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Elasticities of Substitution

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Handbook of Production Economics

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.

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Notes

  1. 1.

    This concept was independently formulated by Cambridge University economist Joan Robinson [69] in her comparably classic book on The Economics of Imperfect Competition. Abba Lerner [60] and A. C. Pigou [68] also contributed to the understanding of the concept at its genesis.

  2. 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 [29] for a clear exposition of this point.

  3. 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. 4.

    The Morishima elasticity is a generalization of Robinson’s [69] characterization of the two-input elasticity and for this reason is called the “Robinson elasticity of substitution” by Kuga and Murota [54] and Kuga [53].

  5. 5.

    See Footnote 2 above.

  6. 6.

    It is dual, not to the Morishima elasticity, but to McFadden’s [61] shadow elasticity of substitution.

  7. 7.

    Separability is also a necessary condition for decentralization of an optimization problem (as in, e.g., two-stage budgeting). See Blackorby et al. [15, Ch. 5] for a thorough exposition of the connection between separability and decentralized decision-making.

  8. 8.

    As we shall see in Digression: Dual Representations of Multiple-Input, Multiple-Output Technologies the homotheticity assumption can be dropped when the elasticity concept is formulated in the dual.

  9. 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. 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. 11.

    The Cobb-Douglas production function was well-known at the time of the SMAC derivation, having been proposed much earlier [25]. The (CES2) production function made its first appearance in Solow’s [73] classic economic growth paper, but the functional form had appeared much earlier in the context of utility theory: Bergson (Burk) [20] 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 et al. [41, p. 13]) as the “mean value of order ρ.”

  12. 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. 13.

    Because the Allen elasticities are now typically exposited in the dual, they are often called the “Allen-Uzawa elasticities,” and I resort to that nomenclature on occasion as well. Uzawa’s approach was later extended by Blackorby and Russell [11, 13].

  14. 14.

    Thorough expositions of duality theory can be found in, e.g., Blackorby et al. [15], Chambers [22], Cornes [26], Diewert [30, 31], Färe and Primont [36], Fuss and McFadden [37], and Russell [71].

  15. 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. 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. 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 [71, footnote 12] for a comparison of these approaches.

  18. 18.

    Whatever is not there can be found in Diewert [30] or the Fuss/McFadden [37] volume.

  19. 19.

    ∇pC(p, y) := 〈 C(p, y)∕∂p1, …, ∂C(p, y)∕∂pn〉.

  20. 20.

    See Färe and Grosskopf [34] and Russell [71] for analyses of the distance function and associated shadow prices.

  21. 21.

    See Blackorby and Russell [11, 13, 14] for a discussion of this asymmetry, which (as pointed out by Stern [76]) was recognized much earlier by Pigou [68] in his analysis of Robinson’s [69] less-formal characterization of this elasticity concept.

  22. 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. 23.

    These notions are referred to as “p-substitutes” and “p-complements” in much of the literature (see Stern [76] and the papers cited there), as distinguished from “q-substitutes” and “q-complements,” which I call dual substitutes and complements in section “Dual Elasticities of Substitutionł::bel sec:dual”.

  24. 24.

    McFadden [61] showed that his direct elasticities 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

    $$\displaystyle \begin{aligned}F(x)=\alpha_0\prod_{s=1}^SF^s(x^s)^{\alpha_s/n^*},\quad \alpha_s>0\ \forall\ s,\quad {\mathrm{and}}\quad n^*=\max_s\{n_s\}\end{aligned} $$

    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. 25.

    Note the “self-duality” of this structure, a concept formulated by Houthakker [46] 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. 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. 27.

    Blackorby and Russell [13] 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. 28.

    See Bertoletti [10], Kim [50], and Stern [76].

  29. 29.

    Yet another possible assignation is “shadow elasticity of substitution,” since this dual concept is formulated in terms of shadow prices.

  30. 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. 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 [71] for a discussion of these issues.

  32. 32.

    See also Hicks [44], Sato and Koizumi [72], and Stern [76] for discussions of these issues.

  33. 33.

    The concept was independently conceived by Leontief [57, 58] and Sono [74]. See Blackorby et al. [15] for a comprehensive development of the concept and its applications and for citations to the literature extending the Leontief-Solo concept.

  34. 34.

    In fact, the concept is abstract: it can be applied to any (multiple variable) function.

  35. 35.

    In the case where m = 1, D(x, F(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. 36.

    Proofs of these and other results in this section can be found in Blackorby et al. [15].

  37. 37.

    Don’t ask. Or if you can’t resist, I refer you to Section 4.6 of Blackorby et al. [15] on “Sono independence” and additivity in a binary partition.

  38. 38.

    Analogous to the case (13), \(\rho (y)={{\hat \rho (y)}/\big ({\hat \rho (y)-1\big )}}\).

  39. 39.

    In fact, as first pointed out by Antras [3], 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. 40.

    Karabarbounis and Neiman [48] estimate an elasticity of substitution greater than one, but Acemoglu and Robinson [1] argues that their use of cross-country data makes their estimates more likely to correspond to endogenous-technology elasticities.

  41. 41.

    Some of this discussion is based on a working paper by Mundra and Russell [65].

  42. 42.

    Follow-ups of the Griliches study can be found in Johnson [47], Kugler et al. [55], and Welch [80].

  43. 43.

    See, e.g., the analysis of the substitutability between a “good” and a “bad” output (in this case, electricity and sulfur dioxide) in Färe et al. [35].

  44. 44.

    This may be an unfair oversimplification: Stern [76], building on Mundlak [63], 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.)

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  1. Department of Economics, University of California, Riverside, Riverside, CA, USA

    R. Robert Russell

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  1. Department of Economics, University of Connecticut, Storrs, CT, USA

    Subhash C. Ray

  2. Department of Agricultural and Resource Economics, University of Maryland, College Park, MD, USA

    Robert Chambers

  3. Department of Economics, Binghamton University, Binghamton, NY, USA

    Subal Kumbhakar

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Russell, R. (2020). Elasticities of Substitution. In: Ray, S., Chambers, R., Kumbhakar, S. (eds) Handbook of Production Economics. Springer, Singapore. https://doi.org/10.1007/978-981-10-3450-3_10-1

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