MakØk3, Fall 2012 (Blok 2)

�Business cycles and monetary stabilization policies�

Henrik JensenDepartment of EconomicsUniversity of Copenhagen

Lecture 2, November 20: A Classical Model (Galí, Chapter 2)

c 2012 Henrik Jensen. This document may be reproduced for educational and research purposes, as long as the copies contain this notice and are retained for personaluse or distributed free.

Introductory remarks

� The standard neoclassical model for (exogenous) economic growth is the simple Solowmodel featuringa �xed savings rate (no microfoundations)

� When extended with optimizing savings behavior, we get the Ramsey model

�Heavily applied model in macroeconomics

� Models have no role for money and monetary policy

� Purpose of this lecture is to introduce money in such a classical model with microfoundations

� How will money and monetary policy a¤ect the economy?

�Does this type of �workhorse�model give implications and predictions that looks like the realworld (so it could be a useful tool)?

� While the end result is not uplifting, we luckily learn a lot of material that are extremely useful innext lectures

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A basic classical model

� Goods market

�Demand side: Households consume (based on utility maximization)

�Supply side: Firms produce consumption good (maximize pro�ts under perfect competition)

� Labor market

�Demand side: Firms hire labor (maximize pro�ts under perfect competition)

�Supply side: Households supply labor (based on utility maximization)

� Financial markets

�Households optimally invest in a one-period risk-less bond

�Households also hold money. Why? We just assume it for now

� All prices are perfectly �exible

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Household behavior

� Household maximize expected utility

E01Xt=0

�tU (Ct; Nt) ; 0 < � < 1: (1)

� Budget constraint:PtCt +QtBt � Bt�1 +WtNt � Tt (2)

(and a No-Ponzi Game constraint ruling out explosive debt)

� Household chooses optimal paths of Ct, Nt and Bt (taking prices Pt, Wt and Qt as given)

� Let�s solve its maximization problem using a Lagrangian method

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@L@Ct

= 0; Uc;t � �tPt = 0; Uc;t = �tPt (*)

@L@Nt

= 0; Un;t + �tWt = 0; Un;t = ��tWt (**)

@L@Bt

= 0; �tQt � �Et f�t+1g = 0; �tQt = �Et f�t+1g (***)

� From (*) and (**):�Un;tUc;t

=Wt

Pt(4)

� From (*) and (***)Qt = �Et

�Uc;t+1Uc;t

PtPt+1

�(5)

� With U (Ct; Nt) � C1��t1�� �

N't

1+', �; ' > 0,

C�N't =

Wt

Pt(6)

Qt = �Et

(�Ct+1Ct

���PtPt+1

)(7)

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� Log-linear versions of households�optimality conditions (lower-case letters are logs or log-deviationsfrom steady state)

� Labor supply:�ct + 'nt = wt � pt (8)

� Bonds pay out 1 the period after the purchase at price Qt. Bond yield is:

yield =1�QtQt

1 + yield = Q�1t

log (1 + yield) = � logQtyield ' � logQt

� it

This is the nominal interest rate

� With this de�nition, consumption-Euler equation becomes:

ct = Et fct+1g � ��1 (it � Et f�t+1g � �) (9)

where �t+1 � pt+1 � pt and � � � log �

� Money demand is postulated (for now):

mt � pt = yt � �it; � > 0 (10)

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Firms�behavior

� Production function for the representative �rm

Yt = AtN1��t ; 0 < � < 1 (11)

� Pro�ts in each period t:PtYt �WtNt (12)

� Firms choose Nt to maximizePtAtN

1��t �WtNt

� First-order condition

(1� �)PtAtN��t = Wt;

(1� �)AtN��t =

Wt

Pt(13)

� Labor demand in logs:log (1� �) + at � �nt = wt � pt (14)

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Equilibrium outcomes

� Goods market clearing (or, resource constraint, or, national account):

yt = ct (*)

� Labor market equilibrium (labor supply equals labor demand):

�ct + 'nt = log (1� �) + at � �nt (**)

� Bond market clearing (household�s intertemporal consumption decision with goods market clearing)

yt = Et fyt+1g � ��1 (it � Et f�t+1g � �) (***)

� Production function in the aggregate:

yt = at + (1� �)nt (****)

� (*), (**) and (****) gives ct, nt and yt

� (***) gives subsequently the real interest rate rt � it� Et f�t+1g

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� Typical classical features: Real variables like consumption, output and employment (and thus thereal wage) are determined from the economy�s supply side

� No role for monetary policy here

�We haven�t used the money demand function at all

�The real interest rate is determined independently of monetary factors

�Any �uctuation in main macro variables will result from �uctuations in at

� We need not even say anything about the nominal interest rate or the price level

�If we want to determine nominal variables, however, we must specify monetary policy

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Monetary policy and pricesThe nominal interest rate as policy instrument

� Most central bank uses the nominal interest rate as policy instrument

� Model Problem: Specifying some exogenous path for the nominal interest rate in the model will notdetermine the price level. Note the �Fisherian relationship�:

Et f�t+1g = it � rtwhere rt is independent of monetary policy

� Hence, (a rational expectations di¤erence equation)

Et fpt+1g = pt + it � rt (*)

� We can now introduce a shock which has nothing to do with the economy, �t+1: Et f�t+1g = 0. Thenany price level satisfying

pt+1 = pt + it � rt + �t+1is consistent with (*).

� The price level is indeterminate

�Likewise the money supply: mt = pt + yt � �it�Likewise the nominal wage

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� An exogenous path for the nominal interest rate does not help determine nominal variables in theclassical model

� The problem is that the exogeneity leaves the economy without a �nominal anchor�

� A solution is to specify the nominal interest rate as a �feedback rule�where it is speci�ed to beadjusted in response to some nominal variable

� One possibility (out of many!)it = � + ���t; �� > 0

� From the Fisher equation:

���t = Et f�t+1g + rt � � = Et f�t+1g + brt (22)

� If �� > 1, we have a unique stationary (i.e., non-explosive) solution

�t =1Xk=0

��(1+k)� Et fbrt+kg (23)

� If �� < 1, we have that any path of �t satisfying (24) is an equilibrium (so, we have indeterminacy)

�t+1 = ���t � brt+k + �t+1 (24)

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� Taylor (1993):

The Taylor Rule:it = � + 1:5 (�t � ��) + 0:5yt

Note the ��� > 1�feature: This is the �Taylor principle�

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The nominal money supply as policy instrument

� In this case the economy has a �nominal anchor�: Consider an exogenous path for mt

� Combine the money demand equation and the Fisher equation

mt � pt = yt � �itit = Et fpt+1 � ptg + rt

pt = �yt + � (Et fpt+1 � ptg + rt) +mt

Hence,

pt =�

1 + �Etpt+1 +

1

1 + �mt + ut

ut �1

1 + �(�rt � yt) (independent of monetary policy)

� Since � > 0 we have a unique stationary (i.e., non-explosive) solution

pt =1

1 + �

1Xk=0

��

1 + �

�kEt fmt+kg + u0t

u0t �1Xk=0

��

1 + �

�kEt fut+kg (independent of monetary policy)

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� In terms of expected future money growth

pt = mt +

1Xk=1

��

1 + �

�kEt f�mt+kg + u0t (25)

(where � is �rst di¤erence operator)

� For given expected money growth, prices move one for one with the money supply

� The nominal interest rate is found by inserting (25) into the money demand function

it = ��1 (yt �mt + pt)

= ��11Xk=1

��

1 + �

�kEt f�mt+kg + u00t

u00t � ��1 (yt + u0t) (independent of monetary policy)

� The nominal interest rate increases with expected future money growth

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An example of a process for nominal money growth

� Let�mt = �m�mt�1 + "

mt

� Assume no real shocks (hence yt and rt does not move; assume they are zero)

� Solution for prices and nominal interest rate

pt = mt +��m

1 + � (1� �m)�mt;

it =�m

1 + � (1� �m)�mt

� If money growth is persistent there are large e¤ects on prices and positive e¤ect on interest rates

�No liquidity e¤ect; i.e., an expansionary money growth shock increases the nominal interest rate(and real money falls equivalently)

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Do all classical models leave monetary policy �useless�?

� No

� Whilemoney neutrality prevails, many models with �exible prices can have e¤ects of monetary policythrough variations in in�ation and the nominal interest rate

�There will not be monetary superneutrality

� Galí presents a model where money enters the utility function

U = U (Ct;Mt; Nt)

� Short cut for money�s liquidity services (e.g., saved leisure)

� The model features a micro-founded money demand relation just as postulated before

� Labor supply can now be a¤ected by changes in the nominal interest rate:

�Un;tUc;t

=Wt

Pt

is a¤ected if marginal rate of substitution is a¤ected by money

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Dynamic e¤ects of money shocks in a classical money-in-the utility function (fromWalsh, 2010)

� To assess the quantitative e¤ects of money shocks, a money-in-the-utility function model is calibratedand simulated

Calibration: Assign empirically plausible values to the parameters of the model

Simulation:

�Perform a linearization of the model�s dynamic equations (everything is expressed as percentagedeviations from steady state);

�solve this system by numerical methods (various simulation programs are available on the inter-net);

�create arti�cial time series data from the system

� From the arti�cial data, one evaluates the properties of the model in terms of:

�Standard deviations of various relevant variables, and their s.d. relative to output

�Correlation coe¢ cients of various variables with output

�Impulse response patterns of variables when shocks hit

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� Main results (when Ucm > 0)

�Steady-state non-superneutrality is of the form of: Higher money growth =) higher in�ationand nominal interest rate =) lower money demand =) lower marginal utility of consumption=) lower labor supply =) output

�If money growth shocks, "mt , shall pay a role, persistence in money growth is necessary (�m > 0is needed). Then, the shock will a¤ects expected next-period in�ation, and thus � through theFisher equation � period t nominal interest rate.

�The e¤ects of money shocks on labor and output are stronger the more persistence in moneygrowth, but the e¤ects are quantitatively very small (nothing compared to what we saw fromVARs)

�Main e¤ects of money shocks are on in�ation and nominal interest rates

�Positive money shocks lead to higher nominal interest rates. In contrast with liquidity e¤ect seenin VARs (and in contrast with usual IS/LM story where nominal rates fall to increase moneydemand). Reason is �exible prices:

� Prices adjust instantaneously so as to reduce real money supply, matching the fall in demandresulting from higher nominal interest rates.

17

Source: Walsh, (2010): Monetary Theory and Policy (The MIT Press) (�u = �m)

18

Source: Walsh, (2010): Monetary Theory and Policy (The MIT Press) (�u = �m)

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Summary� The classical model has little role for money in determination of real variables

� It should be seen as a model of the long run, or, the average hypothetical evolution of the economy

� I.e., it is important as a benchmark for showing the e¢ cient allocation in a micro-founded economy(i.e., an ideal, but unrealistic, world)

� Some, like the money-in-the-utility functionmodel can be used to assess the optimal long-run in�ationrate

�Optimum: Um = 0. This requires 1 � exp f�ig = 0 [cf. eq. (28)], or simply i = 0 and thus� = �r

�The Friedman rule; private opportunity cost of money is equal to social costs (zero)

� Model is not suitable for analyzing the short run implications of monetary shocks as the models, bynature, exhibits monetary neutrality (although not necessarily superneutrality). Impulse responsepatterns are unrealistic, even when there are e¤ects on real variables

� Short and long run are virtually indistinguishable

� To remedy the short-run failure of such models, one must introduce nominal rigidities; This is theaim of New Keynesian Theory

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Next time

Monday, November 26:Lectures: The Basic New Keynesian Model (Galí, 2008, Chapter 3)

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