Question
A firm operating in a perfectly competitive market has a
short-run total cost function given by TC(Q)=20+2Q+0.5Q2. If the market price is P=10, what is the profit-maximizing output level (Q), and what is the resulting profit (π)?Solution
Find Marginal Cost (MC):
- Profit maximization requires MC(Q)=P for a perfectly competitive firm.
- MC is the derivative of the Total Cost (TC) function with respect to Q:
- Set MC equal to the market price (P=10):
- Profit is Total Revenue (TR) minus Total Cost (TC): π=TR−TC.
- TR(Q)=P×Q
- TC(Q)=20+2Q+0.5Q2
- π=TR−TC
- The firm should only produce if P≥AVC.
- VC(Q)=2Q+0.5Q2. AVC(Q)=QVC(Q)=2+0.5Q.
- AVC(8)=2+0.5(8)=6. Since P=10>AVC=6, the firm should continue to operate.
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