Sectoral shocks and aggregate fluctuations

?

Financial support from the Sloan Foundation and National Science Foundation (SBR-9507978)is gratefully acknowledged.I thank Susanto Basu,Larry Christiano,Marty Eichenbaum,Jon Faust,John Fernald,Jonas Fisher,Anna Horvath,Nati Krivatsy,Chad Jones,Andrew Levin,Prakash Loungani,Kiminori Matsuyama,Joseph Mattey,Wolfgang Pesendorfer,John Shea,Mark Watson,and seminar participants at the Federal Reserve Board,International Division,Northwestern,Cornell,Carnegie Mellon,University of Chicago GSB,Stanford,UCSD,Boston University,Dar-tmouth,Yale,Wharton,Pompeu Fabra,and the NBER Summer 1995Economics Fluctuations Conference.Rishi Goyal provided sterling research assistance.I owe a special note of gratitude to Michele Boldrin for his encouragement and many critical comments.All errors are,of course,my own.

*Corresponding author.Tel.:650-723-4116.

E-mail address:mhorvath @https://www.360docs.net/doc/6517866456.html, (M.

Horvath)

Journal of Monetary Economics 45(2000)69}106

Sectoral shocks and aggregate #uctuations ?

Michael Horvath *

Department of Economics,Stanford Uni v ersity,Stanford,CA 94305,USA

Abstract

This paper presents a multisector dynamic general equilibrium model of business cycles with a distinctive feature:aggregate #uctuations are driven by independent sectoral shocks.The model hypothesizes that trade among sectors provides a strong synchroniza-tion mechanism for these shocks due to the limited,but locally intense,interaction that is characteristic of such input trade #ows.Limited interaction,characterized by a sparse intermediate input-use matrix,reduces substitution possibilities among intermediate inputs which strengthens comovement in sectoral value-added and leads to a postpone-ment of the law of large numbers in the variance of aggregate value-added.The chief virtue of this model is that reliance on implausible aggregate shocks is not necessary to capture the qualitative features of macroeconomic #uctuations.Building on Horvath,1998,Review of Economic Dynamics 1,781}808,which establishes the theoretical foundation for the relevance of limited interaction in the context of a stylized multisector model,this paper speci "es a more general multisector model calibrated to the US 2-digit Standard Industrial Code economy.Simulations prove the model is able to match

0304-3932/00/$-see front matter 2000Elsevier Science B.V.All rights reserved.

PII:S 0304-3932(99)00044-6

70M.Hor v ath/Journal of Monetary Economics45(2000)69}106

empirical reality as closely as standard one-sector business cycle models without relying on aggregate shocks. 2000Elsevier Science B.V.All rights reserved.

JEL classi x cation:E1;E32;C67

Keywords:Aggregate#uctuations;Sectoral interaction;Comovement;Input}output

1.Introduction

Explaining aggregate economic#uctuations has occupied theorists and em-piricists throughout this century.To an outside observer of the economic system,it would seem that forces are at play that cause otherwise heterogeneous consumers and producers to simultaneously vary demand and production intensity.Furthermore,once begun,the concerted behavior controls the econ-omy's direction for a signi"cant length of time.These two fundamental regulari-ties of business cycles,sectoral comovement and persistence,have motivated a plethora of models in which exogenous disturbances a!ect all sectors equally, are inherently highly persistent,and serve as the driving force for#uctuations in real aggregate variables.A central problem with this approach is the lack of good candidates for exogenous aggregate shocks that are large enough and persistent enough to account for the volatility in GDP.Consequently,such models seem implausible to many economists.(See Summers,1986;Lucas,1987; Mankiw,1989;McCallum,1989;Cochrane,1994.)

Without relying on aggregate shocks,the outside observer of the economy needs an alternative synchronization mechanism to explain what he or she sees. The organization of production in the economy presents itself as a natural candidate.Most commodities are inputs to the production processes of other commodities.Depending on substitution possibilities,higher production in one sector may necessitate higher production in the sectors supplying its intermedi-ate inputs.If each sector of the economy independently experiences variations in productivity,is it possible that the e!ects of such variations survive at an aggregate level?

This question was asked by Long and Plosser(1983)in one of the earliest Real Business Cycle(RBC)models.They speci"ed a six-sector model of the economy with intermediate input linkages and uncorrelated sector-speci"c shocks.While results at this high level of aggregation were generally encouraging,subsequent attempts to preserve aggregate volatility at lower levels of aggregation were not. Conventional wisdom(e.g.Lucas,1981)suggests an explanation for the limited success of the Long}Plosser model.The e!ect of uncorrelated sector-speci"c disturbances may tend to dissipate through aggregation since,by the law of large numbers,negative variations in some sectors o!set positive variations in other sectors.

M.Hor v ath/Journal of Monetary Economics45(2000)69}10671 The"eld of business cycle research is thus faced with a two-pronged dilemma. Extant theoretical models that account for certain salient features of business cycles are predicated on random disturbances that have little empirical support while the sector-speci"c disturbances that are more readily observable have not been successfully incorporated into theoretical models.The irony in this conun-drum is that most of the research e!ort has been focused on the former class of models and few attempts have been made to formulate arguments for why the law of large numbers may not apply in the latter class of models.This paper entertains the Long}Plosser sectoral shocks hypothesis again,with a di!erent model,in order to explore what forces may limit the law of large numbers. Recent literature has focused on several mechanisms that may contribute to the weakening of the law of large numbers.Examples include asymmetries or threshold e!ects,limited interaction,external economies,and monopolistic competition. The analysis presented here builds on results in Horvath(1998) which explores law of large numbers properties in a highly stylized multi-sector dynamic stochastic general equilibrium(DSGE)model similar to the one originally used by Long and Plosser(1983).

The main insight from Horvath(1998)is that sectoral comovement and aggregate volatility at a given level of disaggregation are increased,and the rate at which aggregate volatility declines upon disaggregation is decreased if the input-use matrix is characterized by limited interaction of a special form:few full rows and many sparse columns.Full rows in the input-use matrix indicate sectors that are important inputs in the production processes of many sectors.If there are few of these key input-sectors,the e!ects of their speci"c shocks are less likely to cancel upon aggregation.Sparse columns indicate that most sectors' production processes are highly speci"c with regard to intermediate inputs.This lack of substitutability among intermediate inputs forces sectors to react to shocks to the key-input sectors in like fashion.It is shown analytically in Horvath(1998)that the rate at which the law of large numbers applies in that highly stylized multi-sector model is proportional to the rate of increase in the number of predominantly full rows in the input-use matrix,rather than being proportional to the rate of increase of the number sectors.Examination of the US input-use matrices at di!erent levels of disaggregation reveals that the rate of increase in the number of predominantly full rows is signi"cantly slower than the rate of disaggregation.Roughly,out of M sectors,only(M are broadly used as inputs.

See Bak et al.(1993),Boldrin and Woodford(1990),Scheinkman(1990),Scheinkman and Woodford(1994),Jovanovic(1984),Shea(1994),Murphy et al.(1989),Basu and Fernald(1997),Basu et al.(1999),and Verbrugge(1996).

Dupor(1996)presents conditions for the same model under which the nature of linkages between sectors is irrelevant for the volatility of aggregate variables.Horvath(1998)directly compares and discusses the two sets of results.

72M.Hor v ath/Journal of Monetary Economics45(2000)69}106

In Horvath(1998),preferences and technologies are speci"ed to yield an analytical solution to the social planning problem of allocating resources across the sectors in response to idiosyncratic technology shocks. The advantage of having an analytical solution to the model's equilibrium is that it directly reveals the channel by which sectoral interaction a!ects the stochastic properties of the model's aggregates.The disadvantage is that the restrictions placed on the model separate it from the economy we actually observe.In particular,labor market#uctuations are absent from the model since preferences over consump-tion and leisure are assumed to be logarithmic.In this paper restrictions on preferences and technologies in the model are https://www.360docs.net/doc/6517866456.html,ing numerical ap-proximation techniques,the more general model is shown to exhibit the same features as the model in Horvath(1998).

A complete calibration exercise in the spirit of Kydland and Prescott(1982) and King et al.(1987)is also undertaken here.The model is parameterized to the US economy at roughly the two-digit Standard Industrial Code(SIC)level of disaggregation.Simulation reveals the time-series properties of the multisector model's aggregates are qualitatively similar to that of the data and to one-sector business cycle models without relying on aggregate shocks.Interestingly,the aggregate Solow residual series estimated from the simulated data is quite variable,with a standard deviation of roughly one-half that of aggregate value-added,as it is in US data.The interpretation of this"nding is that empirically observed shocks to aggregate multifactor productivity may simply be an artifact of aggregation and need not be interpreted as evidence of aggregate shocks.

The chief virtue of a model that can generate aggregate#uctuations from sector-speci"c shocks is that such shocks are constantly being observed and noted in the academic and trade press.Some recent examples of signi"cant, identi"able,sector-speci"c shocks include the introduction of automated teller machines in the banking sector,the severe1993southern drought and Midwes-tern#ood in the agricultural sector,the1996strike at a General Motors brake plant in Ohio,digital compression in cable television transmission,and the development of enhanced oil recovery methods in oil drilling.

Two caveats should be mentioned before proceeding;the goal of this research is not to prove that aggregate shocks are irrelevant to the study of macroeco-nomic#uctuations.It is simply to reduce economists'reliance on them by identifying a role for smaller shocks in generating aggregate movements.At the same time,the goal of this research is not to drive the location of stochastic shocks to the individual"rm level and still generate aggregate#uctuations.The results contained herein and in Horvath(1998)do not obviate the Law of Large Log preferences and full depreciation of capital stocks within one period are assumed to achieve analytical solutions.

M.Hor v ath/Journal of Monetary Economics45(2000)69}10673 Numbers,they just postpone it,making it possible for a multi-sector model parameterized to the2-digit SIC level of disaggregation to preserve su$cient aggregate volatility to match business cycle facts.

The rest of the paper is organized as follows.Section2describes the model. Section3calibrates parameters to match the US economy.Section4presents simulation results from the model that replicate the law of large numbers postponement present in the stylized model of Horvath(1998).Section5reports the performance of the model via moment comparisons with the US data and a popular one-sector DSGE model.Section6summarizes the main"ndings and brie#y comments on future research in the same vein.

2.The model

The model presented is the decentralized competitive equilibrium problem and solution.While the social planning problem is equivalent,aggregation of the multisector model is most naturally carried out using Divisia indices con-structed from both price and quantity data.Since the decentralized model carries prices explicitly,it is the more natural speci"cation to use.

2.1.En v ironment

The economic system consists of M distinct sectors,indexed by h"1,2,M, each producing a di!erent good.The production of each sector is controlled by "rms that operate so as to maximize their expected present discounted value to shareholders.Firms operate constant returns-to-scale production technologies that use capital,labor,and intermediate goods purchased from other sectors as inputs.The technologies are distinct across the sectors.There is one form of uncertainty in the economy.Multi-factor productivity in each sector is subject to stochastic innovations that are not perfectly correlated(and may be mutually independent)across sectors.

The output of each sector goes to potentially di!erent three uses.Some goods are used as intermediate inputs in the production of other goods;sectors do not necessarily use the same intermediate inputs.Some goods are built into the capital stocks of the sectors in the economy;each sector has a distinct capital stock.Finally,a portion of output in each sector is supplied to a"nal consump-tion market.The consumer-shareholders allocate labor resources to the various productive activities and make savings decisions that return investment funds to the"rms.

A note on timing is appropriate here.In the model it is assumed that intermediate inputs are delivered and either used within one period or built into the capital stock of the purchasing sector.While the real world may not conform exactly to this speci"cation,it seems like a reasonable place to start for several

reasons.First,the period length is parameterized to one quarter-year.Any delivery lags in the data less than three months long will not show up with this de"nition of the period length.Second,the labeling of intermediate inputs as durable and non-durable in the model is not arbitrary,but is guided by the intermediate input-use and capital input-use tables for the US economy.How-ever,other timing assumptions have been employed in previous research.For example,Long and Plosser(1983)assume that intermediate inputs arrive with a one period lag and then depreciate fully if not used.Kydland and Prescott (1982)specify a one-sector model with capital adjustment costs that generate lagged responses of investment to exogenous shocks.Clearly it would be interesting to explore the e!ects that holding inventories of non-durable inter-mediate goods and allowing for delivery lags in intermediate good orders have on the behavior of the system.Unfortunately space limitations do not permit these modi"cations.

2.2.Agents and preferences

Preferences of agents in the economy closely resemble those used in Spence (1976),Dixit and Stiglitz(1977),Kiyotaki(1987),Blanchard and Kiyotaki(1987), and Gali(1996).There are N identical consumers in the economy who own all shares of all"rms.Each sector has N identical"rms.Without loss of generality,

N and N are normalized to unity and the remainder of the exposition only considers the representative consumer's and representative"rm's maximization problems.

The representative consumer seeks to maximize his discounted,time separ-able utility stream given in(1).His income is composed of wages earned in each of the M production processes and the net change in the value of his stock portfolio.

max E

R

R[C R?Q R] \R!1

1!

50, (1,

s.t.

+

Q

p Q R c Q R4

+

Q

p L Q R n Q R#

+

Q

(d Q R#q Q R)s Q R!

+

Q

q Q R s Q R> ,a R.

(1)

In(1), is a discount factor,C R is an aggregate consumption index,and?R is an aggregate leisure index at time t.The parameter controls the degree of risk aversion and is inversely proportional to the elasticity of intertemporal substitu-tion.The parameter controls intratemporal substitution between consumption and leisure.The consumption aggregate is a function C(c R)where c R is a vector of consumption quantities.The leisure aggregate is a function?(n R)where n R is a vector of hours worked in each sector at time t.

The consumer's budget constraint is represented by the sum of goods pur-chased,c Q R,valued at their respective prices,p Q R,equaling total income in period t. 74M.Hor v ath/Journal of Monetary Economics45(2000)69}106

M.Hor v ath/Journal of Monetary Economics45(2000)69}10675 The maximization problem is also constrained by an initial share condition s Q given for s"1,2,M,and a condition that states that the price of all shares must stay bounded away from zero.Other variables subscripted by t denote time t values:p L Q R denotes hourly wage in sector s;d Q R denotes the dividend paid on one share held in sector s;q Q R denotes the share price of one share in sector s; s Q R denotes share holdings in sector s;and a R denotes the income#ow after establishing share holding positions,s Q R> ,for period t#1at current share prices.

The consumer's aggregate consumption index C(c R)has the constant elasticity of substitution(CES)form

C(c R)" + Q Q(c Q R) N\ N N N\ ,

for an elasticity of substitution '1and aggregation weights Q.This form implies that the worker will choose c F R to satisfy(2).

c F R" p F R P R \N a R P R( F)N,h"1,2,M.(2) The aggregate price index P R is given by

P R" + Q ( Q)N(p Q R) \N \N .

Substitution and algebraic manipulation yield the result:C(c R)"a R/P R:The aggregate consumption index is equal to nominal consumption expenditure divided by the price level.

The representative consumer is endowed with one unit of time in each period. The aggregate leisure index is assumed to take the form

?(n

R)" 1! + Q (n Q R) O> O O O> , '0.(3) At "R,labor hours are perfect substitutes as far as the worker is concerned. This would imply that the worker would devote all time to the sector paying the highest wage.Hence,at the margin,all sectors pay the same hourly wage.For (R,hours worked are not perfect substitutes for the worker.An interpreta-tion of this is that the worker has a preference for diversity of labor and hence would prefer working closer to an equal number of hours in each sector even in the presence of wage di!erences across sectors.Under standard Walrasian labor market clearing,the consumer takes p L F R,the hourly wage in sector h,as given

See Appendix A for derivation of"rst-order conditions and solution proceedures.

and chooses n F R to equate the marginal rate of substitution between consumption and leisure to this wage.This implies the condition for labor supply in Eq.(4)

n F R"(p L F R)O ?O

R(1!?R)

( a R)

.(4)

A further discussion on leisure preferences in(3)is warranted.The motivation for this speci"cation is the desire to capture some degree of sector speci"city to labor while not deviating from the representative consumer/worker assumption. Considering(4),sectors in which nominal wages are high,in steady state or in a#uctuating economy,draw more of the representative consumer's time endow-ment due to the ordinary substitution e!ect.Wage di!erences across sectors,in steady state or in a#uctuating economy,arise from parametrically imposed di!erences in labor productivity across sectors,made explicit in Section2.3. Optimal share holdings are achieved by satisfying the intertemporal condi-tion in Eq.(5)

q F R

P R a R P R \R?Q \R R

" E

R (d F R> #q F R> )

P R> a R> P R> \R?Q \R R> ,h"1,2,M.(5)

2.3.Firms and production

A quantity,y F R,of good h is produced by combining capital in the sector,k F R, labor devoted to the sector,n F R,and an index of intermediate inputs M F R in a production process described by

y F R"A F R(k F R)?F(n F R)@F(M F R)A F.(6) The index of intermediate inputs for sector h is given by a CES index of intermediate input quantities,in(7),with elasticity of substitution K.The associated price index for intermediate input index M F R is given in(8).The notation B+F denotes the set of sector indices that are inputs to the production of good h,the intermediate`buy-from a set for sector h.

M F R, Q Z+F x QF(m F R Q) N K\ N K N K N K\ ,(7) P+F R, Q Z+F(x QF)N K(p Q R) \N K \N K .(8) 76M.Hor v ath/Journal of Monetary Economics45(2000)69}106

Therefore,the model does not impose equal steady state hours and wages in each sector.Of course,for su$ciently close to R this result obtains.

The matrix K is time-invariant only under the Cobb }Douglas assumption ( K "1)used in the baseline simulations.For values of K di !erent from unity, K denotes the steady-state cost shares for non-durable intermediate goods.In what follows,the time-invariant intermediate input-use matrix is K . Let GH be the ij th element of K ,denoting the cost share of total expenditure on intermediate goods in sector j due to purchases of intermediate goods from sector i .Let H denote the sum of the j th column of K .The weights in (7)are normalized to satisfy the following conditions: Q Z +F x QF "1and x QF " QF / F .When K

"1,(7)corresponds to the Cobb }Douglas speci "cation:M F R , Q Z +F (m F R Q )V QF with price index P +F R , F Q Z +

F p Q R QF

V QF .In (6)A F R represents the state of technology in sector h .It is assumed that A F R

follows a stochastic process described by ln(A F R )" F

ln(A F R \ )# F R .(9)In (9) F R is a normally distributed mean zero random variable with E [ R R ]" .Section 3provides a fuller description of the distributional assumptions on R .Several comments are appropriate here.First,the stochastic process for technology does not incorporate a deterministic trend.The model is interpreta-ble as the detrended version of another model that does allow for trending multifactor productivity.Second,the assumption of constant returns to scale implies that F # F # F "1.Third,note that when x GH "0(equivalently GH "0),good i is not used as an intermediate input in the production process for good j ,hence m H R G "0and i ,B +H .Finally,factor share vectors , ,and ,and the input-use matrix K are assumed to be time invariant.As in Kiyotaki (1987)and Gali (1996)capital accumulation is accomplished through an investment process described by k F R> !(1! F )k F R

" (i F R ),(10)where F is a sector speci "c depreciation rate inside the unit interval and the composite investment good is created by combining inputs according to

(i F R )" Q Z 'F

x QF (i F R Q ) E \ E

E E \ .(11)The notation i

F R Q denotes the quantity of good s purchased by sector h for investment purposes and Q Z 'F denotes the sum over all s in the investment `buy-from a set of sector h ,the set of sectors from whom sector h purchases durable intermediate goods.x QF

is the weight that good s receives in the M.Hor v ath /Journal of Monetary Economics 45(2000)69}10677

As with K , 'is time invariant only under the Cobb }Douglas assumption ( "1)used in the baseline simulations.For values of di !erent from unity, 'denotes the capital goods cost shares in steady state.

production of capital in sector h .With this speci "cation and that in (7),"rms buy goods from a subset of sectors for capital accumulation purposes and buy goods from a subset of sectors for intermediate good purposes.Let the time-invariant capital input-use matrix be 'with typical element GH denoting the cost share out of total expenditure on capital goods in sector j due to purchases of capital goods from sector i . Note that

Q Z '

F QF "1.The weights x QF are related to QF by x QF " E QF .Firms in sector h will maximize e !ective investment (i F R )for a given level of investment expenditure,

z F R , Q Z '

F p Q R i F R Q .This gives rise to investment-related demands of the form

i F R Q "x E QF p Q R F R \E (i F R ),s 3B 'F ,h "1,2,M (12)

and an e !ective investment level of (i F R )"z F R / F R where the investment price index for sector h is given by

F R " Q Z 'F x E QF (p Q R ) \E \E .

The notation has a simple interpretation if one considers (i F R )as the investment good for sector h and F R as its price.The total expenditure on investment,z F R ,is equal to (i F R ) F R ,a result that can be obtained by manipulating (12).Firms in sector h are instructed by the shareholders to maximize the present discounted value of real dividends as in Eq.(13).Note that dividends are discounted by the representative shareholder 's marginal utility of consumption.Absent complete contingent claims markets,this asset pricing kernel is required to equate the competitive equilibrium solution proposed here with the optimal control solution.

max E R R d F R P R a R P R \R ?Q \R R .(13)

78M.Hor v ath /Journal of Monetary Economics 45(2000)69}106

The maximization problem is constrained by a given initial condition for k F and by Eqs.(6)and(14).The latter describes dividends in sector h at time t.

d F R"p F R y F R!p L F R n F R!(k F R> !(1! F)k F R) F R!P+F R M F R.(14) Total demand for good h is given by Eq.(15).Th

e notation S'F and S+F on the summation operators denote the`sell-to a sets for investment and intermediate input use,respectively,the set o

f sectoral indices to which sector h sells a portion of its output for these uses.

y F B R,c F R#

Q Z1'F i Q R F#

Q Z1+F

m Q R F.(15)

2.4.Perfectly competiti v e equilibrium

Here I assume that sectors behave in a competitive fashion and charge prices equal to marginal cost.It should be noted that demand in sector h is a decreas-ing function of the price in sector h.This has led Kiyotaki(1987),Blanchard and Kiyotaki(1987),and Gali(1996)to consider the e!ects of monopolistic competi-tion among"rms in di!erent sectors.Basu et al.(1999)discuss the e!ect of monopolistic competition and non-constant returns to scale on aggregation in an otherwise identical model setting to the one presented here.

Price is equal to the marginal cost of production in each sector.The marginal cost function,taking prices as given,is given by

(x F R;p R,2,p+R,p L F R,A F R)

"g

S F(A F R) \ \? F(p L F

R)

@ \? F y F R k F R ? \? F(P+F R) A \? F.(16)

The notation( /(1! ))F,for example,denotes( F/(1! F))and g S

F is a constant

function of the technological coe$cients.Note that the price in sector h is an increasing function of the prices in the sectors from which it purchases inter-mediate products and a decreasing function of the state of technology in sector h,ceteris paribus.

Labor demand is determined by Walrasian market mechanisms.Firms take the wage rate as given and equate labor's marginal product to the wage to determine demand.Consumers,also taking the wage as given,balance the disutility from reduced leisure with the bene"t of increased labor income to determine supply.The equilibrium condition is summarized by

n F R" F p F R y F R a R?R(1!?R) O O >O .(17)

Sector h demands intermediate goods from potentially all sectors,even its own.The optimal level of the intermediate good index in sector h equates its per M.Hor v ath/Journal of Monetary Economics45(2000)69}10679

unit cost,P+F R,with its marginal product resulting in Eq.(18).The optimal amount of each input purchased to achieve M F R is given by(19).Again, B+F denotes the set of sectors from which sector h purchases intermediate goods (corresponding to the row indices of the non-zero cells in column h of K).

P+F R" F p F R y F R

M F R

,(18)

m F R Q" QF p F R y F R p R N K(M F R) \N K,s3B+F.(19)

The"rms'intertemporal decision at time t concerns the level of capital stock that is desired at time t#1.Di!erentiating(13)combined with(14)with respect to k F R> results in the"rst-order condition for optimal capital stock in sector h:

F R

P R a R P R \R?Q \R R

" E

R 1P R> a R> P R> \R?Q \R R> p F R> y F R> k F R> F#(1! F) F R> .(20) Finally,the model is closed by specifying market clearing for good h y F R"y F B R.(21) Simple manipulation of"rst-order conditions reveals that consumers'total income is independent of share holdings and is given by

a R"+

Q

p Q R y Q R( Q# Q)!(k Q R> !(1! Q)k Q R) Q R.(22)

De x nition:A perfectly competiti v e equilibrium consists of shocks vectors+ R, R , price vectors+p R, R,p L R, R ,and quantity vectors+k R,n R,M R,c R,i R,y R, R such that 1.productivity levels+A R, R follow their log-autoregressive laws of motion subject to shocks+ R, R ,

2."rms maximize present discounted value of dividends to shareholders,

3.consumers maximize lifetime utility,

4.prices clear labor markets and goods markets.

The dynamic program as speci"ed is non-linear in its state and co-state variables.Except for a special case of the parameter set,described in Horvath (1998),analytical solutions are not possible.Therefore,a suitable approximate solution technique is required.Because the sta te space is very large,I elect to use the solution method of log-linearizing the system of equations around their 80M.Hor v ath/Journal of Monetary Economics45(2000)69}106

Note,the price in sector M can,without loss of generality,be normalized to unity every period.The dimension of the state space is reduced to 3M !1when the coe $cient controlling the consumer 's risk aversion equals zero since then a R does not appear in the "rm 's intertemporal "rst-order condition for optimal choice of k R> . See Sato (1975)for the relative merits of the Divisia method.

steady state values with "rst-order Taylor Series expansions.A formal justi "ca-tion for this solution method is presented in Woodford (1986),Theorem 1.A derivation of the linearized solution appears in Appendix A.

The solution to the linear expectational di !erence equations is determined by state space methods in X R and X R> ,where denotes a variable 's percent deviation from its steady-state value and X R ,+a R ,p G R ,A H R ,k H R ,+\ +G H .This implies 3M state variables. The model then has a state space representation given by X R " X R \ where is a 3M ;3M matrix that depends on the parameters of the model,including the input-use matrix K

.2.5.Aggregation

Since this exercise intends to say something about aggregate quantities it is necessary to de "ne the method of aggregation from nominal sectoral output to real aggregate value added.In doing so I am drawing heavily on the accounting methods advocated by Basu and Fernald (1997).Time subscripts are dropped for the remainder of this section to reduce notational clutter.

In nominal terms,the de "nition of sectoral and aggregate value added is quite straightforward.Nominal value added in sector h is de "ned as the di !erence between the value of gross output in that sector and the cost of the intermediate inputs used to produce it as in

p T F v F "p F y F !P +F M F .(23)Aggregate nominal value added is simply the sum over all sectoral nominal value added amounts:p T v , +G p T G v G .Straightforward manipulation reveals that p T v "a from (22).From the perspective of the representative consumer who owns shares in sector h ,nominal value added represents the income generated in sector h ,above the cost of the intermediate inputs,that can be used to increase this consumer 's expenditure on "nal goods.

To arrive at real value added,it is necessary to pick an accounting method that is internally consistent so that the national accounting identity that holds in nominal prices also holds in constant prices.I de "ne value added growth rates with Divisia indices which are de "ned in terms of growth rates.The appendix contains a detailed description of the construction of Divisia indices for this model economy.

M.Hor v ath /Journal of Monetary Economics 45(2000)69}10681

82M.Hor v ath/Journal of Monetary Economics45(2000)69}106

3.Parameterization

The parameters that must be speci"ed to complete the model are summarized below:

E Preferences: , , , , ,+ F,+F

E Production:M,+ F, F, F, F,+

F , K, ', , K,

E Shocks:+ F,+

F , .

3.1.Preference and production parameters

Most of these parameters are set to empirical estimates from the US economy. For simulations in Section5,the level of disaggregation is set to M"36.This conforms closely to the2-digit SIC level,the sectoral de"nitions used by Jorgenson and Fraumeni(1987).The production parameters, F, F,and are derived from cost share data on the US economy presented in the latter work in the following manner.Time-average cost shares for capital,labor,and inter-mediate inputs are calculated for36sectors of the private US economy using annual data from1948to1985by dividing the cost of inputs at producer prices by the value of output at producer prices(hence,this assumes perfect competi-tion and constant returns to scale).The Cobb}Douglas coe$cients , ,and used in the simulations are the time-series averages from these calculations. The share F is divided across all interacting sectors using the fraction that the purchases from these sectors represent out of total intermediate purchases by sector h.The mean value of F across sectors in the simulations is0.16,the mean value of F is0.32,and the mean value of F is0.52.Table1lists these parameter values along with the sectoral de"nitions.

The1977detailed intermediate input-use matrix is used to parameterize K. This matrix can be aggregated up to various useful levels.In particular,law of large numbers properties of the model are explored in Section4using the matrices K , K , K ,and K containing77,36,21,and6sectors, respectively.Simulations in Section5use K .

Data for the investment-use matrix 'come from the capital#ow table from 1977,presented and described in Silverstein(1985).The capital#ow table was converted to 'for77,36,21,and6sectors by aggregating as with K and dividing columns by their sums at each level of aggregation.

The time period considered here is the quarter year.Consistent with previous business cycle models,the discount factor, ,is chosen to be(1.03)\ implying an annual discount rate of3%;the sectoral depreciation rates, F,are those used in Jorgenson and Fraumeni(1987),and are given in Table1.

I thank John Fernald for kindly providing the 's.

Constant and trend terms were included in the regression,but results are largely the same if they are not included.

See Ashenfelter and Altonji (1980)and Altonji (1982)for an extended discussion and further references.

The parameters and control the volatility of hours and the fraction of time spent working in steady state,respectively.The parameter is estimated from the Jorgenson data on annual sectoral outputs,labor inputs,and labor share coe $cients as follows.Manipulating (17)and (3)yields

n F R

"(b F R )O O > (1!?R )(24)where b F R is the fraction of labor 's share of aggregate output accumulating to labor in sector h :

b F R , F p F R y F R +Q Q p Q R y Q R

.(25)Expressing (24)in percentage changes (denoted by above variables)and adding an estimation error term results in the M estimation equations

(n F R ! b Q R \ n Q R )" #1b F R #error F R h "1,2,M .(26)

The interpretation of (26)is that relative labor hour percentage changes in sector h (the left-hand side)are related to relative labor 's share percentage changes in sector h by the elasticity /( #1).

Estimating (26)by SUR with restricted to be identical across sectors results in a point estimate of 0.9996with a standard error of 0.0027. Hence a value of "1.00is used the baseline simulations reported below.This value compares favorably with micro-level studies on the wage elasticity of labor supply.The typical "nding in these studies is that labor supply elasticity is low. The uncompensated labor supply elasticity implied by "1is 0.5.It should be noted that,since (R ,wage di !erences across sectors persist in equilibrium.For comparison purposes,simulations are also presented below for the case "100and "2.The parameter is set so that total hours spent working in steady state represent one-third of the worker 's total time endowment.Contingent on "1,2and 100,this requires "13.4,7.1and 1.93,respectively.The elasticity parameters K , , ,and are worth considering in terms of their respective e !ects on output,consumption,investment and labor volatility.The value of K determines the elasticity of substitution among intermediate inputs for all sectors.It is hypothesized that lower values of K engender greater sectoral comovement and hence greater aggregate output volatility by reducing the ability of sectors to &avoid 'the shocks of their input supplying sectors.The M.Hor v ath /Journal of Monetary Economics 45(2000)69}10683

Using consumption shares in this manner actually hinders the model 's ability to generate aggregate volatility from independent sectoral shocks relative to a parameterization that assumes all sectors have equal weights in the consumer 's utility function.This is due to the fact that several of the least volatile sectors (e.g.food,wholesale and retail trade,and apparel)account for a disproportion-ate share of consumption expenditure in the data.

parameter controls the degree of substitutability between di !erent consump-tion goods.A larger value of implies goods are more substitutable and therefore consumption of a given good is more responsive to variations in its relative price.The parameter a !ects sectoral investment demand for a given investment good in an analogous fashion.The parameter controls both the degree of risk aversion and the degree of intertemporal substitution.As in-creases,workers become less willing to substitute utility intertemporally.This tends to decrease output,investment,and consumption volatility.Workers also become more risk averse and hence less willing to substitute consumption across productivity states.Again this reduces consumption volatility.At "1prefer-ences are logarithmic in the consumption and leisure aggregates.

While it would be desirable to calibrate K , , ,and to some empirical regularities of the US economy,there is little to guide such an exercise.Intuition and precedence suggest K 41and 51while even less information guides selection of reasonable ranges for and .Accordingly,the following baseline set of parameter values are used: K "1.00, "1.00, "1.00, "1.00.All elasticities of substitution are set to unity and the consumer has logarithmic preferences in the consumption and leisure aggregates.Variations from this baseline are then considered in the next section to explore the e !ects these parameters have on the simulation results.

Preference weights are related to empirically observed relative sectoral de-mand intensities as follows.At "1,the nominal consumption expenditure share of sector h in total consumption is constant and is given by "p F c F / p Q c Q .Consumption expenditure shares are calculated for the 36sectors using con-sumption data from the National Income and Product Accounts.Shares are averaged over the period 1959}1995.Values for calibrated in this manner are reported in Table 1.

3.2.The stochastic process for shocks

Completing the parameter set requires values for + F ,+F and .Employing the Jorgenson dataset used to construct F , F ,and F a sectoral productivity series could be constructed for each sector as the residual of outputs minus weighted factors inputs according to (6)(in logs).

ln(A F R ),ln(y F R )! F ln(k F R )! F ln(n F R )! F ln(m F R ).(27)84M.Hor v ath /Journal of Monetary Economics 45(2000)69}106

M.Hor v ath/Journal of Monetary Economics45(2000)69}10685 Values for ln(A F R)could be"t to(9)yielding regression estimates for+ F,+F and residuals,+ F R,+F ,that could be used to calculate .

Several authors including Burnside et al.(1996)(hereafter BER),Basu(1996), and Basu and Kimball(1994),have noted that variations in capital utilization will a!ect the measurement of total factor productivity under(27). In particu-lar,the variance of productivity shocks is likely to be reduced once they have been`corrected a for variations in capital utilization.These authors recommend several methods for accounting for variable capital utilization.The method of BER is particularly easy to implement with the Jorgenson dataset.BER assume that it is the#ow of services from capital stock k G that enters into the production function given in(6)and that this#ow is linearly proportional to energy usage. They recommend correcting ln(A F R)with data on sectoral energy usage as follows:

ln(A F R),ln(y F R)! F ln(e F R)! F ln(n F R)! F ln(m F R),(28) where e F R denotes the time t level of energy usage in sector h.

In the present analysis,correcting for capital utilization is important because it is likely that the sectoral covariance properties of R di!er with the correction in a systematic manner.By their nature,aggregate shocks should a!ect capital utilization in the same direction for a majority of sectors while the "rst-order e!ect of uncorrelated sectoral shocks should be uncorrelated move-ments in capital utilization across sectors.Therefore,failing to correct for varying capital utilization would overstate the cross-sector correlation in sec-toral total factor productivity growth.This indeed turns out to be the case.The average pairwise correlation(excluding the diagonal)among the from(27)is 0.14while from(28)it is0.09suggesting that sectoral shocks would appear to be more important relative to aggregate shocks after correcting for variations in capital utilization.

In the simulations reported below it is important to distinguish between model volatility generated by uncorrelated sectoral shocks versus aggregate shocks,something which depends on the exact structure of .To pin down the structure of ,further assumptions on are needed.Suppose that shocks to sector h were the sum of idiosyncratic shocks, F R,and aggregate shocks, R as in

F R, F R#b F R,h"1,2,M,(29) where b F controls the response of sector h to aggregate shocks,E( R R)"A for a diagonal matrix A,and E( R)" J.It will not be possible to decompose R as in(29)without additional restrictions on the model.For example,

Variable utilization may a!ect other input factors,as these and other authors have noted. However,in the present analysis only variable capital utilization is considered.

Developing an identi "cation scheme for (29)is beyond the scope of this paper but is the subject of Horvath and Verbrugge (1999).

King et al.(1987)report and use this standard deviation for aggregate technology shocks.

normalization of (29)for each h will not help much since there are a total of 2M #1unknowns.

Even without exact identi "cation of aggregate and sectoral shocks,it is possible to investigate the importance of uncorrelated sectoral shocks using the present setup.Loosely,the size of the diagonal elements of relative to the o !-diagonal elements indicate the size of uncorrelated sectoral shocks relative to aggregate shocks.Let denote the diagonal matrix of ,setting o !-diagonal elements to zero.Results will be reported for simulations based on both and .The "rst set of results provides a gauge on how well the model works in replicating the empirical moments of the US economy without relying on aggreg-ate shocks .The second set of results indicates how close the model lies relative to the data when both aggregate and uncorrelated sectoral shocks are present.This comparison is an upper bound on the likely importance of sectoral shocks since the typical diagonal element of is given by GG "A GG #b G J ,which contains a weighted aggregate shock variance.Therefore,this analysis would show conclusively that sector-speci "c shocks are unimportant if the model did a poor job of matching the data when parameterized the productivity shock distri-bution,while it would suggest that sector-speci "c shocks may be an important source of aggregate volatility if the model performs about as well under as under .

The values for F and the square roots of the diagonal elements of are reported in Table 1.The mean value for F is 0.93and that of ( is 0.022;however,there is considerable sectoral variation.It should be noted that F are shocks to gross output in sector h .They must be scaled up by 1/(1! )to gauge their size relative to sectoral value added.One statistic that can be used to assess the plausibility of these parameter values is the volatility of measured total factor productivity in the model simulations.The shocks should generate su $cient variation in aggregate total factor productivity residuals to match the empirically observed estimate of 0.0078. Statistics on aggregate total factor productivity are reported in the discussion of the simulation results.

Table 1tabulates all the parameters of the model discussed above.

4.Limited substitution ,limited interaction and the law of large numbers

At what rate does the volatility of aggregate value added decline in the model as the level of disaggregation increases?The purpose of this section is to 86M.Hor v ath /Journal of Monetary Economics 45(2000)69}106

M.Hor v ath/Journal of Monetary Economics45(2000)69}10687 Table1

Baseline parameter values for the2-digit SIC(M"36)model economy

Aggregate "0.993, "1.0, "1.0, "13.4

"1.0, "1.0, K"1.0

Sector 100;

( Agricultural Products0.570.310.130.010.91 3.600.01 Agricultural Services0.440.450.120.010.91 3.600.01 Metal Mining0.590.260.150.020.95 5.260.01 Coal Mining0.430.410.160.020.99 3.940.01 Petroleum and Natural Gas0.520.130.350.020.97 6.180.01 Nonmetallic Mining0.390.330.270.020.95 3.260.01 Construction0.570.370.060.040.980.800.01 Food and Kindred Products0.770.160.060.020.64 1.390.12 Tobacco Products0.510.200.290.020.96 2.340.01 Textile Mill Products0.640.280.080.020.88 1.810.01 Apparel0.640.320.040.020.990.630.04 Paper0.510.350.140.020.93 2.490.02 Printing and Publishing0.570.360.070.020.88 1.380.01 Chemicals0.610.250.140.020.94 1.450.03 Petroleum and Coal Products0.500.390.110.020.980.800.03 Rubber and Misc.Plastics0.600.220.180.020.96 2.370.01 Leather0.780.110.110.010.96 6.350.01 Lumber and Wood0.550.360.100.020.95 1.670.01 Furniture and Fixtures0.610.340.050.020.89 2.990.02 Stone,Clay,Glass0.510.350.140.020.960.930.01 Primary Metal0.660.230.110.020.92 1.660.01 Fabricated Metal0.520.380.090.020.90 1.020.01 Non-elec.Machinery0.490.390.130.020.91 1.320.01 Elec.Machinery0.530.370.100.020.960.970.02 Motor Vehicles0.610.340.050.020.93 2.320.04 Transportation Equipment0.670.200.130.020.90 1.910.01 Instruments0.360.500.140.020.84 1.590.01 Misc.Manufacturing0.550.360.090.020.90 2.340.01 Transportation Services0.590.270.140.010.97 1.550.03 Communication Services0.210.430.360.020.98 2.170.02 Electric Utilities0.240.520.250.020.98 1.540.02 Gas Utilities0.410.210.380.010.98 2.740.01 Wholesale and Retail Trade0.300.540.160.020.93 1.260.25 Finance,Insurance,and Real Estate0.560.210.230.010.97 1.420.01 Water and Sanitary Services0.610.250.140.030.95 1.190.13 Other Services0.240.420.340.030.97 2.370.01 Note:See text for calibration of these parameters and for descriptions of K and '.

establish that the results found in Horvath(1998)still hold in the more general model presented above.Speci"cally,Horvath(1998)presents a special case of the above model without preferences over leisure and with parameters set to "1, "0, "1, '"I K,and K"1.This permits analytical results

Having no full rows (no intermediate links at all)would not promote the maximum degree of aggregate volatility.In this case the law of large numbers applies exactly:sectors are distinct production economies so aggregate output moves with the average of all the independent shocks. In principle,these two measures of substitutability can both be involved in postponing the law of large numbers.However,it should be obvious that when connectivity is extremely low (for example,if most sectors use only one intermediate input with substantial intensity)the elasticity of substitution between intermediate inputs is irrelevant.Similarly,when the elasticity of substitution is very high,the degree of connectivity may matter less.

showing that the sparseness of K of a particular form results in greater volatility of aggregates for a given level of disaggregation and a slower decline in aggregate volatility for increasing levels of disaggregation.This postponement of the law of large numbers relies on K being characterized by few full rows and mostly sparse columns.The full rows represent &key input 'sectors,sectors that sell inputs broadly to many other sectors. The sparseness of the columns indicates the lack of substitution possibilities.The end result:the rate of decline of aggregate volatility depends on the rate of increase of the number of predomi-nantly full rows in K rather than on the rate of increase in the total number of rows (sectors).Examination of the US input-use matrices at di !erent levels of disaggregation reveals that the rate of increase in the number of predominantly full rows is signi "cantly slower than the rate of disaggregation.Roughly,out of M sectors,only (M are broadly used as inputs.Horvath (1998)also presents a comparison of these results with the results in Dupor (1996)who shows conditions under which the nature of intermediate input linkages is irrelevant for the volatility of aggregate variables.

The lack of factor substitutability is chie #y responsible for this result.The obvious measure of substitutability of intermediate inputs is the elasticity of substitution between such inputs in the production function.However,for models with intermediate goods trading there is another measure of substituta-bility,namely connectivity.Every good produced in the economy may not be suitable for use as an intermediate input in the production of every other good.Connectivity is used to describe the number of input links the typical sector has with other sectors.A low degree of connectivity,characterized by a sparse input-use matrix,implies few possibilities of substitution among intermediate inputs.

This section explores,through simulation,how disaggregation of K and 'a !ects the volatility of aggregate value added in the general model presented above.Simulations are in order rather than analytical investigations for two distinct reasons.First,as mentioned above,analytical solutions for the model in Section 2are only available for a special case of the parameter set,and second,we are interested in the properties of a nonlinear function of the variables in X R ,namely aggregate value added.

88M.Hor v ath /Journal of Monetary Economics 45(2000)69}106