Smartphones, tablets, and notebooks are attention killers.Neuroscientists and many didacts warn against the (excessive)use in the classroom. In fact, there is no need for it. It ismuch more important to follow my explanations actively andwith attention, to ask if something is unclear, to comment ifyou have important additional insights or if you are ofopposing opinion.

I Moreover, the use of electronic devices also detracts attentionof your classmates (negative externality).

I So I will follow a no-screen-policy, and I expect that youcommit yourself to this policy as well – to your own benefit.

p.7

moodle.uni-jena.de

I For this course there exists a moodle room where you find allcourse materials, past exams, interesting links, a discussionforum, short self-tests etc. (to be developed step by step).

I Participants who have registered in Friedolin for this courseautomatically have access to this moodle room (enter moodlewith your URZ login).

p.8

moodle.uni-jena.de

1. The Empirical Picture of Growth1.1 Some Stylized Facts

Literature:

I Barro, R.E., Sala-i-Martin, X. (2004), Economic Growth.Chapter 1.1-1.2 (chapter 10-12 for a deep empirical analysis)

I Sims, E., Garin, J., Lester, R. (2017). IntermediateMacroeconomics. Chapter 4.

Symbols:

Y = A · F (K ,N) = real output or incomeK = capital stockN = employed laborA = total factor productivityr = real interest ratew = real wages

p.9


1. Income per capita y = Y /N is growing with a constant rate (butdeclining growth rate since the 1970’).

2. The capital/output ratio (capital coefficient) K/Y is stationary.

3. The capital/labor ratio (capital intensity) K/N is increasing. This isjust an implication of a growing Y /N and a stationary K/Y .

4. The real rate of return to capital r = ∂Y /∂K is stationary (but hasa certain decline in highly developed countries).

5. The income distribution is stationary in the long run (measured byV = rk/wN or by wN/Y , rK/Y ).

6. The real rate of return to labor w = ∂Y /∂N is increasing. This isjust an implication of stationary distribution, stationary K/Y andgrowing Y /N.

7. The per capita growth rates differ much across countries.

8. Strong correlation between human capital and per capita income.

p.10


A note on growth rates:

I Growth with a constant rate g means that the variable growsexponentially:

y(t) = y(0)eγt

I Taking the log and differentiating with respect to time yields:

ln y(t) = ln y(0) + γt ⇒ gy ≡d ln y(t)

dt=

1

y· dydt

= γ

I For empirical data we use the first differences ∆ ln y(t) todetermine the growth rate gy .

I Growth with a constant rate means that we have a lineartrend of ln y(t) in a figure with absolute scale, or,alternatively, a linear trend of y(t) in a figure with alogarithmic scale.

p.11


A note on cross-country income measurement:

I To be comparable, income has to be measured using the sameunit (e.g. Dollar). Therefore we have to multiply the valueswith the exchange rate.

I As we are interested in the real income, we have to divide theGDP by the price index. However, price levels and inflationrates differ across countries. Therefore we have to account forthe different PPP of currencies (PPP adjusted exchangerates).

⇒ Penn World Tables: www.ggdc.net/pwt

I Methodological problems cannot be avoided completely.Interpret numbers with care!

p.12


Some illustrating empirical facts on growth dynamics:

I From 500 (roman imperium) to 1500: no significant economicgrowth!

I 1500-1800 about 0.1% annual growth rate.

I Moderate growth rates during the industrial revolution1800-1900, increasing in the late 19th century.

I Massive acceleration of economic activity in the 20th century,especially in the post war period.

I Decline of growth rates (in developed countries) starting fromthe 1970ies.

p.13


Real GDP per capita development (1 – 2016):

200

400

600

800

1000

1200

1400

1600

1800

2000

0

2

4

·104

WW II

IR

Year

USA

France

UK

China

Nigeria

The Maddison-Project, http://www.ggdc.net/maddison/maddison-project, 2018 version.

p.14


Real GDP per capita developmentfrom end of midage to end of 1. industrial revolution:

1500

1550

1600

1650

1700

1750

1800

1850

0

2000

4000

Year

USA

France

UK


p.15


Real GDP per capita development:

1860

1880

1900

1920

1940

1960

1980

2000

0

2

4

·104

Year

USA

France

UK

Japan

China

Nigeria

Congo


p.16


Real GDP per capita development:

1860

1880

1900

1920

1940

1960

1980

2000

0

2

4

·104

Year

USA

USA trend, 1.7%

The Maddison-Project, http://www.ggdc.net/maddison/maddison-project, 2018 version; own calculation.

p.17


However, some of these stylized facts could be challenged:

I See some “distortions” in long-term trend since the 1980’ inSims, E., Garin, J., Lester, R. (2017). IntermediateMacroeconomics (chapter 4).

I Functional income (and wealth) distribution seem not to bedetermined by stationary variables. Power, secular trends, andcrises play a role ⇒ discussion provoced by T. Piketty (2014).

I Long-term trend of “secular stagnation” (L. Summers, P.Krugman, R.J. Gordon, B. Bernanke):

I Higher income, aging population ⇒ higher accumulation ofsavings (“savings glut”) while investment opportunities arelimited

⇒ Decline of real interest rate; investment shifts to financialassets (“financialization”)

⇒ Lower real interest rate implies that investment projects withlower productivity are realized ⇒ slowdown of productivitydevelopment

p.18


Growth rates:

(by Alex1011 (Own work), CC BY-SA 3.0)

p.19


Development of interest rates of 10-year governmental bonds:

Source: Behrendt, S. (2017), Low Long-Term Interest Rates - An alternative View. Jena Economic Research

Papers No. 2017-001.)

p.20

1. The Empirical Picture of Growth1.2 Distribution and Convergence

Some illustrating empirical facts on distribution (base = 2014):

I The richest country is Quatar with more than $150,000 percapita, the poorest country is Central African Republic withless than $600 (= factor 253!)

I If Bangladesh ($2900) grows with its average post war growthrate of 1.4% then it approaches the 2014 level of per capitaincome of the USA ($52,000) in more 200 years.

p.21


Feenstra, R. et al. (2015), The Next Generation of the Penn World Table. American Economic Review 105(10),3150-3182, available for download at www.ggdc.net/pwt

p.22


Are less developed countries growing faster (“catching-up”)?

Measuring convergence:

I β-convergence: Negative relationship between per capitaincome y = Y /N and growth rate gy .

I σ-convergence: Decline of a dispersion measure (likestandard deviation of (logarithmic) per capita income, Ginicoefficient etc.)

p.23


−0.02

0.00

0.02

0.04

0.06

0.08

5,000 10,000 15,000y1960

g y

AfricaAsiaOECD

Feenstra, R. et al. (2015), The Next Generation of the Penn World Table. American Economic Review 105(10),3150-3182, available for download at www.ggdc.net/pwt

p.24


−0.02

0.00

0.02

0.04

0.06

0.08

5,000 10,000 15,000y1960

gy

North AmericaOECDSouth America

Feenstra, R. et al. (2015), The Next Generation of the Penn World Table. American Economic Review 105(10),3150-3182, available for download at www.ggdc.net/pwt p.25


Specific results: There is β-convergence within a group ofcountries which are “similar” regarding properties like high humancapital endowment, political institutions etc.

⇒ conditional β-convergence

⇒ “convergence clubs”: many developing and emergingcountries are catching up, some others do not

p.26

1. The Empirical Picture of Growth1.3 Environmental Impact

Economic Growth has impact on the environment:

I Using renewable and non-renewable natural resources.

I Using the absorptive capacity of the environment for pollution(e.g. greenhouse gas, water pollution)

I Erosion of soil, destruction of forests, extinction of species,destabilization of eco-systems

⇒ Problem of sustainability of economic development.

Possible limits of growth and possibilities of de-linking growth fromenvironmental use are discussed in chapter 4!

p.27


I Idea of bundling different forms harmful impact on theenvironment in one aggregated index:

Ecological Footprint

I Methodology: www.footprintnetwork.org; sketch inchapter 4

I An ecological foorprint (related to biocapacity) larger thanone means that human mankind needs more than one globe inorder to sustain the current economic activities.

I Country-specific or individual footprint calculation see website.

p.28

www.footprintnetwork.org


Source: www.footprintnetwork.org

p.29

1. The Empirical Picture of Growth

Final remark:

I The purpose of growth theory is to explain (per capita)growth. Growth models should be consistent with stylizedempirical facts.

I The aim is not to “justify” or to “propagate” growth!

I Policy implications can be derived if the growth path ispareto-inefficient. The normative approach is then to enhanceefficiency and wealth, not (necessarily) to accelerate growthrates.

I Policy implications can be dervived if the growth path is notsustainable from an ecological point of view (could be seen asa specific aspect of ineffeiciency?)

p.30

2. Preliminaries of Growth Theory2.1 The Basic Solow Model

Literature:

I Barro, R.E., Sala-i-Martin, X. (2004), Economic Growth. New York:McGraw-Hill (2nd ed.)

I Solow, R.M. (1956), A Contribution to the Theory of EconomicGrowth. Quarterly Journal of Economics 70, 65–94.

I Sims, E., Garin, J., Lester, R. (2017). IntermediateMacroeconomics. (Chapter 5) for a version of the Solow model indiscrete time!

The latter source discusses the model very extensively and is easyto read. However, we will stick to the continous-time versionbecause most theoretical models int he literature ar in arecontinous-time. For empirical questions the discrete-time versionsare more appropriate.

The mathematical stability analysis for discrete and continous timeare different!

p.31


Assumptions:

I Closed economy without government.

I Identical profit-maximizing firms are producing a homogenous goodY which can either be consumed or invested Y = C + I gross .

I Perfect competition on goods and factor markets, full-employment,flexible goods and factor prices.

I Labor supply L (and due to full employment also the demand forlabor N) is growing with the rate n:

gL =L

L= gN = n =

N

N

p.32


I Constant saving rate: S = Y − C = sY , s ∈ (0, 1)

I There is no investment function. Since we have goods marketequilibrium, it is always I = S . By definition we have

K = I = I gross − δK , δ ∈ (0, 1) depreciation rate

I There is a production technology Y = F (K ,N) with thefollowing properties:

I FK ,FN > 0,FKK ,FNN < 0,FKN > 0I Linear homogeneity: λY = F (λK , λN).

Then the output per capita can be expressed by

y =Y

N= F

(K

N, 1

)≡ f (k)

with k = K/N, fk > 0, fkk < 0.I Inada conditions: limk→0 f (k) = 0, limk→∞ f (k) =∞,

limk→0 fk(k) =∞, limk→∞ fk(k) = 0

p.33


Logic of the model:

Labor N ↑ Y = f (N,K ) ↑

S = s · Y ↑(saving)

I = ∆K ↑(investment)

S = I

p.34


Derivation of the dynamic equation:

From derivation of k with respect to time we have (quotient rule)

k =K

N− nk

From Y = C + I gross = C + I + δK = C + K + δK we have

K = Y − C − δKInserting K into k (with y = Y /N = f (k)) we have

k =Y − C − δK

N− nk

⇒ k = sf (k)− (n + δ)k (1)

For the per capita income we have

y = f (k(t))

y = fk k = fk(k) · (sf (k)− (n + δ)k) (2)p.35


k

y = f (k)

s · f (k)

(n + δ) · k

k∗k

k

p.36


Growth equilibrium (steady state):

I The steady state k∗ is defined as an equilibrium where allvariables grow with a constant rate (and all per capita valuesare constant).

I Steady state condition k = 0 leads to

sf (k∗) = (n + δ)k∗ (3)

I Since k = KN doesn’t change in time, we have gK = gN = n

and due to linear homogeneity we have also gY = n. Hencethe per capita output y = Y /N is constant in steady state (asit can also seen directly in (2)).

p.37


Income distribution in the model:

I Recall, that we have perfect competition so that w = FN andr = FK are the real returns of the factors.

⇒ In per capita terms we have r = fk and the capital income perworker is k · fk .

I For a linear homogenous production function the Eulerequation holds true: Y = wN + rK .

⇒ In per capita terms we have thus y = w + k · fk orw = f (k)− k · fk .

I Obviously the income distribution is constant in thesteady-state.

p.38


Example: Cobb-Douglas production function

Y = Kα · N1−α

In per capita terms:

Y

N= y =

Kα · N1−α

N

= Kα · N−α =Kα

Nα= kα

Therefore the dynamic equation is

k = skα − (n + δ)k

and the steady-state (k = 0) is then given at

k∗ =

(s

n + δ

) 11−α

⇒ y∗ =

(s

n + δ

) α1−α

p.39


Existence and uniqueness of the equilibrium:

I The linear function (n + δ)k is starting in the origin and has apositive finite slope (n + δ).

I Due to the Inada condition the saving function sf (k) alsostarts in the origin but has an infinite positive slope near tothe origin. With k →∞ the slope of the saving functiondeclines to zero. Both functions are monotonously increasing.

⇒ Hence there must exist a unique intersection point with thelinear function (n + δ)k.

p.40


Stability of the equilibrium:

The equilibrium is stable if dk(k∗)/dk < 0:

dk(k∗)

dk= sfk − (n + δ)

Inserting the steady state condition (3)

= sfk −sf (k)

k< 0

⇒ fk <f (k)

k

This is ensured by the concavity of the function(see assumption fk > 0, fkk < 0).

p.41


Convergence:

I Recall that (see eq. (2)):

y =df

dkk

gy =y

y=

df

dk

k

y· kk

=

(df

dk

k

y

)︸︷︷︸

α

· kk

with α as the income share of capital (constant in theCobb-Douglas case). So the output growth dynamics is proportionalto the capital accumulation dynamics.

I For the latter we have (see graphic)

k

k= s · f (k)

k− (n + δ)

I Hence we have higher growth rates the more the economy is apartfrom the steady state, and it is slowing down when approaching thesteady state.

p.42


k

kk = sf (k)

k − (n + δ)

(n + δ)

s · f (k)/k

k∗

kk> 0

kk< 0

p.43


Some consequences:

I The steady-state value for k∗ and therefore for y and c aredetermined by the behavioral parameters s (saving) and n(reproduction), and the technological parameters δ(depreciation rate) and α (capital production elasticity).

I The growth rate of Y ,N,K is given by the exogenously givenpopulation growth rate.

I Countries with the same technological and behavioralcharacteristics should converge towards the same steady-statevalues. Or the other way round: cross-cuntry differences haveto be explained by differences of these characteristics.

p.44


Compatible with stylized facts?

I Growing per capita income y = Y /N cannot be explainedwithout technical progress!

I Growing capital intensity k = K/N cannot be explained.

I Constant ratio K/Y is compatible with the model.

I Constant income distribution is compatible with the model.

I In a transient phase (before approaching the steady state) weshould observe growing per capita income, growing K/N, andconditional β-convergence, but a changing incomedistribution.

p.45


Optimal growth in the basic Solow model:

I Saving rate s is assumed to be exogenously given, i.e. thehouseholds do not maximize their utility (problem of missing“microfoundation”).

I However, the saving rate determines the equilibrium value k∗

and therefore y = f (k∗).

I Idea: households maximize their utility from per capitaconsumption in the steady state. We can avoid introducingutility functions because utility will be maximized if and only ifthe per capita consumption in the steady state is maximized.

p.46


k

y = f (k)

(n + δ) · kf ′ = (n + δ)

kopt

C/Y

S/Y

sopt f (k)

p.47


From the steady state condition (k = k∗(s)) we have

sf (k) = (n + δ)k (4)

⇒ f (k)− c = (n + δ)k

maxs

c = f (k)− (n + δ)k

⇒ dc

ds=

dk

ds(fk − (n + δ)) = 0 (FOC)

Bracket term must be zero. Using condition (4) yields

fk =sf (k)

k

⇒ s =fkk

f (k)(= α in case of Cobb-Douglas) (5)

which is known as the “golden rule” of optimal growth.

p.48


Excourse: Solow model in discrete time(Sims/Garin/Lester 2017)

Production function Yt = F (Kt ,Nt)Population growth Nt = (1 + n)Nt−1

Consumption Ct = (1− s)Yt

Savings = Investment It = sYt

Capital accumulation Kt+1 = It + (1− δ)Kt

In per capita terms:

Yt

Nt= yt = F (Kt/Nt , 1) = f (kt)

and thus

Kt+1

Nt= sf (kt) + (1− δ)kt

Kt+1

Nt+1

Nt+1

Nt= sf (kt) + (1− δ)kt

⇒ kt+1 =1

1 + n(sf (kt) + (1− δ)kt)

p.49


Excourse: Cambrdige-Cambridge Controversy

Piero Sraffa, Joan Robinson, and Luigi Pasinetti (among others)pointed to some severe methodological problems of neoclassicalcapital and production theory, based on the aggregation problemand the fallacy of composition.

Pasinetti, L., Scazzieri, R. (1987), Capital Theory: Paradoxes, The NewPalgrave: A Dictionary of Economics, London and New York: Macmillan andStockton, pp. 363-68.

Stiglitz, J. (1974), The Cambridge-Cambridge Controversy in the Theory of

Capital – A View from New Haven: A Review Article. Journal of Political

Economy 82(4), pp. 893-903.

p.50


Excourse: Cambrdige-Cambridge Controversy (cont.)

I Is it feasible to represent the production apparatus of aneconomy by just one aggregated production function?

I If multiple technologies exist, many of them with limitationalrather than substitutional relationship of input factors, thechoice (mix) of technologies might depend on factor pricesbut there is no monotonous relationship between interest rateand technology mix (“re-switching problem”).

p.51


Excourse: Cambrdige-Cambridge Controversy (cont.)

I Moreover, if a set of different limitational inidividualproduction functions “add up” to an aggregated substitutionalproduction function, the microeconomic calculus (payingcapital according to its marginal output) for the aggregatedoes not hold for the individual production processes!

I Labor can be measured e.g. by hours. Physical capital goodsare evaluated by their capital price in order to get an“aggregated capital stock”. The “real return” of capital andhence the income distribution thus depends on capital goodprices although it should be a pure technical issue⇒ paradoxon: in order to determine the profit rate of capitalwe need to evaluate the price of capital which depends on theinterest rate.

p.52

2. Preliminaries of Growth Theory2.2 Basic Model with Exogenous Technological Change

I In a widely used form, technical progress enters the productionfunction by enhancing the total factor productivity A:

Y = A · F (K ,N)

I In the “old” growth theory, the sources and economicmechanisms driving the technical progress (TP) are not partof the model.

I TP is modeled as an exogenously determined process

A(t) = eγt

I The total factor productivity results from TP affecting labor,capital, or both. Depending on the type of TP we will write

Y = F (AK ,AN) or Y = F (K ,AN) or Y = F (AK ,N)

p.53


TP – Hicks concept:

I TP affects the productivity of both, capital and labor:Y = F (AK ,AN). The productivity growth has the sameimpact on the output like an augmentation of both inputfactors.

I As the growth of (marginal) productivity affects both factorsuniformly, the TP does not affect the relation between factorprices (wages, interest rate)!

I TP is called Hicks-neutral, if the income distributionV = rK/wN remains unchanged. Since TP does not changethe ratio r/w this implies that capital intensity K/N does alsonot change. Hicks-neutral TP is then “factor augmenting”.

I TP is called labor augmenting if K/N and V increase, and itis called capital-augmenting if K/N and V decrease.

p.54


N

K

tanβ = w/r

Y = x

tanα = K/N

Y TP = x

technical progress shifts isoquant

V = tanαtan β = rK

wN

p.55


Growth rates in case of Hick-neutral TP and a linearhomogenous Cobb-Douglas production function:

TP is measured by an efficiency factor A(t) = eγt which ismultiplied with capital and labor

Y = F (AK ,AN) = (AK )α(AN)1−α

= AKαN1−α = eγtKαN1−α

lnY = γt + α lnK + (1− α) lnN

gY = γ + αgK + (1− α)gN

Since Hicks-neutrality implies gK = gN it follows

gY = γ + gN

p.56


Compatibility with stylized facts?

I Per capita income grows with the positive rategy = gY − gN = γ.

I Income distrbution is constant.

I The constant capital intensity K/N is inconsistent withstylized facts.

I With gY > gN = gK the capital coefficient K/Y declines.Inconsistent with stylized facts.

An increasing capital intensity K/N would require Hicks laboraugmenting TP. But then we would have a trend in incomedistribution which is inconsistent with stylized facts.

p.57


TP – Harrod concept:

I TP affects the productivity of labor (“labor aumenting”):Y = F (K ,AN). The marginal productivity of labor increasesand hence the factor prices ratio r/w decreases due to TP.

I Requires to measure labor in efficiency units.

I TP is called Harrod-neutral if the income distributionV = rK/wN remains unchanged. Since r/w decreases, K/Nmust increase with the same rate. Furthermore,Harrod-neutrality implies a constant capital coefficient K/Y .

p.58


N

K

tanβ = w/r

Y = x

tanα = K/N

tanβ′

Y TP = x

technical progress shifts isoquant

tanα′

V = tanαtan β = tanα′

tan β′ = rw ·

KN

r/w declining with rate x

K/N increasing with rate x

p.59


Growth rates in case of Harrod-neutral TP and a linearhomogenous Cobb-Douglas production function:

TP is measured by an efficiency factor A(t) = eγt which ismultiplied with labor

Y = F (K ,AN) = Kα(AN)1−α

= A1−αKαN1−α = e(1−α)γtKαN1−α

lnY = (1− α)γt + α lnK + (1− α) lnN

gY = (1− α)γ + αgK + (1− α)gN

Since Harrod-neutrality implies gK = gY it follows

(1− α)gY = (1− α)γ + (1− α)gN

gY = γ + gN

p.60


The Solow model with Harrod-neutral technical progress:

Labor is measured in efficiency units. Thus the capital intensity isdefined as

k =K

AN⇒ ˙k =

K

AN− γk − nk

The dynamic equation is thus

˙k =K

AN− (γ + n)k

=Y − C − δK

AN− (γ + n)k

˙k = sy − (δ + γ + n)k (6)

with y = Y /AN.

p.61


Steday state in Cobb-Douglas case:

k∗ =

(s

δ + γ + n

) 11−α

⇒ y∗ =

(s

δ + γ + n

) α1−α

or

y∗ =Y

N= A

(s

δ + γ + n

) α1−α

Taking this in logs...

lnY

N= lnA +

α

1− α[ln s − ln(δ + γ + n)]

... allows for an empirical investigation (Mankiw, Romer, Weil(1992)):

lnYi

Ni= a + b ln si + c ln(0.05 + ni ) + εi

p.62


Compatibility with stylized facts?

I Per capita income grows with the positive rategy = gY − gN = γ.

I Income distrbution is constant.

I Increasing capital intensity K/N since gK = gY > gN .

I Constant capital coefficient K/Y .

⇒ many stylized facts are compatible with Harrod-neutral TP.

p.63


Remarks:

I In practice it is difficult to discriminate which part of outputgrowth is due to capital or due to labor augmenting TP.

I If we interpret growing output as a result of inreased laborproductivity and therefore increase real wages and hence w/r(e.g. as a result of “productivity-oriented wage policy”) thenwe treat TP as if it is Harrod-neutral.

I It is unsatisfactory that the TP itself is not explained, i.e. TPis not generated by economic activity which requires someresource input.

p.64

2. Preliminaries of Growth Theory2.3 Basic Model with Human Capital (Mankiw-Romer-Weil)

Literature:I Mankiw, N.G., Romer, D., Weil, D.N. (1992), A Contribution to the

Empirics of Economic Growth. Quarterly Journal of Economics 107,407–437.

Human capital:

I Stock of skills, education, competency, and other productivityenhancing characteristics embedded in labor ⇒ efficiencyunits of labor embedded in raw labor hours

I Individuals invest in their skills, competencies, and earningcapacities as firms invest in their physical capital

Augmented production function:

Y = F (K ,H,AN) (7)

where all previous assumptions also apply to H:

FH > 0, FHH < 0, limH→0 FH =∞, limH→∞ FH = 0p.65


Assumptions:I Households save a fraction sk of their income to invest in K

and a fraction sh to invest in H.I Depreciation rates δk and δh. However, we will assumeδk = δh = δ.

I Constant population growth and constant rate ofHarrod-neutral technological progress A(t).

I Effective human and physical capital ratios as:

k(t) =K

ANand h(t) =

H

AN

Thus,

y =Y

AN= F

(K

AN,H

AN, 1

)= f

(k , h)

p.66


Laws of motion: similarly derived as in the basic model

˙k = sk f (k , h)− (δk + γ + n)k (8)

˙h = shf (k , h)− (δh + γ + n)h (9)

Steady state equilibrium:

sk f (k∗, h∗)− (δk + γ + n)k∗ = 0 (10)

shf (k∗, h∗)− (δh + γ + n)h∗ = 0 (11)

p.67


Cobb-Douglas case:

Y = KαHβ(AN)1−α−β ⇒ y = kαhβ

Using y in (10) and (11) we obtain:

˙k = sk kαhβ − (δ + γ + n)k

˙h = shkαhβ − (δ + γ + n)h

with the steady state solution

k∗ =

(s1−βk sβh

δ + γ + n

) 11−α−β

, h∗ =

(sαk s

1−αh

δ + γ + n

) 11−α−β

with the equilibrium output per effective worker:

y∗ =

(sαk s

βh

(n + γ + δ)α+β

) 11−α−β

p.68


Graphical reresentation:

I Phase diagram: (k , h)-space, each point (vector) is a certainstate of the model. The dynamic equations determine howthis state evolves in time. For a marginal time step this couldbe represented by a vector field in the (k , h)-space.

I Trajectory: Time path of (k(t), h(t)) starting from anyinitial value.

I Isocline: The implicit function of all (k , h)-combinations

where ˙k = 0 or ˙h = 0. The intersection point of both isoclinesis the steady state.

I Stability: Compute derivatives ∂ ˙k/∂h and ∂ ˙h/∂k in order todetermine what happens with the variable growth apart fromthe isocline (see red arrows in graphic). Indicates the directionof the vector field.

p.69


h

k

˙k = 0

˙h = 0h∗

k∗

p.70


I Thus, we can conclude that the steady state (fixpoint) of theMRW model is stable ( a detailed stability analysis will bediscussed latern on).

I Both, greater sk and greater sh will translate into highernormalized output per capita y∗.

I Cross-country: those with higher propensity to invest in Kand H will be relatively richer, assuming the same rate oftechnological progress γ

I Mankiw, Romer, and Weil estimated the model empirically(see section 2.5).

p.71

2. Preliminaries of Growth Theory2.4 Basic Model with Intertemporal Optimization (Cass-Koopmans-Ramsey)

Motivation:

I Major shortcoming of the basic Solow model: nomicrofoundation of household’s behavior.

⇒ Households maximize net present value of utility fromconsumption.

I This requires a lot of harsh assumptions (Walrasian prices,rational expectations, existence of future markets etc.)

I Resulting growth path is an equilibrium growth path wherealways all optimal plans are consistent ex ante.

p.72


Intertemporal Optimization in continous time(very brief intoduction)

I The household has a control variable c(t). The decisionabout consumption implies a decision about savings an hencecapital accumulation.

I The state of the economy is represented by a state variablek(t).

I In each time the present value of the utility e−ρtu(c(t))with ρ > 0 as the time preference.

p.73


I The agent’s goal in t = 0 is to maximize the present value ofutility flow : ∫ ∞

0e−ρtu(c(t))dt

which requires that utility is additive-separable in time.

I Maximization subject to the constraint that the statevariable develops according to a differential equation (law ofmotion, transition equation):

k = f (k(t))− c(t)− δk(t)

I For the state variable we have to define the initial value:k(0) = k0 > 0.

p.74


The complete problem:

maxc(t)

∫ ∞0

e−ρtu(c(t))

subject to k(t) = g(k(t), c(t))

k(0) = k0 > 0 given

For solving this problem we build the Hamiltonian function:

H(c(t), k(t), µ(t)) = e−ρtu(c(t)) + µ(t) · g(c(t), k(t))

where µ(t) is a Lagrangian multiplier for each t.

p.75


Economic interpretation of the multiplier:

I In each t the agent consumes c(t) and holds k(t).I Both affects the utility:

I Choice of consumption enters directly the utility function.I Choice of consumption affects the savings and hence the

development of k(t) according to the law of motion. Thisaffects the future output/income and hence futureconsumption and therefore the present value of utility.

⇒ The multiplier µ(t) is the shadow price (or opportunity cost)of a unit of capital in t expressed in units of utility at time t.

p.76


Solution of the problem:

Let c∗(t) a solution (time path) of the optimization problem, andk∗(t) is the associated time path of the state variable.

Then there exists a function µ∗(t) (so-called costate variable) sothat for all t following statements hold true: [next slide]

p.77


Solution of the problem:

a) First order condition (FOC):

∂H

∂c(t)= 0

b) Canonical equations (CE):

∂H

∂µ(t)= g(c(t), k(t)) = k(t)

− ∂H

∂k(t)= µ(t)

c) Transversality condition (TVC):

limt→∞

µ(t)k(t) = 0

p.78


Interpretation of the TVC:

This means that if the final state variable k(T ) has a positivevalue, then its shadow price must be zero. Otherwise the agentwould leave a positive capital stock unused which could contributepositively to the present utility. Hence, the TC is an dynamicefficiency condition!

p.79


How to proceed:

I From the CE we obtain differential equations for state variablek and the costate variable µ.

I Since the FOC relates c to µ it is possible to eliminate µ andto derive a differential equation c instead (Euler equation).

I Both differential equations c and k have steady state(c∗, k∗) where k = c = 0.

I Depending on the initial conditions, it is usually not clearwhether the system converges to the steady state. Since theinitial conditions are chosen by the optimizing agents, theywill choose c(0) (for a given k(0)) which is consistent with theFOC, CE and the TVC. This ensures that the system will beon a saddle-path to the steady state (saddle-path stability).

p.80


Literature:I Barro, R.J., Sala-i-Martin, X. (2004), Economic Growth. New York:

McGraw-Hill (2nd ed.)

I Cass, D. (1965), Optimum Growth in an Aggregate Model of CapitalAccumulation. Review of Economic Studies 32 (3), 233–240.

I Koopmans, T.C. (1965), On the Concept of Optimal Growth. In: TheEconometric Approach to Development Planning, 225–287,North–Holland, Amsterdam.

The basic idea is to provide a microfoundadtion for the neoclassicalSolow model by assuming an intertemporal maximizing household.

All other assumptions of the basic Solow model still hold true.

p.81


a) The household:

I The household has a time-separable utility function u(c) withuc > 0, ucc < 0. We omit the time index to simplify the notation.

maxc

U(0) =

∫ ∞0

u(c)e−ρtentdt =

∫ ∞0

u(c)e−(ρ−n)tdt

subject to k = w + rk︸︷︷︸y

−c − (n + δ)k

k(0) > 0

where w is the wage, r the interest rate. Therefore w + rk is theper capita income from labor and capital. Subtracting consumption,w − rk − c is the (gross) saving per capita which increases thecapital stock. However, depreciation δ and the growth of thepopulation diminishes the capital per capita.

I In the objective function, ρ is the time-preference rate. Therepresentative household has to take into account that the“members” of the household grow with the rate n. We mustassume ρ > n, otherwise the integral diverges. p.82


Solution:

The Hamiltonian is

H(c , k , µ) = u(c)e−(ρ−n)t + µ · (w + rk − (n + δ)k − c)

The conditions for an optimum (FOC and CE) are

∂H

∂c= uc(c)e−(ρ−n)t − µ = 0 (12)

−∂H∂k

= −(r − n − δ)µ = µ (13)

∂H

∂µ= w + rk − (n + δ)k − c = k (14)

p.83


Differentiating (12) with respect to time

ucc(c)ce−(ρ−n)t − (ρ− n)uc(c)e−(ρ−n)t = µ

Substituting µ (r.h.s.) by condition (13):

ucc(c)ce−(ρ−n)t − (ρ− n)uc(c)e−(ρ−n)t = −(r − n − δ)µ

Substituting µ by condition (12) finally eliminates µ:

ucc(c)ce−(ρ−n)t − (ρ− n)uc(c)e−(ρ−n)t = −(r − n − δ)uc(c)e−(ρ−n)t

p.84


(Continued)

Dividing by e−(ρ−n)t and rearranging leads to

ucc(c)c = uc(c)(r − (ρ+ δ))

Dividing by ucc(c)c yields the Euler equation:

gc =c

c= − uc(c)

ucc(c) · c︸︷︷︸σ

(r − ρ− δ)

The expression −uc/(ucc · c) = σ is the intertemporal elasticity ofsubstitution of the utility function.

p.85


I In many growth models it is assumed that the utilitiy functionis isoelastic (constant σ). Examples:

u(c) =c1−θ − 1

1− θ, θ > 0, σ = 1/θ

u(c) = log(c) σ = 1

I The Euler equation implies that we have a positive growthrate of per capita consumption as long as the net return tocapital r − δ exceeds the time-preference rate ρ. Since thereare decreasing returns to capital and hence a decreasingr = fk , the growth rate will necessarily decrease until zero(steady state).

p.86


b) The Firm:

I The representative firm is a price taker and maximizes itsperiod (real) profit:

maxK ,N

π = N · [f (k)− rk − w ]

I From the first order conditions we have

r = fk(k)

w = f (k)− fk(k)k

In the optimum there are zero profits and the factors are paidby their marginal product.

p.87


c) Market equilibrium:In equilibrium all produced goods are demanded either asconsumption or as investment goods:

y = f (k) = k + (n + δ)k︸︷︷︸gross investment

+c

Summing up:

Solow model with intertemporal maximization (CKR):

c = − uc(c)

ucc(c)(fk(k)− (ρ+ δ)) (15)

k = f (k)− (n + δ)k − c (16)

p.88


The CKR model has three fixpoints:

(a) c∗ = k∗ = 0. This is the trivial solution will not be discussed

(b) c∗ = 0, k∗ = k with f (k) = (n + δ)k . In this case the outputis used only to maintain the capital stock, there is noconsumption. This violates the TVC.

(c) c∗, k∗ as the solution of c = k = 0.

Equalizing (15) and (16) with zero yields the steady state:

fk(k∗) = ρ+ δ (17)

c∗ = f (k∗)− (n + δ)k∗ (18)

p.89


c

kk∗

c = 0

k = 0

c∗

saddle path

p.90


I The isoclines separate the regions with k 6= 0 and c 6= 0.

I We have

∂c

∂k= − uc(c)

ucc(c)fkk < 0

∂k

∂c= −1 < 0

Hence, we obtain the arrow directions of the vector field forthe development of an arbitrary trajectory.

I We see the trivial solution c∗ = 0, k∗ = 0 as well as theTVC-violating solution c∗, k∗ = k in the diagram.

I Since the isoclines have a unique intersection point (steadystate) which is a “saddle point”.

p.91


Stability of the steady state:

The standard analysis of stability is based on linear systems.Therefore, we linearize the nonlinear Cass-Koopmans-Ramseymodel around the steady state.

k = k(k , c)

≈ k(k∗, c∗) +∂k

∂k(k − k∗) +

∂k

∂c(c − c∗)

(analogoulsy for c = c(k, c))

The Taylor approximation of the original system at (c∗, k∗) is then:[kc

]=

[∂k/∂k ∂k/∂c∂c/∂k ∂c/∂c

]·[k − k∗

c − c∗

]

p.92


From (16) and (17) we have

∂k

∂k= fk(k∗)− (n + δ) = (ρ+ δ)− (n + δ) = ρ− n > 0

∂k

∂c= −1 < 0

∂c

∂k= − uc(c∗)

ucc(c∗)· fkk(k∗) < 0

∂c

∂c=

[ucc(c∗)]2 − uccc(c∗) · uc(c)

[ucc(c∗)]2· [fk(k∗)− (ρ+ δ)]︸︷︷︸

=0, see (17)

= 0

Thus we have[kc

]=

[ρ− n −1

− uc (c∗)ucc (c∗) · fkk(k∗) 0

]︸︷︷︸

J

·[k − k∗

c − c∗

]

p.93


The determinant of the Jacobian matrix J is

det J = − uc(c∗)

ucc(c∗)· fkk(k∗) < 0

The characteristic polynom is

λ2 − (ρ− n)λ+ det J

with the roots (eigenvalues)

λ1,2 =ρ− n

2± 1

2

√(ρ− n)2 − 4 det J

Since the determinant det J is negative the square-root is takenfrom a positive term (real valued ⇒ non-cyclical behavior) and wehave two different real-valued roots.

p.94


Cases:

I λ1, λ2 < 0: steady state globally stable

I λ1, λ2 > 0: steady state globally unstable

I λ1 and λ2 have different signs: saddle point equilibrium

The last case can be proven to hold true:

λ1λ2 = det J < 0

With the eigenvalues it is now possible to provide a solutionk(t), c(t) for the linearized model (will not be treated in thiscourse).

p.95


Consequence of saddle-point stability:

I In the intertemporal maximization problem we have an initial valuek(0) > 0. To determine a starting point we need a value c(0). Asthe vector field shows, an in-appropriate choice of c(0) will let thetrajectory diverge from equilibrium! So for every given k(0) thereexists one specific c(0)∗ which leads the trajectory along the saddlepath to the steady state.

I The transitory dynamic in case of c(0) 6= c(0)∗ are depicted in thefollowing graphic by the black lines (example). The transitorydynamic for c(0) = c(0)∗ (saddle path) is depicted by the green line.

I A choice of the initial c(0) 6= c(0)∗ either contradicts the Eulerequation or it contradicts the TVC. By rationality assumption, therepresentative agent will hence properly choose c(0)∗ and thereforethe saddle-point stability of the steady state is ensured.

p.96


stable saddle−pathc=0

k=0

k

c

(with f (k) = k0.6, u(c) = log(c), (n + δ) = (ρ+ δ) = 0.2)

p.97


Further properties of the Cass-Koopmans-Ramsey model(details see Barro/Sala-i-Martin, chapter 2)

I Pareto-Optimality: Since markets are perfect and there are noexternalities, the intertemporal decisions and hence the growth path ofthe model is pareto-optimal. Due to the time preference rate the savingratio in the steady state is below the “golden rule” in the standard Solowmodel.

I Transitory dynamics: The saddle point stability of the steady stateimplies a certain function c∗(k), i.e. for each k the function ensures thatthe economy is on the saddle path to the steady state. It describes thetransitory dynamnics on the saddle path.

I Convergence: Compared to the Solow model the saving rate is nowendogenously determined but we have two additional strutcturalparameters: intertemporal elasticity of substitution σ and time preferencerate ρ. These parameters shape the rate of convergence but the Solowresults for β-convergence also hold true for the CKR- model.

I Policy implications: Policy may change preference parameters (taxinghousehold income and governmental expenditures = changing the savingratio). This affects only the per capita income level, not the growth rate!

p.98

2. Preliminaries of Growth Theory2.5 Empirical Evidence

Literature:

I Acemoglu, D. (2009), Introduction to Modern Economic Growth.Princeton University Press, Chapter 3.

I Barro, R.J., Sala-i-Martin, X. (2004), Economic Growth. New York:McGraw-Hill (2nd ed.), Chapter 12.

I Mankiw, N.G., Romer, D., Weil, D.N. (1992), A Contribution to theEmpirics of Economic Growth. Quarterly Journal of Economics 107,407–437.

Agenda:

I Idea of “Growth accounting”

I Regression analysis of cross-country differences in output andgrowth

I Empirical investigation of the augmented Solow model

p.99


Growth Accounting:

I Aggregated production function:

Y (t) = A(t) · F (K (t),N(t))

I Differentiating with respect to time and expressing the equation ingrowth rates:

Y

Y=

A

Y

A

A+

FKK

Y

K

K+

FNN

Y

N

NgY = x + αKgK + αNn

where: gY , gK , n are the growth rates, αK , αN are the factor incomeshares, and x is the contribution of TP to growth.

I Problem: TFP growth x not observable ⇒ we have to “estimate” it.

I Using estimated x in a regression model.

p.100


Growth rate Contribution Contribution TFP GrowthCountry of GDP from Capital from Labor Rate

Canada 0.0369 0.0186 0.0123 0.0057(51%) (33%) (16%)

France 0.0358 0.0180 0.0033 0.0130(53%) (10%) (38%)

Germany 0.0312 0.0177 0.0014 0.0132(56%) (4%) (42%)

Italy 0.0357 0.0182 0.0035 0.0153(51%) (9%) (42%)

Japan 0.0566 0.0178 0.0125 0.0265(31%) (22%) (47%)

UK 0.0221 0.0124 0.0017 0.0080(56%) (8%) (36%)

US 0.0381 0.0117 0.0127 0.0076(37%) (40%) (24%)

Sample Period: 1960-95

Source: Barro/Sala-i-Martin (2004) p.101


Estimation issues:

I TFP growth x : measure of ignorance, unexplained residual.I If we underestimate contribution of labor and capital we will

arrive at inflated estimates of xI Effective labor hours ⇒ how to account for changes in human

capital of workers?I In the theoretical model: Y can be consumed or invested ⇐⇒

in practice: capital comprises equipment and infrastructure(buildings, . . . ) wich cannot be consumed.

I Measuring the capital stock is based on heuristics.I Change in relative prices of capital goods over time.

p.102


Solow Model with exogenous TP:

y(t) = A(t)f (k(t)) (19)

k

k=

sf (k(t))

k(t)− (δ + γ + n) (20)

Recap: As long as the economy is not in the steady state, thereare two sources of growth in output per capita

I rate of technological progress (γ)

I Convergence towards the steady state (the further below itssteady state value a country is, the faster it will grow)

We could test this idea by using the following regression equation:

gt = b0 + b1 log yt−1 + εi ,t , gt is growth rate of y (21)

p.103


gt = b0 + b1 log yt−1 + εi ,t

I Using a sample of OECD countries for the post WWII period,b1 would be indeed negative ⇒ convergence.

I But, looking at the whole world, there is no suchunconditional convergence.

Predictions of the Solow model:

I There should be a tendency for the income gap between twocountries to decline if they share the same structuralparameters (technology, investment behavior, institutionalcharacteristics, . . . )

I Each country i has an idiosyncratic steady state y∗i toconverge to.

p.104


Barro and Sala-i-Martin (1992, 2004): conditional convergence

gi ,t = Xi ,tβ + b1 log yi ,t−1 + εi ,t (22)

I Vector Xi ,t contains control variables which are country-specific. Examples: schooling rate, fertility rate, investmentrate, terms of trade, rule of law and democracy . . .

I Conditional convergence (b1 < 0) if control variables areallowed to vary across countries.

I Empirical estimations of (22) indeed show a negative b1, buta quite small point estimate.

I However, equation (22) could be also used to identify thedeterminants of economic growth (which variables in Xi ,t aresignificant).

p.105


Potential pitfall of these kind of regressions:

I Endogeneity bias: most of the variables in Xi ,t , as well aslog yi ,t−1 are econometrically endogenous ⇒ they are jointlydetermined with gi ,t . Thus, the effects represented by β maynot be interpreted as causal effects.

I Measurement errors may lead to a downward bias in theestimate of b1 .

I Closed economy : The idea behind equation (22) was derivedassuming a closed economy. But countries trade goods,exchange ideas, and borrow and lend money on internationalfinancial markets ⇒ there is no reason why I = S has to hold.

p.106


Source: Barro/Sala-i-Martin (2004)p.107


Source: Barro/Sala-i-Martin (2004)

p.108



p.109


Source: Barro/Sala-i-Martin (2004)p.110



p.111


Empirical Investigation of the MRW (1992) model:

Aggregated production function of country j (Cobb-Douglas):

Yj = Kαj H

βj (AjNj)

1−α−β ⇒ yj = kjαhjβ

(23)

Assumptions:

I Countries differ in sh,j sk,j , and nj (eventually also in γj).

I Focus is on cross-country differences in income, neglectingconvergence dynamics ⇒ we assume that all countries are inthe neighborhood of their steady state.

We have seen already that in the steady state the output pereffective worker is

y∗j =Yj

Nj= Aj

(sαkjs

βh,j

(nj + γ + δ)α+β

) 11−α−β

(24)

p.112


Estimation equation:

I Assuming common technology advances: Aj(t) = Ajexp(γt)⇒ countries differ according to their technology level, butthey share the same common technology growth rate.

I Taking logs of (24) we get

ln y∗j (t) = lnAj +α

1− α− βln(sk,j) +

β

1− α− βln(sh,j)

− α + β

1− α− βln(nj + γ + δ)

I Assuming: δ + γ = 0.05 the regression equation is

lnYj

Nj= a + b ln sk,j + c ln sh,j + d ln(nj + 0.05) + εj

p.113


MRW Updated data

1985 1985 2000

log(sk) 0.69 0.65 0.96(0.13) (0.11) (0.13)

log(n + γ + δ) -1.73 -1.02 -1.06(0.41) (0.45) (0.33)

log(sh) 0.66 0.47 0.70(0.7) (0.07) (0.13)

Adjusted R2 0.78 0.65 0.6Implied α 0.30 0.31 0.36Implied β 0.28 0.22 0.26Number of observations 98 98 107

Source: Acemoglu (2009)

p.114


Results (MRW original):

I Large R2

I As expected: b > 0, c > 0, d < 0 and significant.

I Implied factor shares α ≈ 0.3, β ≈ 0.28 are “realistic”.

I Theoretically, it should be b + c = −d which could not berejected on 5% level.

Also conditional convergence seems to be explained quite well.

p.115


Major pitfalls of the MRW regression approach:I Critical assumption: technology differences between countries

are unrelated to other variablesI omitted variables bias - technology differences are also the

outcome of investment decisions; same factors drive sh and skdecision

I reverse causality - in countries with high Aj firms andindividuals find it more beneficial to increase their stock ofhuman and physical capital stock

I Consequence: independent variables are correlated with εj ⇒upwardly biased OLS estimates of α and β, and R2 as well

I Estimate for the coefficient on the investment rate in humancapital sh,j appears too large relative to microeconometricevidence

p.116

3. Innovation and Endogeneous Growth3.1 Overview: Sources of Growth

I In the basic Solow model we have no steady state growthneither of per capita income nor of labor productivity.

I Technically spoken, the absence of steady state of per capitagrowth is a result from decreasing returns of capital. In atransitory phase we have an incentive to accumulate capitalbut with decreasing r = fk(k) (Inada conditions) the percapita growth rates diminish and fall to zero in the steadystate (see Euler equation).

I Extending these models with Harrod-neutral technologicalprogress lacks an explanation of such a progress. Progresstakes place without any economic activities and withoutspending ressources to create this progress.

p.117


Looking for models, preferably ...

I with non-diminishing returns of capital

I with endogenous explanation for technological progress

I with policy advice

p.118


Some sources of endogenous growth:

(A) (Technical) Knowledge:

I May be embodied in humans (→ human capital) ordisembodied (“blue prints”, knowledge stock)

I In case of disembodied knowledge: non-rival in use,(non-) disclosure regulated by

I intellectual property rights (patents)I high firm specifityI limited absorbability

p.119


Continued:

I To the extent where we have disclosure and free access toknowledge there are positive spillover effects (externalities)⇒ externalities imply that price system is incomplete andallocation is pareto-inefficient

I To the extent of non-disclosure there is a private return fromproducing knowledge and hence an incentive for R&D in amonopoly which is also pareto-inefficient.

I Increasing knowledge regarding: (a) new products (varietyapproaches), (b) higher product quality (quality approaches),(c) higher production efficiency.

p.120


Continued:

(B) Human Capital:

I skills and specific knowledge of human beings

I rival in use, non-disclosure ⇒ private good with a positivereturn ⇒ incentive to invest into HC.

I Accumulation of HC byI learning by doingI by schooling (investment)

I Not all effects of HC may be appropriatable, positiveexternalities possible.

I One-sector versus two-sector models.

p.121


What we will do in this chapter:

I Embodied knowledge: the Human Capital model byUzawa/Lucas

I Disembodied knowledge:I with disclosure (knowledge externalities): models by

Rebelo/King and ArrowI with non-disclosure (R&D based):

I increasing product variety: model by RomerI increasing product quality: models by Grossman/Helpman and

Aghion/Howitt

(the former has a deterministic innovation process, the latterhas a stochastic one where the quality leader erodes themonopoly power of the precedent supplier ⇒ “Schumpeterianmodels”)

I Interaction between knowledge diffusion and human capital:model by Nelson/Phelps (with a link to “North-Southmodels”, Brezis/Krugman/Tsiddon)

p.122

3. Innovation and Endogeneous Growth3.2 A Model with Knowledge Spillovers

Literature:

I King, R.G., Rebelo, S. (1990), Public Policy and EconomicGrowth: Developing Neoclassical Implications. Journal ofPolitical Economy 98 (5), S126–S150.

I Barro/Sala-i-Martin (chapter 4.1)

⇒ so-called AK model!

In all models of endogenous growth we assume n = 0, i.e. there isno population growth!

p.123


a) Households maximize:

maxc

U(0) =

∫ ∞0

u(c(t))e−ρtdt (25)

conditional to k = f (k)− δk − c

k(0) > 0

and furthermore the TVC holds true:

limt→∞

[µ(t)k(t)] = 0

The solution leads to the Euler equation

gc = σ(r(t)− (ρ+ δ))

where σ is assumed to be constant.p.124


b) Firms produce the output only with capital (constant laborforce is neglected here). Capital includes physical as well as humancapital (“broad measure of capital”, Romer (1989))

y = Ak, A > 0

Hence we have r = fk(k) = A for all t (non-diminishing retuirns ofcapital).

The Euler equation thus reads

gc = σ(A− ρ− δ)

and gc > 0 if net return to capital A− δ exceeds the timepreference rate ρ.

p.125


All values are growing with a constant steady state rate

gy = gc = gk = σ(A− δ − ρ)

Observe that the Euler equation implies a time-independent growthrate for c(t) (and henceforth for k(t)).

Therefore there is no transitory dynamic!

p.126


Convergence:

I Since there is no transitory dynamic, there is no “catchingup”.

I Similar countries (technology, time preference, intertemporalelasticity of substitution) grow with the same rate.

I Growth rate differences have to be explained by differentstructural parameters.

p.127


How to justifiy such an AK technology?

I Arrow, Kenneth J. (1962), The Economic Implications of Learningby Doing. Review of Economic Studies 29, 155–173.

I Romer, Paul M. (1986), Increasing Returns and Long–Run Growth.Journal of Political Economy 94, 1002–1037.

I Basic idea: There is no explicit “investment” into HC and noexplicit income share for this production factor. HC ismodelled as an external effect or as a by-product of physicalinvestment. Operating with physical capital goods leads to“learning by doing” effects which increase human capital K .

p.128


I Here HC/knowledge is non-rival in use and there is discloosure(public good). Each investor also contribute to a public good.

I As for a small firm the influence on the human capital stock ismarginal, it takes K as given.

I Profit maximizing implies that the capital cost equals theprivately appropriable marginal returns of capital (ignoring theexternal effect). Social return exceeds private return of capital.

p.129


Production function (Cobb-Douglas technology):

Y = f (K , K ,N)

In case of Arrow (1962):

y = f (k , K ) = K ηkα = Nηkηkα

(where η + α = 1 yields the standard AK model)

In case of Romer (1986):

Y = f (K , K · N) = Kα(KN)1−α

⇒ y = kαK 1−α = N1−αkαk1−α

p.130


a) Households maximize (25) and we have the Euler equation

gc = σ(r(t)− (ρ+ δ))

where σ is assumed to be constant.

b) Firms maximize

maxK ,N

π(k) = N · [kαK 1−α − rk − w ] (26)

From the first order conditions we have (with K = Nk)

r = αkα−1K 1−α = αN1−α (27)

w = (1− α)kαK 1−α = (1− α)kN1−α (28)

The marginal returns depend on firm specific k as well as on thegiven human capital stock K .

p.131


c) Decentral planning (market solution):

I With a given labor force N the return to capital r in (27) isconstant.

I The Euler equation with decentralized planning reads

gc = σ(αN1−α − (ρ+ δ))

which is also the steady state growth rate for k.

I Since there are positive externalities = the social returns ofcapital by inducing growing human capital are neglected in thefactor price r . Hence, the growth path is pareto-inefficient.

p.132


d) Social planner:

I A social planner is aware of the externalities, she does nottake K as given. Hence the profits according to (26) reads

maxK ,N

π(k) = N ·[Kα

NαK 1−α − rk − w

]= N ·

[K

Nα− rk − w

]I She calculates the FOC as

r = N1−α

and hence the Euler equation is

g∗c = σ(N1−α − (ρ+ δ)) > gc

Higher payment for capital than in market-based solution:N1−α > αN1−α.

p.133


Policy implications:

I Since the decentralized planning leads to pareto-inefficientsteady state growth rates, there is room for welfare increasingpolicy.

I Generally, incentives for economic activities with positivespillovers must be increased (e.g. by subsidies), the incentivesfor activities with negative spillovers have to be reduced (e.g.by taxes).

I In each case it has to be taken into account that subsidieshave to be financed and taxes generate expenditures. Bothhas an economic impact on welfare.

p.134


Since physical investment have positive knowledge spillovers, thereshould be subsidies θ to increase the incentive to invest. Themarginal return is then:

r = α(1 + θ)N1−α

and the Euler equation is

g∗∗c = σ(α(1 + θ)N1−α − (ρ+ δ))

By the “method of eyeballing” it is obvious that the optimal rateof subsidies is

θ∗ =1− αα

because then g∗∗c = g∗c .

p.135


How to finance this subsidy?

I Income tax: In most democratic systems such a tax is perceived as“fair”. However, it lowers the marginal returns of the productionfactors. As a response, an intertemporally maximizing agent wouldthen shift his consumption expenditures from the future to thepresence = lower saving = lower capital accumulation = lowersteady state growth rate!

I Per capita tax: This tax is perceived as “unfair” because it doesn’tregard the agent’s ability to pay taxes. However, such a tax doesnot affect allocation and has no negative impact on the steady stategrowth rate.

I Consumption tax: This would not affect the intertemporal decisionbetween consumption and saving, but it would affect the decisionbetween working and leisure time. In our model (unelastic laborsupply) this doesn’t play a role.

p.136


Convergence:

I There is no transitory dynamic.

I Countries with similar characteristics grow with the samegrowth rate.

I Countries with different scale of labor force N grow withdifferent rates: Large countries are growing faster than smallcountries (see Euler equation!). There is no (or only weak)empirical support for this effect.

I This scale effect could be avoided by assuming that theexternal effect depends on the average human capital K/N.

p.137

3. Innovation and Endogeneous Growth3.3 Models with Human Capital Accumulation

Literature:

I Lucas, R.E. (1988), On the Mechanics of Economic Development.Journal of Monetary Economics 22, 3–42.


Basic idea:

I HC is a specific producable factor. It is produced in a separateschooling sector⇒ 2-sector model (production sector, schooling sector).

I Producing HC requires ressources ⇒ allocation between physicalproduction and human capital accumulation.

I HC is a private good. Investments into HC yield a positive marginalreturn. Extension: also positive externalities.

p.138


There is strong empirical evidence that human capital (measured by yearsof schooling) not only affects the level of income but also the growth rate:

Hanushek, E.A., Woessmann, L. (2008), The Role of Cognitive Skills in

Economic Development. Journal of Economic Literature 46(3), 607-668

p.139


Role of cognitive skills resulting from education (“test score” proxy):

Hanushek, E.A., Woessmann, L. (2008), The Role of Cognitive Skills in

Economic Development. Journal of Economic Literature 46(3), 607-668

p.140


Uzawa-Lucas 2-sector model:

Y = C + K = F (K ,m · H) (29)

H = G ((1−m) · H) (30)

I HC is produced by means of HC. Therefore a fraction m ofHC is allocated to physical prodction while (1−m) isallocated to schooling.

I The representative household decides aboutI consumption/saving rate (s(t))I allocation of human capital to both sectors (m(t))

p.141


production

education

h

m · h

(1− m) · h

ky s

c

(1− s)

p.142


Simplifying assumptions:

I Schooling sector does not require physical capital. Thisassumption can be relaxed but then the household has todecide about the allocation of physical capital across the twosectors.

I To avoid too much notation, we assume no population growthand no depreciation of physical and human capital. Theconstant labor force is normalized to one (N = 1) so it dropsout from equations.

I Human capital H is a private good. Hence it is possible todefine the per capita human capital as h = H/N.

p.143


The two sectors:

I Schooling sector:

h = A · (1−m)h, A > 0, m ∈ [0, 1] (31)

where A is the constant productivity of the sector, and m is thefraction of human capital which is allocated to physical production.HC (output) is produced only with the factor HC (input).

Therefore, H = mH is the human capital stock used in physicalproduction.

I Production sector:

Y = KαH1−α

⇒ y = kα(mh)1−α

p.144


Remark:

I We can see already that from (31) that for an equilibriumgrowth we will obtain

y

y=

k

k=

h

h= A(1−m∗)

andm

m= 0

I We see already that sustained per capita growth is possiblebecause of non-diminishing marginal returns in the schoolingsector. (Recall that declining marginal returns from physicalcapital, fk(k) = r , leads to declining growth rates according tothe Euler equation.)

I The equilibrium value of m∗ will be determined later on.

p.145


I The capital stock evolves according to

k = y − c = [kα(mh)1−α]− c (32)

I Income is used for consumption or for saving. There are noexpenditures for schooling (schooling fees), but these will beincluded in the model later on.

I Household has to decide about savings (s), and human capitalallocation (m).

I Two differential equations for h and k which are constraintsfor the household’s optimization problem!

p.146


a) Households have the following optimization problem:

maxc,m

U(0) =

∫ ∞0

u(c)e−ρtdt

conditional to k = kα(mh)1−α − c

h = A(1−m)h

m ∈ [0, 1], k(0) > 0, h(0) > 0

The Hamiltonian is

H = u(c)e−ρt + µ1[[kα(mh)1−α]− c] + µ2[A(1−m)h]

p.147


The optimality conditions are

∂H

∂c= uc(c)e−ρt − µ1 = 0 (33)

∂H

∂m= µ1(1− α)kαh1−αm−α − µ2Ah = 0 (34)

−∂H∂k

= µ1 = −µ1αkα−1(mh)1−α (35)

−∂H∂h

= µ2 = −µ1(1− α)kαm1−αh−α − µ2(1−m) (36)

The partial derivatives to µ1 and µ2 yields the known differentialequation for k and h. The transversality conditions for k(t) andh(t) are defined in the usual way.

p.148


I Again, we derive the growth rate for consumption (Eulerequation): Differentiating (33) with respect to time andinserting (35) to substitute µ1 we have

gc = σ(r − ρ) (37)

I A steady state is defined as gc = gk = gh = gy = const > 0and gm = 0 (constant human capital allocation betweenproduction and schooling).

I Easier representation: Define q = c/k and z = k/h (capitalstructure). Equilibrium growth implies

gq = gz = gm = 0 ⇐⇒ gy = gc = gh = gk

p.149


Using the new variables, the marginal return to capital can berewritten as

y = kα(mh)1−α

⇒ r = yk = αkα−1(mh)1−α

= αkα−1(mk/z)1−α = α(m/z)1−α (38)

This can be plugged into the the Euler equation (39):

gc = σ(r − ρ) = σ(αm1−αz−(1−α) − ρ) (39)

As long as m and z have not yet reached their equilibrium values,this equation describes the transitory dynamics which will beanalysed later on.

p.150


From the differential equation k and h (using the new variables)we have

gk = m1−αz−(1−α) − q

gh = A(1−m)

and in addition (see previous slide)

gc = σ(αm1−αz−(1−α) − ρ)

Obviously, gq = gc − gk and gz = gk − gh holds true in steadystate.

p.151


We have not yet discussed the optimal choice of m:

I Differentiating (34) with respect to time and then inserting(35), (36) and the differential equations (32) and (31) inorder to substitute µ1, µ2, k and h leads to a differentialequation for m.

I The resulting dynamic system is:

Uzawa-Lucas model

gq = (σα− 1)m1−αz−(1−α) + q − σρgz = m1−αz−(1−α) − q − A(1−m)

gm =(1− α)A

α+ mA− q

p.152


I An equilibrium growth path with gq = gz = gm = 0 leads tothe steady state:

q∗ = σ(ρ− A) +A

α(40)

z∗ =(αA

) 11−α ·

(σρA

+ 1− σ)

(41)

m∗ =σρ

A+ 1− σ (42)

I Taking equilibrium values z∗,m∗ and plugging them into (38),the equilibrium return to physical capital is r = A.

⇒ Equilibrium growth rate is determined by the productivity ofthe educational sector A:

gc = σ(A− ρ) = gy = gk = gh

I Using (42) we have gh = A(1−m∗) = σ(A− ρ).p.153


The Lucas model has a transitory dynamic:

I Initial capital stock might be below k∗ which inducesaccelerated savings.

I The marginal returns of human capital in schooling andproduction may differ in the starting point!

⇒ This leads to a re-allocation of human capital (gm 6= 0) andtherefore to different (and non-constant) growth rates gh andgk (and gq, gz , respectively).

p.154


I The dynamic systems is 3-dimensional, nonlinear, andcomplicated to analyze. It is convenient to operate with atransformed version of the model. Let

x = m1−αz−(1−α)

i.e. z is substituted by x .

I Using the equilibrium values (42) and (41) for m and z wehave the equilibrium value

x∗ =A

α

p.155


I The transformed model is (linearized around the steday state):

gq = (σα− 1)(x − x∗) + (q − q∗) (43)

gx = −(1− α)(x − x∗) (44)

gm = A(m −m∗)− (q − q∗) (45)

I Instead of system (40) – (42) where gq and gz dependnonlinearly on q, z ,m, we have now a linear system ofdifferential equations!

I The steady state value of the new variable x is stable sincegx > 0 ⇐⇒ x < x∗ and vice versa.

I Since gq does not depend on m, and gm does not depend onx , it is possible to show the isoclines in a 2-dimensionalgraphic.

p.156


m x

q

x = 0q = 0m = 0

m∗ x∗

q∗ saddle path

p.157


One famous implication of the Lucas model:

I The growth rate for consumption c (and also for y and for thecapital stock K ) depends negatively on the capital structure term z(see eq. (39)).

I This implies that a disequilibrium z < z∗ – too less physical capitalcompared to human capital – leads to higher (transitory) growthrates. The marginal return of the remaining physical capitalincreases and this stimulates capital accumulation.

I A disequilibrium z > z∗ – too less human capital compared tophysical capital – leads to lower (transitory) growth rates. If HC isscarce, the return (wages) in the production sector is quite large.This prevents households from accumulating human capital becauseof the high opportunity costs of HC in the schooling sector. Thisretards the production of the scace factor HC.

I Policy implication for less developed countries: more important tosupport the local HC rather than physical investments. The Lucasmodel emphasizes the importance of education.

p.158


A version with positive externalities:

I Positive external effects of HC are modelled by

y = kα(mh)1−αhη, η ∈ (0, 1)

where a single firm treats h as exogenously given. Hence themarginal return from physical and human capital arecalculated, neglecting the external effect.

I It can be shown that with decentralized planning the steadystate growth rates are (with σ = 1!):

gy = gc = gk =1− α + η

1− α(A− ρ)

gh = A− ρ < gy

Remark: the growth rate gc is larger than in the model without the

externality. Because of the additional term hη, steady-state growth

requires that h grows less fast than k if gc = gy = gh should hold true.

p.159


I The growth rates gh and gy are constant but different. Theexternal effect of human capital enlarges the returns in thephysical production. Hence, the households work too muchbut learn too less!

I Therefore, gz = gk − gh = η1−α(A− ρ) > 0 increases, i.e.

physical assets accumulate faster than intellectual assets.

I A social planner treats h = h not as exogenously given andincludes the external effect when maximizing the welfare. Shecalculates the social return of human capital.

p.160


Solution with a social planner:

g∗y = g∗c = g∗k =1− α + η

1− αA− ρ

g∗h = A− 1− α1− α + η

ρ

The policy advice is to change the incentives in order to reallocatea part of human capital from the physical to the education sector.This could be done by a tax-transfer-system.

p.161


A design for a tax-transfer system:

I Since we have two production factors with a specific return,we have two income taxes:

I interest rate tax τr ≥ 0 for physical capitalI wage tax τw ≥ 0 for human capital

I Furthermore the incentive to allocate human capital to theeducation sector depends on the opportunity cost w(1−m)h.The government defines fees/grants for education which areproportional to the opportunity cost

ω = θw(1−m)h

where θ > 0 means that the household has to pay fees ω > 0and θ < 0 means that the household receive grants ω < 0.

p.162


I The intertemporal budget constraint can now written as

k = (1− τr )rk + (1− τw )wmh − θw(1−m)h︸︷︷︸income

−c

I Also the government has a budget constraint:

τr rk + τwwmh + θw(1−m)h = 0

p.163


Solving the model with these additional assumptions leads to:

g∗∗y = g∗∗c = g∗∗k =1− α + η

1− α

(1− τw

1− τw + θA− ρ

)g∗∗h =

1− τw1− τw + θ

A− ρ

Observe that τr has no influence on these growth rates!

p.164


Result:

I For θ > 0 (schooling fee) it is gy > g∗∗y for all τw . Thedeparture from the pareto-efficient solution increases!

I For θ = 0 we have the same situation like in the unregulatedcase.

I For θ < 0 (schooling grants) the pareto-efficiency is improveddue to the incentive to allocate more human capital to theeducation sector.

Optimal tax-transfer system:

There is a continuum of (θ, τw )-combinations which internalize theexternalities of human capital and lead to pareto-efficiency:

θ∗ = (τ∗w − 1) · ηρ

(1− α + η)A + ηρ

p.165

3. Innovation and Endogeneous Growth3.4 R&D based growth with increasing product variety

Literature:

I Romer, P.M. (1990), Endogenous Technological Change.Journal of Political Economy 98 (5), 71–102.


Basic Idea:

I In the previous models the aim was to uphold a persistentincentive for capital accumulation by preventing that themarginal return of capital declines. This has been achieved by

I knowledge spillover effects (externalities)I accumulation of human capital (with constant returns)

I Now: innovation activities of firms in a R&D sector.

I Here: Innovation = development of new product varieties.

p.166


I A firm will invest into R&D only if it could earn profits bygenerating innovative products:

I Requires intellectual property rights protection (like patents)which guarantees monopolistic power.

I Due to monopoly we have static efficiency losses. Hencepareto-improving governmental regulation is possible.

I New products are assumed to be intermediate goods = inputsfor the final homogeneous good Y .

I Three-sector model: R&D sector, sector for intermediategoods, production sector (final good)

I All intermediate good firms have identical technology, nopopulation growth.

p.167


The logic of the Romer model:

R&Dintermediate

goods Xjfinal goods

labor

assets

income

monopoly price

invent input

input

wages

profits

interestsavingas

consumption

p.168


Overview of the R&D part of the model:

I At any point in time, level of technology determines howmany varieties (“blue prints”) n can be produced

I R&D is costly but increases the number of variant n.New variant can be sold at monopoly price.

I Decision problem of firms:

1. Devoting R&D resources to invent new blue prints. Theinventor is also the producer of this variant.

2. Determining the optimal price at which to sell the newlyinvented blue prints to the final good producer.

⇒ Backward solution of the model.

p.169


Stage 2: Determining the price

I All intermediate goods are substitutes. Determine the finalgood sector’s demand for the new variant.

I Choose the profit maximzing price (Cournot solution) as thefirm is the monopolist for this variant.

Stage 1: Decision to enter the R&D business

I Calculate the the net present value (NPV) of profits byconsidering the monopoly price.

I Compare the NPV of profits with the fixed R&D costs: ifdifference is positive ⇒ perform R&D!

p.170


An alternative:

I Grossman, G.M., Helpman, E. (1991b), Innovation andGrowth in the Global Economy. MIT Press, Cambridge, MA.


I R&D increases the variety of consumption goods.

I Hence the utility function could not depend on aggregatedconsumption but has to account for product variety (varietypreference). This also affects the Keynes-Ramsey rule for thegrowth of aggregated consumption.

I We will not discuss this approach since the basic logic couldalso be studied in the Romer approach (R&D generatesmonopoly profits = increasing firm value ⇒ persistentstimulus for investing a constant share of (increasing) incomeinto R&D).

p.171


Final good sector:

Y (t) = N1−α∫ n(t)

0X (i , t)αdi , α ∈ (0, 1) (46)

I n(t) is the “number” of intermediate goods (inputs) at time t.

More precisely, there is a continuum of intermediate goods[0, n(t)] with i ∈ [0, n(t)] as the index and X (i , t) as thequantity of the intermediate good i .

I There is no physical capital, and labor supply N(t) is inelastic.

I The production function has constant returns to scale.

I The price of the final good is normalized to 1.

I The price for labor is w , the price for each intermeaite good iis P(i).

p.172


Firms in the final good sector:

Firms are price takers. They maximize

maxN,X (i)ni=0

π = N1−α∫ n

0X (i)αdi − wN −

∫ n

0P(i)X (i)di

From the FOC we have

w = (1− α)Y

N(47)

and∂π

∂X (i)= αN1−αX (i)α−1 − P(i) = 0

⇒ X (i) = N

(α

P(i)

) 11−α

(48)

This is the demand function for intermediate goods which has aconstant price elasticity η = −1/(1− α).

p.173


Firms in the intermediate good sector:

Monopolistic price setting firms (production costs normalized to 1):

maxP(i)

π = (P(i)− 1)X (i) = (P(i)− 1)N

(α

P(i)

) 11−α

(49)

∂π

∂P(i)= 0 ⇒ P(i) =

1

α> 1

I Hence the monopoly price exceeds the marginal costs whichare normalized to one (markup: (1− α)/α).

I Price P(i) is constant and identical for all i , given the cost ofproduction is the same for all goods.

⇒ each good enters symmetrically into the production of thefinal good.

p.174


R&D sector:

I The innovation process is deterministic!

I Developing a new intermediate good has constant costs θ.There are no economies of scale and no synergy effects.

I Firms have an unlimited patent for the innovativeintermediate good. Hence we have n(t) monopolies in theintermediate good sector.

I The incentive to innovate (= being an entrepreneur) dependson the net present value of monopoly profits compared to thecosts of R&D (market entry costs).

p.175


Net present value of profits:

V (i , t) =

∫ ∞t

(P(i)− 1)X (i , t)︸︷︷︸π(t)

e−r(s)(s−t)ds (50)

where r(s) is the average interest rate in the time interval [t, s].

Using the expressions for X (i , t) and P(i) as we have calculatedabove, we obbtain the NPV of profits (see slides later on).

p.176


Remark:

I All intermediate goods are close substitutes (see productionfunction).

I This restricts the monopoly power of a single firm!

⇒ Monopolistic competition! (Dixit/Stiglitz (1977))

I The increasing number of substitutes leads c.p. to a decline ofthe residual demand of each single firm. Therefore, themonopoly price converges to average cost (= zero profit,so-called “Chamberlin solution”).

I This effect does not take place in this model since thegrowing aggregate demand prevents that the demand for asingle intermediate good decreases.

p.177


I Recall, that all intermediate good firms are identical, henceP(i) = P,X = nX (i).

I Inserting the price P(i) = 1/α into the demand function (48)yields

X (i) = Nα2

1−α (51)

I Inserting P(i) and X (i) into the present value of profits (50):

V (t) = N(1− α)α1+α1−α

∫ ∞t

e−r(s)(s−t)ds

p.178


Incentive to innovate:

I If V (t) > θ the net present value of profit exceeds theconstant cost of innovation. Hence there is an incentive tore-allocate all resources in favor of the R&D sector bydetracting them from other sectors. This could not be anequilibrium.

I If V (t) < θ then there is no incentive to innovate at all.

I If V (t) = θ then innovation activities are on an equilibriumlevel. The resource allocation between the sectors is constant.The creation of innovative products has a positive constantgrowth rate gn = n/n > 0.

p.179


The equilibrium condition V (t) = θ for all t implies (after sometedious math):

r(t) =π

V=π

θ

⇒ Interest rate r(t), equals the rate of return to R&D investments( πV = profit rate)

Using profit function (49) for π:

r =N

θ(1− α)α

1+α1−α (52)

I Rate of return is constant (determined by underlyingtechnology and labor force).

I Old and new firms receive the same flow of monopoly profits,the market value of every firm (V (t)) equals θ.

p.180


Households:

Recall that firms in the final good sector have zero profits. Total

assets in the economy in t are therefore∫ n(t)

0 V (i , t)di , so assetsper capita (households are the owner of the firms) are

v(t) =

∫ n(t)0 V (i , t)di

N(t)

and the intertemporal budget constraint is then

v(t) = w(t) + r(t)v(t)︸︷︷︸income

−c(t)

p.181


Households face the usual maximization program

maxU(t) =

∫ ∞0

u(c)e−ρtdt

subject to v(t) = w(t) + r(t)v(t)− c(t)

v(0) > 0

and also the TVC limt→∞ λ(t)v(t) = 0 holds true.

The result is the Keynes-Ramsey rule

gc = σ(r − ρ)

and using (52)

gc = σ

(N

θ(1− α)α

1+α1−α − ρ

)≡ γ

There is no transitory dynamics!p.182


We turn back to the final good production function:

I We assumed symmetry of intermediate good firms. Thus wehave the same prices and input quantities in the final goodsector. Hence

Y = N1−α∫ n

0X (i)αdi

= n · N1−αX (i)α

and using the demand function for X (i)

= α2α

1−αNn

I Therefore the output growth is directly determined by thegrowth of n:

gY = gn = gc = γ

(Recall that there is no population growth)

p.183


Remark:

I It can be shown (for details see Barro/Sala-i-Martin) that thetotal real income can be written as

GDP = Y − X

I Recall that a part of the resources have to be used forproducing intermediate goods. As we assumed unit costs = 1,the total costs of intermediate goods are simply X (for moretechnical details see Barro/Sala-i-Martin).

I Therefore savings = investment into the development of newproduct variants is

θn = Y − X − C

p.184


Determinants of the growth rate γ

gY = gn = gc = σ

(N

θ(1− α)α

1+α1−α − ρ

)= γ

I Lower R&D costs θ lead to a higher rate of return, r , andthus raises the γ.

I Moreover, the model contains a scale effect: a larger laborendowment, N, raises the growth rate γ.

p.185


Summary:

I The economy grows with the same rate as the variety ofintermediate goods. This requires a constant incentive toinvest into R&D and innovation. The interest rate must bekept on a level so that households are willing to financemonopolistic entrepreneurs in the market of intermediategoods. Therefore the net present value of monopolistic profitsmust equal the R&D costs. The markets for intermediategoods grows with the same rate as the aggregated demand.

I There is no transitory dynamic.

I There are scale effects (dependency on N).

p.186


Optimality:

I Since the price of intermediate goods exceed the marginalcost (monopoly due to patents), the result cannot bepareto-efficient!

I Higher price = lower demand for intermediate goods = lowerproduction of final good.

I The efficient interest rate can be calculated (by a socialplanner) as

r∗ =N

θ(1− α)α

α1−α

(since α ∈ (0, 1) it is r∗ > r and hence g∗c > gc)

p.187


Governmental regulation:

Government is able to change the relative prices (incentives) bytaxing or paying subsidies. Each tax-transfer structure requiresthat the governmental budget is balanced, e.g. subsidies have tobe financed by allocation-neutral per capita taxes.

a) Subsidies for the demand for intermediate goods:A subsidy ξ = 1− α would decrease the price to the level ofmarginal cost. Static efficiency is enhanced since the demandfor X and hence the output increases. This also enhances theflow of profits and therefore the interest rate to its sociallyoptimal level. This induces incentives to invest into R&D.Therefore also the dynamic efficiency is increased.

p.188


b) Subsidies for producing the final good:This provides an incentive to expand the production Y andtherefore the demand for X . The results are the same as in a).

c) Subsidies for R&D:This would lower the cost of R&D and thus enhance theinterest rate. The dynamic efficiency increases. But this is nosolution for the static efficiency loss due to monopolisticpricing.

p.189


Critique regarding R&D sector:

I Deterministic: the outcme of the R&D process is sure and known exante.

Alternative: Stochastic outcome. The probability of a successfulinnovation might depend on the amount of spent R&D resouces.

I One innovator = one producer of the intermediate good:

Alternative: Multiple R&D projects by one firm with synergy/scaleeffects; licensing to multiple intermediate good firms.

I Eternal monopoly

Alternative: Limited monopoly power for T periods.

I During the monopoly period the demand for the product ispre-determined and not fundamentally endangered.

Alternative: new entrepreneurs might supply better products whichdrives the old firm out of the market (“creative destruction”,Schumpeterian models)

p.190

3. Innovation and Endogeneous Growth3.5 R&D based growth with increasing product quality

Literature:

I Grossman, G., Helpman, E. (1991), Quality Ladders in theTheory of Growth. Review of Economic Studies 58, 43–61.

I Aghion, P., Howitt, P. (1992), A Model of Growth throughCreative Destruction. Econometrica 60 (2), 323–351.


I Schumpeter, J.A. (1912), Theorie der wirtschaftlichen Entwicklung.Leipzig: Duncker & Humblot.

p.191


Basic Idea:

I Romer : increasing variety = “horizontal innovation”,

now : increasing quality = “vertical innovation”

I If R&D leads to a better product then the “quality leader” isthe monopolist, the previous incumbent has to leave themarket (Schumpeter’s “creative destruction”). The profit flowfrom innovation terminates if a quality-leading entrepreneurenters the market.

I In contrast to the Romer model, innovation is a stochasticprocess.

p.192


I There is a fixed number of intermediate goods i = 1..n.

I The quality of each good is measured by a discrete qualityindex ki = 0, 1, 2, ....

I Successful R&D leads to an incremental increase of theprevalent quality index ki + 1.

I This implies that a potential entrepreneur (follower) “standson the shoulders” of the preceding innovator. The previousknowledge is revealed by the patent, and is the basis forfurther innovations.

⇒ This is an important intertemporal spillover effect(externality).

p.193


quality index ki

interm. good i1 2 3 ... n0

1

2

3

4

p.194


Stochastic evolution of the quality of an intermediate good:

quality index ki

0tt1

1

t2

2

t3

3

t4

4

... tk tk+1

k

k+1

...

p.195


From the quality index to the quality adjusted input of good i :

I Index ki = 0, 1, 2, ...

I Current quality is qki , that means quality evolves with1, q, q2, ..., qki

I A quality adjusted input of an intermediate good i is qkiXi .

p.196


Final good sector:

Y = N1−αn∑

i=1

[qkiXi ,ki ]α (53)

Firms in the competitive final good sector maximize profits(price normalized to 1):

maxN,Xini=1

π = N1−αn∑

i=1

[qkiXi ,ki ]α − wN −

n∑i=1

Pi ,kiXi ,ki (54)

where w is the wageand Pi ,ki is the price for input i with quality ki .

p.197


From FOC we have (similar to the Romer model)

w = (1− α)Y

N(55)

∂Y

∂Xi ,ki

= αN1−αqαkiXα−1i ,ki− Pi ,ki = 0 (56)

⇒ Xi ,ki = N

(αqki

Pi ,ki

) 11−α

(57)

which is the demand function for intermediate goods. Observe,that without quality improvement (ki = 0) this is the same resultas in the Romer model (equation (48)).

p.198


Intermediate good sector:

I The current quality leader is the monopolist. As in the Romermodel we assume constant marginal cost which arenormalized to 1. Again, maximization of the profits leads tothe optimal price

Pi ,ki =1

α

I Employing this price into the demand function yields theoptimal inputs of intermediate goods:

Xi ,ki = Nα2

1−α qkiα

1−α

(with ki = 0 this is the same result as in Romer)

p.199


I Substituting Xi ,ki in the production function by its optimalinput levels leads to

Y = α2α

1−αNn∑

i=1

qkiα

1−α

I Let Q be an aggregated quality measure defined as

Q =n∑

i=1

qαki

1−α

I Then we can write:

Y = α2α

1−αNQ

X =n∑

i=1

Xi ,ki = α2

1−αNQ

I Since labor force N is constant, it follows

gY = gX = gQ p.200


Profits and present value of the intermediate good firm:

I Inserting equilibrium prices and quantities into the profitfunction leads to the momentum profits

πi ,ki = N

(1− αα

)α

21−α q

kiα

1−α (58)

I Recall, that the monopolist earns profits only until a newquality leader with ki + 1 enters the market.

I The time duration of the monopoly is therefore

Ti ,ki = ti ,ki+1 − ti ,ki

p.201


In equilibrium there will be a constant (average) interest rate.The net present value of the profit flow is then

Vi ,ki =

∫ Ti,ki

0πi ,ki e

−rtdt = πi ,ki ·1− e−rTi,ki

r

Note: Duration Ti ,ki is unknown and depends on a stochasticinnovation process!

p.202


Modeling the R&D process:

I In this version of the model, the incumbent does not engagein R&D! He will be replaced by an entrepreneur which is thenew quality leader.

I R&D requires a resource input Zi ,ki (measured in units of Y ).

I The probability of achieving a higher quality level ki + 1(successful innovation) depends on the input Zi ,ki :

pi ,ki = Zi ,kiφ(ki ) (59)

where dφ/dki < 0 (since ki is an index number, this is a slight abuse

of notation!) denotes that with growing quality, the probabilityof further improvements decrease.

p.203


I With these assumptions about the stochastic innovationprocess it is possible to determine the expected value of Vi ,ki

(for details see Barro/Sala-i-Martin, chapter 7.2.2):

E [Vi ,ki ] =πi ,ki

r + pi ,ki

I The higher the R&D effort of all firms in sector i , the higheris the probability pi ,ki of a successful innovation and the loweris the expected duration of the monopoly (and therefore thepresent value of profits).

I We have not yet determined the optimal R&D effort!

p.204


Incentives for R&D effort:

I We assume risk neutrality, i.e. firms respond to the expectedvalue of profits, not to its variance.

I There is free market entry. This implies that firms enter themarket as long as there is a positive expected profit. Hence,in equilibrium the zero profit condition must hold true.

pi ,kiE [Vi ,ki+1]− Zi ,ki = 0

pi ,kiπi ,ki+1

r + pi ,ki+1− Zi ,ki = 0 (60)

Using (59) and (58) and re-arranging leads to

r + pi ,ki+1 = N

(1− αα

)α

21−α · φ(ki ) · q

α(ki+1)

1−α (61)

p.205


I To make things more convenient we will now adopt a specificform of φ(·):

φ(ki ) =1

ξ· q−α(ki+1)

1−α (62)

(Observe the negative dependency on ki ).

I Using this specific form of φ(ki ) in the free-entry condition(61) the very last term is canceled out and we have:

r + p =N

ξ·(

1− αα

)α

21−α (63)

p.206


Now we are able to calculate R&D effort in equilibrium:

I Recall p = Zi ,kiφ(ki ).

I Solving for Zi ,ki and inserting p from (63) and φ(ki ) from(62) we have

Zi ,ki = qα(ki+1)

1−α

(N

(1− αα

)α

21−α − rξ

)(64)

and aggregating all R&D expenditures:

Z =n∑

i=1

Zi ,ki = Q · qα

1−α

(N

(1− αα

)α

21−α − rξ

)(65)

I Hence,gZ = qQ = gY = gX

p.207


Using (63) for p the expected firm value is

E [Vi ,ki ] = ξ · qαki

1−α

and aggregation of all firms leads to

E [V ] = ξ · Q

Therefore, also the expected value of total assets grows with thesame rate:

gV = gQ = gY = gX = gZ

p.208


Households optimize their present value of utility flow under theintertemporal budget restriction (like in the Romer model).

I As we assumed that the intermediate good sector produceswith unit costs, the resource constraint is given by

C = Y − X − Z

I Inserting the calculated expressions for Y ,X ,Z we find thatalso C is proportional to Q, so that gC = gQ .

I In absence of population growth the overall growth rate ishence given by the Keynes-Ramsey rule.

p.209


Important to note:

I While the variables Q, Y , X , Z , and C all grow at the sameconstant growth rate, the realized growth rate in each sectordepends on the random outcomes of research efforts.

I Thus, the relative quality positions of the sectors evolve in arandom fashion.

p.210


Optimality:

I Since R&D requires patents and monopoly power, the staticefficiency condition price = marginal cost cannot hold true.

I Furthermore, there are two externalities:I The fact that the entrepreneur’s R&D effort yields an

incremental quality step from ki to ki + 1 implies that healready possess the knowledge how to produce the existingquality ki . This is an external knowledge spillover effect of thepreceding innovator.

I The R&D effort leads to a higher quality index and enhancestherefore the labor productivity in the final good sector.

⇒ Creates possibilities of welfare-improving governmentalactivities (not discussed in this lecture).

p.211


Extensions (Barro/Sala-i-Martin, chapter 7.4):

I Incumbents may also engage in R&D as a monopolyresearcher.

I Incumbents as well as outsiders engage in R&D. In this case itis reasonable to assume that the quality leader has betterinformation about the current quality level and has thereforelower R&D costs.

Further extensions:

I Variable (endogenously determined) step size in quality.

I Co-existence of quality-leading and older products.

I Including imitation as an alternative to innovation.

p.212

3. Innovation and Endogeneous Growth3.6 Notes on Technology Diffusion and North-South Models

Literature:

I Nelson, R.R., Phelps, E.S. (1966), Investment in Humans,Technological Diffusion, and Economic Growth. AmericanEconomic Review 56, 69-75.

I Benhabib J., Spiegel, M.M. (2003), Human Capital andTechnology Diffusion. Federal Reserve Bank of San Francisco,Working Paper No. 2003-02.

This section is not really about endogenous growth models. We discuss

some mechanisms of technology adoption which could be plugged into

endogenous growth models.

p.213


Basic Idea:

I Technological progress is exogenous, but diffusion ofinnovation depends on human capital (and may henceendogenously determined).

I All previously discussed models assume that new knowledge,new products or better technologies instantanously determinethe production, i.e. there is no diffusion process.

I Technology leader = best practice, imitating followers; anincrease in TFP does not necessarily reflect technologicalprogress, but also improved diffusion of the best practicetechnology.

I Regional models of growth (“North-South” models,leader-follower structures)

p.214


I Here: Human Capital plays not a role as a production factor.

I HC is needed to absorb the knowledge about newtechnologies, and to employ new technologies in theproduction process.

I The diffusion or “catching up” process is therefore notcostless (as in previously discussed models), but it depends onHC investment.

I This mechanism could also be applied to an endogenousgrowth framework: Adoption or imitation cost of the followercreate an incentive to innovate (as an alternative mechanismto patenting and monopoly power).

p.215


The Approach of Nelson/Phelps:

I Exogenous Harrod neutral technological progress:

Y (t) = F (K (t),A(t) · N(t))

where A is the “average” TFP index.

I Best-practice level of technology (technology frontier) growswith rate λ:

T (t) = T0 · eλt , λ > 0

I Human capital is denoted by h and does not enter theproduction function directly.

p.216


Diffusion process:

A(t) = φ(h)(T (t)− A(t))

gA =A

A= φ(h)

(T (t)− A(t)

A(t)

)(66)

where the bracket term is the “technology gap”.

I The term φ(h) with dφ/dh > 0 denotes the strength of thecatching-up dynamic which depends on human capital h.

I The average TFP growth is faster in case of a large technologygap, and becomes zero when the gap declines to zero.

I Since the frontier technology T grows with the rate gT = λ,the gap will never be closed.

p.217


I In equilibrium gA = gT we have the equilibrium technologygap T−A

A = λφ(h) .

T−AA

AA

γ TT

λφ(h)

p.218


I In equilibrium the gap is

T − A

A=

λ

φ(h)(67)

I In a stagnating economy λ = 0 the gap will be closed in finitetime.

I Differentiating (67) with respect to h an rearranging leads to

dA

dh· hA

=

(hφ′(h)

φ(h)

)(λ

φ(h) + λ

)> 0

The effect of an increased education on the TFP is higher themore technologically progressive the economy is (λ).

p.219


(Source: Benhabib/Spiegel)

p.220


The Benhabib/Spiegel approach:

I Modification of the Nelson/Phelps model.

I Innovative country “North” develops the technology frontier,imitating country “South” is catching up.

Variant A:

AN

AN= g(HN) (68)

AS

AS= g(HS) + c(HS)

(AN

AS− 1

)(69)

with g(·), c(·) as increasing functions.

I It is g(Hi ) the base rate of technical progress where the North isendowed with more human capital (HN > HS) and henceforthg(HN) > g(HS). In the starting point there is AN > AS .

I c(HS)(AN/AS − 1) like in Nelson/Phelps approach.

p.221


I Let HN ,HS be constant (ceteris paribus), and thereforegN = g(HN), gS = g(HS), cS = c(HS).

I The solution of the differential equation (69) is given by

AS(t) = (AS(0)− ΩAN(0))e(gS−cS )t + ΩAN(0)egN t

withΩ =

cScS − gS + gN

> 0

I It can be shown that (similar to the Nelson/Phelps approach)there is a balanced growth path with

limt→∞

AS(t)

AN(t)= Ω

(constant relative distance in TFP)

p.222


Variant B:

I In the economics of innovation a broadly used concept is alogistic diffusion process:

I The catching-up dynamic is low when the technology gap islarge, it accelerates with a declining gap, and it slows downagain when the technology gap becomes small: We replace(69) by

AS

AS= g(HS) + c(HS)

(AS

AN

)(AN

AS− 1

)(70)

I It may be the case that the South fails to catch up if theSouth is very low endowed with human capital.

p.223


I The following result could be derived:

limt→∞

AS(t)

AN(t)=

cS−gs+gN

cSif cS + gS − gN > 0

AS (0)AN(0) if cS + gS − gN = 0

0 if cS + gS − gN < 0

I The last case describes a poorly HC endowed South withcS + gS < gN so that the technology gap becomes infinitelylarge.

⇒ “convergence clubs” or “poverty trap” (see stylized facts)

p.224


Some empirical evidence:

Teles, V.K. (2005), The Role of Human Capital in EconomicGrowth. Applied Economics Letters 12, 583-587.

I The Lucas model “satisfactorily explains” the human capitalbased endogenous growth in rich countries, but cannot explainthe fact of convergence clubs or poverty traps.

I Nelson/Phelps type models could explain poverty traps but donot properly describe the growth process in rich countries.

p.225


From “catching up” to “leapfrogging”

I Brezis, E.S., Krugman, P.R., Tsiddon, D. (1993): Leapfrogging inInternational Competition: A Theory of Cycles in NationalTechnological Leadership. American Economic Review 83,1211-1219.

Basic idea:

I Two countries, two goods (food F , manufatures M), Ricardiantrade model

I Dynamic scale effects in the manufacturing sector.

I Specialization of the “technology leader” on M.

I Invention of more productive M-technology creates incentive toadopt this technology by the other country.

I With the change of the “technology leader” position we have also achange of the specialization pattern.

p.226


Model setup:

I Two sectors: food F and manufactures M. Food is thenumeraire.

I Two countries: Great Britain and USA (USA variables denoted

with “*”). Both have the same labor force L.

I Perfect competition

I Production function with labor productivity = 1 in the foodsector:

QF = LF

Q∗F = L∗F

p.227


I Production function with labor productivity Ai in themanufacturing sector:

QM,i (T ) = LM,iAi (T )

Q∗M,i (T ) = L∗M,iAi (T )

with Ai (T ) as a function of the past cumulative output(dynamic scale effects, “learning curve”) which is anexternality for each firm:

Ai (T ) = Ai

(T∑−∞

QM,i

)with A′i > 0 and A′′i < 0. As both countries might differ intheir cumulative output we write Ai and A∗i , respectively.

I The index i denotes the technology level. Technical progressmeans Ai+i (T ) > Ai (T ) (for all T ).

p.228


I The preferences in both countries are characterized by

U = DµMD1−µ

F

where it is assumed µ > 0.5.

I As the dynamic scale effects are externalities, the firms takethe current productivities Ai as given.

I Then we have a standard Ricardian trade model!

p.229


Short recapitulation of Ricardian theory:

I From profit maximization in the F sector we obtain real wagesequals productivity: w = w∗ = 1. The same for the M sector:w = Ai and w∗ = A∗i

I The relative prices (prices = marginal cost = wages = laborproductivities) are 1/Ai and 1/A∗i .

I In the initial situation we assume that Great Britain is thetechnology leader Ai > A∗i and thus has a comparativeadvantage in the M sector.

I Ricardian theory does not necessarily imply full specialization(depends on world demand).

p.230


Is it possible that both countries produce food?

I As labor is homogenous (uniform wage in a country), and thelabor productivity is identical in the F -sector of bothcountries, we have w = w∗ = 1.

I The world income and expenditures is thereforeE = 2wL = 2L.

I According to preferences the expenditure share spent for M isµ. Therefore the total expenditures for M are 2µL which islarger than L = the income of one country.

I This leads to a contradition as both couintries want to spentmore than 50% (recall µ > 0.5) of their income for M.

⇒ We can rule out an equilibrium where both countries producefood. There must always be specialization.

p.231


Full specialization (initial situation)

I We assumed that Great Britain is the technology leader (Ai > A∗i ,comparative price advantage for M), so it will fully specialize on Mwhile the USA specializes fully on F .

I Recall that both countries have identical size. Thus: LM,i = L = L∗F

I World expenditures for M are µE = wL (income of Great Britain).World expenditures for F are (1− µ)E = w∗L (income of the USA).Therefore the relative wages are w/w∗ = µ/(1− µ).

I While Great Britain produces M, the cumulative output in thissector and thus also the productivity increases.

I Wages in Great Britain increase accordingly, and the higher demandfor both goods induce also higher wages in the USA. In fact bothwages grow proportionally (!) due to the productivity effects in onecountry and trade.

p.232


Partial specialization:

I Note that the wage relation must not exceed Ai/A∗i . If this

happens, the USA would be able to produce M at least at thesame cost as Great Britain.

I Average cost are total cost divided by output:

ACi =wLM,i

QM,i=

wLM,i

AiLM,i=

w

Ai

so thatw

w∗=

Ai

A∗i⇐⇒ ACi = AC ∗i

Then USA would start also to produce M as well.

I Situation of partial specialization: Great Britain produces M,and the USA produce F and M.

p.233


How could this happen?

I Assume a technology change from A1 to A2. The productivityof the new technology is higher for each cumulative output;the learning curve is below the old one (see graphic).

Will Great Brtain adopt the new technology?

I No, not necessarily because AC of the old technology is stilllower due to the cumulative output (T > 0) while theexperience with new technology is zero (no learning effects).Therefore: AC1(T ) < AC2(0).

I Recall, that these scale effects are a positive externality,creating a “prisoner’s dilemma”: individually it is rational tostay at the old technology although collectively it would bebetter to switch.

p.234


(Continued)

I However, in the initial situuation the USA have lower wages(recall w/w∗ = µ/(1− µ) > 1).

I If the USA adopt the new technology, the average cost couldbe lower even though there are no dynamic scale effects yet:

AC ∗2 =w∗

A∗2(0)≤ w

A1(T )= AC1

I We will then have partial specialization! The USA starts theproduction even though A∗2(0) might be lower than A1(T )because of the lower wages in the USA.

I The relative wages are then given by

w

w∗=

A1

A∗2

(Note that also the worker in the food sector benefit from this progress)

p.235


cumulative output

w/A1 = AC1

w/A2 = AC2

T

w∗/A2 = AC∗2

p.236


Further change of the specialization pattern:

I Recall A′i > 0 and A′′i < 0. This implies that both countrieswill now experience increasing productivity but the marginaleffect for the new technology in the USA is stronger.

I Therefore the relative productivities in the M-sector andhenceforth the wage relation will change!

I Initially we had w/w∗ = A1/A∗2 > 1 but due to the

catching-up process in the USA regarding scale effects, thesituation will turn to w/w∗ = A1/A

∗2 < 1.

I The USA is then definitely the technology leader and willspecialize fully on M while Great Britain will therefore start toproduce food.

I Finally we will end up in the reverse situation than at thebeginning, namely full specialization (Great Britain: F , andUSA: M).

p.237


Sequence Great Britain USA w/w∗

Equilibrium 1 M F µ1−µ > 1

Equilibrium 2 M F + M A1A∗2> 1

Equilibrium 3 M + F M A1A∗2< 1

Equilibrium 4 F M 1−µµ < 1

p.238


Summary:

I Dynamic scale effects induce productivity (and henceforthincome) growth.

I Via trade, income growth in one country increases wages andincome also in the other country which does not experiencedynamic scale effects.

I A new technology with overall higher productivity might notbe adopted by the technology leader but – thanks to lowerwages – by the other country.

I Due to dynamic scale effects the other country will becometechnology leader (“leapfrogging”), not just “catching up”.

I The “lock-in” of the technological leader is his weakness.Leadership positions might change.

I By this process also the specialization pattern changes.

p.239


Empirical evidence:

I Evidence that innovative industries thrive in geographicallyconcentrated districts (Porter (1990)), and benefit fromclustering effects (sort of positive externality).

I Historical examples for “technologically leading countries” (atleast in specific sectors), e.g. Great Britain during theindustrialization ⇒ replaced by the USA in the 19th century⇒ partially replaced (e.g. semiconductor industry) by Japanin the 20th century etc. etc.

I Currently China aims to get a leading position in somehigh-technology sectors.

p.240

3. Innovation and Endogeneous Growth3.7 Empirical Evidence and Critique

Empiricial growth literature, see e.g. overviews in

I Aghion, P., Akcigit, U. Howitt, P. (2013), What Do We LearnFrom Schumpeterian Growth Theory, in Handbook ofEconomic Growth, Volume 2B, 515–563.

I Acemoglu, D. (2008), Introduction to Modern EconomicGrowth. Princeton University Press.

I will give only few remarks.

p.241


Durlauf, S.N., Kourtellos, A., Tan, C.M. (2008), Are Any GrowthTheories Robust? The Economic Journal 118, 329–346.

“[We] find little evidence that so-called fundamental growththeories play an important role in explaining aggregate growth. Incontrast, we find strong evidence for macroeconomic policy effectsand a role for unexplained regional heterogeneity, as well as someevidence of parameter heterogeneity in the aggregate productionfunction. We conclude that the ability of cross-country growthregressions to adjudicate the relative importance of alternativegrowth theories is limited.”

p.242


Capolupo, R. (2009), The New Growth Theories and TheirEmpirics after Twenty Years Economics: The Open-Access,Open-Assessment E-Journal Vol. 3, 2009-1

“The author [...] argues that: (i) causal inference drawn from theempirical growth literature remains highly questionable, (ii) thereare estimates for a wide range of potential factors but theirmagnitude and robustness are still under debate.”

In order to let endogenous growth theory not to lose out toomuch...

“Her conclusion, however, is that, if properly interpreted, thepredictions of endogenous growth models are gathering increasingempirical support.”

p.243


Parente, S.L. (2000), The Failure of Endogenous Growth.Knowledge, Technology and Policy 13(4), 49-58

“My own assessment is that this line of research has not provenuseful for understanding the most important question faced byeconomists today, namely, why isn’t the whole world rich.Exogenous growth theory, in contrast, is. Endogenous growth mayprove useful for understanding growth in world knowledge overtime, but it is not useful for understanding why some countries areso poor relative to the United States today.”

p.244


A broader view of the empirics of endogenous growth:

I Take the mathematical models as “thought experiments”which shed some light on some basic mechanisms.

I For endogenous growth, R&D as well as human capital seemsto be a driving force.

I In general, investments into physical or human capital or intoR&D have an uncertain outcome:

I General uncertainty about future returns of investment.I Specific risk of “expropriation” due to weak or not enforced

property rights (especially free-riding problem in case of R&Doutcome)

I These sources of uncertainty are linked to the quality ofpolitical and economic institutions.

I This might explain the empirical nexus between quality ofinstitutions and economic performance of economies (seeAcemoglu (2008)).

p.245


Risk of ex-propriation

Uncertaintyof economicconditions

Institutional conditions:• Enforcable property rights• Rule of law• Low corruption• etc.

Investmentinto R&D

Investmentinto HC

Growth

p.246


Some methodological objections:

I Endogenous growth theory has a rigorous methodological basewhich is broadly accepted in mainstream economics andsometimes considered as a “prerequisite” for economicreasoning.

I representative agentsI intertemporal optimizationI perfect markets, general equilibriumI etc.

I Without doubt, these models shed light on the determinantsand mechanisms of economic growth.

I The question arises whether the little plus of explanatorypower compared to exogenous growth models is worth theprice of high artificiality (if not counterfactuality) ofassumptions, and of mathematical effort.

p.247


Solow, R.M. (2007), The last 50 years in growth theory and thenext 10. Oxford Review of Economic Policy 23(1), 3–14.

“I suspect that the most valuable contribution of endogenous growththeory has not been the theory itself, but rather the stimulus it hasprovided to thinking about the actual ’production’ of human capital anduseful technological knowledge.”

“Instead, the main argument for this modelling strategy has been a moreaesthetic one: its virtue is said to be that it is compatible with generalequilibrium theory, and thus it is superior to ad hoc descriptive modelsthat are not related to ‘deep’ structural parameters. The preferrednickname for this class of models is ‘DSGE’ (dynamic stochastic generalequilibrium). I think that this argument is fundamentally misconceived.”

“The cover story about ‘micro-foundations’ can in no way justify recourseto the narrow representative-agent construct. Many other versions of theneoclassical growth model can meet the required conditions; it is onlynecessary to impose them directly on the relevant building blocks.”

p.248


Some methodological objections in more detail:

I Perfect Rationality, Intertemporal OptimizationI Robust empirical and experimental evidence against perfect

rationality. Microfoundation of macro growth model could alsobe based on boundedly rational decision making.

I Innovation – by nature – is not predictable and should be“fundamentally uncertain”. Fundamental uncertainty cannotbe reconciled within the common rationality approach.

I Walrasian Equilibrium Growth Paths

Artificial (Arrow-Debreu) economy for normative theoreticalinvestigations only (e.g. to prove the existence and stability ofgeneral equilibria). It is not clear why this should be the basisfor an explanatory positive theory which is used for empiricalmodels if we consider that economies typically evolve “out ofequilibrium”.

p.249


I Schumpeterian innovation process?

So-called “Schumpetarian” models with “creativedestruction” on an Walrasian equilibrium path is a very highquestionable concept!

Alcouffe A., Kuhn, T. (2004), Schumpeterian endogenousgrowth theory and evolutionary economics. Journal ofEvolutionary Economics 14, 223–236.

“We find endogenous growth theory far from carryingSchumpeter’s idea of an evolutionary approach to change anddevelopment.”

p.250

4. Growth and the Environment – Growth Limits4.1 Economic-ecological interaction

We will discuss:

I Limits of growth due to environmental constraints

I How environmental issues could be implemented in growthmodels

I Some arguments in the debate about limits to growth,sustainability, and de-growth

Main text:

I Xepapadeas, A. (20015), Economic Growth and theEnvironment. In: Handbook of Environmental Economics,Volume 3. Edited by K.-G. Maler and J.R. Vincent. Elsevier.

p.251


Recapitulation from chapter 1:

I Using renewable and non-renewable natural resources.

I Using the absorptive capacity of the environment for pollution(e.g. greenhouse gas, water pollution)

I Erosion of soil, destruction of forests, extinction of species,destabilization of eco-systems

⇒ Debate initiated by the “Club of Rome”: Meadows, D. et al.(1972), The Limits to Growth.

⇒ Debate about de-linking economic growth from environmentalimpact, “green growth” and “de-growth”.

p.252


Birth of Ecological Economics, addressing the shortcomings ofneoclassical approach to the environment:

I Taking more seriously the complexity of ecological systemsand the physical dimension of production and consumptioninto account.

I Shiftiing focus from growth to “sustainability” concepts.

I Pointing out the problematic role of international trade and“outsourcing of environmental pressure”, imbalances betweenprosperity and environmental damages.

I Ethical discourse regarding the capabilities of futuregenerations.

I Drawing into question that higher consumption = higherwelfare or subjective well-being.

Also mainstream economic growth theories acknowledged these

criticisms.p.253


A) Limited natural inputs (resources)

A.1) Non-renewable resources:

I Production can be maintained if use of resource declines over time(so that it is “asymptotically exploited”). Requires a substitution byother inputs. Question whether permanent and unlimitedsubstitutability is possible.

I Structural change in production technology might make theproduction independent from this specific resource ⇒ Backstoptechnologies: non-renewable resources could be extractedcompletely if a backstop technology is available for substitution(typically based on renewable resources).

I Price system serves for appropriate incentives.

I Special focus on changing the material flow from a linear flow(extraction – production – consumption – waste) to closed cycles.

p.254


A.2) Renewable resources:

I Although being renewable, these resources can be“over-harvested” (example: fishery), thus the ecologicaldynamics impose a limit to economic utilization.

I Complex ecological feedback dynamics possible, limitedknowledge about dynamics of reproduction and long-termdestabilization dynamics (e.g. climate change).

I Limits of renewability : erosion of soils, deforestation – longtime horizons of recovery. Possible breakdown of ecosystemand extinction of species possible.

One general result: social planner’s solution differs from marketsolution (market inefficiency).

p.255


B) Pollution output of the economic process

I Limitd absorptive capacity of ecosystems for pollution (air,water, soil, but also noise, radiation).

I Conceptually similar to the use of renewable resources if welook to “absorptive capacity” as a resource.

I Typically modeled as a “negative external effect”.I However:

I Externality perspective limited to existing agents (not futuregenerations)

I Causal links between economic activities and damage might becomplex and controversial.

I Global, cross-border externalities are hard to be internalized bystandard solutions from economic theory.

I How to evaluate “externalities” like extinction of species orhigher vulnerability of ecosystems?

p.256


Material flows: economic system as a part of the globalmetabolism

production

consumption

recyclingecological

dynamics

solar

energy

resource stocks

pollution absorption capacity

A.2

A.1

B

p.257


A) and B) impose Limits to Growth. Moreover:

I On the one hand, economic growth ceteris paribus is harmful for theenvironment as the negative impact is directly linked to theeconomic process.

⇒ This process is fostered by world-wide population growth.

⇒ Potentially negative role of globalization and trade, as thispromotes growth, requires additional resource inputs fortransportation, and might increase cross-border externalities.

I On the other hand, growth happens due to technological change,and is necessarily implying structural change.

⇒ Technological and structural change could reduce environmentalimpact, thus the overall relationship between growth andenvironmental pressure is more ambiguous and complex.

I However, any sort of de-coupling growth from environment throughtechnological and structural change has to be facilitated bygovernment.

p.258


I All these ecological restrictions are summaized as “planetaryboundaries”.

I All economic (growth) processes can be sustained only withinthese boundaries which thus describe the limits of growth andthe possibilities of green grpwth ⇒ but difficult tooperationalize.

Stoknes, P.E, Rockstrom, J. (2018), Redefining green growth within planetaryboundaries. Energy Research & Social Science 44, 41-49

Butler, C.D. (2017), Limits to growth, planetary boundaries, and planetaryhealth. Current Opinion in Environmental Sustainability 25, 59-65

Dearing, J.A. et al. (2014), Safe and just operating spaces for regional

social-ecological systems. Global Environmental Change 28, 227-238.

p.259

4. Growth and the Environment – Growth Limits4.2 Some empirics

Environmental Kuznets Curve (EKC):I Overview:

Dinda, S. (2004), Environmental Kuznets Curve Hypothesis: A Survey.Ecological Economics 49, 431–455.

I Original:Kuznets, S. (1955), Economic Growth and Income Inequality. TheAmerican Economic Review 45(1), 1–28 Income inequality increases withgrowing prosperity but then declines from a certain turning point.

Observations: for somecountries it seems thatpollution (e.g. SO2,NOXetc.) increases with the GDP(per capita) up to a “turningpoint”, and is then declining(imverse U-shaped curve).

SO2 (e.g.)

GDP p.c.

p.260


Explanations:

I With increasing income the demand for clean environmentand thus for environmental policy regulations increase.

I Income growth is driven by tachnological change. Theeco-efficiency of technology improves (due to policy).

I Richer countries are able to externalize pollution-intensiveindustries to other (lower income) countries with a more laxenvironmental policy and re-import the products (“PollutionHaven Hypothesis”, Reinert et al. (2009))

I More eco-sensitive consumption style in very prosperouseconomies?

The empirical evidence for the EKC is mixed (see Dinda (2004))but a couple of studies provide evidence also fo emergingeconomies: Apergis, N., Ozturk, I. (2015), Testing Environmental Kuznets

Curve hypothesis in Asian countries. Ecological Indicators 52, 16-22p.261


Ecological Footprint: (EF)

See: www.footprintnetwork.org,Wackernagel and Rees (1996), Borucke et al. (2013)

I Not focusing on just one environmental variable such like CO2

or SO2 but a bundle of diverse economic influences on theenvironment, including fishery, de-forestation, using land forbuildings etc.

I Each type of environmental impact is translated into globalhectares (gha) of land (to be needed to sustain theenvironmental use).

I The calculation includes equivalence factors to account fordifferent productivities of various types of land, and yieldfactors to account for variations in biological productivityamong nations (details not discussed here). By translatingeverything in gha, a lot of information get lost, but on theother hand, aggregation is possible.

p.262

www.footprintnetwork.org


(cont.)

I For each counry the biocapacity is calculated (in gha).Because of changing productivities, the world biocapacity isnot given and fixed but can change in time.

I For each country the production activities located in this arealead to various ecological impacts, measured by the aggreaedgha relative to the biocapacity, or the EF of production.

I For each country the environmental impact of theconsumption is measured as well. Due to specialization andtrade, this EF of consumption differs from the EF ofproduction.

I The difference is the EF embodied in trade.

p.263


(cont.)

I Aggregating all countries around the world, the global EF ofproduction and EF of consumption should be identical. Forassessing the sustaiability of the country’s economy, the EFof consumption is used.

I As we have seen in chapter 1, the EF of consumption in ghaexceeds the biocaacity in gha, indicating non-sustainablegrowth. The main reason for non-sustainability is the CO2

emission.

I Critical debate of EF in the literature is briefly summarized in

Dam, T.A., Pasche, M., Werlich, N. (2017), Trade Patterns and the

Ecological Footprint – a theory-based Empirical Approach. Jena

Economic Research Papers No. 2017-005.

p.264

https://ideas.repec.org/p/jrp/jrpwrp/2017-005.html

https://ideas.repec.org/p/jrp/jrpwrp/2017-005.html


(For the following see Dam et al. (2017).)

I Positive correlation of GDP per capita and EcologicalFootprint: richer countries contribute much more tounsustainability in absolute terms.

I However, the eco-productivity (GDP/EF) also increases withthe per capita income: richer countries are ecologically moreefficient. This is in line with the EKC hypothesis.

I One explanation is the stricter (and more strictly enforced)environmental policy in richer countries.

I This also applies to the relation GDP and EF of consumption(therefore not just an “outsourcing” effect). However, there isalso evidence for the “Pollution Haven” effect.

I This is consistent with the rebound idea (dicussed later on):higher eco-efficiency is overcompensated by growth.

p.265


Human Development Index and Ecological Footprint

Source: www.globalfootprint.org

p.266


Source: Dam, Pasche, Werlich (2017)

p.267


The Rebound effect:

Binswanger, M. (2001), Technological progress and sustainabledevelopment: what about the rebound effect? Ecological Economics36(1), 119-132

Sorrell, A., Dimitropoulos, J. (2008), The rebound effect: Microeconomicdefinitions, limitations and extensions. Ecological Economics 65(3),636-649

I De-linking output and pollution makes production andconsumption more “green” / eco-effiient.

I This might have behavioral consequences:I Less resource input or less pollution taxes reduces the price.

Demand increases according to price elasticity (“Jevonsparadox”, W.S. Jevons (1865), “The Coal Question”)

I Consumer might feel “absolved” due to the greening effect,and compensate this by higher consumption level.

I Even without these effects, overall growth ceteris paribuscountervails the effect of greening.

p.268


Is there there green growth in the OECD countries?

(de-coupling GDP growth from Ecological Footprint)

Pasche, M. (2018), Is there green growth in OECD countries?MPRA Working Paper No.87726 (downloadable).

p.269

https://mpra.ub.uni-muenchen.de/87726/


Ecological Footprint development 1961-2014 (in gha)

1960 1970 1980 1990 2000 2010

0

0.5

1

1.5

2

·1010

Year

Eco

log

ica

lF

oo

tpri

nt

World

OECD + China

OECD

p.270


Eco-productivity in OECD 1961-2014 (1961=100)

1960 1970 1980 1990 2000 2010

100

200

300

400

Year

pro

du

ctiv

ity

eco-productivity prod.

eco-productivity cons.

labor productivity

1960 1970 1980 1990 2000 2010

−2

0

2

4

6

Year

eco-productivity growth

labor productivity growth

p.271


Environmental taxes and eco-productivity since 1994

5.5 6 6.5 7 7.5

·105

5

6

7

env. tax volume

eco

effici

ency

(sca

led

)

p.272


Eco-productivity and net EF imports in OECD (1961 = 100)

1960 1970 1980 1990 2000 2010

100

200

300

Year

real GDP/EF production

real GDP/EF consumption

net EF import

p.273


The role of net EF imports in OECD

1960 1970 1980 1990 2000 2010

0

2

4

6

·109

Year

EF production

EF consumption

net EF exports

p.274


Reduction of EF due to GDP growth slowdown

−4 −2 0 2 4 6

−5

0

5

GDP growth

EF

gro

wth

before 2005

after 2005

1960 1970 1980 1990 2000 20103

4

5

6

7

·109

Year

Eco

log

ica

lF

oo

tpri

nt

with constant GDP growth

with GDP growth slowdown

p.275


Goal that EF in 2030 should 40% less than in 1990:

−2 0 2 4 6−6

−4

−2

0

2

4

6

Eco-efficiency growth

GD

Pg

row

th

p.276


Summary of the paper:

I OECD countries are successful in de-coupling both,consumption and production from environmental presuure(EF).

I However, this process is too slow (eco-productivity gainssmaller than GDP growth rates).

I Absolute decline of EF only because of growth slowdown.

I Globally main driver of increasing EF is now China.

I Achieving the goal that in 2030 the EF should be 40% lessthan in 1990 would require either massive eco-productivitygains or de-growth.

p.277

4. Growth and the Environment – Growth Limits4.3 Innovation, Structural Change, and the Environment

Non-renewable resources and growth: Hotelling model

Bergstrom, J.C. (2016), Resource Economics. An Economic Approach to

Natural Resource and Environmental Policy, 4th ed. Edward Elgar.

Is growth possible if output depends on non-renewable resources?

Household maximizes

maxC ,R

U =

∫ ∞0

e−ρtu(C (t))dt

subject to

K = F (K ,R)− C

S = −R

where R is the resource extraction, S(0) > 0 is the given stock ofresources. Also K (0) > 0 is given.

p.278


I Dynamic efficiency requires that the entire stock of resourcesmust be used: ∫ ∞

0

R(t) = S(0)

I The Hamiltonian is

H = u(C (t)) + λ1(F (K ,R)− C )− λ2R

I From the FOC and CE (costate variables K and S) we have

U ′(C ) = λ1

λ1FR = λ2 ⇒ λ1 + FR = λ2

λ1 = −λ1FK ⇒ λ1 = −FK

λ2 = 0 ⇒ λ2 = 0

I Using these equations we obtain the famous Hotelling Rule:

FR = FK (= r)

p.279


Interpretation:FR = FK (= r)

The marginal productivity of the resource – and hence the resourceprice – must permanently grow according to the interest rate.

I Whether such an optimal path for K and R exists, depends onthe elasticity of substituion between these factors becauseresource input has to be substituted by accumulated capital.

I This is possible in case of a standard neoclassicalCobb-Douglas function with substitution elasticity σ = 1 bute.g. not for a limitational production function (σ = 0).

p.280


For σ = 1:

K

R

K(0)A

R(0)

A: optimal R(0), given K(0)

B

B: transition determined by Hotteling rule and A

increasing capital intensity

CC : transition determined by Hotteling rule and B

C ′ C ′ is inefficient and leads to∫R(t)dt > S(0)

p.281


I If we assume that at least for high values of K/R thesubstitution elasticity is σ < 1, then the Hotelling rule requiressubstituion rates which are impossible to achieve.

I If resource input is not reduced according to the optimalityconditions, the resource is overexploited :

∫R(t)dt > S(0)

which means that the stock will decline to zero in finite time.

I This holds true for any growth rate. Thus, permanent growthis not possible due to limited substitutability.

I Technological change, i.e. the emergence of “backstoptechnologies” based on renewable resources (reproducablefactors) would be necessary.

p.282


For σ < 1:

K

R

K(0) A

R(0)

B

CKC

RC

C : optimal path to C not possible,

RC requires impossible amount of K ,

or KC requires larger amount of R...,

C ′

C ′: ...,which leads to∫R(t)dt > S(0)

⇒ no permanent growth possible

p.283


Pollution in economic growth models:

Xepapadeas, A. (20015), Economic Growth and the Environment. In:

Handbook of Environmental Economics, Volume 3. Edited by K.-G. Maler and

J.R. Vincent. Elsevier. (Sections 3.1 and 4.1)

The Solow model with pollution

I Consider the Solow model with exogenous labor-augmentingtechnological progress (with g as rate of progress).

I Pollution is a side-effect of production but there is no impactof pollution on production or consumption!

I All per-capita terms are measured in labor efficiency units.I The dynamic equation from the Solow model in per capita

terms is given by

k = sf (k)− (δ + n + g)k

p.284


I Stock of pollution, P, is proportional to output Y :

P = φY −mP (71)

where m denotes a pollution decay (a sort of “recoverydynamics” within the biosphere). This is somehowover-simplistic as there might be complex nonlinear ecologicaldynamics (e.g. captured by an additional factor h(P), notconsidered here).

I Writing pollution also in per capita terms p = P/AL with Aas the efficiency factor which develops according toA(t) = egt , we obtain

p = φf (k)− (m + g + n)p

p.285


I The model is described by two dynamic equations k and pwhich have a stable steady-state (k∗, p∗).

I This implies that absolute values Y ,C ,K but also P aregrowing with the same rate n + g .

I If environmental limits require not to exceed a threshold P,the economic growth has to stop at the level Y = mP/φ.

I Sustained growth is then possible only if we extend the modelby introducing abatement technologies: part of the outputhas to be invested into cleaning/abatement (which thenreduces consumption and/or capital accumulation, of course).

p.286


I Assume that capital can be invested into the production ofoutput (ky ) and into abatement (ka).

I Abatement investments reduce the proportionality betweenoutput and pollution: φ = φ(ka) with φ′ < 0 (for furthertechnical assumptions see Xepapadeas (2015)).

I Households now save a fraction sy of their income forproductive capital accumulation, and sa for abatement. Theresulting dynamic equations are

ky = sy f (ky )− (δ + n + g)ky (72)

ka = saf (ky )− (δ + n + g)ka (73)

p = φ(ka)f (ky )− (m + n + g)p (74)

p.287


I If Inada conditions hold true for ky , i.e. limky→∞ f ′(ky ) = 0there exists a steady-state where all per capita variables don’tgrow but absolute pollution level P grows with the rate n + g .

I However, if the marginal product of capital is bounded to

limky→∞

f ′(ky ) >δ + g + n

sy

then the steady-state does not imply zero growth of allper-capita variables, and we obatin ka/ka > 0. This meansthat we have a steady-state growth of per capita abatementcapital and therefore elimination of pollution.

I Transitory dynamics of the model exhivits an EnvironmentalKuznets Curve: with increasing output, the absolute pollutionlevel first grows accordingly. Form a certain threshold kc itdeclines, and pollution is elimiated in the long-run.

I However, results depends on strong technical assumptions.p.288


The Cass-Koopman-Ramsey model with pollution

I Now we consider optimzing behavior of the household.I For simplicity, we assume no exogenous technical change, no

depreciation, and no population growth (n = g = δ = 0).I We assume that pollution negatively affects the utility of the

household:

u(t) = u(c(t),P(t)), uc > 0, uP < 0

and utility is strictly concave in consumption and convex inpollution.

I Again, we consider the pollution dynamics eq. (71).I The houshold maximizes∫ ∞

0e−ρtu(c,P)dt

subject to the usual intertemporal budget constraint.p.289


I Recall, that for the derivation of the Keynes-Ramesey rule wehave to calculate the time derivative of the FOC. Thereforewe now obtain an additional term because the pollution levelP depends on time:

c

c= σ

(r − ρ+

uc,Puc

P

)I As pollution negatively affects marginal utility of consumption

(uc,P < 0), the household will consume less compared to theoriginal CKR model. If the utility function is fully separable(uc,P = 0) then there is no difference to CKR.

p.290


(cont.)

I As we have assumed n = g = 0, there is a steady-state withc = k = P = 0 and the solution (c∗.k∗,P∗) is essentially thesame as in the CKR model (recall that with P = 0 theadditional term in the Euler equation vanishes ). Thetransitory dynamics, however, is different.

I The result changes if we consider technical change g > 0, aswe have seen above: the per capita values are still constantbut the absolute values – and thus also P – are growing ⇒not consistent with stable environment.

I Again, we can escape from this by assuming boundaries of themarginal capital productivity (like in the Solow model).Pollution will then vanish (also unrealistic).

p.291


So what can be learned from the CKR model with pollution?

I The household optimizes with respect to the variablescontrolled by him. He does not consider the pollutionexternality according to eq. (71).

I Thus, a social planner will take the externality into account,i.e. maximizing utility s.t. both constraints.

I The Hamiltonian is therefore (recall n = δ = 0):

H = u(c ,P) + λk(f (k)− c) + λp(φf (k)− P)

The second costate variable λp is interpreted as the “shadowcost of pollution”.

p.292


I The FOC and the CE are given by

uc(c ,P) = λk

λk = (ρ− f ′(k))λk − λpφf ′(k)

λp = (ρ+ m)λp − uP(c ,P)

plus the dynamic equations for k and P.I The resulting Euler equation is now

c

c= σ

f ′(k)︸︷︷︸r

[1 +

λpφ

uc(c ,P)

]− ρ+

uc,Puc

P

I Bracket term is additional to the market solution. As this

term contains c , the isocline is not a vertical line at point k∗

any longer (see graphic in section 2.4) but downwards sloped ⇒intersection point of both isoclines (steady-state) is at a lowerlevel than in the original CKR model without pollution.

p.293


I Social optimum has to be implemenetd by a tax-revenuescheme (not considered here).

I Further extensions:I Including abatement technologiesI Nonlinear pollution accumulationI Endogenous growth etc.

p.294


Some general results:

I No environmentally sustainable growth (P = 0) if there is agiven emission coefficient φ and no permanent investmentinto abatement.

I If there is an abatement technology which helps to de-linkgrowth and pollution, it depends on the properties of theabatement technology whether permanent growth with stablepollution is possible. It should have non-dimishing returns inorder to enable environmentally sustainable growth.

I As pollution is an externality, social planner demands forpolicy intervention.

p.295


Limits to reasonable growth:

Pasche, M. (2001), Technical Progress, Structural Change, and the

Environmental Kuznets Curve. Ecological Economics 42 (3), 381 – 389.

A) Growth and structrual change:

I Consider that output is composed of pollution-intensive and“green” products. Thus emissions are

E = γpYp + γgYg , γp > γg (75)

I Sustainable growth with E ≤ 0 is possible as long as

Yg ≤ −γpγg

Yp

The green sector can grow if and only if thepollution-intensive sector is shrinking accordingly. The overalloutput level Y = Yp + Yg can grow without increasingpollution as long as this inequality holds true.

p.296


However, this process must come to an end latest when there is nooutput Yp any more (see graphic). Henceforh, structural changealone cannot guarantee growth with constant pollution:

Yp

Yg

E = γgYg + γpYp

growth = shift of production frontier

T0

T1

T2

T3

T4

p.297


Even worse, the process becomes unreasonbale if growth meetsthe inequality condition but leads to lower welfare! The thresholdfor this is when the tangential point of transformation curve andwelfare indifference curve is on the constraint (75) (see graphic).Then further pollution preserving growth reduces welfare :

Yp

Yg

E = γgYg + γpYp

T0

T1

T2

T3

p.298


B) Technological change and abatement:

I Consider the following simple structure:

Emissions: E = γY . It follows the sustainability condition

E = γ + Y ≤ 0 (76)

Pollution stock: P = E − δP, 0 < δ < 1Income composition: Y = C + A + IHere, A are current abatement expenditures while I areinvestments into the stock of eco-efficiency enhancingproductive technologies K with K = I .We consider γ = F (A,K ).

I Therefore, we have from (76):

Y ≤ −γ = −εK K − εAA

with εK = ∂F∂K

KF < 0 and εA = ∂F

∂AKA < 0.

p.299


I It is reasonbale to assume that within a given technoloicalregime there are physical limits of enhancning eco-efficiency,i.e. that the effort to reduce fuel consumption or pollutionemission has permanently to increase. This effect is modelledby declining returns to scale: |εK |+ |εA| < 1.

I A fraction of income has to be spent for A = aY and I = sY(with s + a ≤ 1) so that sustainability condition reads

Y ≤ −(

εK1 + εA

)K −

(εA

1 + εA

)a

I The assumption of decreasing returns imply that the twocoefficients are between 0 and 1. It can be shown that thereexists no constant positive growth rate Y which is consistentwith the sustaiability condtion (76). This result is in line withthe literature discussed above.

p.300


I Moreover: Growth of Y which is consistent with (76) isreasonable if it leads to higher consumption, C ≥ 0. Thisimplies

Y ≥ s + a

1− (a + s)

In case of s + a > 0 the r.h.s. would approach infinity and isthus inconsistent with sustained growth. In case of s + a ≤ 0it can be shown that this violates the sustainability condition(75). This happens in finite time.

⇒ Therefore, under these conditions, the EKC will not be along-run phenomenon. In the long-run we could expect aN-shaped curve (rebound).

p.301


Implication for welfare:

I For keeping the sustainability condition the economy mustspent a growing part of the GDP into abatement andeco-efficiency. Once when this growth exceeds GDP growth,consumption must decline. More than GDP growth has to bespent in order to compensate the nagtive environmentalimpact of this GDP increase. This leads to lower consumptionand welfare and is thus unreasonable.

I The threshold of growth “running idle” happens in finitetime before the limits of sustainability are reached!

I Note that “environmental industries” create jobs and incomewhich are counted as a part of the GDP. So the greeneconomy might be seen as a sign of prosperity. However, oneshould acknowledge that this indicates a huge effort tocompensate the negative impact of growth.

p.302


C) Radical technological change:

I Crucial assumption of decreasing returns of abatement andeco-efficiency enhancing technology is debatable but appearsreasonbale within a technological regime.

I Technological regimes might change due to technologicalchange (basic innovations, disruptive innovations).Example: from Otto motor based automotive sector to electicvehicles (or fuel cells).

I Such disruptive technological change is possible (and likely)but cannot be planned and predicted.

I We cannot know in advance whether the properties of a newtechnlogical regime are really better for the environment (e.g.unknown risks in case of nano particles or geneticallymanipulated plants).

I Possible de-linking growth and pollution is then a “gift” fromtime to time which we cannot rely on.

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4. Growth and the Environment – Growth Limits4.4 Sustainability and De-Growth

Growth, consumption, and subjective well-being:

I All discussed growth models are based on a very common, but alsovery critical assumption: u(c) with u′(c) > 0 for all c . Welfarestems from consumption only, and utility is characterized bynon-saturation. The existence of a utility function is based on givenand fixed preferences.

I The assumption of non-saturation is unrealistic.However, as a proxy it might be justified by e.g.:

I Love for variety : behind the aggregated consumption variablethere is a bundle of different goods with increasing variety.Assume that people prefer to choose among a larger set ofvarieties (Lancaster 1979, Dixit/Stiglitz 1977).

I Status consumption: the additional utility does not comedirectly from consuming more than before but from consumingmore than the neighbour (or something which symbolizes ahigher social status). So utility is only indirectly related to theconsumption good as such.

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I A broader (evolutionary) approach argues that consumer do nothave given preferemces but “learn to consume” in a social context.

Witt, U., 2001. Learning to consume – a theory of wants and the growth

of demand. Journal of Evolutionary Economics 11, 23–36.

I As it is known from Behavioral Economics, the nexus betweenincome or consumption and “happiness” is not linear: happinessincreases up to a certain threshold, and is then independent fromfurther income growth.

Easterlin, R.A. (2001), Income and Happiness: Towards a Unified TheoryAuthors. The Economic Journal 111, 465–484.

Deaton, A. (2008), Income, health, and well-being around the world:

Evidence from the Gallup World Poll. Journal of Economic Perspectives

22, 53-72.

I Contrary findings:

Lindqvist, E., Ostling, R., Cesarini, D. (2018), Long-run Effects of Lottery

Wealth on Psychological Well-being IFN Working Paper No. 1220p.305


What happens if consumers do not prefer to consume permanentlymore (“volunrtary stagnation”):

I Preference for a more simple and lean lifestyle.

I Technical progress is used for having more leisure instead ofmore consumption.

I Investment and technical progress used for reduction ofenvironmental pressure.

Example: Bilancini, E., D’Alessandro, S., 2012. Long-run welfare under

externalities in consumption, leisure, and production: a case for happy

degrowth vs. unhappy growth. Ecological Economics 84, 194–205.

p.306


I Firms are under competitive pressure and have a strongincentive to invest into R&D in order to gain temporarymonopolistic profits (see chapter 3). To some extent they canuse (i) own savings, (ii) demanding loans. They do notdepend on pre-existing savings from households.

I Once, when new products are developed which might be animprovement to older products, and if there is enoughmarketing, this might overcome the “voluntary stagnation”effect (if it exists).

⇒ Is thus growth an unwarrantable “inherent pressure” of thecapitalistic system? Or an unavoidable consequence of humancuriosity and creativity?

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Sustainability:

Costanza, R., Patten, B.C. (1995), Defining and predicting sustainability.Ecological Economics 15, 193-196

Daly, H.E. (1996), Beyond Growth: the Economics of SustainableDevelopment. Beacon Press.

I Ability to maintain defined operations within anenvironmental system indefinitely.

I Environmental sustainability: maintaining the rates ofharvetsing renewable resources, pollution emission, andnon-renewable resource depletion; maintining the stability ofecosystems and their “resilience” to human impact.

I Economic sustainability: maintaining a defined level ofeconomic activities indefinitely.

p.308


I Limitations:I Really “indefinitely”? What is the time horizon?I What is the relevant environmental system (local, global,...)?I What do we know about the (eventually changing) conditions

of being able to maintain an activity?I Why should we accept sustainability as a normative concept?

I Questionable whether any sort of exponential growth processcould be consistent with sustainability requirements.

Hueting, R. (2010), Why environmental sustainability canmost probably not be attained with growing production.Journal of Cleaner Production 18(6), 525–530.Ayres, R.U. (1996), Limits to the Growth Paradigm. EcologicalEconomics 19(2), 117-134Daly, H. (2013), A further critique of growth economics.Ecological Economics 88, 20-24

p.309


De-Growth debate:

I National governments as well as all international organizations(OECD etc.) are seeking for growth.Is there a “Growth Paradigm”?

I After pessimistic “Limits to Growth” debate (in the afterneathof Club of Rome), more optimistic “Green Growth” debate,based on the ideas of structural change, innovation, andde-linking from environmental pressure.

I However:I Empirical picture shows that rebound effects prevent from

global de-linking. Growth is too fast for making de-linkingwork. Too many countries are in need for “catching-up”.

I Greening the economic process requires increasing effeortwhich might lead to unreasonbale (“idle”) growth.

I Limits of substitutability of non-renewable resources byre-producable capital or renewable resources.

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I More fundamentally: Whatever we are producing, there isnecessarily a material transfomation of matter and energy,leading to thermodynamic dissipation, making sustainable andexponential growth impossible.

Georgescu, N. (1971), The Entropy Law and the EconomicProcess.Daly, H.E., Farley, J. (2004), Ecological Economics: Principlesand Applications. Island Press.Daly, H.E. (1991), Steady-State Economics, 2nd Ed. IslandPress, Washington D.C.

I Concept remains a bit vague: What should de-grow?GDP, consumption, physical acitivites (materialtransformation), etc.

⇒ Physical activities – but GDP is a value (not quantity)measure

p.311


Debate about de-growth (addressing primarly highly developedcountries, not poor countries):

I In order to achieve sustainability, economic activities first haveto be reduced, implying negative GDP and consumptiongrowth rates:

Martınez-Alier, J. et al. (2010), Sustainable de-growth: mapping the

context, criticisms and future prospects of an emergent paradigm.

Ecological Economics 69 (9), 1741–1747.

I GDP is anyway not a good indicator for subjective welfare,and thus GDP growth is a questionable goal:

Stiglitz, J.E. (2009), GDP Fetishism. The Economists’ Voice, 6(8), article

5.

p.312


From de-growth to a-growth:

van den Bergh, J.C.J.M. (2011), Environment versus growth –A criticism of

“degrowth” and a plea for “a-growth”. Ecological Economics 70(5), 881-890

I As growth is driven by technological progress and structuralchange, these sources of growth are also the sources ofde-linking economic activities from environmental pressure!

I Public and political resistance against degrowth strategieshave to be acknowledged.

I Idea of “a-growth”: does not mean to propagate a zero ornegative growth rate but being indifferent regarding thegrowth of specific variables (e.g. GDP) as long assustainability criteria are met.

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Further problem:

I Other goals of economic and social policy are linked togrowth, e.g. sustainable public finance:

Pasche, M. (2018), Degrowth and sustainable public finance. MPRA

Working Paper No. 87109 (downloadable).

I If we we have to consider ecological limits to growth as animperative contraint, and thus growth is possible only to theextent to which it is de-coupled from environmental pressure,the oher economic and social policy measure should not relyon ever preserving GDP growth rates (becoming independentfrom growth).

p.314

https://mpra.ub.uni-muenchen.de/87109/

chair of macroeconomics - growth and innovation · 1.1 some stylized facts however, some of these...

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