Profit Maximization Problem

Definition

Consider a firm:

• buy inputs
• convert inputs to outputs using technology
• sell outputs

Given price vector , production vector , the profit maximization problem of the firm is:

19. t.

or using transformation function:
s.t.

We can calculate:

• supply correspondence (optimal production correspondence) at p y(p): element of production set Y that maximizes profit
• profit function : total profit at optimum

by solving the optimization of profit.

Note: here the profit function is the "function" from optimal aspect, not from general aspect ( )

Relationship between UMP, EMP, PMP and CMP

From utility maximization problem (UMP)

• Malrasian demand
• indirect utility function
• Roy's identity

From expenditure minimization problem (EMP)

• Hicksian demand
• expenditure function
• Shepard's lemma

From profit maximization problem (PMP)

• Supply correspondence
• Profit function
• Hotelling's Lemma

From cost minimization problem (CMP)

• Conditional factor demand
• cost function
• Shepard's lemma

In the case of production function, the profit maximization problem becomes:

Suppose that is the profit function of the production set Y and that is the associated supply correspondence.
Assume also that Y is closed and satisfies the free disposal property. Then:
(1) is homogeneous of degree one
(2) is convex
(3) If Y is convex, then
(4) is homogeneous of degree zero.
(5) If Y is convex, then y(p) is a convex set for all p. Moreover, if Y is strictly convex, then y(p) is single-valued (if not empty)
(6) Hotelling's lemma: if consists of a single point, then is differentiable at and , or
(7) If is a function differentiable at , then is a symmetric and positive semi definite matrix with

Dy(p) is the supply substitution matrix.

If the price of an output increases, the supply of the output increases;
if the price of an input increases, the demand for the input decreases.

Note

Example