The generalized Alchian-Allen theorem: a slutsky equation for relative demand.

We generalize the Alchian-Allen theorem so as to account for income and endowment effects and provide two versions of a Generalized Alchian-Allen theorem: one for a unit cost component and one for a proportional cost component. Both versions provide a decomposition of an uncompensated change in the demand ratio of two goods into a substitution effect and an income-endowment effect-and may thus be regarded as extensions of the familiar Slutsky equation for relative demand. Finally, we apply our results to the choice of real estates and to parental time allocation decisions, the latter providing implications for child care policies. UEL Dll, H21, J22, R21)

I. INTRODUCTION

The Alchian-Allen theorem (see Alchian and Allen 1964, 74-75) suggests that an increase in the prices of two goods by the same amount leads to a decrease in the relative price of the more expensive good, and hence to a relative increase in the compensated demand for that good. As this pure substitution effect applies to any pair of goods with a common but abstract unit cost component, the Alchian-Allen theorem has a broad range of applications: for example, the common unit cost component may be interpreted as a specific tax, as a fixed transportation cost, as a flat transaction cost, or as the wage forgone in favor of different leisure time activities. Accordingly, the Alchian-Allen theorem has been discussed in various fields of both theoretical and empirical economics. (1) Above all, Borcherding and Silberberg (1978) formulated the Alchian-Allen theorem for compensated demand in the standard consumer model with three goods. Then, Hummels and Skiba (2004) extended the analysis of Borcherding and Silberberg by introducing an ad valorem tariff on the two goods under consideration and showed that the effect of a change in the ad valorem tariff is basically opposite to the effect of a change in the unit cost.

In this paper, we generalize the two formulas for substitution effects-one for demand changes resulting from the unit cost components and one resulting from the proportional cost components (2)-within the Alchian-Allen framework so as to incorporate income and endowment effects. We believe that this generalization constitutes an important result, as these effects are arguably significant for a broad variety of goods. In this way, we arrive at two versions of the generalized Alchian-Allen theorem (GAAT), which essentially provides a decomposition of an uncompensated change in the demand ratio of two goods into a substitution effect (compensated effect) and an income-endowment effect.

Because of the apparent similarity of our decomposition of the change in the demand ratio with the renowned Slutsky equation, the two versions of the GAAT we provide here may be regarded as the proper versions of the Slutsky equation for relative demand.

The Alchian-Allen result (i.e., an increase in the demand for the more expensive good relative to the demand for the less expensive good resulting from an increase of a common unit cost component of both goods) has been examined empirically. For example, Bertonazzi, Maloney, and McCormick (1993) interpreted the Alchian-Allen theorem within a household production framework (3) and found that football fans traveling farther tend to purchase higher quality tickets. Hummels and Skiba (2004) presented the hypothesis that an increase in unit transport costs increases the share of the higher quality goods and therefore the share weighted average of prices, and confirmed this using international trade data.

Moreover, some researchers have indirectly provided findings that are consistent with the Alchian-Allen result. Sobel and Garrett (1997) found that an increase in specific taxes is estimated to increase the market share of premium-brand cigarettes, and Nesbit (2007) found that an increase in specific taxes is estimated to increase the market share of premium-grade gasoline. On the other hand, Lawson and Raymer (2006) found that a parallel increase in the prices of three grades of gasoline sees the consumers "switching to mid-grade gasoline from premium-grade gasoline leaving the market share of regular gasoline unchanged" (see Abstract and Conclusion therein). Consequently, the market share of premium-grade gasoline decreases in response to higher (overall) gasoline prices. To see why the empirical findings are seemingly contradictory, it is important to realize that in due compliance with the Alchian-Allen theorem, these empirical tests are based on the assumption that substitution effects dominate.

However, because price increases induce not only substitution effects but also income effects and as there are no a priori restrictions on income effects, it is necessary to derive a generalized version of the Alchian-Allen theorem taking into account both effects, and to clarify its implications. In fact, Gould and Segall (1969) already showed, by using a diagram, that the Alchian-Allen result can be reversed by the inclusion of income effects. (4)

Although a few studies have addressed income effects, hardly any studies have taken into account endowment effects, which are, however, essential whenever individuals own a significant amount of the good considered. For endowment effects work in opposition to the usual income effect: while a higher price of a consumption good induces a negative income effect (provided the good is normal), a higher price of a good in which the consumer has some initial endowment induces a positive income effect. There are many goods in which individuals may possess significant amounts of worth-real estate, artworks, articles of virtu, shares, gold, wine, classic cars, etc.-and all of them may be analyzed within a suitably generalized Alchian-Allen framework. In particular, we focus on the household's choice between two different types and prices of real estate. As the value (or the cost) of a real estate may be considered to be composed of the price of the lot and the price of the building, a rise in either the price of the lot or of the building increases the wealth of the owner. Yet, as we shall see, this type of example may apply to many other pairs of goods (such as the ones listed above) whose prices can be decomposed into two cost components.

Another particularly fine and important example is time: the disposable daily time, the initial time endowment, may be used for various types of working and leisure time activities, and the wage rate represents a common component of the opportunity costs of all types of leisure activities. Consequently, as the price of pure leisure is higher than the price of parental child care by the cost of external child care, a rise in wages lowers the relative price of pure leisure, and therefore raises the relative demand for leisure. Recently, Minagawa and Upmann (2013b) acknowledged the applicability of the Alchian-Allen framework for time allocation problems (of parents with young children) and derived a specific version of the Alchian-Allen theorem with income and endowment effects tailored to this particular application.

In sum, the literature documents a long tradition of endeavor to extend the well-known Alchian-Allen theorem to more than two goods, to proportional cost components, and to incorporate income and endowment effects. In this paper, we demonstrate that all this can be performed in a systematic way, which leads us to identify a fundamental structure in the Alchian-Allen framework. To this end, we first derive a generalized version of the Alchian-Allen theorem so as to account for all of the crucial determinants of observed economic behavior: income effects, endowment effects, and substitution effects, as well as their mutual interactions. To achieve this, we build on the work of Gould and Segall (1969), Borcherding and Silberberg (1978), and Bauman (2004), who have previously extended the Alchian-Allen theorem to the case of three or more goods, in terms of compensated demand functions. (5) In our approach, however, we use ordinary instead of compensated demand functions to capture both income and endowment effects and show under which conditions the Alchian-Allen result for compensated demand continues to hold when income and endowment effects are taken into account. As a result, our generalized Alchian-Allen formula (for unit cost components) concurs with the pure substitution version provided by Borcherding and Silberberg (1978) if income and endowment effects are absent, if the income elasticities for the goods under consideration coincide, or if "on average" (that is, aggregated over all goods), a consumer consumes his/her endowments.

While in conformance with the original approach of Alchian and Allen, our first version of the GAAT is concerned with a common unit cost component, our second version of the GAAT takes into account common proportional cost components. Then, by incorporating income and endowment effects, we arrive at a generalized version of the result derived by Hummels and Skiba (2004). Naturally, our second GAAT version concurs with the formula of Hummels and Skiba when income and endowment effects are absent. In addition, we demonstrate that the GAAT for proportional cost components coincides with the pure substitution version of Hummels and Skiba if either both goods possess identical income elasticities or "on average" consumers consume the value of their initial endowment.

In sum, both the GAAT versions for unit and for proportional cost components provide a decomposition of the cost effects into a pure substitution effect and an income-endowment effect. Furthermore, the conditions for unit cost components that make the income and endowment effects cancel out, so that the GAAT (for ordinary demand) reduces to the Alchian-Allen theorem (for compensated demand), are similar to the corresponding conditions for proportional cost components. Yet, irrespective of this remarkable parallelism of the two versions of the GAAT, they provide conflicting predictions for the effects of unit and proportional cost components on relative demand. This phenomenon, observed for substitution effects in the literature, is fully characterized by the inclusion of incomeendowment effects. In particular, when applied to the taxation of goods and services, (6) this result casts substantial doubts on the widely celebrated equivalence result between unit (specific) and proportional (ad valorem) taxes, which has been derived under conditions of competitive behavior and in the absence of uncertainty. (7)

From a more general perspective, while the familiar Slutsky equation decomposes the effect of a price change on (uncompensated) demand into a substitution and an income (and endowment) effect, the GAAT provides a similar decomposition for relative (uncompensated) demand. In this sense, the GAAT concomitantly also establishes a generalization of the Slutsky equation, and the two versions of the GAAT provided here may be legitimately referred to as proper generalizations of the Slutsky equation. In fact, because of this similarity, we can transform both versions of the GAAT into elasticity forms: the uncompensated elasticity of relative demand equals the sum of (1) the compensated elasticity of relative demand and (2) the net share of aggregate cost components of the two goods in total income multiplied by the income elasticity of relative demand. The elasticity forms of the GAAT imply that both the compensated and uncompensated elasticities of relative demand coincide if either the net share of the (unit or proportional) cost of both goods in total income is zero, or if the income elasticity of relative demand is zero.

Finally, our results, when applied to specific economic frameworks, may be used to derive important implications. For example, when applied to time allocation problems, the GAAT for unit cost components provides predictions for the change of the ratio of any two types of leisure activities in response to an increase in the wage rate. In particular, as mentioned above, Minagawa and Upmann (2013b) addressed the time allocation decisions of a parent with young children and explored the wage effect on the demand ratio of pure leisure and parental child care. As we shall see, their result may be regarded as a limiting case of the GAAT (for unit cost components) when the consumer possesses a positive initial endowment of only one good (time) and cannot consume this good (leisure time) in excess of his/her initial endowment. Moreover, we extend the time allocation model and present, among others, a straightforward, although new, application of the GAAT for proportional cost components. Most importantly, then, the analysis offers insights on child care and labor market policies.

The remainder of this paper is structured as follows. Section II sets up the Alchian-Allen framework to be used throughout the paper. Our main results are presented in Section III, where Propositions 2 and 4 provide the GAAT for unit cost and proportional cost components, respectively. We demonstrate the broad applicability of our results in Section IV, where we (1) analyze a household's choice between different types of real estates and (2) use parental time allocation decisions to derive policy implications. Finally, we summarize and conclude in Section V.

II. THE MODEL

We consider the standard model of consumer behavior with three goods. Suppose that the consumer's preferences may be represented by a well-behaved utility function u : [R.sup.3.sub.+] [right arrow] R : ([x.sub.1],[x.sub.2],[x.sub.3]) [??] u ([x.sub.1], [x.sub.2],[x.sub.3]). While the familiar Alchian-Allen approach assumes that goods 1 and 2 are (close) substitutes, we do not impose such a restriction here, but allow for good 1 and good 2 to be either substitutes or complements (or being neither). The consumer may possess nonnegative initial endowments, denoted by [[omega].sub.1], [[omega].sub.2], and [[omega].sub.3] for the three goods respectively, and a positive money income m. Moreover, let [p.sub.1], [p.sub.2], and [p.sub.3] denote the respective (net) prices, where by assumption [p.sub.1] [greater than or equal to] [p.sub.2]. For the purpose of a more compact notation, we define the following vectors x := [(x:=([x.sub.1],[x.sub.2],[x.sub.3]).sup.T] and [omega] :=[([[omega].sub.1], [[omega].sub.2]. [[omega].sub.3].sup.T] (We write commodity vectors as column vectors.)

While, by assumption, the third good is free of additional cost, such as tax rates, transportation costs, and user charges, these types of cost apply to the first two goods. More precisely, the first two goods are subject to a unit cost (specific cost) t as well as to a proportional cost (value cost) [tau] > 0. Accordingly, gross consumer prices amount to [q.sub.1] [equivalent to] [tau][p.sub.1], + t, [q.sub.2] [equivalent to][tau][p.sub.2] + t, and [q.sub.3] [equivalent to] [p.sub.3]; or in vector notation q:=([q.sub.1],[q.sub.2],[q.sub.3]), where consumer prices are assumed to be always positive, i.e., q [member of] [R.sup.3.sub.++. (Under this condition, we allow pj to be negative.) (8) Consequently, as [p.sub.1] > [p.sub.2] by assumption, we have [q.sub.1] > [q.sub.2], as [tau] > 0. The consumer's total income (or wealth) level given by q x [omega] = + m depends on the price vector and the money income and thus we may define a function I(q, m) := q x [omega] + m. Then, as usual, the consumer's utility maximization problem is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

yielding ordinary (or Marshallian) demand functions [x.sup.0](q, /(q, m)). Then, taking into account that consumer prices depend on the unit cost t and the proportional cost [tau], we define demand as a function of t and x (and exogenous income m) rather than of prices (and total income I): [[??].sup.0] (t, [tau], m) : = [x.sup.0] (q(t,[tau]), I (q(t,[tau]),m)).

Correspondingly, the expenditure minimization problem may be expressed as

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the solution of which is given by the compensated (or Hicksian) demand function [x.sub.*](q,v).

Similarly, it is helpful to write the compensated demand as a function of the unit cost t and the proportional cost [tau] (and of the utility level v): [??](t, [tau], v) : = [x.sub.*] (q (t, x), v). In general, our notation is as follows: The superscript "o", e.g., [x.sub.0], refers to ordinary demand; the superscript e.g., [x.sub.*], to compensated demand; and the superscript "[LAMBDA]", e.g., [??], to demand in terms of the unit cost t and the proportional cost [tau].

In the remainder of our analysis, we focus on interior solutions and use the following notation. The compensated price elasticity of good i with respect to the consumer price of good j is defined as [[epsilon].sup.*.sub.ij] (q, v) : = ([q.sub.j]/[x.sup.*.sub.i] (q, v)) ([partial derivative][x.sup.*.sub.i] (q, v) /[partial derivative][q.sub.j]); the income elasticity of good i, as [[epsilon].sub.iI] (q,I) : = (I/[x.sup.0.sub.i] (q, I)) ([partial derivative][x.sup.0.sub.i] (q, I)/[partial derivative]I). Whenever it is clear at which point of the domain these elasticities are evaluated, we suppress their arguments; a similar hint applies to consumer prices where we frequently suppress the arguments t and [tau] and simply write [q.sub.i] or q. Likewise, in order to save notational effort, we shall henceforth simply write [x.sub.i] and x to denote the demand level (or the image) of the demand function under consideration, rather than the generic consumption level. As in the subsequent analysis, we exclusively deal with the solution of the consumer choice problem only, so that no ambiguity should arise.

III. RESULTS

A. The Effect of a Unit Cost

Before we derive our central result, Proposition 2, it is advantageous to reformulate a result of Borcherding and Silberberg (1978) in our setting, which can also be found in Hummels and Skiba (2004, eq. 3). Following our notational convention, we here write [x.sub.i] = [[??].sup.*.sub.i] (t, [tau], v), [e.sup.*.sub.ij] instead of [[epsilon].sup.*.sub.ij] (q (t, [tau]), v) and [q.sub.i] instead of [q.sub.i] (t, [tau]). With this note of caution, we arrive at

PROPOSITION 1 (Borcherding and Silberberg 1978). The change in the compensated demand for the more expensive good relative to the compensated demand for the less expensive good in response to an increase of the unit cost of both goods is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof of Proposition 1

As long as compensated demand functions are concerned, the demonstration of Borcherding and Silberberg holds without essential modification except for the use of consumer prices rather than net prices.

Proposition 1 implies that if the two goods are not perfect complements (i.e., [[epsilon].sup.*.sub.11] < [[epsilon].sup.*.sub.21]) and good 1 is not a much stronger substitute for the third good than is good 2 (i.e., [[epsilon].sup.*.sub.23] - [[epsilon].sup.*.sub.13] > -[alpha] for some small positive [alpha]), then the Alchian-Allen result, that the partial derivative ([partial derivative]/[partial derivative]t)([[??].sup.*.sub.1] / [[??].sup.*.sub.2]) is positive, generalizes to this framework. (9) (Recall that we assumed [p.sub.1] > [p.sub.2] and thus [q.sub.1] > [q.sub.2].) In addition, if the price difference between good 1 and good 2 is small and therefore the term [absolute value of (1/[q.sub.1])-(1/[q.sub.2])] is small in magnitude, the first product term within the brackets, representing the direct substitution effect, will be small, ceteris paribus.10 Conversely, given that [[epsilon].sup.*.sub.11] < [[epsilon].sup.*.sub.21], the Alchian-Allen result, ([partial derivative]/[partial derivative]t) ([[??].sup.*.sub.1]/ [[??].sup.*.sub.2]) > 0, fails to hold only if good I is a much stronger substitute for good 3 than is good 2, i.e., [[epsilon].sup.*.sub.13] [much greater than] [[epsilon].sup.*.sub.23]. Borcherding and Silberberg (1978) argue that'if the two goods under consideration are close substitutes for each other--an assumption that we do not impose here--then both goods should be similarly related to the third good, thus, the term |[[epsilon].sup.*.sub.23] - [[epsilon].sup.*.sub.13]| should be small. (11,12)

In order to formulate Proposition 2, it is expedient to write the demand as a function of q and m: [[??].sup.0] (q, m) : = [x.sup.0] (q, I (q, m)). Similarly, we define the expenditure function E(q, v) := q x [x*(q, v) - [omega]]. It then follows as a familiar duality result that [x.sub.i] = [x.sup.*.sub.i] (q, v) [equivalent to] [[??].sup.0.sub.i] . (q, E (q, v)), or, more compactly, x = [x.sup.*] (q, v) [equivalent to] [[??].sup.0] (q, E (q, v)). Alternatively, this identity may also be expressed as x = [??]* (t, [tau], v) [equivalent to] [[??].sup.0] (t, [tau], E (q (t, [tau]), v)). We are now prepared to derive a version of the Alchian-Allen theorem, which takes into account both income and endowment effects. (13)

PROPOSITION 2 (The Generalized Alchian-Allen Formula for a Unit Cost). (14) The change in the ordinary demand for the more expensive good relative to the ordinary demand for the less expensive good in response to an increase of the unit cost of both goods is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the last derivative can be expressed as

[partial derivative]/[partial derivative]I [x.sup.0.sub.1]/[x.sup.0.sub.2] (q, I(q,m))= [x.sub.1]/[x.sub.2] 1/I ([[epsilon].sub.1I] - [[epsilon].sup.2I]).

The proof of Proposition 2 makes use of the following lemma, which is a generalization of the well-known Slutsky equation when endowment effects are present, that is, when the consumer's income is given by the value of their initial endowment (in some or all of the commodities) in addition to money income.

LEMMA 1 (The Slutsky equation with endowment effects).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof of Lemma 1

Differentiating both sides of [x.sup.*.sub.i] (q, v) [equivalent to] [[??].sup.0.sub.i]. (q, E (q, v)) with respect to [q.sub.j] yields

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Applying Shephard's lemma: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (see Cornwall (1984, 747)) and noting that [partial derivative][[??].sup.0.sub.i]/[partial derivative]m = [partial derivative][x.sup.0.sub.i]/ [partial derivative]I, we obtain the required result.

Proof of Proposition 2

By definition, we have [[??].sup.*.sub.i] (t, [tau], v) = [x.sup.*.sub.i] (q (t, [tau]), v), and thus

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Using [partial derivative][q.sub.j] (t,[tau])/[partial derivative]t = 1 and applying Lemma 1, we obtain

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Now, it is straightforward to calculate the partial derivative of the demand ratio

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Substituting for the partial derivatives [partial derivative][[??].sup.*.sub.i] (t, [tau], v)/[partial derivative]t, i = 1,2 and rearranging terms, we arrive at

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As [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we know that the first part on the right-hand side equals ([partial derivative]/[partial derivative]t) ([[??].sup.0.sub.1] (t, [tau], m)/ ([[??].sup.0.sub.2] (t, [tau], m)). Finally, transposing the last term to the left-hand side and expressing it in terms of the income derivative of relative demand ([partial derivative]/[partial derivative]I) ([x.sup.0.sub.1] (q, I (q, m))/[x.sup.0.sub.2] (q, I (q, m))) (resp. in terms of income elasticities), we have the required result.

Note that from our formulation of Proposition 2 and Lemma 1, it is apparent that our model covers the baseline scenario where the consumer has no initial endowments and income is thus equal to exogenous monetary income (i.e., I [equivalent] m). In this case, endowment effects do not appear and only ordinary income effects are present. If, however, income is price dependent (i.e., I [equivalent to] q x [omega] + m), endowment effects arise, reflecting the fact that higher prices of these goods directly increase the consumer's income.

Exogenous Monetary Income. We first consider the basic case where income exclusively consists of exogenous monetary income: [[omega].sub.j] = 0 [[for all].sub.j] and thus I = m. Then, Proposition 2 implies that if good 1 does not have much stronger income effects than good 2 (i.e., [epsilon]2I - [epsilon]1I > - [beta] for some small positive [beta]), the Alchian-Allen result for compensated demand, ([partial derivative]/[partial derivative]t) ([[??].sup.*.sub.1] / ([[??].sup.*.sub.2]) > 0, continues to hold, under the conditions mentioned above (see Section III.A), after the inclusion of income effects; that is, we have ([partial derivative]/[partial derivative]t) ([[??].sup.0.sub.1]/ [[??].sup.0.sub.2] > 0. Conversely, in order for the compensated Alchian-Allen result to be reversed, namely, ([partial derivative]/[partial derivative]t) ([[??].sup.0.sub.1]/ [[??].sup.0.sub.2] < 0 when ([partial derivative]/[partial derivative]t) ([[??].sup.*sub.1]/ [[??].sup.*.sub.2]> 0, it must be true that good 1 has much stronger income effects than good 2 (i.e., [[epsilon].sub.1I] [much greater than] [[epsilon].sub.2I]) [much greater than] [[epsilon].sub.2I]). Gould and Segall (1969) illustrate this by a diagram where there are only two goods and good 1 is normal while good 2 is inferior. The income effect in this case is intuitive: an increase in the unit cost t causes a decrease in real income and therefore leads to a decrease in the demand for good 1 and an increase in the demand for good 2: which means a decrease in the demand ratio. (15)

Price-Dependent Income. Next, we turn to the general case, where total income consists of exogenous monetary income and price-dependent income (the market value of the initial endowment): I [equivalent to] q x [omega] T- m. If the household's aggregate excess supply, the term [[summation].sup.2.sub.j=1] ([[omega].sub.j] - [x.sub.j]), is negative, the implications of Proposition 2 are basically similar to those of the first case with exogenous monetary income. However, if this sum is positive, then this gives rise to contrasting implications. To see this, suppose that [[summation].sup.2.sub.j=1] ([[omega].sub.j] - [x.sub.j) > 0. Then, Proposition 2 implies that if good 2 does not have much stronger income effects than good 1 (i.e., [[epsilon].sub.1I] - [[epsilon].sub.2I] > - [gamma] for some small positive [gamma]), then the Alchian-Allen result for compensated demand under the conditions above, ([partial derivative]/[partial derivative]t) ([[??].sup.*.sub.1]/ [[??].sup.*.sub.2]) > 0, continues to hold for uncompensated demand, that is, (([partial derivative]/[partial derivative]t) ([[??].sup.0.sub.1]/ [[??].sup.0.sub.2]) > 0. Conversely, in order for the compensated Alchian-Allen result to be reversed, namely ([partial derivative]/[partial derivative]t) ([[??].sup.0.sub.1]/ [[??].sup.0.sub.2]) < 0 when ([partial derivative]/[partial derivative]t) ([[??].sup.*.sub.1]/ [[??].sup.*.sub.2]) > 0, it must be that good 2 has much stronger income effects than good 1 (i.e., [epsilon]2I [much greater than] [epsilon]1I).

Moreover, it should be noticed that when both goods have identical income elasticities, i.e., [[epsilon].sub.1I] = [[epsilon].sub.2], the generalized Alchian-Allen formula, given in Proposition 2, coincides with the substitution result of Borcherding and Silberberg (1978) stated in Proposition 1, irrespective of whether consumption falls short of or exceeds initial endowment. This concurrence materializes, for example, when the consumer's preferences are homothetic or quasilinear (linear in the third good), because in either case, the income elasticities are identical (viz unity or zero, respectively). Similarly, if "on average" the household consumes its initial endowment, in the sense that [[summation].sup.2.sub.j=1) ([[omega].sub.j] - [x.sub.j] = 0 then both formulas also coincide--and thus whether the Alchian-Allen result continues to hold depends on the same qualifications as given above (see our discussion of Proposition 1 in Section III.A).

We can transform the generalized Alchian-Allen formula for unit cost into an elasticity form. To this end, assuming t > 0, we define the elasticity of relative compensated demand with respect to the unit cost t by [[epsilon].sup.*.sub.12,t := [t/ ([x.sub.1]/[x.sub.2)[[partial derivative]/[partial derivative]t) ([[??].sup.*.sub.1]/ [[??].sup.*.sub.2])] and the elasticity of relative ordinary demand with respect to the unit cost t by [[epsilon].sub.12,t] := [t/[V([x.sub.1]/[x.sub.2])] [partial derivative]/[partial derivative]t) ([[??].sup.0.sub.1]/ [[??].sup.0.sub.2]). We also define the income elasticity or relative demand by [[epsilon].sub.12,I] := [I/([x.sub.1]/[x.sub.2])] [partial derivative]/[partial derivative]I) ([[??].sup.0.sub.1]/ [[??].sup.0.sub.2]). Moreover, we define the net share of the unit cost of good 1 and good 2 in total income I by [[eta].sub.12,t] := t [[summation].sup.2.sub.j=1] ([[omega].sub.j]- [x.sub.j]) /I. the share of the unit cost of good j in the consumer price of good j by [[theta].sub.j] := t/[q.sub.j], and the net budget share of good j by [[kappa].sub.j] := [q.sub.j] (([[omega].sub.j] - [x.sub.j])/I.

Now, multiplying the formula given in Proposition 2 through by t/([x.sub.1]/[x.sub.2]) and the last term of the formula by I/I and rearranging terms, we obtain

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Now, using the definition of the elasticities, [[epsilon].sup.*.sub.12,t], [[epsilon].sub.12,t], and [[epsilon].sub.12,I], and of the share [[eta].sub.12,t], we obtain the elasticity form of the generalized Alchian-Allen formula. Similarly, multiplying the formula given in Proposition 1 through by t/([x.sub.1]/[x.sub.2]) and rearranging the terms yields

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Now, using the definition of the elasticity [[epsilon].sup.*.sub.12.t] and of the share [[theta].sub.j];, we obtain the elasticity form of the formula. Finally, we observe that

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We summarize these results as follows.

REMARK 1. The elasticity form of the generalized Alchian-Allen formula for a unit cost is given by

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The elasticity form of the generalized Alchian-Allen formula for the unit cost implies that the difference between the uncompensated and compensated elasticities of relative demand with respect to the unit cost t will be small, ceteris paribus, if either the net share of the unit cost of both goods in total income, [absolute value of [[eta].sub.12,t]] = [absolute value of [[summation].sup.2.sub.(j=1)] [[theta].sub.j] [[kappa].sub.j]| or the income elasticity of relative demand, [absolute value of [[epsilon].sub.12,I]] = [absolute value of [epsilon]1I - [[epsilon].sub.2I]], small. (16)

B. The Effect of a Proportional Cost

Hummels and Skiba (2004, eq. 4) show that the effect of a change in the proportional cost on the compensated demand ratio is basically opposite to the effect of a change in the unit cost, i.e., the compensated Alchian-Allen effect, provided in Proposition 1.

PROPOSITION 3 (Hummels and Skiba 2004). The change in the compensated demand for the more expensive good relative to the compensated demand for the less expensive good in response to an increase of the proportional cost of both goods is given by

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Proof of Proposition 3

As the proof of Proposition 3 is very similar to the proof of Proposition 1, we omit it.

Notice that the terms within the brackets are similar to those of Proposition 1 but that the term ([p.sub.1]/[q.sub.1]) - ([p.sub.2]/[q.sub.2]) is positive if the unit cost f is positive, and zero if t is zero, as ([P.sub.1]/[q.sub.1])-([p.sub.2]/[q.sub.2]) = ([p.sub.1] - [P.sub.2])t/([q.sub.1][q.sub.2]) and [p.sub.1] > [p.sub.2]. Thus, for 1 > 0, Proposition 3 implies that the partial derivative ([partial derivative]/[partial derivative]t)([[??].sup.*.sub.1] / [[??].sup.*.sub.2]) is negative if the two goods are not perfect complements (i.e., [[epsilon].sup.*.sub.11] < [[epsilon].sup.*.sub.21]) and if good 2 is not a much stronger substitute for the third good than is good 1 (i.e., [[epsilon].sup.*.sub.13] - [[epsilon].sup.*.sub.23] > - [gamma] for some small positive [gamma]) when [p.sub.2] > 0 (and vice versa when [p.sub.2] < 0). Conversely, given that [[epsilon].sup.*.sub.11] < [[epsilon].sup.*.sub.21], the above result, ([partial derivative]/[partial derivative]t)([[??].sup.*.sub.1] / [[??].sup.*.sub.2]) < 0, fails to hold only if good 2 is a much stronger substitute for good 3 than is good 1, i.e., [[epsilon].sup.*.sub.23] [much greater than] [[epsilon].sup.*.sub.23], provided [p.sub.2] > 0 (and vice versa, provided [p.sub.2] < 0). Consequently, the derivatives ([partial derivative]/[partial derivative]t) ([[??].sup.*.sub.1]/ [[??].sup.*.sub.2]) (see Proposition l) and ([partial derivative]/[partial derivative]t) ([[??].sup.*.sub.1]/ [[??].sup.*.sub.2]) (see Proposition 3) have opposite signs if both goods enjoy identical relations of substitution with the third good, i.e., [[epsilon].sup.*.sub.23] = [[epsilon].sup.*.sub.13].

We are now prepared to provide a version of Proposition 3, which accounts for both income and endowment effects.

PROPOSITION 4 (The Generalized Alchian-Allen Formula for a Proportional Cost). The change in the ordinary demand for the more expensive good relative to the ordinary demand for the less expensive good in response to an increase of the proportional cost of both goods is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the last derivative can be expressed as

[partial derivative]/[partial derivative]I [x.sup.0.sub.1]/ [x.sup.0.sub.2](q, I(q,m)) = [x.sub.1]/[x.sub.2] 1/I [[epsilon].sub.1I] - [[epsilon].sub.2I].

Proof of Proposition 4

The proof of Proposition 4 is very similar to the proof of Proposition 2, only noting [partial derivative][q.sub.j](t, [tau])/[partial derivative][tau] = [p.sub.j].

Proposition 4 implies that the result of Proposition 3 under the conditions above, ([partial derivative]/[partial derivative][tau]) ([[??].sup.*.sub.1]/ [[??].sup.*.sub.2]) < 0, continues to hold for uncompensated demand, that is, ([partial derivative]/[partial derivative][tau]) ([[??].sup.0.sub.1]/ [[??].sup.0.sub.2]) <0, if good 2 does not have much stronger income effects than good 1 (i.e., [[epsilon].sub.1I] - [[epsilon].sub.2I] > - [delta] for some small positive [delta]), provided that the aggregate value of the excess supply, the term [[summation].sup.2.sub.j=1] [p.sub.j]([w.sub.j] - [x.sub.j]), is negative (and vice versa, if the term is positive). Conversely, in order for the compensated result to be reversed, namely, ([partial derivative]/[partial derivative][tau])([[??].sup.0.sub.1]/ [[??].sup.0.sub.2]) > 0 when ([partial derivative]/[partial derivative][tau]) ([[??].sup.*.sub.1]/ [[??].sup.*.sub.2]) < 0, it must be true that good 2 has mucn stronger income effects than good 1 (i.e., [[epsilon].sub.2I] [much greater than][[epsilon].sub.1I]), provided that the aggregate value of the excess supply is negative (and vice versa, if the term is positive).

In addition, similar to the case of the generalized Alchian-Allen formula for the unit cost, we have: when both goods have identical income elasticities, i.e., [[epsilon].sub.1I] = [[epsilon].sub.2I], the uncompensated formula, given in Proposition 4, coincides with the compensated version of Hummels and Skiba (2004), stated in Proposition 3. Similarly, if "on average" the household consumes the value of its initial endowment in the sense that [[summation].sup.2.sub.j=1] [p.sub.j]([w.sub.j] - [x.sub.j]) = 0, then both formulas also coincide.

Finally, we transform the generalized Alchian-Allen formula for proportional cost into an elasticity form. To this end, we define the elasticity of the relative compensated demand with respect to the proportional cost [tau] by [[epsilon].sup.*.sub.12,[tau]] : = [[tau]/([x.sub.1]/ [x.sub.2])] ([partial derivative]/[partial derivative][tau])([[??].sup.*.sub.1]/ [[??].sup.*.sub.2])] and the elasticity of the relative ordinary demand with respect to the proportional cost [tau] by [[epsilon].sup.*.sub.12,[tau]] : = [[tau]/([x.sub.1]/ [x.sub.2])] ([partial derivative]/[partial derivative][tau])([[??].sup.0.sub.1]/ [[??].sup.0.sub.2])]. We also define the net share of the proportional cost of good 1 and good 2 in total income I by [[eta].sub.12,[tau]] :=[tau] [[summation].sup.2.sub. (j=1)] [p.sub.j] ([w.sub.j] - [x.sub.j]) /I and the share of the proportional cost of good j in the consumer price of good j by [v.sub.j] := [tau][p.sub.j]/[q.sub.j]. (17)

By a procedure similar to that used for Remark 1, we arrive at the corresponding results, only noting that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

REMARK 2. The elasticity form of the generalized Alchian-Allen formula for a proportional cost is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The elasticity form of the generalized Alchian-Allen formula for the proportional cost implies that the difference between the uncompensated and compensated elasticities of relative demand with respect to the proportional cost x will be small, ceteris paribus, if either the net share of the proportional cost of both goods in total income, [absolute value of [[eta].sub.12.[tau]]] = [absolute value of [[summation].sup.2.sub.(j=1)] [v.sub.j][[kappa].sub.j]], or the income elasticity of relative demand, [absolute value of [[epsilon].sub.12.I]] = [absolute value of [[epsilon].sub.1I]] - [[epsilon]2.sub.I]], is small. Now, comparing the results given by Remarks 1 and 2 clarifies the conflicting predictions for the effects of unit and proportional costs. Note, in particular, that the direct substitution effect of the proportional cost, represented by the term ([[epsilon].sup.*.sub.11], - [[epsilon].sup.*.sub.21])([v.sub.1] - [v.sub.2]) = -[[epsilon].sup.*.sub.11], [[epsilon].sup.*.sub.12] ([[theta].sub.1] - [[theta].sub.2]) is exactly opposite to the direct substitution effect of the unit cost.

IV. APPLICATIONS

A. Real Estate Choice

Consider the choice between two different types of real estate, e.g., houses or apartments, which may differ in various characteristics, such as location, or style and quality of the building. Any real estate is composed of two goods: a lot (plot of land) and the building. Correspondingly, the value (or the cost) of a real estate consists of two components: the price of the lot and the price of the building. Let [x.sub.1] and [x.sub.2] denote the quantities of the two types of real estate, measured in square meters (or square feet), and [q.sub.1] and [q.sub.2] their respective prices. Then, the price of real estate, [q.sub.i], i = 1,2, consists of the price (cost) of the lot and the price (cost) of the building.

We can decompose the price of the real estates in two ways, according to which of the costs is regarded as a common cost component. First, if we consider a building of a certain type and quality at different locations, the common unit cost component t is represented by the building cost. The real estates then differ according to the lot (land) prices, [p.sub.1] and [p.sub.2], reflecting the fact that the value of a real estate ceteris paribus varies with its location. For example, the price of a given type of a house in a preferred urban area typically exceeds the price of the same house located in rural area. Second, if we consider the cost of the lot (land) as a common unit cost component t, the real estates differ according to the building cost, [p.sub.1] and [p.sub.2], reflecting two different styles or qualities of building. For example, given the size and the location of the lot, the buildings may differ according to the quality of the house. Then, the price of a real estate located in a given district varies with the quality of the house.

Adding some composite commodity as good 3, we can model the household's choice between the two real estates (and the composite good) as a direct application of the model in Section II. Consequently, if we focus on the effects of either the building cost or the lot (land) cost as a common cost component on the housing demand ratio, the results and the discussion provided in Section III directly apply. The compensated Alchian-Allen formula, i.e., Proposition 1, suggests, under the conditions above (see Section III.A), that an increase in the building cost or the land (lot) cost leads to a relative increase in the compensated demand for a more expensive real estate. On the other hand, by applying the uncompensated Alchian-Allen formula, i.e., Proposition 2, to this example, we may acknowledge income and endowment effects. Both of these effects are presumably significant, unless the household is extraordinary rich (such that the income effect vanishes) or the household does not possess any real estate (such that the endowment effect vanishes). In other words, Proposition 2 (and correspondingly Remark 1) applies to all those middle- and upper-class households for which their belongings in real properties represent a non-negligible part of their wealth, and for those we conclude: provided that the income elasticity of the more expensive real estate exceeds the income elasticity of the less expensive real estate, (18) the income effects work in the opposite direction to the substitution effect, while the endowment effects work in the same direction to it. Whether the income-endowment effects (in total) are positive or negative depends on whether the household's aggregate excess supply of real estates is positive or negative.

While the building cost is arguably most often independent of the location of a real estate and may thus be viewed as a unit cost component f, this need not be the case for all locations. If preferred sites (lots) are located on a mountaintop, on an island, etc., the construction cost could be higher than usual. In that case, the construction cost may be roughly proportional to the price of the land (lot) and may then be formalized as a proportional cost component x. Thus, provided that there is some common unit cost (such as design cost, material cost, interior cost, etc.), the results and the discussion for proportional cost components presented in Section III immediately apply.

We like to emphasize that this type of an example applies beyond the present specification: similar arguments and a similar analysis apply to many other cases, where the prices of two similar goods can be decomposed into two cost components. To see this, we may consider jewelry made of two materials, for example, rings made of gem (like diamond) and precious metal (like gold). Then, the price of the rings is composed of the price of the gem and the price of the metal, and either the gem cost or the metal cost may be used as a common unit cost component, depending on the specification of the second good. In the simplest example, the second good is either gold or diamonds, but we may also consider watches made of gold, or jewelry with a different type of gem. In all of these cases, an increase in either the price of gold or of the gem affects the relative demand for the two goods by means of the induced income and endowment effects.

B. Labor-Leisure-Child-Care Choice

Consider a familiar labor-leisure-child-care choice model, as presented, for example, in Minagawa and Upmann (2013b). Here, we extend this model by introducing an ad valorem tax (or subsidy) for external child care.

Suppose that a mother with young children allocates her disposable time [bar.T] to either labor, pure leisure time (leisure time without children), or maternal child care (leisure time with children), which are denoted by h, l, and c, respectively. Suppose that she derives utility from leisure, maternal child care, and the consumption level z of some composite good; and her preferences can be represented by a utility function u : [R.sup.3.sub.+] [right arrow] R: (l, c, z) [??] u(l,c,z). Let w be the after-tax wage rate, i.e., the wage rate after deduction of labor taxes and social contributions. Let [p.sub.ec] be the price of (private or public) external child care per unit time. Let the external child care be subject to an ad valorem tax [tau]; then the cost of external child care is given by [tau][p.sub.ec]. (19) Accordingly, her income is composed of the amount of time devoted to working, h = [bar.T] -l- c, multiplied by the net wage w - [tau][p.sub.ec] and exogenous non-labor income m. (Observe that the child, by assumption, must be under supervision while the mother is either at work or consuming leisure time.) Denoting the price of the composite good by [p.sub.z], the budget constraint is

[tau][p.sub.ec]l + [p.sub.z]z [??] (w - [tau][p.sub.ec]) ([bar.T]-l-c)+m. The mother's utility maximization problem is then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Correspondingly, the expenditure minimization problem is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Observe that by means of suitable substitutions, viz., [x.sub.1] = l, [x.sub.2] = c, [x.sub.3] = z, [w.sub.1], = 0, [w.sub.2] = [bar.T], [w.sub.3] = 0, [p.sub.1] = 0, [p.sub.2] = -[p.sub.ec], [p.sub.3] = [p.sub.z], t = w, [q.sub.1] = w, [q.sub.2] = w - [tau][p.sub.ec], and [q.sub.3] = [p.sub.z], the labor-leisure-child-care choice problem can be put into the form described in Section II.

In the following, we use the subscript j = l,c,z instead of j = 1,2,3 when denoting the respective consumer prices, elasticities, and shares, and write the compensated (or Hicksian) demand for leisure and for maternal child care as functions of the after-tax wage w, the ad valorem tax x, and the utility level v: l = [??] (w, [tau], v) : = [l.sup.*] ([q.sub.l], (w), [q.sub.c] (w, [tau]), [q.sub.z], v) and c = [[??].sup.*](w, [tau], v) := [c.sup.*] ([q.sub.l](w), [q.sub.c] (w, [tau]), [q.sub.z], v), respectively. Similarly, we write the ordinary (or Marshallian) demand for leisure and for maternal child care as functions of the after-tax wage w, the ad valorem tax [tau], and the non-labor income [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively.

We are now prepared to obtain the following results for the effects of the after-tax wage w on the demand ratio l/c, which correspond to the compensated and uncompensated Alchian-Allen formula, i.e., Propositions 1 and 2, in this labor-leisure-child-care choice model.

COROLLARY 1 (Minagawa and Upmann 2013b). The change in the compensated demand for pure leisure time relative to the compensated demand for parental child care in response to an increase in the (after-tax) wage is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof of Corollary 1

This follows immediately from an application of Proposition 1. Alternatively, the direct derivation of Minagawa and Upmann holds without essential modification except for the use of the after-tax wage rather than the net wage (here, w - [tau][p.sub.ec]).

COROLLARY 2 (Minagawa and Upmann 2013b). The change in the ordinary demand for pure leisure time relative to the ordinary demand for parental child care in response to an increase in the (after-tax) wage is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof of Corollary 2

This follows immediately from an application of Proposition 2. Alternatively, the direct derivation of Minagawa and Upmann holds without essential modification except for the use of the after-tax wage rather than the net wage. (20) We briefly summarize the results provided in Corollaries 1 and 2: The Alchian-Allen result for compensated demand, ([partial derivative]/[partial derivative]w) ([[??].sup.*]/[[??].sup.*]) > 0, holds under the conditions that leisure and maternal child care are not perfect complements (i.e., [[epsilon].sup.*.sub.ll] < [[epsilon].sup.*.sub.cl]) and leisure is not a much stronger substitute for the composite good than is maternal child care (i.e., [[epsilon].sup.*.sub.cz] - [[epsilon].sup.*.sub.lz] > -[alpha] for some small positive a). Now, given tfie empirical evidence that a rise in wages decreases leisure and increases maternal child care (see Kimmel and Connelly 2007, and also Guryan, Hurst, and Kearney 2008), implying that ([partial derivative]/[partial derivative]w) ([[??].sup.0]/[[??].sup.0]) < 0, the GAAT requires that the income elasticity of maternal child care must be substantially higher than that of leisure, [[epsilon].sub.cI] [much greater than] [epsilon]lI, provided that the Alchian-Allen result for compensated demand, ([partial derivative]/[partial derivative]w) ([[??].sup.*]/[[??].sup.*]) >0, holds true. (21) Notice that the term [bar.T] - l - c equals the labor supply h and can thus not be negative-that is, this is a special case of our GAAT for the unit cost.

From Remark 1 and Corollaries 1 and 2, we obtain

REMARK 3. The elasticity form of the generalized Alchian-Allen formula for the (after-tax) wage is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, [[eta].sub.lc,w] = [[theta].sub.l],[[kappa].sub.l], + [[theta].sub.c][[kappa].sub.c] = w ([bar.T] - l - c)/I represents the share of (gross) labor income in total income.

By Remark 3, the Alchian-Allen result for compensated demand is likely to be observed for mothers whose labor income share is sufficiently small. In addition, unlike the traditional example of apples, as the labor income share [[eta].sub.lc,w] (and thus income-endowment effects) could be large enough for some mothers, the theory provided in this paper seems to be consistent with the seemingly opposite finding in the empirical literature that an increase in wages decreases leisure and increases maternal child care.

Similarly, corresponding to Propositions 3 and 4, we derive the following results for the effects of an ad valorem tax [tau] on the demand ratio, which indicate, as shown in Section III, basically opposite effects of the after-tax wage w on the demand ratio.

COROLLARY 3. The change in the compensated demand for pure leisure time relative to the compensated demand for parental child care in response to an increase of the ad valorem tax on external child care is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof of Corollary 3

Applying Proposition 3 and noting that [p.sub.1] = 0 in this case, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using Hicks's "third law of consumer theory" That [[epsilon].sup.*.sub.ll] + [[epsilon].sup.*.sub.lc] + [[epsilon].sup.*.sub.lz] = 0 and [[epsilon].sup.*.sub.cl] + [[epsilon].sup.*.sub.cc] + [[epsilon].sup.*.sub.lz] = 0 completes the proof.

COROLLARY 4. The change in the ordinary demand for pure leisure time relative to the ordinary demand for parental child care in response to an increase of the ad valorem tax on external child care is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof of Corollary 4

Applying Proposition 4, we obtain the required result.

Corollary 3 implies that the partial derivative ([partial derivative]/[partial derivative][tau]) ([[??].sup.*]/ [[??].sup.*] is negative if leisure and maternal child care are not perfect complements (i.e., [[epsilon].sup.*.sub.cc < [[epsilon].sup.*.sub.lc]. On the other hand, Corollary 4 implies that the result of Corollary 3 under the conditions above, i.e., ([partial derivative]/[partial derivative][tau]) ([[??].sup.*]/ [[??].sup.*]), continues to hold for uncompensated demand, that is, ([partial derivative]/[partial derivative][tau]) ([[??].sup.0]/ [[??].sup.0]) < 0, unless maternal child care has much stronger income effects than leisure (i.e., [[epsilon].sub.cI] [much greater than] [[epsilon].sub.lI]).

From Remark 2 and Corollaries 3 and 4, we obtain

REMARK 4. The elasticity form of the generalized Alchian-Allen formula for an ad valorem tax on external child care is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Here, the absolute value of [[eta].sub.lc,[tau]] = [v.sub.c] [[kappa].sub.c] = -[tau][p.sub.ec] ([bar.T]-c)/I represents the share of external child care cost in total income.

As mentioned above, contrary to the Alchian-Allen result for compensated demand, some empirical analyses indicate that an increase in wages decreases leisure and increases maternal child care, implying that ([partial derivative]/[partial derivative][tau]) ([[??].sup.0]/ [[??].sup.0]) < 0.

Applying the GAAT to these observations, we can therefore infer that the income elasticity of maternal child care substantially exceeds that of leisure, that is, [[epsilon].sub.cI][much greater than] [[epsilon].sub.lI]. (Note that since w is defined in this paper as the wage after deduction of labor taxes and social contributions, any change in either labor taxes or social contributions that increases the wage has the same effect on time allocation as does a change in w.) Then, most importantly, we deduce the following policy implication: if the inferred relation, [[epsilon].sub.cl] [much greater than] [[epsilon].sub.lI], holds, we will expect, contrary to the prediction of the compensated version, that an increase in an ad valorem tax increases the ordinary demand for leisure relative to maternal child care, that is, ([partial derivative]/[partial derivative][tau]) ([[??].sup.0]/ [[??].sup.0]) > 0. Reversely, by interpreting a decrease in the ad valorem tax x as a subsidy for external child care, ([partial derivative]/[partial derivative][tau]) ([[??].sup.0]/ [[??].sup.0]) > 0 implies that a child care subsidy has qualitatively the same effects on parental time allocation as has the wage. In particular, this implication is likely to be valid for mothers whose budget share of external child care cost is large.

V. CONCLUSION

In this paper, we derived two versions of a generalization of the well-known Alchian-Allen theorem-one for a change in a unit cost component (Proposition 2) and one for a change in a proportional cost component (Proposition 4). Notably, both versions of our GAAT provide a decomposition of an uncompensated change in the demand ratio of two goods into a substitution effect (compensated effect) and an income-endowment effect. For this reason, both versions may be regarded as proper extensions of the renowned Slutsky equation, to price effects on relative demand. In fact, by focusing on the formal similarity of the GAAT with the Slutsky equation, we also obtained the elasticity forms of both versions of the GAAT. Thus, we may say that we provided the generalized Alchian-Allen-Slutsky equation.

In particular, based on the work of Borcherding and Silberberg (1978), we were able to derive a version of the Alchian-Allen theorem, which uses uncompensated rather than compensated demand, and, which, in addition to the usual income effects, also takes into account endowment effects. Our generalization of the Alchian-Allen theorem (Proposition 2) shows that the Alchian-Allen result for compensated demand (i.e., a positive effect of a unit cost component on the compensated demand for a more expensive good relative to the compensated demand for a less expensive good) continues to hold for uncompensated demand unless the income elasticity of the lower priced good substantially exceeds the income elasticity of the higher priced good, provided that consumption does not exceed initial endowment (and vice versa, if consumption exceeds the initial endowment).

Based on the work of Hummels and Skiba (2004), we were also able to derive a version of the generalized Alchian-Allen formula for the substitution effects of a proportional cost component on the demand ratio of two goods, which takes into account both income and endowment effects. Remarkably, the effect of a proportional cost is basically the opposite of the effect of a unit cost. Our generalized Alchian-Allen formula (Proposition 4) indicates that the result for compensated demand (i.e., a negative effect of a proportional cost on the compensated demand for a more expensive good relative to the compensated demand for a less expensive good) continues to hold for uncompensated demand unless the income elasticity of the lower priced good substantially exceeds the income elasticity of the higher priced good, provided that the value of consumption exceeds the value of initial endowment (and vice versa, if the value of consumption does not exceed the value of initial endowment).

Finally, in order to illustrate the applicability of our results (especially with endowment effects), we applied our results to specific economic frameworks. We first considered, as a direct example of our model, the household choice between two different types of real estates--and by the same procedure, this example can be used to accommodate for many other choices of consumption. We then applied our results to the labor-leisure-child-care allocation decision of a parent with young children. As the literature finds a negative effect of an increase in wages on the uncompensated demand for leisure relative to parental child care, Corollary 2 (Proposition 2) requires that given a positive substitution effect, the income elasticity of parental child care must be substantially higher than that of leisure. Given this, we expect, in view of Corollary 4 (Proposition 4), a positive effect of an increase in ad valorem taxes for external child care on the uncompensated demand for leisure relative to parental child care. In reverse, this implies that an ad valorem subsidy for external child care decreases pure leisure time compared with parental child care. This policy implication is important, because parental time allocation may have a serious impact on children's development. Thus, the examination of the elasticities becomes an empirical matter.

ABBREVIATION

GAAT: Generalized Alchian-Allen Theorem

doi: 10.1111/ecin.12205

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(5.) For another extension of the Alchian-Allen theorem, see Liu (2011), who attempted to analyze the case of three goods, all of which are subject to a unit cost component. Yet, because the demand ratio of two goods is no longer a unique measure in this case, Liu considered the demand for one good compared with the total demand for all goods instead.

(6.) While it is perfectly appropriate to think of unit and proportional cost components as specific and ad valorem taxes, respectively, especially when a consumer has no initial endowments of goods (as in the application to tax presented in fn. 15), we prefer to maintain the more general terms in order to accentuate that the applicability of our results is not limited to problems in the realm of public finance, but covers a broader range-as the reader will see.

(7.) To our knowledge, Suits and Musgrave (1953) were the first who formally proved the nowadays widely accepted result of the equivalence between ad valorem and unit taxes in a competitive world. Subsequently, this insight has been refined in various ways. (A list of references for these refinements is provided, for example, by Goerke, Herzberg, and Upmann 2014, fn. 1.)

(1.) See, for example, Gould and Segall (1969), Borcherd ing and Silberberg (1978), Umbeck (1980), Leffler (1982), Bertonazzi, Maloney, and McCormick (1993), Cowen and Tabarrok (1995), Sobel and Garrett (1997), Razzolini, Shughart, and Tollison (2003), Bauman (2004), Hummels and Skiba (2004). Lawson and Raymer (2006). Nesbit (2007), Saito (2008), Liu (2011), Minagawa (2012), Minagawa and Upmann (2013a), and Minagawa and Upmann (2013b).

(2.) While common unit cost components may include specific taxes, fixed transportation fees, service charges, upgrading costs, etc., common proportional cost components may represent ad valorem taxes, iceberg transportation costs, mark-up rates, proportional service charges, etc.

(3.) For this interpretation, see also Cowen and Tabarrok (1995).

(4.) Saito (2008) provided a version of the Alchian-Allen theorem with (zero) income effects in a two-stage budgeting framework under the assumption of homogeneous utility function. Using a two-stage budgeting process, Hallak (2006) also addressed income effects.

(9.) Notice that, unlike the original Alchian-Allen theorem in a two-goods case, the compensated price elasticity [[epsilon].sup.*.sub.21] may be of either sign in this three-goods case.

(8.) Note that we speak of proportional cost (components), even though [tau] is not a proportional cost, but a factor of proportional cost. Thus, if the good is not subject to a proportional cost supplement, x equals unity, not zero. Only if [tau] exceeds unity, are additional cost imposed on the good (and [tau]--1 represents the proportional cost added to the net price), while [tau] < 1 means that the good becomes cheaper, i.e., that the charge is negative (and the good is subsidized), provided that [p.sub.j] > 0. (And vice versa, when [p.sub.j] < 0.)

(10.) On this implication, see also Bauman (2004), who argues that if the two goods are not close complements and do not have similar prices, then the Alchian-Allen result will hold. Similarly, in our formulation, we may say: if the price difference between good 1 and good 2 is large, the direct substitution effect will be large, ceteris paribus.

(11.) Minagawa (2012) points out that there may be an exception to this argument.

(12.) If [[epsilon].sup.*.sub.13] = [[epsilon].sup.*.sub.23] holds, then both elasticities must be nonnegative. See Minagawa and Upmann (2013a, fn. 3).

(13.) We here remark that by the duality approach, we can equally work with the indirect utility function [u.sup.0] (q, m) : = u ([[??].sup.0] (q, m)) instead of the expenditure function.

(14.) The generalized Alchian-Allen formula can be extended to the n-goods case where the first two goods are subject to the unit cost t. In fact, the formula and its proof hold without any changes in the n-goods case. Yet, it should be noted that the income elasticities obey the law that their share weighted sum must be unity (see, for example, Silberberg and Suen 2001, 292). Also, see Bauman (2004) for the compensated Alchian-Allen formula in this kind of n-goods case.

(15.) Nesbit (2007) finds that an increase in income is estimated to increase the market share of premium-grade gasoline (good 1) and to decrease the market share of regular-grade gasoline (good 2). This finding may be interpreted as ([partial derivative]/ [partial derivative]I) ([x.sup.0.sub.1]/ [x.sup.0.sub.2]) > 0 and thus [[epsilon].sub.1I] > [[epsilon].sub.2I]. Nevertheless, Nesbit finds that an increase in specific taxes is estimated to increase the market share of premium-grade gasoline. Because this finding may be interpreted as ([partial derivative]/[partial derivative]t) ([[??].sup.0.sub.1]/ [[??].sup.0.sub.2]) > 0, our GAAT suggests that ([partial derivative]/[partial derivative]t) ([[??].sup.*.sub.1]/ [[??].sup.*.sub.2]) > 0 must hold in this case. Also, Coats, Pecquet, and Taylor (2005) find moderate support for the Alchian-Allen theorem, claiming from their estimates that the income effect in the relative demand for premiumgrade gasoline is likely to be weak.

(16.) The aggregate net income share [absolute value of [[eta].sub.12,t]] may be small in the traditional example of high and low quality apples (for this example, see, e.g., Silberberg and Suen 2001, 335-41). Thus, in such an example, the Alchian-Allen result for compensated demand is likely to be observed.

(17.) As ([tau]/[q.sub.j])([partial derivative][q.sub.j]/[partial derivative][tau]) = [tau][p.sub.j]/[q.sub.j], the terra [v.sub.j] equals the elasticity of the consumer price of good j with respect to the proportional cost t.

(18.) While many studies have attempted to estimate the income elasticity of housing demand, there are few empirical studies on the income elasticity of housing demand when households have multiple real estates. An exceptional study is Belsky, Di, and McCue (2007) whose estimates show that the income elasticity of demand for all houses owned by second home owners is lower than the income elasticity of demand for their primary house, suggesting that the income elasticity of demand for their second house is lower.

(19.) For consistency, we specify t as a taxation factor (cf., fn. 8).

(20.) Yet, their derivation is different from the proof of Proposition 2 in the sense that it does not explicitly use the Slutsky equation with endowment effects (Lemma 1).

(21.) Although Ramey and Ramey (2010) found no empirical support for income effects as a possible explanation for why educated parents spend more child care time, several researchers have criticized their methodology (cf., the Comments and Discussion section of Ramey and Ramey 2010). See also the Discussion section of Minagawa and Upmann (2013b).

JUNICHI MINAGAWA and THORSTEN UPMANN *

* We are grateful for the valuable comments and suggestions of the Co-Editor and anonymous referees. Junichi Minagawa acknowledges financial support from the Japan Society for the Promotion of Science.

Minagawa: Faculty of Economics, Chuo University, 742-1 Higashinakano, Hachioji, Tokyo 192-0393, Japan. Phone +81-(0) 42-674-3343, Fax +81-(0) 42-674-3425, E-mail minagawa@tamacc.chuo-u.ac.jp

Upmann: Bielefeld University, Faculty of Business Administration and Economics, Postfach 10 01 31, Bielefeld, NRW 33501, Germany. Phone +49-(0) 521-106-4862, Fax +49 521 106-89005, E-mail TUpmann@wiwi.unibielefeld.de
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