The pprpose of compensated demand functions is to try to isolate the substitution and income effects on demand arising from a change in price. Hicksian compensation
holds the consumer to a given ·level of utility, so that we have the Hicksian demand function, h(p, u), as a function of the level of prices p and the level of utility
u to which we wish to hold the consumer. An alternative is Slutsky compensation, where we hold the consumer to the income needed to purchase a given bundle of goods.
Formall~ define s(p, x) as the demand of the consumer at prices p if the consumer is given just enough income to purchase the bundle x at the prices p. We learned that
the matrix of Hicksian substitution terms, whose (i,j)th element is ∂hi/ ∂pj, is symmetric and negative semi-definite. Suppose we looked at the matrix of Slutsky
substitution terms, whose (i, j)th entry is ∂si / ∂pj. This matrix is also symmetric and negative semi-definite. Prove this. (Hint: Express s(p, x) in terms of the
Marshallian demand function. Then see what this tells you about the matrix of Slutsky substitution terms.)
