Question

A local electric utility provider is a considered by regulators to be a natural monopoly. It has fixed costs of $100 million and a constant marginal cost of $0.25 per KWH. Its demand curve is linear: ?=160−0.00001? where ? is the price per KWH and Q is the quantity demanded by consumers in KWH per year.

a. Confirm that this utility provider is a natural monopoly. [HINT: It might be helpful to use Excel for this exercise.]

b. Find the profit-maximizing price and output, PM and QM, that the monopoly would choose if unregulated.

c. Calculate consumer surplus, the monopoly’s profits, and deadweight loss when the firm operates at the unregulated monopoly equilibrium.

d. If the regulator were to constrain the price to be equal to marginal cost, would the monopoly firm still make a profit, or would it operate at a loss? If it makes a loss, how might the regulator approach this problem?

Thank you for the help

Answer 1

(a)

Total cost (TC) = Fixed cost (FC) + Total variable cost = FC + Q x MC = 100,000,000 + 0.25Q

Average total cost (ATC) = TC/Q = (100,000,000/Q) + 0.25

A natural monopoly arises if ATC is falling with increase in Q.

Q	ATC
10,000	10,000.25
20,000	5,000.25
30,000	3,333.58
40,000	2,500.25
50,000	2,000.25
60,000	1,666.92
70,000	1,428.82
80,000	1,250.25
90,000	1,111.36
1,00,000	1,000.25
1,10,000	909.34
1,20,000	833.58
1,30,000	769.48
1,40,000	714.54
1,50,000	666.92
1,60,000	625.25
1,70,000	588.49
1,80,000	555.81
1,90,000	526.57
2,00,000	500.25

As is seen, ATC continuously decreases with increase in Q, so it is a natural monopoly.

(b)

Profit is maximized when MR = MC.

Total revenue (TR) = P x Q = 160Q − 0.00001Q²

MR = dTR/dQ = 160 − 0.00002Q

160 − 0.00002Q = 0.25

0.00002Q = 159.75

Q = 7,987,500

P = 160 − (0.00001 x 7,987,500) = 160 - 79.875 = 80.125

(c)

From demand function, when Q = 0, P = 160 (vertical intercept of demand curve).

Consumer surplus (CS) = Area between demand curve and price = (1/2) x (160 - 80.125) x 7,987,500 = 3,993,750 x 79.875

= 319,000,781.3

TR = P x Q = 7,987,500 x 80.125 = 639,998,437.5

TC = 100,000,000 + 0.25 x 7,987,500 = 100,000,000 + 1,996,875 = 101,996,875

Profit = TR - TC = 639,998,437.5 - 101,996,875 = 538,001,562.5

In efficient outcome, P = MC.

160 − 0.00001Q = 0.25

0.00001Q = 159.75

Q = 15,975,000

P = MC = 0.25

Deadweight loss = (1/2) x Change in P x Change in Q = (1/2) x (80.125 - 0.25) x (15,975,000 - 7,787,500)

= (1/2) x 79.875 x 7,987,500 = 319,000,781.3

(d)

If P = MC = 0.25, Q = 15,975,000.

TR = 0.25 x 15,975,000 = 3,993,750

TC = 100,000,000 + 0.25 x 15,975,000 = 100,000,000 + 3,993,750 = 103,993,750

Profit = TR - TC = 3,993,750 - 103,993,750 = - 100,000,000 (Loss of 100 million)

To compensate for the loss, government may subsidize the monopolist to induce it to continue operations.