Question

Prove that for any t > 0  x_i(tp,tw) = x_i(p.w), i.e. that if all prices are multiplied by the same positive number the factor inputs that maximize profits will not change, or that the factor demand functions are homogeneous of degree zero.

Answer 1

Multivariate functions that are “homogeneous” of some degree are often used in economic theory. A function is homogeneous of degree k if, when each of its arguments is multiplied by any number t > 0, the value of the function is multiplied by tk. For example, a function is homogeneous of degree 0 if, when all its arguments are multiplied by the same number t > 0, the value of the function is multiplied by the same number t.

Here is a precise definition. Because the definition involves the relation between the value of the function at (x1, ..., xn) and its values at points of the form (tx1, ..., txn) where t is any positive number, it is restricted to functions for which (tx1, ..., txn) is in the domain whenever t > 0 and (x1, ..., xn) is in the domain.

Suppose that a consumer's demand for goods, as a function of prices and her income, arises from her choosing, among all the bundles she can afford, the one that is best according to her preferences. Then we can show that this demand function is homogeneous of degree zero: if all prices and the consumer's income are multiplied by any number t > 0 then her demands for goods stay the same.