Examkin

Question

Difficulty: Very hardMonopoly: Short-Run and Long-Run Price and Output Determination

A monopolist faces a market demand function given by P=180−3QP = 180 - 3Q, where PP is the price in Naira and QQ is the output quantity. The firm operates with a total cost function of TC=100+20Q+Q2TC = 100 + 20Q + Q^2. If a regulatory authority forces the monopolist to adopt marginal cost pricing (P=MCP = MC) to achieve economic efficiency, by how many units will the firm's output increase compared to its unregulated profit-maximizing output?

  1. 1212 unitsAnswer
  2. B
    3232 units
  3. C
    2020 units
  4. D
    88 units

Answer

The firm's output will increase by 12 units.
Under unregulated monopoly profit maximization, the firm sets MR=MCMR = MC, yielding an output of Q=20Q = 20 units. Under marginal cost pricing (P=MCP = MC), output expands to Q=32Q = 32 units to eliminate deadweight loss. The difference between these two output levels is 32−20=1232 - 20 = 12 units.

Step-by-Step Solution

1
Derive the Marginal Revenue (MR) and Marginal Cost (MC) functions.
Total Revenue TR=P×Q=(180−3Q)Q=180Q−3Q2TR = P \times Q = (180 - 3Q)Q = 180Q - 3Q^2, so MR=dTRdQ=180−6QMR = \frac{dTR}{dQ} = 180 - 6Q. Given TC=100+20Q+Q2TC = 100 + 20Q + Q^2, MC=dTCdQ=20+2QMC = \frac{dTC}{dQ} = 20 + 2Q.
MR and MC are required to find the profit-maximizing output condition for the unregulated firm.
2
Determine the unregulated monopoly output (QMQ_M).
Set MR=MC  ⟹  180−6Q=20+2Q  ⟹  8Q=160  ⟹  QM=20MR = MC \implies 180 - 6Q = 20 + 2Q \implies 8Q = 160 \implies Q_M = 20 units.
A profit-maximizing monopolist produces where marginal revenue equals marginal cost.
3
Determine the output under marginal cost pricing (QCQ_C).
Set P=MC  ⟹  180−3Q=20+2Q  ⟹  5Q=160  ⟹  QC=32P = MC \implies 180 - 3Q = 20 + 2Q \implies 5Q = 160 \implies Q_C = 32 units.
Marginal cost pricing forces price to equal marginal cost, replicating the competitive socially optimal output level.
4
Calculate the increase in output.
Increase in output =QC−QM=32−20=12= Q_C - Q_M = 32 - 20 = 12 units.
Subtract the unregulated output from the regulated output to find the change.

Key Concept

Monopoly Output Determination vs Socially Optimal Output
Rate this question

Topics

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

More questions from this topic