dicecio 2005 - comovement it's not a puzzle

8/4/2019 DiCecio 2005 - Comovement It's Not a Puzzle

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Comovement: its not a puzzle.

Riccardo DiCecio

Federal Reserve Bank of St. Louis

First Draft: November 2003This Draft: April 2005

Abstract

A dening feature of business cycles is the comovement of inputsat the sectoral level with aggregate activity. Standard models cannotaccount for this phenomenon. This paper develops and estimates atwo-sector dynamic general equilibrium model which can account forthis key regularity. My model incorporates three shocks to the econ-omy: monetary policy shocks, neutral technology shocks, and embod-ied technology shocks in the capital producing sector. The estimatedmodel is able to account for the response of the US economy to allthree shocks. Using this model, I argue that the key friction underly-ing sectoral comovement is rigidity in nominal wages.

JEL: E20, E32, O41, O42Keywords: Comovement; Businsess cycles; Sticky wage

Mail: P.O. Box 442, St. Louis, MO 63166-0442. E-mail: [email protected] views expressed are my own and do not necessarily reect the views of the FederalReserve Bank of St. Louis or the Federal Reserve System. I am indebted to LarryChristiano and Marty Eichenbaum for their suggestions, support and guidance. I amgrateful for comments from L. Barseghyan, F. Braggion, H. Braun, F. De Fiore, G. Favara,J. Fisher, N. Jaimovich, F. Molinari, A. Monge, P. Sakellaris, F. Smets and R. Vigfussonand from seminar participants at the Board of Governors of the FRS, the EuropeanCentral Bank, the Kansas City FRB, Northwestern University, the Philadelphia FRB, theSt. Louis FRB, University of Virgina and the 2004 SED conference. All errors are myown.

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1 Introduction

Comovement refers to the fact that, empirically, the level of economic ac-tivity across dierent sectors of the economy tends to be similar along thebusiness cycle. In particular when the economy is in a boom, the majorityof sectors tends to use more inputs (capital and labor), and produce moreoutput.

This paper develops and estimates a two-sector dynamic general equilib-rium model which can account for this key regularity. My model incorporatesthree shocks to the economy: neutral technology shocks in the consumptionand investment sector, embodied technology shocks in the capital producingsector, and monetary policy shocks.

Wage stickiness is the crucial feature of the model for obtaining comove-ment in response to technology shocks of either kind. Both consumptionand investment increase in response to technology shocks. The householdsdesire to smooth consumption translates into a relative shift of demand fromconsumption to investment goods. Labor demand increases, but wage rigidi-ties prevent the wage from fully adjusting. Firms in the consumption goodssector face an increase in demand and a (relatively small) increase in thewage rate. The rst eect dominates and they hire more workers.

Standard models cannot account for comovement in response to tech-nology shocks. In standard real business cycle (RBC) models1 the nominalwage fully adjusts to the increase in labor demand. The eect of the higherwage osets the increase in demand for consumption goods, and rms in the

consumption goods sector hire less workers. Economy-wide competitive fac-tor markets imply that the capital-to-labor ratio is equated across sectors.Hence, capital is also relocated to the investment goods sector in response toa positive technology shock. In other words, standard RBC models predictthat inputs in the consumption goods sector comove negatively with inputsin the investment goods sector and with aggregate output.2

The third source of uctuations I consider is monetary policy. Monetarypolicy shocks generate an increase in economic activity in both sectors andwould produce comovement even if wages were exible. However, nominalwage rigidities play an important role in generating a persistent response ofeconomic activity to monetary policy shocks.

A quantitative assessment of the role of sticky wages in generating co-movement requires a rich model. My model incorporates frictions that are

1 For an exposition of the standard RBC model see Hansen (1985) and Prescott (1986).2 Despite using fewer resources, the consumption good sector produces more output

because of the increased productivity.

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standard in the literature studying the eects of monetary policy.3 The real

side of the model incorporates investment adjustment costs, variable capi-tal utilization and habit formation preferences in consumption. Moreover,rms must borrow working capital to nance the wage bill. The model in-corporates nominal price rigidities, in the form of sticky prices la Calvo(1983).

The estimated model generates comovement of sectoral labor inputs inaccordance to the data. Furthermore, my model is able to account for theresponse of the U.S. economy to all three shocks. The parameter estimatesimply a plausible wage rigidity (3.7 quarters) and a modest degree of pricestickiness. Estimates of the other parameters of the model, when compa-rable, and the model responses to neutral technology and monetary policy

shocks are consistent with results in the literature.Sticky wages deliver plausible empirical implications for my model bycreating a countercyclical wedge between the real wage and the marginal rateof substitution between consumption and leisure. The existence of a wagemarkup over the marginal rate of substitution between consumption andleisure has been extensively documented empirically (see Gal, Gertler, andLpez-Salido, 2002).4 My model generates a wage markup whose cyclicalcomponent has properties similar to its empirical counterpart.

The following section presents comovement in the U.S. data, the coun-terfactual implications of standard business cycle models and shows howsticky wages generate comovement. Section 3 describes the model economyin detail. Section 4 is devoted to the model estimation and to the analysis

of the comovement properties of the estimated model. Section 5 concludes.Appendices provide details on the model solution method and on the dataused.

2 The Comovement Puzzle

Comovement is a dening feature of business uctuations. According toBurns and Mitchell (1946, pg. 3),

... a cycle consists of expansions occurring at about thesame time in many economic activities, followed by similarly

3 See, for example, Christiano, Eichenbaum, and Evans (2005), Altig, Christiano,Eichenbaum, and Lind (2002), Smets and Wouters (2003) and Smets and Wouters (2005).The last three papers also examine the eect of other shocks to the economy.

4 See also Hall (1997) and Chari, Kehoe, and McGrattan (2004).

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general recessions, contractions, and revivals which merge into

the expansion phase of the next cycle ...,

a denition endorsed by Gordon (1986).5 Lucas (1977), Long and Plosser(1983) and Christiano and Fitzgerald (1998) dene business cycles in a sim-ilar way.6

In this section I document comovement of sectoral inputs and outputswith aggregate output at business cycle frequencies for U.S. data. I thenillustrate why the standard RBC model implies negative comovement, andI show how sticky wages solve the comovement puzzle.

2.1 Comovement in the U.S. Data7

Figure 1 displays the cyclical components of output, consumption, invest-ment and of the corresponding labor inputs for the U.S. over the period1964:Q1-2001:Q4.

Sectoral outputs (consumption and investment) and labor inputs comovewith aggregate output. Table 1 reports correlations of sectoral outputs andlabor inputs with aggregate output at business cycle frequencies8 (thirdcolumn). The second column displays the standard deviations of sectoraloutputs/inputs relative to the standard deviation of aggregate output. Con-sumption is half as volatile as output. Investment is 3.7 times more volatilethan output. Both consumption and investment are highly correlated withoutput. The corresponding labor inputs display similar properties. They

are both highly correlated with output. Labor in the consumption goodssector is less volatile than output; labor in the investment goods sector istwo and a half times more volatile than output.

5 The NBER Business Cycle Dating Committee (Hall, Feldstein, Frankel, Gordon,Romer, Romer, and Zarnowitz, 2003), denes a recession as follows:

A recession is a signicant decline in activity spread across the economy,lasting more than a few months, visible in industrial production, employ-ment, real income, and wholesale-retail sales.

6 Other authors, such as Schumpeter (1939, pg. 200), Prescott (1986, pg. 10) and Sar-gent (1987, pg. 282), focus on the comovement of economy-wide variables in deningbusiness uctuations.

7

See Christiano and Fitzgerald (1998) for a more disaggregated analysis of the comove-ment properties of the US economy.

8 The business cycle component is extracted using an HP1600 lter applied to the seriesin logs. See appendix B for a description of the data used.

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Q1-64 Q3-70 Q1-77 Q3-83 Q1-90 Q3-96

-0.04

-0.02

0

0.02

Output

Q1-64 Q3-70 Q1-77 Q3-83 Q1-90 Q3-96

-0.04

-0.02

0

0.02

Hours (aggregate)

Q1-64 Q3-70 Q1-77 Q3-83 Q1-90 Q3-96

-0.02

-0.01

0

0.01

Consumption

Q1-64 Q3-70 Q1-77 Q3-83 Q1-90 Q3-96

-0.02

0

0.02

Hours (consumption)

Q1-64 Q3-70 Q1-77 Q3-83 Q1-90 Q3-96

-0.1

0

0.1

Investment

Q1-64 Q3-70 Q1-77 Q3-83 Q1-90 Q3-96

-0.1

-0.05

0

0.05

Hours (investment)

Figure 1: Cyclical components of aggregate output, sectoral outputs andcorrepsonding labor inputs.

Variable x=~y x;~y~c 0.53 0.70

~lC 0.74 0.85

~{ 3.69 0.92~lI 2.44 0.88

Table 1: Second moments of the cyclical components of consumption, in-vestment and corresponding labor inputs

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2.2 The Puzzle

The standard RBC model can be interpreted as a two-sector model whereconsumption and investment goods are produced using the same technology:

Ct = ezt (kC;t)

(lC;t)1 ;

It = ezt (kI;t)

(lI;t)1 ;

where kx and lx are capital and labor employed in sector x = C; I. Neutraltechnology shocks are denoted by z.

Firms in the consumption goods sector hire labor to the point where themarginal cost, the real wage, equals the marginal product of labor:

wt = (1 )CtlC;t (1)

The households labor supply is determined by equating the marginalrate of substitution between consumption and leisure to the real wage rate:

wt = Ct (lt) (2)

where and are nonnegative constants.Equations (1) and (2) imply that lC;t (lt)

is constant. In response to

technology shocks aggregate labor, lt, increases. In order for lC;t (lt) to re-

main constant, lC;t must decrease. Also, since the two sectors have identical

capital-to-labor ratios,

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capital will be moved from the consumption goodssector to the investment goods sector.10

Christiano and Fitzgerald (1998) show that the result is robust to thechoice of functional form for the utility function within the class of pref-erences consistent with balanced growth (see King, Plosser, and Rebelo,1988). Christiano and Fitzgerald (1998) also show that a non-unit elasticityof subsitiution between capital and labor in the production function couldgenerate comovement. However, they argue that for plausible values of theelasticity of substitution a standard model does not generate comovement.

2.3 Sticky wages and comovement

In a model with sticky wages, households are not allowed to equate themarginal rate of substitution between consumption and leisure to the real

9 This follows from the fact that there are economy-wide factor markets.10 For a review of the literature which addresses the comovement puzzle in real models,

see Christiano and Fitzgerald (1998).

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wage. Let H;t denote the wedge between the two:

H;t =wt

Ct (lt)

: (3)

Sticky wages create a time varying markup of the price on the marginalcost of the good supplied. The real wage can be interpreted as the priceof leisure, and the marginal rate of substitution between consumption andleisure is the marginal cost, measured in consumption units. In response toa technology shock, the denominator increases but sticky wages prevent thenumerator from increasing by the same amount, and H;t decreases.

Combining the households intratemporal equation with the rst ordercondition for labor for rms in the consumption goods sector, (1), and rear-

ranging:

1

1 lC;t (lt)

=1

H;t: (4)

A countercyclical wage markup implies that lC;t (lt) is procyclical. Thus

lC;t can be procyclical as well, and the puzzle is solved.To evaluate empirically the relevance of nominal wage rigidities in gen-

erating comovement I present in the following section a two sectors DSGEmodel which incorporates sticky wages and several other departures from thebasic RBC model. One sector models featuring similar frictions have beenrecently used to study business cycles and/or the eects of monetary policy

by Christiano, Eichenbaum, and Evans (2005), Altig, Christiano, Eichen-baum, and Lind (2002), Smets and Wouters (2003), and Smets and Wouters(2005).

3 The Model

The economy is populated by a unit measure continuum of households. Eachhousehold makes consumption, investment, capital utilization, cash holdingand deposit decisions. Also, each household supplies monopolistically a dif-ferentiated labor service to a competitive labor market intermediary. Thehomogenous composite labor is supplied to monopolist intermediate goods

rms in the consumption and investment goods sectors. An intermediaterm in either sector combines labor services and capital services into anintermediate input for the production of the nal good in the respectivesector. Intermediate good producers need to borrow money from a compet-itive nancial intermediary to pay for the wage bill before production takes

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place. Final good producers are perfectly competitive. Finally, the mon-

etary authority implements monetary policy by setting the money growthrate.

In every period the economy is aected by three dierent shocks. Aneutral (disembodied) technology shock increases the productivity of inter-mediate good producers in either sector. An investment specic (embodied)technology shock increases the productivity only of intermediate goods pro-ducers in the investment goods sector. Monetary policy shocks aect thegrowth rate of money supply, beyond the endogenous response of monetarypolicy to realizations of the other shocks.

3.1 Timing

At the beginning of every period, embodied and disembodied technologyshocks are realized. Then, prices and wages are set and households maketheir consumption, investment and capital utilization decisions. After this,the monetary policy shock is realized. Then, households make their portfoliodecision, goods and labor markets meet and clear, and production, invest-ment and consumption occur. To reect the fact that dierent decisions arebased on dierent information, let t be the information set including all theshock realizations up to and including time t, and mt be the information setincluding all the shock realizations up to t excluding the time t realizationof the monetary policy shock. The corresponding conditional expectationsoperator are denoted by:

E(jt) = Et () ; E(jmt ) = E

mt () :

The timing described above ensures that the identication assumptionsused to identify monetary policy shocks in the data hold by construction inthe model.

3.2 Final goods rms

In either sector, nal goods output is produced by competitive rms accord-ing to the technology:

Yx;t = Z10

(Yx;j;t)1f f ; f 2 [1; +1); x = C;I: (5)

where Yxj;t denotes the time t input of intermediate good j for sector x.

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Consumption and investment goods producers solve:

maxYx;t;fYx;j;tgj2[0;1]

Px;tYx;t Z10

Px;j;tYx;j;tdj; x = C;I: (6)

where Px;t is the price of nal good x.The demand for intermediate goods in the two sectors is given by:

Px;t

Px;j;t

ff1

=Yx;j;tYx;t

; x = C;I: (7)

3.3 Intermediate Goods Firms

Intermediate good j 2 [0; 1] for the consumption goods sector is producedaccording to:

YC;j;t = maxn

ezN;th

(KC;j;t) (lC;j;t)

1 C(Zt1)i

; 0o

; (8)

where 2 (0; 1), C> 0, KC;j;t and lC;j;t denote capital and labor services,and zN;t represents a neutral technology shock. The term e

zN;tC(Zt1)

is a xed production cost which ensures that prots are zero in a non-stochastic steady state.11

Intermediate investment goods j 2 [0; 1] is produced according to:

YI;j;t = maxn

ezN;t+zI;th

(KI;j;t) (lI;j;t)

1 I(Zt1)i

; 0o

; (9)

where KI;j;t and lI;j;t denote capital and labor services, and zI;t representsa capital embodied shock. The term ezN;t+zI;tI (Zt1)

is a xed pro-duction cost which will ensure that prots are zero along a non-stochasticbalanced growth path.

Technology shocks evolve as follows:

zx;t = zx;t1 + x;t,

x;t = (1 x) x + xx;t1 + "x;t; (10)jxj < 1; "x;t

iidN

0; 2x

, x = N;I:

11 The "xed" cost is assumed to grow at the same rate as sectoral output, to ensurethat it does not become negligible.

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Let Rkt and Wt denote the nominal rental rate on capital services and

the wage rate. Firms need to borrow money to pay the wage bill in advanceof production. Thus labor has a unit cost of RtWt.

The rst order conditions with respect to capital and labor imply thatthe capital-to-labor ratios across rms in each sector are identical, and theyare equalized across sectors:

Kx;j;tlx;j;t

=Kx;tlx;t

=Ktlt

,8j 2 [0; 1] , x = C;I:

The typical intermediate goods rms prots are:

(Px;j;t Px;tsx;t) Yx;j;t; x = C;I:

In each period, with constant probability 1 p;x, a rm in sectorx = C; I is allowed to reoptimize its nominal price.12 If a rm is not allowedto reoptimize the following equations hold:13

PC;j;t = C;t1PC;j;t1, C;t1 PC;t1PC;t2

PI;j;t = eI;tI;t1PI;j;t1, I;t1

PI;t1PI;t2

eI;t1 .

Firms choose ~PC;t and ~PI;t in order to maximize14 :

Emt

+1Xl=0

p;Cl t+l h ~PC;tXC;t;l sC;t+lPC;t+liYCj;t+l;Emt

+1Xl=0

p;I

lt+l

h~PI;te

(zI;t+lzI;t)XI;t;l sI;t+lPI;t+l

iYIj;t+l;

where

Xx;t;l =

l1j=0x;t+j for l 1

1 for l = 0; x = C;I:

t+l is the marginal value of a dollar for the household, which rms takeas given; sx;t denotes the marginal costs for rms in sector x = C; I.

12 Independently across rms, within and between sectors, and time.13 See Christiano, Eichenbaum, and Evans (2005) for a justication of this updating rule,

as opposed to the standard rule Px;j;t = Px;j;t1, where is the unconditional averageof ination.

14 All the rms in either sector that can reoptimize will choose the same price, as shownin Woodford (1996).

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The aggregate prices in the two sectors can be expressed as:

PC;t =

1 p;C

~PC;t

11f + p;C(C;t1PC;t1)

11f

1f; (11)

PI;t =

1 p;I

~PI;t

11f + p;I

eI;tI;t1PI;t1

11f

1f:(12)

3.4 Households

Households rank consumption, leisure and real balances streams according

to:15

E0

+1Xt=0

t

264log(Ct bCt1) l (hj;t)22 + qqt#t

1q1 q

375 , (13)2 (0; 1) ; 2 [0; +1); l 2 (0; +1) ,

q 2 (0; +1) , q 2 [0; +1);

where qt Qt=PC;t are real money balances, #t = eN;t+I;t

1 is a scalingvariable, and hj;t is household js labor supply.

The households budget constraint is:

Mt+1 Rt [Mt Qt + Mt+1 (t 1)] + Qt + Aj;t + Dt

PI;tIt PC;tCt PC;ta (ut)#tZt

Kt + Rkt ut Kt + Wj;thj;t (14)

where Mt is the households beginning of period stock of money, (t 1) Mt+1is a lump sum transfer from the monetary authority, Qt denotes the nomi-nal cash balances the household carries from the previous period, Dt prots,and Aj;t the net cash ow from state-contingent securities.

16 The amount[Mt Qt + (t 1) Mt+1] is deposited with a nancial intermediary.

15 The only functional form for utility consistent with identical consumption acrosshouseholds and with balanced growth is logarithmic in consumption.

16 State contingent securities allow the households to insulate consumption and assetholdings from the realizations of idyosincratic Calvo uncertainty.

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Kt denotes the physical stock of capital. The quantity of capital services

is given by the product of physical capital and the utilization rate, ut:

Kt = ut Kt: (15)

The cost of utilizing capital is measured in consumption units, and isproportional to the quantity of physical capital. In steady state utilizationis normalized to one, and it is costless, i.e. a (1) = 0.

The households stock of capital evolves according to:

Kt+1 = (1 ) Kt + F(It; It1) ; (16)

F(It; It1) =

1 S

It

It1

It;

SeN+I1 = S0 eN+I1 = 0; { e2N+I1 S00 eN+I1 > 0:The function S() gives the adjustment cost to be paid if the investment

growth rate is changed from its previous period level.17

3.4.1 Labor Decision

The household labor supply decision is modeled as in Erceg, Henderson,and Levin (2000). Households are monopoly suppliers of dierentiated laborservices, hj;t. These services are sold to a representative, competitive rmwhich aggregates them into homogeneous labor input according to:

nt =Z1

0

h1wj;t dj

w , w 2 [1; +1):The demand curve for hj;t is:

hj;tnt

=

Wt

Wj;t

ww1

; (17)

where Wt is the aggregate wage rate, and Wj;t is the individual householdwage rate.

The aggregate wage is related to the individual households wage ratesas follows:

Wt = Z10

W1

1wj;t dj1w : (18)

17 See Christiano, Eichenbaum, and Evans (2005) for a detailed discussion of this spec-ication of investment adjustment costs and a comparison to adjustment costs in theinvestment rate.

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In each period a household is allowed to reoptimize its nominal wage,

with constant probability (1 w),18

taking Wt and nt as given. If thehousehold is not allowed to reoptimize:

Wj;t+1 = C;teN+I

1 Wj;t.

The rst order condition associated with the wage choice,19 ~Wt, of ahousehold that is reoptimizing is:

Emt

+1Xl=0

(w)l hj;t+l

Ct+l

"exp

N+ I

1 l

~WtXC;t;lPC;t+l

whj;t+lCt+l

#= 0:

(19)

The aggregate wage can be expressed as:

Wt =

"(1 w)

~Wt 1

1w + w

eN+I

1 C;t1Wt1 1

1w

#1w: (20)

3.5 Loan market clearing

Loan market clearing requires that the demand for loans from rms to -nance their working capital is equal to money deposited by the households:

Wtlt = tMt Qt: (21)

3.6 Sectoral resource constraints

The unweighted average of consumption goods is:

YC;t =

Z1

0

YC;j;tdj = ezN;t

KC;tl

1C;t CZ

t1

=

PC;tPC;t

! ff1

Ct + a (ut)

#tZt

Kt

;

where KC;t and lC;t are the total of the capital and labor services usedin the consumption goods sector, and

PC;t = Z10

P

f

1f

C;j;t dj!1ff

:

18 Independently across households, sectors and time.19 All the households that can reoptimize will choose the same wage.

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Hence, the resource constraint for the consumption goods sector can be

expressed as:

Ct + a (ut)#tZt

Kt = ezN;t

PC;tPC;t

ff1

KC;tl

1C;t CZ

t1

: (22)

In a similar way, one can show that:

It = ezN;t+zI;t

PI;tPI;t

ff1

KI;t l

1I;t IZ

t1

; (23)

where

PI;t =Z1

0

P

f1f

I;j;t dj! 1ff :

A measure of employment in the economy is given by:

ht =

Z1

0

hj;tdj = (lC;t + lI;t)

WtWt

ww1

:

where

Wt =

Z1

0

Ww

1wj;t dj

1ww

:

Hence:

Ct + a (ut)#tZt

Kt = ezN;t

PC;tPC;t

ff1

24WtWt

w(1)w1

KC;th1C;t CZ

t1

35 ;(24)

It = ezN;t+zI;t

PI;tPI;t

ff1

24WtWt

w(1)w1

KI;th1I;t IZ

t1

35 : (25)As shown in Yun (1996), the terms

Px;tPx;t

ff1 and

WtWt

w(1)w1 do not

have a rst-order eect, and they will disappear in the linearized versions ofthe above expressions.

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3.7 Monetary Policy

Monetary policy is modelled as a money growth rule:

t p;t + N;t + I;t ; (26)

p;t = pp;t1 + "p;t; "p;t iidN

0; 2p

N;t = NN;t1 + cN"N;t + lN"N;t1;

I;t = II;t1 + cI"I;t + lI"I;t1;p ; N ; I < 1Here, "p;t represents a monetary policy shock. The terms N;t and

I;t

capture the response of monetary policy to innovations in neutral andcapital embodied technology, respectively. The dynamic response of x;t to"x;t, x = I; N, is characterized by ARMA(1,1) processes.

A textbook sequence-of-markets notion of equilibrium applies.

4 Econometric Methodology

I estimate the key parameters of the model with a standard limited in-formation approach. The parameters are chosen in order to minimize thedistance of the impulse responses of the model to embodied, disembodiedand monetary policy shocks from the impulse responses of a structural VARrepresentation of U.S. data.

4.1 A VAR Representation of the Data

The variables included in the VAR analysis are the growth rate of the realinvestment price, the growth rate of average labor productivity, the ina-tion rate, capacity utilization, the ratio of the real wage to average laborproductivity, the consumption and investment shares of output, the fed-eral funds rate, and the money growth rate. The sample covers the period1959:Q2-2001:Q4. The variables are required to be covariance stationary.The estimated VAR coecients corroborate the stationarity assumption.

Consider the following reduced form VAR:20

Yt = A (L) Yt1 + ut; Eutu0t = V;

A (L) = A1 + A2L + ::: + ApLp; p < 1

20 In the estimation four lags (i.e. p = 3) and a constant were included. For ease ofexposition the constant has been omitted in what follows.

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In the analysis that follows Yt 2 R10 is dened as:

Yt =

24 l n (pI;t) ; lnYtlt ; t; ln wtYt=lt ; ln (ut) ;ln (lt) ; ln

CtYt

; ln

ItYt

; Rt; l n (Mt)

350 :The reduced form residuals, ut, are related to the structural shocks, t,

by t = A0ut. Also, the structural shocks are orthogonal to each other,i.e. Et

0t = I. The rst two elements of t are the investment specic and

neutral technology shocks, respectively. The last element is the monetarypolicy shock. The remaining elements of t are not identied.

Following Fisher (2003), I identify embodied and disembodied technol-ogy shocks using long run restrictions. As in the model, investment specic

technology shocks are the only shocks to have a long run eect on the rela-tive price of investment. Neutral technology shocks and investment specictechnology shocks are the only shocks to aect average labor productivityin the long run. The long run eects of the structural shocks are given by:

Y1 = t;

[IA (1)]1 A10

The identifying assumptions described above boil down to assuming thatthe rst two rows of matrix have the following structure:

(1 : 2; :) = 11 0 01821 22 018 :Monetary policy shocks are identied as in Christiano, Eichenbaum, and

Evans (1999) by assuming that the 9th column ofA0 has the following struc-ture:21

A0 (:; 9) = [018; a0; a1]0 :

This identication assumption can be interpreted as the monetary au-thority following a Taylor rule like policy, which responds to all the variablesordered before the interest rate in the VAR.

Figures 2-4 report the impulse response functions (IRFs) of the VARto the three structural shocks. The shaded areas are bootstrapped 95%

condence intervals around the point estimates. The short run behavior ofthe variables in response to technology shocks is consistent with the evidence

21 Notice that there are two overidentifying restrictions. The rst two elements of twould be just identied by imposing the long run restrictions. The identication of mon-etary policy shocks poses two additional zero restrictions.

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reported in Fisher (2003). Monetary policy shocks produce the pattern of

responses well documented in the literature. In particular, they generatepersistent output and ination responses, and hump-shaped responses ofconsumption, investment and hours.

4.2 Estimation

The model parameters are collected in the vector :

=

24 ; ; ; ; l; l; q; N; I; q;b; f; p;C; p;I; w; w; ; a; p; N;cN; lN; I; cI; lI; N; I; p ; N; I

350 == [1; 2]

0

The elements of1 have been xed at the same values as in Altig, Chris-tiano, Eichenbaum, and Lind (2002) (table 2).

Parameter in 1 Value

(1:03)1=4

0:36

0:025

1:017

l 1

l s.t. l = 1

q s.t. qm = 0:25w 1:05

N 0:002

I 0:003

Table 2: Fixed parameters

The elements of2, are chosen to minimize the weighted distance of themodel IRFs, (2), from the VAR IRFs, :

2 = arg min2

( (2))0 V1 ( (2)) ;

where V is a diagonal matrix containing the variances of the estimated VARimpulse response functions. The loss function above attributes more weightto the impulse response functions estimated more precisely.

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0 5 10 15

0

0.2

0.4

0.6

Output

0 5 10 15

-1

0

1

MZM Growth

0 5 10 15

-0.2

0

0.2

0.4

Inflation

0 5 10 15-0.4

-0.2

0

0.2

0.4Fed Funds

0 5 10 15

-0.5

0

0.5

Capacity Util

0 5 10 15

-0.2

0

0.2

0.4

0.6Hours

0 5 10 15-0.4

-0.2

0

Wage

0 5 10 15

0

0.2

0.4

0.6

Consumption

0 5 10 15

0

1

2

3

Investment

0 5 10 15

-1

0

1

MZM Velocity

0 5 10 15-1

-0.8

-0.6

-0.4

-0.2

Price of Inv.

Figure 2: Model () and VAR based responses to an embodied technologyshock

18


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0 5 10 15

0.2

0.4

0.6

0.8

Output

0 5 10 15-1

0

1

MZM Growth

0 5 10 15

-0.8

-0.6

-0.4

-0.2

0

Inflation

0 5 10 15-0.4

-0.2

0

0.2

0.4Fed Funds

0 5 10 15-1

-0.5

0

0.5

Capacity Util

0 5 10 15

-0.20

0.20.40.60.8

Hours

0 5 10 150

0.2

0.4

0.6Wage

0 5 10 15

0.2

0.4

0.6

0.8

Consumption

0 5 10 15-1

0

1

2

Investment

0 5 10 15

-2

-1

0

MZM Velocity

0 5 10 15-0.6-0.4-0.2

00.20.4

Price of Inv.

Figure 3: Model () and VAR based responses to an neutral technologyshock

19


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0 5 10 15

-0.4

-0.2

0

0.2

0.4

Output

0 5 10 15

-2

0

2

MZM Growth

0 5 10 15

-0.2

0

0.2

Inflation

0 5 10 15

-0.6-0.4-0.2

00.20.4

Fed Funds

0 5 10 15

-0.5

0

0.5

1Capacity Util

0 5 10 15-0.4

-0.2

0

0.2

0.4

Hours

0 5 10 15-0.1

0

0.1

Wage

0 5 10 15

-0.2

0

0.2

Consumption

0 5 10 15

-1

0

1

Investment

0 5 10 15

0

1

2

MZM Velocity

0 5 10 15-0.4

-0.2

0

0.2

Price of Inv.

Figure 4: Model () and VAR based responses to a monetary policy shock

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q b f p;C p;I w a

10.46 0.76 1.21 0.54 0.77 0.73 3.54 0.08(2.98) (0.03) (0.06) (0.06) (0.07) (0.02) (0.71) (0.04)

Table 3: Estimated economic parameters - std. errors in parentheses

p N cN lN I cI

0.13 0.30 6.96 2.70 0.69 0.53(0.335) (0.230) (1.086) (2.754) (0.096) (0.46)

lI N I p N I

1.22 0.91 -0.53 0.24 0.03 0.08

(0.67) (0.01) (0.52) (0.010) (0.003) (0.032)

Table 4: Estimated shocks parameters - std. errors in parentheses

In order to impose the constraints on the parameters in 2 the elementshave been transformed as follows:

xj =

8>>>>>>>>>>>:

ln

2;j

if 2;j 2 [0;1)

ln12;j2;j

if 2;j 2 (0; 1)

ln12;j1+2;j

if 2;j 2 (1; 1)

ln

2;j 1

if 2;j 2 [1;1)

2;j if 2;j 2 (1;1)

; (27)

and the following equivalent optimization has been solved:

x = arg minx

( (x))0 V1 ( (x)) :

2 is computed by inverting (27) :The parameter estimates are reported in tables 3 and 4. They are broadly

consistent with the estimates reported by Altig, Christiano, Eichenbaum,and Lind (2002). The estimated Calvo parameters imply an average pricecontract duration of 2.17 and 4.26 quarters for rms in the consumption andinvestment goods sectors, respectively. The average wage contract duration

is 3.72 quarters.The model impulse responses to a monetary policy shock are remarkablyclose to the VAR ones. The IRFs to technology shocks are reasonably closeto the data (see gures 2 and 3).

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x x=~y x;~y

-2 -1 0 1 2Model ~lC 0.80 0.75 0.85 0.86 0.74 0.54

[0.75,0.92] [0.48,0.89] [0.56,0.96] [0.58,0.97] [0.48,0.84] [0.34,0.65]~lI 1.68 0.61 0.75 0.83 0.78 0.67

[1.54,2.12] [0.55,0.79] [0.69,0.90] [0.77,0.95] [0.73,0.93] [0.60,0.83]

U.S. data ~lC 0.74 0.79 0.87 0.85 0.68 0.46~lI 2.44 0.80 0.89 0.88 0.71 0.48

Table 5: Comovement statistics - 95% C.I. in paretheses

4.3 Comovement

Table 5 displays comovement statistics for the simulated model and thecorresponding statistics for U.S. data. I generate 10,000 samples of the samesize as the U.S. data from the estimated model, and I report the averagesand the corresponding 2:5th and 97:5th percentiles for logged, HP1600 lteredseries.

The contemporaneous correlations of sectoral labor inputs with outputfor the model are positive and close to the corresponding values for the data.The model understates the relative volatility of labor in the investment goodssector.

Table 622 reports the relative volatility of sectoral labor with respect tooutput and the contemporaneous correlation with output for various versionsof the model. The baseline model refers to the model where all frictions areoperative. The other versions of the model are characterized as follows:

no habits: b = 0;

exible prices: p;C= p;I = 0;

exible wages: w = 0;

small investment adjustment costs: = 0:05;

no variable capital utilization: a = 10; 000;

no xed production costs: f = 1;

22 The statistics are the averages of 10,000 model simulations for logged and HP lteredseries.

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no working capital: rms in the intermediate good sectors do not

need to borrow their working capital, so that the unit labor cost issimply the wage rate; the loan market clearing equation is modiedaccordingly.

Model Embodied tech. shocks Disembodied tech. shocks

~lC=~y ~lC;~y ~lI=~y ~lI;~y ~lC=~y ~lC;~y ~lI=~y ~lI;~yBaseline 0.86 0.95 2.27 0.98 0.79 0.80 0.86 0.77

No habits 1.13 0.94 1.89 0.90 1.17 0.75 0.27 -0.20

Flexible prices 0.88 0.96 2.12 0.98 0.80 0.82 1.05 0.79

Flexible wages 1.47 0.52 5.55 -0.14 0.47 0.37 2.55 0.05

Small investment adj. costs 0.54 0.93 3.14 0.99 0.66 0.77 1.35 0.58

No variable capital util. 0.94 0.95 3.22 0.97 0.59 0.67 1.43 0.64

No xed prod. cost 1.22 0.96 2.75 0.98 1.04 0.76 0.75 0.68

No working capital 0.85 0.96 2.18 0.99 0.85 0.95 1.97 0.98

Model Monetary policy shocks All shocks

~lC=~y ~lC;~y ~lI=~y ~lI;~y ~lC=~y ~lC;~y ~lI=~y ~lI;~yBaseline 0.83 0.95 2.23 0.97 0.81 0.86 1.69 0.83

No habits 1.15 0.94 1.70 0.87 1.13 0.83 0.27 0.59

Flexible prices 0.84 0.96 2.19 0.99 0.81 0.88 1.72 0.86

Flexible wages 0.94 0.61 3.12 0.31 0.53 0.38 2.82 0.06

Small investment adj. costs 0.53 0.92 3.10 0.99 0.57 0.84 2.55 0.84

No variable capital util. 0.84 0.96 2.53 0.96 0.69 0.76 1.95 0.75No xed prod. cost 1.16 0.96 2.67 0.97 1.09 0.84 1.90 0.76

No working capital 0.85 0.96 2.15 0.99 0.85 0.96 2.08 0.98

Table 6: Statistics for various versions of the model

Demand shocks (i.e. monetary policy shocks) generate comovement in-dependently of the frictions included in the model. A lower interest ratestimulates demand for both consumption and investment goods. With anunchanged technology, the only way to satisfy the increased demand is touse more inputs in both sectors.

The key friction in generating comovement in response to technology

shocks is sticky wages. The model with exible wages generates signicantlylower correlations of lC and lI with output in response to any shock. Theexible wages model simulated with the three shocks together displays acorrelation oflCwhich is less than one third of the correlation for the baselinemodel, and a correlation for lI close to zero.

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~H ~H=~y Cross correlation with output

-2 -1 0 1 2Model 1.51 -0.81 -0.94 -0.98 -0.87 -0.68

[1.44,1.57] [-0.70,-0.88] [-0.90,-0.97] [-0.97,-0.99] [-0.83,-0.90] [-0.58,-0.75]

U.S. data 1.53 -0.48 -0.68 -0.84 -0.82 -0.71

Table 7: BC component of the wage markup and of the MRS consumption-leisure

Habit formation plays a minor role in delivering comovement in responseto embodied technology shocks. However, the model without habits displaysa substantial amount of comovement when all the shocks are considered.

This is in contrast with the results in Boldrin, Christiano, and Fisher (2001)who present a two-sector model drive by neutral technology shocks withimmobile capital and labor, and habit persistence. In response to technol-ogy shocks labor cannot be relocated from the consumption goods to theinvestment goods sector. In subsequent periods, agents are addicted to therelatively higher level of consumption.

Jin and Zeng (2002) focus on the working capital channel in a limitedparticipation model as a rationalization of the comovement puzzle. Thenominal interest rate decreases both in response to monetary policy shocks,and to neutral technology shock. The general equilibrium eect of technol-ogy shocks on the nominal interest rate is due to an increase in the desired

amount of savings. The reduced unit labor cost for rms in both sectors isresponsible for comovement in their model. A similar channel operates inmy model, but it is inessential in order to generate comovement.

4.3.1 Wage markup

The model generates a countercyclical wedge between the wage rate and themarginal rate of substitution between consumption and leisure:23

~H;t = ~wt

~ct + ~lt

;

where ~w is the real wage, ~c is real consumption, and ~l denotes hours worked,

all log HP1600 ltered. The business cycle properties of the models wedgeare similar to the properties of its empirical counterpart (table 7).

23 The model implies a more complicated expression for the wedge, due to the presenceof habit persistence. Taking this into account does not change the results in table 7.

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5 Conclusions

The analysis in this paper is very similar, in spirit, to the RBC literature. Inthe RBC literature a model is parametrized using independent information,and it is evaluated by assessing its capability to reproduce second momentsof aggregate data.

My model has been parametrized by estimating the relevant parameters,instead of calibrating them, without using sectoral input variables. Themodel is consistent with the literature in terms of its ability to produceresponses to neutral technology shocks and monetary policy shocks that aresimilar to those obtained in a VAR representation of U.S. data. In addition,the model is an empirically plausible account of the eects of investmentspecic technology shocks. The estimated model is then simulated to assessits ability to reproduce second moments of sectoral inputs which we observein the data.

The model presented in this paper succeeds in generating comovement,a major problem for standard models. Sticky wages are the key frictionin generating comovement. Nominal wage rigidities creat a countercyclicalmarkup of the real wage on the marginal rate of substitution between con-sumption and leisure with the same business cycle properties as its empiricalcounterpart. The wage markup, by keeping the cost eect of productivityshocks smaller than the demand eect, is responsible for comovement ofsectoral inputs.

The modelling of sticky wages is admittedly very stylized. I interpret the

evidence on the importance of wage rigidity as a starting point for furtherresearch.24

24 Other authors, in dierent contexts, have pointed to the importance of wage rigidities.Christiano, Eichenbaum, and Evans (2005) argue that sticky wages are important toobtain a persistent response of ination to monetary policy shock without having to relyon inplausible amounts of price stickiness. Hall (2005) shows how wage rigidities deliverplausible volatility for unemployment, job-nding rates and vacancies.

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References

Altig, D., L. J. Christiano, M. Eichenbaum, and J. Lind (2002):Technology Shocks and Aggregate Fluctuations, manuscript, North-western University.

Anderson, G., and G. Moore (1985): A Linear Algebraic Procedurefor Solving Linear Perfect Foresight Models, Economics Letters, 17(3),247252.

Boldrin, M., L. J. Christiano, and J. D. M. Fisher (2001): HabitPersistence, Asset Returns, and the Business Cycle, American EconomicReview, 91(1), 149166.

Burns, A. F., and W. C. Mitchell (1946): Measuring Business Cycles.National Bureau of Economic Research, New York, NY.

Calvo, G. A. (1983): Staggered Prices in a Utility-Maximizing Frame-work, Journal of Monetary Economics, 12(3), 38398.

Chari, V. V., P. J. Kehoe, and E. R. McGrattan (2004): BusinessCycle Accounting, Federal Reserve Bank of Minneapolis, Sta Report328.

Christiano, L. J. (2002): Solving Dynamic Equilibrium Models by aMethod of Undetermined Coecients, Computational Economics, 20(1-

2), 2155.

Christiano, L. J., M. Eichenbaum, and C. Evans (1999): MonetaryPolicy Shocks: What Have We Learned, and to What End?, in Handbookof macroeconomics, ed. by J. B. Taylor, and M. Woodford, VOL 1B, pp.10511135. Elsevier Science, North-Holland, Amsterdam, New York andOxford.

(2005): Nominal Rigidities and the Dynamic Eects of a Shockto Monetary Policy, Journal of Political Economy, 113(1), 145.

Christiano, L. J., and T. J. Fitzgerald (1998): The Business Cycle:Its Still a Puzzle, Federal Reserve Bank of Chicago Economic Perspec-tives, 22(4), 5683.

Erceg, C. J., D. W. Henderson, and A. T. Levin (2000): OptimalMonetary Policy With Staggered Wage and Price Contracts, Journal ofMonetary Economics, 46(2), 281313.

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Fisher, J. D. M. (2003): Technology Shocks Matter, Federal Reserve

Bank of Chicago, Working Paper 2002-14.

Gal, J., M. Gertler, and J. D. Lpez-Salido (2002): Markups,Gaps, and the Welfare Costs of Business Fluctuations, NBER WorkingPaper 8850.

Gordon, R. J. (ed.) (1986): The American Business Cycle. The Universityof Chicago Press, Chicago, IL.

Hall, R., M. Feldstein, J. Frankel, R. Gordon, C. Romer,

D. Romer, and V. Zarnowitz (2003): The NBERs Recession DatingProcedure, http://www.nber.org/cycles/recessions.html.

Hall, R. E. (1997): Macroeconomic Fluctuations and the Allocation ofTime, Journal of Labor Economics, 15(1), S22350.

(2005): Employment Fluctuations with Equilibrium Wage Stick-iness, American Economic Review, 95(1), forthcoming.

Hansen, G. D. (1985): Indivisible Labor and the Business Cycle, Journalof Monetary Economics, 16(3), 309327.

Jin, Y., and Z. Zeng (2002): The Working Capital Channel and Cross-Sector Comovement, Journal of Economics, 28(1), 5165.

King, R. G., C. I. Plosser,and

S. T. Rebelo (1988): Production,Growth and Business Cycles: I. The Basic Neoclassical Model, Journalof Monetary Economics, 21(2-3), 195232.

Long, J. B., and C. J. Plosser (1983): Real Business Cycles, Journalof Political Economy, 91(1), 3969.

Lucas, R. E. J. (1977): Understanding Business Cycles, in Stabilizationof the Domestic and International Economy, ed. by K. Brunner, and A. H.Meltzer, vol. 5 of Carnegie-Rochester Series on Public Policy, pp. 729,Amsterdam, Holland. North-Holland Publishing Company.

Prescott, E. C. (1986): Theory Ahead of Business Cycle Measurement,

Federal Reserve Bank of Minneapolis Quarterly Review, 10(4), 922.

Sargent, T. J. (1987): Macroeconomic Theory. Second Edition. AcademicPress, Inc., San Diego, CA.

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Schumpeter, J. A. (1939): Business Cycles. McGraw-Hill Book Company,

Inc., New York, NY.

Smets, F., and R. Wouters (2003): An Estimated Stochastic DynamicGeneral Equilibrium Model of the Euro Area, Journal of the EuropeanEconomic Association, 1(5), 11231175.

(2005): Comparing Shocks and Frictions in US and Euro AreaBusiness Cycles: A Bayesian DSGE Approach, Journal of AppliedEconometrics, 20(2), 161183.

Woodford, M. (1996): Control of the Public Debt: A Requirement forPrice Stability?, NBER Working Paper No. 5684.

Yun, T. (1996): Nominal Price Rigidity, Money Supply Endogeneity, andBusiness Cycles, Journal of Monetary Economics, 37(2), 34570.

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A Solving the Model

A.1 Scaled Model

Consider the following random variables:

Zt = ezN;t+zI;t

1 ; #t = ezN;t+zI;t

1 ;

Their expected growth rate is given by

Z = E

Z;t E

ln

Zt

Zt1

=

N+ I1

;

# = E#;t Eln#t

#t1 =N+ I

1

:

It is easily veried that the following scaled variables are stationary:

ct Ct#t

; it ItZt

; kC;t KC;tZt1

; kI;t KI;tZt1

;

lC;t; lI;t ; lt;

wt =WtPC;t#t

; rkt

RktZtPC;t#t

;

pK0t

PK0tZt#t

; pI;t

PI;tPC;t

exp(zI;t) ;

C;t PC;tPC;t1

; I;t PI;tPI;t1

mt MtPC;t#t

; qt QtPC;t#t

The households rst order conditions can be expressed as:

pK0 ;tU#C;t = E

mt

(U#C;t+1Z;t+1

hrkt+1ut+1 a (ut+1) +pK0 ;t+1 (1 )

i); (28)

pI;tU#C;t = U

#C;tpK0 ;tF1;t + E

mt

U#C;t+1Z;t+1

pK0 ;t+1F2;t+1

!; (29)

U#C;t = Et

Rt+1U

#C;t+1

C;t+1#;t+1

!; (30)

Emt hrkt a0 (ut)i = 0; (31)q (qt)

q = (Rt 1) U#C;t: (32)

29


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where

F1;t = 1 SZ;tit

it1

Z;titit1

S0Z;titit1

;F2;t =

Z;tit

it1

2S0

Z;tit

it1

;

U#C;t #tUC;t =#;t

#;tct bct1

b

#;t+1ct+1 bct: (33)

Physical capital is related to capital services as follows:

kt = utkt:

Given the assumed functional forms, the marginal utility of leisure is

given by:zh = lhj;t: (34)

Also, scaling the labor demand (17), and imposing the labor marketclearing condition, nt = lt:

ht+l = lt+l

wt+lC;t+lwtC;t ~wt

#t+l#t#

ww1

: (35)

Hence, the wage equation (19) can be written as:

+1

Xl=0 (w)l hj;t+lU

#C;t+l "

~wtwtC;t

C;t+l

#t#

#t+l

wzh;t+l

U#

C;t+l# = 0: (36)Dividing (20) by #tPC;t :

wt =

"(1 w) ( ~wtwt)

11w + w

#

#;t

C;t1C;t

wt1

11w

#1w: (37)

The scaled marginal cost and rst order conditions for rms in the con-sumption goods sector are:

sC;t = 1

1 1

1

rkt

(Rtwt)

1 ; (38)

Rtwt = (1 ) kC;t

Z;tlC;t

sC;t; (39)

rkt =

kC;t

Z;tlC;t

1sC;t: (40)

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The corresponding scaled equations for rms in the investment goods

sector are given by:

pI;tsI;t =

1

1

1 1

rkt

(Rtwt)

1 ; (41)

Rtwt = (1 )

kI;t

Z;tlI;t

pI;tsI;t ; (42)

rkt =

kI;t

Z;tlI;t

1pI;tsI;t : (43)

The pricing equation for rms in the consumption goods sector is scaled

as follows:+1Xl=0

p

lU#C;t+l

YCj;t+l#t+l

!~pC;tC;tC;t+l

fsC;t+l

= 0: (44)

The pricing equation for rms in the investment goods sector is scaledas follows:

+1Xl=0

p;I

lU#C;t+l

YIj;t+lZt+l

!pI;t+l

~pI;tI;tI;t+l

fsI;t+l

= 0: (45)

Scaling the aggregate price level in sector x = C; I:

1 =

"1 p;x

(~px;t)

11f + p;x

x;t1

x;t

11f

#1f: (46)

The ratio of the growth rate of prices in the two sectors is related to therelative price of investment goods at time t and t 1 as follows:

pI;tpI;t1

=I;tC;t

: (47)

The scaled loan market clearing equation is:

wtlt = tmt qt: (48)

The money growth denition equation is:

mtmt1

=t1

#;tC;t: (49)

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The scaled resource constraint for the consumption goods sector is:

ct + a (ut)kt

Z;t=

1

Z;t

PC;tPC;t

ff1

24WtWt

w(1)w1

(kC;t) l1C;t C

35 :(50)

The scaled resource constraint for the investment goods sector is:

it =1

Z;t

PI;tPI;t

ff1

24WtWt

w(1)w1

(kI;t) h1I;t I

35 : (51)The scaled law of motion for capital is:

kt+1 = (1 )kt

Z;t+

1 SZ;tit

it1

it: (52)

Aggregate labor is simply the sum of labor in the two sectors:

lt = lC;t + lI;t

Aggregate output, measured in consumption units, is given by:

yt = ct +pI;tit

A.2 Steady StateThis subsection shows how to determine the steady state for the scaledmodel, which corresponds to the balanced growth path for the model inlevels.

From the money growth denition equation:

C =

#:

By construction, the CPI and PPI ination rates coincide in steady state:

I = C= :

The households FOC for Mt+1 in steady state is:

R =#

:

32


33/40

The scaled sectoral resource constraints in steady state become:

c =1

f

kCZ

l1C ;

i =1

f

kIZ

l1I =

1

(1 )

Z

(kC+ kI) :

The intertemporal Euler equation simplies to:

rk = Z

1

(1 )

Z

;

taking into account that in steady state pK0 = pI = 1.

The marginal utility of consumption in steady state is given by:

U#C =# b

# b

1

c:

In steady state, the rms rst order conditions imply that the capital-to-labor ratios in the two sectors are equalized, and they coincide with theeconomy-wide ratio:

kClC

=kIlI

=

1

Rw

rkZ =

k

1:

Firms charge a constant mark-up over marginal cost:1

1

1 1

rk

(wR)1 =1

f

Real money balances, q, are pinned down by the loan market clearingcondition:

q =w

mq 1:

The utility function parameters l and q are determined by:

l =wU#C

w

q = qq (R 1) U#C:

33


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A.3 Linearization

Let xt xtxx be the percentage deviation of variable xt from its steadystate value.

Linearizing around the non-stochastic steady state equations (28)(33):

^pK0 ;t + U#C;t = U

#C;t+1

1

1

N;t+1 + I;t+1

+

rk

rk + 1 rkt+1 +

1

rk + 1 ^pK0 ;t+1

(53)

^pK0;t = ^pI;t + {({t {t1) {({t+1 {t) +

+{

1 N;t + I;t

N;t+1 + I;t+1

(54)

U#C;t = Et Rt+1 C;t+1 + U#C;t+1 + N;t+1 + I;t+1 1 (55)qt =

1

q

R

R 1Rt + U

#C;t

(56)

rkt a

kt bkt = 0 (57)

[e# b] [e# b] U#C;t be# ct+1 +

e2# + b2

ct+

be# ct1 be#

1

N;t+1 + I;t+1

+

be#

1

N;t + I;t

= 0; (58)

where { 2ZS

00 (Z) ; and a a00 (1) =a0 (1).

Linearizing (35):

ht+l = lt+l +w

w 1

wt+l + C;t+l b~wt wt C;t+

+ 11N;t+l +

1N;t+l

Linearizing (36):

+1Xj=0

(w)j

b~wt + wt + C;t C;t+l 11

N;t+j

1 N;t+j

+

+1

Xj=0 (w)j ht+l +

+1

Xj=0 (w)j U#C;t+j = 0:

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Substituting for ht+l and quasi-forward dierencing:1 + ww 1

11 w

hb~wt + wt + C;t w b~wt+1 + wt+1 + C;t+1i+

1 +

ww 1

1

1

N;t+1 + I;t+1

ww 1

wt

1 +

ww 1

C;t + U

#C;t lt = 0 (59)

Linearizing and rearranging (37) :

b~wt+ wt =

1

1 wwt+

w1 w

C;t +

1

1 N;t +

1 N;t

w1 w

[( wt1 + C;t1)] :

Substituting from the previous expression for b~wt + wt into (59) andrearranging,

$1 ( wt+1 + C;t+1) + $2wt 1 +

$1C;t +

$1

( wt1 + C;t1)

1

1

$1

N;t+1 + I;t+1

+

$11

N;t + I;t

+ U#C;t lt = 0;

(60)

where

$1 =w

(1 w) (1 w) 1 +

ww 1

;

$2 = 1 + 2w

w$1

ww 1

:

Notice that when wages are exible, i.e. w = 0, the linearized wageequation simplies to the standard intratemporal Euler equation:

wt = lt U#C;t;

which equates the real wage to the MRS between consumption and leisure.Linearizing the marginal cost and rst order conditions for rms in the

consumption goods sector:

sC;t = rkt + (1 )Rt + wt (61)

Rtwt =

kC;t lC;t

+ sC:t +

1

N;t + I;t

rkt = ( 1)

kC;t lC;t

+ sC;t +

N;t + I;t

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Combining the rst and third equations above and taking into account

that kC;t lC;t = kt lt:rkt = lt kt + Rt + wt +

1

1

N;t + I;t

(62)

Linearizing the marginal cost for rms in the investment goods sector:

^pI;t + sI;t = rkt + (1 )

Rt + wt

(63)

Linearizing the aggregate price level in sector x = C; I, equation (46):

b~px;t = p;x1 p;x (x;t x;t1)Linearizing the pricing equation for consumption goods rms, substitut-

ing for b~pC;t from the previous equation, quasi-dierencing, and substitutingfor sC;t from (61):

p;C1 p;C

1 p;C

C;t+1 p;C(1 + )1 p;C

1 p;C

C;t + p;C1 p;C

1 p;C

C;t1++h

rkt + (1 )

Rt + wti

= 0: (64)

Repeating the same steps for the price equation for investment goods

rms:

p;I1 p;I

1 p;I

I;t+1 p;I (1 + )1 p;I

1 p;I

I;t + p;I1 p;I

1 p;I

I;t1+h

rkt + (1 )

Rt + wt ^pI;t

i= 0: (65)

Linearizing (47):I;t C;t = ^pI;t ^pI;t1: (66)

The loan market clearing equation (48) boils down to:

wl wt + lt = m (t + mt) qqt: (67)The money growth equation is:

mt mt1 = t1 C;t +

N;t + I;t

1

: (68)

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The log-linearized versions of the sectoral resource constraints are:

ct +rkk

Zc

kt

bkt = 1

N;t + I;t

+ fkC;t + f (1 ) lC;t; (69)

{t =

1

N;t + I;t

+ fkI;t + f (1 ) lI;t : (70)

Linearizing the law of motion for capital:

bkt+1 = 1 ik

bkt 11

N;t + I;t

+

i

k{t: (71)

Total labor is related to labor in the two sectors as follows:

lt = lClC;t + (1 lC) lI;t : (72)

Finally,

yt =c

c + ict +

i

c + i({t + ^pI;t) (73)

A.4 Solving the Model

Let zt 2 R17 be dened as:

zt = "C;t; I;t ; ^pI;t ; r

kt ; wt U

#C;t b

kt+1; ^pK0 ;t; {t;

qt; ct; mt; Rt; lC;t; lI;t ; kt; lt #0

Lett = t1 + et; (74)

where

t =

p;t "p;t p;t1 "p;t1 N;t N;t1 I;t I;t1 ;

N;t "N;t N;t1 "N;t1 I;t "I;t I;t1 "I;t1

0;

et =

"p;t "p;t 0 0 cN"N;t 0 cI"I;t 0 "N;t "N;t 0 0 "I;t "I;t 0 00

:

and is a conforming matrix constructed in the obvious way.

The linearized equations (53) (58), (60), (62) and (64) (72), togetherwith the appropriate transversality conditions, characterize the local dy-namics of the model economy and can be written in matrix form as:

Et (0zt+1 + 1zt + 2zt1 + 0t+1 + 1t) = 0; (75)

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where i 2 R17 R17, i = 0; 1; 2; i 2 R

17 R16, i = 0; 1; and Et is the

expectation operator, reecting the timing assumptions embodied in thelinearized equations.

The unique solution to (75) is of the form:

zt = A () zt1 + B () t; (76)

I compute the feedback part of the solution, the matrix A (), usingthe algorithm described in Anderson and Moore (1985). Given the matrixA (), I use the method of undetermined coecients in Christiano (2002) tocompute the feedforward part of the solution, i.e. the matrix B ().

The solution of the model is a mapping from a set of model parameters to the matrices A () and B (). Equations (76) and (74) can be used to

simulate the model and compute statistics of interest, such as IRFs, whichcan be viewed as functions of the underlying model parameters.

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Variable Units Haver (USECON)

Civilian Noninstitutional Population Thousands LN16NNominal GDP Bil. $, SAAR GDP

Real GDP Bil. Chn. 2000 $, SAAR GDPH

GDP: Chain Price Index Index, 2000=100, SA JGDP

PCE: Nondurable Goods Bil. $, SAAR CN

PCE: Services Bil. $, SAAR CS

PCE: Durable Goods Bil. $, SAAR CD

GPDI Bil. $, SAAR I

Govt Cons. Exp. & Gross Inv. Bil. $, SAAR G

Federal Funds (eective) Rate % p.a. FFED

Hours of all persons (Nonfarm Bus. Sector) Index, 1992=100, SA LXFNH

Avg. weekly hours (Durable Goods Mfg.) Hours, SA LRDURGAAvg. weekly hours (Nondurable Goods Mfg.) Hours, SA LRNDURA

Avg. weekly hours (Priv. Service-providing ind.) Hours, SA LRPSRVA

All Employees (Durable Goods Mfg.) Thousands, SA LADURGA

All Employees (Nondurable Goods Mfg.) Thousands, SA LANDURA

All Employees (Priv. Service-providing ind.) Thousands, SA LAPSRVA

Compensation per hour (NBS) Index, 1992=100, SA LXNFC

Capacity utilization (manufacturing) % of capacity, SA CUMFG

Money Stock: MZM Bil. $, SA FMZM

Money Stock: M2 Bil. $, SA FM2

Money Stock: Small Time Deposits Bil. $, SA FMSTT

Table 8: Raw data

B Data

Table (8) describes the raw data used in the paper, and provides the cor-responding Haver mnemonics. The data are readily available from othercommercial (e.g. DRI-WEFA) and non-commercial (e.g. the St. Louis FRBdatabase FREDII) databases, as well as from the original sources (BEA,BLS, Board of Governors of the FRS).

The price of investment goods is measured as in Fisher (2003).25 Theprice of investment is converted in real terms by dividing it by the GDPdeator (JGDP).

The monetary aggregate used is money zero maturity (after 1974), spliced

25 I am grateful to Jonas Fisher for sharing with me his investment price data, which Iupdated to 2001:Q4.

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with M2 minus small time deposits (before 1974).26

The remaining variables used in the VAR analysis, are constructed fromthe raw data in the obvious way:

Y

l=

GDPH/LN16N

LXFNH/LN16N; = 4 log (JGDP) ;

w

Y =l=

LXNFC

JGDP

1

Y=l;

u = CUMFG, l =LXFNH

LN16N;

C

Y=

CN+CS+G

GDP;

I

Y=

CD+I

GDP:

The variables used for constructing comovement statistics in table 1 areconstructed as follows:

c = CN+CS+GJGPD*LN16N

, lC=LRDURGA*LADURGA

LN16N;

i =GCD+GPI

JGPD*LN16N; lI =

LRNDURA*LANDURA+LRPSRVA*LAPSRVA

LN16N;

y =GDPH

LN16N, l = lC+ lI:

26 I am grateful to Larry Christiano for suggesting the use of this monetary aggregate.

40

dicecio 2005 - comovement it's not a puzzle

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