An Expository Note on Alchian-Allen Theorem When Sub-Utility Functions are Homogeneous of Degree n > 0 with Two-Stage Budgeting

This expository note shows a proof of Alchian and Allen's conjecture broadly known as a phrase `shipping the good apples out'' meaning that consumers purchase fine quality relatively more than coarse one if a fixed charge is imposed. Their statement is often referred as the Alchian-Allen theorem (or effect). The proof requires conditions about homogeneity, inner solution and substitutability and they also justify two-stage budgeting. In order this work to be an exposition, I emphasize graphical representations, however, a comparison with the proof of Borcherding and Silberberg (JPE 1978) is also considered. This comparison clarifies difference between proofs using Hicksian demand functions and using Marshallian demand functions (with some specific conditions). Extending sequential budgeting procedures, we also discuss perspectives toward multiple-quality analysis. That turns out a definition based on sequential budgeting may open a way of experimental study about the effect.