A "One Line" Proof of the Slutsky Equation

One focus of the usual classroom discussion of consumer theory is the demonstration that the individual consumer's reaction to a change in the market price of a commodity can be usefully broken down into vectors of substitution effects and income effects. The Slutskv equation relating the price effect to the substitution and income effects can be simply nmotivated by J. R. Hicks' graphical presentation, p. 31, but the usual proof (see Paul Samuelson) is very tedious and nonintuitive. If the instructor includes a discussion of the expenditure function in his curriculum, however, he has available a concise, intuitively appealing proof of the Slutsky equation. 1 Suppose a consumer with income y faces a vector of commoditv prices p. His Marshallian demand curve for commodity j is given by xj = Dj(y, p). The minimum expenditure necessary for the consumer to achieve any utility level u is given by his expenditure function, y = m(u, p) (here y is in units of the jth good). His Hicksian income-compensated demand for commodity j is represented xj= hi(u, p); if m is differentiable, we have the well-known result that