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Elasticities of substitution and complementarity

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Abstract

This paper develops a classification scheme of the many different definitions of elasticities of substitution and complementarity in the production case based on primal and dual representations of technology and their related direct and inverse demand functions, gross and net substitution, elasticity type, and three different basic concepts of substitution and complementarity. The ten elasticities of substitution are derived from the cost, profit, input distance, and revenue functions. All the elasticities are equally valid for single and multi-output technologies. The classic Berndt-Wood dataset is used to show the considerable variation across the elasticities.

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Notes

  1. De La Grandville (1997) points out that the standard mathematical definition of curvature does not have any relation to the elasticity of substitution. Economists have used the term curvature very loosely in this literature. Shape of the isoquant is another synonym for this concept. I prefer “difficulty of substitution”. From an economic perspective that is what actually matters.

  2. Various other synonyms have been used including aiding (mesayy’im in Hebrew) and competing (mitharim) (Blumenthal et al. 1971), cooperant and rival (Hicks 1936), and complementary and anti-complementary (e.g. Hirschleifer and Hirschleifer 1997). An increase in the use of one q-substitute input reduces the marginal product of the other input, while an increase in the use of a q-complement input raises the marginal product of the other input.

  3. Two inputs are p-substitutes in Hicks’ (1970) terminology if when the price of one rises the quantity of the other increases and p-complements if the quantity decreases as a result.

  4. Throughout the paper capital letters will refer to functions and lower case letters to variables. Derivatives of the objective functions (cost, profit, input distance, revenue, production) are indicated by subscripts. Subscripts on the demand functions indicate which input is demanded. Bold indicates a vector. Where arguments of functions are not specified this is either because a general class of function is proposed which could have different arguments depending on the problem involved or because these are defined in the accompanying text.

  5. Samuelson developed two further measures of complementarity in the consumption case which I will ignore as this paper only discusses the production case.

  6. There are no citations to Pigou’s paper in the period 1954–2008 in the ISI Citation Index. The point was rediscovered many years later by Mundlak (1968), Morishima (1967), and Blackorby and Russell (1975).

  7. These constant cost demand functions can be derived from the indirect production function via Roy’s identity (Kim 1988). Therefore, the relevant elasticities of substitution could be derived from the indirect production function.

  8. This follows from the definition of the elasticity of substitution in the two input case as (Uzawa 1962): \( \sigma_{ij} = {\frac{{Y_{i} Y_{j} }}{{YY_{ij} }}} \) implying that \( Y_{ij} > 0 \) for p-substitutes and, therefore, that the two inputs must be q-complements as well.

  9. Slutsky independently developed these ideas in 1915 (Samuelson 1974).

  10. As Bertoletti (2005) points out, this depends on the "size properties" of the inputs involved, and in particular on whether the inputs are normal or inferior.

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Acknowledgments

I thank Catherine Morrison-Paul, Donald Siegel, Donald Vitaliano, James Adams, Ken Simons, Astrid Kander, Fernando de Almeida Martins, Heather Anderson, and anonymous referees for very useful comments and suggestions.

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Authors and Affiliations

  1. Arndt-Corden Department of Economics, Crawford School of Economics and Government, Australian National University, Canberra, ACT, 0200, Australia

    David I. Stern

  2. Centre for Applied Macroeconomic Analysis, Australian National University, Canberra, ACT, 0200, Australia

    David I. Stern

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Appendix: computing the elasticities from the cost function estimates

Appendix: computing the elasticities from the cost function estimates

I computed the sample mean Allen elasticities of substitution from the cost function parameters. The SES, AEC, HEC, HLES, and MES were then computed from the AES and the parameters of the translog cost function. The remaining elasticities were derived from the HLES, HEC, and AEC according to the following schema:

From equation (24) in Broer (2004) we have the following method of deriving the AEC from the AES:

$$ \left[ {\begin{array}{*{20}c} {{\mathbf{AEC}}} & \iota \\ {\iota '} & 0 \\ \end{array} } \right] = \left[ {diag(S_{1} , \ldots ,S_{n} ,1)\left[ {\begin{array}{*{20}c} {{\mathbf{AES}}} & \iota \\ {\iota '} & 0 \\ \end{array} } \right]diag(S_{1} , \ldots ,S_{n} ,1)} \right]^{ - 1} $$
(21)

where AEC and AES are the matrices of the elasticities of complementarity and substitution. I replace the cost shares with the respective first order parameters from the translog cost function to obtain the sample mean AEC. Bertoletti (2005) shows how to derive the HLES from the AES:

$$ {\hbox{HLES}}_{ij} = \left( {{\frac{\partial \ln C}{\partial \ln y}} - 1} \right)\left( {\hbox{AES}}_{ij} - {\frac{{{{\partial \ln X_{i} }/ {\partial \ln y}}}}{{{{\partial \ln C} \mathord{\left/ {\vphantom {{\partial \ln C} {\partial \ln y}}} \right. \kern-\nulldelimiterspace} {\partial \ln y}}}}}\,\frac{{\partial \ln X_{j} }/{\partial \ln y}}{{\partial \ln \lambda }/{\partial \ln y}} \right) $$
(22)

where λ is marginal cost. For the translog cost function:

$$ {\frac{\partial \ln C}{\partial \ln y}} = \beta_{y} + \beta_{yy} \ln y + \sum\limits_{i = 1}^{m} {\beta_{iy} \ln p_{i} } + \beta_{y\tau } \tau $$
(23)

and, therefore:

$$ {\frac{\partial \ln \lambda }{\partial \ln y}} = - 1 + {\frac{\partial \ln C}{\partial \ln y}} + {\frac{{\beta_{yy} }}{\partial \ln C/\partial \ln y}} $$
(24)

\( \partial \ln X_{i} /\partial \ln y \) is the elasticity of the demand for input i with respect to output y. For the translog cost function:

$$ X_{i} = {\frac{C}{{p_{i} }}}\left[ {\beta_{i} + \beta_{iy} \ln y + \sum\limits_{j = 1}^{m} {\beta_{ij} \ln p_{j} } + \beta_{i\tau } \tau } \right] $$
(25)

and, therefore:

$$ {\frac{{\partial \ln X_{i} }}{\partial \ln y}} = {\frac{\partial \ln C}{\partial \ln y}} + {\frac{{\beta_{iy} }}{{S_{i} }}} $$
(26)

From Syrquin and Hollender (1982) we have the following method of deriving the HEC:

$$ {\hbox{HEC}}_{ij} = {\frac{{\Upsigma_{ij} }}{{S_{i} S_{j} }}} - {\frac{\partial \ln \lambda }{\partial \ln y}} $$
(27)

where λ is marginal cost and:

$$ \Upsigma = \left[ {\begin{array}{*{20}c} {{\mathbf{AES}}} & {\left[ {\nabla_{\ln y} \ln {\mathbf{X}}} \right]'} \\ {\nabla_{\ln y} \ln {\mathbf{X}}} & 0 \\ \end{array} } \right]^{ - 1} $$
(28)

The MEC, PEC, MGES, and MES are easily derived from the AEC, HEC, HLES, and AES respectively as they are functions of the cross-price elasticities. By substituting the formula for the AES into that for the SES and of the AEC into that for the SEC, the following expressions can be derived:

$$ {\hbox{SES}}_{ij} = {\frac{{S_{i} S_{j} }}{{S_{i} + S_{j} }}}\left( { - {\hbox{AES}}_{ii} + 2{\hbox{AES}}_{ij} - {\hbox{AES}}_{jj} } \right) $$
(29)
$$ {\hbox{SEC}}_{ij} = {\frac{{S_{i} S_{j} }}{{S_{i} + S_{j} }}}\left( { - {\hbox{AEC}}_{ii} + 2{\hbox{AEC}}_{ij} - {\hbox{AEC}}_{jj} } \right) $$
(30)

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Stern, D.I. Elasticities of substitution and complementarity. J Prod Anal 36, 79–89 (2011). https://doi.org/10.1007/s11123-010-0203-1

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