The preceding example is more familiar where the firm produces a single output and we distinguish inputs and outputs. Assume that a competitive firm produces a single output y from n inputs x = (x1, x2, . . . , xn) according to the production function y = f(x) so as to maximize profit

Assume that there is a unique optimum for every p and w. Show that the input demand x*I (w, p) and supply y*(w, p) functions have the following properties:

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Π (w, p) max pf(x)-WTX Upward sloping supply Downward sloping demand Dwxw,p Dww, p Symmetry Dex ,PD Dwy , p Reciprocity