Cyclicality and Sectoral Linkages: Aggregate Fluctuations from Independent Sectoral Shocks

review of economic dynamics 1, 781–808 (1998) article no. RD980028

Cyclicality and Sectoral Linkages: Aggregate Fluctuations from Independent Sectoral Shocks* Michael Horvath Department of Economics, Stanford University, Stanford, California 94305 E-mail: [email protected] Received January 22, 1997

The traditional argument against the relevance of sector-specific shocks for the aggregate phenomenon of business cycles invokes the law of large numbers: positive shocks in some sectors are offset by negative shocks in other sectors. This paper hypothesizes that cancellation of sector-specific shocks via the law of large numbers is affected by interactions among producing sectors. The analysis is performed within the context of a multisector model similar in spirit to that of Long and Plosser [J. Polit. Econ. 91 (1983), 39–69]. It is shown that the rate at which the law of large numbers applies is controlled by the rate of increase in the number of full rows in the input-use matrix rather than by the rate of increase in the total number of sectors. Investigations of actual input-use matrices from the U.S. economy reveal that the number of full rows increases much slower than the total number of rows upon disaggregation, and when these input-use matrices are used to parameterize the model, aggregate volatility from sectoral shocks declines at less than half the rate implied by the law of large numbers. This finding leaves open the possibility that a sizeable portion of aggregate volatility is caused by “smaller” shocks to individual sectors. Simple statistics calculated from the model indicate that as much as 80% the volatility in U.S. gross domestic product growth rates could be the result of independent shocks to two-digit Standard Industrial Code sectors. Journal of Economic Literature Classification Numbers: E1, E32, C67. © 1998 Academic Press Key Words: aggregate fluctuations; sectoral interaction; comovement; input– output.

* Financial support from the Sloan Foundation and the National Science Foundation (SBR9507978) is gratefully acknowledged. I thank Bill Dupor, Chad Jones, Ken Kasa, and Fabiano Schivardi for their interest and suggestions. Three anonymous referees provided numerous valuable comments and suggestions. I owe a special note of gratitude to Michele Boldrin for his encouragement and many critical comments. All errors are, of course, my own. 781 1094-2025/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.

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michael horvath 1. INTRODUCTION

The traditional argument against the relevance of sector-specific shocks for the aggregate phenomenon of business cycles invokes the law of large numbers: positive shocks in some sectors are offset by negative shocks in other sectors. In my Ph.D. dissertation [18], I considered a setting similar to that in Long and Plosser [23] in which sectors produced different goods using sector-specific capital as well as intermediate goods purchased from other sectors. The intuition I sought to formalize was that the nature of intermediate goods flows between sectors has a direct effect on the rate at which shocks cancel by the law of large numbers. More precisely, I tried to show that limited interaction among sectors, characterized by a sparse input-use matrix, tended to lead to greater aggregate volatility from sectoral shocks since it reduced substitution possibilities in production. Dupor [12] considered the aggregate effects of sector-specific shocks to production in the context of the same multisector model and concluded that the law of large numbers applies in a sufficiently broad class of parameterizations of this model to obviate its use in the study of business cycles. In particular, Dupor was able to show that under a certain set of assumptions the manner in which sectors interact, as characterized by the inputuse matrix, was irrelevant to the behavior of aggregate volatility induced by sector-specific shocks: as definition of sectors became more disaggregate, aggregate volatility converged to zero at the rate implied by the law of large numbers. These results suggested that whether the input-use matrix was sparse or full, substantial aggregate volatility could not be the result of sector-specific shocks to production alone. Dupor concluded, if sectorspecific shocks are indeed important for aggregate fluctuations, some other mechanism such as the production complementarities studied in [13] must be at play.1 Upon further investigation, it turns out that it is possible to prove other theorems from the same model which leave the original intuition intact: limited interaction, albeit of a special form, gives rise to greater aggregate volatility from sector-specific shocks. These results are the subject of this paper. To understand how both Dupor’s results and the results contained herein can be correct, one must understand the difference in the assumptions we make on the nature of intermediate goods flows. Simply put, Dupor assumes that every sector sells intermediate inputs to some other sector(s): that sectors are roughly equally important as input suppliers. Theorem 1 of this paper repeats the main theoretical result from Dupor [12] on the ir1 Recent literature has focused on several mechanisms that may contribute to the weakening of the law of large numbers. Examples include asymmetries or threshold effects, limited interaction in a nonlinear setting, and monopolistic competition (see [1, 3, 6, 22, 24, 28–31]).

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relevance of sectoral input-use linkages for the volatility of aggregates with independent sectoral shocks. Conversely, I assume that some sectors are important input suppliers, while other sectors are not. Analytical results in this paper show that aggregate volatility is heightened when the input-use matrix is characterized by only a few full rows and many sparse columns. Such an input-use matrix implies that a handful of sectors serve as key input factors in the production processes of most of the sectors in the economy. Idiosyncratic shocks that cause factor price changes in these sectors will be important to the aggregate economy because there are few such sectors. More precisely, Theorem 2 reveals that the rate at which the law of large numbers applies is controlled by the rate of increase in the number of full rows in the inputuse matrix rather than by the rate of increase in the total number of rows in the matrix (number of sectors in the model economy). Since the relevance or irrelevance of the pattern of intermediate input flows depends on assumptions about the intensity with which sectors serve as intermediate input suppliers, this paper presents empirical evidence on factor demand intensities among disaggregated sectors. Investigations of actual input-use matrices from the U.S. economy reveal that the number of full rows increases much slower than the total number of rows upon disaggregation. At the one-digit Standard Industrial Code (SIC) level, all sectors look like important inputs in the economy. At the four-digit SIC level, only a fraction of the sectors supply inputs broadly. Therefore, when the actual input-use matrices are used to parameterize the model, aggregate volatility from sectoral shocks declines at less than half the rate implied by the law of large numbers. This finding leaves open the possibility that a sizeable portion of aggregate volatility is caused by “smaller” shocks to individual sectors. Simple statistics calculated from the model indicate that as much as 80% the volatility in U.S. gross domestic product growth rates could be the result of independent shocks to two-digit SIC sectors. The rest of the paper is organized as follows. Section 2 describes the model. Section 3 provides analytical and simulation results from the model that illuminate when and how limited interaction affects aggregate volatility and the law of large numbers. Section 4 describes empirical regularities of input flows in the U.S. economy. Section 5 discusses empirical implementation of the model and assesses the ability of sectoral shocks to explain U.S. aggregate output movements. Section 6 summarizes the main findings and briefly comments on related research. 2. THE MODEL The economy consists of M distinct sectors of production, indexed by h = 1; : : : ; M, each producing a different good. Each sector, in turn, con-

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sists of a continuum of identical firms with unitary mass operating production technologies which use capital and intermediate goods purchased from other sectors as inputs. The technologies are distinct across sectors and are subject to stochastic productivity variation. It is assumed that shocks to sectoral productivity are uncorrelated across sectors and across time. There is an infinitely lived representative consumer who derives utility from consuming the M types of goods. A benevolent social planner determines allocations of capital and intermediates in the M production processes and the amount of each good consumed so as to maximize the expected lifetime utility of the representative consumer. The model is essentially a multisector version of the classic Brock–Mirman [7] one-sector model. Because of its one-sector nature, the latter did not involve intermediate goods. 2.1. Production and Consumption The representative firm in sector h produces a quantity Yth of good h according to Yth = Aht Kth ‘αh Mth ‘γh ; Aht ,

Kth ,

h = 1; : : : ; M:

(1)

Mth

and denote, respectively, the period t, sector h The variables values of productivity level, capital stock, and aggregate intermediate input. It is assumed that, αh + γh < 1, for h = 1; : : : ; M. The stochastic process for Aht is given by ln Aht = ρh ln Aht−1 + ht ;

Žρh Ž ≤ 1; h = 1; : : : ; M:

(2)

ht

are assumed to be mean zero and i.i.d. across The productivity shocks time with diagonal variance–covariance matrix : An input-use matrix for this economy is an M × M matrix 0 with typical element γij . The list of intermediate inputs for sector h are summarized by the set of indices Bh (for the “buy-from” set of sector h) defined as Bh = ”sx γsh 6= 0• P for h = 1; : : : ; M. Hence, the hth column sum of 0 is defined by γh = s∈Bh γsh . The aggregate intermediate input in each sector is a constant (unitary) elasticity of substitution function of specific goods produced in other sectors, Y Mt;h s ‘xsh ; h = 1; : : : ; M: (3) Mth = s∈Bh

The coefficients in (3) are constructed according to xsh = γsh /γh . The capital stock in sector h evolves according to h = Ith ; Kt+1

h = 1; : : : ; M;

(4)

where Ith denotes investment in sector h in period t. In other words, capital depreciates fully after one period. Furthermore, assume that only the output from sector h is suitable for use as capital in the production of good

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h. Both of these rather drastic assumptions are necessary to achieve the analytical solution described below. The representative consumer has lifetime utility given by ∞ X t=0

δ

t



M X i=1

θi ln Cti

 ;

0 < δ < 1;

(5)

where Cti denotes consumption of good i in period t, θi denotes the relative preference for good i, and δ is a discount factor. The assumption of logarithmic preferences is also necessary for the analytical solution described below. 2.2. The Social Planner’s Problem and Solution Given initial capital stocks Ko ; the social planner seeks to maximize (5) subject to (1), (4), and the resource constraints X s Mt;h ; h = 1; : : : ; M; (6) Yth = Cth + Ith + s∈Sh

where the set Sh (for “sell-to”) contains the indices of sectors that purchase intermediate inputs from sector h. Formally this set is defined by Sh = ”sx γhs 6= 0• for h = 1; : : : ; M. To accomplish this objective, given a sequence of productivity levels ∞ ”At •∞ t=0 , the social planner chooses vector sequences ”Ct ; Mt ; Kt+1 •t=0 , 1 M 0 1 M 0 1 M 0 where At = At · · · At ‘ , Ct = Ct · · · Ct ‘ , Kt = Kt · · · Kt ‘ , and Mt = Mt1 · · · MtM ‘0 . Given the assumptions made on technologies and given preferences, a closed form solution exists for the planner’s problem (see Appendix A.1 for derivation of the policy function, the planner’s value function, and the dynamic response function for kt+1 ).2 Let Z ≡ I − 0‘−1 with typical element zij . The policy function determining capital (investment) in sector i at time t as a function of (Kt , At ) is given by (in vector form) kt+1 = h + Z 0 αd kt + Z 0 at ;

(7)

where lowercase letters denote natural logarithms of their uppercase counterparts, the vector h is a function of model parameters, and αd is an M × M diagonal matrix with the vector α on its diagonal. 2 Horvath [19] examines a more general model with general CES utility, labor supply, lessthan-full depreciation of capital, and sectoral capital stocks that are constructed from goods from potentially several other sectors. In this case an exact analytical solution is impossible and an approximation strategy is adopted instead.

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It is possible to show that the vector autoregressive form for sectoral outputs differs from the policy function for sectoral capital only in the constant term.3 Hence, the stochastic process for yt+1 is given by yt+1 = h˜ + Z 0 αd yt + Z 0 at+1 ;

(8)

where h˜ is another vector of constants. The present model differs from Long and Plosser [23] in that production involves both capital and nonstorable intermediate goods. In the Long and Plosser model, there are no capital stocks and intermediate goods are delivered with a one-period lag. In this case, the associated equilibrium stochastic process for yt+1 is given by yt+1 = hˆ + 00 yt + at+1 ;

(9)

where hˆ denotes yet another vector of constants. Comparing (8) with (9) it is clear that, in the Long and Plosser model, sector-specific shocks only have direct own-sector effects in the period of the shock, while in the present model, sector-specific shocks have both contemporaneous direct own-sector effects and contemporaneous indirect cross-sector effects. However, it can be verified that the results presented in Section 3 apply to the Long and Plosser model as well. 3. LIMITED INTERACTION AND THE LAW OF LARGE NUMBERS Two questions will be asked of the model in this section. Question 1. Does the extent of factor substitutability, represented by the sparseness of 0; affect the time-series behavior of the model’s aggregates? Question 2. As the level of disaggregation increases, at what rate does the law of large numbers apply (how fast is the aggregate effect of sectorspecific shocks being diminished), and is this rate affected by the extent of interaction among sectors? Following Dupor [12], it turns out to be convenient to transform the time-series relationship for capital in (7) into the frequency domain. Let L denote the lag operator. The infinite-order moving average representation for kt+1 is given by
−1
−1 (10) kt+1 = I − Z 0 αd h + I − Z 0 αd L Z 0 at : 3 This implies that all results derived for the behavior of sectoral and aggregate capital apply to the behavior of sectoral and aggregate output as well.

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Furthermore, one can write at = I − ρd L‘−1 t , where ρd is a diagonal matrix with the elements of ρ = ρ1 ; : : : ; ρM ‘0 on its diagonal and t = 0 4 1t ; : : : ; M t ‘ . As long as kt+1 is stationary the population spectrum for k, denoted Sk ω‘, is given by (see Hamilton [17, p. 276])
−1
−1 Sk ω‘ ≡ 2π‘−1 I − Z 0 αd e−iω Z 0 I − ρd e−iω
−1
−1 ×  I − ρd eiω Z I − αd eiω Z :

(11)

The spectrum is a useful function of the parameters embodied in 0 and αd since it contains all the information of the population autocovariances of kt . This will be used below to assess how pairwise sectoral correlation is affected by the sparseness of 0. For the sake of simplicity, assume that  = IM and ρh = 0 for every h = 1; : : : ; M. In this case (11) simplifies to Sk ω‘ ≡ 2π‘−1 I − αd e−iω − 00


−1


−1 I − αd eiω − 0 :

(12)

The effect of 0 on Sk ω‘ can be traced by examining (12). Furthermore, the spectrum of aggregate capital, denoted Sω‘, can be analyzed by aggregating the information in Sk ω‘. This is defined by Sω‘ ≡ w 0 Sk ω‘w;

(13)

where w is a vector of aggregation weights for log-aggregate capital stock. Dupor [12] restricts attention to a particular class of input-use matrices in answering Questions 1 and 2. The class is defined by the following assumptions. Assumption D.1. Let capital’s share be the same across sectors. For 0 < α < 1, let αd = αI. Assumption D.2. 0 ∈ 0γ‘. 0γ‘ defines a class of matrices which satisfy 0l = γl, where l is a length M vector of ones. Assumption D.2 implies that the row sums of 0 are identical. The row sums of 0 indicate the importance of each sector’s output as intermediate inputs to all other sectors. Therefore, Assumption D.2 implies that all sectors are equally important as intermediate input suppliers. Theorem 1 restates the main result of Dupor [12] regarding the invariance of Sω‘ to changes in 0 under Assumptions D.1 and D.2. 4 This requires the moving average coefficients to be absolutely summable, which, in turn, requires each sector to exhibit decreasing returns to scale and less than unitary persistence in productivity.

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Theorem 1 (Dupor). All parameterizations of the M sector model described by Equations (1)–(6), and which satisfy Assumptions D.1 and D.2 for fixed α and γ and aggregation weights w = 1/M‘l have the univariate spectral density Sω‘ = 2π‘−1

1/M : 1 − αe−iω − 㑐1 − αeiω − γ‘

(14)

See Appendix A.2 for the proof. As long as all sectors are equally important as input suppliers, varying factor share among sectors while keeping total factor share constant will have no effect on the volatility of aggregates in the model. Dupor interprets this finding as both an aggregation result and as an irrelevance proposition. As an aggregation result it suggests an observational equivalence between the single aggregate sector economy and the multisector economy where the variance of sector-specific shocks scales by the level of disaggregation (M). As an irrelevance proposition it shows that different input-use matrices generate exactly the same spectrum for aggregate capital as long as Assumptions D.1 and D.2 hold and w = 1/M‘l. From (14) it is clear what the law of large numbers has to say about the aggregate effects of sector-specific shocks. As in Dupor [12], let ”0M •∞ M=1 be a sequence of matrices each of dimension M such that, for each 0, Assumption D.2 holds. Then Sω‘ → 0 at rate M for all ω, independently of how the factor shares are distributed inside each element of the sequence ”0M •∞ M=1 . These results appear damaging to the intuition and results put forth in [18]. Indeed, they contradict them. When I first encountered Dupor’s work in an earlier draft, I turned to the simulations I had carried out in the process of writing my dissertation. I had calibrated the model presented above with the actual input-use matrices from the U.S. economy (this exercise is described in Section 5.1) and calculated the rate at which the spectral density of aggregate capital declined as disaggregation increased, holding the variance of sectoral shocks constant. The rate at which aggregate volatility declined was less than half that implied by (14). Clearly, the simulation results did not match those predicted by Theorem 1. Since the U.S. input-use matrices did not satisfy Assumption D.2, I began investigating whether this assumption was critical for the irrelevance result. It turns out that if one assumes that all sectors are not equally important as input suppliers, but rather that some sectors sell inputs while other sectors do not, then Theorem 1 no longer holds. I refer to this condition on 0 as row sparseness. Definition 1. a zero vector.

The matrix 0 is row-sparse if one or more of its rows is

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Assumption H.2 establishes an alternate class of input-use matrices to those in Assumption D.2. Assumption H.2 permits 0 to be row-sparse whereas Assumption D.2 does not. Assumption H.2. 0 ∈ 0γ; x‘. 0γ; x‘ defines a class of matrices which ˜ 0 , where l˜ is a length M vector with M − N‘ zeros and N satisfy 0 = lv ones, and v is a length M vector satisfying v0 l = γM/N, v0 l˜ = x for all M; N and Nvi + α < 1 for i = 1; : : : ; M: Clarifying, Assumption H.2 imposes M − N rows of zeros on 0, representing sectors that supply no inputs to other sectors. The other N rows of 0 are equal to the vector of share weights v. Hence, all input-supplying sectors are equally important, but some sectors are not input suppliers. Furthermore, the nonzero row sums of 0 are proportional to M/N. The proportionality assumption is sensible since it implies an average value of vi proportional to 1/N and hence column sums that neither shrink to zero nor transgress their upper bound of 1 − α as M and N get large. Note that the row sums of 0 under Assumption H.2 satisfy 0l = γ

M˜ l; N

(15)

indicating that 0γ; x‘ ⊂ 0γ‘ for N = M. The next theorem establishes the relationship between the spectral density of aggregate capital and the row sparseness of 0. Theorem 2. All parameterizations of the M sector model described by Eqs. (1)–(6), and which satisfy Assumptions D.1 and H.2 for fixed α, γ, and x and aggregation weights w = 1/M‘l, have the univariate spectral density   1 2 aω‘ 1 2 bω‘ − γ ‘ + γ ; (16) Sω‘ = 2π‘ M N where aω‘ ≡

1 − αeiω ‘−1 1 − αe−iω ‘−1 1 − αeiω − x‘1 − αe−iω − x‘

and bω‘ ≡ 1 − αeiω − x + 㑐1 − αe−iω − x + γ‘: See Appendix A.3 for the proof. Under Assumption H.2 the spectrum of aggregate capital depends on M as in Dupor’s case, but it also depends on N. More precisely, let ”0M •∞ M=1 be a sequence of matrices each of dimension M such that, for each 0 Assumption H.2 holds. Then Sω‘ → 0 at rate N for all ω. This suggests that law of large numbers effects are postponed if, in disaggregating 0, the

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number of full rows grows at a slower rate than M. That is, the law of large numbers is postponed if 0 becomes increasingly row-sparse through disaggregation. In the extreme case, if N converges to a positive constant N¯ as M increases, then the law of large numbers is broken completely. In ¯ this case the spectrum converges to aω‘γ 2 ‘/2π N. When all rows in the input-use matrix are full (N = M), the second term inside the brackets in (16) vanishes and the spectrum converges to zero at the rate 1/M. Dupor [12, Theorems 1–4], only considers input-use matrices with the characteristic that all sectors sell inputs to some other sector(s); in other words, N = M so that the matrices are not row-sparse. This explains why Dupor finds no effect of varying the extent of sectoral interaction in 0 on the volatility of aggregate variables in the model setting. From inspection of (16) and the arguments presented above it should be clear that row sparseness in 0 certainly does affect the aggregate volatility generated by sectoral shocks.5 Pairwise correlation statistics for sectoral capital stock can be calculated from (11). The correlation in capital stocks between sectors i and j depends upon the position of i and j and is given by  1; i = j;     s  1   ; i; j ≤ M − N; i 6= j;   s + s2  1    s1 + s3 (17) ρij = p ; i > M − N and j ≤ M − N √  s + s  1 2 + 2s3 s1 + s2    or j > M − N and i ≤ M − N;       s1 + 2s3   ; i; j > M − N; i 6= j; s1 + s2 + 2s3 where s1 = γ 2 /N‘/1 − α − x‘2 , s2 = 1/1 − α‘, and s3 = γ/N‘/1 − ᑐ1 − α − x‘‘. The important feature of sectoral comovement in this case is that it is decreasing in N and independent of M. More row sparseness in the input-use matrix (lower N) increases all pairwise correlations. Intuition suggests the reason why an input-use structure of sparse columns and relatively few full rows with large share weights contributes to increased volatility, sectoral comovement, and a retardation of the law of large numbers. A primary supplier of inputs will transmit the same signal, either positive or negative, to many sectors, thus increasing the likelihood that the responses of the purchasing sectors will be positively correlated. If the share weights are large, then the transmitted signal is important to the receiving sector. If there are only few of these supplying sectors, their 5

Note, also, that when there are no input-use links, Sω‘ is given by (14).

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signals may not average to zero. For the law of large numbers to operate on the system in (13), cancellation of the effects of shocks must occur along both the supply and demand dimensions, along both the rows and the columns of the input-use matrix. This is less likely to occur if only a few of the rows are full.

4. EMPIRICAL REGULARITIES OF U.S. INPUT-USE MATRICES The previous section has established that the rate at which the law of large numbers applies on sector-specific shocks in the model depends crucially on the nature of intermediate goods flows. Contrasting Assumption D.2 and Theorem 1 with Assumption H.2 and Theorem 2 begs the question, which assumption is closer to empirical truth? This section describes the empirical regularities of the U.S. input-use matrices at different levels of disaggregation. The main finding is that Assumption H.2 is supported by the data, while Assumption D.2 is not. The rows of the raw input-use matrix show the dollar value in producer prices of materials, semifinished goods, and services that are consumed by industries indexed in the columns.6 The least aggregated matrix available is the detailed input-use matrix constructed by the Department of Commerce. The detailed matrix (denoted below as 0523 ) is of dimension 523 and includes mostly four-digit SIC definitions.7 Using the SIC definitions, 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 077 conforms to the two-digit SIC definitions, while 06 conforms to the one-digit SIC definitions. Matrix 036 conforms to the sectoral definitions used by Jorgenson et al. [21], while 020 is a slightly more aggregated version. Appendix B contains sectoral definitions for each of these matrices. For use in what follows, the raw input-use matrices are converted into share weights by dividing each column by its column sum times the chosen value for γ. Analyzing the input-use matrices, I find that they become increasingly sparse with disaggregation. While 06 has no zero elements, the detailed input-use matrix has 224,410 zeros and only 49,119 nonzero elements. 6 Structures and equipment sold between industries represent capital investment and are not included in the input-use table, but instead in the capital flow table. 7 I use the 1977 matrix. The detailed input-use matrix is actually of dimension 537. However, the import, scrap, second-hand goods, government industry, and household sectors are dropped from the present analysis (they use no inputs). The government demand sectors are also dropped (they supply no intermediate goods), as is the inventory valuation adjustment column and row.

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Moreover, most of the nonzero elements are trivial in size. For example, only 2,272 elements of the detailed matrix constitute input uses larger than 5% of a sector’s total input use. More pertinent to the present analysis, it turns out that the row sparseness of the U.S. input-use matrices is also affected by aggregation. Table I presents the number of rows in the U.S. input-use matrices 06 –0523 which are completely full, > 2/3 full, > 1/2 full, and > 1/3 full. For example, only two rows of 0523 contain all nonzero elements. The disaggregate input-use matrices 036 ; 077 ; and 0523 are characterized by relatively few full or > 2/3 full rows, while the aggregate input-use matrices 06 and 021 have nearly all full rows. The number of rows with > 2/3 nonzero elements increases from 6 in 06 to 59 in 0523 . Translating this increase √ into the model from Section 3 implies that N is increasing roughly with M . Taking the model seriously implies that the sectors that contribute significantly to aggregate volatility are the sectors which supply inputs to many other sectors, the sectors with full rows in the input-use matrix. Table II identifies the sectors with the fullest rows in 0523 and the number of sectors they supply for different tolerance levels for “zero”. The table is sorted on the tolerance = 0:05 column, which counts only links larger than 0:05. For example, engineering/architectural services are used by 414 four-digit sectors although only 30 of these use such services in excess of 5% of their total input purchases. Topping the list by a wide margin is wholesale trade, which is an input into the production of every sector. Besides wholesale trade, the other top five input suppliers are blast furnaces and steel mills, real estate, plastics products, and petroleum refining. At least two of these sectors, real estate and petroleum, have long been hypothesized to be a major source of macroeconomic fluctuations (see Hamilton [16] and Davis [11]). While it is difficult to think of historical events in the other sectors on the list that were as widely publicized as the oil-price shocks of the 1970s and 1980s and the real estate crisis of the early 1990s, this is not necessary. The point of Theorem 2 is not that large shocks need to occur in these sectors for large TABLE I Number of Full Rows in U.S. Input-Use Matricesa

Full > 2/3 Full > 1/2 Full > 1/3 Full

06

021

036

077

0523

6 6 6 6

18 20 20 20

21 28 30 31

19 42 47 56

2 59 71 93

a Entries indicate the number of rows in the matrices (columns) which satisfy the fullness criterion (rows).

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TABLE II The Primary Suppliers of Inputs in the U.S. and Their Total Row Linksa Toleranceb Sector definition

0.00

0.01

0.05

0.10

Wholesale trade Blast furnaces and steel mills Real estate Plastics products Petroleum refining Industrial chemicals Advertising Electrical utilities Motor freight/warehousing Paperboard containers Engineering/architectural services Plastics materials Sawmills Broadwoven fabric mills Paper mills Electronic components Pipes and valves

523 262 523 383 521 376 395 517 522 443 414 144 190 77 103 70 152

513 199 277 230 266 143 191 344 365 201 63 75 60 45 36 41 62

393 117 78 71 68 62 48 44 41 30 30 28 28 25 22 18 15

92 91 38 25 20 33 18 13 19 10 19 15 16 17 17 10 9

a

The table lists the sectors which have the fullest rows in the U.S. detailed input-use matrix and the number of sectors they supply with inputs. b Tolerance refers to the zero tolerance used when counting nonzero links.

aggregate effects to be observed. Rather, even if their productivities vary independently, it is likely that most of these sectors receive shocks of the same sign in any given period since there are relatively few of them. Hence it is unlikely that their shocks cancel. The size distribution of the nonzero elements in the input-use matrix is altered by aggregation as well. Table III displays Gini coefficients for the TABLE III Input-Share Gini Coefficients—Column Means and Standard Deviationsa

06 021 036 077 0523

G¯

S.D.

0.54 0.72 0.79 0.86 0.96

(0.08) (0.08) (0.07) (0.05) (0.02)

a Entries correspond to the mean and standard deviation (in parentheses) of Gini coefficients for input-use shares in the columns in U.S. input-use matrices.

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share weights averaged across the columns of the matrices 06 , 021 , 036 , 077 , and 0523 . The Gini P coefficient for the jth column of the matrix is P given by Gj ≡ 1/2γ¯ j M 2 ‘ i k Ž γij − γkj Ž, where γ¯ j denotes the average share weight in the column. The Gini coefficient resides in the unit interval and measures the dispersion of the share weights in the column. Values close to 0 imply nearly equal share weights, while values close to 1 imply very disparate share weights. The results are by now familiar: dispersion of input-shares is substantially higher at higher levels of disaggregation. At the one-digit SIC level (06 ) G¯ = 0:54, while at the four-digit SIC level (0523 ) G¯ = 0:96. The empirical evidence from U.S. input-use matrices presented in this section suggests that production is highly specialized even at mild levels of disaggregation. Disaggregated input-use matrices are row-sparse: few sectors serve as important inputs to all other sectors and most sectors use few intermediate inputs in production.

5. COMPARING THE MODEL TO DATA FROM THE U.S. ECONOMY What remains is to implement the model. In Section 5.1 I investigate law of large number properties in the model parameterized with the actual input-use matrices from the U.S. economy. In Section 5.2 I assess the model’s ability to explain data on aggregate output fluctuations. 5.1. Law of Large Numbers Properties Figure 1a plots Eq. (13) using 0523 , 077 , 036 , 020 , and 06 . In generating these curves, the aggregation weight vector was assumed to be w = 1/M‘l in each case implying that each sector gets an equal weight in aggregate capital. The coefficients in production where set to α = 0:475 and γ = 0:475 in all sectors.8 Figure 1b plots the rate of decline in the spectrum at a particular frequency, ω = 0:625. Results are identical for all frequencies. Figure 1a clearly shows that the spectrum of aggregate capital declines with disaggregation. However, Fig. 1b reveals that the rate of decrease in the spectrum is slower than that implied by the law of large numbers. In disaggregating between 6 and 20 sectors,9 the aggregate spectrum does decline at the rate of 1/M. However, in disaggregating between 20 and 523 sectors, the ag8 9

Law of large numbers results do not depend critically on the values of w, α, or γ. This is not shown on the graph to improve resolution at M = 36–523.

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FIG. 1. LLN properties of parameterized model. (a) Effect of disaggregation on aggregate spectral density spectra at different levels of disaggregation. (b) Effect of disaggregation on aggregate spectral density rate of decline in spectrum for w = 0:0625.

√ gregate spectrum declines at a rate slower than 1/ M .10 This corresponds to the empirical regularities presented above on the row-sparseness of U.S. input-use matrices at √ different levels of aggregation: The number of full rows increases with M from 036 to 0523 . 5.2. Quantifying the Importance of Sectoral Shocks The results presented above suggest that in the limit, the effects of idiosyncratic shocks to atomistic units of production do not produce aggregate fluctuations, but the rate at which “the limit” is achieved is slower than the law of large numbers suggests. Therefore, the interesting question in taking 10 Dupor [12] presents a similar graph to Fig. 1b. His results match those presented here: at the lowest level of aggregation the empirically generated spectrum is more than twice as large as that implied by the law of large numbers. His interpretation of these results focuses on the magnitude of the spectrum rather than on its rate of decline, arguing that, though the empirical curve lies above the curve 1/M, both are quite small. Whether sufficient aggregate volatility can be achieved solely from independent sectoral shocks remains to be seen. However, this is a separate issue from that of the rate at which the law of large numbers applies in the model.

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the model as a candidate explanation for aggregate fluctuations is: What percent of aggregate fluctuations can be attributed to uncorrelated sectoral productivity shocks at some interesting level of disaggregation, say the twodigit SIC level? A related question of interest is: To what extent are sectoral linkages responsible for propagation of both sector-specific and aggregate shocks. The following analysis attempts to shed light on both of these questions. However, the model in this paper is not a good starting point for a full-blown calibration and simulation exercise in the real business cycle tradition because of its highly stylized form. For example, simulation results from the model in the present paper would be silent on labor market fluctuations, arguably the most prominent feature of aggregate business cycles. A thorough calibration and simulation exercise from a more complete multisector model can be found in Horvath [19]. In order to proceed, the model in Section 2 requires realistic parameter values for αd , 0, and . Since the analysis will be carried out at the twodigit SIC level of disaggregation, the input-use matrix used is 036 . I assume that ρh = 1 for all h. Thus the model is stationary in first differences. Accordingly, the solution to the planner’s problem is given by yˆt+1 = Z 0 αd yˆt + Z 0 t ;

(18)

where yˆt denotes the log first difference of the vector of sectoral outputs and t is the vector of sectoral productivity shocks. These are assumed to be i.i.d. across time. However, since I want to allow for the possibility of aggregate shocks, no restriction is imposed requiring  = Et 0t ‘ to be diagonal. The variance–covariance matrix of sectoral output growth rates is given by Eyt yt0 ‘ = Z 0 Z + Z 0 αd ‘Z 0 Zαd Z‘ + Z 0 αd ‘2 Z 0 Zαd Z‘2 + · · · : (19) The variance of some aggregate statistic based on (18), such as yˆ¯t ≡ w 0 yˆt , can be calculated given a vector of aggregation weights w. The variance of yˆ¯t is given by σ 2 = w 0 Eyt yt0 ‘w:

(20)

Two comparative dimensions will be explored from here. The first is to assess to what extent the sectoral interactions in the model generate heightened aggregate volatility above what would obtain in the case of M independent sectors each producing according to Yti = Ai Kti ‘αi /1−γi ‘‘ . The assumption implicit in this formulation is that Mth = Yth in (1) so that all intermediate inputs come from own-sector output.11 Let α˜ d denote the 11 Results are similar if it assumed that Mth is constant across time in all sectors and hence nets out of (18). In this case αd is used instead of α˜ d

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diagonal matrix with nonzero elements αi /1 − γi ‘‘. The solution to the model in the no linkages case is given by (18) with Z = I and αd replaced by α˜ d . Let the standard deviation of aggregate output growth rates in a model with no linkages be given by σNL . This can be compared to σ from the model with 0 = 036 from the U.S. economy. The extent to which σ exceeds σNL indicates the importance of sectoral linkages in propagating shocks, both sectoral and aggregate. The second comparison made is between model volatility generated by uncorrelated sectoral shocks versus aggregate shocks, something which depends on the exact structure of . To pin down the structure of , further assumptions on  are needed. Suppose that shocks to sector i were the sum of idiosyncratic shocks, ηit , and aggregate shocks, νt , as in it ≡ ηit + βi νt ;

i = 1; : : : ; M;

(21)

where βi controls the response of sector i to aggregate shocks, Eηt η0t ‘ = A for a diagonal matrix A, and Eνt2 ‘ = σν2 . It will not be possible to decompose t as in (21) without additional restrictions on (18). For example, normalization of (21) for each i will not help much since there is a total of 2M + 1 unknowns. Developing an identification scheme for (21) is beyond the scope of this paper, but is the subject of Horvath and Verbrugge [20]. Even without exact identification of aggregate and sectoral shocks, it is possible to form an upper bound on the importance of uncorrelated sectoral shocks using the present setup. Loosely, the size of the diagonal elements of  relative to the off-diagonal elements indicates the size of uncorrelated sectoral shocks relative to aggregate shocks. Then, σ 2 could be calculated by (20) using only the diagonal of , setting off-diagonal elements to zero. Let this measure of aggregate variance from the model be denoted σd2 . Let the variance of output growth rates from the U.S. economy be denoted by 2 . The size of σd relative to σUS is an upper bound on the contribution σUS of sector-specific shocks to measured aggregate volatility.12 The ratio is an upper bound since the typical diagonal element is given by ii = Aii + β2i σν2 , which contains a weighted aggregate shock variance. Therefore, this analysis would show conclusively that sector-specific shocks are unimportant if σd /σUS were small while it would suggest that sector-specific shocks may be an important source of aggregate volatility if σd /σUS were large. The matrix  is estimated as the sample average variance–covariance matrix of sectoral total factor productivity residuals. The residuals (t ) are 12 In principle, this ratio can be expressed in terms of the variances rather than the standard deviation. In practice, the literature favors expressing relative volatilities in terms of the ratio of standard deviations since this represents the fraction of the average deviation in the actual economy explained by the model economy. See Plosser [25].

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calculated using annual data between 1947 and 1989 on outputs and inputs at the two-digit SIC level of disaggregation13 using the methodology pioneered by Hall [14, 15] and improved on by Basu and Fernald [4]. Accordingly, it could be calculated by ˆ it ; (22) it ≡ yˆti − ciK kˆ it − ciL lˆti − ciM m where ciX denotes the factor-X share of total cost of producing gross output in sector i. Several authors including Burnside, Eichenbaum, and Rebelo [8] (hereafter BER), Basu [2], and Basu and Kimball [5], have noted that variations in capital utilization will affect the measurement of total factor productivity under (22).14 In particular, the variance of productivity residuals and their procyclicality with measures of economic activity are likely to be reduced once they have been “corrected” 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 data set. BER assume that it is the flow of services from capital stock K i that enters into the production function given in (1) and that this flow is linearly proportional to energy usage. They recommend correcting it with data on sectoral energy usage as ˆ it ; (23) ˜ it ≡ yˆti − ciK eˆ it − ciL lˆti − ciM m where eˆ it denotes the time t log first difference of energy usage in sector i. In the present analysis, correcting for capital utilization is important because it is likely that the sectoral covariance properties of t differ from those of ˜ t in a systematic manner. By their nature, aggregate shocks should affect capital utilization in the same direction for a majority of sectors, while the first-order effect of uncorrelated sectoral shocks should be uncorrelated movements in capital utilization across sectors. Therefore, failing to correct for varying capital utilization would overstate the correlation in sectoral total factor productivity growth. This indeed turns out to be the case. The average pairwise correlation (excluding the diagonal) among the  is 0.083 while for the ˜ it is 0.038, suggesting that sectoral shocks would appear to be less important in generating aggregate fluctuations without correcting for variations in capital utilization. All that remains is to parameterize αd and w. The Jorgensen data set is used again to calculate α as capital’s share of sectoral gross output and wi as sector i’s output as a share of total output.15 13 The data set is commonly referred to as the Jorgenson data. See Jorgenson et al. [21]. I thank John Fernald for kindly providing it. 14 Variable utilization may affect other input factors, as these and other authors have noted. However, in the present analysis only variable capital utilization is considered. 15 Results are virtually identical if wi = 1/M for all i.

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TABLE IV Aggregate Volatility Measures from the Model and the U.S. Economya

Uncorrected Corrected

σ (%)

σd (%)

σ/σNL

σd /σUS

1.21 0.89

0.75 0.86

2.0 2.2

0.70 0.80

a See text for definition of column headings. “Uncorrected” denotes estimates from sectoral productivity residuals that have not been corrected for variable capital utilization while “corrected” denotes estimates estimated from residuals that have received such corrections.

Table IV presents the results of the decomposition exercise, reporting σ and σd in percentage terms on a quarterly basis, and the ratios σ/σNL and σd /σUS for both the uncorrected residuals, , and the corrected residuals ˜ . The value for σUS is estimated at 1:07% from quarterly GDP growth rates over the time period 1947:2–1996:4.16 Concerning the impact of sectoral linkages, the results (column 3 of the table) show that the model with intermediate input flows generates about twice the aggregate volatility compared with a model with no intermediate input flows. For both the corrected and uncorrected residuals, σNL is roughly half the size of σ. Sectoral linkages are very important in the propagation of shocks in the model. Turning to the assessment of the importance of sectoral shocks, based on the uncorrected residuals, it appears that sectoral shocks alone account for at most 70% of aggregate volatility in the data. Based on the corrected residuals, sectoral shocks account for at most 80% of aggregate volatility in the data. Even though both aggregate and sectoral shocks combined account for a smaller portion of aggregate volatility after correcting for varying capital utilization, the (maximum) amount of volatility attributed to independent sectoral shocks with the utilization correction is greater than in the uncorrected case. Furthermore, inspection of the off-diagonal elements in , estimated with or without the capacity utilization correction, provide little evidence of large aggregate shocks in the residuals t and ˜ t . The average estimated off diagonal element is roughly 1/50 the size of the average diagonal element in the uncorrected case and 1/100 the size in the corrected case. Furthermore, in the typical column (sector) of  there are 13 slightly negative elements in the uncorrected case and 15 in the corrected case, indicating that aggregate shocks have opposing effects on nearly half the sectors (many βi s are negative in (21)).17 Bearing this 16

Data come from the National Income and Product Accounts. Of course, the small negative off-diagonal elements in  could also result from sectoral shocks that are slightly negatively correlated. Still, this does not give evidence in favor of large aggregate shocks since in this case the off-diagonal elements of  would be uninformative about the size of aggregate shocks. 17

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in mind it is unlikely that the upper bound estimate on the importance of sectoral shocks presented above is much larger than the true value. Since the model seems to be capturing a significant portion of the variability in the data, it is reasonable to ask whether it matches other second moments. One aspect of the data on output fluctuations that have received recent attention are the autocorrelation of output growth rates. Cogley and Nason [10] find that standard one-sector business cycle models such as can be found in Christiano and Eichenbaum [9] have a difficult time in generating autocorrelated output growth rates. While the standard RBC models can generate output fluctuations, the movements in terms of growth rates are not sufficiently persistent and hence do not look much like the fluctations in the data. In the U.S. GDP data used here, output growth has a one-period autocorrelation statistic of 0.58. This statistic from the model can be calculated as w 0 Z 0 αd Eyt yt0 ‘w‘/w 0 Eyt yt0 ‘w‘ and equals 0.34 based on both the corrected and uncorrected residuals. This number obtains for aggregate output despite the fact that own-sector growth rate autocorrelations statistics are substantially lower, ranging from 0.06 to 0.44 with a mean of 0.22. Presumably, additional autocorrelation for the aggregate growth rate is the result of cross-sector autocorrelation induced by input linkages. Indeed, the one-period autocorrelation of aggregate output growth in the model without input links is a much lower 0.17. 6. CONCLUDING COMMENTS This paper has investigated the role that limited interaction among producing sectors plays in preserving aggregate volatility from idiosyncratic sectoral shocks. Using a model of a social planner choosing to allocate sectoral production among intermediate uses, capital stock, and final consumption, it was shown that, contrary to Dupor’s [12] irrelevance result, certain characteristics of input-use matrices affect the rate at which aggregate volatility declines for increasing levels of disaggregation. The key feature of intermediate input-use matrices that achieves these results is row sparseness: some sectors supply inputs broadly, while others do not. It is shown that the rate of decrease of the volatility of aggregate variables through disaggregation is proportional to the rate of increase in the number of full rows in the input-use matrix. Several concluding comments are in order. First, the model presented here is highly stylized with functional forms chosen to guarantee closed form solutions to the optimal control problem. Labor supply is not considered and therefore the model is silent on perhaps the largest puzzle in business cycle research: explaining the size and timing of labor market fluctuations. Horvath [19] performs a similar analysis in the context of a more general, fully calibrated model in the spirit of the real business cycle

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tradition. Simulation results confirm the analytical results presented here: appropriately sparse input-use matrices heighten the volatility of aggregate variables and result in a postponement of law of large numbers effects. Furthermore, the simulation results presented suggest that the multisector model calibrated to the two-digit SIC level of disaggregation is capable of matching empirical moments from U.S. aggregate time-series data as well as “good” one-sector models without relying on aggregate shocks. Second, the main lesson to extract from the preceding analysis is that the Long and Plosser [23] sectoral shocks hypothesis may prove to be viable even in the face of the law of large numbers. In particular, the model suggests two-digit SIC sectoral shocks alone can account for as much as 80% of the volatility in aggregate output growth rates. However, more research is certainly required since this number is only an upper bound on the importance of sectoral shocks. Consequently it is incumbent upon advocates of the sectoral shocks hypothesis to identify major sector-specific shocks associated with modern era business cycles. This is a challenging problem since data on sectoral production are tainted with the “observational equivalence” between an own-sector shock and the cross-sector effects of other sectors’ shocks. Horvath and Verbrugge [20] use the model setting in this paper to motivate a set of identifying assumptions in a structural vector autoregression of sectoral growth rates in an effort to solve the observational equivalence problem and permit association of particular cycles with particular sectoral shocks. Finally, the spirit of this research is to identify the role that smaller shocks play in generating aggregate fluctuations. The goal is not to show the irrelevance of aggregate shocks; it is simply to reduce the reliance on them within the field of macroeconomics. Several decades ago macroeconomics disowned its reduced-form past and embraced the notion that macro models should have at their core microeconomic foundations. It is ironic that the exogenous component of most of the ensuing micro-based models remained highly aggregated and largely unobservable in the data. Sufficient research may prove that business cycles are aggregate manifestations of some combination of sector-specific shocks and aggregate shocks. How much the former and how much the latter remains to be seen. But the only way to find the answer to this question is to entertain the hypothesis that both may be at play. APPENDIX A: PROOFS AND DERIVATIONS A.1. Solution for the Multisector Brock–Mirman Economy The method of solving for the closed form solution to the Brock–Mirman multisector problem is analogous to the method used in the one-sector ver-

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sion. This is known as the Howard policy improvement algorithm (see Sargent [27, p. 47]). The algorithm postulates a functional form for the policy function, implements the policy in the value function, maximizes the value function using the postulated policy, and then updates the policy based on the first-order condition for the maximization problem. Once a candidate solution is achieved, the postulated functional form can be checked to verify that it is true. Begin by postulating that Mt;i j = Gij Yti , for constants 0 ≤ Gij < 1, and i = Hoi Yti 1 − that K i is a linear function of output in sector i, Kt+1 PM t+1 i 18 Let Z = j=1 Gij ‘, where Ho is a constant strictly between 0 and 1. −1 I − 0‘ . Letting lowercase letters denote logarithms of uppercase variables, log output can then be written as (in vector form) yt0 = a0t Z + k0t αd Z + d10 Z;

(24)

where αd denotes an M × M diagonal matrix with the vector α on the diagonal and d1 is a vector of constants depending on G. With this, vector log capital becomes k0t = hˆ 0o + a0t−1 Z + k0t−1 αd Z;

(25)

where hˆ 0o denotes a constant vector. Substituting recursively into (25) results in k0t = hˆ 0o ’I + αd Z + · · · + αd Z‘t−1 “ + a0t−1 Z + a0t−2 Zαd Z‘ + · · · + a00 Zαd Z‘t−1 + k00 αd Z‘t :

(26)

Now form the value function for the first iteration using the vectors ct = lnCt ‘ and θ, as Jo Ko ; Ao ‘ = Eo

∞ X t=0

δt ct0 θ;



d2i = ln ’1 −

where cti = Yti + d2i and 

Hoi “

1−

X s∈Sh

 Gis :

(27)

Substituting in for yt using (24) and for k0t using (26) and taking expectations at time zero results in Jo Ko ; Ao ‘ = R0 + a0o R1 + k0o R2 ;

(28)

where R0 = ’hˆ 0o αd + d10 + d20 “Zθ, R1 = Zθ, and R2 = αd Zθ. Note that R0 is a relatively uninteresting vector of constants that affects the steady state of the economy but not its dynamics. 18 Throughout, where equations are written for representative sector i it is assumed that they hold for i = 1; : : : ; M.

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The end is near. Note that the social planner’s problem can be formulated recursively in the Bellman form as  JKo ; Ao ‘ = max θ0 ct + δE’JK1 ; A1 ‘“ : K1

(29)

Now form the next iteration’s value function, written in Bellman form, using (28), as J1 Ko ; Ao ‘ = max

M X

K1i i=1

θi lnY˜ oi − K1i ‘ + δER0 + a01 R1 + k01 R2 ‘;

(30)

P where Y˜ ti = Yti 1 − s∈Sh Gis ‘. Solving the first-order condition from (30) results in (31) where the initial postulate for the policy function is verified since θi > 0: K1i =

δR2i Y˜ i ≡ H1i Y˜ oi : θi + δR2i o

(31)

Further iterations on the policy rule using the algorithm would evidently result in the same solution since H1i satisfies the requirement that the constant coefficient be between 0 and 1. Hence, (28) is the planner’s value function. Using the definition for R2 above it is evident that H1i =

δαi θi +

PM

j=1 Zij θj : P δαi M j=1 Zij θj

Investment in sector i capital is increasing in its productivity αi ‘ and decreasing in the representative consumer’s relative preference for good i θi ‘. One detail remains: solving for Gkj . This is straightforward. From the first-order condition for choosing Mkj one can show that Gkj is given by Gkj =

θj 1 − H1k ‘1 − θk 1 −

j H1 ‘1

−

PM

l=1

Gkl ‘

l=1

Gjl ‘

PM

γkj :

Given the equilibrium policy rule (7) for investment at time t it is easy to calculate the percentage change in investment or output in sector i at time t + p in response to a supply shock in sector j at time t. The gradient of (7) at time t + p with respect to at is ∇at kt+p ≡ Rp = Z 0 αd ‘p−1 Z 0 for p > 0, where Z 0 αd ‘s is the s-fold product of the matrix Z 0 αd ‘. This is denoted as Rp since the ijth element has the interpretation of an impulse j response function for the percent deviation of Kt from its steady state level p periods after a supply shock in sector i.

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A.2. Proof of Theorem 1 The proof is taken from Dupor [12] and follows directly by noting that l is an eigenvector of the matrices I, αd , and, under Assumption D.2, 0. Let 8eiω ‘ ≡ I − αd eiω − 0‘ and 2eiω ‘ ≡ 8eiω ‘−1 . Since matrix inversion preserves eigenvectors, 2eiω ‘l = 1/1 − αeiω − γ‘l. The spectrum for aggregate capital with aggregation weights w = 1/M‘l is given by Sω‘ =

1 l0 2eiω ‘0 2eiω ‘l; 2πM 2

(32)

where the transpose of a complex matrix denotes the transpose conjugate. Substituting in for 2eiω ‘l completes the proof. A.3. Proof of Theorem 2 Under Assumption H.2, the input-use matrix can be written as 0 = ˜ 0 and let 2eiω ‘ ≡ 8eiω ‘−1 . Using the ˜ 0 . Let 8eiω ‘ ≡ 1 − αeiω ‘I − lv lv Sherman–Morrison formula for matrix inverses (see Press et al. [26, p. 73]) it is possible to write 2eiω ‘ as iω

iω −1

2e ‘ = 1 − αe ‘



 1 0 ˜ lv : I+ 1 − αeiω − x‘

(33)

The spectrum for aggregate capital is given by Sω‘ = 2π‘−1 w 0 2eiω ‘0 2eiω ‘w: When w = 1/M‘l, this simplifies to (16), repeated here: Sω‘ =

1 − αeiω ‘−1 1 − αe−iω ‘−1 2π1 − αeiω − x‘1 − αe−iω − x‘   1 1 ’1 − αeiω − x + 㑐1 − αe−iω − x + γ‘ − γ 2 “ + γ 2 : × M N

When N = M, this simplifies further to Sω‘ =

1/2πM‘ : 1 − αeiω − 㑐1 − αe−iω − γ‘

cyclicality and sectoral linkages APPENDIX B Sectoral Classifications for Input-Use Matrices

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Notes. The table presents the sectoral definitions at the levels of disaggregation resulting in 6, 21, 36, and 77 sectors for the U.S. Economy. Definitions for the 77-sector level are taken from the Department of Commerce. Definitions at the 36-sector level are taken from Jorgenson, et al. (1987), as described in Section 4. Definitions at the 21-sector level are created by the author. Definitions at the 6-sector level correspond to one-digit SIC definitions. Definitions at the 523-sector level were not included to conserve space. They are available from the author’s website: http: //www-leland.stanford. edu/;mhorvath/.

REFERENCES 1. P. Bak, K. Chen, J. Scheinkman, and M. Woodford, Aggregate fluctuations from independent sectoral shocks: Self organized criticality in a model of production and inventory dynamics, Ricerche Econom. 47 (1993), 3–30. 2. S. Basu, Procyclical productivity: Increasing returns or cyclical utilization, Quart. J. Econom. 111 (1996), 719–752. 3. S. Basu and J. G. Fernald, “Aggregate Productivity and Aggregate Technology,” manuscript, Department of Economics, University of Michigan, 1997. 4. S. Basu and J. G. Fernald, Returns to scale in U.S. production: Estimates and implications, J. Polit. Econ. 105 (1997), 249–283. 5. S. Basu and M. S. Kimball, “Cyclical Productivity with Unobserved Input Variation,” manuscript, Hoover Institution, 1994. 6. M. Boldrin and M. Woodford, Endogenous fluctuations and chaos: A Survey, J. Monetary Econom. 25 (1990), 189–222. 7. W. Brock and L. Mirman, Optimal economic growth and uncertainty: The Discounted Case, J. Econom. Theory 4 (1972), 479–513. 8. C. Burnside, M. Eichenbaum, and S. T. Rebelo, Sectoral Solow residuals, Eur. Econom. Rev. 40 (1996), 861–869. 9. L. J. Christiano and M. Eichenbaum, Current real-business-cycle theories and aggregate labor-market fluctuations, Amer. Econom. Rev. 82 (1992), 430–450. 10. T. Cogley and J. M. Nason, Output dynamics in real-business cycle models, Amer. Econom. Rev. 85 (1995), 492–511.

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11. S. Davis, Allocative disturbance and specific capital in real business cycle theories, Amer. Econom. Assoc. Papers Proc. 77 (1987), 326–332. 12. W. Dupor, “Aggregation and Irrelevance in Multi-Sector Models,” manuscript, University of Chicago, 1996. 13. W. Dupor, “Aggregate Fluctations and Production Complementarities,” manuscript, University of Chicago, 1996. 14. R. E. Hall, The relation between price and marginal cost in U.S. industry, J. Polit. Econ. 96 (1988), 921–947. 15. R. E. Hall, Invariance properties of Solow’s productivity residual, in “Growth/Productivity/ Unemployment: Essays to Celebrate Bob Solow’s Birthday” (P. Diamond, Ed.), MIT Press, Cambridge, MA. 16. J. D. Hamilton, “Oil and the macroeconomy since World War II,” J. Political Econ. 91 (1983), 228–248. 17. J. D. Hamilton, “Time Series Analysis,” Princeton Univ. Press, Princeton, NJ, 1994. 18. M. T. K. Horvath, “New Mechanisms in Macroeconomics,” Ph.D. dissertation, Northwestern University, 1994. 19. M. T. K. Horvath, Sectoral shocks and aggregate fluctuations, J. Monetary Econom. forthcoming. 20. M. T. K. Horvath and R. Verbrugge, “Shocks and Sectoral Interactions: An Empirical Investigation,” manuscript, Department of Economics, Stanford University, 1997. 21. D. Jorgenson, F. Gollop, and B. Fraumeni, “Productivity and U.S. Economic Growth,” Harvard Univ. Press, Cambridge, MA, 1987. 22. B. Jovanovic, “Micro Uncertainty and Aggregate Fluctuations,” manuscript, C.V. Starr Center for Applied Economics, 1984. 23. J. Long and C. Plosser, Real business cycles, J. Polit. Econ. 91 (1983), 39–69. 24. K. M. Murphy, A. Shleifer, and R. W. Vishny, Building blocks of market clearing business cycle models, in “NBER Macroeconomics Annual 1989,” (O. Blanchard and S. Fischer, Eds.), MIT Press, Cambridge, MA, 1989. 25. C. I. Plosser, Understanding real business cycles, J. Econom. Persp. 3 (1989), 51–77. 26. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical Recipes in C,” 2nd ed., Cambridge Univ. Press, 1992. 27. T. J. Sargent, “Dynamic Macroeconomic Theory,” Harvard Univ. Press, Cambridge, MA, 1989. 28. J. Scheinkman, Nonlinearities in economic dynamics, Econom. J. 100 (1990), 33–47 (suppl.). 29. J. Scheinkman and M. Woodford, Self-organized criticality and economic fluctuations, Amer. Econom. Assoc. Papers Proc. 84 (1994), 417–421. 30. J. Shea, “Complementarities and Comovements,” working paper 9402, Social Systems Research Institute. University of Wisconsin, 1994. 31. R. J. Verbrugge, “Aggregate Fluctutations from Local Market Interactions,” manuscript, Department of Economics, Virginia Polytechnic Institute, 1996.