Hicks and Marshall Decompositions

We contrast two alternative decompositions of the calculation of the conjugate of a concave function—the “Hicks decomposition” and the “Marshall decomposition.” Each decomposes the calculation into two steps. One obtains an economic interpretation by viewing the function as the utility of consumption. Economists refer to expenditure-minimizing consumption as Hicks demand (compensated demand) (Hicks, J. R. (1946). Value and capital (2nd ed.). Oxford: Clarendon) and to utility-maximizing consumption as Marshall demand (consumer demand) (Marshall, A. (1890). Principles of economics). In the Hicks decomposition, consumption minimizes the expenditure required for a given utility. In the Marshall decomposition, consumption maximizes utility subject to the budget constraint. The expenditure function E is the minimum cost of obtaining utility u. The indirect-utility function V is the maximum utility possible, subject to the budget constraint that expenditure is less than or equal to income y. Located midway between utility and its conjugate, either the expenditure function or the indirect utility function determines the conjugate. The relationship between the expenditure function and the indirect-utility function is simple: reversing the axes in the closure of the epigraph of the expenditure function obtains the epigraph of the indirect-utility function. Suppress the price from the arguments of E and V. That E and V are nondecreasing is the key to axis reversal: for a given price, $$\delta _{\mathrm {epi\ }V}\left ( y,u\right ) =\delta _{\mathrm {epi}\left ( \mathrm {cl\ }E\right ) }\left ( u,y\right ) \!.$$ We prove this axis-reversal theorem via the duals of expenditure minimization and utility maximization.