A monopolist faces a market demand function given by P = 180 - 3Q, where P is the price in… — JAMB UTME

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

$12$ unitsCevap
B
$32$ units
C
$20$ units
D
$8$ units

Cevap

The firm's output will increase by 12 units.

Under unregulated monopoly profit maximization, the firm sets

MR = MC

, yielding an output of

Q = 20

units. Under marginal cost pricing (

P = MC

), output expands to

Q = 32

units to eliminate deadweight loss. The difference between these two output levels is

32 - 20 = 12

units.

Adım Adım Çözüm

Derive the Marginal Revenue (MR) and Marginal Cost (MC) functions.

Total Revenue

TR = P \times Q = (180 - 3Q)Q = 180Q - 3Q^2

, so

MR = \frac{dTR}{dQ} = 180 - 6Q

. Given

TC = 100 + 20Q + Q^2

MC = \frac{dTC}{dQ} = 20 + 2Q

MR and MC are required to find the profit-maximizing output condition for the unregulated firm.

Determine the unregulated monopoly output (

Q_M

Set

MR = 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.

Determine the output under marginal cost pricing (

Q_C

Set

P = 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.

Calculate the increase in output.

Increase in output

= Q_C - Q_M = 32 - 20 = 12

units.

Subtract the unregulated output from the regulated output to find the change.

Anahtar Kavram

Monopoly Output Determination vs Socially Optimal Output

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

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