Shipping the Good Apples Out: A New Perspective

I. INTRODUCTION The Alchian and Allen substitution theorem posits that a per unit tax or shipping fee applied to similar goods will increase the relative consumption of the higher-quality good. Originally formulated in Alchian and Allen's 1964 textbook University Economics, the theorem is often called the Shipping the Good Apples Out theorem because of the empirical observation that supermarkets in apple-importing areas (such as Indiana) have a higher proportion of high-quality apples (relative to low-quality apples) than supermarkets in apple-growing areas, such as Washington State. A Washington resident on holiday in Indiana might well conclude that the good apples are getting shipped out. The theoretical basis for the Alchian and Allen result was questioned by Gould and Segall (1969), who demonstrate that the result holds unequivocally only in a two-good world. Borcherding and Silberberg (1978) defend the Alchian and Allen result in an n-good world, but only when the two taxed goods are close substitutes. This special case appears to be all that can be salvaged in terms of theory. (For further discussion, see Umbeck 1980). The purpose of this article is to reinterpret the Alchian and Allen result in an n-good world. This reinterpretation shows that their result holds more broadly than suggested by Borcherding and Silberberg and indeed more broadly than (though not as robustly as) originally claimed by Alchian and Allen. II. BACKGROUND Consider a world with n goods, [x.sub.1], [x.sub.2], ..., [x.sub.n], the first two of which can be thought of as, respectively, the high-quality and standard-quality versions of some product (e.g., good apples and bad apples). By assumption, then, [p.sub.1] > [p.sub.2] > 0. Following Borcherding and Silberberg, I phrase the Alchian and Allen thesis as [[differential]([x.sub.1]/[x.sub.2])]/[differential]t > 0, where [x.sub.1]([p.sub.1],[p.sub.2], ..., U) and [x.sub.2] ([p.sub.1],[p.sub.2] ..., U) are Hicksian (income-compensated) demand functions and t is a per unit charge applied to both goods. (1) The chain rule gives us ([differential][x.sub.i] /[differential]t)=([differential][x.sub.i]/[differential][p.sub.1]) +([differential][x.sub.i]/[differential][p.sub.2]), and combining this with the quotient rule I get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] Substituting in the compensated elasticities, [[epsilon].sub.ij] = ([p.sub.j]/[x.sub.i]) * ([differential][x.sub.i] /[differential][p.sub.j]), I arrive at [[differential]([x.sub.1] /[x.sub.2])/[differential]t] = ([x.sub.1]/[x.sub.2])([epsilon].sub.11] /[p.sub.1]] + [[epsilon].sub.12]/[p.sub.2]] - [[epsilon].sub.21]/[p.sub.1]] - [[epsilon].sub.22]/[p.sub.2]]). The first term here is always positive, so I will focus attention on the second term, (1) ([[epsilon].sub.11]/[p.sub.1]) + ([[epsilon].sub.12]/[p.sub.2] - ([[epsilon].sub.21]/[p.sub.1]) - ([[epsilon].sub.22]/[p.sub.2]). The Alchian and Allen claim is that (1) is positive. III. A TWO-GOOD WORLD With only two goods, Hicks's (1946, 310-11) third law [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] reduces to [[epsilon].sub.ij] = [[epsilon].sub.ii] and one can substitute for [[epsilon].sub.11] and [[epsilon].sub.21] in (1) to get (2) ([[epsilon].sub.12] - [[epsilon].sub.22])([1/[p.sub.2]] - [1/[p.sub.1]]). The first term here is positive because the two goods in a two-good world must be substitutes ([[epsilon].sub.12] > 0) and own-price elasticities are negative ([[epsilon].sub.22] [p.sub.2] > 0). I therefore get the Alchian and Allen result: [[differential]([x.sub.1]/[x.sub.2])]/[differential]t > 0. The intuitively compelling story is that consumers are substituting out of bad apples and into good apples. …