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Asymmetric substitutability: theory and some applications.

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). Section III gives two potential explanations for asymmetric gross substitutability in terms of Lancaster's (1966) approach to consumer theory and in terms of Gilley and Karels' (1991) constraints approach. Section IV gives four potential examples/applications of asymmetric gross substitutability. We end with a discussion in Section V.

II. (A)SYMMETRIC SUBSTITUTABILITY: DEFINITIONS AND EXISTENCE

Our focus is on the two-good case. For i = 1, 2, denote by [x.sub.i] the uncompensated, Marshallian demand for good i, by [p.sub.i] the price of good i, and by m the consumer's income. Good 1 is a gross substitute for Good 2 if [partial derivative][x.sub.1](pl, [p.sub.2], m)lO[p.sub.2] > 0 and Good 1 is a gross complement to Good 2 if O[x.sub.1](pl, [p.sub.2], m)/O[p.sub.2] < 0. For the two-good case, microeconomic textbooks usually without further ado describe goods as being gross substitutes or being gross complements. This suggests then that goods are either what we call weak symmetric gross substitutes, meaning that sgn [partial derivative][x.sub.1]([p.sub.1],[p.sub.2], m)/ [partial derivative][p.sub.2] = sgn [partial derivative][x.sub.2]([p.sub.1], [p.sub.2], m)/ [partial derivative][p.sub.2], or even what we call strong symmetric gross substitutes, meaning that [partial derivative][x.sub.1]([p.sub.1], [p.sub.2], m)/[partial derivative][p.sub.2] = [partial derivative][x.sub.2]([p.sub.1], [p.sub.2], m)/ [partial derivative][p.sub.1], as summarized in the first two columns of Table 1. (3)

However, two goods can also exhibit asymmetric gross substitutability, which we define in the following way. A strong criterion for asymmetric gross substitutability is that sgn [partial derivative][x.sub.1]([p.sub.1],[p.sub.2], m)/ [partial derivative][p.sub.2][not equal to] sgn [partial derivative][x.sub.2]([p.sub.1], [p.sub.2], m)/[partial derivative][p.sub.1]; this we refer to as strong asymmetric gross substitutability (see again Table 1 for a summary). While our main focus is on strong asymmetric substitutability, we also introduce a weak criterion for asymmetric gross substitutability, where it suffices that [partial derivative][x.sub.1]([p.sub.1], [p.sub.2], m)/ [partial derivative][p.sub.2] [not equal to] [partial derivative][x.sub.2]([p.sub.1],[p.sub.2], m)/[partial derivative][p.sub.1]; we refer to this as weak asymmetric gross substitutability. The interest of the latter concept of asymmetric substitutability is that it can directly be related to income effects. Note that goods that exhibit strong asymmetric gross substitutability necessarily exhibit weak asymmetric gross substitutability, while the reverse is not true. We now consecutively derive claims that give insight into the circumstances under which weak and strong asymmetric gross substitutability occur. We show that these circumstances are familiar. Weak asymmetric gross substitutability fully coincides with the case where one good is a luxury and the other good is a necessity; strong asymmetric gross substitutability fully coincides with the case where one good is price elastic (the complement) and the other good is price inelastic (the substitute).

A. Weak Asymmetric Gross Substitutability

We start by showing that there is a one-to-one relation between weak asymmetric substitutability and nonhomothetic preferences, where the latter are synonymous with asymmetric income elasticities. Homothetic utility functions have indifference curves that are "blown-up" versions of one another. Concretely, for the case where a consumer consumes N goods, take bundle ([x.sub.1], [x.sub.2], ..., XN). Next, take bundle (t[x.sub.1], t[x.sub.2], ..., [tx.sub.N]) with t > 1. Then, the indifference curve has the same marginal rates of substitution at this bundle. This means that, when income is increased by a factor t, consumption of each good increases by a factor t. In other words, the income elasticity [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is equal to 1 for each good i.

Note that the case where all income elasticities are equal to 1 is the only case where all income elasticities of the consumer are equal, so that (non-)homothetic preferences are synonymous with (a)symmetric income elasticities. To see why, denote as [s.sub.i] = ([p.sub.i][x.sub.i])/m, the consumer's budget share spent on good i, where [SIGMA][s.sub.i] = 1. By totally differentiating the budget constraint [Summation][p.sub.i][x.sub.i] = m with respect to m, one obtains that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all goods, then [bar.[epsilon]][SIGMA][s.sub.i] = [[bar.[epsilon]] = 1. Nonhomothetic preferences are thus synonymous with a case where not all income elasticities are equal. In the particular case where the consumer consumes only two goods, by the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], nonhomothetic preferences imply that Good 2 is a luxury ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) and Good 1 is a necessity ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) and/or an inferior good ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).

Claim 1 now shows that there is a one-to-one relationship between weak asymmetric gross substitutability and nonhomothetic preferences. (4)

CLAIM 1. Preferences that exhibit weak asymmetric substitutability are nonhomothetic preferences. In particular, [partial derivative][x.sub.1]([p.sub.1], [p.sub.2], m)/ [partial derivative][p.sub.2] > [partial derivative][x.sub.2]([p.sub.1], [p.sub.2], m)/[partial derivative]pl iff [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For the two-good case, this' means that [partial derivative][x.sub.1]([p.sub.1],[p.sub.2], m)/ [partial derivative][p.sub.2] > [partial derivative][x.sub.2]([p.sub.1] [p.sub.2], m)/[partial derivative][p.sub.1] iff Good 1 is a necessity and/or an inferior good and Good 2 is a luxury. (5)

One way to show Claim 1 is to look at the Hicksian decomposition of the cross-price effects (6):

[partial derivative][x.sub.1]([p.sub.1],[p.sub.2], m)/[partial derivative][p.sub.2] = [partial derivative][h.sub.1]([p.sub.1],[p.sub.2], m)/[partial derivative][p.sub.2] - [[partial derivative][x.sub.1]([p.sub.1],[p.sub.2], m)/[partial derivative]m][x.sub.2] (1)

[partial derivative][x.sub.2]([p.sub.1],[p.sub.2], m)/[partial derivative][p.sub.1] = [partial derivative][h.sub.2]([p.sub.1],[p.sub.2], m)/[partial derivative][p.sub.1] - [[partial derivative][x.sub.2]([p.sub.1],[p.sub.2], m)/[partial derivative]m][x.sub.1], (2)

where [h.sub.1]([p.sub.1], [p.sub.2], m) denotes compensated, Hicksian demand for Good 1 (=demand compensating the consumer for the loss in purchasing power from the price increase such that his or her utility remains constant). Good 1 is a net substitute for Good 2 [partial derivative][h.sub.1]([p.sub.1],[p.sub.2], m)/ [partial derivative][p.sub.2] > 0 and Good 1 is a net complement for Good 2 if [partial derivative][h.sub.1]([p.sub.1], [p.sub.2], m)/[partial derivative][p.sub.2] < 0. As is well known, the symmetry of the Slutsky matrix tells us that [partial derivative][h.sub.1]([p.sub.1],[p.sub.2], m)/[partial derivative][p.sub.2] = [partial derivative][h.sub.2]([p.sub.1], [p.sub.2], m)/[partial derivative][p.sub.1], meaning that we always have strong symmetric net substitutability. Subtracting Equation (2) from Equation (1) and using the equality of the Hicksian cross-price effects, we obtain that

[partial derivative][x.sub.1]/[partial derivative][p.sub.2] - [partial derivative][x.sub.2]/[partial derivative][p.sub.1] = ([partial derivative][x.sub.2]/[partial derivative]m)[x.sub.1] - ([partial derivative][x.sub.1]/ [partial derivative]m)[x.sub.2]. (3)

It follows by Equation (3) that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thus, weak asymmetric substitutability is a synonym for asymmetric income elasticity. Intuitively, in the two-good case, if a necessity and a luxury are mutual substitutes, then the necessity is a stronger substitute for the luxury than the luxury is for the necessity; bread can be a substitute for yachts, but yachts cannot easily substitute for bread. (7)

To get a graphical insight into the one-to-one relation between weak asymmetric gross substitutability and nonhomothetic preferences, it is useful to focus on the two-good case and consider the following. Let us look at the effect of an increase in income on ([x.sub.2]/[x.sub.1] With homothetic preferences, this effect must be zero; if Good 1 is a necessity and Good 2 is a luxury, then this effect must be positive; if Good 1 is a luxury and Good 2 is a necessity, this effect must be negative. An increase in income can now be decomposed into a decrease of both prices with the same percentage. Thus, to increase income, we decrease both prices such that the relative price remains the same. In particular, fix [p.sub.2] = v[p.sub.1],, so that we can rewrite ([x.sub.2]/[x.sub.1] as [x.sub.2]([p.sub.1], [vp.sub.1], m)/[x.sub.1]([p.sub.1], [vp.sub.1], m). Then,

[partial derivative]([x.sub.2]/[x.sub.1])/[partial derivative]m = - [partial derivative]([x.sub.2]/[x.sub.1])/ [partial derivative][p.sub.1] -v[[partial derivative]([x.sub.2]/[x.sub.1])/[partial derivative][p.sub.2]], (4)

where

[partial derivative]([x.sub.2]/[x.sub.1])[partial derivative][p.sub.1] = - [x.sub.1]( [partial derivative] [x.sub.2]/ [partial derivative][p.sub.1]) -[x.sub.2]([partial derivative][x.sub.1]/[partial derivative][p.sub.1])]/ [x.sup.2.sub.1] (5)

-v[[partial derivative]([x.sub.2]/[x.sub.1])/[partial derivative][p.sub.2]] = -v[[x.sub.1] [partial derivative][x.sub.2]/[partial derivative][p.sub.2] -[x.sub.2][partial derivative][x.sub.1]/[partial derivative][p.sub.2]]/[x.sup.2.sub.1]. (6)

Given that we have only two goods, using the budget line, one obtains that

(7) [partial derivative][x.sub.1]/[partial derivative][p.sub.1] = - ([x.sub.1]/[p.sub.1]) - ([p.sub.2]/[p.sub.1]) [partial derivative][x.sub.2]/[partial derivative][p.sub.1]

(8) [partial derivative][x.sub.2]/[partial derivative][p.sub.2] = -([x.sub.2]/[p.sub.2]) - ([p.sub.1]/[p.sub.2]) [partial derivative][x.sub.1]/[partial derivative][p.sub.2].

Substituting these values into Equations (5) and (6), using the fact that v = [p.sub.2][p.sub.1] and reworking, one obtains that

-[partial derivative]([x.sub.2]/[x.sub.1])/[partial derivative][p.sub.1] = - [m[partial derivative][x.sub.2]/ [partial derivative][p.sub.1] + [x.sub.1][x.sub.2]]/[p.sub.1][x.sup.2.sub.1] (9)

-v[[partial derivative]([x.sub.2]/[x.sub.1])/[partial derivative][p.sub.2] = [m[partial derivative][x.sub.1]/ [partial derivative][p.sub.2] + [x.sub.1][x.sub.2]]/[p.sub.1][x.sup.2.sub.1]. (10)

Plugging Equations (9) and (10) into Equation (4), it follows that

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This is illustrated in Figure 1, where it is shown that Equation (4) can be seen as decomposing an income decrease into an increase of both prices that leaves the slope of the budget line unchanged. Figure la represents homothetic preferences. With strong symmetric gross substitutability (meaning that the cross-price effects, indicated as [DELTA][x.sub.1] and [DELTA][x.sub.2], are equal), ([x.sub.2]/[x.sub.1]) is unchanged, meaning that the income offer curve is a line through the origin and that we have homothetic preferences. With weak asymmetric gross substitutability, where ([partial derivative][x.sub.1]/[partial derivative][p.sub.2]) > ([partial derivative][x.sub.2]/ [partial derivative][p.sub.1]) (reflected in the fact that [DELTA][x.sub.1] > [DELTA][x.sub.2]), Good 1 is a necessity and Good 2 is a luxury, as represented in Figure 1b.

In Figure lb, the indifference curves "fan out." For given levels of Good 1, the distance between indifference curves does not change that much as one moves from one indifference curve to the other. However, for given levels of Good 2, the distance between indifference curves changes sharply as one moves from one indifference curve to the other. This is formalized in the Appendix to this article, which shows that in the case of weak asymmetric gross substitutability (where ([partial derivative][x.sub.1]/[partial derivative][p.sub.2]) > ([partial derivative][x.sub.2]/[partial derivative][p.sub.1])), the curvature of the utility function in terms of Good 1 must be large relative to the curvature of the utility function in terms of Good 2. Intuitively, the marginal utility of a necessity (Good 1) decreases more sharply compared to the marginal utility of a luxury (Good 2).

[FIGURE 1 OMITTED]

B. Strong Asymmetric Gross Substitutability

The following claim gives insight into the phenomenon of strong asymmetric gross substitutability (Table 1):

CLAIM 2. Consider the case where Goods 1 and 2 exhibit strong asymmetric gross substitutability, namely where Good 1 is a substitute for Good 2, while Good 2 is a complement to Good 1.

(i) A necessary but not sufficient condition for such strong asymmetric substitutability is that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For the two-good case, a necessary but not sufficient condition is that Good 1 is a necessity or an inferior good and that Good 2 is a luxury.

(ii) Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, all else equal strong asymmetric gross substitutability is more likely to be obtained the smaller the Hicksian cross-price elasticities. Also, all else equal, strong asymmetric gross substitutability is more likely to be obtained the smaller the budget share of Good 2. Finally, all else equal, strong asymmetric substitutability is more likely to be obtained the smaller the income elasticity of good 1 is compared to the income elasticity of good 2.

(iii) In the two-good case, a sufficient condition for strong asymmetric gross substitutability is that Good 1 is price inelastic and Good 2 is price elastic.

As all goods that exhibit strong asymmetric gross substitutability also exhibit weak asymmetric gross substitutability, it follows that it is a necessary condition for strong asymmetric gross substitutability that one good is a necessary or an inferior good and that the other good is a luxury. To see that this is not a sufficient condition, we now rewrite Equations

(1) and (2) as elasticities:

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the uncompensated cross-price elasticity of good i, for i = 1, 2 and j [not equal to] i; similarly, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the compensated (Hicksian) cross-price elasticity. Note that our measure for asymmetric substitutability continues to be in terms of the levels of the cross-price effects and not the elasticities. However, for the concept of asymmetric gross substitutability, where it is the sign of the cross-price effects that matters, it does not matter whether we use levels or elasticities. The point of Equations (12) and (13) is that looking at cross-price elasticities leads to useful additional insights. It is clear from Equations (12) and (13) that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 1 is not a sufficient condition for strong asymmetric gross substitutability. When the income effect on Good 1 is weaker than the income effect on Good 2, strong asymmetric gross substitutability is more likely as the budget share of Good 2 gets smaller and as the Hicksian cross-price elasticities get smaller.

To see part (iii) of Claim 2, rewrite Equations (7) and (8) as elasticities to obtain:

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the uncompensated own-price elasticity of good i. It follows directly from Equations (14) and (15) that, in the two-good case, strong asymmetric substitutability, where Good 1 is a substitute for Good 2 and Good 2 is a complement to Good 1, is synonymous with Good 1 being price inelastic and Good 2 being price elastic. Intuitively, the demand for a good tends to be price inelastic if there are no good substitutes available for it and the demand for a good tends to be price elastic if good substitutes are available for it. Unfortunately, this result is only clear-cut for the two-good case. (8)

III. STRONG ASYMMETRIC GROSS SUBSTITUTABILITY: INTUITIONS

The main result of Section II is that there is a one-to-one relation between asymmetric income elasticities (nonhomothetic preferences) and weak asymmetric substitutability. Yet, asymmetric income elasticity is not a sufficient condition for strong asymmetric gross substitutability. Moreover, the standard intuition about two goods being substitutes or complements relates to some physical aspect of the goods at hand and not to income elasticities. Blue pencils and black pencils are perfect substitutes because people do not care too much whether they write in black or blue. Left shoes and right shoes are perfect complements because one shoe is of little use. Thus, to complete our intuition about asymmetric substitutability, it is important to find an intuition about physical aspects of goods that would lead them to present strong asymmetric gross substitutability.

We now provide two such intuitions for strong asymmetric gross substitutability, which will turn out to be useful for analyzing some potential examples in Section IV. The intuitions are based on Lancaster's (1966) approach to consumer theory, which argues that we do not directly have preferences over goods, but instead have preferences over certain characteristics, which are produced using goods. In particular, let the consumer's utility be a function of two characteristics, [C.sub.1] and [C.sub.2], and let Goods 1 and 2 be inputs in producing each of these characteristics:

(16) U = u{[C.sub.1]([x.sub.1],[x.sub.2]),[C.sub.2]([x.sub.1],[x.sub.2])}.

To make the comparison with perfect substitutes and complements easier, we consider the case where the utility function and characteristics functions take the functional form either of perfect substitutes or of perfect complements. Consumers Consume Two Complementary Characteristics section shows that nonhomothetic preferences are approximated by the following discontinuous utility function. The consumer consumes two perfectly complementary characteristics. Good 1 (2) is better at producing Characteristic 1 (2). A minimum level of Characteristic 1 must be achieved (subsistence constraint), and in the neighborhood of this constraint, Characteristic 1 becomes the main concern on the consumer's mind. Good 1 Substitute for Good 2 Only for Low Utility section treats the case where the consumer normally uses Good 1 to produce Characteristic 1 and Good 2 to produce Characteristic 2. However, Good 1 can also be used to produce Characteristic 2. The consumer uses Good 1 to produce Characteristic 2 only when the availability of this characteristic threatens to dip below an acceptable level.

[FIGURE 2 OMITTED]

A. Consumers Consume Two Complementary Characteristics

Consider a utility function where indifference curves have the form U = min{[alpha][C.sub.1], (1 - [alpha])[C.sub.2]}, so that the two characteristics are perfect complements. (9) The weight [alpha], with 0 < [alpha] < 1, determines the relative importance that the consumer attaches to Characteristic 1; note that, as utility is equal to the minimum of [alpha][C.sub.1] and (1 - [alpha])[C.sub.1], Characteristic 1 is relatively less important to the consumer as [alpha] increases. Let the two goods be perfect substitutes in producing the two characteristics, with [C.sub.1] = (a[x.sub.1] + b[x.sub.2]) and [C.sub.2] = (c[x.sub.1] + d[x.sub.2]). Let Good 1 be better at producing Characteristic 1 and let Good 2 be better at producing Characteristic 2, that is, let a > c and d > b. Moreover, let [alpha]a > (1 - [alpha])c and let (1 - [alpha]) d > [alpha]b, meaning that the contribution to utility by Good 1 is larger through Characteristic 1 than through Characteristic 2; for Good 2, the reverse is the case. Finally, let [alpha] be a function of U, [alpha](U), with [alpha]'(U) > 0, meaning that Characteristic 1 becomes less important to the consumer as utility increases. The utility function expressed over the goods is implicitly given by:

u = min{ [alpha](u)[a[x.sub.1] + b[x.sub.2]], (1 - [alpha](u))[c[x.sub.1] + d[x.sub.2]]}. (17)

To get insight into the shape of the preference mapping, let us first look at a single indifference curve for a fixed level of [alpha]. As [alpha]a > (1 - [alpha])c and (1 - [alpha])d > [alpha]b, for large [x.sub.2] and small [x.sub.1], the indifference curve is fully determined by u = [alpha](a[x.sub.1] + b[x.sub.2]) (steep, as Good 1 is then a good substitute for Good 2), for small [x.sub.2] and large [x.sub.1], it is fully determined by the u = (1 - a[alpha] (c[x.sub.1] + d[x.sub.2]) (less steep, as Good 2 is then a good substitute for Good 1). The indifference curve has a kink where (1 - [alpha]) (c[x.sub.1] + d[x.sub.2]) = [alpha][x.sub.1] + b[x.sub.2]) or [x.sub.2] = [[alpha]a - (1 - [alpha])c] [(1 - [alpha])d-ab] l[x.sub.1]. Simply, the discontinuous utility function described by Equation (17) presents a rough approximation of convexity: if the consumer has little of Good 1 (2), Good 1 (2) becomes a very good substitute for Good 2 (1). (10)

If [alpha] is fixed, the line connecting all the kinks across different indifference curves goes through the origin and we have homothetic preferences. However, we assume that [alpha]'(U) > 0. It can be checked that [[alpha]a - (1 - [alpha])c] [[(1 - [alpha])d -ab].sup.-1] (the proportion of [x.sub.2] to [x.sub.1] at a kink) is an increasing function of [alpha] as long as db > ac, which is indeed the case given our assumptions. It follows that, for lower utility levels, the kinks of the indifference curves lie at bundles with a lower and lower proportion of [x.sub.2] to [x.sub.1]; in terms of income expansion paths and Engel curves, this means that Good 1 is a necessity and/or an inferior good. One thus obtains nonhomothetic preferences. At the same time, as utility decreases, it happens sooner than before that the consumer considers having too little of Good 1, meaning that Good 1 gets to be considered as a good substitute for Good 2. Two indifference curves for the case where [alpha]'(U) > 0, [U.sub.1] and [U.sub.2], are sketched in Figure 2. Note that the shape of this indifference mapping is in accordance with a relatively sharply decreasing marginal utility of Good 1 (see end of Weak Asymmetric Gross Substitutability section).

To see how having nonhomothetic preferences can cause strong asymmetric gross substitutability, let us consider the following two situations. In the first situation, utility is high and [alpha] is relatively high (e.g., [alpha] = 0.5). As b and c are small, [alpha]b and (1 - [alpha])c become negligible. In essence, Good 1 (2) is used to produce Characteristic 1 (2) and Good 2 is a complement to Good 1 (indifference curve [U.sub.1] in Figure 2). In the second situation, utility is low and [alpha] is small, meaning that it is the level of the first characteristic, that typically determines utility. Good l is a substitute for Good 2 in producing Characteristic 1 and, therefore, also a substitute for Good 2 in producing total utility (cf. indifference curve [U.sub.2] in Figure 2; note that for sufficiently low utility, the indifference curve will be entirely linear).

[FIGURE 3 OMITTED]

Strong asymmetric gross substitutability is now obtained if price changes make the consumer switch between the two above situations. Graphically, if starting from Bundle I in Figure 2, the price of Good 2 increases, utility decreases, and Characteristic 1 becomes dominant on the consumer's mind. Good 1 is a very good substitute for Good 2 when it comes to producing Characteristic 1. Therefore, the consumer will substitute away from Good 2 toward Good 1. If, starting from Bundle II, the price of Good 1 decreases, utility increases, and the two characteristics again become equally important on the consumer's mind. Therefore, the consumer will spend her increased purchasing power to obtain more of both Characteristics 1 and 2. Good 2 is better at producing Characteristic 2 and therefore more of Good 2 will be consumed.

It should be stressed that, in accordance with the analysis in Strong Asymmetric Gross Substitutability section, Figure 2 shows that nonhomothetic preferences are a necessary, but not a sufficient condition for strong asymmetric gross substitutability: this phenomenon only occurs if [alpha]'(U) is large, meaning that the increase in the extent to which Characteristic 1 becomes more important as utility decreases is considerable.

[FIGURE 4 OMITTED]

But why should it be the case that Characteristic 1 becomes relatively more important as utility decreases and why would the priorities of the consumer shift so radically? A simple intuition can be provided using Gilley and Karels' (1991) constraint approach. (11) Suppose that the consumer needs a certain minimum amount of Characteristic 1 to survive, meaning that

(18) a[x.sub.1] + b[x.sub.2] [greater than or equal to] [[bar.C].sub.1].

Equation (18) can be seen as a subsistence constraint (the lines in Figures 3b and 4b indicated as S). The consumer's choice now proceeds in two steps. The first and standard step is to look for the bundle such that the highest indifference curve is reached given the budget line (Bundle I for budget line [B.sub.1] and Bundle II for budget line [B.sub.2] in Figures 3a and 4a). The second step is to look whether the bundle lies above or below the subsistence constraint S. If it lies above, then it is the optimal bundle (as is the case for Bundle I in Figures 3 and 4). If not (as is the case for Bundle II in Figures 3a and 4a, which lies below S), then the consumer looks for the bundle such that the highest indifference curve is reached, given that the bundle should not lie above the budget constraint, and should not lie below the subsistence constraint. For budget line [B.sub.2] in Figures 3b and 4b, allowable bundles are those below budget line [B.sub.2] and above constraint S, yielding the gray area. The consumer then chooses Bundle III. Thus, in Figures 3a and 4a, Goods 1 and 2 appear to be mutual gross complements. However, as illustrated in Figure 3b, close to the subsistence constraint, Good 1 becomes a gross substitute for Good 2 and serves to assure that the consumer can subsist. As illustrated in Figure 4b, Good 2 remains a gross complement to Good 1.

The relation between the analysis in Figure 2, on the one hand, and Figures 3 and 4, on the other hand, is the following. If we want to integrate the subsistence constraint directly into the consumer's preferences, (12) then subsistence needs to receive more weight as utility decreases. Only then do the consumer's optimal bundles allow her to maintain the same level of subsistence. The analysis in Figure 2 can thus be seen as a more general case, where subsistence becomes relatively more important as utility decreases and some approximate subsistence level is thereby achieved.

A final note is due concerning the preference mapping in Figure 2. As is clear from Figure 2b, Good 1 is a Giffen good. Indeed, for the utility function in Equation (17), Giffenity is synonymous with strong asymmetric gross substitutability. However, this is due to the discontinuity in the preferences. In general, neither Giffenity nor inferiority of one of the goods is a necessary condition for strong asymmetric gross substitutability. An indifference mapping exhibiting strong asymmetric gross substitutability, but where Good 1 is an ordinary and a normal good, is represented in Figure 5. The consumer initially consumes Bundle I. When the price of Good 1 rises, in Bundle II, less of Good 2 is consumed, so that Good 2 is a complement to Good 1; less of Good 1 is consumed, so that Good 1 is an ordinary good. When the price of Good 2 rises, in Bundle III, more of Good 1 is consumed, so that Good 1 is a substitute for Good 2. When income decreases, in Bundle IV, less of Good 1 is consumed, so that Good l is a normal good.

[FIGURE 5 OMITTED]

B. Good 1 Substitute for Good 2 Only for Low Utility

Let U take on the form U = min {[C.sub.1], [C.sub.2]} and let characteristics [C.sub.1] and [C.sub.2] each be linear in Good 1 and Good 2. In particular, consider two utility levels, [U.sub.1] and [U.sub.2], with [U.sub.1] < [U.sub.2]. Consider the following two indifference curves. For utility level [U.sub.2], we have

(19) min{[ax.sub.1], [dx.sub.2]},

where a > d. For utility level [U.sub.1], we have

(20) [U.sub.1] = min{[ax.sub.1], [cx.sub.1] + [dx.sub.2]},

where a > c. The left part of Figure 5 shows that Good l can be a gross substitute for Good 2: if the price of Good 2 rises, the consumer consumes more of Good 2. The right part of Figure 6 shows that Good 2 is a gross complement to Good 1: if the price of Good 1 rises, the consumer buys less of Good 2.

To explain why Good 1 becomes a substitute for Good 2 only when utility is low, consider the following case. Assume that Good 1 can produce both characteristics, but Good 2 can only produce Characteristic 2. Contrary to what was implicitly assumed in all previous cases in this section, assume that a unit of a good used to produce Characteristic 1 does not at the same time contribute to the production of Characteristic 2. Denote by [x.sub.1]. I, the amount of Good 1 used to produce Characteristic 1 and let

(21) [C/sib/1] = [alpha][x.sub.1, I]

(22) [C.sub.2] = [gamma]([x.sub.1] - [x.sub.1,I]) + [delta][x.sub.2].

The two characteristics [C.sub.1] and [C.sub.2] are perfect complements. While it is also possible to produce Characteristic 2 by using Good 1, the consumer as such prefers to use Good 1 exclusively for the production of Characteristic 1. Thus, the consumer also considers Goods 1 and 2 as perfect complements:

(23) U = min{[alpha][x.sub.1], [delta][x.sub.2]}.

In Figures 6 and 7, indifference curve U, is flat to the right of [x.sub.1]. Extra units of Good 1 are not normally used for the production of Characteristic 1 and therefore do not normally increase utility. However, assume that the consumer at the same time wants to achieve at least a minimal level of Characteristic 1 ([C.sub.1] [greater than or equal to] [[bar.C].sub.1]) as well as a minimal level of Characteristic 2 ([C.sub.2] [greater than or equal to] [C.sub./2]).

The consumer's choice can now again be seen as consisting of two steps. The first step consists of looking for highest indifference curve that can be achieved for given budget lines such as [B.sub.1] and [B.sub.2] in Figures 6a and 7a, leading to candidate optimal Bundles I and II where again Good 1 (2) is only used to produce Characteristic 1 (2). From this perspective, Goods 1 and 2 are mutual gross complements, as indicated in Figures 6a and 7a. The second step consists of checking whether the procedure of using Good 1 (2) only for the production of Characteristic 1 (2) produces at least the minimal levels of each good, that is, to check whether [x.sub.1] [greater than or equal to] [[bar.C].sub.1]/[alpha] (denoted as constraint [S.sub.1]) and [x.sub.2] [greater than or equal to] [C.sub.2]/[delta] (denoted as constraint [S.sub.2]). If so, the candidate optimal bundle is indeed optimal. This is the case for Bundle I in Figures 7 and 8 and for Bundle II in Figure 7, as these bundles each lie above constraints [S.sub.1] and [S.sub.2]. As is clear from Figure 7, Good 2 is thus a gross complement to Good 1.

[FIGURE 6 OMITTED]

If the minimal levels are not achieved, as is the case for Bundle II in Figure 8a, the consumer does take into account that Good 1 can also be used to produce Characteristic 2. Thus, by Equations (21) and (22), we now have the constraints [x.sub.1] > [x.sub.1], I (where [x.sub.1], I [greater than or equal to] [[bar.C].sub.1]/[alpha] and denoted as [S'.sub.1]) and [S'.sub.2] [greater than or equal to] ([[bar.C].sub.2]/[delta])- ([gamma]/[delta])[[x.sub.1] - [x.sub.1, I]] (denoted as [S'.sub.2]). One can represent these three-dimensional constraints in the [x.sub.1] - [x.sub.2] space for fixed levels of [x.sub.1], I. Two such constraints, for [x.sub.1], I = ([[bar.C].sub.1]/[alpha]) (Figure 9a) and for [x.sub.1], I > ([[bar.C].sub.1]/[alpha]) (Figure 9b), are represented in Figure 9. The feasible bundles are those in the gray area below the budget constraint and above the constraints [S'.sub.1] and [S'.sub.2]. We assume that the consumer continues to have the preferences defined by Equation (23), reflecting the assumption that the consumer continues to dislike using Good 1 to produce Characteristic 2. Figure 9 now makes clear that such a consumer always prefers to put [x.sub.1, I] = ([[bar.C].sub.1]/[alpha]); in Figure 9b, the consumer who puts [x.sub.1], I > ([[bar.C].sub.1]/[alpha]) is worse off than in Figure 9a. It follows that the relevant constraints are [x.sub.1] [greater than or equal to] ([[bar.C].sub.1] /[alpha]) (denoted as constraint_[S'.sub.1] in Figure 8b) and constraint [x.sub.2] [greater than or equal to] ([[bar.C].sub.2]/[delta]) - ([gamma]/[delta]) [[x.sub.1] - ([[bar.C].sub.1]/[alpha])] (denoted as constraint [S'.sub.2] in Figure 8b). The consumer chooses Bundle III in Figure 8b, and Good 1 is a gross substitute for Good 2 instead of a gross complement to it.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Admittedly in this example, there is a sudden change in consumer choice, where suddenly the constraint [S'.sub.2] becomes relevant instead of constraint [S.sub.2], reflecting the assumption that Good 1 is suddenly also used to produce Characteristic 2. In a more realistic example, as the consumer gets more and more tight on Characteristic 2, she considers Good 1 as a better and better substitute for Good 2 in producing Characteristic 2. To conclude that the preference mapping in Figure 6 can be interpreted as having an approximate constraint for Characteristic 2 directly built into it as well as an approximate constraint for Characteristic 1. The latter constraint assures that extra units of Good 1 get used to produce Characteristic 2 when the consumer gets tight on Characteristic 2. When this constraint is built directly into the preferences, it is reflected in indifference curves that lie close to one another and become nearly vertical for small levels of Good 1. This in turn is in accordance with indifference curves that fan out (see Weak Asymmetric Gross Substitutability section).

IV. POTENTIAL EXAMPLES AND APPLICATIONS

A. Rice and Meat

Let a consumer consume two goods and let Good 1 be a Giffen good. Then, Good 1 is necessarily an inferior good, and by Equation (1), Good 1 is a substitute for Good 2. By Equation (14), Good 2 is a complement to Good 1. In the case of two goods, Giffenity thus necessarily implies strong asymmetric gross substitutability. (13) This is no surprise, as the characteristics approach in Consumers Consume Two Complementary Characteristics section has been used by Jensen and Miller (2007) to explain the possibility of Giffen behavior and the constraint approach explained in Consumers Consume Two Complementary Characteristics section has been used by Gilley and Karels (1991) with the same purpose.

In particular, in the analysis of Consumers Consume Two Complementary Characteristics section, let Good 1 be rice and let Good 2 be meat; let Characteristic 1 be subsistence and let Characteristic 2 be taste. Rice is better at producing subsistence and meat is better at producing taste. Let subsistence become relatively more important as the consumer's utility decreases. A rough approximation of this lies in Gilley and Karels' (1991) subsistence constraint, where consumers maximize utility with respect to the constraint that they must at least attain some minimum level of subsistence. When such a constraint is translated into preferences, it means that subsistence must become relatively more important when utility decreases. In such a case, it may be that, as meat becomes more expensive, the consumer substitutes rice for meat to subsist. As rice becomes cheaper, however, the consumer consumes both more rice and more meat to go with it.

[FIGURE 9 OMITTED]

Jensen and Miller (2007) recently estimate that, among the moderately poor in the Chinese province of Hunan, a 1% increase in the price of rice causes a 0.45% increase in the consumption of rice. Thus, rice is a Giffen good in this case, and also, as confirmed by the authors' estimates, an inferior good. By Equation (14), this means that rice is a substitute for other goods, including for meat. At the same time, the authors find that a 1% increase in the price of rice decreases the consumption of meat by 1.13%.

We end this section by admitting that many readers will consider Giffenity an exceptional case. However, as noted in Weak Asymmetric Gross Substitutability section, one does not need Giffenity and not even inferiority of one of the goods to have strong asymmetric gross substitutability. Giffenity is a special case within the wider class of preferences presenting strong asymmetric gross substitutability. This is why we now look for some additional potential applications where Giffenity does not apply.

[FIGURE 10 OMITTED]

B. Primary Goods and Secondary Goods

Marketing research indicates the existence of borderline cases between weak and strong gross asymmetric gross substitutabilities, for pairs of goods consisting of a primary good and a secondary good. Walters (1991), and Mulhern and Leone (1991) estimate the cross-price effects of cake mix and cake frosting. For a sample of retail stores, Walters (1991) studies the effect of price promotions for three brands of cake mix on the sales of two brands of cake frosting and the effect of price promotions of the two brands of cake frosting on the sales of the three brands of cake mix. He finds that price promotions on cake mix brands result in significant increases in sales of cake frosting brands in 50% of the possible cases, whereas price promotions on cake frosting result in significant increases in sales of cake mix in only 17% of the possible cases. Using similar data, Mulhern and Leone (1991) find that price promotions on cake mix result in significant increases in sales in 26% of the possible cases, whereas price promotions on cake frosting brands result in significant increases in sales of cake mix brands in only 3% of the possible cases.

These observations can be explained by the analysis in Consumers Consume Two Complementary Characteristics section. Let a consumer of cakes value the two characteristics "satisfaction from eating" (Characteristic 1) and "decoration" (Characteristic 2). The consumer buys two goods. Good 1, cake mix, is better at producing the characteristic "satisfaction from eating"; Good 2, cake frosting, is better at producing "decoration." As the consumer's utility decreases, "satisfaction from eating" becomes relatively more important. This is because the consumer wants a certain more or less fixed level of "satisfaction from eating." Such a fixed level can only be achieved if this characteristic becomes relatively more important as utility decreases. Cake mix is a necessity and cake frosting is a luxury. Cake mix is neither a gross substitute nor a gross complement for cake frosting (Figure 10a). Cake frosting is a gross complement to cake mix (Figure 10b). Intuitively, when the price of cake frosting changes, the consumption of cake mix may not change because the consumer wants to maintain about the same "satisfaction from eating." However, when the price of cake mix decreases, the consumer may buy both more cake mix and more cake frosting.

Asymmetric gross substitutability has concrete consequences for the effectiveness of sales promotions. Clearly, for complements, one sales promotion can do the job of two and managers can save costs by promoting only one of a pair of goods. But the possibility of asymmetric substitutability reveals that it may be that sales promotions need to be concentrated on a particular good within a pair of goods. For instance, the sales promotion intended at boosting the sales of both cake mix and cake frosting should be focused on cake mix and not on cake frosting (Waiters 1991).

C. Asymmetric Substitutability of High-Skilled and Low-Skilled Labors

While labor is the subject of production theory and not consumer theory, for expositional reasons, it is useful to look at an example that lies closer to labor economics than to consumer theory before treating an example in All-in-One-Devices and Special Purpose Devices section that unambiguously lies in the realm of consumer theory. Moreover, labor is one of the few subjects where asymmetric substitutability is explicitly mentioned in the literature (e.g., Azariadis 1976).

In accordance with the analysis in Good 1 Substitute for Good 2 Only for Low Utility section, let the dictatorial dean of a faculty maximize her utility given the faculty budget. The dean's utility function is defined over two characteristics, namely academic output (Characteristic 1) and administrative output (Characteristic 2). Academic output can only be produced by means of academics (Good 1). Administrative output can be produced by means of both administrative personnel (Good 2) and academics, even though the marginal administrative output of administrative personnel is higher. Note also that one working hour of an academic used for administration cannot at the same time be used for academic work.

As such, from the dean's perspective, academics and administrative personnel are mutual complements. However, at the same time, she wants to make sure that a minimum of academic output and of administrative work gets produced. If complementary use of administrative personnel and academics, where they each do the task at which they are best, causes a situation where not enough administrative output is produced, the dean may let academics do some of the administrative work. Thus, as illustrated in Figures 7 and 8, when the price of administrative personnel (Good 2) increases, the dean may employ more academics (Good 1), such that academics are a gross substitute for administrative personnel. However, when the price of academics decreases, the dean employs more academics and more administrative personnel to go with it, such that administrative personnel is a gross complement to academic personnel.

Many studies of labor demand report that the demand for low-skilled labor is more elastic than the demand for high-skilled labor (e.g., Falk and Koebel 2001, for short-run labor demand). This seems to point toward asymmetric gross substitutability. (14) Similar evidence lies in the tendency of employers to fire unskilled workers first during recessions. Skilled workers may be substituted for unskilled workers, where skilled workers during recessions would take over some of the tasks that would otherwise be done by unskilled workers (Reder 1964, p. 315).

D. All-in-One-Devices and Special-Purpose Devices

Strong asymmetric gross substitutability has recently been observed by Garbacz and Thompson (2007) for mobile phone services versus fixed phone services in developing countries. The authors find that an increase in the price of mobile phones of 1% decreases demand for fixed phones by about 0.1% (fixed phones services are a complement to mobile phones services); an increase in the price of fixed phones services of 1% causes an increase in the demand for mobile phones of about 0.5%: mobile phone services are a substitute for fixed lines services.

A similar intuition to the one developed in Asymmetric Substitutability of High-Skilled and Low-Skilled Labor section, but now applied to a case that lies unambiguously in the realm of consumer theory, may help explain this phenomenon. Let the utility of a consumer of phone calls depend on the characteristics "phone calls made outside" (Characteristic 1) and "phone calls made at home" (Characteristic 2). Let the consumer be able to produce these characteristics by means of mobile phone services (Good 1) and of fixed phone services (Good 2). Fixed phone services cannot produce calls outside and thus do not contribute to the production of characteristic 1. One can, however, use a mobile phone to call at home. This does not mean, however, that one will actually use a mobile phone both for calling outside and at home. In fact, as reported by Campbell (2001), the poorer the country, the higher the proportion of mobile phones with respect to fixed phones. This is in line with the characteristics model in Good 1 Substitute for Good 2 Only for Low Utility section. When the consumer is relatively well off, he or she uses mobile phones outside and fixed phones inside. However, when the consumer has less income, a minimum required number of phone calls at home may not be achieved if the consumer uses mobile phones only outside and fixed phones only inside. To achieve a minimum number of calls at home, the consumer may then start to use her mobile phone more often at home. But why does the consumer then not use the fixed phone more often and the mobile phone less often? The reason is that such behavior would again push the number of phone calls made outside, which can only be made with the mobile phone, below a minimum. (15)

In general, candidates for pairs of goods that exhibit strong asymmetric gross substitutability are all-in-one devices versus special-purpose devices. This distinction appears to become more and more relevant. Consumers face the choice between buying a mobile phone with built-in MP3 player and/or digital camera versus designated and higher quality MP3 players and digital cameras.

The possibility of strong asymmetric gross substitutability raises important questions regarding the concept of market power and of the market itself. For a given good, the degree of market power in the provision of that good depends on the substitutes that are available for it (Church and Ware 2000, p. 605). Thus, the fact that there is only one producer of cellophane is not a problem if close substitutes for cellophane are available, that is, if the cross-price elasticity of a goodlike aluminum foil with respect to the price of cellophane is positive. However, this article illustrates that substitutability need not be symmetric. Indeed, as argued by Banerjee (2007, pp. 16-17), in terms of the mobile phone/fixed phone services example, concentration of production of fixed phone services may not be problematic, as mobile phone services are a substitute. Mobile phone services are thus a part of the market for fixed phone services. But concentration of production of mobile phone services may be problematic, as there are no substitutes for mobile phone services. Fixed phone services are not part of the market for mobile services. Banerjee (2007) shows how in the United States the Federal Communications Commission made the mistake of concluding this from the fact that mobile phone services are part of the market for fixed phone services and that fixed phone services are part of the market for mobile phone services. However, competition authorities in Europe do take into account the possibility of asymmetric substitutability in their definition of the market (Report by Bird & Bird for the European Commission, Directorate General Competition 2002, p. 267).

To our knowledge, the theory of industrial organization has not yet analyzed competition under asymmetric substitutability. The dominant model of competition under product differentiation is the one by Singh and Vives (1984). In their duopoly model, they assume that the representative consumer's three-good utility function is quasilinear in the third good, taking on the form U([x.sub.1], [x.sub.2]) + [x.sub.3]. The authors study Cournot and Bertrand competition between Goods 1 and 2; the function of the third good is to assure that there is zero income effect on both goods, so that the consumer surplus can be calculated for both the demand for Goods 1 and 2. By Equations (1) and (2), the Goods 1 and 2 are necessarily strong symmetric gross substitutes (Table 1) with Singh and Vives' utility function. Thus, not only does the dominant model exclude strong asymmetric gross substitutability, it even excludes weak asymmetric gross substitutability. As shown in Weak Asymmetric Gross Substitutability section, any attempt to model competition in the case of weak or strong asymmetric gross substitutability will need to give up on the assumption of zero income effects, as there is a one-to-one relationship between weak asymmetric gross substitutability and asymmetry of the income effects. Thus, introducing a third good into the consumer's utility function when treating competition between two products has no sense in this case. Given that additionally the concept of consumer surplus cannot be used, a welfare analysis of competition between asymmetrically substitutable goods should make use of explicit utility functions. Some two-good utility functions that exhibit asymmetric substitutability are treated in De Jaegher (2008). (16)

IV. DISCUSSION

We started this article by distinguishing between weak and strong asymmetric gross substitutabilities. In its weak form, asymmetric substitutability just means that cross-price effects are not the same. In its strong form, it means that cross-price effects have a different sign (Good 1 is a substitute for Good 2 and Good 2 is a complement to Good 1). It was shown that asymmetric substitutability occurs in circumstances that seem quite plausible. Weak asymmetric gross substitutability occurs as soon as one good is a necessity and the other good is a luxury; in the two-good case, strong asymmetric gross substitutability occurs as soon as one good is price inelastic and the other good is price elastic.

Still, while weak asymmetric substitutability seems intuitive, strong asymmetric gross substitutability seems at first sight unintuitive. We hope to have shown that there are clear intuitions for such a phenomenon and to have attached plausible potential empirical examples to these intuitions. A first intuition is that the consumer consumes a primary good (e.g., cake mix) and a secondary good (e.g., cake frosting). It is the secondary good that is a complement to the primary good; the primary good is not a complement to the secondary good. On the contrary, the primary good may be a substitute for the secondary good. This is because the purpose served by the primary good (e.g., the purpose of satisfaction from eating, served by the cake mix) gains larger importance relative to the purpose served by the secondary good (e.g., the purpose of decoration served by cake frosting) as the utility of the consumer decreases; the reason for this is that the consumer wants to maintain a minimal level of satisfaction from eating. When the price of the secondary good increases, it may be that this can only be achieved by consuming less of the secondary good and more of the primary good.

A second intuition is that Good 1 is best at doing Job 1, while Good 2 is best at doing Job 2; however, Good 1 is also able to do the job that Good 2 normally does, even though it can do this job less well (Good 2 is not able to do the job that is normally done by Good 1). In normal circumstances, each good does its own job, and the two goods are complements. However, when the price of Good 2 increases, it may be that too little output is obtained for the job normally done by Good 2 and that Good 1 is then exceptionally also used to do the job of Good 2. Good 1 is thus a substitute for Good 2. If the price of Good 1 increases, enough of the job of Good 2 gets done, so that Good 2 continues to be a complement to Good 1. An example may be the following. If the price of mobile phone services increases, less mobile phone services and less fixed phone services are consumed. This is because fixed phones cannot substitute for mobile phones, as the former cannot be used to call outside of home. However, if the price of fixed phone services increases, more mobile phone services may be bought. This is because mobile phones will then also be used to call at home, a job that was otherwise done by fixed phones.

Applied microeconomic models often treat cases of weak symmetric gross substitutability (both goods are mutual gross substitutes or mutual gross complements) or even of strong symmetric substitutability (the cross-price effects are equal, e.g., Singh and Vives 1984). The analysis in this article suggests that applied microeconomics should also pay attention to the possibility of asymmetric gross substitutability. From a pedagogical point of view, when exposing students of microeconomics to the consequences of asymmetric income elasticities, it may be better to expose them to asymmetric gross substitutability rather than Giffenity, as the former concept would seem to be more empirically plausible. (17)

doi: 10.1111/j.1465-7295.2008.00158.x

APPENDIX

Denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the determinant of the bordered Hessian and denote by [lambda], the Lagrangian multiplier. Applying Cramer's rule, we obtain that

(24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Weak asymmetric substitutability, where [partial derivative][x.sub.1]/[partial derivative][p.sub.2] > ***/ [partial derivative][p.sub.1], is obtained if

(26) -[x.sub.1][ - [p.sub.2][u.sub.11] + [p.sub.1][u.sub.21]] < [x.sub.2][ - [p.sub.2][u.sub.12] + [p.sub.1][u.sub.22]]

[??]

(27) - [p.sub.1][x.sub.1][u.sub.21] + [p.sub.2][x.sub.2][u.sub.12] < [p.sub.1][x.sub.2][u.sub.22] - [p.sub.2][x.sub.1][u.sub.11]

(28) -([x.sub.2][u.sub.22])/[u.sub.2] + ([x.sub.2][u.sub.12])/[u.sub.1] < - ([x.sub.1][u.sub.11])/[u.sub.1] + [x.sub.1][U.sub.21]/[u.sub.2],

where Equation (28) is obtained from the fact that [u.sub.2]/[u.sub.1] = [p.sub.2]/[p.sub.1]; -([x.sub.1][u.sub.ii])/[u.sub.1] is nothing, but the relative risk averseness of the consumer for good i (with i = 1, 2), which again is a unit-free measure of the curvature of the utility function in good i for given levels of good j. Thus, weak asymmetric substitutability is more likely to be obtained when the curvature of the utility function as a function of Good 1 is larger than the curvature of the utility function as a function of Good 2 (where the second term of the right- and the left-hand sides of Equation (28) is assumed fixed). This is in line with Figure lb, where the indifference curves lie at a higher distance from one another for changes in the consumption of Good 1 than for changes in the consumption of Good 2.

For an interpretation of the expressions on the right-and left-hand sides of Equation (28), note that

(29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, the right-hand side of Equation (28) is the absolute value of the elasticity of the marginal rate of substitution of Good l for Good 2 with respect to a change in consumption of Good 1 (keeping consumption of Good 2 fixed), while the left-hand side of Equation (28) is the elasticity of the marginal rate of substitution of Good 2 for Good 1 with respect to a change in consumption of Good 2 (keeping consumption of Good 1 fixed). With homothetic preferences, these elasticities are necessarily equal. It is also clear from this that weak asymmetric gross substitutability is obtained when the indifference curves "fan out."

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Singh, N., and X. Vives. "Price and Quantity Competition in a Differentiated Duopoly." Rand Journal of Economics, 15, 1984, 546-54.

Walters, R. G. "Assessing the Impact of Retail Price Promotions on Product Substitution, Complementary Purchase, and Interstore Displacement." Journal of Marketing, 55, 1991, 17-28.

Wichers, R. "In Search of Giffen Behavior: Comment." Economic Inquiry, 32, 1994, 166-67.

(1.) Nicholson (2005, pp. 168-171), an exception among handbooks in mentioning the possibility of asymmetric gross substitutability, takes over the argument of Hicks and Allen (1934).

(2.) For an overview of the history of the complementarity and substitutability in economic theory, see Lenfant (2003, 2006).

(3.) In our concept of symmetric and asymmetric substitutability, we use cross-price level effects rather than cross-price elasticities. As we are comparing two level effects, we do not need elasticities to make our concept of cross-price effects unit free (see Hicks and Allen 1934, p. 213, for this argument). In a similar argument, Sethuraman, Srinivasan, and Doyle (1999) argue that it is better to compare absolute cross-price effects than to compare cross-price elasticities, as the latter are affected by budget shares.

(4.) Allenby and Rossi (1991) provide a similar link between asymmetric switching between brands and nonhomothetic preferences.

(5.) This point generalizes to the case of more than two goods, with the exception that two goods that are asymmetric gross substitutes may both be necessities and may both be luxuries. To have asymmetric substitutability, it suffices that one luxury is a luxury to a lesser extent than the other luxury or that one necessity is a necessity to a lesser extent than the other necessity.

(6.) For the two-good case, an alternative proof that does not involve Slutsky equations is the following. We know that, to have unit income elasticity, the demand for Good 1 must take the form [x.sub.i] - m[florin]([p.sub.1] ,[p.sub.2]). We also know that this demand should be homogenous of degree 1 (otherwise, it would matter in which units we measure demand), meaning that we are able to rewrite the demand as [x.sub.1] - (m/[p.sub.1])[florin](l, [p.sub.2]/ [p.sub.1]) or in short [x.sub.1] = (m/[p.sub.1])[florin]([p.sub.2]/[p.sub.1]). It follows that [partial derivative] [x.sub.1]/ [partial derivative] [p.sub.2] = (m/[p.sub.1])[florin]'([p.sub.2]/[p.sub.1])(1/[p.sub.1]). Using the budget constraint and the above demand function, we have [x.sub.2] = (m/[p.sub.2])[1 - [florin]([p.sub.2]/ [p.sub.1])]. It follows that [partial derivative][x.sub.2]/[partial derivative][p.sub.1] =(m/[p.sub.2])[- [florin]'([p.sub.2]/[p.sub.1])] (-[p.sub.2]/[p.sup.2.sub.1]). But this is the same result as obtained above for [partial derivative][x.sub.1]/[partial derivative][p.sub.2].

(7.) This result reflects the Hicksian decomposition of cross-price effects into substitution and income effects. From the point of view of Hicks and Allen (1934), only the substitution effect, which is necessarily symmetric, measures the substitutability between goods. This is why weak asymmetric gross substitutability is synonymous with asymmetric income elasticity; since net substitutability is symmetric, asymmetric gross substitutability is purely a product of the income effects. Yet, for all practical purposes, the degree to which substitutes are available for a good is measured by the uncompensated cross-price effect and the income effect makes out an integer part of this effect.

(8.) Consider the case of three goods. Equations (14) and (15) now become [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let Goods 1 and 2 be weak or strong symmetric substitutes (see Table 1 for these concepts). However, let Good 3 be a complement to Good 1 ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) and let Good 3 be a substitute for Good 2 ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). Then, even though Goods 1 and 2 are symmetric substitutes, it may be that the demand for Good 1 is price inelastic (because it has a complement in Good 3) and that the demand for Good 2 is price elastic (because it has a substitute in Good 3). Thus, in the multiple-good case, the argument that for a pair of Goods 1 and 2, relatively inelastic demand for Good 1 and elastic demand for Good 2 point toward asymmetric substitutability is only maintained if other goods are all substitutes to Goods 1 and 2 or are all complements to Goods 1 and 2.

(9.) All the examples treated in this section are kinked indifference curves. This assumption is taken for simplicity, and it is in fact perfectly possible to construct utility functions that involve at least weak asymmetric substitutability. This can, in particular, be done by letting the weight that goods get in the utility function vary with the utility level itself, just as in the case in the discontinuous example in the body of the article.

(10.) A procedure that can be used to turn such discontinuous utility functions into continuous ones is presented by Moffatt (2002).

(11.) Lipsey and Rosenbluth ( 1971 ) present a Lancasterian approach, where there is satiation for one characteristic. This has the same effect as a subsistence constraint.

(12.) Wichers (1994) criticizes Gilley and Karels' (1991) approach on the grounds that the subsistence constraint should be directly part of the consumer's preferences.

(13.) As noted by Silberberg and Walker (1984), for the two-good case, the fact that the non-Giffen good needs to be a complement to the Giffen good has not been generally realized.

(14.) An alternative explanation of the fact that low-skilled labor is more elastic than high-skilled labor involves a third input, capital. For low-skilled labor, a close substitute is available in the form of capital, making low-skilled labor price elastic. On the other hand, capital is a complement to high-skilled labor, making high-skilled labor less elastic. See Hamermesh (1993). Footnote 8 above provides an analysis of this case, where Good 1 is high-skilled labor, Good 2 is low-skilled labor, and Good 3 is capital. However, our explanation of asymmetric substitutability between high- and low-skilled labors continues to be relevant in the short run, where capital is fixed.

(15.) An alternative explanation is that mobile phone consumers and fixed phone consumers are different breeds. At least historically, mobile phone consumers would be innovators, who do not switch back to fixed phones once they have bought a mobile phone. However, fixed phone consumers can follow the innovators and switch to mobile phones as they get cheaper.

(16.) For a utility function that exhibits at least weak asymmetric substitutability, consider any quasilinear utility function of the form u([x.sub.1], [x.sub.2]) = v([x.sub.1] + [x.sub.2]. Given that one good has a zero income effect, there cannot be symmetric substitutability. For a utility function on the borderline between weak and strong asymmetric substitutabilities, consider the utility function u([x.sub.1], [x.sub.2]) = -ln w([x.sub.1] + In [x.sub.2]. It is easy to show for such a utility function that Good 2 is neither a substitute for nor a complement to Good 1, while Good 1 may be a substitute for Good 2 or a complement to it. Finally, consider the utility function u([x.sub.1], [x.sub.2]) = [ z([x.sub.1]) + [x.sub.2]]/[x.sub.1]. In this case, there is a strong asymmetric substitutability, with Good I a substitute for Good 2 and Good 2 a complement to Good 1. A particular feature of this utility function is that there is satiation in Good 1, which eventually becomes a bad.

(17.) I am indebted for this point to one of the referees.

KRIS DE JAEGHER *

* I would like to thank two anonymous referees and the editor for helpful comments and the Tjalling C. Koopmans Institute for financial support.

de Jaegher: Assistant Professor, Utrecht School of Economics, Utrecht University, Janskerkhof 12. 3512 BL Utrecht, The Netherlands. Phone 31-30-253-9964; Fax 31-30-253-7373; Email k.dejaegher@econ.uu.nl
TABLE 1

Symmetric and Asymmetric Gross Substitutability

 Strong Symmetric Weak Symmetric
 Gross Substitutability Gross Substitutability

[partial derivative][x.sub.1] sgn [partial derivative][x.sub.1]
 ([p.sub.1], [p.sub.2], m)/ ([p.sub.1], [p.sub.2], m)/
 [partial derivative][p.sub.2] = [partial derivative][p.sub.2] =

[partial derivative][x.sub.2] sgn [partial derivative][x.sub.2]
 ([p.sub.1], [p.sub.2], m)/ ([p.sub.1], [p.sub.2], m)/
 [partial derivative][p.sub.1] [partial derivative][p.sub.1]

 Weak Asymmetric Strong Asymmetric
 Gross Substitutability Gross Substitutability

[partial derivative][x.sub.1] sgn [partial derivative][x.sub.1]
 ([p.sub.1], [p.sub.2], m)/ ([p.sub.1], [p.sub.2], m)/
 [partial derivative][p.sub.2] [partial derivative][p.sub.2]
 [not equal to] [not equal to]
[partial derivative][x.sub.2] sgn [partial derivative][x.sub.2]
 ([p.sub.1], [p.sub.2], m)/ ([p.sub.1], [p.sub.2], m)/
 [partial derivative][p.sub.1] [partial derivative][p.sub.1]
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