Web Appendix for “Tricks for Hicks: The EASI Demand System”* I. General Cost Functions for Implicit Marshallian Demands

Theorem 5 provides a very weak sufficient condition for existence of implicit Marshallian demands w = ω[ p, z, x − G (w, p, z)] for some function G. for example, this condition is satisfied if any good or combination of goods has a budget share that is strictly increasing or strictly decreasing in total expenditures (and hence in utility). Moreover, this condition is sufficient but not necessary, e.g., if preferences are homothetic so C ( p, u, z) = u + t ( p, z) for some function t, then even though the sufficient condition need not hold now, we can define G (w, p, z) = − t ( p, z) to obtain implicit marshallian demands. So the only kinds of situations where G might fail to exist is when all budget shares are independent of u for some, but not all, ranges of values of u. To give an example of Theorem 5, suppose that the hicksian budget share function for good 1, w1 = ω1 ( p, z, u), is invertible in u, so u = ω 1 −1 ( p, z, w1 ). Then a deflator G that makes u = y is G (w, p, z) = C ( p, ω 1 −1 ( p, z, w1 ), z) − ω 1 −1 ( p, z, w1 ). The function G satisfying equation (I.2) is not unique, e.g., it can be freely translated by an ordinal transformation of u, and in the previous example, different invertible budget shares could be used to obtain different expressions for G. for applications, the main point of implicit Marshallian demands is not existence or uniqueness, but rather convenience for demand estimation, which is why this paper focuses on examples where G has a simple closed form, namely, a translation of the log Stone index.