Examples and exercises on short-run profit maximization

Procedure

• Find the minimum of the AVC.
• Find the SMC.
• For p less than this minimum of the AVC the firm produces 0. For p at least equal to this minimum the firm produces y such that p = SMC(y); to get the formula for the supply curve you need to isolate y in this equation.

Example: a production function with fixed proportions

The short run cost function is

STCk(y) =	w1y + w2k	if y k
		if y > k.

AVCk = SMCk(y) =	w1	if y k
		if y > k.

0	if p < w1
all outputs from 0 to k	if p = w1
k	if p > w1.

Example

• We have AVC(y) = y + 20, so the minimum of the AVC is 20 (attained at the output 0).
• SMC(y) = 2y + 20.
• Thus for p < 20 the firm produces 0; for p  20 it produces y such that SMC(y) = p, or p = 2y + 20, or y = (1/2)p 10.

0	if p < 20
(1/2)p  10	if p  20.

Example

• We have AVC(y) = y² 60y + 1200. The derivative is 2y 60, so the minimum of AVC occurs at y = 30. (You can check that this is in fact a minimizer, not a maximizer.) The minimum value of the AVC is thus (30)² (60)(30) + 1200 = 300.
• SMC(y) = 3y² 120y + 1200.
• Thus if p  300 the firm's short run supply is 0. If p  300 then the firm's short run supply function satisfies p = SMC(y) = 3y² 120y + 1200 = 3[y²  40y + 400] = 3(y  20)². Isolating y we get y = 20 + (p/3).

0	if p < 300
0 or 30	if p = 300
20 + (p/3)	if p > 300