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      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 (Ï€)?
      A Q=8; π=12 Correct Answer Incorrect Answer
      B Q=12; π=28 Correct Answer Incorrect Answer
      C Q=8; π=44 Correct Answer Incorrect Answer
      D Q=12; π=20 Correct Answer Incorrect Answer

      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:
      TC(Q) = 20+2Q+0.5Q2 MC(Q)= dTC/dQ = 2+(2×0.5)Q=2+Q Find Optimal Output (Q):
      • Set MC equal to the market price (P=10):
      MC=P 2+Q=10 Q=8 Calculate Total Profit (Ï€):
      • Profit is Total Revenue (TR) minus Total Cost (TC): Ï€=TR−TC.
      • TR(Q)=P×Q
      TR(8)=10×8=80
      • TC(Q)=20+2Q+0.5Q2
      TC(8)=20+2(8)+0.5(8)2 TC(8)=20+16+0.5(64)=20+16+32=68
      • Ï€=TR−TC
      π=80−68=12 Short-Run Condition Check: (Shut-down rule)
      • 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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