The Generalized Alchian-Allen Theorem: A Slutsky Equation for Relative Demand

Article excerpt

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. …

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