Contributions to Macroeconomics Volume , Issue   Article  Is the U.S. Aggregate Production Function Cobb-Douglas? New Estimates of the Elasticity of Substitution ∗ Pol Antr`as ∗Harvard University, [email protected] Copyright (cid:13)c2004 by the authors. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, bepress. Contributions to Macroeconomics is one of The B.E. Journals in Macroeconomics, produced by The Berkeley Electronic Press (bepress). http://www.bepress.com/bejm. Is the U.S. Aggregate Production Function Cobb-Douglas? New Estimates of the ∗ Elasticity of Substitution Pol Antra`s Abstract Ipresentnewestimatesoftheelasticityofsubstitutionbetweencapitalandlaborusing data from the private sector of the U.S. economy for the period 1948-1998. I first adopt Berndt’s (1976) specification, which assumes that technological change is Hicks neutral. Consistentlywithhisresults,Iestimateelasticitiesofsubstitutionthatarenotsignificantly different from one. I next show, however, that restricting the analysis to Hicks-neutral technological change necessarily biases the estimates of the elasticity towards one. When Imodifytheeconometricspecificationtoallowforbiasedtechnicalchange,Iobtainsignif- icantly lower estimates of the elasticity of substitution. I conclude that the U.S. economy is not well described by a Cobb-Douglas aggregate production function. I present esti- mates based on both classical regression analysis and time series analysis. In the process, I deal with issues related to the nonsphericality of the disturbances, the endogeneity of the regressors, and the nonstationarity of the series involved in the estimation. KEYWORDS: Capital-Labor Substitution, Technological Change ∗Harvard University, NBER and CEPR. Email: [email protected] Antràs: Is the U.S. Aggregate Production Function Cobb-Douglas? 1 1 Introduction The elasticity of substitution between capital and labor is a central parameter in economic theory. Models investigating the sources of economic growth and the determinants of the aggregate distribution of income have been found to deliver substantially di⁄erent implications depending on the particular value of the elasticity of substitution. Perhaps to a larger extend than in other (cid:133)elds, the elasticity of substitu- tionbetweencapitalandlaboriscentralingrowththeory, bothtraditionaland new. On the one hand, in the framework of the Neoclassical growth model, the sustainability of long-run growth in the absence of technological change depends crucially on whether the elasticity of substitution is greater than or smaller than one.1 On the other hand, the recent literature on induced tech- nical change has developed models that deliver very di⁄erent implications de- pending on the particular value of the elasticity of substitution. For instance, Acemoglu (2003) builds on the assumption of a lower-than-one elasticity of substitution to construct a model that rationalizes the coexistence of both purposeful labor- and capital-augmenting technological change in the transi- tional dynamics of an economy that converges to a balanced growth path in which technical change is purely labor-augmenting. Furthermore, as pointed outbyHsieh(2000),thevalueoftheelasticityofsubstitutionisalsorelevantto the empirical debate on the sources of economic growth (cf., Mankiw, Romer and Weil, 1992).2 In the (cid:133)eld of public (cid:133)nance, the value of the elasticity of substitution constitutes an important determinant of the response of investment behavior to tax policy. The late 1960(cid:146)s witnessed a lively debate between those who perceived (cid:133)scal policy as an e⁄ective tool for in(cid:135)uencing investment behavior (e.g., Hall and Jorgenson, 1967) and those who recognized only minor bene(cid:133)ts from tax incentives (e.g., Eisner and Nadiri, 1968). Their debate revolved around the issue of whether the elasticity of substitution between capital and labor was signi(cid:133)cantly below one, with a lower elasticity being associated with 1If the elasticity of substitution is greater than one, the marginal product of capital re- mainsboundedawayfromzeroasthecapitalstockgoestoin(cid:133)nity. Undercertainparameter restrictions,thisviolationoftheInadaconditioncanyieldlong-runendogenousgrowtheven in the absence of technological progress (cf. Barro and Sala-i-Martin, 1995, p. 44). 2Following Mankiw, Romer and Weil (1992), most studies trying to disentangle the rela- tiveroleoftechnologicalchangeandfactoraccumulationinexplainingcross-countryincome di⁄erences have assumed the elasticity of substitution to be equal to one. Hsieh (2000), shows that relaxing this assumption and allowing for biased technical change may alter substantially the results of these studies. Produced by The Berkeley Electronic Press, 2004 2 Contributions to Macroeconomics Vol. 4 [2004], No. 1, Article 4 a lower response of investment to tax bene(cid:133)ts.3 Soon after the explicit derivation of the Constant Elasticity of Substitu- tion (CES) production function by Arrow et al. (1961), a wealth of articles appeared trying to estimate this elasticity for the U.S. manufacturing sec- tor.4 Cross-sectional studies at the two-digit level tended to (cid:133)nd elasticities insigni(cid:133)cantly di⁄erent from one (e.g., Dhrymes and Zarembka, 1970). Lucas (1969), however, discussed several biases inherent in the use of cross-sectional data in the estimation of the elasticity. He suggested the use of time series data instead. Time series studies generally provided much lower estimates of the elasticity. Lucas (1969) himself estimated the elasticity of substitution to be somewhere between 0.3 and 0.5, while Maddala (1965), Coen (1969) and Eisner and Nadiri (1968) also computed estimates signi(cid:133)cantly below one. In a widely cited contribution, Berndt (1976) illustrated how the use of higher quality data translated into considerably higher time-series estimates of the elasticity, thus leading to a reconciliation of the time-series and cross- sectional studies. In particular, a careful construction of the series involved in the estimation led himto obtain time series estimates for the period 1929-1968 insigni(cid:133)cantly di⁄erent from one. It has become customary in the literature to cite Berndt(cid:146)s paper as providing evidence in favor of the assumption of a Cobb-Douglas functional form for the aggregate production function (e.g., Judd, 1987, Trostel, 1993). In this paper, I will start by following closely the approach suggested by Berndt (1976), which assumes that technological change is Hicks neutral. Us- ing time-series data from the private sector of the U.S. economy for the period 1948-1998, the (cid:133)rst result of this paper will be a con(cid:133)rmation of Berndt(cid:146)s (cid:133)nding of a unit elasticity of substitution between capital and labor when technological change is assumed to be Hicks neutral. The second and more substantive contribution of this paper will consist in demonstrating that in the presence of non-neutral technological change, Berndt(cid:146)s approach leads to estimates of the elasticity that are necessarily biased towards one. When the econometric speci(cid:133)cation is modi(cid:133)ed to allow for biased technical change, I generally obtain signi(cid:133)cantly lower estimates of the elasticity of substitution. The source of the bias is rather simple to illustrate. Suppose that U.S. aggregate output can be represented by a production function of the form: Y = A F(K ;L ), t t t t characterized by constant returns to scale in the two inputs, capital and labor. 3See Chirinko (2002) for more details. 4For a thorough review of this literature see Nerlove (1967) and Berndt (1976, 1991). http://www.bepress.com/bejm/contributions/vol4/iss1/art4 Antràs: Is the U.S. Aggregate Production Function Cobb-Douglas? 3 The parameter A is an index of technological e¢ ciency, which is neutral in t Hicks(cid:146)sense, i.e., in the sense that it has no e⁄ect on the ratio of marginal products for a given capital-labor ratio. Pro(cid:133)t maximization by (cid:133)rms in a competitive framework delivers two optimality conditions equalizing factor prices with their marginal products. Combining these conditions delivers r K f (k )k =f(k ) t t 0 t t t = , w L 1 f (k )k =f(k ) t t 0 t t t (cid:0) where f(k) is output per unit of labor, k is the capital-labor ratio, and r and w are the rental prices of capital and labor, respectively. As is well-known, in the United States, the value of the left-hand side of this expression has been remarkably stable throughout the post-WWII period, while the capital-labor ratiohassteadilyincreased. Itfollowsthatthisequationcanbeconsistentwith U.S. data only if f (k )k =f(k ) is not a function of k , i.e., only if F(K ;L ) is 0 t t t t t t a Cobb-Douglas production function.5 In words, when technological change is Hicks neutral and the capital-labor ratio grows through time, the only aggre- gateproductionfunctionconsistentwithconstantfactorsharesisonefeaturing a unit elasticity of substitution between capital and labor. As I will discuss in section 2, the approach of Berndt (1976) consists of running log-linear speci(cid:133)- cations closely related to the expression above. In light of this discussion, his (cid:133)nding of a unit elasticity of substitution should not be surprising. The main problem with Berndt(cid:146)s approach is that when technological change is allowed to a⁄ect the ratio of marginal products, the Cobb-Douglas production function ceases to be the only one consistent with stable factor shares. In particular, a well-known theorem in growth theory states that, in the presence of exponential labor-augmenting technological change, any well- behaved aggregate production function is consistent with a balanced growth path in which factor shares are constant. In a similar vein, Diamond, McFad- den and Rodriguez (1978) formally proved the impossibility of identifying the separaterolesoffactorsubstitutionandbiasedtechnologicalchangeingenerat- ing a given time series of factor shares and capital-labor ratios. The literature has generally circumvented this impossibility result by imposing some type of structure on the form of technological change. As I will discuss in section 5, when technological e¢ ciency grows exponentially these two e⁄ects can be separatedandthe elasticityof substitutioncanbe recoveredfromthe available data. Furthermore, myempiricalresultsbelowsuggestthatallowingforbiased technological change leads to estimates of the elasticity of substitution that 5Solving the di⁄erential equation f (k )k =f(k ) = (cid:11) yields y = Ck(cid:11), where C is a 0 t t t t constant of integration. Produced by The Berkeley Electronic Press, 2004 4 Contributions to Macroeconomics Vol. 4 [2004], No. 1, Article 4 are, in general, signi(cid:133)cantly lower than one. I conclude from my results that the U.S. aggregate production function does not appear to be Cobb-Douglas. This is not the (cid:133)rst paper to estimate the elasticity of substitution while takingintoaccountthepresenceofbiasedtechnologicalchange. Amongothers, David and van de Klundert (1965) and Kalt (1978) ran regressions analogous to those in section 5 below, and estimated elasticities equal to 0.32 and 0.76, respectively. Thispaperaddstothisliteratureinatleastthreerespects. First, by explicitly discussing and correcting the bias inherent in the assumption of Hicks-neutral technological change, I am able to reconcile the traditional low estimates of Lucas (1969) and others with the widely cited ones of Berndt (1976).6 Second, by focusing on a more recent period, I am able to bene(cid:133)t from the higher-quality data made available by the work of Herman (2000), Krusell et al. (2000), and Jorgenson and Ho (2000). Finally, my empirical analysis incorporates recent developments in the econometric analysis of time series that permit a better treatment of the nonstationary nature of the series involved in the estimation.7 The rest of the paper is organized as follows. In section 2, I follow Berndt (1976)inderivingsixalternativespeci(cid:133)cationsfortheestimationoftheelastic- ityofsubstitutionundertheassumptionofHicks-neutraltechnologicalchange. Section 3 discusses the data used in the estimations. Section 4 presents es- timates of the elasticity based on both classical econometrics techniques and modern time series analysis. Section 5 discusses the crucial misspeci(cid:133)cation in Berndt(cid:146)s (1976) contribution and presents estimates that correct for it by allowing for biased technical change. Section 6 concludes. 2 Model speci(cid:133)cation IbeginbyassumingthataggregateproductionintheU.S.privatesectorcanbe representedbyaconstantreturnstoscaleproductionfunctioncharacterizedby a constant elasticityof substitution betweenthe two factors, capital andlabor. 6Kalt(1978),forinstance,incorrectlydismissedBerndt(cid:146)sresultsbyclaimingthathehad estimated the elasticity (cid:147)without regard to technological change(cid:148)(p. 762). 7Following the dual cost function approach pioneered by Nerlove (1963) and Diewert (1971), a separate branch of the literature has provided estimates of the elasticity based on (cid:133)rst-order conditions derived from cost minimization rather than pro(cid:133)t maximization. For instance, Berndt and Christensen (1973) (cid:133)tted a translog cost function to the U.S. manufacturing sector for the period 1929-68, obtaining elasticities of substitution between capital equipment and labor and between capital structures and labor slightly higher than one. Nevertheless, their estimates should be treated with caution because, like Berndt (1976), the authors failed to deal properly with technological change. http://www.bepress.com/bejm/contributions/vol4/iss1/art4 Antràs: Is the U.S. Aggregate Production Function Cobb-Douglas? 5 Arrow et al. (1961) showed that the assumption of a constant elasticity of substitution implied the following functional form for the production function: (cid:27) Y = A (cid:14)K(cid:27)(cid:0)(cid:27)1 +(1 (cid:14))L(cid:27)(cid:0)(cid:27)1 (cid:27)(cid:0)1 , t t t t (cid:0) h i where Y is real output, K is the (cid:135)ow of services from the real capital stock, t t L is the (cid:135)ow of services from production and nonproduction workers, A t t is a Hicks-neutral technological shifter, (cid:14) is a distribution parameter, and the constant (cid:27) is the elasticity of substitution between capital and labor.8 Following Berndt (1976), it is useful to de(cid:133)ne the aggregate input function F Y =A , which given the assumption of Hicks-neutral technological change t t t (cid:17) isindependentofA . Pro(cid:133)tmaximizationby(cid:133)rmsinacompetitiveframework t implies two (cid:133)rst-order conditions, equating real factor prices to the real value of their marginal products. These conditions can be rewritten and expanded with an error term to obtain: log(F =K ) = (cid:11) +(cid:27)log(R =P )+" (1) t t 1 t t 1;t log(F =L ) = (cid:11) +(cid:27)log(W =P )+" , (2) t t 2 t t 2;t where R , W , and P are the prices of capital services, labor services, and t t t aggregate input F , respectively, and (cid:11) and (cid:11) are constants that depend on t 1 2 (cid:14).9 A third alternative speci(cid:133)cation can be obtained by subtracting (1) from (2) log(K =L ) = (cid:11) +(cid:27)log(W =R )+" . (3) t t 3 t t 3;t Following Berndt (1976) one can also rearrange equations (1) through (3) to obtain the following three reverse regressions: log(R =P ) = (cid:11) +(1=(cid:27))log(F =K )+" (4) t t 4 t t 4;t log(W =P ) = (cid:11) +(1=(cid:27))log(F =L )+" (5) t t 5 t t 5;t log(W =R ) = (cid:11) +(1=(cid:27))log(K =L )+" . (6) t t 6 t t 6;t I hereafter denote the estimates of (cid:27) based on equations (1) through (6) by (cid:27) , i = 1;:::6.10 As pointed out by Berndt (1976), in this bivariate setting, i 8The elasticity of substitution between capital and labor is de(cid:133)ned as (cid:27) = dlog(K=L)=dlog(F =F ), where F and are F the marginal products of capital and L K K L labor, respectively. 9A simple way to justify these disturbance terms is to appeal to optimization errors on the part of (cid:133)rms (cf., Berndt, 1991, p. 454). 10In the presence of imperfect competition in the product market, the markup becomes Produced by The Berkeley Electronic Press, 2004 6 Contributions to Macroeconomics Vol. 4 [2004], No. 1, Article 4 the following equalities will necessarily hold for the OLS estimates: (cid:27) (cid:27) (cid:27) 1 = R2 = R2 ; 2 = R2 = R2 ; 3 = R2 = R2, (7) (cid:27) 1 4 (cid:27) 2 5 (cid:27) 3 6 4 5 6 where R2 refers to the R-square in equation i. These equalities in turn imply i the inequalities (cid:27) (cid:27) , (cid:27) (cid:27) and (cid:27) (cid:27) . More importantly, it follows 1 0 4 2 0 5 3 0 6 from (7) that the larger the R-square in the OLS regressions, the closer will the standardandreverse estimates be. It shouldbe emphasized, however, that these results hold only for the OLS estimates.11 On the other hand, nothing can be predicted on statistical grounds about the relative size of the estimates (cid:27) , (cid:27) and (cid:27) , although previous studies led 1 2 3 Berndt (1976) to point out that estimates based on the marginal product of labor equation (2) seem to yield higher estimates of the elasticity of substi- tution than estimates based on the marginal product of capital equation (1). One could therefore expect the estimates to satisfy (cid:27) > (cid:27) .12 2 1 3 Data Construction and Sources Estimation of equations (1) through (6) requires data on the (cid:135)ow of labor services L , the nominal price of these labor services W , the (cid:135)ow of capital t t services K , the rental price of capital R , and the aggregate input index F , t t t as well as its associated price P . To illustrate the e⁄ect of data quality on t the estimates of the elasticity, I experiment with di⁄erent methods in the construction of these variables. I initially assume that labor services are proportional to employment and an omitted variable in equations (1), (2), (4), and (5), but equations (3) and (6) remain valid. On the other hand, in the presence of imperfect competition in the factor markets, even equations (3) and (6) may produce biased estimates if the wedge between marginal products and factor prices is di⁄erent for di⁄erent factors. 11As pointed out by a referee, the error terms " , i = 1;::;6, are likely to be correlated i;t across equations. I have experimented with running equations (1) through (3) and (4) through (6) as a seemingly unrelated regression (SUR) system. Because little e¢ ciency is gained by doing so, I only present single-equation estimates, which are easier to compare with previous studies. 12One point that was not explicitly described in Berndt (1976) is the derivation of the standarderrorsfor(cid:27) ,(cid:27) and(cid:27) . ByadirectapplicationoftheDeltaMethod,theestimated 4 5 6 variance of these elasticities can be computed as follows: 1 2 1 1 2 Est:Var((cid:27) )= (cid:0) Est:Var( ) (cid:0) i=4;5;6 i (cid:27) (cid:1) (cid:27) (cid:1) (cid:27) (cid:18) i (cid:19) i (cid:18) i (cid:19) http://www.bepress.com/bejm/contributions/vol4/iss1/art4 Antràs: Is the U.S. Aggregate Production Function Cobb-Douglas? 7 proxy the (cid:135)ow of these services by total private employment, de(cid:133)ned as the sum of the number of employees in private domestic industries and the num- ber of self-employed workers.13 For the regressions including the public sector (data con(cid:133)guration A below), the total number of government employees was added to the labor input measure. Jorgenson has argued repeatedly that to- tal employment is not an appropriate measure of the (cid:135)ow of labor services because it ignores signi(cid:133)cant di⁄erences in the quality of the labor services provided by di⁄erent workers. Jorgenson and collaborators have also provided quality-adjusted measures of labor services in several contributions by com- bining individual data from the Censuses of Population and from the Current Population Survey. Their measure of labor input re(cid:135)ects characteristics of individuals workers, such as age, sex and education, as well as class of em- ployment and industry. In particular, their measure is a weighted sum of the supply of the di⁄erent types or categories of labor input, where the weights are the share of overall labor compensation captured by a particular type. I con- sider here the most recent series reported in Jorgenson and Ho (2000), which considers 168 di⁄erent categories of workers. I take the nominal price of labor services to equal the total compensation of employees divided by L . Compensation of employees was obtained from t the National Income and Production Accounts (NIPA) and includes wage and salary accruals, as well as supplements to wages and salaries (e.g., employer contributions for social insurance). Following the approach in Krueger (1999), I next correct this wage measure by adding two-thirds of proprietors(cid:146)income to the overall compensation of employees.14 As is standard in the literature, I assume that the (cid:135)ow of capital services is proportional to the U.S. capital stock.15 The nominal capital stock data was obtained from Herman (2000) and is de(cid:133)ned as the sum of nonresidential private (cid:133)xed assets and government assets, the latter being left out when only the private sector is considered. The real capital stock K is simply de(cid:133)ned t as the nominal capital stock divided by the price of capital.16 I (cid:133)rst construct 13These series were obtained from the Bueau of Econonomic Analysis website. 14Gollin(2002)suggeststreatingall proprietors(cid:146)sincomeaslaborincome. Thisalternative adjustmentturnsouttohaveonlyamarginale⁄ectontheestimates(detailsavailableupon request). 15An interesting literature (Burnside et al., 1995; Basu, 1996) casts some doubts on this assumptionbyemphasizingtheimportanceofvariationsincapitalutilizationforexplaining theprocyclicalnatureofproductivity. Anexplicitcorrectionforfactorutilizationisbeyond the scope of this paper. 16As a robustness test, I employed the perpetual inventory method to construct an al- ternative measure of the real private capital stock using investment data from NIPA and depreciation data from Fraumeni(1997). The resultingcapital stocks were remarkably sim- Produced by The Berkeley Electronic Press, 2004 8 Contributions to Macroeconomics Vol. 4 [2004], No. 1, Article 4 the price of capital using nonresidential private investment de(cid:135)ators obtained from the NIPA. The NIPA de(cid:135)ator for capital equipment has been criticized for not adjusting for the increasing quality of capital goods, thereby system- atically overstating the price of capital equipment. Krusell et al. (2000) have constructed an alternative de(cid:135)ator for equipment, building on previous work by Gordon (1990). They also suggest the use of the implicit price de(cid:135)ators for nondurable consumption and services when de(cid:135)ating the nominal stock of capital structures. I employ their price indices to construct an alternative de(cid:135)ator for private nonresidential (cid:133)xed assets using Tornqvist(cid:146)s discrete ap- proximation to the continuous Divisia index.17 In particular, letting PE be t the adjusted price of equipment and PS the price of structures, the price of t capital PK is constructed as follows: t logPK logPK = sE logPE logPE + 1 sE logPS logPS , t (cid:0) t 1 t t (cid:0) t 1 (cid:0) t t (cid:0) t 1 (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) wheresE isthearithmeticmeanoftheexpendituresharesincapitalequipment t in the two periods, i.e., 1 PE KE 1 PEKE sE = t 1 t 1 + t t . t 2PE KE(cid:0)+P(cid:0)S KS 2PEKE +PSKS t 1 t 1 t 1 t 1 t t t t (cid:0) (cid:0) (cid:0) (cid:0) Capital income is de(cid:133)ned as the sum of corporate pro(cid:133)ts, net interest, and rental income of persons, and is taken from the NIPA. The rental price of capital services R is computed as the ratio of total capital income to the real t capital stock K .18 As shown by Hulten (1986), if capital is the sole quasi- t (cid:133)xedinputinproductionandthereisperfectcompetition, thisapproachyields unbiased estimates of the unobserved shadow rental rate of capital, whereas thealternativeHallandJorgenson(1967)formulaeproducebiasedestimates.19 Finally, we are left with the construction of the aggregate input index F t anditspriceP . Unfortunately, thereisnoclearcounterpartforthesevariables t inthedata. GiventhatF = Y =A ,onealternativewouldbetoconstructF by t t t t de(cid:135)ating value added Y by some index of Hicks-neutral technical e¢ ciency. t Berndt (1976) instead suggested constructing a measure of F based on the t available data on capital and labor services. In particular, he computed the ilar to those obtained by Herman (2000). 17The de(cid:135)ator for structures is also constructed as a Tonqvist index using the NIPA implicit price de(cid:135)ators for nondurable consumption and services. 18Assuming instead that the rental price of capital is proportional to the price of capital PK leads to very similar results. t 19I am grateful to an anonymous referee for pointing this out. http://www.bepress.com/bejm/contributions/vol4/iss1/art4