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Zorluk: Çok zorMonopoly: Short-Run and Long-Run Price and Output Determination

A monopolist faces a market demand curve given by P=70−2QP = 70 - 2Q and operates with a short-run total cost function TC=50+10Q+Q2TC = 50 + 10Q + Q^2, where PP is the price in Naira (N\mathbb{N}) and QQ is the output level in units. What is the firm's profit-maximizing output level and its resulting short-run economic profit?

  1. 1010 units of output and a profit of N250\mathbb{N}250Cevap
  2. B
    1515 units of output and a profit of N175\mathbb{N}175
  3. C
    1010 units of output and a profit of N200\mathbb{N}200
  4. D
    1010 units of output and a profit of N50\mathbb{N}50

Cevap

The profit-maximizing output is 1010 units and the short-run economic profit is N250\mathbb{N}250.
To find the equilibrium output for a monopolist, set Marginal Revenue equal to Marginal Cost (MR=MCMR = MC). From the demand equation P=70−2QP = 70 - 2Q, Total Revenue is TR=70Q−2Q2TR = 70Q - 2Q^2, yielding MR=70−4QMR = 70 - 4Q. The derivative of Total Cost TC=50+10Q+Q2TC = 50 + 10Q + Q^2 gives MC=10+2QMC = 10 + 2Q. Equating 70−4Q=10+2Q70 - 4Q = 10 + 2Q yields 6Q=606Q = 60, so Q=10Q = 10 units. Substituting Q=10Q = 10 into the demand curve yields P=70−2(10)=N50P = 70 - 2(10) = \mathbb{N}50. Total Revenue is 10×50=N50010 \times 50 = \mathbb{N}500 and Total Cost is 50+10(10)+102=N25050 + 10(10) + 10^2 = \mathbb{N}250. Subtracting Total Cost from Total Revenue gives an economic profit of N250\mathbb{N}250.

Adım Adım Çözüm

1
Derive Total Revenue (TRTR) and Marginal Revenue (MRMR) functions from the demand curve.
TR=P×Q=(70−2Q)Q=70Q−2Q2TR = P \times Q = (70 - 2Q)Q = 70Q - 2Q^2, so MR=dTRdQ=70−4QMR = \frac{dTR}{dQ} = 70 - 4Q.
Monopoly pricing power implies MRMR declines at twice the rate of the linear demand curve.
2
Derive the Marginal Cost (MCMC) function from Total Cost (TCTC).
MC=dTCdQ=10+2QMC = \frac{dTC}{dQ} = 10 + 2Q.
Marginal cost is the first derivative of the total cost function with respect to output.
3
Equate MRMR and MCMC to find the profit-maximizing equilibrium output (QQ).
70−4Q=10+2Q  ⟹  60=6Q  ⟹  Q=1070 - 4Q = 10 + 2Q \implies 60 = 6Q \implies Q = 10 units.
All profit-maximizing firms produce where marginal revenue equals marginal cost.
4
Substitute equilibrium output into the demand function to find the selling price (PP).
P=70−2(10)=50P = 70 - 2(10) = 50 Naira.
The price a monopolist can charge is determined by consumer demand at the profit-maximizing output level.
5
Calculate Total Revenue (TRTR), Total Cost (TCTC), and Economic Profit (π\pi).
TR=50×10=500TR = 50 \times 10 = 500 Naira; TC=50+10(10)+(10)2=250TC = 50 + 10(10) + (10)^2 = 250 Naira; π=TR−TC=500−250=250\pi = TR - TC = 500 - 250 = 250 Naira.
Economic profit equals total revenue minus total cost.

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Monopoly Short-Run Price and Output Determination
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Monopoly: Short-Run and Long-Run Price and Output Determination

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