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Accounting far Intermedlate Input: The Lluk Between Sectored and Aggregate Measures of Productlv~y Orowtb P~K H. o0~0P An economies macro-economic o~ecdve ~ to maxlmlze aggregate out- put given supples of primal Actors of producdon. Open this char- acterlzatlon of the macro-economic problem, neither lutermedlate lupus nor deUverles to lntermedlate demand need be considered. Only psalms inputs and deNverles destined far Anal demand are relevant. Aggregate productlv~y analysis historically has had this macro-economic character. The earn nadona1 lucome approach adopted by Kuzuets (1941j, the production ~ncdon model introduced by Solow (1957), and the cared empldca1 research reported by Denlson (1962, 1974) abstract Mom both lnter-ludust~ transactions and shlOs in pectoral productlv~y. Kendrlck's pathbreaking sectora1 research published in 1961 shares The mateda1 reposed in this paper ~ ~ theoretical result derived as paw of a much larger Rudy supposed by the Nabona1 Sconce Foundation under grant APR75-18713AO1. The senior research ~"onnel include Dale Jorgenson, Charles Hulten, pinko Nishimizu, and me, who ae at Harvard University. the Urban Institute, Princeton University, and Unive[- d~ ~ Wh~nsin, Sly. Any opinions, endings, conclusions, or Commendations expressed in this paper are those of the Search group and do not necessarHy reject [he Hews of the National Sconce foundation. Though I have wrhten 1hk paper. 1 should not be vised as the materials sole originator. Though 1 assume ~N msponsibUi~ far 1hk papillar potion, all Cur fins ~nthbuted ~ the de~lopmen1 ~ 1hk maria. 1 wrote this paper on behalf of my research colleagues. ~ 0 Ago

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Accounting for Intermediate Input 319 the macro-economic orientation of the aggregate studies. Adopting Solow's model of aggregate production, Kendrick (1961, 1973) focuses on sectoral net output, defined as real value added. Hendrick (1973, p. 17) says: From a macroeconomic viewpoint, there are persuasive arguments for using real product estimates (i.e., real value added, in which real intermediate costs are deducted from the real value of total gross output). The real industry prod- uct estimates in relation to real factor (labor and capital) costs alone indicate changes in the efficiency with which the basic factors resident in the industry are used to add value to the intermediate products purchased from other in- dustries. Intermediate input plays no role in Kendrick's macro-economic char- acterization of sectoral productivity. The correct measurement of the rate of growth in economy-wide pro- ductivity is no less important today than it has ever been, but the events of the current decade impress us with the importance of the micro- economic foundations of aggregate productivity change. The OPEC shock in 1973 and the subsequent production bottlenecks resulting from the shortage of critical raw materials have led to a growing awareness that long-term economic trends can be understood and projected only by relating economy-wide developments to changes in economic activity at the level of industrial sectors. Advances in productivity in an individual industry contribute to aggregate economic growth both directly through increased deliveries to final demand and indirectly through increased deliveries to sectors dependent on its output as intermediate input. Recognition of the important role of intermediate input is in itself not a novel insight. References to this notion appear in the economics literature as early as 1940. Fabricant (1940, 1942), for example, dis- cusses the importance of intermediate input and interindustry relations for understanding changes in factory output and employment. How- ever, the wholehearted incorporation of intermediate input into sectoral accounts has been hampered by the absence of a formal proof dem- onstrating that the economy-wide measure of productivity is in fact some aggregate over the sectoral productivity measures. Domar (1961), Watanabe (1971), and Star (1974) have previously addressed this aggregation problem. Domar and Watanabe begin from accounting frameworks; Star bases his model on the theory of produc- tion. However, the proposed aggregating procedures rely on inappro- priate and/or unnecessarily restrictive assumptions. Domar begins with a Cobb-Douglas index of productive inputs and excludes consideration

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320 PAPERS of all intermediate inputs provided by an industry to itself. Watanabe's economy-wide analysis is based on the gross production possibility frontier. This conflicts with the correct interpretation of the aggregate production account which is measured net of all intermediate produc- tion. Star assumes that the ratio of sectoral intermediate input to output is constant over time. Nishimizu (1974) provides an excellent discussion of the limitations of the Domar, Watanabe, and Star models and dem- onstrates the biases introduced by their application. The primary objective of this paper is to motivate and demonstrate the proper theoretical link between productivity change in the micro- sectors with productivity change in the macro-economy. The particular restrictions introduced into a sectoral analysis by failing to incorporate intermediate input are described in the following section. In the section on aggregation over sectors, I formally demonstrate how, through ap- propriate aggregation, the sectoral production accounts can be used to derive the economy-wide production account. The former accounts in- clude both intermediate input and deliveries to intermediate demand; the latter, consistent with Solow's original framework, excludes the pro- duction and use of intermediate input. INTERMEDIATE INPUT The micro-economic theory of production provides the appropriate point of departure for a sectoral analysis of technical change. The familiar production function can be used to model output as a function of cap- ital, labor, intermediate input, and time. For purposes of this paper, I abstract from a discussion of the model's particular functional form. All that is required is that the production function include among its argu- ments all primary and intermediate inputs and time. Failing to include intermediate input (or, for that matter, any input) in a sectoral produc- tion function unnecessarily restricts the generality of the basic model of production. Moreover, should the restrictions prove inappropriate, a model excluding intermediate input maintains an incorrect description of not only producer behavior but technical change as well. PRODUCER BEHAVIOR As a general model, micro-economic theory requires a production func- tion that includes among its arguments all factors of production, both primary and intermediate inputs. The unrestricted production function must also encompass substitution possibilities among all inputs and im

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Accounting for Intermediate Input 321 pose no restrictions on the substitution possibilities among the individ- ual inputs. The elasticity of substitution between any pair of inputs is not necessarily constant over time, nor need it be equal to the elasticity of substitution between any other pair of inputs. The unrestricted sec- toral production account can be represented by the functionf, Q=f[K,L,X,t], where Q is output, t represents time, and K, L, and X are capital, labor, and intermediate inputs, respectively. Excluding intermediate input is equivalent to assuming the existence of the sub-function g: Q = f [g(K, L, t), X], where g is separable from X. If g exists, then the properties of g, includ- ing technical change, can be analyzed in isolation from intermediate in- put. This notion of separability underlies the familiar sectoral value- added ~ V) model, V=g(K,L, t). Maintaining the separability of g from X significantly restricts the generality of the sectoral model of production. In particular, value- added separability implies that the marginal rate of substitution between any pair of arguments within the value-added subfunction g is indepen- dent of intermediate input. As Berndt and Christensen (1973) dem- onstrate, separability imposes certain equality restrictions on the Allen partial elasticities of substitution ~ air). Weak or groupwise separability of g from X requires that (JKX = ~LX; a stronger version of the same hy- pothesis, strong or explicit groupwise separability, additionally requires _ _ ~ AX - (JLX - I The importance of including intermediate goods and services in a study of sectoral productivity is easily demonstrated. A supermarket manager can choose to have his own employee display frozen ice cream products in the frozen foods cabinet or he may contract with the raw materials supplier to have its delivery person display the product. The former is a direct labor cost to the supermarket; the latter is an expense related to intermediate input. Presumably, the store manager makes this choice such that the ratio of marginal products equals the corresponding ratio of factor prices. Imposing value-added separability unnecessarily restricts the production function describing the supermarket's operation,

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322 PAPERS the characterization of the manager's rational behavior, and the neces- sary conditions for producer equilibrium. TECHNICAL CHANGE In addition to its static restrictions on the model of production, main- taining value-added separability also limits the role of technical change. Stated more formally, value-added separability unnecessarily restricts the characterization of technical change in the sectoral model of produc- tion. Maintaining value-added separability is equivalent to assuming that intermediate input cannot be a source or medium of productivity growth. If technical change occurs, it can affect output only through g. If technical change is of the factor augmenting variety, it can augment only capital and/or labor. Both Hicks-neutral and Leontief-neutral tech- nical change are precluded. Value-added separability significantly restricts the static and dynamic properties of the sectoral model of production. The micro-economic theory of production, the associated principles of rational producer be- havior, and the necessary conditions for producer equilibrium can be most defensibly applied only if all inputs are included in the sectoral production account. The unnecessary separability restrictions can be avoided by simply incorporating intermediate input into the model, treating it symmetrically with capital and labor inputs. At most, value- added separability should be a testable hypothesis, not a maintained as- sumption. 2 AGGREGATION OVER SECTORS The discussion in the present section is unavoidably technical, but the use of mathematics permits the concise demonstration of three critical propositions. Proposition 1 The rate of sectoral technical change derived from a model treating all inputs symmetrically is less than the corresponding rate resulting from a model maintaining value-added separability. The Hicks-neutrality maintains that technical change equally augments all inputs. Leontief- neutrality implies that intermediate input is the only source of technical advance. 2For a discussion of how value-added separability can be formally tested as part of an econometric model of sectoral production, see Gollop and Jorgenson (1979).

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Accounting for Intermediate Input 323 two rates differ by an amount that is proportional to the sector's share of intermediate input purchases in its total factor payments. Proposition 2 Given estimates of sectoral technical change derived from a model incorporating both primary and intermediate inputs, the economy-wide measure of productivity growth equals a weighted sum of the sectoral rates of technical change. The appropriate weights are the ratios of the value of each sector's gross output (deliveries to final de- mand plus deliveries to intermediate demand) to the value of aggregate output (the economy-wide sum of value added). Proposition 3 Given estimates of sectoral technical change derived from a model excluding intermediate input, the correct measure of economy-wide productivity growth equals a weighted average of the sec- toral rates of technical change with weights equal to the ratios of the value of each sector's net output (value added) to the value of aggre- gate output. The demonstration of these propositions follows closely the arguments developed in Nishimizu (19741. However, two important amendments have been incorporated into the following discussion. First, Hicks-neu- trality is not assumed. Second, the output price used in measuring the current-dollar value of a sector's output is not equivalent to the price associated with that sector's output when used as intermediate input into a purchasing sector's production process. The former is measured as a producers' price; the latter is measured as a consumers' price. The former is net of all sales and excise taxes and gross of subsidies received by producers; the latter is gross of sales and excise taxes attributed to the output of the sector supplying the intermediate input but net of any subsidies to producers. The formal derivation of the three propositions has been divided into four sections. Sectoral measures of technical change corresponding to production functions respectively including and excluding intermediate input are derived in the first section. Proposition 1 follows directly. The development of the aggregate measure of productivity growth is de- scribed in the second section. The third section develops Proposition 2. The sector specific weights required to form the aggregate measure from correctly specified sectoral estimates are derived. Finally, the construction of the aggregate measure from sectoral estimates derived from models maintaining value-added separability is described in the fourth section. Proposition 3 follows directly.

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324 SECTORAL MODELS PAPERS As motivated in the section above on intermediate input, the appropriate specification of an industry's technology is a production function incor- porating all primary and intermediate inputs and time: Xj = fj(K,j, , K`ej, , Kmj; L~j, where , L`j, , Lrj; Xjj, , Xij, , Xnj; t) Xj quantity of the jth industry's gross output; Kkj kth capital input used in the jth industry; L'j Ith labor input used in the jth industry; Xij quantity of the ith intermediate input used in the jth industry. (1) Totally differentiating (1) logarithmically with respect to time, the total rate of growth in output can be decomposed into its source components: X,j Xj k ~ in K where ej = blnX; Kkj blnX; + Knj , ~ lnL, Xj Xj Kkj d ln K,r K'rj dt L,j d ln L' L`j dt Xij = d ln X Xij dt blnX ~t d ln X dt . k = 1, 2, , m, 1= 1,2, ~.,r, i= 1,2, ~.,n, L,j + <, ~ lnX L, i d ln X (2) Assuming competitive equilibrium in all output and input markets implies that each input is paid the value of its marginal product. Conse- quently, the output elasticities appearing in (2) can be characterized by factor income shares: ~ ln Xj P ki Kki = - = l k bln K,~j qjXj J k = 1,2, ~.,m,

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Accounting for Intermediate Input dlnX dlnL dlnX dlnX qjX p _ eij 325 ED 1=1,2,---. i = 1, 2, A, n, where qj is the price received by producers of the jth output and Pej, Pa, and Pij are the prices paid by consumers in sector j for the kth cap- ital, Ith labor, and ith intermediate inputs, respectively. Maintaining constant returns to scale assures consistency with the ac- counting identity: ~ em + ~ en + ~ eij = 1. Equation (2) can then be written in equivalent form, . . . . e = X, _ ~ Gil -- ~ Eli T. ~ Oil Xy ~ =,j _ ~L,j i (3) The rate of technical change ej for each sector can be expressed as the rate of growth of the corresponding sector's output less a weighted average of the rates of growth of capital, labor, and intermediate inputs into the sector. The appropriate weights are given by factor income shares with output and all input prices valued in terms relevant to sector j. The current-dollar value of output is measured in producers' prices; inputs are valued in consumers' prices. If the sectoral model of production is restricted so as to maintain value-added separability, then, following the arguments developed in the intermediate input section, the relevant model of production becomes the net production function, Vj = gi(K~j, K2j, , Kmj; Lid, L2j, ~ Lrj; t), (4) where Vj is the jth sector's net output or, equivalently, quantity of value added. Paralleling the derivation developed for the gross production function fi, the total rate of growth in net output can be similarly decomposed into its source components, Vj * + ~ bin Vj Kkj + ~ bin V; Lo; Vj i k ~ ln Kaj K#j ~ ~ in LO Lij (5)

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326 PAPERS where ej* = ~ in Vj/8t. Under constant returns to scale and in com- petitive equilibrium, the implied rate of technical change for each sector j is defined as the rate of growth of the sector's net output less a weighted average of the rates of growth of the primary inputs ej* = V - ~ Ski* K ~ I* where (6) eki - v ~k = 1, 2, , m, e,j 5 qjVv, 1 = 1, 2, I, r, with qjV, the price of the jth sector's net output, measured as a pro- ducers' price. The demonstration of Proposition 1 follows directly from a compar- ison of ej and ej*. The formal comparison begins with the definition of the value of a sector's net output, qj v Vj = qj X,;- ~ pjjXij; (7) value added is defined as the value of gross output less the value of all intermediate input purchases. Totally differentiating (7) with respect to time yields an expression that can be written in the form Vj_ qjXj Xj _ . _ ~ _ Vj q,jV V,j X,j ~ Substituting (8) into (6) and using (3) yield pijXij Xi; qeiV V.j Xj,j ~ qjX,j ~ ~ qjXj-~ p jjXij] " (8) (9) where, since the ratio of gross output to value added is necessarily greater than one, qjXj q jXj-~ p jjXij

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Accounting for Intermediate Input It necessarily follows that ej < ej* and, verifying Proposition 1, ej = (1- ~ eij' ej*; 327 (10) ej is less than ej* by an amount that is proportional to the income share of intermediate input in total factor payments. AGGREGATE MODEL An economy's macro-economic objective is to maximize aggregate out- put given supplies of primary factors of production. The appropriate macro-economic characterization of aggregate technical change be- gins with defining aggregate output as a proportion of all sectoral quan- tities of value added. The maximum value of aggregate ouput (^y) can then be expressed as a function of all quantities of value added, all sup- plies of primary inputs, and time: ~ = F(V~, V2, . Vn; Kit, Kit, , Kmn; Late Lit, , Lrn; t). (11) Consistent with earlier analyses of aggregate productivity growth, the model abstracts from both intermediate inputs and output deliveries to intermediate demand. Only primary inputs and quantities of value added are relevant in an aggregate model. The model of aggregate output is characterized by constant returns to scale. This is required by the constraints of the maximization prob- lem, the sectoral value-added functions, which are themselves linear homogeneous functions in the primary factors of production. The func- tion F is therefore homogeneous of degree minus one in the quantities of value added, homogeneous of degree one in the factor supplies, and homogeneous of degree zero in quantities of value added and factor supplies together. The measure of aggregate productivity growth is derived by first fixing the level of aggregate output at unity: ~ = F(V~, V2, ., V,,; K,,, K,2, ~ K,,~,,; L,,. L,2. - . Urn; t). (12) This expression now defines the production possibility frontier for the economy as a whole. The rate of aggregate technical change (E) is de- termined by taking the total logarithmic derivative of the function F with respect to time and solving for E:

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328 .i PAPERS E ~5 In F V, _ ~ ~ In F Ink; _ ~ ~ ~ In F . By, (13) ., where E = ~ in F/3t. a, Necessary conditions for producer equilibrium at the aggregate level maintain that the aggregate output elasticities appearing in (13) can be characterized by value shares: 3 ~ lnF = _,7 VV/~ I've bin F lnF nK lnF in L,j A, .,,~ V/ Vj ., pki Kki/ qiv V; ./ pljL'j/E qiv Vj ., j = 1, 2, , n, k = 1,2, i, m;j= 1,2, 1= 1,2, . . . , n, , r;j= 1,2, ~., n. The expression for the rate of aggregate technical change can then be written in equivalent form: E = ~ (qiv Vj/[ qiv Vj) V, _ ~ ~ (pkiKki/[ qivVj) ~ -~ ~ (p/jL/j/5qjVVj) (14) Given this definition of aggregate technical change, the important issue is how E can be formed by aggregating over the sectoral measures ej and ej*. Since ej is generated from a model incorporating inter-indus- try transactions, the aggregating algorithm must capture the contribu- tions of each sector's productivity growth through deliveries of its output to both final demand and intermediate demand. In contrast, since the sectoral model assuming value-added separability abstracts from all 3Under constant returns to scale, the elasticities with respect to all quantities of value added sum to minus unity; the corresponding value shares sum to unity. Similarly, the sum of the elasticities with respect to all components of capital and labor inputs and the sum of the corresponding value shares equal unity.

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Accounting for Intermediate Input 329 inter-industry flows, aggregating over ej* must capture sectoral advances in productivity transmitted only through value-added contributions to aggregate output. AGGREGATION OVER e ,j The value of each sector's output is equal to the sum of the values of intermediate demand for that output by all n sectors and the value of final demand for that output: q jXj = ~ q,jiX,;i + q,jY Yi, (15) where {qji} is the set of prices received by the jth sector for output de- livered to intermediate demand and q jy is the price received by the jth sec- tor for output delivered to final demand. Totally differentiating (15) logarithmically with respect to time leads to an expression decomposing the growth rate of sectoral output into the growth rates of its delivered components: X ~ qjiXji X,i + qjy Yj Yj Xj i q jXj Xj; q jXj Yj (16) Substituting (16) into (3) permits the measure of sectoral productivity to be expressed in terms identifying the immediate uses of the sector's out- put: qjiXji Xji + qjy Yj Yj i , q jXj Xj; q,Xj Yi pkiKki Kkj peal'; Lo p,jX,, X,j - - ~ (17) k q jXj Kkj / q jXj Lo; i q jXj X,; This formulation makes clear that an analysis of sectoral technical change accounts for deliveries to both final and intermediate demands. A similar set of substitutions permits the rate of aggregate technical change E in (14) to be decomposed into terms identifying the transmis- sion of advances in sectoral productivity through both inter-industry sales and direct deliveries to final demand. Using (8), the growth rate of each sector's quantity of value added in (14) can be replaced by an expression in terms of gross output and intermediate inputs:

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330 E = PAPERS [(qixj/~qjVvj)~x-~ ~(pijxij/~qjVvj)- j j j , . j Xij -~ ~ (pkjKkj/[qjV Vj) ~ _ ~ ~ ~ljLlj/~qjv Vj)-L b (18) Furthermore, the growth rate of each sector's gross output can be de- composed into deliveries to final and intermediate demands using (16): E =E [(qjixji/EqjVvj) X~ + ~ (qjyyj/~qjvVj) yet (pkjKkj/~qjv~j) ~-~ ~ (p,].L,j/ EqjvVj) Lay (pijxij/E qjV Vj' Xi/ (19) Summing (17) over all n sectors and substituting the result into (19) produce the relation between the aggregate and sectoral rates of tech- nical change: E = ~ (qjXj/EqjVVj)ej. (20) The aggregate measure of productivity growth can be expressed as a weighted sum of the sectoral rates of technical change where each rate is derived from a sectoral model appropriately including all primary and intermediate inputs. Inspection of the weights (20) verifies Proposition 2. The weights are given by the ratios of the values of sectoral gross output to the value of aggregate output. The sum of these weights exceeds unity since each sector contributes to the rate of technical change for the aggregate econ- omy through its deliveries to 'ooth final demand and intermediate de- mand. A heuristic but equally persuasive case can also be developed for the weights associated with ej in (201. Consider an individual sector that experiences an advance in its rate of technical change. Holding constant all primary and intermediate inputs, the sector can provide the economy with increased output. The objective of creating the appropriate weight for this sector's technological advance is to correctly assign causal re

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Accounting for Intermediate Input 331 sponsibility to this sector for any effect on aggregate technical change. Since the ultimate macro-economic concern is the effect of this tech- nical change on aggregate output, the appropriate denominator in the weight is the sum of all n sector contributions to aggregate output. Since the individual sector transmits the benefits of its productivity growth to final consumers both directly through deliveries to final de- mand and indirectly through deliveries to intermediate demand, the appropriate numerator in the weight is the sector's total output,i.e., the sum of its deliveries to final and intermediate demands.4 AGGREGATION OVER ej* The macro-economic orientation of the sectoral model assuming value- added separability abstracts from all inter-industry transactions. The relevant measure of sectoral output is the quantity of value added. The aggregate production possibility frontier similarly abstracts from all inter-industry deliveries. The relevant measure of aggregate output is defined in terms of each sector's quantity of value added. The link be- tween the sectoral and aggregate rates of productivity growth is direct. Substituting (6) into (14) leads to the equivalent expression E = ~ (qjV Vj/[ qjV All ej* j J (21) The rate of aggregate technical change equals a weighted average of the sectoral rates of productivity growth where each rate is derived from a sectoral model maintaining value-added separability.5 4Note that an increase in ej does not necessarily imply that the intermediate input Xji will be more productive in sector i. Sector i has an independent rate of technical change ei. An increase in ej implies only that the economy now has more output Xj available for delivery to final demand and to vertically higher sectors purchasing Xj as intermediate input. sIt is important to emphasize that the measure of E used in the development of these arguments is based on a production possibility frontier (12), which distinguishes both primary inputs and quantities of value added by sector. Conventional economy-w~de productivity studies suppress the sectoral detail explicit in (12). The relevant analog of (12) becomes V = F(KI, K2, A, Km; LI, L2, . , Lr; I). The appropriate prices associated with V and each Kk and Ll would be economy-wide prices_q v, PKk ~ and PLI, respectively. The aggregate rate of technical change has a new interpretation. Instead of equalling E defined in (14), it now equals E' where . . E, = _ _ ~ PKk k . k _ . PLI I . I. V k q V Kk l qVv L! (Continued Overleaf)

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332 PAPERS Proposition 3 follows directly from (211. The appropriate weights are given by each sector's share in aggregate value added. Consequently, the weights sum to unity. It is important to emphasize the corollary that follows from (20) and (21~: E = ~ (qjxj/5 qjv Vj) ej = ~ (qjv Vj/[ qjv Vj) ej* J J J J (22) The appropriately weighted sum contribution of sectoral advances in technology to aggregate productivity growth is identical whether the sectoral rates of technical change are developed from models including or excluding intermediate input. CONCLUSION Incorporating inter-industry transactions into an analysis of productivity growth conforms with both sectoral and economy-wide principles of economic accounting and production. First, including intermediate in- put in the sectoral model of production preserves the full integrity of rational producer behavior. Value-added separability is not required. Neither Hicks-neutral nor Leontief-neutral technical change is precluded. Second, the interdependence of economic activity among micro-economic sectors is explicitly recognized. The model identifies the important role played by intermediate input in transmitting the benefits of sectoral advances in productivity throughout the economy. Third, incorporating intermediate input permits the derivation of the appropriate link be- t~reen sectoral and aggregate measures of productivity. The measure of aggregate technical change derived from a macro-economic model Beginning from such a model changes none of the substantive results of this paper. How- ever, the reader should be aware that defining aggregate technical change as E' requires that three terms be added to both (20) and (21): ~ ~ - + ~ ( ( q V-q j V ) V; / ~ q j V Vj ) - ~ ~ ~ i(( p kj -p Kk )K kj /E qj v Vj ) _ - ~ ~ (( p/j-p~/)L/j/~ qi VVj) / / / L/j L/j These terms reflect the contribution to aggregate productivity growth of changes in the distribution of value added and primary factors of production among sectors.

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Accounting for Intermediate Input 333 focusing only on aggregate output and primary inputs can equivalently be derived by appropriately weighting over the sectoral measures based on micro-economic models of production that treat capital, labor, and intermediate inputs symmetrically. Advances in productivity in indi vidual industries contribute to aggregate economic growth both directly through increased deliveries to final demand and indirectly through increased deliveries to sectors dependent on inter-industry transactions for intermediate input. REFERENCES Berndt, Ernst, and Christensen, Laurits R. (1973) The internal structure of functional relationships: separability, substitution, and aggregation. The Review of Economic Studies 40(3):403-410. Denison, Edward F. (1962) Sources of Economic Growth in the United States and the Alternatives Before Us. New York: Committee for Economic Development. Denison, Edward F. (1974) Accounting for United States Economic Growth, 1929-1969. Washington, D.C.: Brookings Institution. Domar, Evsey D. (1961) On the measurement of technological change. The Economic Journal 71(December):709-729. Fabricant, Solomon (1940) The Output of Manufacturing Industries: 1899-1937. New York: National Bureau of Economic Research. Fabricant, Solomon (1942) Employment in Manufacturing, 1899-1939: An Analysis of Its Relation to the Volume of Production. New York: National Bureau of Economic Research. Gollop, Frank M., and Jorgenson, Dale W. (1979) U.S. Economic Growth, 1947-1973. Unpublished manuscript. Department of Economics, University of Wisconsin. Kendrick, John W. (1961) Productivity Trends in the United States. National Bureau of Economic Research. Princeton: Princeton University Press. Kendrick, John W. (1973) Postwar Productivity Trends in the United States. 1948-1969. New York: National Bureau of Economic Research. Kuznets, Simon (1941) National Income and Its Composition, 1919-1938. New York: National Bureau of Economic Research. Nishimizu, Mieko (1974) Total Factor Productivity Analysis: A Disaggregated Study of the Postwar Japanese Economy with Explicit Consideration of Intermediate Inputs, and Comparison with the United States. Unpublished PhD dissertation, Johns Hopkins University. Solow, Robert M. (1957) Technical change and the aggregate production function. Review of Economics and Statistics 39(3):312-320. Star, Spencer (1974) Accounting for the growth of output. American Economic Review 61(1): 123-135. Watanabe, Tsunehiko (1971) A note on measuring sectoral input productivity. Review of Income and Wealth 17(4):335-340.