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Difficulty: Very hardPerfect Competition: Price and Output Determination in Short and Long Run

A price-taking firm operating in a competitive market has a short-run total cost function given by TC=Q3−6Q2+25Q+100TC = Q^3 - 6Q^2 + 25Q + 100, where QQ represents the quantity of output produced. If the prevailing market price is $25\$25 per unit, what is the firm's profit-maximizing output and its corresponding economic profit or loss?

  1. 44 units with an economic loss of $68\$68Answer
  2. B
    44 units with an economic profit of $32\$32
  3. C
    22 units with an economic loss of $84\$84
  4. D
    66 units with an economic loss of $100\$100

Answer

The firm maximizes profit at an output of 44 units, resulting in an economic loss of $68\$68.
Under perfect competition, a price-taking firm maximizes short-run profit or minimizes loss where P=MCP = MC on the upward-sloping segment of MCMC. Taking the first derivative of TC=Q3−6Q2+25Q+100TC = Q^3 - 6Q^2 + 25Q + 100 gives MC=3Q2−12Q+25MC = 3Q^2 - 12Q + 25. Equating MCMC to market price P=25P = 25 yields 3Q2−12Q=03Q^2 - 12Q = 0, giving Q=4Q = 4 units. Substituting Q=4Q = 4 into the cost and revenue equations yields TR=100TR = 100 and TC=168TC = 168, resulting in an economic loss of $68\$68. Since price (P=25P = 25) exceeds average variable cost (AVC(4)=42−6(4)+25=17AVC(4) = 4^2 - 6(4) + 25 = 17), the firm minimizes losses by continuing production in the short run.

Step-by-Step Solution

1
Derive the Marginal Cost (MCMC) function from Total Cost (TCTC).
MC=dTCdQ=3Q2−12Q+25MC = \frac{dTC}{dQ} = 3Q^2 - 12Q + 25
Profit maximization under perfect competition requires setting market price equal to marginal cost (P=MCP = MC).
2
Set market price P=25P = 25 equal to MCMC and solve for output QQ.
25=3Q2−12Q+25  ⟹  3Q2−12Q=0  ⟹  3Q(Q−4)=025 = 3Q^2 - 12Q + 25 \implies 3Q^2 - 12Q = 0 \implies 3Q(Q - 4) = 0. Since Q>0Q > 0, Q=4Q = 4 units.
Equating PP and MCMC identifies the output level where profit is maximized or loss is minimized.
3
Calculate Total Revenue (TRTR) and Total Cost (TCTC) at Q=4Q = 4.
TR=P×Q=25×4=100TR = P \times Q = 25 \times 4 = 100. TC=43−6(4)2+25(4)+100=64−96+100+100=168TC = 4^3 - 6(4)^2 + 25(4) + 100 = 64 - 96 + 100 + 100 = 168.
Evaluating TRTR and TCTC at the optimal output level allows determination of overall economic profit or loss.
4
Compute economic profit or loss.
Profit=TR−TC=100−168=−68\text{Profit} = TR - TC = 100 - 168 = -68 (an economic loss of $68\$68).
Subtracting total cost from total revenue yields the firm's financial outcome.

Key Concept

Profit Maximization and Loss Minimization in Perfect Competition
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Topics

Perfect Competition: Price and Output Determination in Short and Long Run

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