Abstract
This chapter lays out the theoretical foundation of the measurement of the degree of substitutability among inputs utilized in a production process. It proceeds from the well-settled (Hicksian) notion of this measure for two inputs (typically labor and capital) to the more challenging conceptualization for technologies with more than two inputs (most notably, Allen-Uzawa and Morishima elasticities). Dual elasticities of substitution (also called elasticities of complementarity) and gross elasticities of substitution (measuring substitutability for non-homothetic technologies taking account of output changes) are also covered. Also analyzed are functional representations of two-input technologies with constant elasticity of substitution (CES) and of n-input technologies with constant and identical elasticities for all pairs of inputs. Finally, the chapter explores the relationship between elasticity values and the comparative statics of factor income shares and the relationships between certain elasticity identities and separability conditions rationalizing consistent aggregation of subsets of inputs.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
- 2.
While the use of the word “curvature” in this quote conveys the appropriate intuition, it is nevertheless technically incorrect, in part because curvature, formally defined, is a unit-dependent mathematical concept. See de la Grandville [30] for a clear exposition of this point.
- 3.
As pointed out by Blackorby and Russell [14, p. 882], “[O]nly if the two variables were separable from all other variables would [this elasticity] provide information about shares; if we were to require all pairs to have this property, the production function would be additive. When combined with homotheticity (an assumption maintained in all these studies …) this implies that the production function is CES, in which case [the elasticities] are constant for all pairs of inputs.”
- 4.
- 5.
See footnote 2 above
- 6.
It is dual, not to the Morishima elasticity, but to McFadden’s [60] shadow elasticity of substitution.
- 7.
Separability is also a necessary condition for decentralization of an optimization problem (as in, e.g., two-stage budgeting). See Blackorby, Primont, and Russell [16, Ch. 5] for a thorough exposition of the connection between separability and decentralized decision making.
- 8.
As we shall see in Section 3, the homotheticity assumption can be dropped when the elasticity concept is formulated in the dual.
- 9.
We could extend our analysis to all of \({\mathbf {R}}^2_{+}\) by employing directional derivatives at the boundary but instead leave this technical detail to the interested reader.
- 10.
Subscripts on functions indicate differentiation with respect to the specified variable. The relation, = , should be interpreted as an identity throughout this chapter (i.e., as holding for all allowable values of the variables). Also, A := B means the relation defines A, and A =: B means the relation defines B.
- 11.
The Cobb-Douglas production function was well known at the time of the SMAC derivation, having been proposed much earlier (Cobb and Douglas [26]). The (CES2) production function made its first appearance in Solow’s [72] classic economic growth paper, but the functional form had appeared much earlier in the context of utility theory: Bergson (Burk) [1936] proved that additivity of the utility function and linear Engel curves (expenditures on individual goods proportional to income for given prices) implies that the utility function belongs to the CES family. As the SMAC authors point out, the function (CES2) itself was long known in the functional-equation literature (see Hardy, Littlewood, and Pólya [41, p. 13]) as the “mean value of order ρ”.
- 12.
The SMAC theorem is easily generalized to homothetic technologies, in which case the production function is a monotonic transformation of (CES2) or (CD2); in the limiting case as σ → 0 it is a monotonic transformation of (L2).
- 13.
- 14.
- 15.
These assumptions are stronger than needed for much of the conceptual development that follows, but in the interest of simplicity I maintain them throughout.
- 16.
Vector notation: \(\bar y\ge y\) if \(\bar y_j\ge y_j\) for all j; \(\bar y> y\) if \(\bar y_j\ge y_j\) for all j and \(\bar y\ne y\); and \(\bar y\gg y\) if \(\bar y_j> y_j\) for all j.
- 17.
We restrict the domain of the distance function to assure that it is globally well defined. An alternative approach (e.g., Färe and Primont [36]) is to define D on the entire non-negative (n + m)-dimensional Euclidean space and replace “max” with “sup” in the definition. See Russell [70, footnote 12] for a comparison of these approaches.
- 18.
- 19.
∇pC(p, y := 〈 C(p, y)∕∂p1, …, ∂C(p, y)∕∂pn〉.
- 20.
See Färe and Grosskopf [1994] and Russell [70] for analyses of the distance function and associated shadow prices.
- 21.
- 22.
“Own” Morishima elasticities are identically equal to zero and hence uninteresting, as one might expect to be the case for a sensible elasticity of substitution.
- 23.
These notions are referred to as “p-substitutes” and “p-complements” in much of the literature (see Stern [75] and the papers cited there), as distinguished from “q-substitutes” and “q-complements”, which I call dual substitutes and complements in Section .
- 24.
McFadden [60] showed that his direct elasticites of substitution are constant and identical if and only if
$$\displaystyle \begin{aligned}F(x)=\Bigg(\sum_{s=1}^S\alpha_sF^s(x^s)\Bigg)^{1/\rho},\quad 0\ne \rho<1/n^*,\quad 0\le\rho<1/n^*,\quad \alpha_s>0\ \forall\ s,\end{aligned}$$or
$$\displaystyle \begin{aligned}F(x)=\alpha_0\prod_{s=1}^SF^s(x^s)^{\alpha_s/n^*},\quad \alpha_s>0\ \forall\ s,\quad {\textrm and}\quad n^*=\max_s\{n_s\}\end{aligned}$$where
$$\displaystyle \begin{aligned}F^s(x^s)=\prod_{i=1}^{n_s}x_i,\quad s=1,\ldots , m;\end{aligned}$$that is, if and only if the production function can be written as a CES or Cobb-Douglas function of (specific) Cobb-Douglas aggregator functions.
- 25.
Note the “self-duality” of this structure, a concept formulated by Houthakker [45] in the context of dual consumer preferences: the cost-function structure in prices mirrors the CES/Cobb-Douglas structure of the production function in input quantities.
- 26.
I am unaware of similar explorations of possible generalizations of the results on constancy of the Allen-Uzawa elasticities, but intuition suggests that similar results would go through there as well.
- 27.
Blackorby and Russell [14] proved that the dependence on y of the corresponding coefficients, βi, i = 1, …n, in (18) leads to a violation of positive monotonicity of the cost function in y. Thus, generalization to non-homothetic technologies does not expand the Cobb-Douglas technologies consistent with constancy of the MES.
- 28.
- 29.
Yet another possible assignation is “shadow elasticity of substitution,” since this dual concept is formulated in terms of shadow prices.
- 30.
Of course, the Allen and Morishima elasticities of complementarity are identical when n = 2, as is the case with Allen and Morishima elasticities of substitution.
- 31.
While shadow prices and dual elasticities are well defined even if the input requirement sets are not convex, the comparative statics of income shares using these elasticities requires convexity (as well, of course, as price-taking, cost-minimizing behavior), which implies concavity of the distance function in x. By way of contrast, convexity of input requirement sets is not required for the comparative statics of income shares using dual elasticities, since the cost function is necessarily concave in prices. See Russell [70] for a discussion of these issues.
- 32.
- 33.
- 34.
In fact, the concept is abstract: it can be applied to any (multiple variable) function.
- 35.
In the case where m = 1, D(F(x), x) = 1 on the isoquant for output F(x). Differentiate this identity with respect to xi and xj and take the ratio to obtain this equivalence.
- 36.
Proofs of these and other results in this section can be found in Blackorby, Primont, and Russell [16].
- 37.
Don’t ask. Or if you can’t resist, I refer you to Section 4.6 of Blackorby, Primont, and Russell [16] on “Sono independence” and additivity in a binary partition.
- 38.
Analogously to the case (13), \(\rho (y)={{\hat \rho (y)}/\big ({\hat \rho (y)-1\big )}}\).
- 39.
In fact, as first pointed out by Antras [4], the pre-1980 constancy of income shares does not imply unitary elasticity of substitution when one takes into account the empirical evidence of aggregate labor-saving technological change, which would tend to increase the share of capital, offsetting its declining share owing to a increasing capital intensity and an elasticity of substitution below one.
- 40.
Karabarbounis and Neiman [47] estimate an elasticity of substitution greater than one, but Acemoglu [2014] argues that their use of cross-country data makes their estimates more likely to correspond to endogenous-technology elasticities.
- 41.
Some of this discussion is based on a working paper by Mundra and Russell [64].
- 42.
- 43.
See, e.g., the analysis of the substitutability between a “good” and a “bad” output (in this case, electricity and sulphur dioxide) in Färe, Grosskopf, Noh, and Weber [35].
- 44.
This may be an unfair oversimplification: Stern [2011a], building on Mundlak [1968], proposes a related but somewhat different and more comprehensive taxonomy of the elasticities. (Nevertheless, I’m reminded of a (private) comment made by a prominent social choice theorist back in the heyday of research in his area: “The problem with social choice theory is that there are more axioms than there are ideas.” Well, perhaps we have reached the point where there are more elasticity-of-substitution concepts than there are ideas.)
References
Acemoglu, D. [2015], “The Rise and Decline of General Laws of Capitalism,” Journal of Economic Perspectives 29: 3–28.
Allen, R. G. D. [1938], Mathematical Analysis for Economists, London: Macmillan.
Allen, R. G. D., and J. R. Hicks [1934], “A Reconsideration of the Theory of Value, II,” Economica 1, N.S. 196–219.
Antras, P. [2004], “Is the U.S. Aggregate Production Function Cobb-Douglas? New Estimates of the Elasticity of Substitution,” Contributions to Macroeconomics 4: 1–34.
Arrow, K. J., H. P. Chenery, B. S. Minhaus, and R. M. Solow (SMAC) [1961], “Capital-Labor Substitution and Economic Efficiency,” Review of Economics and Statistics 63: 225–250.
Barnett, W. M., D. Fisher, and A. Serletis [1992], “Consumer Theory and the Demand for Money,” Journal of Economic Literature 30: 2086–2119.
Berndt, E. R., and L. Christensen [1973a], “The Internal Structure of Functional Relationships: Separability, Substitution, and Aggregation,” Review of Economic Studies 40: 403–410.
Berndt, E. R., and L. Christensen [1973b], “The Translog Function and the Substitution of Equipment, Structures, and Labor in U.S. Manufacturing 1929–68,” Journal of Econometrics 1: 81–114.
Berndt, E. R., and D. O. Wood [1975], “Technology, Prices, and the Derived Demand for Energy,” The Review of Economics and Statistics 57: 259–268.
Bertoletti, P. [2001], “The Allen/Uzawa Elasticity of Substitution as a Measure of Gross Input Substitutability, ” Rivista Italiana Degli Economisti 6: 87–94.
Bertoletti, P. [2005], “Elasticities of Substitution and Complementarity A Synthesis,” Journal of Productivity Analysis 24:183–196.
Blackorby, C., and R. R. Russell [1975], “The Partial Elasticity of Substitution,” Discussion Paper No. 75-1, Department of Economics, University of California, San Diego.
Blackorby, C., and R. R. Russell [1976], “Functional Structure and the Allen Partial Elasticities of Substitution: An Application of Duality Theory,” Review of Economic Studies 43: 285–292.
Blackorby, C., and R. R. Russell [1981], “The Morishima Elasticity of Substitution: Symmetry, Constancy, Separability, and Its Relationship to the Hicks and Allen Elasticities,” Review of Economic Studies 48: 147–158.
Blackorby, C., and R. R. Russell [1989], “Will the Real Elasticity of Substitution Please Stand Up? A Comparison of the Allen/Uzawa and Morishima Elasticities,” American Economic Review 79: 882–888.
Blackorby, C., D. Primont, and R. R. Russell [1978], Duality, Separability, and Functional Structure: Theory and Economic Applications, New York: North-Holland.
Blackorby, C., D. Primont, and R. R. Russell [2007], “The Morishima Gross Elasticity of Substitution,” Journal of Productivity Analysis 28: 203–208.
Borjas, G. J. [1994], “The Economics of Immigration,” Journal of Economic Literature 32: 1667–1717.
Borjas, G. J., R. B. Freeman, and L. F. Katz [1992], “On the Labor Market Effects of Immigration and Trade,” in G. Borjas and R. Freeman (eds.), Immigration and The Work Force, Chicago: University of Chicago Press.
Borjas, G. J., R. B. Freeman, and L. F. Katz [1996], “Searching for the Effect of Immigration on the Labor Market,” AEA Papers and Proceedings 8: 246–251.
Burk (Bergson), A. [1936], “Real Income, Expenditure Proportionality, and Frisch’s ‘New Methods of Measuring Marginal Utility’,” Review of Economic Studies 4: 33–52.
Cass, D. [1965], “Optimum Growth in an Aggregative Model of Capital Accumulation,” Review of Economic Studies 32: 233–240.
Chambers, R. [1988], Applied Production Analysis, Cambridge University Press, Cambridge.
Chirinko, R. S. [2008], “σ: the Long and Short of It,” Journal of Macroeconomics 30: 671–686.
Chirinko, R. S., and D. Mallick [2017], “The Substitution Elasticity, Factor Shares, and the Low-Frequency Panel Model,” American Economic Journal: Macroeconomics 9: 225–253.
Cobb, C. W., and P. H. Douglas [1928], “A Theory of Production,” American Economic Review 18: 23–34.
Cornes, R. [1992], Duality and Modern Economics, Cambridge: Cambridge University Press.
Davis, G. C., and J. Gauger [1996], “Measuring Substitution in Monetary-Asset Demand Systems,” Journal of Business and Economic Statistics 14: 203–209.
Davis G.C, and C. R. Shumway [1996], “To Tell the Truth about Interpreting the Morishima Elasticity of Substitution,” Canadian Journal of Agricultural Economics 44: 173–182.
de la Grandville, O. [1997], “Curvature and the Elasticity of Substitution: Straightening It Out,” Journal of Economics 66: 23–34.
Diewert, W. E. [1974], “Applications of Duality Theory,” in Frontiers of Quantitative Economics, Vol. 2, edited by M. Intriligator and D. Kendrick, Amsterdam: North-Holland.
Diewert, W. E. [1982], “Duality Approaches to Microeconomic Theory,” in Handbook of Mathematical Economics, Vol. II, edited by K. Arrow and M. Intriligator, New York: North-Holland.
Dixit, A. K., and J. E. Stiglitz [1977], “Monopolistic Competition and Optimum Product Diversity,” American Economic Review 67: 297–308.
Ewis, N. A., and D. Fischer [1984], “The Translog Utility Function and the Demand for Money in the United States,” Journal of Money, Credit, and Banking 16: 34–52.
Färe, R., S. Grosskopf, D-W Noh, and W. Weber [2005], “Characteristics of a Polluting Technology: Theory and Practice,” Journal of Econometrics 126: 469–492.
Färe, R., and D. Primont [1995], Multi-Output Production and Duality Theory: Theory and Applications, Boston: Kluwer Academic Press.
Fuss, M., and D. McFadden (eds.) [1978], Production Economics; A Dual Approach to Theory and Applications, Amsterdam: North-Holland.
Griliches, Z. [1969], Capital-Skill Complementarity, Review of Economic and Statistics 51: 465–468.
Grossman, J. [1982], “The Substitutability of Natives and Immigrants in Production,” Review of Economics and Statistics 64: 596–603.
Hall, R. E. [1988], “Intertemporal Substitution in Consumption,” Journal of Political Economy 96: 339–357.
Hardy, G. H., J. E. Littlewood, and G. Pólya [1934], Inequalities, Cambridge University Press, Cambridge.
Hicks, J. R. [1932], The Theory of Wages, MacMillan Press.
Hicks, J. R. [1970], “Elasticity of Substitution Again: Substitutes and Complements,” Oxford Economic Papers 22: 289–296.
Hicks, J. R., and R. G. D. Allen [1934], “A Reconsideration of the Theory of Value, Part II,” Economica 1, N.S. 196–219.
Houthakker, H,. S [1965], “A Note on Self-Dual Preferences,” Econometrica 33: 797–801.
Johnson, G. E. [1970], “The Demand for Labor by Educational Category,” Southern Economic Journal 37: 190–204.
Karabarbounis, L., and B. Neiman [2014], “The Global Decline of the Labor Share,” The Quarterly Journal of Economics 129: 61–103.
Kohli, U. [1999], “Trade and Migration: A Production Theory Approach,” in Migration: The Controversies and the Evidence, edited by R. Faini, J. de Melo, and K. F. Zimmermann, Cambridge: Cambridge University Press.
Kim, H. Y. [2000], “The Antonelli versus Hicks Elasticity of Complementarity and Inverse Input Demand Systems,” Australian Economic Papers 39: 245-261.
Klump, R., and O. de la Grandville [2000], “Economic Growth and the Elasticity of Substitution: Two Theorems and Some Suggestions,” American Economic Review 90: 282–291.
Koopmans, T. C. [1965], “On the Concept of Optimal Economic Growth,” in The Economicric Approach to Development Planning, Amsterdam, North Holland (for Pontificia Academic Science).
Kuga, K. [1979], On the Symmetry of Robinson Elasticities of Substitution: the General Case,” Review of Economic Studies 46: 527–531.
Kuga, K., and T. Murota [1972], “A Note on Definition of Elasticity of Substitution,” Macroeconomica 24: 285–290.
Kugler, P., U Müller, and G. Sheldon [1989], “Non-Neutral Technical Change, Capital, White-Collar and Blue-Collar Labor,” Economics Letters 31: 91–94.
Lau, L. [1978], “Applications of Profit Functions,” in M. Fuss and D. McFadden (eds.), Production Economics: A Dual Approach to Theory and Applications, Amsterdam: North-Holland, 133–216.
Leontief, W. W. [1947a], “A Note on the Interrelation of Subsets of Independent Variables of a Continuous Function with Continuous First Derivatives,” Bulletin of the American Mathematical Society 53: 343–350.
Leontief, W. W. [1947b], “Introduction to a Theory of the Internal Structure of Functional Relationships,” Econometrica 15: 361–373.
Leontief, W. W. [1953], “Domestic Production and Foreign Trade: The American Capital Position Re-examined,” Proceedings of the American Philosophical Society 97: 331–349.
Lerner A.P. [1933], “Notes on the Elasticity of Substitution II: the Diagrammatical Representation,” Review of Economic Studies 1: 68–70.
McFadden, D. [1963], “Constant Elasticity of Substitution Production Functions,” Review of Economic Studies 30: 73–83.
Morishima, M. [1967], “A Few Suggestions on the Theory of Elasticity” (in Japanese), Keizai Hyoron (Economic Review) 16: 144–150.
Mundlak, Y. [1968], “Elasticities of Substitution and the Theory of Derived Demand,” Review of Economic Studies 35: 225–236.
Mundra, K. [2013], “Direct and Dual Elasticities of Substitution under Non-homogeneous Technology and Nonparametric Distribution,” Indian Growth and Development Review 6: 204–218.
Mundra, K., and R. R. Russell [2004], “Dual Elasticities of Substitution,” Discussion Paper 01-26, Department of Economics, University of California, Riverside.
Murota, T. [1977], “On the Symmetry of Robinson Elasticities of Substitution: A Three-Factor Case,” Review of Economic Studies 42:79–85.
Parks, R. W. [1971], “Price Responsiveness of Factor Utilization in Swedish Manufacturing, 1870–1950,” Review of Economic Studies, 53: 129–139.
Pigou A.C. [1934], “The Elasticity of Substitution,” Economic Journal 44: 23–241.
Robinson, J. [1933], Economics of Imperfect Competition, London: MacMillan.
Russell, R. R. [1976], “Functional Separability and Partial Elasticities of Substitution,” Review of Economic Studies 42: 79–85.
Russell, R. R. [1997], “Distance Functions in Consumer and Producer Theory,” Essay 1 in Färe, R., S. Grosskopf, Index Number Theory: Essays in Honor of Sten Malmquist, Boston: Kluwer Academic Publishers, 7–90 .
Sato, R., and T. Koizumi [1973], “On the Elasticities of Substitution and Complementarity,” Oxford Economic Papers 25: 44–56.
Solow, R. M. [1956], “A Contribution to the Theory of Economic Growth,” Quarterly Journal of Economics 65: 65–94.
Sono, M. [1945, 1961], “The Effect of Price Changes on the Demand and Supply of Separable Goods,” International Economic Review 2: 239–271 (Originally published in Japanese in Kokumin Keisai Zasshi, 74: 1–51.).
Stern, D. I. [2010], “Derivation of the Hicks, or Direct, Elasticity of Substitution from the Input Distance Function,” Economic Letters 108: 349–351.
Stern D. I. [2011a], “Elasticities of Substitution and Complementarity,” Journal of Productivity Analysis 36: 79-89.
Stern D. I. [2011b], “The Role of Energy in Economic Growth,” Annals of the New York Academy of Sciences, 1219: 26–51.
Syrquin, M, and G. Hollender [1982], “Elasticities of Substitution and Complementarity: the General Case,” Oxford Economic Papers 34: 515–519.
Thompson, P., and T. G. Taylor [1995], “The Capital-Energy Substitutability Debate: A New Look,” Review of Economics and Statistics 77: 565–569.
Uzawa, H. [1962], “Production Functions with Constant Elasticities of Substitution,” Review of Economic Studies 29: 291–299.
Welch, F. [1970], “Education in Production,” Journal of Political Economy 78: 35–59.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Singapore Pte Ltd.
About this entry
Cite this entry
Russell, R.R. (2022). Elasticities of Substitution. In: Ray, S.C., Chambers, R.G., Kumbhakar, S.C. (eds) Handbook of Production Economics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3455-8_10
Download citation
DOI: https://doi.org/10.1007/978-981-10-3455-8_10
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-3454-1
Online ISBN: 978-981-10-3455-8
eBook Packages: Economics and FinanceReference Module Humanities and Social SciencesReference Module Business, Economics and Social Sciences