Monetary Policy in a Stochastic Equilibrium Model with Real And

Home , Charles Plosser

Monetary Policy in a Sto chastic Equilibrium

Mo del with Real and Nominal Rigidities

Jinill Kim

First draft: July 1995

This draft: March 25, 1996

Abstract

A dynamic sto chastic general-equilibrium (DSGE) mo del with real and

nominal rigidities succeeds in capturing some key nominal features of U.S.

business cycles. Monetary p olicy is sp eci ed following the developments in

the structural vector autoregression (VAR) literature. Four sho cks, including

b oth technology and monetary p olicy sho cks, a ect the economy. Interaction

between real and nominal rigidities is essential to repro duce the liquidity e ect

of monetary p olicy. The mo del is estimated by maximum likeliho o d on U.S.

data. The mo del's t is as go o d as that of an unrestricted rst-order VAR and

that the estimated mo del pro duces reasonable impulse resp onses and second

moments. An increase of interest rates predicts a decrease of output two

to six quarters in the future. This feature of U.S. business cycles has never

b een captured by previous research with DSGE mo dels. Lastly, the p olicy

implications are discussed.

Dept. of Economics, Yale University,Box 208268 Yale Station, New Haven, CT 06520-8268.

E-mail: [email protected]. Sp ecial thanks are due to Christopher Sims for his guidance and

supp ort. Imp ortant discussions with William Brainard, Giancarlo Corsetti, Jordi Gali, Federico

Galizia, Mark Gertler, Rob ert Shiller and T. N. Srinivasan are gratefully acknowledged. I also

thank the participants in job market seminars at the Board of Governors of the Federal Reserve

System, Federal Reserve Banks of New York and Richmond, the University of Cambridge, Univer-

sitat Pomp eu Fabra, SUNY at Stony Bro ok, Washington UniversityinSt. Louis, and the Wharton

Scho ol. 1

Contents

1 Intro duction 3

2 The Mo del 8

2.1 The Aggregator : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 9

2.2 Households : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10

2.2.1 Preferences : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10

2.2.2 Real Rigidities via Capital Adjustment Costs : : : : : : : : : 12

2.2.3 Wage Rigidities via Wage Adjustment Costs : : : : : : : : : : 13

2.2.4 Budget Constraints and First Order Conditions : : : : : : : : 13

2.3 Firms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15

2.3.1 Technology : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16

2.3.2 Price Rigidities via Price Adjustment Costs : : : : : : : : : : 17

2.3.3 Value of Firms and First Order Conditions : : : : : : : : : : : 18

2.4 The Government : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20

2.5 The Equilibrium : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 22

3 E ects of Monetary Policy 26

3.1 Impulse Resp onses : : : : : : : : : : : : : : : : : : : : : : : : : : : : 26

3.2 Estimation Pro cedure : : : : : : : : : : : : : : : : : : : : : : : : : : : 29

3.3 Estimation Results : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31

3.4 Assessing the Fit : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 35

3.5 Variance Decomp ositions : : : : : : : : : : : : : : : : : : : : : : : : : 39

3.6 Cyclical Implications : : : : : : : : : : : : : : : : : : : : : : : : : : : 42

3.7 Policy Exp eriments : : : : : : : : : : : : : : : : : : : : : : : : : : : : 43

4 Conclusion 46

A App endix 48

A.1 The Data : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 48

A.2 Derivatives and Elasticities : : : : : : : : : : : : : : : : : : : : : : : : 48

A.3 The Equilibrium : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 49

A.4 Steady State : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 50

A.5 Log-linearization : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 51 2

1 Intro duction

The comovement of monetary and real aggregates and the inverse relation b etween

the movements of money growth and nominal interest rates are two prominent nom-

inal features of business cycles in the United States and many other countries.

The strong and stable covariation of monetary aggregates and aggregate output has

prompted several economists to fo cus on monetary instability as a cause of the Great

Depression of the 1930s. The negative correlation of money growth and nominal in-

terest rates is also one of the stylized facts in monetary economics. In this pap er,

we will try to explain these features through the two channels of monetary p olicy:

the output e ect and the liquidity e ect.

The output e ect, de ned here as the p ositive resp onse of aggregate output to

expansionary monetary p olicy, has b een a key question for economists who have

searched for a purely monetary explanation of the business cycle. Explaining the

strong relationship b etween money and real activity in a general equilibrium theory

faces two challenges. The rst is to provide a theory in which money is valued in

equilibrium. The second and more dicult one is to showhow monetary p olicy has

real e ects in a world where economic agents are b ehaving rationally without simply

asserting some ad hoc form of money illusion. In this pap er, the nonneutrality of

money stems from menu costs.

The liquidity e ect, de ned as the decrease in interest rates in resp onse to mon-

etary expansion, has b een an imp ortant issue in empirical macro economics. Re-

cently, pursuing alternative assumptions for identifying monetary p olicy sho cks, re-

searchers provide strong empirical evidence in supp ort of the liquidity e ect. They

argue that innovations in monetary aggregates re ect sho cks to money demand

rather than to money supply, or p olicy. This pap er intro duces this recent develop-

Even if the rst feature is universally accepted, the presence of the second feature is somewhat

controversial. It dep ends on the choice of monetary aggregates and trending mechanisms. See

Chari, Christiano and Eichenbaum (1995) as an example. Co oley and Hansen (1995) summarize

additional stylized facts of the nominal features.

In a static IS-LM framework, an increase in money supply moves interest rates down to induce

larger money demand. The decrease of interest rates moves investment and output up. Interpreting

the two features as the e ects of monetary p olicy is, of course, not unanimous. Plosser (1990),

following King and Plosser (1984), interprets the two features as a consequence of endogenous

inside money.

Friedman and Schwartz (1963) do cumented a strong asso ciation between p erio ds of severe

economic decline and sharp declines in the sto ck of money.

Three frameworks have b een develop ed: money in the utility function, transaction cost tech-

nology, and cash-in-advance constraints.

This de nition of the liquidity e ect as a causal relation is, of course, not universal. For

example, Ohanian and Sto ckman (1995) use the term to refer to the statistical correlation. 3

mentinto a general equilibrium mo deling.

Stimulated by Kydland and Prescott (1982) and Long and Plosser (1983), dy-

namic sto chastic general-equilibrium (DSGE) mo dels have b ecome a useful to ol for

macro economic analysis, esp ecially for business cycle analysis. Previous work using

a exible-price, comp etitive DSGE mo del has provided a reasonable description of

the data on real variables. Recently, the prototyp e DSGE mo dels have b een en-

riched by the intro duction of non-newclassical features and sources of uctuations

other than technology sho cks. Furthermore, many recent mo dels are characterized

by sub optimal equilibria, which generally provide a rationale for activegovernment

intervention.

One stream of recentwork incorp orates outside money in a exible-price comp et-

itive DSGE mo del. Money is intro duced in a cash-in-advance economy by Co oley

and Hansen (1989) to study the e ects of in ation. Sims (1989, 1994) intro duces

money through a transaction-cost framework. Using a simple money-in-the-utility-

function mo del, Benassy (1995) shows analytically that the dynamics of the real

variables are exactly the same as those in the mo del without money. Such mo dels

do not provide a good description of the money-output correlation and cannot re-

pro duce reasonable impulse resp onses to the sho cks in monetary p olicy, b ecause of

the following generic implication. If money growth displays a p ositive p ersistence,

then sho cks to the growth rate of money drive output down and nominal interest

rate up through an anticipated in ation e ect.

To generate the output e ect, nominal rigidities are intro duced into DSGE

mo dels. There are two alternative ways of intro ducing nominal rigidities. The

DSGE mo dels broadly refer to real business cycle (RBC) mo dels, equilibrium business cycle

mo dels, and neo classical growth mo dels. They are regarded as a paradigm for macro analysis.

Prescott (1986) compares a DSGE mo del to the supply and demand construct of micro economics.

Despite the e ort to analyze monetary phenomenon to b e describ ed b elow, DSGE mo deling has

not b een used so frequently to analyze the e ects of monetary p olicy. Friedman (1995) categorize

the empirical metho dologies for monetary p olicy into three: partial equilibrium structural mo dels,

VARs based on observed prices and quantities, and VARs based incorp orating non-quantitative

information. DSGE mo deling is not included.

The two pap ers fo cus on rather di erent p oints. Sims (1989) shows by simulation that a purely

classical equilibrium mo del with monetary p olicy misses certain asp ects of the actual b ehavior of

the data. Meanwhile, Sims (1994) considers various forms of monetary p olicy with an emphasis

on the relation with scal p olicy and discussed the existence and the uniqueness of equilibrium.

Besides intro ducing nominal rigidities, another way to pro duce the output e ect is to treat

monetary sho cks as a source of confusion that makes it dicult for agents to separate relative

price changes from aggregate price changes, as in the \Lucas island mo del." Co oley and Hansen

(1995) nd that monetary sho cks do not app ear to play a quantitatively imp ortant role in driving

business cycles in a mo del based on this theory. Ball, Mankiw and Romer (1988) also provides 4

rst is to replace an equilibrium equation matching demand and supply with an

equation describing the determination of price and/or wage. The staggered con-

tract theory of Fisher (1977) is usually adopted. Cho (1993), Co oley and Hansen

(1995), and Cho and Co oley (1995) study the implications of nominal wage contracts

for the transmission of monetary sho cks, and Yun (1994) explains the comovement

of in ation and output with a staggered multi-p erio d price setting. Leep er and Sims

(1994) also exp eriment with b oth price and wage rigidities by p ostulating equations

describing price and wage movements. Another way of intro ducing nominal rigidi-

ties is via adjustment costs. For an economic agenttohave control over the price

and/or wage, some form of imp erfect comp etition is needed. Following Blanchard

and Kiyotaki (1987), the monop olistic comp etition framework has b een widely used.

Hairault and Portier (1993) show that monetary sho cks are necessary to repro duce

some stylized facts of business cycles, and Rotemb erg (1994) presents a mo del that

is consistent with a variety of facts concerning the correlation of output, prices and

hours of work. All the ab ove mo dels of nominal rigidities do indeed generate the

output e ect of monetary p olicy. However, as shown in Kimball (1995), the pres-

ence of nominal rigidities by themselves do es not pro duce the liquidity e ect. To

generate the liquidity e ect, I assume real rigidities in the form of adjustment costs

for capital. This conjecture is found in King (1991) and implemented in King and

Watson (1995) and Dow (1995).

Note that the imp ortance attributed to the di erent kind of nominal rigidities

has shifted over time. From Keynes until the early 1980s, nominal wage rigidities

were p opular in Keynesian macro economics. Later price rigidities gained p opularity

among New-Keynesians. Recently, Cho (1993) considers b oth typ es of rigidities

characterized by nominal contracts and concludes that nominal wage contracts are

a b etter way of incorp orating nominal rigidities than nominal price contracts. This

pap er is ambivalent ab out the two nominal rigidities by incorp orating b oth of them.

Meanwhile, the research on imp erfect comp etition is partly motivated by Hall

(1990) who interprets the rejection of the invariance hyp othesis as the consequence

of imp erfect comp etition and/or increasing returns to scale. Hornstein (1993)

evidence against this mo del.

However, these quantitative equations have no microfoundations in the mo del.

Rotemb erg (1987) surveys the literature on adjustment costs and Hairault and Portier (1993)

incorp orate price adjustment costs in a DSGE framework.

In a p erfectly comp etitive market, a rm do es not set its price, but accepts the price quoted by

the Walrasian auctioneer. Only under imp erfect comp etition, when rms set prices, do es it make

sense to ask whether a rm adjusts its price to a sho ck.

Rotemb erg (1994) neglects the empirical b ehavior of interest rates and monetary aggregates,

since their b ehavior seems to hinge on asp ects of the mo del that are more incidental.

The invariance hyp othesis p ostulates that under p erfect comp etition and constant returns to 5

and Rotemb erg and Wo o dford (1992, 1995) discuss the consequences of intro ducing

imp erfectly comp etitive pro duct markets and increasing returns into exible-price

DSGE mo dels. Hornstein (1993) concludes that pro ductivity sho cks still account

for a substantial fraction of observed output volatility. On the contrary, Rotemb erg

and Wo o dford (1992, 1995) conclude that imp erfect comp etition changes the way

in which the economy resp onds to various sho cks and that demand sho cks are im-

p ortant for explaining business cycles.

Outside the DSGE framework, some researchers use structural vector autore-

gression (VAR) mo dels to isolate the e ects of monetary p olicy. Traditionally, the

stance of monetary p olicy was measured by the growth rate of a monetary aggre-

gate. This identi cation created a problem, called a \liquidity puzzle": monetary

expansions are asso ciated with rising rather than falling interest rates. This was

do cumented by Melvin (1983), Reichenstein (1987) and Thornton (1988), based

on single-equation distributed-lag mo dels of interest rates. Recently, Leep er and

Gordon (1992) also nd that the relationship b etween innovations in monetary ag-

gregates and interest rates has a sign opp osite of that predicted by the liquidity

e ect.

This problem with money-based measures of monetary p olicy has led Sims (1992)

to identify monetary p olicy with innovations in interest rates. Bernanke and Blinder

(1992) also argue for identifying federal funds rate innovations as p olicy sho cks,

supp orting their time series analysis with institutional details. This identi cation

scheme, while successful in pro ducing reasonable results in some dimensions, leaves

theoretical and empirical problems. For example, an expansionary monetary p olicy

is asso ciated with a strong and p ersistent drop in the price level.

Recently, Gordon and Leep er (1994), Strongin (1995) and Sims and Zha (1995)

identify monetary p olicy in such a way that the sto ck of money dep ends on the

innovations in money demand as well as in monetary p olicy. My sp eci cation of

scale, the Solow residual is uncorrelated with all variables known neither to be causes of nor to

be caused by pro ductivity sho cks. Another source of the rejection is the factor hoarding, as in

Burnside and Eichenbaum (1994). Meanwhile, Evans (1992) shows that the Solow residual is

Granger-caused by nominal variables such as money and interest rates using U.S. quarterly data.

Compared with DSGE mo dels which put strong restrictions on the data through optimizing

b ehavior, structural VAR mo dels are weakly identi ed time series mo dels. They are also called

identi ed VAR mo dels.

Traditional sp eci cation of monetary p olicy created another problem. While monetary ag-

gregates Granger-cause output in a VAR without interest rates, monetary aggregates no longer

Granger-cause output in the sp eci cations with interest rates. Sims (1980) shows that interest rate

innovations absorb most of the predictivepower of money for output when they are included in

the multivariate system.

Co chrane (1994) and Pagan and Rob ertson (1995) survey this recent literature. 6

monetary p olicy hinges on this typ e of identi cation, by including b oth monetary

aggregates and interest rates in the p olicy rule. The empirical results of this pa-

per show that this sp eci cation is imp ortant in explaining the nominal features of

business cycles.

In this pap er, I prop ose a DSGE mo del extended to allow for real and nominal

rigidities. The mo del features the two e ects of monetary p olicy. It also cap-

tures many interesting U.S. business cycle facts. Section 2 presents the mo del and

the solution. Households maximize their utility and rms maximize their pro t.

Government action is characterized by monetary and scal p olicies. Four sho cks,

including b oth technology and monetary p olicy sho cks, a ect the economy. The

equations describing the equilibrium are log-linearized and the solution is achieved

following Sims (1995).

Section 3 illustrates the mo del's qualitative and quantitative prop erties. Impulse

resp onses from various versions of the mo del are provided as a way to understand

the roles of various rigidities. Because the time-series and p olicy implications crit-

ically dep end on the choice of the parameters, maximum likeliho o d estimates are

computed. The impulse resp onses evaluated at the estimated parameters feature

the two e ects of monetary p olicy: the output and liquidity e ects. The t of the

mo del is comparable to that of an unrestricted rst-order VAR and the variance

decomp ositions are similar to those in the structural VAR literature. Cyclical im-

plications are considered by comparing the second moments of the mo del forecasts

with those of the data. Finally, one of the challenges in King and Watson (1995)

is satisfactorily met by the estimated mo del: an increase in interest rates in the

current p erio d predicts a decrease of real economic activity two to six quarters in

the future. None of their three mo dels captures this feature of the U.S. business

cycle. Section 4 concludes.

For details on this identi cation scheme, see Bernanke and Mihov (1995) and the references

therein. This p oint will b ecome clear when the equation of monetary p olicy is formulated.

Christiano and Eichenbaum (1992) prop ose a DSGE mo del with an alternative transmission

mechanism of monetary p olicy. The mo del generates b oth the output e ect and the liquidity

e ect due to participation constraints in the nancial market. Dotsey and Ireland (1995) provide a

skeptical view of the mo del. Beaudry and Devereux (1995) also construct a mo del which features

the two e ects by assuming predetermined prices as a way of resolving indeterminacy.

For the issue of estimation versus calibration, see Kydland and Prescott (1996) and Sims

(1996). See Anderson, Hansen, McGrattan and Sargent (1995) on the estimation of a DSGE

mo del by maximum likeliho o d. Recent empirical work has also used other estimation metho ds:

generalized metho d of moments and simulated moments. 7

2 The Mo del

The economy consists of a government and three typ es of private agents: an aggre-

gator, I households indexed by i; and J rms indexed by j . The aggregator serves

two functions in this economy. In the lab or market, it transforms heterogeneous

lab or into a \comp osite lab or" usable for rms' pro duction. In the go o ds market, it

collects di erent goods to make a \comp osite go o d" which households can consume

and invest. Its demand for an input dep ends on the relative price of the input.

Each household has monop oly p ower over its own typ e of lab or, facing the demand

by the aggregator. In b oth capital and goods markets, it is a price-taker. It also

accumulates capital and rents it to rms. Each rm has monop oly power over its

own pro duct, facing the demand by the aggregator. It acts comp etitively in factor

markets. The government derives revenue from issuing money and debt and exp ends

its revenue through transfers and interest payments on outstanding debt.

The assumptions on monop oly p ower of households ( rms) in the lab or (go o ds)

market allows nominal rigidities to arise in the form of adjustment costs for wages

(prices). Following the literature on menu costs, assume that it is costly for rms to

adjust their output price. Similarly,wage adjustment costs on the part of households

are assumed to capture wage rigidities. Meanwhile, adjustment costs for capital give

rise to real rigidities.

Four sho cks a ect the economy. In the part of the households, a preference

sho ck is identi ed as a sho ck in money demand. The pro duction function of the

rms involves two sho cks: a technology sho ck and a xed-cost sho ck. The last one

is a sho ck in monetary p olicy. A word on information structure: variables dated t

are always known at t.

I and J are held xed in the mo del, at least on the balanced growth path. Trade theories

emphasizing the role of increasing returns to scale and imp erfect comp etition, Helpman (1984) and

Helpman and Krugman (1985) among others, endogenize the numb er of di erentiated go o ds. On

the consumption side, households have utility functions which reward pro duct diversity. Rotemb erg

and Wo o dford (1995) also set up a mo del where the steady-state numb er of di erentiated go o ds

grows at the same rate as real variables, while the output of each rm is held constant. In my

mo del, the output of each rm increases at the same rate as real variables.

Because of the heterogeneity of inputs, households and rms act in a monop olistic-comp etition

environment. Oligop olistic pricing of households and rms pro duces similar b ehavior, but the issue

of collusion and punishment should b e considered, as in Rotemb erg and Wo o dford (1992).

The lab or market could b e another source of real rigidities via eciency wages, risk sharing

contracts, or lab or hoarding. 8

2.1 The Aggregator

Recall that there are heterogeneous households and rms. In principle, various

typ es of lab or are b ought by rms for pro duction and the goods they pro duce

are b ought by households, rms, and the government. To simplify the mo del and

to make it comparable to the standard p erfectly comp etitive mo del, all of these

ultimate demanders are assumed to be interested in a \comp osite good" and a

\comp osite lab or" which are supplied by an arti cial agent, called the aggregator.

Consumption and investment are measured in terms of the unit of the comp osite

good. Firms' output dep ends on the quantity of the comp osite lab or.

The aggregator's b ehavior is describ ed as follows. The aggregator purchases dif-

ferentiated inputs which are describ ed byaN-dimensional vector, (H ;H ;;H ),

1 2 N

and transforms them into H units of the comp osite output, where the functional

form is:

( 1)

(1 )

H = N : (1) H

n=1

This is a constant returns to scale (CRS) and constant elasticity of substitution

(CES) pro duction function which has b een used in the literature on monop olistic

comp etition, e.g. Dixit and Stiglitz (1977). The parameter is the elasticity of

substitution b etween the di erent inputs, p ossibly di erent for go o ds and lab or. To

guarantee the existence of an equilibrium, is restricted to be greater than unity.

The supplier of each di erentiated input sets a price for it; the collection of these

prices describ es a price vector conformable with the vector of inputs purchased. The

pro t of the aggregator is:

=PH P H ; (2)

n n

n=1

where P is the price index and P is the price of k th input. The rst order condition

The presence of the aggregator can b e avoided. In this case, every agentcho oses the goods

and lab or index comp osition optimally, as in Blanchard and Kiyotaki (1987) and Hairault and

Portier (1993). Since this choice is static and the same for all agents, notation is simpli ed when

the aggregator solves the problem instead. This device for aggregation is a standard feature of

general equilibrium trade mo dels, for example Backus, Keho e and Kydland (1994).

(1 )

The co ecient term, N , is for zero pro t at the symmetric equilibrium with the price

(1 )

: The co ecient term and the price index could be index de ned as P = P

n=1

disp ensed with if cost-minimizing b ehavior, instead of pro t-maximizing b ehavior, were analyzed.

However, there would remain the problem of who o ccupies the pro t.

The following results could b e obtained without the global assumptions of the CES pro duction

function. See Rotemb erg and Wo o dford (1995) for details. 9

with resp ect to H reduces to a constant-elasticityinverse demand function:

P = P : (3)

In the case of lab or market where the elasticity of substitution is , (3) is

interpreted as follows:

; (4) W = W

it t

where W is the wage of household i, W is the wage index, L is the lab or supply of

it t it

household i, and L is the amount of the comp osite lab or supplied by the aggregator,

all at time t. Note that this is a relation b etween the relative price and the relative

quantity. Likewise, in the goods market where the elasticity of substitution is ,

(3) is written as follows:

P = P ; (5)

jt t

where P is the output price of rm j , P is the price index, Y is the output supply

jt t jt

of rm j , and Y is the amount of the comp osite go o d supplied by the aggregator, all

at time t. In the following, I use the same symb ols for every agenttoavoid the need

for separate equations sp ecifying market clearing conditions of the equilibrium.

2.2 Households

Households are identical, except for the heterogeneity of lab or. Having monop oly

power over one unit of its own lab or, each household enters at time t with predeter-

mined capital sto ck, money holdings and b ond holdings. The household receives its

rental income, its wage income, a lump-sum transfer, and a constant share of pro ts.

It also pays the costs of adjusting its capital and wage. One source of sho cks lies in

the sp eci cation of preferences.

2.2.1 Preferences

Preferences are given by the utility function U which represents the exp ectation

of the discounted sum of instantaneous utilities, conditional on the information at

time zero:

" #

U = E U C ; ;L (6)

i0 0 it it

t=0

Or equivalently, the demand function is H = : Note that the functions have

N P

constant elasticity, which simpli es the rst order conditions of households and rms. 10

2 3

a 1a

(C ) (1 L )

6 7

; = E

4 5

t=0

where

=( 1)

2 2

( 1)=

2 2

( 1)=

2 2

C = C + b : (7)

The discount factor is between 0 and 1. Consumption, C ; and real bal-

; interact through a CES function. The CES parameter, ( 0), decides ances,

the elasticity of money demand. Instantaneous utility is a constant relative risk

aversion (CRRA) transformation of a Cobb-Douglas function of the CES bundle,

C , and the amount of leisure, (1 L ). The CRRA co ecient, , is assumed

it 1

to be p ositive. The framework of money-in-the-utility-function is adopted here.

However, this mo del can be converted into a transaction-cost framework without

altering any implications, where the variable representing consumption is not C

but C :

The only sto chastic element of households' preferences is in b . The variable

decides the imp ortance of consumption relativeto real balances and it follows:

log (b )= log (b )+(1 ) log (b)+" ; (8)

t b t1 b bt

where the " 's are i.i.d. random variables distributed N (0; ): Since b decides the

bt t

level of the money demand, the innovations in b are identi ed as money demand

sho cks; as a matter of fact, the other sho cks a ect money demand only indirectly.

Another element app earing in the money demand equation is , which is the elas-

ticity of money demand. Money demand sho cks could b e intro duced via , but it is

more reasonable to intro duce a randomness in the level, rather than in the elasticity.

Utility maximization is sub ject to several constraints.

For convergence of the utility function at the steady state, need not b e less than 1. Since

consumption and real balances grow at a rate of G ( 1), the discount factor in the stationary

a(1 )

economyis G , which is less than , when > 1. A sucient condition for well-de ned

a(1 )

utility at the steady state is that 0 < G < 1: However, assuming that households prefer

the present to the future, lies b etween 0 and 1.

If I assumed the additive separabilityof C and L , the only parametric class of preferences

consistent with stationary lab or supply would be the logarithmic utility over C , i.e. log C +

it it

h (L ). See the technical app endix of King, Plosser and Reb elo (1988). This corresp onds to the

sp eci cation of = 1 in this mo del.

1 11

2.2.2 Real Rigidities via Capital Adjustment Costs

Each household accumulates capital and rents it to rms. The accumulation tech-

nology is given by the following equation:

K = I +(1)K ; (9)

it i;t1 i;t1

where K is the amount of capital sto ck and I is the amount of investment, b oth

it it

at time t. The existing capital depreciates at a constant rate of . Note that

investment is pro ductive next p erio d and so the capital sto ck is predetermined.

Mo dels of investment without adjustment costs result in to o volatile investment.

This suggests the imp ortance of recognizing real rigidities via adjustment costs.

Adjustment costs for capital have b een used to provide a rigorous foundation for

the q {theory of investment. In this pap er, adjustment costs are internal to the

households and given by:

K it

AC = I ; (10)

2 K

where is the adjustment cost scale parameter for capital. This functional form,

satisfying the assumptions in Ab el and Blanchard (1983), pro duces strictly p ositive

steady-state adjustment costs. To invest I units, the extra amount I

it it

2 K

is used up in transforming goods to capital. These costs are interpreted as for-

gone resources within households. The presence of adjustment costs makes current

investment dep end on the future through exp ectations.

Adjustment costs for capital play an imp ortant role in determining nominal and

real interest rates. More investment makes installed capital more valuable due to

adjustment costs. However, the price of capital is exp ected to return to the steady

state in the future. This exp ected decrease of capital price moves real interest rate

down. This mechanism is crucial in generating the liquidity e ect. Expansionary

monetary p olicy increases investment and thus decreases real interest rate. If this

o sets the anticipated in ation e ect, nominal interest rate go es down: the liquidity

e ect.

In a continuous time mo del of investment without adjustment costs, investment is either in-

nitely p ositive or in nitely negative, if not zero.

31 K

An alternative formulation would make installation costs a function of net investment: AC =

I . This formulation pro duces zero steady-state adjustment costs. Quadratic costs

2 K

are justi ed on the ground that it is easier to absorb new capacityinto the rm at a slow rate.

Recent literature on investment considers discontinuous adjustment costs. In sp ecial cases, the

optimal rule takes an (S; s) recursive form. See Bertola and Caballero (1991) and Ab el and Eb erly

(1994). Recently, Ab el, Dixit, Eb erly, and Pindyck (1995) link q -theory to option pricing.

1+ when the price of output is normalized to 1. The price of capital is

2 K

it 12

2.2.3 Wage Rigidities via Wage Adjustment Costs

Following Blanchard and Kiyotaki (1987), a monop olistic-comp etition market struc-

ture is adopted in b oth lab or and goods markets. As the elasticity of substitution

between heterogeneous lab or entering the aggregating function is not in nite, each

household has market power over the market for its own lab or. Therefore, each

household cares ab out its wage level relative to the aggregate wage index. The

numb er of households is assumed to b e large enough to ensure that each household

has a negligible e ect on aggregate variables. Households play a Nash game on the

lab or market, and each one cho oses its wage and lab or knowing its lab or demand

function and taking aggregate variables as given.

Now, wage rigidities are intro duced through the cost of adjusting nominal wages.

To obtain a simple solution to the problem, these costs are assumed to b e quadratic

and zero at the steady state. The real total adjustment cost for household i is given

by:

P W

W t it

AC = W ; (11)

2 P W

t1 i;t1

where is the adjustment cost scale parameter for the wage and is the steady-

state growth rate of money. The multiplicative term, W , makes the costs grow

at the growth rate of real wage index. This real cost, AC ; enters the budget

constraintofthe household in a similar way to adjustment costs for capital.

Wage rigidities are a source of the output e ect of monetary p olicy. Facing

the increased lab or demand induced by expansionary monetary p olicy, households

would increase wages prop ortionately if it were not for adjustment costs. However,

households increase wages less than prop ortionately b ecause the costs of adjustment

are quadratic. Their supply of lab or increases and so do es the output. This intuition

will b ecome clear with a graph when price rigidities are intro duced.

2.2.4 Budget Constraints and First Order Conditions

The budget constraint of household i is given by:

I B M

K it it it

C + I 1+ + +AC (12) +

it it

2 K P P

it t t

Adjustment costs for wages are a device to capture the imp erfection in the lab or market. The

framework contains the elements of search pro cess and it is as realistic as overlapping contracts

theory. Regarding the question of whyitiswages, not the amount of lab or, which incur adjustment

costs, see b elow when price rigidities are intro duced via adjustment costs for prices.

There is also a discussion of this functional form b elow when price rigidities are intro duced.

Since adjustment costs for nominal rigidities are null at the steady state, the quadratic form with

one co ecient do es not lose any generality. 13

r B M

t1 i;t1 i;t1

+ + s ; = W L + Z K + T +

ij jt it it t it it

P P

t t

j =1

where B is government debt, Z is the rental rate, T is the government lump-sum

it t it

transfer payment, r is the gross nominal interest rate, and s is its constant share

t ij

of real pro ts of rm j . Government debt earns nominal interest rate and

money is not an interest b earing asset. A No-Ponzi-Game condition is imp osed on

households' b orrowing: it requires debt not to increase asymptotically faster than

the interest rate. This prevents households from b orrowing to the level that makes

the marginal utility of consumption zero at every p erio d.

Noting that W is a function of L according to (4), the rst order conditions

it it

are:

0 = ; (13)

# "

t t+1

; (14) 0 = + E

@M P P

it t+1 t

@U 1

0 = ; (15) + W 1

it t

@L e

0 = Z + Q + (1 )E [Q ] ; (16)

t K t t t t+1

3 I

K it

0 = 1+ + E [Q ] ; (17)

t t t+1

2 K

" #

t+1 t

0 = r E ; (18)

t t

P P

t+1 t

L 37

where e is the lab or demand elasticity augmented with the adjustment cost.

and Q are the Lagrangian multipliers of the budget constraint and the capital

accumulation equation and so they are interpreted as marginal utility of resources

(or consumption) and marginal utility of capital, resp ectively. All other constraints

Note that s is assumed to be a constant beyond the choice of household i. This makes

calculations easier. If s wereachoice variable, a rst order condition with resp ect to s would

ij ij

put a restriction b etween households' and rms' discount factors. In this pap er, the assumption

of a complete market connects the two discount factors in the equilibrium.

M2 is used as the data for money. Even if M2 earns interest, it p erforms b etter than narrower

monetary aggregates from the transaction p ersp ective. Note that I incorp orate the linear trends

inside the mo del and the trends of various variables are related each other in the equilibrium.

37 L

The formul for e and the derivatives are given in the App endix.

The multiplier Q is slightly di erent from the one in a standard q -theory. It relates capital

(or investment) to utility, not to pro t. 14

are eliminated via substitution.

Combining (13), (14) and (18), a standard money demand equation is derived.

P C

t it

r =1b : (19)

The elasticity of real money balances with resp ect to the net interest rate is approx-

imately ( ) : In the following, (19) replaces (14).

Optimal capital accumulation generally involves two eciency conditions on

the part of households. The rst, (16), stems from the optimal intertemp oral

choice of capital. If rented, the marginal utility of capital (Q ) is the sum of

three terms: the discounted and depreciated marginal utility of next p erio d's capital

( (1 )E [Q ]), the marginal utility of the rental rate (Z ), and the marginal

t t+1 t t

utility of the gain in adjustment costs . Note that the accumulation

K t

of capital reduces the adjustment cost in the next p erio d due to a larger scale. The

second, (17), is an arbitrage condition between consumption and investment. It

sets the relative price of capital 1+ to the marginal utilityof capital

2 K

( E [Q ]) divided by the marginal utility of consumption ( ). The marginal util-

t t+1 t

ity of capital is the same as the discounted marginal utility of next p erio d's capital.

In a sense, (17) can be interpreted as a demand for capital that relates investment

to its marginal shadow value.

Also, to a rst-order approximation, the following relation holds between the

nominal interest rate and the rental rate:

0 1

2 3

I I

t+1 t+1

(1 )E 1+ +E Z +

t t t+1 K K

2 K K

B C

t+1 t+1 P

t+1

B C

r = E : (20)

t t

@ A

3 I

2 K

It is clear from this relation that the exp ected decrease of investment price implies

alower nominal interest rate and that real rigidities are imp ortant in generating the

liquidity e ect.

2.3 Firms

Firms are identical, except for the heterogeneity of outputs. Each go o d is pro duced

by one rm only, and this rm takes all other output prices as given. In the market

This expression is obtained byinvoking certainty equivalence, i.e. E[f ( Y )] = f (E [ Y ]) , which

is justi ed by the assumption that the variances are very small. This could b e achieved by manip-

ulating deterministic rst order conditions and restoring the exp ectational op erators. From this

expression, the real interest rate is read from the formula in the big parenthesis. 15

for its inputs, capital and lab or, these rms b ehave comp etitively. The decision on

inputs results in a certain amount of output and the amount, in turn, determines

the output price from the demand curve of the aggregator. The rm also pays the

cost of adjusting its output price.

2.3.1 Technology

The rm j pro duces Y units of net output under a common increasing-returns-to-

scale technology. The pro duction function is:

t 1 t

K (g L ) G ; (21) Y = A

jt t jt t

1 41

with the restrictions that 1 ;g1;and G 1. K is the capital sto ck

and L is the quantity of comp osite lab or. A and , common to all rms, follow

jt t t

the sto chastic pro cesses

log(A ) = log(A )+(1 ) log(A)+" ; (22)

t A t1 A At

log ( ) = log ( )+(1 ) log () + " ; (23)

t t1 t

where A (> 0) and ( 0) are the steady-state values, and " and " are b oth i.i.d.

At t

2 2

variables distributed N (0; ) and N (0; ), resp ectively. Since b oth of these two

sho cks app ear in the pro duction function, " and " are assumed to b e correlated

At t

with a correlation co ecient . Any other pair of errors in the mo del is assumed

to b e uncorrelated.

The variable represents a \ xed cost" comp onent. It is essential to intro duce

xed costs, as they have crucial implications for pro ductivity and pro tability. In

each p erio d, the amount G is used up for administration purp oses just to keep

pro duction going and this is indep endent of how much output is pro duced. This

\ xed cost" implies increasing returns to scale. An additional source of increasing

returns to scale is emb o died in , whenever > 1.

Firms pro duce di erent pro ducts using the same technology, which is \costless pro duct

di erentiation."

G is the growth rate of nonstationary real variables. To guarantee the existence of a balanced

(1 )

growth path, G = g : This functional form pro duces a steady state path with a geometric

trend, and so the data are not pre ltered for the estimation. Incorp orating a trending mechanism

inside a mo del as ab ove, the mo del integrates b oth growth and business cycles. See Co oley and

Prescott (1995) on this p oint.

The presence of xed cost makes it p ossible for the rms to earn zero pro t in the long run.

Hairault and Portier (1993) interpret as pro t rather than xed cost and they do not imp ose a

zero pro t condition in the long run. A third p ossible interpretation of is an additive technology

sho ck.

There are other routes of increasing returns to scale than these two. Baxter and King (1991)

use a pro duction externality to generate increasing returns to scale. 16

2.3.2 Price Rigidities via Price Adjustment Costs

Each rm sells its output to the aggregator in a monop olistically comp etitive market.

Without any other assumptions, the ab ove rm problem is essentially a static one.

I now intro duce price rigidities through price adjustment costs. As with wage

rigidities, the real adjustment cost for rm j is given by:

P jt

AC = Y ; (24)

2 P G

j;t1

is the steady-state where is the adjustment cost scale parameter for price and

gross in ation rate.

This functional form of adjustment cost do es not match the original idea of menu

costs exactly. Here a rm pays the cost if the increase in its output price is di erent

from the steady-state in ation rate, while it pays costs if it changes the price at

all in the menu cost literature. However, my sp eci cation may be rationalized as

follows. In a stable economy, the agents will adapt themselves to a stable in ation

rate. Therefore, it is costly to increase the price di erently from this rate b ecause

this involves costs of advertising and also b ecause an erratic pricing causes consumer

dissatisfaction.

Likewage rigidities, price rigidities are a source of the output e ect of monetary

p olicy. Figure 1 presents an intuitive explanation using an economy where the

steady-state in ation rate is 1:0. The [-shap ed dashed graph is the price adjustment

cost and the two \-shap ed graphs represent gross pro ts excluding adjustment cost,

b efore and after a sho ck. Supp ose that a rm is maximizing its pro t without

changing its price. In the gure, the maximum is achieved when the ratio of prices

is 1:0. Pro t is represented by the distance AB. Supp ose that a p ositive demand

sho ck moves the gross pro t to the right (the dotted line) so that the maximum

is achieved at 1:1. If there were no adjustment costs, the rm would increase the

price by 10% and would not change the quantity of output. However, b ecause of

the adjustment cost, the rm increases the price only by 5% and also increases the

One may ask why it is not quantities but prices which incur adjustment costs. A p ossible

answer is that changing prices involves information cost. The decision by a rm to change the

price should b e known to its consumers, whereas the decision on quantitychanges is made within

a rm and need not be known to the consumers. This makes the cost of changing prices larger

than that of changing quantities. The same explanation holds for adjustment costs for wages.

Parkin (1986) intro duces lump-sum adjustment costs and considers di erent monetary p olicies

to study the implications for pricing b ehavior.

46 P

The alternative sp eci cation following the menu cost literature is AC = 1 Y .

2 P

j;t1

Note also that my sp eci cation pro duces \second-order private costs and rst-order business

cycles" along the stationary path. The same argument holds for adjustment costs for wages. 17 1 B F D

0.8

0.6

0.4 gross profit and adjustment cost

0.2 E

C 0 A 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3

new price/old price

Figure 1: Price Rigidities and the Output E ect

quantity of output. Note that CD is longer than EF in the gure. It is also clear

from the gure that bigger adjustment-cost scale parameter would imply bigger

output e ect.

2.3.3 Value of Firms and First Order Conditions

Once price adjustment costs are intro duced, the problem of rm j is dynamic: the

rm maximizes its value which is the exp ectation of the discounted sum of its pro t

ows, conditional on the information at time zero:

" #

=E P ; (25)

j 0 0 t t jt

t=0

The three graphs are quadratic with the same second derivative, which makes the rm increase

the price by 5%.

The problem could b e divided into two parts and solved recursively. The static part is min-

imizing the cost sub ject to a given output. This part pro duces the conditional demand function

for capital and lab or. The dynamic part is maximizing the pro t with capital and lab or replaced

by the conditional demand functions. Following this metho d, we could obtain the formul for

marginal cost and markup. See Hornstein (1993) for details. 18

where

P = P Y P Z K P W L P AC ; (26)

t jt jt jt t t jt t t jt t

P = P ;

jt t

t 1 t

K (g L ) G ; Y = A

jt t jt t

log(A ) = log(A )+(1 ) log(A)+" ;

t A t1 A At

log ( ) = log ( )+(1 ) log () + " ;

t t1

P jt

AC Y : =

2 P G

j;t1

The rm discount factor is given by a sto chastic pro cess, f g. In the equilibrium,

it represents a pricing kernel for contingent claims.

Noting that P is a function of Y and that Y is a function of (K ;L ), the

jt jt jt jt jt

rst order condition with resp ect to K is:

! !

Y + G 1

jt t

P Z = P 1 (27)

t t jt

K e

Y 50

where e is the output demand elasticity augmented with the adjustment cost.

Likewise, the rst order condition with resp ect to L is:

! !

1 Y + G

jt t

P W = (1 ) P 1 : (28)

t t jt

L e

Markup, the ratio of price to marginal cost, is inversely prop ortional to 1 .

With in nite elasticity of substitution between the di erent go o ds, the markup is

constant at unity. If the elasticity is nite, it will measure the market power of the

rm. With price rigidities, b oth technology and demand sho cks a ect the markup

and the markup a ects the real variables. When a p ositive technology sho ck shifts

50 Y

The formula for e is given in the App endix. If marginal cost is decreasing, the solution to (27)

and (28) might not b e a pro t-maximizing choice. The restriction that is nonnegative guarantees

that the choice de ned by (27) and (28) satis es a sucient condition for pro t maximization.

With exible prices, the markup do es not change with either sho ck. However, a technology

sho ck a ects real quantities directly, whereas a demand sho ck do es not. When a p ositive technology

sho ck o ccurs, the marginal cost curve shifts downward, and the rms adjust their price fully,

resulting in a higher supply at a lower price according to (27) and (28). When a p ositive demand

sho ck o ccurs, rms adjust their price to equalize marginal cost and marginal revenue, without

changing the equilibrium quantity. 19

the marginal cost curve downward, the rm do es not adjust its price fully and so

the markup increases. If a p ositive demand sho ck moves marginal revenue upward,

prices increase less than in the fully exible case and output increases, which leads

to a cut in the markup. Therefore, the cyclicality of the markup dep ends on the

dominant source of sho cks.

2.4 The Government

The budget constraint of the government is:

M M B r B

t t1 t t1 t1

+ = T ; (29)

P P

t t

where

I I I

X X X

M = M ; B = B ; and T = T :

t it t it t it

i=1 i=1 i=1

Among the ve variables in the constraint, (M ;P ;B ;r ;T ), the government has

t t t t t

two degrees of freedom since the remaining three are determined by its budget con-

straint and the optimizing b ehavior of households and rms. Government b ehavior

is describ ed by monetary p olicy and scal p olicy.

Monetary p olicy is sp eci ed following the developments in the structural VAR

literature describ ed and referenced in the Intro duction. This literature suggests that

monetary p olicy is b est describ ed by a combination of interest rate and monetary

aggregate measures. The equation can b e interpreted either as one deciding money

supply or as one deciding the interest rate.

Interpreted in the rst way, the money growth rate uctuates around a convex

combination of last p erio d's money growth rate and the steady state money growth

rate and it is also a ected by a sho ck. Another element of monetary p olicy concerns

the role of the interest rate. It enters the p olicy equation as a function of this

p erio d's interest rate, two previous p erio ds' interest rates, and its steady state.

The resulting sp eci cation constrains the government when setting its two p olicy

variables as follows:

0 1

( + )

M M r r r

t t1 t

t1 t2

@ A

= ( ) ; (30)

(1 )(1 )

M M r

t1 t2

The Federal Reserve usually refers to money growth as an intermediate target and the nominal

interest rate, the federal funds rate in practice, as an op erating pro cedure of monetary p olicy. This

view is rep eatedly stated in various issues of the Federal Reserve Bulletin. High output growth

and stable prices are referred to as the ultimate ob jective of monetary p olicy.

Tomy knowledge of DSGE literature, a monetary p olicy equation includes b oth measures only

in Sims (1989). 20

log = log + " ; (31)

t t1 t

where r is the steady state value of r and the " 's are i.i.d. variables distributed

t t

N (0; ).

The traditional money-based measure of monetary p olicy corresp onds to the

sp eci cation of = 0. Most DSGE research on money follows this sp eci cation.

Even if my sp eci cation is richer than the traditional one, it is still limited in con-

sidering the feedback from the real side: the feedback is only through the interest

rate. It would be more realistic to include the measures of real activities, income

or consumption, in the monetary p olicy reaction function. For example, monetary

p olicy could react to or . This limitation turns out to b e imp ortantin

Y Y

t1 t2

the analysis of the variance decomp ositions.

The parameter (> 0) emb o dies the sensitivity of the monetary authority to

interest rate movements. The higher the parameter, the smo other the interest rate.

If it is 0, the monetary authority do es not care ab out the interest rate. If it is 1, the

monetary authority increases the interest rate by 1% in resp onse to a 1% increase

of the money growth rate. If it is 1, the interest rate b ecomes the only instrument

of monetary p olicy.

The parameters, and ; determine the targeting level of the interest rate. If

r R

b oth are 0, the interest rate is targeted at the steady state. If one is 0 and the other

is 1, the monetary authority smo oths the interest rate by targeting last p erio d's

interest rate. If b oth are 1, the di erence of the interest rates is targeted at the

di erence in the last p erio d.

The interpretation of the sho ck in monetary p olicy, " , is not straightforward.

If we follow an extreme view of the rational exp ectations theory that there is a

common information set in the economy, the sho ck should b e purely random and so

beyond the choice of the monetary authority. If a variable is chosen by an agent,

then the variable is not random and known to every agent. However, an alternative

Bayesian view allows for the di erence of information sets. Having an information

advantage, the monetary authority may resp ond to other variables that it observes

but the private agents do not observe. This view do es not exclude the randomness

of the sho ck, for example an institutional randomness. In this pap er, the sho ck

represents b oth the randomness and the variables observed only by the monetary

authority.

Note that B and T have app eared only in the households' budget constraint.

it it

Aggregating the households' budget constraints and combining it with the gov-

ernment budget constraint, we have B and T only in the government budget

t t

constraint. This characteristic of the mo del is a version of Ricardian equivalence,

In describing the equilibrium, I combine an aggregate version of (12) with (29). 21

if the subsystem without the government budget constraint and the scal p olicy ex-

hibits a unique equilibrium. Even if Ricardian equivalence holds, scal p olicy should

satisfy certain restrictions for the equilibrium to exist and be unique. Sims (1994)

shows analytically which combinations of monetary and scal p olicies pro duce a

unique equilibrium, not only lo cally but also globally, in a more simple mo del. In

my mo del, scal p olicy is sp eci ed as follows:

T = TG ; (32)

where T and are constant co ecients. The conditions for a unique equilibrium

are checked lo cally and numerically.

2.5 The Equilibrium

Recall that using the same notations for every agent automatically guarantees mar-

ket clearing conditions. The main part of the equilibrium is given by the optimizing

b ehavior of the households and the rms, and the b ehavior of the government. How-

ever, those do not make the system complete; the number of equations is less than

that of the variables by 1. It is b ecause there has b een no relation between the

sto chastic discount factor of the households and that of the rms. To make the

system complete, I assume that every agent in the economy has access to a com-

plete and comp etitive market for contingent claims. That is, the rms maximize

their market value. Then there is a unique market discount factor, which implies

the following equation at all states:

t+1 t+1

= : (33)

t t

We can rationalize the ab ove equation in two di erent ways. First, supp ose that

the households, as owners of the rms, instruct the management of the rms. Then

it would direct the rms to discount future pro ts according to its own discount

factor. This is equivalent to (33). Second, as fo otnoted b efore, if an household can

buy and sell the shares of the rms, then rst order conditions with resp ect to the

shares would pro duce (33).

There might p otentially b e equilibria in which identical agents b ehave di erently.

However, in view of the symmetry of the environment, it is b oth reasonable and

practical to scrutinize a symmetric equilibrium. Note that the equations describing

Note that price adjustment costs make the rm problem dynamic. Without price adjustment

costs, the rm problem is reduced to a static one so there is no need to sp ecify the rm discount

factor. 22

the demand of the aggregator, (4) and (5), disapp ear since they b ecome identities in

a symmetric equilibrium. For analytic tractability, b oth I and J are normalized to

1. Ricardian equivalence, if it holds, enables (29) and (32) to disapp ear from the

system without a ecting anyvariables other than B and T . Three sets of equations

t t

describing the b ehavior of households, rms and the government are given in the

App endix.

The variables are divided into ve groups by their growth rate at the steady state.

P W

The variables that grow at a rate of G are Y ;K ;I ;AC ; ;W ;AC ;and C :M

t t t t t t t

t t

grows at a rate of and P grows at a rate of . The two Lagrangian multipliers,

a(1 )1

and Q ; grow at a rate of G which is less than 1. All other variables are

t t

constant at the steady state. I use lower case letters to represent the transformed

stationary variables. That is, if is the growth rate of X , x = X = : Now it is

X t t t

p ossible to transform every variable into a stationary variable.

To express the system in the transformed variables as an VAR(1) pro cess, I de ne

the following two new variables: the gross in ation rate and the growth rate of real

wages,

P p

t t

f = = ;

P Gp

t1 t1

W Gw

t t

v = = :

W w

t1 t1

The equations describing the problem of the households are transformed into the

57 a(1 )1

following equations. For notational convenience, replaces G .

ac = (f v ) w ;

t t t

K t

; c = w L + Z k + ac i 1+

t t t t t t t

2 k

Gk = i +(1 )k ;

t t1 t1

a 1a

(c ) (1 L )

u = ;

0 1

=( 1)

2 2

( 1)=

2 2

( 1)=

2 2

@ A

c + b c = ;

= ;

Consider a situation where b oth households and rms are distributed continuously on the real

line over ( 0; 1). Integrating on the real line reduces to this normalization.

The formul for the derivatives are given in the App endix. 23

p c

t t

; r = 1 b

2 3

1 (f v )(f v )+

t t t t

h i

4 5

e = ;

t+1

(f v )(f v ) E

t+1 t+1 t+1 t+1 G t

t t

1 @u

1 = w ;

t t

@L e

0 = Z + q + (1 )E [q ] ;

t K t t G t t+1

3 i

K t

1+ 0 = E [q ] ;

t G t t+1

2 k

" #

t+1

= r E ;

t G t t

t+1

log (b ) = log (b )+(1 ) log (b)+" :

t b t1 b bt

The equations describing the problem of the rms are transformed as follows.

; y = A k L

t t t t

log(A ) = log(A )+(1 ) log(A)+" ;

t A t1 A At

log ( ) = log ( )+(1 ) log () + " ;

t t1 t

ac = f y ;

t t

2 G

= y Z k w L ac ;

t t t t t t

2 3

1 f f +

P t t

4 h i 5

e = ;

t+1

E f f

G P t t+1

t+1

1 ; Z k = (y + )

t t t t

w L = (1 ) Z k :

t t t t

The equations for government b ehavior are transformed, to o.

0 1

( + )

m m r r r

t t1 t

t1 t2

@ A

= ( ) ;

(1 )(1 )

m m r

t1 t2

log( ) = log( )+" :

t t1 t

Let us denote the steady state values with an upp er bar. That is, x is the steady-

state value of x . The steady state values are recursively calculated and the analytic

t 24

formul are given in the App endix. Two p oints need to be mentioned ab out the

steady state. First, it is plausible to assume that there are zero pro ts in the steady

state. This assumption is equivalent to determining the value of as follows:

" !#

= 1 1 A k L : (34)

The restriction that is nonnegative requires that ( 1) ( 1) 1: In words,

if there are large increasing returns to scale (high ), there should b e a high degree

of monop olistic comp etition in the output sector (low ). Second, m cannot be

determined from the ab ove equations, so it is treated as a parameter. In words,

money is neutral in the steady state. Since the steady-state money growth rate,

, a ects the steady state through the nominal interest rate, money is not sup er-

neutral.

Since the equilibrium cannot b e solved for analytically, I log-linearize the system

around the steady state. The equations describing the log-linearized mo del are given

in the App endix. Now the mo del can b e cast in the form,

x^ = x^ + " + (^x E [^x ]) ; (35)

0 t 1 t1 2 t 3 t t-1 t

where x^ is the p ercentage deviation of x from its steady state. Note that the

t t

co ecient matrices, ( ; ; ; ), are nonlinear functions of the deep parameters.

0 1 2 3

The system is solved following Sims (1995), whose metho d is a generalization of

Blanchard and Khan (1980). The metho d, based on the QZ decomp osition, is

analytically more general and numerically more stable. The solution, if there is a

unique equilibrium, takes the following form:

x^ = x^ + " ; (36)

t 1 t1 2 t

where there is no exp ectational term. The solution is a restricted VAR in the

sense that the co ecient matrices, ( ; ), are functions of the deep parameters.

1 2

Note that the solution is equivalent to the following relation of the non-transformed

variables:

log(X )= [(I ) logx + log ]+[(I ) log ] t + log(X )+ " ;

t 1 1 X 1 X 1 t1 2 t

where the log of a vector is the vector containing logs of the comp onents. This

relation is a rst-order VAR with a constant and a time trend. The t of the DSGE

mo del is compared with an unrestricted VAR of this form.

x x x

t t

That is, x^ = ' log :

x x

I use a mo di ed version of the MATLAB program gensys.m written by Christopher Sims.

The program reads ( ; ; ; ) from (35) as inputs and writes ( ; ) in (36) as outputs.

0 1 2 3 1 2 25

Table 1: Restrictions for Rigidities

( ; ; )

L Y K W P

Prototyp e DSGE (1; 1; 1) 0 0 0

Monop olistic Comp. free 0 0 0

Real Rigidities (1; 1; 1) free 0 0

Wage Rigidities free 0 free 0

Price Rigidities free 0 0 free

All Rigidities free free free free

3 E ects of Monetary Policy

The structural VAR literature on the identi cation of monetary p olicy convention-

ally uses four variables. They are the interest rate, money sto ck, the price level and

output: (r;M;P;Y ): Restricting the analysis to the ab ove four variables preserves

comparabilityofmy mo del with the ones in the literature. This said, note that the

data should not contain more than four variables. Since the error structure of the

mo del comprises four sho cks, using more than four variables almost always make

the covariance matrix of the data to b e singular: a degenerate likeliho o d.

3.1 Impulse Resp onses

Before analyzing the quantitative implications of the estimated mo del, it is inter-

esting to study the impulse resp onses for several restricted versions. These impulse

resp onses are drawn with resp ect to a 1% expansionary temp orary sho ck in monetary

p olicy. The qualitative role of rigidities in pro ducing the business-cycle prop erties of

monetary p olicy b ecomes clearer in the exercise. The parameters are set to arbitrary

but plausible values and then restricted as follows.

Six sp eci cations of the mo del are considered, each summarized by a row in

Table 1. The rst row describ es a prototyp e DSGE mo del without monop olistic

comp etition or any rigidities. The second corresp onds to a mo del featuring mo-

nop olistic comp etition but no rigidities. The third intro duces real rigidities alone,

without monop olistic comp etition or nominal rigidities. Then, monop olistic comp e-

tition and wage rigidities are considered in the fourth mo del, without real rigidities.

Analogously, the fth sp eci cation has monop olistic comp etition and price rigidities,

without real rigidities. Finally, the full mo del with all parameters free has all the

rigidities.

These four variables lie at the heart of analysis also in King and Watson (1995). 26 −3 −3 −3 x 10 x 10 x 10 4 4 4 0.1 0.5 0

2 2 2 0 0

interest rate 0 0 0 −0.1 −0.5 −0.05 0 5 10 0 5 10 0 5 10 0 5 10 0 5 10 0 5 10 2 2 2 2 2 2

1 1 1 1 1 1 money

0 0 0 0 0 0 0 5 10 0 5 10 0 5 10 0 5 10 0 5 10 0 5 10 2 2 2 2 0.4 2

1 1 1 1 0.2 1 price

0 0 0 0 0 0 −8 −7 −8 0x 10 5 10 0x 10 5 10 0x 10 5 10 0 5 10 0 5 10 0 5 10 5 0 5 10 50 2

0 −2 0 5 0 1 output

−5 −4 −5 0 −50 0 0 5 10 0 5 10 0 5 10 0 5 10 0 5 10 0 5 10

RBC M.C. Real R. Wage R. Price R. All R.

Figure 2: Rigidities and Impulse Resp onses*

* The resp onses are with resp ect toa1% expansionary sho ck in monetary p olicy. 27

Each column of Figure 2 corresp onds to a row in the table. The rst three

columns illustrate how the economy resp onds to a monetary sho ck when there are

no nominal rigidities. In these cases, expansionary monetary p olicy decreases output

and increases the nominal interest rate, which is the reverse of the output e ect and

the liquidity e ect|the two e ects of monetary p olicy de ned in the Intro duction.

In the rst column, called RBC in the gure, a monetary sho ck has a p ositive

e ect on the interest rate and a negative e ect on output. Both e ects are quanti-

tatively very small. This results from the fact that anticipated in ation, through a

substitution e ect, moves the interest rate up and output down. The resp onses of

the second column (M.C.) are similar to those of the prototyp e DSGE mo del, except

for the di erent magnitude of the output movement. The resp onses in the Real R.

column are almost the same as the prototyp e DSGE mo del, even if there are real

rigidities. The only di erence is that the magnitude of the output movement is a

little larger than that in the prototyp e DSGE mo del. Thus, real rigidities by them-

selves do not seem to signi cantly impact the b ehavior of the nominal variables.

However, as we shall see, when real rigidities are combined with nominal rigidities,

the b ehavior of the mo del resp onses is drastically di erent.

The next three columns consider how the economy resp onds to a monetary sho ck

when there are nominal rigidities. Nominal rigidities pro duce the output e ect, but

the liquidity e ect is likely to app ear only when real rigidities are added to nominal

rigidities.

In the fourth column, called Wage R., wage rigidities are intro duced together

with monop olistic comp etition. The impulse resp onses display the output e ect

of monetary p olicy. After an expansionary monetary sho ck, the households adjust

wages only gradually due to the presence of adjustment costs. Therefore, they

partially increase hours of work and this, in turn, increases output. Also note

that prices do not adjust instantaneously. However, the liquidity e ect do es not

app ear. In the fth column (Price R.), price rigidities replace wage rigidities. This

mo del also exhibits the output e ect of monetary p olicy, through the b ehavior of the

In an IS-LM framework, an increase in money supply decreases the `real' interest rate. However,

the exp ectation of future in ation pushes the `nominal' interest rate upwards.

Recall the result of Benassy (1995) that the movements of real variables are not a ected by

nominal sho cks at all. There is no ro om for the substitution e ect since the utility function is

separable among consumption, real balances, and lab or.

This result is robust to the size of , since the e ect of real rigidities through the price of

investment is to o small compared with the anticipated in ation e ect. This is not true if there are

nominal rigidities.

Ball and Romer (1990) also show that real rigidities, in the forms of imp erfect information

and eciency wages, play a crucial role in explaining nominal rigidities and the nonneutralityof

nominal sho cks. 28

rms. The rms will not adjust their price fully and would rather pro duce more

output. The movement of prices is slower than in the mo del with wage rigidities.

The liquidity e ect do es not app ear, either.

A common feature of columns 4 and 5 is the signi cantly p ositive resp onses of

the interest rate. This is due to a dramatic increase in investment, not matched

by an equal increase in savings. Investment increases since higher output demand in

subsequent p erio ds is exp ected, which then increases the rental rate. Savings do

not increase as much, despite the fact that there is a temp orary increase in income.

Furthermore, since prices adjust only gradually, exp ectations of future in ation also

push the nominal interest rate up.

The last column (All R.) corresp onds to the full mo del with all the rigidities.

Combining real and nominal rigidities, we nally capture the liquidity e ect of

monetary p olicy. The output e ect is also quite sizable. The instantaneous increase

of output is much smaller than in the mo dels with only nominal rigidities.

3.2 Estimation Pro cedure

For the mo del to be well-de ned, the parameters are restricted within the region

sp eci ed in Table 2. Also, recall here a few restrictions already mentioned. The

existence of a balanced growth path needs the following:

(1 )

G = g :

The next restriction makes the problem of the rms well-de ned:

1 ;

This mo del is not in accord with the argument preferring wage rigidities to price rigidities.

Cho (1993) intro duces nominal rigidities via nominal contracts and concludes that nominal wage

contracts give a b etter picture of the economy. His argument is based up on the fact that the

technology sho ck has a negative e ect on output in a mo del with price rigidities. My mo del do es

not exhibit such an anomaly. Yun (1994) also shows that the presence of this anomaly dep ends on

the degree of price rigidities and the sp eci cation of monetary p olicy.

This feature is, of course, not insensitive to the parameter values. Imp ortant parameters are

the serial correlation co ecients of monetary p olicy, and . However, assigning plausible

values to these parameters do not pro duce the liquidity e ect. Note also that the p ositive resp onse

of the interest rate in Price R. induces the oversho oting b ehavior of the money sto ck.

This increase of the real interest rate is not a generic result of a sticky price mo del. In a

cash-in-advance mo del of Ohanian and Sto ckman (1995), monetary expansion decreases the real

interest rate.

To describ e this in terms of an IS-LM framework, the two exp ectational e ects of an increase

in money supply shifts out the IS curvesomuch that it overwhelms the more usual shift of the

LM curve, to raise b oth real and nominal interest rates. 29

Table 2: Parameter Region

parameters region

; ; ;a (0;1)

; G (or g ) ;; ; (1; 1)

L Y

A; ; ; ;b;m; (0; 1)

K W P

; ; ; ; ; ; ; (1;1)

b A A M r R

2 2 2 2

; ; ; ; ; (0; 1)

1 2

b A

1 :

The requirement that the mo del has a unique solution forces additional restric-

tions on the parameter space which cannot be expressed in an analytic form. For

any parameter vector which pro duces no equilibrium or multiple equilibria, the like-

liho o d value is set to a very small number so as not to a ect the maximum of

the likeliho o d function. Therefore, the likeliho o d function is discontinuous on the

b oundary.

For a parameter vector in the region of a unique equilibrium, the solution of the

log-linearized mo del takes the form of (36). This is used in the construction of the

likeliho o d, which is de ned as the log-likeliho o d conditional on the rst observation,

i.e.

@ x^

LH (; X ;X ;X ) = log (p df (x ^ ; x^ ; x^ jx^ )) + log ; (37)

1 2 T 2 3 T 1

t=2

where is a vector of all the parameters. The likeliho o d rep orted b elow excludes

the constant term of the normal density involving and the Jacobian term, since

the same forms app ear in the likeliho o d of a comparable VAR to the logged data.

The evaluation of the likeliho o d is nonstandard in two ways: the variance of

the error term is not full rank and some of the variables are not observed in the

data. Moreover, the dimension of the variance-covariance matrix of the full data is

so large that the matrix cannot be stored on a p ersonal computer. Therefore, we

need a recursive way of computing the likeliho o d which avoids storing the matrix.

From a Bayesian p ersp ective, this is equivalent to putting a zero prior probability on the region

of the parameter space where the mo del do es not exhibit a unique solution. Some researchers,

Farmer (1993) for example, say that a mo del with multiple equilibria is theoretically reasonable

and empirically plausible. Kim (1994) argues that those researchers examined the plausibilityof

indeterminacy without presenting a formal statistical test and prop oses a p osterior o dds ratio test,

which formally tests for the existence of indeterminacy. 30

In this pap er, the likeliho o d is calculated by the Kalman ltering metho d.

The ab ove discontinuities are problematic for usual optimization routines based

on a gradient metho d. Optimization routines which do not require gradient eval-

uation can be used instead, for example Nelder-Meade. However, given the large

dimension of the parameter vector to estimate, it would be inecient to use this

routine. For the estimation of the mo del, I use a routine which is robust to discon-

tinuities even if it is based on a gradient metho d.

Quarterly U.S. data from 1959:I to 1995:I are extracted from Citibase. The

App endix contains the exact data descriptions and Figure 6 plots the data for the

levels, the log-levels, the p ercentage deviations, and the log-di erences. Since the

data set includes only four variables, the estimation pro duces large standard errors

for the parameters which are not very relevant for the movement of those four

variables. Esp ecially, only one out of the four variables is a real variable, so the

parameters which are mainly related with real variables are lo osely estimated. Also,

the discussion of the estimates will be fo cused on the parameters most relevant to

the e ects of monetary p olicy.

3.3 Estimation Results

Let us b egin with the preference parameters given in Table 3. The discount factor,

, is 0.999. It is larger than the usually calibrated value, but similar to other estima-

tion results. For example, the estimate in McGrattan, Rogerson, and Wright (1995)

is 1.001. The share of consumption in utility, a, is 0.672, with a standard deviation

of 0.658. Thus, my estimate is within one standard deviation of the conventionally

calibrated value, 0.4. The standard deviations are given in the parentheses. The

CRRA co ecient, , is 12.18 with a standard error of 7.30, therefore the hyp othesis

of the logarithmic preferences ( = 1) is rejected at a 10% signi cance level, but not

at a 5% signi cance level. Both the elasticityof substitution b etween consumption

An alternative metho d based on approximating an in nite order VAR with a nite order VAR

can b e used instead. Kim (1995) explains b oth metho ds and compares them from a computational

p ersp ective.

The routine is implemented byaMATLAB program csminwel.m written by Christopher Sims.

The standard errors are computed by using the second derivatives of the (log) likeliho o d

function. The Hessian of the likeliho o d function is evaluated at the estimates and then inverted.

The standard errors are read as the ro ots of the diagonal elements. Note that the justi cation

for computing these t-statistics is Bayesian and that they should be interpreted carefully. The

standard errors and test statistics computed here are meantascharacterizations of the shap e of

the likeliho o d function, which under weak regularity conditions are asymptotically normal-shap ed

regardless of the presence of unit ro ots and cointegrations.

The parameters for capital accumulation are explained later together with the technology

parameters. 31

and real balances, , and the steady-state share of real balances in the consumption

bundle, b, are very close to the calibrated values of Hairault and Portier (1993). The

sho ck in money demand, b , is p ersistent, with an autoregressive co ecient 0.99, and

0:008

has an unconditional variance of 0.40 = .

10:99

Table 3: Preference Parameters

Description Function Parameter estimates

C (1L )

utility a =0:672; =12:18;

(( ) )

function (0:658) (7:30)

h i

discount =0:999;

C ; ;L E U

t t 0

t=0

factor (0:011)

=( 1)

2 2

( 1)=

2 2

=0:112; b =610 ;

( 1)=

2 2

C C + b

(0:060) (1 10 )

money

=0:990; =0:008;

b b

demand

= log + " log

b bt

b b

(0:018) (0:008)

sho ck

Bayesian standard errors in parentheses.

The estimated parameters for monop olistic comp etition and the rigidities are in

Table 4. The degree of monop olistic comp etition is higher in the output sector, as

one may exp ect, and so is the rigidity co ecient. Despite that, note that one should

consider also the other co ecients a ecting the households' and the rms' choices

in order to assess the relative imp ortance of the two nominal rigidities. However,

such an assessmentisbeyond the scop e of this pap er. The estimate for real rigidities

( = 312) implies that, at the steady state, the installation of 100 units of capital

is accompanied by 5:6 units as the adjustment cost. The parameter is very sharply

estimated with a standard error of 4.

The estimated parameters for capital accumulation and technology are in Table

5. The value of depreciation, , is 0.019 and so close to the conventionally calibrated

value, 0.015 or 0.020. The parameter for the capital income share, , is to o small and

the parameter for increasing returns, , is to o large. However, since their standard

errors are large, these parameters are within one standard error of plausible values.

Since is so large, the lab or-augmenting growth factor, g ,isvery close to 1. It is 1.0001.

Pegging at a reasonable value, I obtained another lo cal maximum of the likeliho o d function

which attains the same likeliho o d value. At the new estimate, however, the risk-aversion parameter,

, b ecomes to o large. Again, its standard error is large. Meanwhile, the empirical results suchas

impulse resp onses and variance decomp ositions are not much a ected by which estimate is used.

That is, the mo del is weakly identi ed in these dimensions. This calls for further research using

more data on real variables to obtain sharp er estimates. 32

Table 4: Monop olistic Comp etition and Rigidities

Description Function Parameter estimates

L i

W = W

monop olistic =12:37; =1:101;

L Y

comp etition JY (13:38) (0:240)

P = P

P W

t t

AC = W

nominal t =0:153; =0:806;

W P

2 P W

t1 t1

rigidities (0:297) (1:815) P

AC = Y

2 P G

real = 312;

AC = I

2 K

rigidities (4)

Bayesian standard errors in parentheses.

The estimates of by Hall (1990) for U.S. manufacturing industries range from 1:1

to 10.The growth rate of output, G (=1:002), is smaller than the average growth

rate of data on output, 1.003. This is partly b ecause the upward movement of output

maybe caused by p ersistent sho cks in the pro duction function.

The estimated autoregressive pro cesses for the sho cks related with the pro duc-

tion function are also rep orted in Table 5. The technology sho ck is very p ersistent,

with an autoregressive co ecient of 0.981, and has an unconditional variance of

0:0003

0.008 = ,much smaller than usually calibrated values. The xed-cost sho ck

10:981

is also p ersistent, with an autoregressive co ecient of 0.911, and has an uncondi-

0:020

tional variance of 0.11 = . The variance of the xed-cost sho ck is larger

10:911

than that of the technology sho ck. The two sho cks are negatively correlated with

a correlation co ecient of (0:656). The xed-cost sho ck is very imp ortant in the

estimated mo del, whichmay indicate a problem in the conventional sp eci cation of

a pro duction function that do es not include an additive term.

Lastly, the parameters for monetary p olicy are in Table 6. The estimate of the

growth rate of money, , is 1.013 and this is very close to the average money growth

rate of the data, 1.014. The parameter for the sensitivity of the interest rate, ,

is 0.576 and signi cantly di erent from 0. This indicates that the conventional

sp eci cation of monetary p olicy in the DSGE literature that do es not include the

interest rate is not empirically supp orted. The serial correlation of the money

growth rate, , is negative but this is o set by the p ositive serial correlation of

Note that there are near unit ro ots in the interest rate pro cess and that the mo del implies strong

restrictions on the geometric trends of the four variables. In order to check the robustness of the

estimates against the low frequency prop erties of the data, it is worth estimating the transformed

mo del using ltered data to eliminate low frequencies and comparing the estimates with those of

this pap er. 33

Table 5: Technology Parameters

Description Function Parameter estimates

=0:225; =3:808;

t 1

pro duction Y = A (K (g L ) ) (0:260) (5:905)

t t t

function G G =1:002;

(0:002)

capital =0:019;

K = I +(1 )K

t t1 t1

accumulation (0:055)

) log(

technology =0:981; =0:0003;

sho ck (0:025) (0:0002)

= log( )+"

A At

=0:911; =0:020;

log

(0:029) (0:008) xed-cost

= 0:656; sho ck

= log + " A

(0:298)

Bayesian standard errors in parentheses.

the sho ck in monetary p olicy, . This sho ck is mildly serially correlated with a

0:00002

co ecient of 0.487 and has an unconditional variance of 0.00005 = .

10:487

Table 6: Parameters for Monetary Policy

Description Function Parameter estimates

=0:576; = 0:158;

= ( )

M M

t1 t2

monetary (0:266) (0:085)

( )

r r r

t1 t2

p olicy =0:999; =0:980;

(1 ) 1

( )

(3:771) (3:582)

monetary

=0:487; =0:00002;

p olicy

log( )= log( )+"

t t1 t

(0:006) (0:00001)

sho ck

Bayesian standard errors in parentheses.

The parameters describing the scal p olicy, T and , are not identi ed if the

economy is a version of Ricardian equivalence. Even if Ricardian equivalence do es

not hold, the parameters are so weakly identi ed that they are not included in the

estimation. 34

3.4 Assessing the Fit

We have seen how the log-linearized rst order conditions pro duce the solution of

which the form is a restricted rst-order VAR to all the variables. Since the VAR

structure allows us to interpret the likeliho o d as a measure of the \normalized mean

squared error of forecasts," it is natural to compare the t of the DSGE mo del with

that of a comparable unrestricted VAR to only the data variables. Previous DSGE

mo dels, Leep er and Sims (1994) for example, ts the data worse than unrestricted

VAR mo dels.

There are 30 free parameters in the DSGE mo del and its likeliho o d value at

the b est t is 2775. A rst-order reduced-form VAR is estimated with a constant

term and a time trend. Its maximum likeliho o d value is 2790 and it has 34 free

parameters. According to the likeliho o d criterion, the DSGE mo del forecasts only

0.9% worse than the VAR.

The time series charts in Figure 3 compare the residuals of the DSGE mo del

with the innovations from the VAR with a time trend. Overall, the DSGE mo del

predicts no b etter than the VAR, but it do es nearly as well and is sup erior over

certain time ranges. If we compare the p erformance over each decade by averaging

the residuals, the DSGE mo del do es worse in the 1960s, b etter in the 1970s, ab out

as go o d in the 1980s, and again worse in the 1990s.

The chart do es not contain the residuals for the rst three p erio ds corresp onding

to the rst year of the data. The DSGE mo del do es esp ecially worse than the VAR

in those early unrep orted p erio ds, which is a generic problem when tting a DSGE

mo del. This is b ecause a DSGE mo del implies an in nite order VAR for the data

variables. Therefore, it is dicult to explain the movements with a few previous

observations. In other words, since our data vector do es not include all the variables

of the DSGE mo del, the rst observation do es not capture either short run or long

run information completely. Note also that the likeliho o d is de ned conditional on

There are 4 constants, 4 time trends, 16 co ecients, and 10 free elements in the covariance

matrix of the innovations.

The maximum likeliho o d value of a VAR without a time trend is the same as that of the DSGE

mo del. The VAR has 30 free parameters. What is more, if the likeliho o d is de ned conditional

on -10th p erio d (10 p erio ds b efore the sample), the DSGE mo del p erforms b etter than the two

VARs. See the following table for the exact likeliho o d values. Meanwhile, mo del selection criteria

which p enalize the numb er of parameters pro duce similar results. Three criteria are used: Akaike

information criterion, Hannan-Quinn criterion, and Schwarz criterion.

observation conditioned on DSGE VAR with a trend VAR without a trend

the 1st observation 2775 2790 2775

the -10th observation 2786 2756 2778

Recall that the data series start at 1959:I. Since the likeliho o d is conditional on the rst

observation, the predictions start at 1959:I I. 35 0.02 0.06

0.04 0.01 0.02 0

money 0 interest rate −0.01 −0.02

−0.02 −0.04 1960 1970 1980 1990 2000 1960 1970 1980 1990 2000

0.03 0.04

0.02 0.02 0.01 0 price 0 output −0.02 −0.01

−0.02 −0.04 1960 1970 1980 1990 2000 1960 1970 1980 1990 2000

dotted line=DSGE solid line=VAR

Figure 3: Prediction Errors of the DSGE and the VAR*

* Rep orted are the residuals in the p ercentage terms. 36

the rst observation. In a rst-order VAR where all variables are available in the

data, the rst observation captures b oth information precisely.

Table 7: Forecast Standard Deviations

standard deviation r M P Y

The DSGE mo del 0.00256 0.00673 0.00422 0.00797

The VAR with a trend 0.00247 0.00740 0.00417 0.00686

The VAR without a trend 0.00259 0.00752 0.00426 0.00693

For convenience, Table 7 rep orts the Cholesky decomp osition of the covariance

matrices, which enables comparisons by variable across the mo dels. The ordering

of the variables for Cholesky decomp osition is (r;M;P;Y ). The rep orted elements

are interpreted as the standard deviation of the one-step-ahead forecast errors. The

DSGE mo del improves on the VAR with a time trend only for money. Compared

with the VAR without a time trend, the DSGE mo del improves on the interest rate,

money and price. The DSGE mo del captures the movement of the nominal variables

pretty well. Overall, we daresay that the DSGE mo del explains the movements of

the nominal variables relatively b etter than those of the real variables.

Another way to assess the t is the steady state of the mo del with the means of

the data. The estimated parameters imply a steady state that matches the means

of the detrended data in some but not all asp ects, as rep orted in Table 8. The

steady state of the interest rate is di erent from the mean by less than 1%. The

steady states of money and output are lower than their means since much of their

movements are explained by p ersistent sho cks. This is consistent with the fact

that the steady state of the price level is higher than the mean by 9%.

For the mo del's implications to b e credible, the estimates must pro duce sensible

dynamic resp onses to the exogenous sho cks. Before computing the impulse resp onses

and the variance decomp ositions, one must decide how to treat the covariation in

any two innovations. In my mo del, there is only one covariation: the technology

sho ck, " , and the xed-cost sho ck, " . To make the case that monetary p olicy dis-

At t

turbances have an imp ortant e ect on real aggregate activity, I make the technology

Rep orted are the diagonal elements of the Cholesky decomp ositions. Since the VAR is unre-

stricted, the maximum likeliho o d covariance matrix is the same as the sample covariance matrix

formed by the innovations. However, since the DSGE mo del corresp onds to a restricted VAR,

those two covariance matrices need not matchin the case of the DSGE mo del. Also, there is a

problem of which information should the forecast b e conditioned on. In this pap er, the maximum

likeliho o d covariance matrix is used to form the covariance matrix of the data.

See Kim (1995) for more discussion on this p oint.

See the graphs plotting the p ercentage deviations of the data in Figure 6. The plots for money

and output are steadily increasing. 37 0 0.2 0 0.4

−0.05 0.1 −0.1 0.2 interest rate −0.1 0 −0.2 0 0 5 10 0 5 10 0 5 10 0 5 10 2 0.1 0.05 0.2

1 0 0 0 money

0 −0.1 −0.05 −0.2 0 5 10 0 5 10 0 5 10 0 5 10 1 0 0 0.5

0.5 −0.5 −1 price

0 −1 −2 0 0 5 10 0 5 10 0 5 10 0 5 10 0.5 0 2 1

1 0.5 output

0 −0.5 0 0 0 5 10 0 5 10 0 5 10 0 5 10

monetary policy money demand technology fixed cost

Figure 4: Impulse Resp onses at the Estimated Parameters*

* The resp onses are with resp ect to one-standard-deviation sho cks. 38

Table 8: Data Means and the Steady State

interest rate money price output

U.S. data means 1.0166 2.869 24.367 19.610

Mo del steady state 1.0259 2.444 26.752 16.231

sho ck as imp ortantas p ossible. For this purp ose, the technology sho ck is assumed

to b e ordered b efore the xed-cost sho ck. This ordering implies that an innovation

in technology a ects not only the technology sho ck but also the xed-cost sho ck.

Figure 4 rep orts the impulse resp onses evaluated at the estimated parameters.

All resp onses are with resp ect to a temp orary sho ck of one standard deviation.

The rst column corresp onds to the resp onses to a p ositive monetary p olicy sho ck.

We can nd the liquidity e ect in the resp onse of the interest rate and also the

output e ect in the resp onse of output. The auto correlation of the sho cks causes

the movement of money to b e smo oth and the nominal rigidities are a source of slow

adjustment of prices.

The next three columns corresp ond to the other sho cks. A p ositive money de-

mand sho ck moves the interest rate up and so output down. A p ositive technology

sho ck decreases the interest rate. This is mainly b ecause the technology sho ck is

negatively correlated with the xed-cost sho ck. A p ositive xed-cost sho ck moves

the interest rate up since it is accompanied by an increase in exp ected in ation.

Exp ected in ation is a source of p ositivemovement of output.

3.5 Variance Decomp ositions

Once estimated, the mo del can b e used to evaluate the underlying exogenous sources

of uctuations over the sample p erio d. The fraction of the variance attributable to

each sho ck is readily computed from the solution, (36), evaluated at the estimated

parameters. In the following four tables, the variances of the interest rate, money,

price, and output are decomp osed into the fractions that are explained by the sho cks

in monetary p olicy, money demand, technology, and xed cost. The p ercentages

of each variable's forecast error variance due to the four sho cks are rep orted for

several forecast horizons. The rep ort is b oth in the short and medium runs: from 1

to 5 quarters, 10 quarters, and 20 quarters. The rst line of Table 9 can b e read as

Impulse resp onses and variance decomp ositions let us analyze only the overall e ects of the

sho cks. In contrast, historical decomp ositions allow us to understand the role of the sho cks p erio d

by p erio d. By historical decomp ositions, the unanticipated movements are attributed to each

structural disturbance at each date, so that the prop ortion of monetary p olicy sho ck explaining

the forecast errors at each date is calculated. 39

follows: 2% of the variance of the interest rate is explained by the monetary p olicy

sho ck when the forecast horizon is 1 quarter and 3% is explained by the sho ck when

the forecast horizon is 2 quarters. Note that the four numb ers in each column add

up to 100%.

Table 9: Interest Rate Decomp ositions

INTEREST RATE 1 2 3 4 5 10 20

monetary p olicy .02 .03 .04 .04 .04 .03 .03

money demand .20 .17 .15 .13 .12 .08 .06

technology .04 .07 .10 .12 .15 .24 .32

xed cost .74 .73 .71 .70 .70 .64 .59

Table 9 gives the results for interest rate decomp ositions. The xed-cost sho ck

accounts for more than 70% of interest rate movement in the short run. As the

forecast horizon lengthens, its imp ortance decreases but is still more than 50%.

This is partly b ecause the xed-cost sho ck strongly a ects exp ected in ation. An

intuitive explanation is that the increase in xed cost is re ected in the price level

slowly over time and that this causes in ation. The second most imp ortant sho ck

dep ends on the forecast horizon. It is the sho ck in money demand in the short run,

and the technology sho ck in the medium run. The sho ck in monetary p olicy do es not

seem to b e imp ortant for the movement of the interest rate. Including a feedback

mechanism of the real activities in the monetary p olicy reaction function would

make monetary p olicy more imp ortant. Under the new sp eci cation, the technology

and xed-cost sho cks account for less of interest rate movement since part of the old

propagation is endogenized by the feedback of the real activities. This conjecture is

con rmed by calibrating the parameter representing the sensitivity of the feedback.

Table 10 shows that the movement of the money sto ck is almost entirely due to

the sho ck in monetary p olicy. Even if money is endogenous in principle, the degree

of the endogeneityisvery small. Other sho cks play little roles. The variance decom-

p osition of money in my mo del is quite di erent from the results in the structural

VAR literature, where the sho ck in money demand plays an imp ortant role. Here,

the money demand sho ck do es not contribute at all, even if its variance is large

relativetothe variance of the monetary p olicy sho ck.

Therefore, my mo del is immune to the p otential criticism that it captures the movementof

the interest rate mainly b ecause of the complicated sp eci cation of monetary p olicy. Even if my

sp eci cation of monetary p olicy is imp ortant in generating a realistic b ehavior of the interest rate,

the variance decomp osition shows that monetary p olicy is not akey determinantof the interest

rate movement. 40

Table 10: Money Decomp ositions

MONEY 1 2 3 4 5 10 20

monetary p olicy .95 .98 .99 .99 .99 .99 .99

money demand .01 .00 .00 .00 .00 .00 .00

technology .00 .00 .00 .00 .00 .00 .00

xed cost .04 .02 .01 .01 .00 .00 .00

Table 11: Price Decomp ositions

PRICE 1 2 3 4 5 10 20

monetary p olicy .17 .17 .17 .17 .18 .19 .20

money demand .12 .12 .12 .12 .12 .12 .12

technology .66 .65 .65 .65 .65 .65 .64

xed cost .05 .05 .05 .05 .05 .05 .05

The most imp ortant factor for price movements is the technology sho ck, which

explains steadily over 60% as shown in Table 11. The sho ck in monetary p olicy

explains 15-20% and the sho ck in money demand explains 12% of price movement.

Fixed cost has some e ect, 5%, on price movements. Here again, the new sp ec-

i cation of monetary p olicy including the feedback of real activities increases the

imp ortance of the monetary p olicy sho ck. Note that the variable decomp osed is the

price level, not the in ation rate. For the in ation movement, the xed-cost sho ck

is an imp ortant factor.

Table 12: Output Decomp ositions

OUTPUT 1 2 3 4 5 10 20

monetary p olicy .21 .20 .19 .17 .15 .07 .03

money demand .20 .19 .17 .15 .13 .07 .03

technology .03 .11 .20 .31 .41 .72 .89

xed cost .55 .50 .44 .38 .32 .14 .05

Table 12 rep orts that the xed-cost sho ck explains 40-55% of output uctuations

in the short run. If the technology and xed-cost sho cks are combined and called

supply sho cks, these explain less than 70%, if the forecast horizon is less than 1

year. However, in the medium run, the technology sho ck accounts for 70-90% of

output uctuations. In the short run, the sho ck in monetary p olicy and the sho ck

Compare my results on short and medium runs with the conclusion in Prescott (1986) that

technology sho cks account for 75% of business cycles. 41

in money demand are also imp ortant in understanding output uctuations. Each

explains 15-20% of output movements in the short run, but much less in the medium

run. This variance decomp osition of output is similar to the results of the structural

VAR literature.

3.6 Cyclical Implications

It is standard in the RBC literature to fo cus on the summary statistics describing rel-

ativevolatility of the series and their correlation with output. The formal maximum

likeliho o d estimation pro cedure obviously do es not place all of the weight on this

small set of statistics, so it is worthwhile to see what the parameter estimates imply

for these commonly studied statistics. Therefore, the standard deviations and the

contemp oraneous correlations of the mo del forecasts are computed and compared

with those of the data. The tth p erio d mo del forecasts are conditioned on the

sample up to t 1, and the parameters are set to the full-sample maximum likeli-

hood estimates. Both the mo del and data series are logged and di erenced b efore

calculation, so these statistics are in p ercentage terms.

Table 13 shows that the standard deviation of output forecasted by the mo del

exactly matches that of the data. For the other three variables, the standard devia-

Table 13: Contemp oraneous Second Moments

mo del forecast

interest rate money price output

(data)

0.0034

interest rate -0.3918 0.3753 0.0198

(0.0026)

0.0097

money (-0.1276) 0.0866 0.1524

(0.0086)

0.0099

price (0.0954) (0.0231) -0.4229

(0.0066)

0.0089

output (0.3488) (0.1360) (-0.2455)

(0.0089)

An alternativeway is to calculate the second moments of the simulated series and to compare

them with those of the data, which is more conventional in the RBC literature.

If the series are logged and detrended, the correlation of money and output is negative and

that of money and the interest rate is p ositive, for b oth the data and the mo del forecasts. This is

incompatible with the cyclical implications from the two e ects of monetary p olicy, so I compare

the statistics of the logged and di erenced series here. Log-di erencing is also adopted in King

and Watson (1995). 42

tions of the mo del forecasts are larger than those from the data. The absolute values

of the correlations of the mo del forecasts are also higher, except for the interest rate

and output pair. One p ossible reason is that the variables are highly p ersistenteven

after logarithmic di erencing and that the estimation may b e tting this feature at

the cost of tting higher frequencies. Overall, the mo del matches all the signs of

the second moments. Considering that the estimation pro cedure is designed to t

all asp ects of the data, it is not obvious ex ante how well the estimates are able to

match the small set of statistics.

King and Watson (1995) analyze the nominal features of business cycles with

three di erent DSGE mo dels. While the mo dels have diverse successes and failures,

none can account for the fact that the nominal and real interest rates are \inverted

leading indicators" of real economic activity. That is, none of their three mo dels

captures the U.S. business cycle fact that a high nominal or real interest rate in the

current quarter predicts a decrease of real economic activity two to four quarters

in the future. To see if my mo del improves on that p ersp ective, serial correlation

co ecients between interest rate and output are computed.

Table 14 shows that my mo del ts the prop erty of the data that the nominal

interest rate is an inverted leading indicator. The serial correlation co ecients from

the mo del repro duce the pattern of the data that an increase in the interest rate in

the current p erio d predicts a decrease of output two to six quarters in the future.

3.7 Policy Exp eriments

We analyze four exp eriments in monetary p olicy. Two regimes of monetary p olicy

are considered. Under each regime, scal p olicy is also mo di ed so that the economy

exhibits a unique equilibrium. The rst regime, called an M -p olicy, is one in which

Using the linear trends, I do not rep ort the comparison with other mo dels. Applying Ho drick-

Prescott ltering to the mo del-generated data, the four data variables in my mo del p erforms b etter

than those in other DSGE mo dels.

The three mo dels are a \real business cycle" mo del, a \sticky price" mo del, and a \liquidity

e ect" mo del. Their \sticky price" mo del is theoretically similar to the mo del presented in this

pap er. However, its empirical implications are rather di erent, mainly b ecause of a di erent sp ec-

i cation of monetary p olicy. Their \liquidity e ect" mo del is usually called a limited participation

mo del.

This fact is consistent with the evidence presented in Bernanke and Blinder (1992).

Here the results only for the nominal interest rate are calculated, since there is no data for

exp ected in ation rate. Using the exp ected in ation rates generated from the DSGE mo del, I also

nd that my mo del ts the idea that the real interest rate is also an inverted leading indicator. King

and Watson (1995) calculate the correlations from the estimated sp ectral density matrix, using only

the business cycle (6-32 quarters) frequencies. However, a di erent metho d of detrending is hardly

likely to change the results. 43

Table 14: Serial Correlations between Interest Rate and Output

Corr ( log r ; log Y )

t t+d

d(quarters) 1 2 3 4 5 6

data 0.072 -0.297 -0.188 -0.209 -0.198 -0.177

mo del 0.042 -0.177 -0.173 -0.193 -0.153 -0.109

the monetary authority controls only the sto ck of money: the elasticity of money

growth to the interest rate in the p olicy function, , is set to zero. The second regime,

called an r -p olicy, is one in which the monetary authority controls only the interest

All other co ecients rate and it corresp onds to the sp eci cation that = 1.

except stay at the value of maximum likeliho o d estimates. Note that the value

of do es not a ect the steady state. Each regime shift can be implemented in two

alternativeways: exp ected and unexp ected. In the rst case, the change in p olicy is

p erceived by the private agents of the economy so they corresp ondingly change their

exp ectations. In the second case, the change is not p erceived by the private agents

so they do not change their exp ectations. Since the p olicy exp eriments are based

up on optimizing b ehavior from a mo del with deep parameters, they are not sub ject

to the criticisms from the Lucas critique. The exp eriments fo cus on how the regime

shift changes the mechanism through which technology sho cks are transmitted, and

so are not a fully edged welfare analysis. However, such analysis gives insight to

the welfare analysis and has never b een done. For this purp ose, impulse resp onses

to a one standard deviation technology sho ck are drawn and compared with those

from the estimated mo del.

The rst two columns of Figure 5 concern an M -p olicy. If the change of the

p olicy is unexp ected (the second column), the resp onses to the technology sho ck

are very similar to those b efore the change. One minor di erence is that money

returns to the steady state faster than b efore the p olicy change. This is b ecause

The economy with an r -p olicy is not a version of Ricardian equivalence, since scal p olicy

a ects variables other than tax and debt through the exp ectation of future prices. See Sims (1994)

on this p oint.

Algebraically, the rst one is to change in the p olicy equation and then to obtain the solution

from scratch. The second is to solve the equilibrium equations under the old p olicy regime and

then to substitute the new p olicy equation for the old one, without changing the other equations

of the equilibrium.

Recent structural VAR mo dels, Co chrane (1995) and Sims and Zha (1995) for example, extract

p olicy-invariant identifying assumptions from optimizing b ehavior. This is partly meanttoavoid

the Lucas critique.

Variance decomp ositions are irrelevant for p olicy exp eriments, since the variance of the mon-

etary p olicy sho ck cannot b e determined in a plausible way. 44 −3 Expected Unexpected x 10Expected Unexpected 0 0 0 0

−0.1 −0.1 −0.5 −0.1 interest rate −0.2 −0.2 −1 −0.2 −15 0x 10 5 10 0 5 10 0 5 10 0 5 10 0 0.05 0 0

−2 0 −10 −0.02 money

−4 −0.05 −20 −0.04 0 5 10 0 5 10 0 5 10 0 5 10 0 0 0 0

−1 −1 −10 −1 price

−2 −2 −20 −2 0 5 10 0 5 10 0 5 10 0 5 10 2 2 0 2

1 1 −2 1 output

0 0 −4 0 0 5 10 0 5 10 0 5 10 0 5 10

M−policy M−policy r−policy r−policy

Figure 5: Policy Exp eriments*

* The resp onses are with resp ect to a one-standard-deviation technology sho ck under

a new p olicy regime. 45

money do es not resp ond to the interest rate. If the p olicy change is exp ected (the

rst column), the resp onses to the technology sho ck are also very similar to those

b efore the p olicy change. Of course, money stays at the steady state. These two

exp eriments show that there is little e ect of changing into an M -p olicy, based on

the parameter estimates. Note that an M -p olicy corresp onds to the sp eci cation

of exogenous money supply. This suggests that the endogeneity of money do es

not have a big in uence on the resp onses to the technology sho ck at the estimated

parameters, even if imp ortant for the propagation of monetary p olicy.

The two right columns of Figure 5 are the resp onses under the r -p olicy regime.

If the change is unexp ected (the fourth column), the resp onses are again similar,

except that money returns to the steady state very slowly. The resp onse of money is

sluggish b ecause the amount of money is not controlled by the monetary authority.

If the change is exp ected (the third column), the resp onses are drastically di erent.

The interest rate do es not move, of course. Money and prices uctuate wildly

and are very unstable. They decrease by around 10% after 10 quarters and these

uctuations are much larger than those found in any other exp eriment. Output do es

not increase but decreases after the p ositive technology sho ck, due to the fact that

money decreases faster than prices. This suggests that an appropriate choice of the

sensitivity-to-interest-rate parameter, , can remove the output uctuations due to

the technology sho ck. UnlikeanM-p olicy, the e ect of changing into an r -p olicy is

quite sizable, esp ecially if the change is exp ected.

To summarize, the mo del can b e used to analyze a wide variety of p olicy issues.

Changes of monetary p olicy can only b e analyzed seriously in a mo del that features

an explicit transmission mechanism of the p olicy. The DSGE mo del in this pap er is

an example.

4 Conclusion

Previous work using a exible-price, comp etitive DSGE mo del has provided reason-

able descriptions of the data on real variables. However, suchwork has not captured

the nominal features of business cycles well. Typically, expansionary monetary p ol-

icy pro duces neither a p ositive resp onse of aggregate output nor a negative resp onse

96 14

Note that there is a scale factor of 10 in the resp onse of money.

Note that there is also a scale factor in the resp onse of the interest rate and that is set not

to in nity but to a very large numb er in the computation.

Using the structural VAR framework, Sims and Zha (1995) exp eriment the changes to an M -

p olicy and an r -p olicy. They also nd that the change into an M -p olicy has mo derate e ects on

output uctuations while the change into an r -p olicy has substantial e ects.

See Dow (1995) for an example of monetary stabilization p olicy. 46

of interest rates.

This pap er aims at lling this gap with a DSGE mo del extended to allow for

real and nominal rigidities. Four sho cks, including b oth technology and monetary

p olicy sho cks, a ect the economy. The exercise with several restricted versions of

the mo del shows that b oth real and nominal rigidities are necessary to pro duce

reasonable impulse resp onses to the sho ck in monetary p olicy. In order to select the

magnitude of the e ects of monetary p olicy in a data-dep endent manner, the mo del

is estimated using maximum likeliho o d on U.S. data. The estimated mo del exhibits

reasonable impulse resp onses and its forecasts pro duce the second moments similar

to those of the data. As a by-pro duct, it also repro duces the fact that an increase

of interest rates in the current p erio d predicts a decrease of real economic activity

two to six quarters in the future, a feature of U.S. business cycles which has never

b een captured by previous research using DSGE mo dels.

It would b e interesting to estimate the mo del for a sub-p erio d and to see howwell

the estimated mo del explains the out-of-sample data. This is particularly interesting

since monetary p olicy regimes are said to have changed several times, e.g. the

Octob er 1979 Volker disin ation. It would b e more helpful to randomize the p olicy

regimes.

For further research, adding more structure into the mo del and using more data

for estimation are likely to pro duce a b etter-b ehaving estimated mo del. For instance,

one may notice that the parameter for real rigidities, ,isvery sharply estimated.

Thus, there maywell b e enough information in the data to enable estimating a more

100

general form of adjustment costs for capital. Also, more real variables need to b e

included in the data to pin down the sources of the volatility of real variables. Since

this pap er deals with only four variables as the data and three out of the four are

nominal variables, it would b e particularly interesting to add more real variables in

101

the estimation and to see howwell this mo del can explain the additional variables.

(1+" )

100 K

" I , where three "'s are The most general form is AC =

3 it

2(1+" )

nonnegative.

101

The interesting features to investigate are the cyclicality of real wages and the volatility of

investment and lab or. The mo di cations in this pap er, real and nominal rigidities, may have

improved or deteriorated the p erformance of DSGE mo dels along these dimensions. 47

A App endix

A.1 The Data

The following time series are extracted from the Citibase database.

Series Title

fyff Interest Rate: Federal Funds (E ective) (% p er annum, NSA)

fm2 Money Sto ck: M2 (Bil. $)

gd Implicit Price De ator: Gross National Pro duct

gnpq Gross National Pro duct (Bil. 1987$) (T.1.6)

p16 Population: Total Civilian Noninstitutional (Thous., NSA)

The variables of the mo dels are de ned as follows.

fyff

; r = 1+

400

fm2

M = 1000 ;

p16

P = gd,

gnpq

: Y = 1000

p16

A.2 Derivatives and Elasticities

The formul for the derivatives and the elasticities are as follows.

U = U (C ;M =P ;L );

it it it t it

@U (1 ) U aC

1 it

= ;

( 1)=

2 2

(C )

(1 ) U ab

1 it

P P

t t

= ;

( 1)=

2 2

(C )

(1 ) U (1 a) @U

1 it

= :

@L (1 L )

it it

3 2

P W

t L

+ 1

L P W P W

t1 t1

it i;t1 i;t1

7 6

; e =

5 4

P W P W

t+1 t+1

i;t+1 i;t+1

t+1

L P W

P W

t t

it it

2 3

1 +

Y P G P

jt j;t1 j;t1

6 7

e = :

4 5

P P P

t+1

j;t+1 j;t+1

t+1

Y P G

jt jt

jt 48

u = U (c ;m =p ;L );

t t t t t

@u (1 ) u ac

1 t

= ;

( 1)=

2 2

(c )

m 1

(1 ) u ab

1 t

p p

t t

= ;

( 1)=

2 2

(c )

(1 ) u (1 a) @u

1 t

= :

@L (1 L )

t t

A.3 The Equilibrium

The rst set of equations describing the equilibrium comes from the problem of the

households.

P W

W t t

AC = W ;

2 P W

t1 t1

K t

; C = W L + Z K + AC I 1+

t t t t t t t

2 K

K = I +(1 )K ;

t t1 t1

= ;

P C

t it

; r = 1 b

2 3

P W P W

W t t t t

1 +

L L P W P W

t1 t1 t1 t1

4 h i 5

e = ;

P W P W

t+1 t+1 t+1 t+1 t+1

L P W P W

t t t t t t

@U 1

= W 1 ;

t t

@L e

0 = Z + Q + (1 )E [Q ] ;

t K t t t t+1

I 3

t K

0 = 1+ E [Q ] ;

t t t+1

2 K

" #

t+1 t

= r E ;

t t

P P

t t+1

log (b ) = log (b )+(1 ) log (b)+" :

t b t1 b bt 49

The second set comes from the problem of the rms.

t 1 t

K (g L ) G ; Y = A

t t t t

log(A ) = log(A )+(1 ) log(A)+" ;

t A t1 A At

log ( ) = log ( )+(1 ) log () + " ;

t t1 t

P t

AC = Y ;

2 P G

= Y Z K W L AC ;

t t t t t t

2 3

P P

t t

1 +

P G P

t1 t1

6 7

e ; =

4 5

P P

t+1 t+1 t+1

P t

P G P

t t t

1 Z K = Y + G ;

t t t t

W L = (1 ) Z K :

t t t t

The third set comes from the b ehavior of the government, with two equations

for the budget constraint and scal p olicy deleted.

0 1

( + )

M r r r M

t t t1

t1 t2

@ A

; =

(1 )(1 )

M M r

t1 t2

log = log + " :

t t1 t

A.4 Steady State

Non-trivial steady-state values are as follows. In what follows, replaces ( +(G 1))

to save space.

r = ;

1 (1 ) 3

2 3

Z = ; 1+

K K

G G

= 1 1+ ;

y Z 2

1 a

(1 ) 1

= ;

c b

(1r )

1 1

L ; y = A 1

Y 50

k = y;

i = k;

w = y;

m 1

p = 1 r ;

c b

1 0

=( 1)

2 2

( 1)=

2 2

( 1)=

2 2

A @

; c = c + b

(c ) 1 L

u = ;

(1 )uac

= ;

( 1)=

2 2

(c )

q = 1+ :

A.5 Log-linearization

First, I have two de nitional equations.

f = p^ p^ ; (38)

t t t1

v^ = w^ w^ : (39)

t t t1

Second, the following equations come from the problem of the households. Due

to the ab ove two de nitional equations, the problem can be written as an VAR(1)

form.

^ ^

c^ = (1 ) w^ + L + k + Z + (40)

t t t t t

y y

! !

i k 3

2 3

i + 1+ k ;

t K K t

G G

y 2 y

^ ^

Gk = i +(1)k ; (41)

t G t1 t1

u^ = (1 ) ac^ (1 a) L ; (42)

t 1 t

1 L

( 1)=

2 2

( 1)=

2 2

c m

c^ = c^ + b (^m p^ ); (43)

t t t

c pc 51

1 1

c^ c^ ; (44) = u^

t t t

2 2

p^ +^c m^

t t t

r^ = (r 1) b + ; (45)

t t

2 2

i h

W W

^ ^

; (46) e^ = f +^v f +^v E

G t t t1 t1 t-1

L L

1 L

u^ = w^ + + e^ L ; (47)

t t t t

1 1 L

(48) qq^ = Z Z +3 i k

t1 t1 K t1 t1

Z + + (1 )qE [^q ] ; +

K t1 G t-1 t

2 2

^ ^

qE [^q ] = 1+ +3 i k ; (49)

G t-1 t t1 K t1 t1

G G

h i

^ ^ ^

= r^ +E f ; (50)

t1 t1 t-1 t t

^ ^

b = b + " : (51)

t b t1 bt

Third, the following equations come from that of the rms. Again, replaces

1 to save space.

^ ^ ^

y^ = A + k + (1 ) L [1 ] ; (52)

t t t t t

^ ^

A = A + " ; (53)

t A t1 At

^ ^

= + " ; (54)

t t1 t

^ ^

=y = y^ Z + k (1 ) w^ + L ; (55)

t t t t t t

h i

Y 2 3

^ ^

e^ = f f f E f ; (56)

P t1 G P t-1 t

^ ^

Z + k = y^ +[1 ] + e^ ; (57)

t t t t

^ ^

w^ + L = Z + k : (58)

t t t t

Fourth, government b ehavior is characterized as the following equations.

m^ m^ = (^m m^ )+(1+)^ (59)

t t1 M t1 t2 t

+ (^r ( + )^r + r^ );

t r R t1 r R t2

^ = ^ + " : (60)

t t1 t 52 Levels Logs % Deviations Log−differences 1.05 0.05 0.05 0.02

0 0 interest rate 1 0 −0.05 −0.02 0 100 200 0 100 200 0 100 200 0 100 200 20 4 0.5 0.05

10 2 0 0 money

0 0 −0.5 −0.05 0 100 200 0 100 200 0 100 200 0 100 200 200 5 0.5 0.02

100 4 0 0 price

0 3 −0.5 −0.02 0 100 200 0 100 200 0 100 200 0 100 200 30 3.5 0.4 0.05

20 3 0.2 0 output

10 2.5 0 −0.05

0 100 200 0 100 200 0 100 200 0 100 200

Figure 6: Data Plots 53

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