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Industry-level Econometric Estimatesof Energy-Capital-Labor Substitution with a Nested CESProduction Function
Yazid Dissou & Lilia Karnizova & Qian Sun
# International Atlantic Economic Society 2014
Abstract Despite substantial interest in the role of energy in the economy, thedegree of substitutability between energy and other production inputs and theway energy should be included in the production function remain unresolvedissues. This study provides industry-level parameter estimates of two-levelconstant elasticity of substitution (CES) functions that include capital, laborand energy inputs and allow for technological change, for Canada. In contrastto many existing studies, we do not impose prior restrictions on the order ofinput nesting, and we report the estimates for three possible specifications. Wefind that a nested production structure, which first combines labor and energyinto a composite good that is then combined with capital, fits the Canadian databest, in terms of respecting the restrictions imposed by cost minimization. Wealso find rather low elasticities of substitution between capital and labor, andlimited evidence of exogenous technological change.
Keywords Energy . Elasticity of substitution . CES function . Industry estimates
JEL E23 . Q41 . Q43 . Q55
Atl Econ JDOI 10.1007/s11293-014-9443-1
Y. Dissou (*)Department of Economics, University of Ottawa, 120 University Private, Ottawa, ON K1N6N5, Canadae-mail: [email protected]
L. KarnizovaDepartment of Economics, University of Ottawa, 120 University Private, Ottawa, ON K1N 6N5, Canadae-mail: [email protected]
Q. SunDepartment of Economics, University of Ottawa, 120 University Private, Ottawa, ON K1N 6N5, Canadae-mail: [email protected]
Introduction
Understanding the effects of energy-related shocks is important for many economic andpolicy questions. For example, the use of energy is directly related to greenhouse gasemissions, and is thus central to the design of environmental policies. The scarcity oftraditional energy sources raises concerns about sustainability and affects long-runproduction plans. In the short run, energy price surges increase production costs andreduce aggregate demand, potentially leading to economic recessions. How an econo-my adjusts to energy-related shocks largely depends on the extent to which energy issubstituted for other inputs of production. This study provides industry-level estimatesof the elasticities of substitution between energy, capital, and labor.
Despite the extensive empirical and theoretical literature on the role of energy in theeconomy, there is no consensus on how energy input enters the production function. Aconstant elasticity of substitution (CES) production function and its special case of theCobb-Douglas production function are the most popular choices among theoreticalmodelers. In this paper, we focus on aggregating capital, labor and energy into a two-level CES function. The two-level CES specification, proposed by Sato (1967), hasseveral advantages. This function has an intuitive economic meaning. It is simple, andyet sufficiently flexible for capturing the substitution possibilities among inputs. Inaddition, it is well suited for formalizing a correspondence between input substitutionand economic growth (e.g. Klump and de La Grandville 2000).
Capital, labor and energy can be nested into a two-level CES function in threedifferent ways: (KL)E, (KE)L and K(LE). The variables in parentheses here and in therest of the paper indicate the input aggregation at the lowest level of nesting. Forexample, in the (KL)E specification, capital and labor are combined first. The resultingcomposite is then combined with energy at the upper level of nesting. Which of thethree specifications is more appropriate for modeling energy in production is subject todebate. Nine out of ten studies on climate policy listed in van der Werf (2008) eithercombine energy with a capital-labor composite or include all three inputs at the samelevel of aggregation. However, the (KE)L nesting is popular in computable generalequilibrium models. A prominent example is the GTAP-E model, discussed byBurniaux, and Truong (2002). Many business cycle models use the (KE)L specification,following Kim and Loungani (1992). Yet, the (KL)E specification is also common (e.g.Carlstrom and Fuerst 2006; Bodenstein et al. 2011).
The values of the elasticities of substitution and the choice of the input nestingstructure can have a significant impact on theoretical predictions and economic infer-ences. For example, the elasticity of substitution between energy and other inputs is akey parameter that determines the sign of output responses to energy price shocks inVinals (1984), the costs of environmental policy in Jacoby et al. (2006) and the rates ofeconomic growth in Schubert and Turnovsky (2011). Lecca et al. (2011) establish thattheoretical results of a computable general equilibrium model can be rather sensitive tothe choice of the input nesting in the production function.
Theoretical modelers often assume a Cobb-Douglas function with unitary elasticitiesof substitution between inputs (e.g. Finn 2000; Popp 2004). This assumption has beenchallenged empirically. Some recent studies find that empirical data do not exhibitproperties that are compatible with the standard neo-classical growth model thatfeatures the Cobb-Douglas production function with technological change (Klump
Y. Dissou et al.
et al. 2007). Others argue that restrictive assumptions on the nature of technologicalchange, embedded in the Cobb-Douglas production function, can lead to an estimationbias in favor of this function (Antràs 2004).
In this study, we estimate the parameters of the CES function for all threepossible nesting structures of capital, labor, and energy, allowing for the possibil-ity of factor-specific or input-neutral technological change. Our estimation isbased on a system of three equations with cross-equations restrictions that arederived from cost minimization. The empirical model allows us to identify theelasticities of substitution at the lower and the upper levels of the CES nesting,and to test the nature of technological change. We also test the validity of the cost-minimization restrictions, to determine which production structure fits the databest.
Our work offers two improvements to previous studies. First, elasticities ofsubstitution between energy and other inputs are often estimated using linear ornon-linear single-equation models (e.g. Chang 1994; Kemfert 1998). However,the estimation of a single-equation factor demand function could be systemati-cally biased (David and van de Klundert 1965). Our system estimation shouldhelp to address this bias. Second, many of the previous empirical studies makespecific assumptions on the nesting of the CES production function or the natureof technological change (e.g. Prywes 1986; Bosetti et al. 2006). However, thechoice of the production structure may affect the value and the significance ofthe estimated elasticities between factors in the same pair. We do not impose anyprior assumptions on the validity of the CES specification. Instead, we conducttests and provide parameter estimates for all three nesting structures.
To the best of our knowledge, Kemfert (1998) and van der Werf (2008) are theonly other studies that directly compare three different input combinations withina CES production function with technological change. Kemfert estimates produc-tion parameters based on aggregate and sectoral data of the German economyusing a non-linear single equation model. Van der Werf works with an unbalancedpanel of seven industries of the OECD countries. Due to data limitations, heprovides estimates only by countries or industries. Industry-level estimates bycountries would be valuable for multi-sector general equilibrium models thatincorporate energy, such as Beausejour et al. (1995), and Bohringer andRutherford (1997).
Our study differs from Kemfert (1998) and van der Werf (2008) in terms ofthe estimation techniques and sample coverage. We focus on Canadianmanufacturing industries, and estimate seemingly unrelated regressions for eachindustry. Canada provides an interesting case study. It is a small open economythat exports both energy and traditional manufacturing goods. The design of anappropriate energy and climate change policy has been at the center of policyand political debate in this country. Additional recent estimates of the elasticitiesof substitution between energy and other inputs could contribute to resolving thisdebate.
Our results for Canadian industries favor more the K(LE) nesting structure, in termsof respecting theoretical restrictions imposed by cost minimization. We find somesupport for the Leontief and the Cobb-Douglas technologies, as well as limitedevidence of exogenous technological change.
Industry-level Estimates of Energy-Capital-Labor Substitution
Theoretical and Empirical Framework
Our model specification is derived from three assumptions about firm behavior: (i) costminimization, (ii) price-taking on markets for capital, labor and energy and (iii) a nestedCES production function. Cost minimization, according to which inputs are chosen tominimize the production costs, is a conventional theoretical assumption. It is a neces-sary condition for profit maximization. Unlike profit maximization, the optimalityconditions for cost minimization remain valid even in the presence of imperfectcompetition on output markets, as long as the input markets are perfectly competitive.1 While the assumption of constant returns to scale is controversial, it is supported, forexample, by the plant-level estimates of Baily et al. (1992).
We consider three possible ways of aggregating three inputs into a CES function:(KL)E, (KE)LandK(LE). In explaining the methodology, we focus on the first case forconcreteness. Theoretical and empirical specifications of the other two cases, (KE)LandK(LE), are derived in a similar way. In the (KL)E production structure, capital K andlabor L are first combined in a CES composite Z, with the elasticity of substitution σK,L.The composite Z is further combined with energy E in an upper level CES aggregator,with the elasticity of substitution σKL,E.
Z tð Þ ¼ φ AK tð ÞK tð Þ½ � σK;L−1ð Þ=σK;L þ 1−φð Þ AL tð ÞL tð Þ½ � σK;L−1ð Þ=σK;Ln oσK;L= σK;L−1ð Þ ð1Þ
Y tð Þ ¼ ϕ AE tð ÞE tð Þ½ � σKL;E−1ð Þ=σKL;E þ 1−ϕð ÞZ tð Þ σKL;E−1ð Þ=σKL;En oσKL;E= σKL;E−1ð Þ ð2Þ
The share parameters are φ∈(0,1) and ϕ∈(0,1). Production can be subject to factor-specific deterministic technological change, represented by Aj(t) = (1 + aj)
t,j∈{K,L,E}.Technological change is defined as Hicks neutral with respect to all inputs if and only ifaK = aL = aE. If Aj(t) grow at different rates, technological change is defined as factor-specific.
The two-level CES production function implies a constant elasticity of substitutionbetween the inputs at the lower level of nesting, but generally not between the other twopairs of inputs. The latter elasticity depends on the relative shares of inputs andthe relative shares of composites in total costs, which typically vary with thelevel of inputs over time. For instance, for the case of the (KL)E specification,the elasticity of substitution between capital and labor σK,L is constant.However, the direct partial elasticities of substitution between these inputs andenergy are harmonic means of σK,LandσKL,E (Sato 1967). Clearly, differentassumptions about input nesting in the CES function generally imply differentelasticities of substitution among inputs. All partial elasticities of substitutionsare constant only when all three inputs can be included in the same nesting, inwhich case σK = σKL,E = σ. The Cobb-Douglas production functions arise as a
1 Empirical effects of non-competitive input markets are difficult to estimate, since this requires the jointmodelling and estimation of factor supply functions. Similar to other studies in the literature (e.g. Berndt andWood 1975; Christensen, Jorgenson, and Lau 1971; León-Ledesma et al. 2010), we treat input prices asexogenous.
Y. Dissou et al.
special case when σK,L = 1 or σKL,E = 1. We estimate three equations modelsderived from theoretical optimality conditions:
Y 1 tð Þ ¼ α1 þ β1X 1 tð Þ þ ε1 tð Þ ð3Þ
Y 2 tð Þ ¼ α2 þ β21X 21 tð Þ þ β22X 22 þ ε2 tð Þ ð4Þ
Y 3 tð Þ ¼ α3 þ β31X 21 tð Þ þ β32X 32 tð Þ þ ε3 tð Þ ð5ÞFor the case of the (KL)E structure, the dependent variables are Y1(t) = e(t)−
y(t),Y2(t) = θK,Z(t), and Y3(t) = θL,Z(t). Lowercase letters here denote the firstdifferences in logarithms, such as e(t) = Δ lnE(t) = lnE(t)− lnE(t−1). Further,θN,M(t) = Δ lnΘN,M(t), where ΘN,M(t) represents the cost share of input N in thecosts of producing M in period t. The explanatory variables are the percentagechanges of the relative prices and cost shares: X1(t) = pY(t)−pE(t),X22(t) = pK(t)−pY(t),X32(t) = pL(t)−pY(t) and X21(t) = θZ,Y(t). Stochastic error terms εj(t),j = 1,2,3,E[εj(t)] = 0, capture differences between theoretical conditions andthe actual data, due to approximations and simplifications of the model and/orpossible measurement errors. We assume that the error terms are uncorrelatedover time, but possibly correlated across the equations. The parameters of theempirical model are related to theoretical parameters as follows (e.g. van derWerf 2008):
α1 ¼ aE σKL;E−1� �
;β1 ¼ σKL;E ð6Þ
α2 ¼ aK σK;L−1� �
;β21 ¼ − 1−σK;L
� �= 1−σKL;E
� �;β22 ¼ 1−σK;L ð7Þ
α3 ¼ aKL σK;L−1� �
;β31 ¼ − 1−σK;L
� �= 1−σKL;E
� �;β32 ¼ 1−σK;L ð8Þ
Cross-equation restrictions among the structural parameters motivate our choice ofthe non-linear seemingly unrelated regression (SUR) method. We first estimate theunrestricted model (3)-(5) for each of the three CES specifications. Testing restrictionsβ21 ¼ β31and β22
¼ β32
allows us to assess which nesting is more favored by thedata. Our main results are based on the restricted system of equations. Given theparameter estimates, we test for the Cobb-Douglas form (σK,L = 1,σKL,E = 1), thenon-nested CES function (σK = σKL,E) and the nature of technological change.
Data
We apply the theory of the firm to the industry-level data. Our data come fromthe productivity program database of Statistics Canada. The database containschained-weighted prices and quantities of capital, labor, and energy, as well as
Industry-level Estimates of Energy-Capital-Labor Substitution
total nominal costs of these inputs, by industry. We use this information toconstruct the KLE-measures of industrial output and its price as Törnqvistaggregates. 2 The data are annual, from 1962 to 1997. This is the lengthiestperiod of consistent detailed Canadian industry data needed for our analysis.This is mainly due to the change in the industrial classification that occurred in1997 in North America.
Our dataset covers ten manufacturing industries, four of which (primary metal, paperand allied products, chemical and chemical products, and non-metallic mineral prod-ucts) are classified as energy-intensive. Our industry choice is determined by theavailability of consistent detailed industry data. Prior to the empirical estimation, wechecked all required relative prices, cost shares and input-to-output ratios (in percentagechanges) for stationarity. All data passed the augmented Dickey-Fuller (ADF) tests forunit root in levels, thus could be considered stationary. Consequently, the level spec-ifications were appropriate for our analysis.
Table 1 summarizes the average cost shares of capital, labor and energy inputs, aswell as the average cost shares of input composites in total production costs. The timeseries of these shares are used in the empirical estimation. Given the values ofelasticities of substitution, the values of the share parameters φ and ϕ in the CESfunctions for Z(t) and Y(t), for use in applied theoretical models, can be calibrated fromthe average shares in Table 1.
Results
We first examine the tests of theoretical cost minimization restrictions. Then we presentthe estimates of the substitution elasticities and of the rates of technological change.
Cost-Minimization Restrictions
How energy, capital and labor enter the production function at the industry level is anunresolved empirical issue. Our comparison of the adequacy of the three possible CESnesting structures is based on the tests of the cost minimization restrictions β21 ¼ β31
and β22 ¼ β32. Table 2 summarizes the results of individual Wald tests for theparameter equality.
The Canadian data favor more the K(LE) specification, in which labor and energyare combined first. Both theoretical restrictions cannot be rejected in nine out of tenindustries at the 5 % significance level. By contrast, the (KE)L specification has theleast empirical support, in terms of the number of accepted restrictions. The restrictionβ21 ¼ β31 holds only for the primary metal industry. The restriction β22 ¼ β32
cannot be rejected in four industries: chemicals, food, machinery, and paper and alliedproducts. For the (KL)E specification, both theoretical restrictions are satisfied forfabricated metal and machinery industries at 5 % significance level.
How can we explain that a popular (KE)L structure seems to fit the data poorlyrelative to the K(LE) specification? The demand for capital and energy may not bedetermined jointly since energy cost is a variable cost, while physical capital comprises
2 See Allen and Diewert (1981) for discussions on the properties of Törnqvist indices.
Y. Dissou et al.
fixed investment. This fixed investment may be less mobile than labor and energy.Consequently, the elasticity of substitution between capital and labor or energy is likelyto be different from the substitution elasticity between energy and labor, as the K(LE)structure suggests.
Let us compare our results with the previous findings in the literature. Van der Werf(2008) rejects the cross-equation restriction β
22¼ β
32for all countries and all
industries for the (KL)E and K(LE) nesting structures, and for 12 out of 19 cases forthe (KE)L specification. He uses the adjusted R2 statistics for the panel regressions tocompare the empirical models. Based on this statistic, the (KE)L specification is inferiorto the other two production structures. It is however important to note that, in general,the goodness of fit measures for the system of equations should be treated with caution.Moreover, the comparison of the results for three possible CES nestings is furtheraffected by the fact that the empirical systems have different independent andexplanatory variables. One cannot easily compare R2 statistics across thesespecifications.
For the German economy, Kemfert (1998) prefers the (KE)L specification at theaggregate level and the (KL)E specification for five out of seven industries. The R2
statistics reported in her study are meaningful since the estimation is based on a singleequation with the same variables. Interestingly, the R2 statistics for the K(LE)
Table 1 Average cost shares in total expenditures (1962–1997)
Industry/Shares ΘK,Y ΘL,Y ΘE,Y ΘKL,Y ΘKE,Y ΘLE,Y
Chemicals 0.425 0.457 0.119 0.881 0.543 0.575
(0.071) (0.082) (0.047) (0.047) (0.082) (0.071)
Fabricated metal 0.282 0.692 0.025 0.975 0.308 0.718
(0.029) (0.028) (0.007) (0.007) (0.028) (0.029)
Food 0.372 0.585 0.043 0.957 0.415 0.628
(0.049) (0.050) (0.007) (0.007) (0.050) (0.049)
Machinery 0.298 0.684 0.018 0.982 0.316 0.702
(0.044) (0.045) (0.005) (0.005) (0.045) (0.044)
Non-metal. mineral products 0.345 0.546 0.109 0.891 0.454 0.655
(0.042) (0.031) (0.026) (0.026) (0.031) (0.042)
Paper and allied products 0.320 0.539 0.141 0.859 0.461 0.680
(0.098) (0.078) (0.042) (0.042) (0.078) (0.098)
Primary metal 0.234 0.580 0.186 0.814 0.420 0.766
(0.078) (0.054) 0.042) (0.042) (0.054) (0.078)
Textile 0.279 0.684 0.036 0.964 0.316 0.721
(0.036) (0.036) (0.012) (0.012) (0.036) (0.036)
Transportation 0.247 0.607 0.146 0.854 0.393 0.753
(0.038) (0.023) (0.037) (0.037) (0.023) (0.038)
Transportation equipment 0.284 0.692 0.024 0.976 0.308 0.716
(0.069) (0.067) (0.005) (0.005) (0.067) (0.069)
Standard errors are in parentheses. The shares do not necessarily add up to one due to rounding. Each valueΘN,Y represents the cost share of the input N in total costs of producing output Y
Industry-level Estimates of Energy-Capital-Labor Substitution
specification in Kemfert (1998) are very close to the ones for the preferred specifica-tions, albeit the differences are not tested statistically.
Elasticities of Substitution
Table 3 reports the estimated elasticities of substitution for three possible nestings of theCES production function. We report the estimates under the imposed cost miminizationrestrictions even when these restrictions are rejected, in order to provide the parametervalues for the CES functions that can be directly used in applied work. It is worthmentioning that while the estimated direct elasticities between the factors pairs in thesame nest are constant, the indirect elasticities between factors that are not in the samenest may be endogenous. Table 4 reports the test results for the Cobb-Douglas andsingle-level CES functional forms.
Several conclusions emerge from our results. First, there is a substantial variationin the parameter estimates across the industries at the lower and higher levels ofnesting. The estimates vary between 0.07 and 1.00. We find no stark differencesbetween energy-intensive and energy non-intensive industries. Second, the elastic-ities of substitution are generally positive. The only exceptions are the food andmachinery industries. We cannot reject perfect complementarity (a Leontief tech-nology) between capital and the composite of labor and energy (σLE,K =0) in theK(LE) specification for the food industry at the conventional levels of significance.For the machinery industry, perfect complementarity between energy input and thecomposite of capital and labor (σKE,L =0) can be rejected at the 5 % significancelevel, but not at the 10 % level.
Third, capital, labor and energy are generally not substitutable with the same ease.Table 4 indicates that the hypothesis that the elasticities of substitution between the
Table 2 Tests of cost minimization restrictions
CPE (KL)E (KE)L K(LE)
β21=β31 β22=β32 β21=β31 β22=β32 β21=β31 β22=β32
Chemicals 0.190 0.012 0.000 0.282 0.864 0.504
Fabricated metal 0.270 0.818 0.005 0.015 0.044 0.225
Food 0.000 0.000 0.003 0.758 0.880 0.879
Machinery 0.069 0.081 0.000 0.998 0.055 0.563
Non-metal. mineral prod. 0.165 0.038 0.000 0.003 0.331 0.612
Paper and allied products 0.769 0.000 0.001 0.671 0.904 0.020
Primary metal 0.041 0.000 0.454 0.000 0.282 0.162
Textile 0.544 0.000 0.000 0.002 0.822 0.284
Transportation 0.380 0.001 0.000 0.005 0.796 0.414
Transportation equipment 0.020 0.000 0.000 0.001 0.227 0.208
Industries with the accepted restrictions 7 2 1 4 9 9
The table reports the p-values for the Wald tests. A p-value below 0.05 indicates that the null of the parameterequality can be rejected at the 5 % significance level. The last row gives the number of industries for which atheoretical restriction cannot be rejected at the 5 % significance level
Y. Dissou et al.
three inputs are equal can be accepted only in five cases: for primary metals, transpor-tation equipment and textile products for the (KL)E specification, and non-metallicmineral products and machinery for the (KE)L case. Note, however, that for these fiveindustries at least one of the restrictions β21 = β31or β22= β32 is rejected, as the statisticsin Table 2 imply.
Fourth, the elasticities of substitution at the upper level of nesting tend to be smallerthan at the lower level of nesting. For example, the values σKE,L and σLE,K are onaverage more than 2 and 4.5 smaller than the values σK,E and σL,E. The property maybe expected a-priori in the aggregation of several inputs in production. As Sato (1967)argues, the input aggregation is justified by the similarity of techno-economic charac-teristics, one of which is the ease of substitutability. Only in three cases (chemicals,transportation and food), the upper nesting elasticity σKL,E exceeds the lower nestingelasticity σK,L for the (KL)E specification.
Finally, the relatively low values of the elasticities σKE,L and σLE,K at the uppernesting suggest low substitutability between labor and capital. By contrast, the largestestimates are for the elasticities of substitution between labor and energy σL,E. Thesevalues vary between 0.60 and 1.00 and suggest that it is easier to substitute towardsenergy. In four cases (primary metals, paper, transportation equipment and machinery),
Table 3 Estimated elasticities of substitution (SUR, restricted model)
Industry/ elasticity (KL)E (KE)L K(LE)
σKL,E σK,L σKE,L σK,E σLE,K σL,E
Chemicals 0.4833*** 0.2514*** 0.2739*** 0.4315*** 0.1485*** 0.6492***
(0.0406) (0.0301) (0.0446) (0.0521) (0.0369) (0.0324)
Fabricated metal 0.2436*** 0.4567*** 0.3144*** 0.4384*** 0.4037*** 0.7803***
(0.0899) (0.0284) (0.0327) (0.0478) (0.0492) (0.0604)
Food 0.7435*** 0.3522*** 0.2900*** 0.6621*** 0.1028 0.7463***
(0.0427) (0.0364) (0.0468) (0.0630) (0.0664) (0.0313)
Machinery 0.2042* 0.4650*** 0.3714*** 0.4401*** 0.3771*** 0.9519***
(0.1068) (0.0217) (0.0277) (0.0431) (0.0515) (0.0445)
Non-metal. mineral 0.3061*** 0.4395*** 0.3582*** 0.4667*** 0.4247*** 0.6631***
products (0.0699) (0.0177) (0.0297) (0.0413) (0.0257) (0.0524)
Paper etc. 0.1529*** 0.4389*** 0.0868*** 0.4743*** 0.1281*** 1.0008***
(0.0286) (0.0102) (0.0187) (0.0171) (0.0463) (0.0038)
Primary metal 0.1026*** 0.1549*** 0.0858*** 0.3034*** 0.0707** 0.9969***
(0.0285) (0.0151) (0.0160) (0.0164) (0.0317) (0.0017)
Textile 0.3397*** 0.3678*** 0.2272*** 0.4598*** 0.2611*** 0.9008***
(0.0673) (0.0301) (0.0297) (0.0387) (0.0698) (0.0333)
Transportation 0.5409*** 0.3605*** 0.1692*** 0.4475*** 0.3712*** 0.6094***
(0.0500) (0.0317) (0.0407) (0.0471) (0.0372) (0.0362)
Transp. equipment 0.3560*** 0.3860*** 0.2662*** 0.4328*** 0.2630*** 0.9754***
(0.0590) (0.0282) (0.0320) (0.0302) (0.0705) (0.0151)
Standard errors are in parentheses. The stars *** , ** and * indicate the significance level of 1, 5 and 10 %
Industry-level Estimates of Energy-Capital-Labor Substitution
we cannot reject the Cobb-Douglas specification between labor and energy at theconventional levels of significance.
To demonstrate the advantage of the SUR technique, we provide two sets ofadditional results. Table 5 reports the correlation coefficients between the residuals ofthe estimation equations implied by the SUR variance-covariance matrix. The statistics
Table 4 Tests for Cobb-Douglas function (unitary elasticities) and non-nested CES function (commonelasticities)
Industry /p-values
(KL)E (KE)L K(LE)
σKL,E=1 σK,L=1 σKL,E=σK,L σKE,L=1 σK,E=1 σKE,L=σK,E σLE,K=1 σL,E=1 σLE,K=σL,E
Chemicals 0.00 0.00 0.00 0.00 0.00 0.04 0.00 0.00 0.00
Fabric. metal 0.00 0.00 0.01 0.00 0.00 0.02 0.00 0.00 0.00
Food 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Machinery 0.00 0.00 0.01 0.00 0.00 0.21 0.00 0.28 0.00
Non-metal.mineral prod.
0.00 0.00 0.04 0.00 0.00 0.06 0.00 0.00 0.00
Paper etc. 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.84 0.00
Primary metal 0.00 0.00 0.11 0.00 0.00 0.00 0.00 0.07 0.00
Textile 0.00 0.00 0.68 0.00 0.00 0.00 0.00 0.00 0.00
Transportation 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Transp. equip. 0.00 0.00 0.61 0.00 0.00 0.00 0.00 0.10 0.00
The table reports probability values for the two-sided Wald tests for the restrictions on the elasticity valuesindicated for each column. A p-value below 0.05 indicates that the null of the parameter equality can berejected at the 5 % significance level
Table 5 Off-diagonal elements of the SUR covariance matrix, converted into correlations (restricted modelunder cost minimization restrictions)
Industry (KL)E (KE)L K(LE)
ρ(Y1,Y2) ρ(Y1,Y3) ρ(Y2,Y3) ρ(Y1,Y2) ρ(Y1,Y3) ρ(Y2,Y3) ρ(Y1,Y2) ρ(Y1,Y3) ρ(Y2,Y3)
Chemicals 0.16 0.27 −0.09 −0.02 −0.66 −0.03 −0.03 −0.02 −0.68
Fabric. metal −0.02 0.64 0.18 −0.55 −0.16 0.03 −0.02 0.07 −0.80
Food 0.00 0.58 −0.24 −0.15 −0.64 −0.31 0.01 −0.13 0.29
Machinery 0.09 0.54 0.17 −0.56 −0.49 0.09 0.02 0.09 −0.73
Non-metal. mineralprod.
0.03 0.37 0.17 0.02 −0.71 −0.18 0.23 −0.03 −0.17
Paper etc. −0.05 0.14 0.94 0.50 0.32 −0.12 −0.03 0.05 −0.97
Primary metal 0.31 0.28 0.96 0.88 0.88 0.91 −0.01 −0.01 −0.99
Textile 0.00 0.33 0.60 0.22 −0.15 −0.27 0.01 0.41 −0.11
Transportation −0.57 −0.48 0.53 −0.06 −0.32 −0.49 −0.34 0.37 −0.47
Transp. equip. 0.03 0.42 0.50 −0.04 −0.23 −0.16 0.25 −0.03 −0.80
Y. Dissou et al.
in Table 5 indicate a substantial interdependence across the equations. Single equationestimation of production function parameters would miss such interdependence. Toillustrate the practical impact of the SUR, Table 6 reports the substitution elasticities atthe lower level of nesting, based on separate estimation of Eqs. (4) and (5). Theequivalence of the single equation and SUR estimates is strongly rejected in manycases, as shown by the t-statistics reported in Table 6.
Technological Change
In our framework, exogenous technological change is input-neutral if the growth ratesat the lower level of nesting are zero, but the growth rate at the upper level of nesting isdifferent from zero. For the (KL)E specification, input-neutrality of technologicalchange corresponds to a null hypothesis that aK = aL=0 and aE ≠ 0. Table 7 reportsthe estimates of aK, aL, and aE for the three specifications of the production function foreach industry. For six industries (fabricated metal, machinery, non-metallic mineral
Table 6 Comparison of the SUR and single equation parameter estimates
Industry σK,L - (KL)E σK,E - (KE)L σL,E - K(LE)
Eq. (4) Eq. (5) Eq. (4) Eq. (5) Eq. (4) Eq. (5)
Chemicals 0.2762 0.2691 0.6146 0.0503 0.7491 2.5151
[−1.530] [−1.982]* [−10.345]*** [−8.777]*** [−2.431]** [113.91]***
Fabricated metal 0.2576 0.5928 0.9317 0.3929 0.4693 2.6699
[−0.551] [−8.175]*** [−11.087]*** [−6.492]*** [−9.762]*** [55.435]***
Food 0.0660 0.3646 0.8508 0.4956 0.5850 3.0234
[−1.162] [−7.538]*** [−15.413]*** [−9.702]*** [−4.807]*** [93.253]***
Machinery 0.4035 0.5811 0.9574 0.6542 0.7805 3.1623
[−0.778] [−8.816]*** [−11.470]*** [−8.497]*** [−6.829]*** [53.822]***
Non-metal. mineral 0.3221 0.5258 0.6579 0.0060 0.4958 2.6896
products [−1.338] [−6.007]*** [−10.963]*** [−13.873]*** [−0.041] [137.87]***
Paper etc. −0.1479 0.6357 0.2355 0.3426 0.7164 1.3908
[0.941] [−1.791]* [−5.119]*** [4.296]*** [−11.191]*** [28.417]***
Primary metal −0.4152 0.7386 −0.1199 0.4532 0.9153 1.3912
[0.448] [−1.135] [1.580] [0.215] [−8.247]*** [40.146]***
Textile 0.0664 0.5976 0.9399 0.3931 0.4963 2.6630
[−1.297] [−6.445]*** [−10.644]*** [−5.382]*** [−6.597]*** [77.742]***
Transportation 0.1845 0.5482 0.4156 0.4174 0.5101 2.3506
[0.402] [−4.073]*** [−5.265]*** [−2.871]*** [−0.678] [134.57]***
Transp. equipment 0.2467 0.5716 0.9681 0.5425 0.8179 1.8520
[−1.584] [−4.232]*** [−1.006] [0.781] [−8.274]*** [33.00]***
The table reports the estimated elasticities of substitution, implied by a separate estimation of Eqs. (4) and (5).The values in the brackets are the t-statistics for the null hypothesis that the single equation parameterestimates are equivalent to the SUR estimates of the unrestricted models without cost minimization restric-tions. The stars *** , ** and * indicate a rejection of the parameter equivalence at the significance level of 1, 5and 10 %
Industry-level Estimates of Energy-Capital-Labor Substitution
Tab
le7
Rates
offactor-specifictechnologicalchange
Industry/rateof
technologicalchange
Capita
la K
Labora L
Energya E
(KL)E
(KE)L
K(LE)
(KL)E
(KE)L
K(LE)
(KL)E
(KE)L
K(LE)
Chemicals
−0.0187*
*−0
.0195*
−0.0147*
0.0128
***
0.0168
***
0.0200
***
−0.0429*
−0.0427
−0.0778*
(0.0089)
(0.0107)
(0.0082)
(0.0038)
(0.0055)
(0.0058)
(0.0226)
(0.0267)
(0.0406)
Fabricated
metal
−0.0092
−0.0069
−0.0027
−0.0011
0.0028
−0.0004
−0.0117
−0.0171
−0.0433
(0.0168)
(0.0133)
(0.0114)
(0.0057)
(0.0038)
(0.0091)
(0.0107)
(0.0184)
(0.0467)
Food
−0.0233*
**−0
.0237*
−0.0192*
**0.0100
***
0.0086
***
0.0082
**0.0022
0.0155
−0.0130
(0.0082)
(0.0127)
(0.0056)
(0.0031)
(0.0033)
(0.0038)
(0.0267)
(0.0108)
(0.0376)
Machinery
−0.0232
−0.0198
−0.0146
0.0031
0.0067
0.0036
−0.0091
−0.0099
−0.0910
(0.0207)
(0.0180)
(0.0140)
(0.0054)
(0.0047)
(0.0295)
(0.0154)
(0.0267)
(0.3069)
Non-m
etal.m
ineral
−0.0044
−0.0035
−0.0019
0.0013
0.0025
0.0019
−0.0079
−0.0085
−0.0139
products
(0.0092)
(0.0086)
(0.0080)
(0.0038)
(0.0045)
(0.0055)
(0.0116)
(0.0189)
(0.0302)
Paperetc.
−0.0179
−0.0153
−0.0220*
*0.0142
0.0127
−0.0480
−0.0182
−0.0271*
0.1809
(0.0643)
(0.0546)
(0.0089)
(0.0204)
(0.0092)
(0.2436)
(0.0118)
(0.0144)
(0.9165)
Prim
arymetal
−0.0180
−0.0163
−0.0150
0.0048
0.0061
0.0895
−0.0167
−0.0196
−2.7878
(0.0450)
(0.0558)
(0.0108)
(0.0376)
(0.0113)
(0.9922)
(0.0106)
(0.0285)
(3.3198)
Textile
−0.0275
−0.0226
−0.0140
−0.0021
0.0046
0.0026
−0.0234*
−0.0266
−0.1641
(0.0219)
(0.0184)
(0.0138)
(0.0102)
(0.0060)
(0.0188)
(0.0128)
(0.0198)
(0.1227)
Transportation
−0.0076
−0.0069
−0.0069
0.0057
0.0056
**0.0070
*−0
.0274*
*−0
.0241*
*−0
.0334*
*
(0.0099)
(0.0097)
(0.0078)
(0.1276)
(0.0025)
(0.0037)
(0.0140)
(0.0118)
(0.0165)
Transp.
equipm
ent
−0.0398
0.0190
−0.0280
0.0057
0.0093
0.0131
−0.0107
−0.0097
−0.3134
(0.0336)
(0.0245)
(0.0234)
(0.0093)
(0.0068)
(0.0317)
(0.0128)
(0.0213)
(0.4414)
Notes:The
values
inparenthesesarestandard
errors.T
hestars***,*
*and
*indicatethesignificance
levelof
1,5,
and10
%
Y. Dissou et al.
products, primary metal, textile and transportation), we find no evidence of exogenoustechnological change, either neutral or factor-specific. For the remaining four industries(chemicals, food, paper, and transportation) we test whether technological change isfactor-specific or input-neutral. Table 8 reports the results.
We cannot reject the null hypothesis that technological change is input-neutral in five cases at the 5 % level of significance: paper and transportationindustries for the (KL)E specification; chemicals, food and paper for the (KE)Lspecification; and paper for the K(LE) specification. It should be noted, how-ever, that in none of these five cases are both theoretical restrictions of costminimization satisfied (see Table 2). The only entry in Table 8, for which bothcost minimization restrictions β21 = β31and β22 = β32 hold, is the K(LE)specification for chemicals. Our results suggest that this industry was subjectto labor-, capital- and energy-augmenting technological change of 2, −1.5 and−7.8 % per year between 1962 and 1997.
Concluding Remarks
This paper contributes to the literature on the CES production function by estimatingthe parameters of nested CES production functions with capital, labor and energyinputs and exogenous technological change. Our estimates of the elasticities of substi-tution for ten Canadian industries can be directly used in applied general equilibriummodels with multiple sectors that incorporate energy. Since our estimation is derivedfrom cost minimization, the use of the estimates can help address criticisms of weakempirical foundations of computable general equilibrium models (Beckman and Hertel2010), and the paucity of industry-level estimates.
We find that the nesting structure K(LE), in which energy and labor are firstcombined using a CES function, and then this composite combined with capital in anupper CES function, fits the Canadian data best, in terms of respecting theoreticalrestrictions imposed by cost minimization. For this structure, we are also able to rejectthe hypothesis that the elasticities are equal for both nests for all ten industries. Hence,our two-level nesting structure cannot be rejected. At the top level, we find evidence of
Table 8 Input-neutral versus factor-specific technological change
Industry / Test results (KL)E (KE)L K(LE)
aK=aL =0(p-value)
aE aK=aE =0(p-value)
aL aL=aE =0(p-value)
aK
Chemicals 0.0007 −0.0429* 0.0871 0.0168*** 0.0027 −0.0147*
Food 0.0001 0 0.0626 0.0086*** 0.0227 −0.0192***
Paper etc. 0.1204 0 0.0558 0 0.9807 −0.0220**
Transport. 0.1612 −0.0274** 0.0219 0.0056** 0.0374 0
For each production function, the first column reports the probability value for the Wald tests that both growthfactors at the lower level of nesting are equal to zero, and the second column reproduces the growth factor atthe upper level of nesting from Table 7. The stars *** , ** and * indicate the significance level of 1, 5, and 10 %
Industry-level Estimates of Energy-Capital-Labor Substitution
the Leontief technology for one industry, and at the low level, we find evidence of theCobb-Douglas technology for four industries.
We generally find relatively low substitutability between capital and labor andrelatively high substitutability between labor and energy. For example, for the K(LE)structure, our elasticity estimates are substantially low for the top level and higher forthe low level. Lower elasticities at the top level imply that it becomes harder tosubstitute away from capital, and the higher elasticities at the low level imply that itis easier to substitute towards energy.
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