# Asymmetric Substitutability: Theory and Some Applications

## Article excerpt

I. INTRODUCTION

Economics textbooks predominantly classify goods as either being (gross) substitutes or being (gross) complements. As is well known, when the cross-price derivative of the uncompensated demand for Good 1 with respect to the price of Good 2 is positive, then Good 1 is a gross substitute for good j. When this cross-price derivative is negative, then Good 1 is a gross complement to Good 2. However, what is less well known is that it is perfectly possible for Good 1 to be a gross substitute for Good 2, while Good 2 is a gross complement to Good 1.

The possibility of cross-price effects with opposite sign in the case of two goods is mentioned by Hicks and Allen (1934, p. 213). In fact, Hicks and Allen use the apparent nonintuitiveness of this case as one more argument for their own concept of net substitutability, which is based on the cross-price derivative of compensated demand (i.e., demand keeping utility fixed and excluding the effect of price changes on purchasing power). (1) With compensated demand, one not only obtains cross-price derivatives with the same sign but also of the same size. Thus, goods are necessarily symmetrically net substitutable: if Good 1 is a substitute for Good 2, then Good 2 is necessarily also a substitute for Good 1 and an equally strong substitute. A single measure, the elasticity of substitution, can then be used to measure the degree of substitutability between two goods.

Hicks and Allen's main reason for proposing the concept of net substitutability is the definition that was predominantly used for substitutes and complements in those days. (2) In the definition of Auspitz and Lieben (1889), two goods are gross complements when [[partial derivative].sup.2]u/[partial derivative][x.sub.1][partial derivative][x.sub.2] > 0 (the marginal utility of Good 1, [partial derivative]u/[partial derivative][x.sub.1], increases when more of Good 2 is bought) and are gross substitutes when [[partial derivative].sup.2]u/[partial derivative][x.sub.1][partial derivative][x.sub.2] < 0 (the marginal utility of Good 1 decreases when more of Good 2 is bought). However, this definition only makes sense within a cardinal view of utility. For instance, take the Cobb-Douglas utility function u([x.sub.1], [x.sub.2]) = a ln [x.sub.1] + b ln [x.sub.2]. We have [[partial derivative].sup.2]u/[partial derivative][x.sub.1][partial derivative][x.sub.2] = 0, and it can be checked that the two goods are indeed neither substitutes nor complements. Yet, this same preference mapping can also be represented by the utility function v([x.sub.1],[x.sub.2]1 =[x.sup.a.sub.1][x.sup.a.sub.2] which has [[partial derivative].sup.2]ul [partial derivative][x.sub.1][partial derivative][x.sub.2] > 0. As [[partial derivative].sup.2]u/[partial derivative][x.sub.1][partial derivative][x.sub.2] represents a move from one indifference curve to another (with an interpretation of cardinal utility attached to it), Hicks and Allen propose the concept of net substitutability, which does not require a shift from one indifference curve to the other.

Yet, the contemporary definition of gross substitutability in terms of the sign of uncompensated demand does not require the concept of cardinal utility. It seems then that the only reason for not at least using gross substitutability side by side with net substitutability is that gross substitutability allows for the apparently unintuitive case of asymmetric substitutability. The purpose of this article is to show that asymmetric gross substitutability is in fact an intuitive phenomenon, leading to deeper insight into consumer theory, and with many potential applications. Thus, while the argument in favor of net substitutability is that gross substitutability allows for asymmetric substitutability, our argument is that the disadvantage of net substitutability is that it does not allow for the concept of asymmetric substitutability.

Section II introduces two concepts of asymmetric gross substitutability, namely, weak asymmetric gross substitutability and strong asymmetric gross substitutability, and relates these two concepts to more familiar classifications of goods (namely, luxuries and necessities and elastic and inelastic goods). …

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