What are the properties of the profit function?

Properties of the Profit Function:

The properties specified below follow solely by the assumption of profit maximization. No assumptions regarding convexity, monotonicity or other sorts of regularity are essential.

The properties of profit function as follows:

a. Non-decreasing in output prices, non-increasing within input prices. When p′_i> pi for all outputs and p′_j < p_j, for each inputs, then π(p′) > π(p).

b. Homogeneous of degree 1 into p. π(tp) > tπ (p) for each t > 0.

c. Convex into p. Assume p′′= tp+(1-t)p′ for 0 < t < 1. After that π(p′′) < tπ (p)+(1-t)π(p′).

d. Continuous into p. This function π(p) is continuous, at least while π(p) is well-described and p_i > 0 for i = 1, ..., n.