Sectoral shocks and aggregate fluctuations

Journal of Monetary Economics 45 (2000) 69}106

Sectoral shocks and aggregate #uctuationsq Michael Horvath* Department of Economics, Stanford University, 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 synchronization 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 postponement 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

q 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, Dartmouth, Yale, Wharton, Pompeu Fabra, and the NBER Summer 1995 Economics 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: [email protected] (M. Horvath) 0304-3932/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 3 2 ( 9 9 ) 0 0 0 4 4 - 6

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empirical reality as closely as standard one-sector business cycle models without relying on aggregate shocks. ( 2000 Elsevier Science B.V. All rights reserved. JEL classixcation: E1; E32; C67 Keywords: Aggregate #uctuations; Sectoral interaction; Comovement; Input}output

1. Introduction Explaining aggregate economic #uctuations has occupied theorists and empiricists 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 economy's direction for a signi"cant length of time. These two fundamental regularities 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 intermediate 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.

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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 conundrum 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.1 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 JM are broadly used as inputs.2 1 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). 2 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.

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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.3 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 consumption and leisure are assumed to be logarithmic. In this paper restrictions on preferences and technologies in the model are removed. Using numerical approximation 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 severe 1993 southern drought and Midwestern #ood in the agricultural sector, the 1996 strike 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 macroeconomic #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

3 Log preferences and full depreciation of capital stocks within one period are assumed to achieve analytical solutions.

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Numbers, they just postpone it, making it possible for a multi-sector model parameterized to the 2-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. Section 2 describes the model. Section 3 calibrates parameters to match the US economy. Section 4 presents simulation results from the model that replicate the law of large numbers postponement present in the stylized model of Horvath (1998). Section 5 reports the performance of the model via moment comparisons with the US data and a popular one-sector DSGE model. Section 6 summarizes 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 constructed from both price and quantity data. Since the decentralized model carries prices explicitly, it is the more natural speci"cation to use. 2.1. Environment 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 consumption 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

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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. However, 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 intermediate 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 separable 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. = [C ¸s]1~t!1 max E + dt t t t50, d(1, 0 1!t t/0 (1) M M M M s.t. + ps cs 4 + pns ns # + (ds#qs )ss! + qs ss ,a . t t t`1 t t t t t t t s/1 s/1 s/1 s/1 In (1), d is a discount factor, C is an aggregate consumption index, and ¸ is an t t aggregate leisure index at time t. The parameter t controls the degree of risk aversion and is inversely proportional to the elasticity of intertemporal substitution. The parameter s controls intratemporal substitution between consumption and leisure. The consumption aggregate is a function C(c ) where c is a vector of t t consumption quantities. The leisure aggregate is a function ¸(n ) where n is t t a vector of hours worked in each sector at time t. The consumer's budget constraint is represented by the sum of goods purchased, cs, valued at their respective prices, ps, equaling total income in period t. t t

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The maximization problem is also constrained by an initial share condition ss given for s"1,2, M, and a condition that states that the price of all shares 0 must stay bounded away from zero. Other variables subscripted by t denote time t values: pns denotes hourly wage in sector s; ds denotes the dividend paid on t t one share held in sector s; qs denotes the share price of one share in sector s; t ss denotes share holdings in sector s; and a denotes the income #ow after t t establishing share holding positions, ss , for period t#1 at current share t`1 prices. The consumer's aggregate consumption index C(c ) has the constant elasticity t of substitution (CES) form

C

D

p@(p~1) M C(c )" + ms(cs)(p~1)@p , t t s/1 for an elasticity of substitution p'1 and aggregation weights ms. This form implies4 that the worker will choose ch to satisfy (2). t ph ~pa t (mh)p, h"1,2, M. ch" t (2) t P P t t The aggregate price index P is given by t 1@(1~p) M . P " + (ms)p(ps)1~p t t s/1 Substitution and algebraic manipulation yield the result: C(c )"a /P : The t t t 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

C D C

D

G C

D H

q@(q`1) M ¸(n )" 1! + (ns)(q`1)@q , q'0. (3) t t s/1 At q"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 q(R, hours worked are not perfect substitutes for the worker. An interpretation 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 pnh , the hourly wage in sector h, as given t 4 See Appendix A for derivation of "rst-order conditions and solution proceedures.

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and chooses nh to equate the marginal rate of substitution between consumption t and leisure to this wage. This implies the condition for labor supply in Eq. (4) ¸q (1!¸ ) t . nh"(pnh )q t (4) t t (sa )q t 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 endowment 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 Section 2.3.5 Optimal share holdings are achieved by satisfying the intertemporal condition in Eq. (5)

AB

qh a ~t t t ¸s(1~t) t P P t t ~t (dh #qh ) a t`1 t`1 t`1 " dE ¸s(1~t) , h"1,2, M. t`1 t P P t`1 t`1

G

A B

H

(5)

2.3. Firms and production A quantity, yh, of good h is produced by combining capital in the sector, kh, t t labor devoted to the sector, nh, and an index of intermediate inputs Mh in t t a production process described by yh"Ah(kh)ah (nh)bh (Mh)ch . (6) t t t t t The index of intermediate inputs for sector h is given by a CES index of intermediate input quantities, in (7), with elasticity of substitution p . The m associated price index for intermediate input index Mh is given in (8). The t notation BM denotes the set of sector indices that are inputs to the production of h good h, the intermediate `buy-froma set for sector h.

C C

D D

Mh, + x (mh )(pm ~1)@pm t sh t,s s|BMh PMh , + (x )pm (ps)1~pm sh t t s|BMh

pm @(pm ~1) ,

1@(1~pm ) .

(7) (8)

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

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In what follows, the time-invariant intermediate input-use matrix is C .6 Let m c be the ijth element of C , denoting the cost share of total expenditure on ij m intermediate goods in sector j due to purchases of intermediate goods from sector i. Let c denote the sum of the jth column of C . The weights in (7) are j m normalized to satisfy the following conditions: + Mh x "1 and x "c /c . s|B sh sh sh h When p "1, (7) corresponds to the Cobb}Douglas speci"cation: m ps xsh t Mh, < (mh )xsh with price index PMh ,c < . t t,s t h M c s|BMh s|Bh sh In (6) Ah represents the state of technology in sector h. It is assumed that t Ah follows a stochastic process described by t (9) ln(Ah)"o ln(Ah )#eh. t t~1 t h In (9) eh is a normally distributed mean zero random variable with E[e e@ ]"X. t t t Section 3 provides a fuller description of the distributional assumptions on e . t Several comments are appropriate here. First, the stochastic process for technology does not incorporate a deterministic trend. The model is interpretable as the detrended version of another model that does allow for trending multifactor productivity. Second, the assumption of constant returns to scale implies that a #b #c "1. Third, note that when x "0 (equivalently h h h ij c "0), good i is not used as an intermediate input in the production process for ij good j, hence mj "0 and iNBM. Finally, factor share vectors a, b, and c, and the j t,i input-use matrix C are assumed to be time invariant. m As in Kiyotaki (1987) and Gali (1996) capital accumulation is accomplished through an investment process described by

A B

(10) kh !(1!k )kh"g(ih), t t`1 h t where k is a sector speci"c depreciation rate inside the unit interval and the h composite investment good is created by combining inputs according to

C

D

g@(g~1) g(ih)" + x8 (ih )(g~1)@g . (11) t t,s sh s|BIh The notation ih denotes the quantity of good s purchased by sector h for t,s investment purposes and + Ih denotes the sum over all s in the investment s|B `buy-froma set of sector h, the set of sectors from whom sector h purchases durable intermediate goods. x8 is the weight that good s receives in the sh 6 The matrix C is time-invariant only under the Cobb}Douglas assumption (p "1) used in the m m baseline simulations. For values of p di!erent from unity, C denotes the steady-state cost shares m m for non-durable intermediate goods.

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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 C with typical element c8 denoting the cost share I ij out of total expenditure on capital goods in sector j due to purchases of capital goods from sector i.7 Note that + c8 "1. sh s|BIh The weights x8 are related to c8 by x8 "c81@g. sh sh sh sh Firms in sector h will maximize e!ective investment g(ih) for a given level of t investment expenditure, zh, + psih . t t t,s s|BIh This gives rise to investment-related demands of the form

CD

ps ~g ih "x8 g t g(ih), t,s sh nh t t

s3BI , h"1,2, M h

(12)

and an e!ective investment level of g(ih)"zh/nh where the investment price index t t t for sector h is given by

C

D

nh" + x8 g (ps )1~g t sh t s|BIh

1@(1~g) .

The notation has a simple interpretation if one considers g(ih) as the investment t good for sector h and nh as its price. The total expenditure on investment, zh, is t t equal to g(ih)nh, a result that can be obtained by manipulating (12). t t 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.

A BA B

= dh max E + dt t 0 P t t/0

a ~t t ¸s(1~t). t P t

(13)

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

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The maximization problem is constrained by a given initial condition for kh and 0 by Eqs. (6) and (14). The latter describes dividends in sector h at time t. (14) dh"phyh!pnh nh!(kh !(1!k )kh)nh!PMh Mh. t t t`1 h t t t t t t t Total demand for good h is given by Eq. (15). The notation SI and SM on the h h summation operators denote the `sell-toa sets for investment and intermediate input use, respectively, the set of sectoral indices to which sector h sells a portion of its output for these uses. yh ,ch# + is # + ms . d,t t t,h t,h s|SIh s|SMh

(15)

2.4. Perfectly competitive 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 decreasing 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 competition 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 u(xh; p1,2, pM, pnh , Ah) t t t t t

AB

yh (a@(1~a))h h h "g h (Ah)(~1@(1~a)) (pnh )(b@(1~a)) t (PMh )(c@(1~a))h . (16) u t t t kh t The notation (c/(1!a)) , for example, denotes (c /(1!a )) and g h is a constant h h h u 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 intermediate 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

A

B

q@(1`q) b phyh nh" h t t ¸ (1!¸ )1@q . (17) t t s a t t 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

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unit cost, PMh , with its marginal product resulting in Eq. (18). The optimal t amount of each input purchased to achieve Mh is given by (19). Again, t BM denotes the set of sectors from which sector h purchases intermediate goods h (corresponding to the row indices of the non-zero cells in column h of C ). m phyh PMh "c t t , t h Mh t

A

(18)

B

phyh pm (Mh)1~pm , s3BM. mh " c t t t h t,s sh ps t

(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 kh results in the "rst-order condition for optimal capital stock in sector h: t`1

AB

nh a ~t t t ¸s(1~t) t P P t t "dE

G A B

A

~t a ph yh t`1 ¸s(1~t) t`1 t`1 a #(1!k )nh t`1 h h t`1 kh P t`1 t`1 t`1 1

t P

BH

.

(20)

Finally, the model is closed by specifying market clearing for good h yh"yh . t d,t

(21)

Simple manipulation of "rst-order conditions reveals that consumers' total income is independent of share holdings and is given by M (22) a " + ps ys(a #b )!(ks !(1!k )ks )ns . t`1 s t t s t t s t s/1 Dexnition: A perfectly competitive equilibrium consists of shocks vectors Me N= , t t/0 price vectors Mp ,n , pnN= , and quantity vectors Mk , n , M , c , i , y N= such that t t t t t t t/0 t t t t/0 1. productivity levels MA N= follow their log-autoregressive laws of motion t t/0 subject to shocks Me N= , t t/0 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

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steady state values with "rst-order Taylor Series expansions. A formal justi"cation 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 XK and XK , where K denotes a variable's percent t t`1 deviation from its steady-state value and XK ,Ma( ,p( i ,AK j,kK j NM~1,M. This implies t t t t t i, j/1 3M state variables.8 The model then has a state space representation given by XK "PXK where P is a 3M]3M matrix that depends on the parameters of t t~1 the model, including the input-use matrix C . m 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 pvh vh"phyh!PMh Mh.

(23)

Aggregate nominal value added is simply the sum over all sectoral nominal value added amounts: pvv,+M pvi vi. Straightforward manipulation reveals i/1 that pvv"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 indices9 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.

8 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!1 when the coe$cient controlling the consumer's risk aversion t equals zero since then a does not appear in the "rm's intertemporal t "rst-order condition for optimal choice of k . t`1 9 See Sato (1975) for the relative merits of the Divisia method.

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3. Parameterization The parameters that must be speci"ed to complete the model are summarized below: f Preferences: p,q,t,d,s,Mm NM h h/1 f Production: M,Mk ,a ,b ,c NM ,C ,C ,g,p , m h h h h h/1 m I f Shocks: Mo NM ,X. h h/1 3.1. Preference and production parameters Most of these parameters are set to empirical estimates from the US economy. For simulations in Section 5, the level of disaggregation is set to M"36. This conforms closely to the 2-digit SIC level, the sectoral de"nitions used by Jorgenson and Fraumeni (1987). The production parameters, a ,b , and c are h h 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 intermediate inputs are calculated for 36 sectors of the private US economy using annual data from 1948 to 1985 by dividing the cost of inputs at producer prices by the value of output at producer prices (hence, this assumes perfect competition and constant returns to scale). The Cobb}Douglas coe$cients a,b, and c used in the simulations are the time-series averages from these calculations. The share c is divided across all interacting sectors using the fraction that the h purchases from these sectors represent out of total intermediate purchases by sector h. The mean value of a across sectors in the simulations is 0.16, the mean h value of b is 0.32, and the mean value of c is 0.52. Table 1 lists these parameter h h values along with the sectoral de"nitions. The 1977 detailed intermediate input-use matrix is used to parameterize C . m This matrix can be aggregated up to various useful levels. In particular, law of large numbers properties of the model are explored in Section 4 using the matrices C , C , C , and C containing 77, 36, 21, and 6 sectors, m,77 m,36 m,21 m,6 respectively. Simulations in Section 5 use C . m,36 Data for the investment-use matrix C come from the capital #ow table from I 1977, presented and described in Silverstein (1985). The capital #ow table was converted to C for 77, 36, 21, and 6 sectors by aggregating as with C and I m 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, d, is chosen to be (1.03)~0.25 implying an annual discount rate of 3%; the sectoral depreciation rates, k , are those used h in Jorgenson and Fraumeni (1987), and are given in Table 1.10

10 I thank John Fernald for kindly providing the k's.

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The parameters q and s control the volatility of hours and the fraction of time spent working in steady state, respectively. The parameter q is estimated from the Jorgenson data on annual sectoral outputs, labor inputs, and labor share coe$cients as follows. Manipulating (17) and (3) yields (24) nh"(bh)q@(q`1)(1!¸ ) t t t where bh is the fraction of labor's share of aggregate output accumulating to t labor in sector h: b phyh h t t . bh, (25) t +M b psys s/1 s t t Expressing (24) in percentage changes (denoted by K above variables) and adding an estimation error term results in the M estimation equations q bK h#errorh h"1,2, M. (n( h!+bs n( s)" t t~1 t t q#1 t

(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 q/(q#1). Estimating (26) by SUR with q restricted to be identical across sectors results in a point estimate of 0.9996 with a standard error of 0.0027.11 Hence a value of q"1.00 is 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.12 The uncompensated labor supply elasticity implied by q"1 is 0.5. It should be noted that, since q(R, wage di!erences across sectors persist in equilibrium. For comparison purposes, simulations are also presented below for the case q"100 and q"2. The parameter s is set so that total hours spent working in steady state represent one-third of the worker's total time endowment. Contingent on q"1,2 and 100, this requires s"13.4,7.1 and 1.93, respectively. The elasticity parameters p , p, g, and t are worth considering in terms of m their respective e!ects on output, consumption, investment and labor volatility. The value of p determines the elasticity of substitution among intermediate m inputs for all sectors. It is hypothesized that lower values of p engender greater m sectoral comovement and hence greater aggregate output volatility by reducing the ability of sectors to &avoid' the shocks of their input supplying sectors. The

11 Constant and trend terms were included in the regression, but results are largely the same if they are not included. 12 See Ashenfelter and Altonji (1980) and Altonji (1982) for an extended discussion and further references.

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parameter p controls the degree of substitutability between di!erent consumption goods. A larger value of p implies goods are more substitutable and therefore consumption of a given good is more responsive to variations in its relative price. The parameter g a!ects sectoral investment demand for a given investment good in an analogous fashion. The parameter t controls both the degree of risk aversion and the degree of intertemporal substitution. As t increases, 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 t"1 preferences are logarithmic in the consumption and leisure aggregates. While it would be desirable to calibrate p , p, g, and t to some empirical m regularities of the US economy, there is little to guide such an exercise. Intuition and precedence suggest p 41 and t51 while even less information guides m selection of reasonable ranges for p and g. Accordingly, the following baseline set of parameter values are used: p "1.00, p"1.00, g"1.00, t"1.00. All m 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 demand intensities as follows. At p"1, the nominal consumption expenditure share of sector h in total consumption is constant and is given by m"phch/+pscs. Consumption expenditure shares are calculated for the 36 sectors using consumption data from the National Income and Product Accounts. Shares are averaged over the period 1959}1995. Values for m calibrated in this manner are reported in Table 1.13 3.2. The stochastic process for shocks Completing the parameter set requires values for Mo NM and X. Employing h h/1 the Jorgenson dataset used to construct a , b , and c a sectoral productivity h h h series could be constructed for each sector as the residual of outputs minus weighted factors inputs according to (6) (in logs). ln(Ah),ln(yh)!a ln(kh)!b ln(nh)!c ln(mh). t t h t h t h t

(27)

13 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 disproportionate share of consumption expenditure in the data.

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Values for ln(Ah) could be "t to (9) yielding regression estimates for Mo NM and t h h/1 residuals, MehNM , that could be used to calculate X. t h/1 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).14 In particular, the variance of productivity shocks is likely to be reduced once they have been `correcteda 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 ki that enters into the production function given in (6) and that this #ow is linearly proportional to energy usage. They recommend correcting ln(Ah) with data on sectoral energy usage as t follows: ln(Ah),ln(yh)!a ln(eh)!b ln(nh)!c ln(mh), t t h t h t h t

(28)

where eh denotes the time t level of energy usage in sector h. t In the present analysis, correcting for capital utilization is important because it is likely that the sectoral covariance properties of e di!er with the t 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 movements in capital utilization across sectors. Therefore, failing to correct for varying capital utilization would overstate the cross-sector correlation in sectoral total factor productivity growth. This indeed turns out to be the case. The average pairwise correlation (excluding the diagonal) among the e from (27) is 0.14 while from (28) it is 0.09 suggesting 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 X. To pin down the structure of X, further assumptions on e are needed. Suppose that shocks to sector h were the sum of idiosyncratic shocks, gh, and aggregate shocks, l as in t t eh,gh#b l , h"1,2, M, t h t t

(29)

where b controls the response of sector h to aggregate shocks, E(g g@ )"A h t t for a diagonal matrix A, and E(l2)"p2. It will not be possible to decompose l t e as in (29) without additional restrictions on the model. For example, t 14 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.

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normalization of (29) for each h will not help much since there are a total of 2M#1 unknowns.15 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 X relative to the o!-diagonal elements indicate the size of uncorrelated sectoral shocks relative to aggregate shocks. Let X denote the diagonal matrix of X, setting o!-diagonal $ elements to zero. Results will be reported for simulations based on both X and $ X. 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 aggregate 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 X is given by X "A #b2p2, which contains i l ii ii 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 X 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 X as $ under X. The values for o and the square roots of the diagonal elements of X are h reported in Table 1. The mean value for o is 0.93 and that of JX is 0.022; h $ however, there is considerable sectoral variation. It should be noted that eh are shocks to gross output in sector h. They must be scaled up by 1/(1!c) 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 0.0078.16 Statistics on aggregate total factor productivity are reported in the discussion of the simulation results. Table 1 tabulates 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

15 Developing an identi"cation scheme for (29) is beyond the scope of this paper but is the subject of Horvath and Verbrugge (1999). 16 King et al. (1987) report and use this standard deviation for aggregate technology shocks.

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Table 1 Baseline parameter values for the 2-digit SIC (M"36) model economy! Aggregate Sector

c

Agricultural Products Agricultural Services Metal Mining Coal Mining Petroleum and Natural Gas Nonmetallic Mining Construction Food and Kindred Products Tobacco Products Textile Mill Products Apparel Paper Printing and Publishing Chemicals Petroleum and Coal Products Rubber and Misc. Plastics Leather Lumber and Wood Furniture and Fixtures Stone, Clay, Glass Primary Metal Fabricated Metal Non-elec. Machinery Elec. Machinery Motor Vehicles Transportation Equipment Instruments Misc. Manufacturing Transportation Services Communication Services Electric Utilities Gas Utilities Wholesale and Retail Trade Finance, Insurance, and Real Estate Water and Sanitary Services Other Services

0.57 0.44 0.59 0.43 0.52 0.39 0.57 0.77 0.51 0.64 0.64 0.51 0.57 0.61 0.50 0.60 0.78 0.55 0.61 0.51 0.66 0.52 0.49 0.53 0.61 0.67 0.36 0.55 0.59 0.21 0.24 0.41 0.30 0.56 0.61 0.24

d"0.993, t"1.0, q"1.0, s"13.4 p"1.0, g"1.0, p "1.0 m b a k o 100] JX $ 0.31 0.13 0.01 0.91 3.60 0.45 0.12 0.01 0.91 3.60 0.26 0.15 0.02 0.95 5.26 0.41 0.16 0.02 0.99 3.94 0.13 0.35 0.02 0.97 6.18 0.33 0.27 0.02 0.95 3.26 0.37 0.06 0.04 0.98 0.80 0.16 0.06 0.02 0.64 1.39 0.20 0.29 0.02 0.96 2.34 0.28 0.08 0.02 0.88 1.81 0.32 0.04 0.02 0.99 0.63 0.35 0.14 0.02 0.93 2.49 0.36 0.07 0.02 0.88 1.38 0.25 0.14 0.02 0.94 1.45 0.39 0.11 0.02 0.98 0.80 0.22 0.18 0.02 0.96 2.37 0.11 0.11 0.01 0.96 6.35 0.36 0.10 0.02 0.95 1.67 0.34 0.05 0.02 0.89 2.99 0.35 0.14 0.02 0.96 0.93 0.23 0.11 0.02 0.92 1.66 0.38 0.09 0.02 0.90 1.02 0.39 0.13 0.02 0.91 1.32 0.37 0.10 0.02 0.96 0.97 0.34 0.05 0.02 0.93 2.32 0.20 0.13 0.02 0.90 1.91 0.50 0.14 0.02 0.84 1.59 0.36 0.09 0.02 0.90 2.34 0.27 0.14 0.01 0.97 1.55 0.43 0.36 0.02 0.98 2.17 0.52 0.25 0.02 0.98 1.54 0.21 0.38 0.01 0.98 2.74 0.54 0.16 0.02 0.93 1.26 0.21 0.23 0.01 0.97 1.42 0.25 0.14 0.03 0.95 1.19 0.42 0.34 0.03 0.97 2.37

m 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.12 0.01 0.01 0.04 0.02 0.01 0.03 0.03 0.01 0.01 0.01 0.02 0.01 0.01 0.01 0.01 0.02 0.04 0.01 0.01 0.01 0.03 0.02 0.02 0.01 0.25 0.01 0.13 0.01

!Note: See text for calibration of these parameters and for descriptions of C and C . m I

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 k"1, t"0, p"1, C "I , and p "1. This permits analytical results I m m

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showing that the sparseness of C of a particular form results in greater m 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 C being characterized by few full rows and m mostly sparse columns. The full rows represent &key input' sectors, sectors that sell inputs broadly to many other sectors.17 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 predominantly full rows in C rather than on the rate of increase in the total number of m 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 JM 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 substitutability, 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.18 This section explores, through simulation, how disaggregation of C and m C a!ects the volatility of aggregate value added in the general model presented I above. Simulations are in order rather than analytical investigations for two distinct reasons. First, as mentioned above, analytical solutions for the model in Section 2 are 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 XK , t namely aggregate value added.

17 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. 18 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.

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The least aggregated matrix available is the detailed input-use matrix constructed by the Department of Commerce.19 The detailed matrix (denoted below as C ) is of dimension 523 and includes both 3- and 4-digit m,523 SIC de"nitions.20 Using the SIC de"nitions, this input-use matrix can be aggregated to various smaller dimensions. I performed aggregations resulting in matrices of dimension 77, 36, 21 and 6 sectors. The matrix C conforms m,77 to the 2-digit SIC de"nitions while C conforms to the 1-digit SIC m,6 de"nitions. Matrix C conforms to the sectoral de"nitions used by Jorgenson m,36 et al. while C is a slightly more aggregated version. Both have a mixture of m,20 1- and 2-digit industries. Capital-use matrices, C at each of these levels of I aggregation are constructed from the capital-use matrix at M"77 reported in Silverstein (1985). The raw input-use and capital-use matrices are then converted into share-weights for use in the model by dividing each column by its column sum. The general model is simulated using the empirically observed input-use and capital-use matrices with elasticity parameters at their baseline values.21 Since the emphasis of the experiment in this section is to illuminate the rate of decline in the variance of aggregate value added, all parameters are held "xed as M increases. In particular, values for o, k, a, b, and c for all M are set to the average values across the 36 sectors reported in Table 1. Preference weights m are set to (1/M)1@p. Finally, the variance}covariance matrix X is assumed to be diagonal with all diagonal elements equal to 0.01 for all M. Hence, if the law of large numbers applies exactly, then the variance of aggregate value added should decline exactly with M. Fig. 1 presents the results from this exercise. The aggregate output variance from simulations with M"21, 36, 77 is plotted relative to the aggregate variance at M"6, which is normalized to unity. The graph also plots curves for 1/M and 1/JM normalized to unity at M"6. Aggregate volatility declines slower than what is implied by the law of large numbers. At M"77, aggregate volatility in the model is twice that implied by the law of large numbers convergence rate. In fact, the rate of zero convergence in aggregate variance is slower than 1/JM when disaggregating from M"36 to M"77. This "nding is consistent with the decreasing rate of increase in the number of predominantly

19 I use the 1977 matrix in what follows. 20 The detailed input-use matrix is actual of dimension 537. However, the import, scrap, second-hand goods, government industry, government demand, and household sectors are dropped from the present analysis. The inventory valuation adjustment column and row is also removed. 21 The value of s required to achieve a one-third, two-third split between time spent working and leisure depends on M as well. When other parameters are set to their baseline values, s"M5.5,11.2,14.6,21.7N for M"M6,21,36,77N.

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Fig. 1. Simulated model versus LLN.

full rows in C in moving from M"36 to M"77 described in Horvath m (1998).22

5. Simulation results from the 2-digit SIC economy Two sets of results are reported in this section. The "rst set of results report simulated moments from the model 2-digit SIC economy (M"36) with elasticity parameters set to their baseline values. The second set of results report the sensitivity of the simulations to variations in the elasticity parameters t, p, g, p , m and q. In all cases, simulations involve 100 replications of a 120 period model economy. Data from the sectoral model are aggregated as described above and the aggregate data are detrended using the Hodrick}Prescott "lter. Tables report statistics calculated from the simulated data, averaged across 22 Presumably, the increasing sparseness of C may contribute to the postponement of the law of I large numbers well. However, results in Horvath (1998) are nearly identical to Fig. 1 even though C "I suggesting that the structure of C is less important to the LLN properties of the model. I m I Perhaps this is due to the fact that C a!ects #ows of goods into sectoral capital stocks which tend to I move slowly and hence have less impact on sectoral value added. Further explorations of the relative contributions of row sparseness in C versus C are beyond the scope of this paper. m I

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replications. The statistics reported are the standard set of business cycle moments: the standard deviation of value added (output), the relative (to output) standard deviations of consumption, investment, hours worked, real wages, and average labor product, the correlation of each of these variables with output, and the correlation of hours worked with wages and average labor productivity. The latter two correlation statistics have received considerable attention in recent work (Christiano and Eichenbaum, 1992) since the standard dynamic stochastic general equilibrium model with spot labor markets counterfactually predicts a high correlation between hours worked and wages and average labor productivity. 5.1. Baseline simulations Table 2 reports simulation results for the baseline parameter values under X , $ where only uncorrelated sectoral productivity shocks drive variations in sectoral output, and under X, where sectoral productivity shocks have the same variance covariance matrix as their empirical counterparts. Also reported on Table 1 for comparison purposes are statistics calculated from the actual US economy as well as simulation statistics reported in Hansen (1985) for his &divisible' labor one-sector economy.23 The latter results are included to permit comparison of simulation results from the multisector model with standard results from a one-sector model. The table also reports statistics from a version of the multisector model where transmission of sectoral productivity shocks via non-durable intermediate input #ows has been shut down, labeled no links in the table. In this case production is still modeled according to (6) but C is modi"ed m so that all sectors purchase all their intermediate goods from a single sector which is not subject to stochastic productivity variation.24 The statistics reported for quarterly, private-sector US data, 1947}1990, are based on standard data sources. Series are taken from Citibase: output is de"ned as real GDP less production associated with housing, government, and farms. Consumption is real private consumption by individuals on non-durables and services; investment is private, non-residential "xed investment on structures and durable equipment plus consumer expenditure on durable goods. Hours worked represent private establishment survey hours and hourly wages are

23 Under Hansen's divisible labor, workers can continuously chose how many hours to work taking wages as given. Under indivisible labor, workers can choose only whether or not to work a "xed number of hours. Comparisons with the indivisible labor case are not relevant to this paper since, in the multisector model, only divisible labor is considered. See also Rogerson (1988) for the theoretical background. 24 Though this is labeled no links in Table 2 the sectors are still &linked' via capital good #ows as well as labor and "nal goods markets.

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Table 2 Simulation results: baseline case! (a) Standard deviations Series

US Data

Model (X ) $

Model (X)

Hansen Divisible

Model (no links)

Output Consumption Investment Hours Real wage Avg. Labor Product

2.25 0.39 1.95 0.84 0.43 0.48

1.51 0.58 3.94 0.57 0.60 0.59

2.44 0.51 3.18 0.54 0.54 0.52

1.35 0.24 2.41 0.40 * 0.39

0.74 0.65 6.62 0.67 0.66 0.65

(b) Key Correlations Output Consumption Output Investment Output Hour Output Real Wage Output ALP Hour Real Wage Hour ALP

0.76 0.81 0.88 0.32 0.55 0.27 0.10

0.86 0.72 0.86 0.88 0.87 0.52 0.49

0.94 0.85 0.95 0.95 0.94 0.81 0.79

0.89 0.99 0.98 * 0.98 * *

0.72 0.61 0.76 0.76 0.75 0.16 0.15

!Note: Standard deviations of variables other than output are reported relative to the standard deviation of output. US Data columns present statistics from US private domestic economy based on quarterly data 1947}1990. Model columns present simulation results from multi-sector model using baseline parameter values. Under the model with X shocks in the are independent across $ sectors while under X o!-diagonal elements from the estimated variance covariance matrix of sectoral productivity residuals are not assumed to be zero. Hansen column reports results from Hansen (1985), divisible labor case. The column Model (no links) reports results from multisector model without non-durable intermediate links using the same innovations to sectoral productivity as under the column Model X . Simulation results are average statistics over 100 simulated $ economies of length 120 quarters. All data are "ltered with the HP "lter. See text for method of aggregating simulated sectoral data.

calculated as total compensation to private sector employees (including employer contributions to social security and bene"ts paid to workers) divided by total labor hours worked by private sector employees. Average labor product is calculated as output divided by hours worked. All data are "ltered with the Hodrick}Prescott "lter. The results show that the multisector model is quite capable of reproducing the stylized macroeconomic facts of economic #uctuations. Furthermore, comparisons with results in Hansen (1985) reveal that the multisector model generates aggregate statistics that are quite similar to those of a one-sector dynamic stochastic general equilibrium model. The model driven entirely by independent sectoral shocks generates a quarterly aggregate output standard deviation of 1.51%. This comes from the aggregation of real value added in the model's 36

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sectors which have a weighted average quarterly standard deviation of 3.20%.25 Absent sectoral interaction that gives rise to comovement among the sectors, the aggregate value added standard deviation would be expected to be on the order of 3.20%/JM or 0.54% per quarter. Indeed, the last column of the table indicates that simulations of the model where sectoral shocks do not travel via C generate aggregate output volatitility of only 0.74%. Instead, the model with m intermediate good #ows following C generates aggregate volatility roughly 36 twice this "gure. While aggregate output volatility under X still falls short of the empirically $ observed output volatility, it exceeds the volatility generated by Hansen's divisible labor model. Of course, this may simply be due di!erences in the parameterization of Hansen's aggregate Solow residual volatility and the amount of volatility driving the multisector model. One way to check that this is not the case is to compare Hansen's aggregate Solow residual volatility with estimates of the same from the multisector model. Hansen parameterizes the aggregate Solow residual standard deviation at 0.00712. The average across simulations of the multisector model (X case) underlying Table 2 is only slightly $ higher at 0.0079 indicating that the output volatilities from Hansen's model and the multisector model can be directly compared. The excess output volatility in the multisector model is not coming at the expense of counterfactually high measured aggregate Solow residual volatility. The relative volatilities of other variables are roughly in line with empirical observation: Output is roughly twice as volatile in simulation as consumption. Investment is too volatile, more than three times as volatile as output. Hours volatility is still too low relative to output volatility, yet it exceeds the relative hours volatility generated by Hansen's divisible labor case. Furthermore, wages and average labor productivity are su$ciently smooth. The correlations with output, investment, hours, wages, and average labor productivity also match the empirical statistics quite well. The relative smoothness of wages and average labor productivity in the multisector simulations are due to an aggregation bias. In each sector, wages equal labor marginal product and the volatility of wages and average labor productivity on the sectoral level is roughly two-thirds that of output. But since wage payments are not as highly correlated across sectors as either output or labor hours, aggregation tends to eliminate much of the variation in wages while it eliminates less of the variation in output and hours. In this manner aggregate wages appear to be about half as volatile as aggregate output. Since average labor product is the ratio of output to hours worked in each sector, and sectoral output and hours worked are highly correlated, aggregate labor product is smoothed through aggregation in a similar fashion. The empirical data from the

25 The weights used in this average are the steady state share of total value added for each sector.

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US economy corroborate this "nding. Table 3a presents estimates of the relative volatility of wages and hours at the sectoral level based on annual data in the Jorgenson dataset. The ratio of the standard deviation of total wages to total hours worked at the aggregate level is 0.41 whereas the average of this ratio across all sectors is 0.98 with a standard deviation of 0.52: wages appear to be more volatile relative to hours at the sectoral level than at the aggregate level. Table 3b of the table presents the same statistics from the simulated data. Wages at the sectoral level are, on average, much more volatile relative to hours than they appear to be at the aggregate level though the cross-sectoral variation is large. With regard to the correlation between labor market variables, the model with independent sectoral shocks is able to match the data better than the one-sector models considered by Christiano and Eichenbaum (1992). Hours and wages have a contemporaneous correlation of only 0.52 in the model under X compared with 0.27 in the data. Hours and average labor productivity have $ a correlation of 0.49 in the model compared with 0.10 in the data. The standard one-sector models generate correlations between these variables near unity. Table 3, reveals that the one-sector model is at odds with the sectoral-level data as well. In Table 3a, the cross-sector average hours-wages and hours-average labor productivity correlations in US data are strongly negative, !0.37 and !0.22 respectively. Therefore, the data seem to indicate that aggregation biases tend to increase, rather than decrease, the correlation of labor market variables. Table 3b reveals that the same is true for the simulated data. In the sectoral model, under the baseline parameterization, average labor productivity di!ers Table 3 Sectoral and aggregate labor market composition biases!

Aggregate

Sectoral Average

Sectoral Std. Dev.

(a) US Data Std. Dev Wages/Std. Dev. Hours Corr(Hours,Wages) Corr(Hours,ALP)

0.41 0.27 0.10

0.98 !0.37 !0.22

0.52 0.43 0.32

(b) Simulated data Std. Dev Wages/Std. Dev. Hours Corr(Hours,Wages) Corr(Hours,ALP)

1.05 0.52 0.44

10.44 0.20 0.20

15.03 0.32 0.32

!Note: Panel (a) reports statistics from US data. Aggregate statistics are calculated from data used to calculate statistics in Table 2 as described in text. Sectoral averages and standard deviations are calculated from the Jorgenson dataset "ltered with the HP "lter. Panel (b) reports statistics from HP "ltered, simulated data under X (uncorrelated sectoral shocks). In both panels, sectoral wages are $ real product wages.

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from sectoral real product wages only by a constant. Therefore, the cross-sector average correlation of (log) hours and wages matches that of hours and average labor productivity at 0.20. These correlations at the aggregate level from the simulations are signi"cantly higher, around 0.50. Sectoral labor markets in the multi-sector model do not replicate the labor market results of the standard one-sector model but do conform closely to their empirical counterparts. Eq. (4) indicates the reason. Sectoral labor supply is upward sloping in sectoral wages but depends negatively on wealth and the leisure aggregate. Increases in the real wage in a sector may be the result of shocks that have a substantial positive e!ect on wealth due to the transmission of these shocks to other sectors. The result is only limited movement in hours in response to these movements in wages. Three additional results not reported in Table 2 are interesting as well. First, the model under X generates aggregate Solow residual series that are quite $ similar to that observed in the data even though there are no persistent aggregate shocks to productivity. The standard deviation of aggregate productivity growth in the simulated economy is 0.79% or roughly half that of aggregate value added growth, as is the case in US data. The interpretation of this "nding is that empirically observed shocks to aggregate multifactor productivity (i.e., Solow residuals) may simply be an artifact of aggregation of the e!ects of sector-speci"c shocks. Second, statistics on the autocorrelations in "rst di!erences of the data are discouraging and are not reported. Positive autocorrelations in "rst di!erences would suggest the model is capable of generating business cycles in the sense that positive (negative) output growth in one period is often followed by positive (negative) output growth in the next period. Estimates on empirical data place the autocorrelation in growth rates of output around 0.52. In the model the statistic is slightly negative and not statistically di!erent from zero. Essentially, the log-linear approximation of the multi-sector model results in an vector autoregression with lag order 1. Finally, simulations exhibit the ability of the model to turn uncorrelated sectoral productivity shocks into correlated output movements. The mean pair-wise correlation of sectoral productivity shocks is 0 by construction while the mean pair-wise correlation between sectoral value-added is 0.032. Clearly this "gure is still too low to match empirical observations on sectoral comovement. However, adding a small amount of aggregate shock comovement improves on this score without altering the positive results reported in Table 2. When the o!-diagonal elements of X are not assumed to be zero, the model generates more output volatility, as expected. Table 2, column Model (X) reports simulation results in this case. The multi-sector model generates aggregate time series that have second moments that lie close to empirically observed aggregates. The average pairwise correlation in the sectoral productivity shocks is 0.08 and consequently, comovement at the sectoral level is substantially higher with

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an average pair-wise output correlation of 0.17. This compares favorably to statistics calculated from the US economy at a disaggregated level.26 Table 4 reports empirical and simulation estimates of sectoral value added volatility at the 2-digit SIC level of disaggregation. Simulation results are reported under both X and X. Overall, the model simulations generate more $ sectoral volatility than is found in the data.27 The excess sectoral volatility is most severe in the agriculture and mining sectors as seen in Table 5. Table 5 reports the same statistics aggregated to the 1-digit SIC level. While the match in each case is not exact, the broad facts of sectoral volatilities are captured by the model. Service sectors have relatively low volatility while manufacturing and mining sectors have relatively high volatility. The model counterfactually predicts a high volatility in the agricultural sectors and generates roughly the same volatility in durable manufacturing as non-durable manufacturing though in the data the latter is signi"cantly less volatile. 5.2. Sensitivity to parameter values Tables 6}9 present statistics from simulations under alternate parameter values. In all four tables, only independent sectoral shocks drive #uctuations, as in Table 2 under ) . Table 6 presents results for alternate values of t. Table $ 7 considers variations in p and g. Table 8 considers variations in p . Table m 9 considers the case of q"100 and q"2. The results are as expected. In Table 6, raising t from 1.0 to 2.0 decreases the volatility of output, hours, investment, and consumption as workers become less willing to substitute intertemporally.28 Lowering t to 0.8 has the opposite e!ect and reduces the correlation of output and consumption considerably. In Table 7, raising p and g to 1.1 decreases the volatility of output and the relative volatilty of investment. This is due to the increased substitution possibilities in the production of capital goods with a higher degree of investment input substitution. Lowering these parameter values to 0.9 increases the volatility of all variables and reduces the output correlation of consumption, real wages, and average labor product. In fact, the simulation results under p"g"0.9 conform even closer to the empirically observed values from US data reported in Table 2. Notably, the ratio of hours to wage volatility rises to 1.5, close to the value of 2.0

26 See Shea (1994). 27 The comparison between the simulated statistics and the data is made murky by the lack of quarterly series on US sectoral value added. The Jorgensen series are annual series. Approximate quarterly standard deviations are calculated by dividing the annual variance by four. 28 Recall that what is being reported in the table is the volatilities of all variables (except output) relative to the volatility in aggregate output. This must be taken into account when considering the e!ects of changes in t.

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Table 4 Simulation results: 2-digit SIC sectoral volatilities! Sector

US Data

Model (X ) $

Model (X)

Agricultural Products Agricultural Services Metal Mining Coal Mining Petroleum and Natural Gas Nonmetallic Mining Construction Food and Kindred Products Tobacco Products Textile Mill Products Apparel Paper Printing and Publishing Chemicals Petroleum and Coal Products Rubber and Misc. Plastics Leather Lumber and Wood Furniture and Fixtures Stone, Clay, Glass Primary Metal Fabricated Metal Non-elec. Machinery Elec. Machinery Motor Vehicles Transportation Equipment Instruments Misc. Manufacturing Transportation Services Communication Services Electric Utilities Gas Utilities Wholesale and Retail Trade Finance, Insurance, and Real Estate Water and Sanitary Services Other Services

2.51 2.36 5.11 3.35 3.12 3.87 2.26 2.35 2.82 3.11 2.07 3.32 3.09 2.48 1.76 2.56 5.94 3.80 2.88 3.50 3.69 3.37 4.19 3.75 7.49 7.37 2.94 3.11 3.04 2.44 2.35 5.12 1.59 1.44 1.60 3.37

6.00 4.62 7.83 5.89 8.85 4.47 2.81 3.11 4.03 3.26 1.54 4.55 2.25 2.73 3.57 3.46 10.72 3.34 3.85 1.78 3.90 2.29 4.51 3.15 6.37 3.37 3.81 4.04 2.67 2.95 2.01 4.33 1.84 3.12 2.07 3.10

6.24 5.15 8.18 6.34 8.96 5.13 3.95 3.00 3.86 3.48 1.72 4.38 2.23 3.62 4.22 4.89 11.67 3.98 3.80 3.02 5.49 3.56 5.67 4.18 7.55 3.82 4.33 5.38 3.17 2.72 1.98 4.98 2.12 2.79 1.98 3.21

Mean

3.31

3.95

4.47

!Note: US Data columns present sectoral value added standard deviation statistics from based on the Jorgenson dataset as described in the text, "ltered with the HP "lter. Model columns present simulated sectoral value added standard deviations from multi-sector model using baseline parameter values, also "ltered with the HP "lter. Under the model with X , shocks are independent $ across sectors while under X o!-diagonal elements from the estimated variance covariance matrix of sectoral productivity residuals are not assumed to be zero.

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Table 5 Simulation results: 1-digit SIC sectoral volatility averages! Sector

US Data

Model (X ) $

Model (X)

Agriculture Mining Manufacturing Nondurable Durable Transportation Services

2.44 3.86 3.60 2.95 4.19 3.04 2.56

5.31 6.76 3.79 3.92 3.67 2.67 2.78

5.70 7.15 4.47 4.31 4.62 3.17 2.83

!Note: Results represent aggregation of sectoral value added standard deviations in Table 4 to 1-digit SIC sectoral de"nitions.

Table 6 Alternate parameter values: varying t Standard deviation

Output correlation

Series

Baseline!

t"2.0

t"0.8

Baseline

t"2.0

t"0.8

Output Consumption Investment Hours Real wage Avg. Labor Product

1.51 0.58 3.94 0.57 0.60 0.59

1.50 0.58 3.38 0.44 0.60 0.58

1.57 0.58 4.17 0.68 0.59 0.58

1.00 0.86 0.72 0.86 0.88 0.87

1.00 0.98 0.68 0.97 0.98 0.98

1.00 0.74 0.75 0.82 0.78 0.75

!Note: Baseline column repeats statistics reported in Table 2. Other columns report statistics under alternate parameter values as shown. All simulations use X which assumes uncorrelated sectoral $ shocks to aggregate productivity.

found in the US data. Limitations to intermediate good substitutability make the aggregation biases discussed above in regards to Table 2 more severe. Table 8 reveals that lower values of p do indeed engender more output m volatility. Also lower values of p generate relatively more volatility in hours m and investment and less volatility in wages and average labor productivity. The increased output volatility is to be expected with lower degree of substitutability in intermediate goods since each sector becomes more tied to their input suppliers and the shocks they receive. The average pairwise correlation increases from 0.032 in the baseline case to 0.045 for p "0.8. Lower values of p also m m reduce the output correlation of consumption, investment, real wages, and average labor product when other parameters remain at their baseline values. Table 9 shows that changing leisure preferences so that wage di!erences largely do not exist across sectors (q"100) lowers the volatility of output and

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Table 7 Alternate parameter values: varying p and g Standard deviation

Output correlation

Series

Baseline!

p,g"1.1

p,g"0.9

Baseline

p,g"1.1

p,g"0. 9

Output Consumption Investment Hours Real wage Avg. Labor Product

1.51 0.58 3.94 0.57 0.60 0.59

1.46 0.62 3.20 0.63 0.63 0.63

1.83 0.50 5.31 0.80 0.51 0.50

1.00 0.86 0.72 0.86 0.88 0.87

1.00 0.79 0.75 0.79 0.81 0.79

1.00 0.58 0.79 0.86 0.66 0.61

!Note: Baseline column repeats statistics reported in Table 2. Other columns report statistics under alternate parameter values as shown. All simulations use X which assumes uncorrelated sectoral $ shocks to aggregate productivity.

Table 8 Alternate parameter values: varying p ! m Standard deviation

Output correlation

Series

Baseline!

p "1.2 m

p "0.8 m

Baseline

p "1.2 m

p "0.8 m

Output Consumption Investment Hours Real wage Avg. Labor Product

1.51 0.58 3.94 0.57 0.60 0.59

1.49 0.80 5.58 0.99 0.79 0.80

1.69 0.49 5.65 0.73 0.51 0.49

1.00 0.86 0.72 0.86 0.88 0.87

1.00 0.40 0.60 0.68 0.45 0.41

1.00 0.67 0.54 0.88 0.76 0.72

!Note: Baseline column repeats statistics reported in Table 2. Other columns report statistics under alternate parameter values as shown. All simulations use X which assumes uncorrelated sectoral $ shocks to aggregate productivity.

the relative volatility of hours worked. As evident from (3), a high value of q means that hours worked in di!erent sectors are nearly perfect substitutes as far as the worker's leisure preferences are concerned. It also increases the relative volatility of consumption, investment, wages, and average labor product. This result obtains despite the fact that, by (24), raising q would seem to #atten the sectoral supply curve thereby making hours worked and presumably sectoral output more volatile. In fact, as q approaches in"nity labor hours become perfect substitutes as far as the worker is concerned (consider (3) and the discussion following). Minor wage di!erences between two sectors due to

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Table 9 Alternate parameter values: varying q Standard deviation

Output correlation

Series

Baseline!

q"100 s"1.93

q"2 s"7.1

Baseline

q"100 s"1.93

q"2 s"7.1

Output Consumption Investment Hours Real wage Avg. Labor Product

1.51 0.58 3.94 0.57 0.60 0.59

1.43 0.56 5.75 0.50 0.66 0.64

1.52 0.56 4.20 0.58 0.59 0.58

1.00 0.86 0.72 0.86 0.88 0.87

1.00 0.72 0.63 0.84 0.91 0.90

1.00 0.83 0.73 0.86 0.87 0.86

!Note: Baseline column repeats statistics reported in Table 2. Other columns report statistics under alternate parameter values as shown. All simulations use X which assumes uncorrelated sectoral $ shocks to aggregate productivity.

sector-speci"c productivity shocks cause large but opposite movements in labor hours in the two sectors as the worker allocates more time to the sector paying higher wages. Consequently, though sectoral hours and output become more volatile with higher q, aggregate hours and output become smoother. Wages become more correlated with higher q and therefore move more relative to output. This results suggests that an intermediate value of q would increase labor hours volatility by #attening sectoral labor supply curves without engendering negative correlation in labor hours through (3). When q"2 in Table 9 this is indeed the case. The volatility of output and the relative volatility of hours rises compared to the baseline case.

6. Conclusions The long history of business cycle theory has tried to explain the persistent aggregate #uctuations observed in macroeconomic time series. The major contribution of this paper to the existing theory is the conclusion that independent sector-speci"c disturbances may have signi"cant aggregate e!ects, as envisioned by Long and Plosser (1983). The model speci"cation began by questioning the aggregate shock models because aggregate shocks with the required persistence and volatility properties are hard to observe in the data. At the end of the analysis, has a plausible alternative been found to explain persistent business cycles and sectoral comovement? I believe that the answer is yes. By the simulation results in Section 5, this multisector model does equally well in quantitatively matching the data compared with one-sector models. Moreover, the model generates an

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explanation for empirically observed volatility in multifactor productivity growth without appealing to aggregate disturbances. If the model generates roughly the same aggregate results as a one-sector model, why go to all the trouble to model these sectoral interactions? I believe that there are two reasonable conjectures to o!er in response to this question. First, this paper establishes that one sector models are not such a bad approximation to reality, so long as economists do not interpret the one-sector shocks as &real' aggregate shocks. When one aggregates to the one sector level, real sectoral shocks get aggregated into the appearance of aggregate shocks. The second response concerns optimal policy. The optimal policy in response to variations in aggregate variables in the one sector model may be very di!erent from the optimal response in the multi-sector model, largely because the set of policy instruments is much richer in the multisector model.

Appendix A. Derivation of 5rst-order conditions for multisector model A.1. Consumer's problem The representative consumer seeks to maximize his discounted, time separable, utility stream subject to a budget constraint, given in (1). Letting h be the t Lagrange multiplier associated with the consumer's budget constraint, di!erentiating (1) with respect to ch results in the following: t C(c )1@p~t(ch)~1@pmh¸~s(1~t)"phh . (A.1) t t t t t Manipulating (A.1) (isolate ch and build up the aggregate consumption index t from this) results in an expression for the Lagrange multiplier in (A.2). (A.2) h "C(c )~t¸s(1~t)P~1. t t t t Then it is trivial to substitute (A.2) into (A.1) to achieve the consumption demand equation in the text. Labor supply equation is derived by di!erentiating (1) with respect to hours worked and substituting in for the Lagrange multiplier using (A.2). A.2. Firm's problem Capital accumulation is accomplished through the investment process described by (10) where the composite investment good has the form in (11). Consider spending zh"+M I ps ih on investment-related demand for other s|Bh t t,s t sector's goods. Then the investment demand equation in the text is derived by maximizing (11) subject to the constraint that expenditure is equal to zh. Let t /h be the Lagrange multiplier associated with this constraint for sector h. This t

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gives rise to investment-related demands of the form in (A.3). g(ih)1@gNI~1@g(ih )~1@g"ps /h. t t t,s h t

(A.3)

Performing analogous manipulations to those used in deriving consumption demands, the Lagrange multiplier in (A.3) can be written as in (A.4). /h"1/nh t t

(A.4)

and substitution into (A.3) yields the investment-related demand for sector h. Derivation of labor supply and intermediate demand is achieved by straightforward di!erentiation of (14) with respect to nh and mh , respectively. Derivat,s t tion of the intertemporal Euler equation for optimal capital stocks is achieved by di!erentiating (13), substituting in (14) and (6), with respect to kh . Goods t`1 market clearing for sector h is achieved by substituting in for consumption demand, investment-related demand and intermediate good related demand.

Appendix B. Log-linearization and state space solution The basic principle (Christiano, 1988) is to log-linearize all equilibrium equations with a "rst-order Taylor series expansion around the model's steady state, and reduce the system of equations to correspond to the state space described in the text. Log-linearization replaces a variable x with its percentage t deviation from steady state, denoted x( which is equal to log x !log x5 . The t t notation x denotes the time t expectation of the variable x . t`1@t t`1 The system of linearized equations can be represented by two sub-systems. The "rst solves for the 3M vector s( ,(y( @ n( @ m( @ )@ in terms of the state and co-state t t t t variables XK ,Ma( ,p( H,AK ,kK N where p( H is the M!1 vector constructed, without t t t t t t loss of generality, by removing the "rst element of the vector p( :29 t s( "M XK . t sX t

(B.1)

The linearized intertemporal equations can be written as in30 A XK #A s( "B XK #B s( #De . X t`1 s t`1 X t s t t

(B.2)

29 Recall that solutions for only M!1 prices are needed since the price in sector 1 (without loss of generality) can always be normalized to unity. 30 Recall that, by Walras' Law only M!1 good market clearing equations are represent in the system.

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Substitution of (B.1) into (B.2) results in (B.3). AXK "BXK #De , t`1 t t A,A #A M , X s sX B,B #B M . X s sX

(B.3)

The "rst-order expectational di!erence equation that yields the behavior of the system around steady state is given by XK "PXK , t`1@t t P,A~1B.

(B.4)

At any time, t, the M unknown elements in XK are its "rst M elements as it is t ordered above: XK u,(a pH{)@. If a unique, stable equilibrium path exists from t t t any non-steady-state point X to the steady state of the system X then the t matrix P has M eigenvalues greater than one in modulus. Let the eigenvectors associated with these &explosive' eigenvalues be (c21 c22) where c22 has dimension M]2M and c21 has dimension M]M. The solution for XK u in terms of the t known values XK k of the vector XK is then given by t t XK u"c~1c XK k. 21 22 t t

(B.5)

This completes the solution method for the model.

Appendix C. Aggregation The growth (in continuous time) of aggregate value added, dv, equals a weighted average of sectoral value-added growth as in M dv, + wi dvi, (C.1) i/1 where wi is no longer the wage paid in sector i, but instead is the ith sector's share of nominal value added: M wi,pvi vi/ + pvi vi. (C.2) i/1 Sectoral value-added growth in sector h is de"ned as a Divisia index of sectoral output growth minus total sectoral intermediate input growth as in phyh PMh Mh dvh, dyh! dMh. pvh vh pvh vh

(C.3)

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Eqs. (C.1), (C.2), and (C.3) perform the necessary transformations from data generated by the model speci"ed above to aggregate value added.31 It is also of primary interest to assess the role of sector speci"c shocks in generating movements in aggregate multi-factor productivity. The residual accounting method of Solow (1957) is consistent with the accounting method employed in the construction of aggregate value added. De"ne aggregate labor and capital as the sum of these quantities across all sectors: M n" + ni, i/1 M k" + ki. i/1 Aggregate productivity growth, dp, is simply aggregate value added growth minus suitably weighted growth rates of capital and labor inputs. The assumptions of constant returns to scale and perfect competition make easy the choice of de"ning these weights for primary inputs to the production of sectoral value added: revenue-based shares are equivalent to cost-based shares. Eq. (C.4) gives these words in mathematical form: dp,dv!s dk!s dl, (C.4) K L where s ,+M wagei ni/pvv (here wagei denotes wages in sector i) and i/1 n s "1!s . k n In the simulations reported in Section 5, the weighted sum of sectoral productivity growth is roughly 10% less variable than aggregate productivity growth. The source of the added volatility in aggregate productivity growth stems from capital and labor reallocation following sector-speci"c technology shocks. While these contributions are small, they are commensurate with the measured reallocation terms in Basu and Fernald (1997). Basu et al. (1999) explore the roles played by monopolistic competition and increasing returns to scale in adding to variability in aggregate productivity growth beyond the contribution coming from sectoral productivity shocks. References Altonji, J.G., 1982. The intertemporal substitution model of labour market #uctuations: an empirical analysis. Revue of Economic Studies 49, 783}824. Ashenfelter, O., Altonji, J.G., 1980. Wage movements and the labour market equilibrium hypothesis. Econometrica 47, 217}245.

31 Levels of variables generated by the model are converted to growth rates by "rst-di!erencing their natural logarithms.

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Bak, P., Chen, K., Scheinkman, J., Woodford, M., 1993. Aggregate #uctuations from independent sectoral shocks: self organized criticality in a model of production and inventory dynamics. Ricerche-Economiche 47, 3}30. Basu, S., 1996. Procyclical productivity: increasing returns or cyclical utilization. Quarterly Journal of Economics 111, 719}752. Basu, S., Kimball, M.S., 1994. Cyclical productivity with unobserved input variation. Manuscript, Hoover Institution. Basu, S., Fernald, J.G., 1997. Aggregate productivity and aggregate technology. Manuscript. Basu, S., Fernald, J.G., Horvath, M., 1999. Aggregate production function failures and real business cycle analysis. Manuscript. Blanchard, O.J., Kiyotaki, N., 1987. Monopolistic competition and the e!ects of aggregate demand shocks. American Economic Review 77, 647}666. Boldrin, M., Woodford, M., 1990. Endogenous #uctuations and chaos: a survey. Journal of Monetary Economics 25, 189}222. Burnside, C., Eichenbaum, M., Rebelo, S.T., 1996. Sectoral solow residuals. European Economic Review 40, 861}869. Christiano, L., 1988. Why does inventory investment #uctuate so much. Journal of Monetary Economics 21, 247}280. Christiano, L.J., Eichenbaum, M., 1992. Current real-business-cycle theories and aggregate labormarket #uctuations. American Economic Review 82, 430}450. Cochrane, J., 1994. Shocks. Carnegie-Rochester Conference Series on Public Policy, Vol. 41, pp. 295}364. Dixit, A., Stiglitz, J., 1977. Monopolistic competition and optimum product diversity. American Economic Review 67, 297}308. Dupor, B., 1996. Aggregation and irrelevance in multi-sector models. Manuscript, University of Chicago. Gali, J., 1996. Multiple equilibria in a growth model with monopolistic competition. Economic Theory 8, 251}266. Hansen, G., 1985. Indivisible labor and the business cycle. Journal of Monetary Economics 16, 309}328. Horvath, M.T.K., 1998. Cyclicality and sectoral linkages: aggregate #uctuations from sectoral shocks. Review of Economic Dynamics 1, 781}808. Horvath, M.T.K., Verbrugge, R.J., 1999. Shocks and sectoral interactions: an empirical investigation. Manuscript, Stanford University. Jorgenson, D.G., Fraumeni, B., 1987. Productivity and US Economic Growth. Harvard University Press, Cambridge. Jovanovic, B., 1984. Micro uncertainty and aggregate #uctuations. Manuscript, C.V. Starr Center for Applied Economics. King, R.G., Plosser, G.I., Rebelo, S., 1987. Production, growth and business cycles: I. The basic neoclassical model. Journal of Monetary Economics 21. Kiyotaki, N., 1987. Multiple expectational equilibria under monopolistic competition. Quarterly Journal of Economics 87, 695}713. Kydland, F., Prescott, E.C., 1982. Time to build and aggregate #uctuations. Econometrica 50, 1345}1370. Long, J., Plosser, C., 1983. Real business cycles. Journal of Political Economy 91, 39}69. Lucas, R.E., 1981. Understanding business cycles. In: Studies in Business Cycle Theory. MIT Press, Cambridge. Lucas, R.E., 1987. Models of Business Cycles: Yrjo Jahnsson Lectures. Basil Blackwell, Oxford. Mankiw, N.G., 1989. Real business cycles: a new Keynesian perspective. Journal of Economic Perspectives 3, 79}90.

106

M. Horvath / Journal of Monetary Economics 45 (2000) 69}106

McCallum, B.T., 1989. On &Real' and &Sticky-Price' theories of the business cycle. Journal of Money, Credit, and Banking 18, 397}414. Murphy, K.M., Shleifer, A., Vishny, R.W., 1989. Building blocks of market clearing business cycle models. In: Olivier B., Fischer, S. (Eds.), NBER Macroeconomics Annual 1989. MIT Press, Cambridge. Rogerson, R., 1988. Indivisible labor, lotteries and equilibrium. Journal of Monetary Economics 21, 3}16. Sato, K., 1975. Production Functions and Aggregation. American Elsevier, New York. Scheinkman, J., 1990. Nonlinearities in economic dynamics. The Economic Journal 100, 33}47 (supp.). Scheinkman, J., Woodford, M., 1994. Self-organized criticality and economic #uctuations. American Economics Association Papers and Proceedings 84.2, 417}421. Shea, J., 1994. Complementarities and comovements. Social Systems Research Institute Working Paper 9402, University of Wisconsin. Silverstein, G., 1985. New structures and equipment by using industries, 1977. Survey of Current Business 65, 26}35. Solow, R.M., 1957. Technical change and the aggregate production function. Review of Economics and Statistics 39, 312}320. Spence, M.A., 1976. Product selection, "xed costs, and monopolistic competition. Review of Economic Studies 43, 217}235. Summers, L., 1986. Some skeptical observations on real business cycle theory. Federal Reserve Bank of Minneapolis. Quarterly Review 10, 23}27. Verbrugge, R.J., 1996. Aggregate #uctuations from local market interactions. Manuscript, Virginia Polytechnic Institute and State University. Woodford, M., 1986. Stationary sunspot equilibria, the case of small #uctuations around a deterministic steady state. Manuscript, University of Chicago.