Suppose a firm sells in a highly competitive market where the going price is \$15 per unit. Its cost equation is C=\$25+.25Q2.

A. Find the profit maximizing level of output for the firm. Determine its level of profit.

Profit, π = TR-TC = (P*Q) – C = 15Q – 25 – 0.25Q2 .

For profit maximization the first order necessary condition is that,

dπ/dQ =0,

i.e. dTR/dQ – dTC/dQ =0,

i.e. MR-MC=0,

i.e. MR=MC.

Here, dπ/dQ =0,

i.e. 15 – 0.5Q= 0,

i.e. Q = 150/5 =30.

Profit = 15Q – 25 – 0.25Q2  = \$200.

Hence Q=30 is the profit maximizing level of output and the firm earns a profit of \$200 in the short run.

B. Suppose that fixed costs rise to \$75. Verify this change does not affect the firms level of output.

Now C=75+0.25Q2.

Profit, π = TR-TC = (P*Q) – C = 15Q – 75 – 0.25Q2 .

For profit maximization the first order necessary condition is that,

dπ/dQ =0,

i.e. dTR/dQ – dTC/dQ =0,

i.e. MR-MC=0,

i.e. MR=MC.

Here, dπ/dQ =0,

i.e. 15 – 0.5Q= 0,

i.e. Q = 150/5 =30.

Profit = 15Q – 75 – 0.25Q2  = \$150.

Hence Q=30 is a profit maximizing level of output and earns a profit of \$150.

The change in fixed costs does not affect the profit maximizing level of output (or loss minimizing) because the firm considers only its marginal cost and marginal revenue while determining the optimum output. It does not consider the fixed costs.