Mosak's Equality and the Theory of Duality

Abstract

Slutsky in the interpretation of the substitution matrix in the Fundamental Equation of Value Theory.2 Slutsky's interpretation involves a compensation of the change in real income caused by a price change .that makes possible the purchase of the same quantities of all the goods that had formerly been bought,13 while Hicks' compensation keeps the consumer on the same utility level as before the price change.4 It is known that when the price change is infinitesimal, the Hicks substitution matrix is equal to the Slutsky matrix. We will call this Mosak's Equality. Mosak's Equality has played an important role in index number theory. The price index that truly reflects the welfare change should be based on the Hicksian compensation. Such an index is difficult to compute, however, because utility levels are not observable. Instead, the Laspeyres index is widely adopted in practice. This index is based on the Slutsky compensation, since it indicates the change in income that would be needed in the current year in order to buy the commodity bundle bought in the base year. Mosak's Equality reveals that for small price changes the Laspeyres index is a good approximation to the ideal index.5 Despite its practical importance, Mosak's Equality has been given only a rather cumbersome proof.6 The aim of the present paper is to give a new proof of Mosak's Equality based on the principles of duality theory. Like the duality proofs of many other theorems, this proof is simpler and more transparent than the traditional one.