A note on CES functions

Accepted Manuscript

A note on CES functions Christian Thoni ¨ PII: DOI: Reference:

S2214-8043(15)00124-X 10.1016/j.socec.2015.10.001 JBEE 157

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Journal of Behavioral and Experimental Economics

Received date: Accepted date:

19 June 2015 1 October 2015

Please cite this article as: Christian Thoni, ¨ A note on CES functions, Journal of Behavioral and Experimental Economics (2015), doi: 10.1016/j.socec.2015.10.001

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Highlights • The article looks at the share parameter (θ) in CES functions. • In typical CES functions θ is not interpretable for all elasticities of substitution.

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• A novel CES utility function is introduced to address this shortcoming.

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A note on CES functions∗ October 8, 2015

Abstract

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Christian Thöni†

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The most commonly used parameterizations of constant elasticity of substitution functions (CES) have the disadvantage that the share parameter has no meaningful interpretation over the range of possible elasticities of substitution. This note introduces a slightly reformulated CES function which avoids the problem.

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Introduction

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JEL-Classification: D12, D24 Keywords: constant elasticity of substitution; CES; share parameter

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Stimulated by experimental results on social dilemma, dictator, or ultimatum games, the past two decades have seen an increasing number of models of social preferences appearing in the literature. Sobel (2005) classifies these models into models of interdependent preferences and models of reciprocity. The former category typically postulates a utility function u(x, y), where x refers to the agent’s own monetary payoff and y refers to the monetary payoff of one or more other players. While early models such as Fehr and Schmidt (1999) typically assume very simple functional forms for u(), a number of recent approaches use the constant elasticity of substitution function (CES).1 The CES function goes back to Solow (1956), and is widely applied in other fields of economics to model production functions or ordinary consumer choice problems. Relative to simpler formulations the CES function ∗ Acknowledgements: I thank the editor, an anonymous referee, and Stefan Buehler for helpful comments and suggestions. † University of Lausanne, Quartier UNIL-Dorigny, Bâtiment Internef, CH1015 Lausanne; phone: +41 21 692 2843; mail: [email protected]; web: http://sites.google.com/site/christianthoeni/ 1 Thöni and Gächter (2015) provide a recent overview of the models of social preferences. The CES function is used by e.g. Andreoni and Miller (2002), Cox, Friedman, and Gjerstad (2007), Fisman, Kariv, and Markovits (2007).

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has the advantage that it covers a much broader spectrum of substitutability between x and y. The most commonly used parameterizations of CES functions is 1

u(x, y) = (xρ + θy ρ ) ρ ,

(1)

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with ρ ∈ (−∞, 1] \ {0} and θ > 0. This function allows for the full range of substitutionability between the two goods X and Y . For ρ = 1 the function collapses to u() = x + θy and models the case of perfect substitutes. For ρ → −∞ the function approximates the case of perfect complements. For ρ → 0 the function converges to the Cobb-Douglas function u() = xy θ . The parameter θ is often referred to as share parameter.

Problem

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The share parameter is in the center of our interest. For example, if θ = 12 we would say that X is more valuable to the individual than Y , and, if both come at the same price, should be consumed in larger quantities (x∗ > y ∗ ). For perfect substitutes the individual should value every unit of X twice as much as every unit of Y and consume exclusively X while for perfect complements the consumer should consume two units of X for every unit of Y , i.e., x∗ = 2y ∗ . If the goods have different prices then the individual should switch to Y as soon as Y costs half as much than X for perfect substitutes, and the price should have no influence on the relative amounts consumed for perfect complements. Unfortunately, the standard formulation of the CES utility function does not have this property. Here is why: Assume an individual maximizes (1) subject to a budget constraint m = x + py. In the optimum, the marginal rate of substitution is equal to the relative price: MRS =

1 xρ−1 =p θ y ρ−1

(2)

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Using the budget constraint we can derive the Marshallian demand functions: x(m, p, θ, ρ) =

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mθ ρ−1 ρ

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p ρ−1 + θ ρ−1

and

y(m, p, θ, ρ) =

mp ρ−1 ρ

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p ρ−1 + θ ρ−1

(3)

For perfect complements (ρ → −∞) both demand functions converge to i.e., irrespective of θ the individual consumes equal amounts of X and Y . Thus the standard CES function shown in equation (1) cannot generate the case where two goods are perfect complements which are consumed at m p+1 ,

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a ratio other than one-to-one.2 This result may come as a surprise, because intuitively one might think that the standard CES function should produce L-shaped indifference curves with the kink at xy = θ.3 On the other hand, this ‘shortcoming’ of the standard CES function was already discussed by Arrow, Chenery, Minhas, and Solow (1961, p. 231) for perfect complements in production functions:

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[. . . ] This represents a system of right-angled isoquants with corners lying on a 45 ◦ line from the origin. But it is clearly more general than that, since the location of the corners can be changed simply measuring K and L in different units.

Remedy

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Their proposed solution (rescaling one of the two axes) seems unsatisfactory, in particular because there are more elegant solutions to the problem. For production functions and their applications to general equilibrium models Senhadji (1997) proposes a CES function where the share parameter
1/ρ is taken to the power of 1 − ρ: u() = xρ + θ1−ρ y ρ . This formulation solves the problem for ρ → −∞. However, for ρ = 1 the function reduces to u() = x + y, i.e., X and Y are perfect one-to-one substitutes and the share parameter becomes ineffective at the other end of the scale of ρ. Another apparent solution is to take the share parameter to the power of ρ: u() = (xρ + θρ y ρ )1/ρ (David & van de Klundert, 1965). In this variant the share parameter is effective in both extreme cases, perfect substitutes and complements. However, the function has the strange property that for θ < 1 good X is favored when substitution is easy (x∗ > y ∗ if ρ > 0), and the opposite is true when substitution becomes difficult (x∗ < y ∗ if ρ < 0, both examples for p = 1).

To the best of my knowledge this is the first proposal of a CES function which combines the following properties

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• The share parameter θ identifies the proportion to which X and Y are consumed in case of perfect complements, i.e., is able to produce L-shaped indifference curves with a kink on the ray y = θx.

• The share parameter identifies preferences for X and Y also for perfect substitutes. If θ < p the consumer demands only X, is indifferent in case of equality, and consumes exclusively Y otherwise.

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To be precise, the standard CES function can produce approximately L-shaped functions with kinks off the 45 degree line if the share parameter approaches zero or infinity. However, in this case θ would not work for prefect substitutes. 3 Indeed, one can find several occurrences where the standard CES function is claimed to have that property, see e.g. Cox et al. (2007, p.822).

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Figure 1: Indifference curves for different elasticities of substitution from perfect substitutes (ρ = 1, linear functions), intermediate cases (ρ = −1, −10) to perfect complements (ρ → −∞, L-shaped functions). Solid lines show the case where X is more valuable than Y (θ = 21 ), dashed lines show the opposite case (θ = 2). The left panel shows the case of the standard CES function (equation 1), the right panel shows the reformulated CES function (equation 4).

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• The share parameter θ has the same interpretation over the entire range of ρ: θ < 1 implies x∗ > y ∗ for p = 1 and for all ρ ∈ (−∞, 1]\{0}.

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The function I propose takes the share parameter to the power of 2 − ρ: 

u(x, y) = xρ + θ2−ρ y ρ

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(4)

ρ

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The reformulated CES function4 comes at very limited additional complexity relative to the standard version shown in equation (1). Like before we can easily derive the marginal rate of substitution as

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MRS =

1 xρ−1 , y ρ−1

(5)

θ2−ρ

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equate the MRS with the relative price and use the budget constraint to derive Marshallian demand functions: 2−ρ

x(m, p, θ, ρ) =

mθ ρ−1 ρ

2−ρ

p ρ−1 + θ ρ−1

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and

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y(m, p, θ, ρ) =

mp ρ−1 ρ

2−ρ

p ρ−1 + θ ρ−1

(6)

This is not a unique solution for the problem. Any transformation θ(ρ) of the share parameter with the properties θ(1) = 1, θ(−∞) → ∞, and θ0 < 0 would do the trick.

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These demand functions have the desired properties. For perfect complements we have lim x() =

ρ→−∞

m pθ + 1

and

lim y() =

ρ→−∞

mθ . pθ + 1

(7)

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Consequently, the proportion between X and Y is equal to θ−1 . For perfect substitutes utility is simply u() = x + θy, like it is the case with the standard CES function. Demand is then non-continuous at p = θ: For lower p only X is consumed, for higher p only Y . Figure 1 shows the differences between the two CES formulations graphically. On the left panel I depict the standard CES function (equation 1) and show that the indifference curves for θ = 12 and θ = 2 converge if we move towards ρ → −∞. The right panel demonstrates that this is not the case for the CES function shown in equation 4.

Discussion

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References

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The reformulated CES function uses a very simple transformation of the share parameter from θ to θ2−ρ . When estimating CES preferences or production functions from data then the two CES functions (1) and (4) are equally restrictive. Thus, if one is not interested in interpreting the value of θ then the differences between the two does not matter. If, however, the aim is to calibrate utility or production functions like e.g. Andreoni and Miller (2002), Fisman, Jakiela, Kariv, and Markovits (2015), or Cunha, Heckman, and Schennach (2010) then it is certainly an advantage to use a functional form where the share parameter allows for a straightforward interpretation.

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Andreoni, J. & Miller, J. (2002). Giving according to GARP: An experimental test of the consistency of preferences for altruism. Econometrica, 70 (2), 737–753. doi:10.1111/1468-0262.00302 Arrow, K. J., Chenery, H. B., Minhas, B. S., & Solow, R. M. (1961). Capitallabor substitution and economic efficiency. Review of Economics and Statistics, 43 (3), 225–250. Cox, J. C., Friedman, D., & Gjerstad, S. (2007). A tractable model of reciprocity and fairness. Games and Economic Behavior, 59, 17–45. doi:10.1016/j.geb.2006.05.001 Cunha, F., Heckman, J., & Schennach, S. (2010). Estimating the technology of cognitive and noncognitive skill formation. Econometrica, 78 (3), 883–931. doi:10.3982/ECTA6551

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David, P. A. & van de Klundert, T. (1965). Biased efficiency growth and capital-labor substitution in the U.S., 1899-1960. American Economic Review, 55 (3), 357–394. Fehr, E. & Schmidt, K. M. (1999). A theory of fairness, competition, and cooperation. Quarterly Journal of Economics, 114 (3), 817–868. doi:10. 1162/003355399556151 Fisman, R., Jakiela, P., Kariv, S., & Markovits, D. (2015). The distributional preferences of an elite. Science, 349 (6254). doi:10.1126/science. aab0096 Fisman, R., Kariv, S., & Markovits, D. (2007). Individual preferences for giving. American Economic Review, 97 (5), 1858–1876. doi:10.1257/ aer.97.5.1858 Senhadji, A. S. (1997). Two common problems related to the use of the Armington aggregator in computable general equilibrium models. Applied Economics Letters, 4 (1), 23–25. doi:10.1080/758521826 Sobel, J. (2005). Interdependent preferences and reciprocity. Journal of Economic Literature, 43 (2), 392–436. doi:10.1257/0022051054661530 Solow, R. M. (1956). A contribution to the theory of economic growth. Quarterly Journal of Economics, 70 (1), 65–94. doi:10.2307/1884513 Thöni, C. & Gächter, S. (2015). Peer effects and social preferences in voluntary cooperation. Journal of Economic Psychology, 48, 1–47. doi:10. 1016/j.joep.2015.03.001

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