Question 3: [40 points] A monopolist faces demand of q(p) = 440 p and has cost f

Question

Question 3: [40 points] A monopolist faces demand of q(p) = 440 p and has cost function C(q) = 40q.

(a) [5 points] What is the monopolist’s maximization problem?

(b) [5 points] What is the first order condition corresponding to this problem?

(c) [5 points] What is the profit-maximizing (price, quantity) chosen by the monopolist?

(d) [5 points] What are the monopolist’s profits, consumer surplus, and deadweight loss of monopoly?

(e) [10 points] How much would consumers be willing to pay to have the monopolist set price equal to its marginal cost? How much would the monopolist be willing to accept in terms of a check/bribe/one-time payment to agree to price at marginal cost?

(f) [10 points] How much would consumers be willing to pay to have the monopolist set price equal to zero? How much would the monopolist be willing to accept in terms of a check/bribe/one-time payment to agree to price at zero?

Explanation / Answer

q = 440 - p

Or,

p = 440 - q

C(q) = 40q

Total revenue, TR = p x q = q x (440 - q) = 440q - q2

(a)

Monopolist's maximization problem will be:

Maximize Profit = Total Revenue - Total Cost

Or,

Maximize Z = 440q - q2 - 40q = 400q - q2

(b)

First order condition will be:

dZ / dq = 0

(c)

dZ / dq = 400 - 2q = 0

400 = 2q

q = 400 / 2 = 200

p = 440 - q = 440 - 200 = 240

(d)

(i) Profit = 400q - q2 = (400 x 200) - (200 x 200) = 80,000 - 40,000 = 40,000

(ii) Deadweight loss:

A competitive firm's equilibrium is found out by setting price equal to marginal cost.

p = 440 - q

MC = dC / dq = 40

Equating,

440 - q = 40

q = 400 [= q*]

When q = 0, p = 440 [Vertical intercept of monopolist's demand curve]

So, deadweight loss = (1/2) x (q* - q) x (440 - 240)

= (1/2) x (400 - 200) x 200 = 20,000

(e)

(i) Consumer surplus = area between demand curve and price

= (1/2) x (440 - 240) x 200 = 20,000

This is the maximum amount that consumers will be willing to pay.

(ii) Producer surplus = Area between supply curve (Cost curve) and price

= (1/2) x (240 - 0) x 200 = 24,000

This is the amount monopolist will want as one-time payment.

(f)

(i) Consumer surplus if price is 0 = (1/2) x (440 - 0) x (200) = 44,000

This is the amount consumers will be willing to pay.

(ii) Producer surplus if price is 0 = (1/2) x 0 = 0

So monopolist will not accept any one-time payment to set price equal to 0 (Since it will not produce any gain to him).

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