Question

(8 points) Suppose a monopolist is the only firm that supplies water for the city of Atlanta The firm faces a market demand curve given by Q- 33-P, where P is the price of water and Q is the market demand for water per day. (1 pt) Derive the monopolist's marginal revenue curve. (2 pts) Suppose the total cost of this firm is described by a. b. TC 0.50+30. What output level will the firm choose to maximize profits? What is the price at this output level? (2 pts) Given your solution from part b), what are the consumer surplus, the producer surplus, and the social welfare? (Hint: producer surplus producer profits.) (3 pts) Now, suppose a regulator requires that the firm must produce at a competitive market level such that P-MC. What are the new output level and new market price? What are the consumer surplus, the producer surplus, and the social welfare? C. d.

Answer 1

(a)

Q = 33 - P

P = 33 - Q

Total revenue (TR) = PQ = 33Q - Q²

Marginal revenue (MR) = dTR/dQ = 33 - 2Q

(b)

Marginal cost (MC) = dTC/dQ = Q + 3

Profit is maximized when MR = MC.

33 - 2Q = Q + 3

3Q = 30

Q = 10

P = 33 - 10 = 23

(c)

From demand function, When Q = 0, P = 33 (Vertical intercept)

Consumer surplus (CS) = Area between demand curve & price = (1/2) x (33 - 23) x 10 = 5 x 10 = 50

When Q = 10, MC = MR = 33 - (2 x 10) = 33 - 20 = 13

From MC function, When Q = 0, MC = 3 (Vertical intercept)

Producer surplus (PS) = Area between MC curve & price = (1/2) x [(Price - Vertical intercept of MC) + (Price - MC)] x Quantity

= (1/2) x [(23 - 3) + (23 - 13)] x 10 = 5 x (20 + 10) = 5 x 30 = 150

Social welfare (SW) = CS + PS = 50 + 150 = 200

(d)

When P = MC,

33 - Q = Q + 3

2Q = 30

Q = 15

P = 33 - 15 = 18

CS = (1/2) x (33 - 18) x 15 = (1/2) x 15 x 15 = 112.5

PS = Area between MC curve and price = (1/2) x (18 - 3) x 15 = (1/2) x 15 x 15 = 112.5

SW = 112.5 + 112.5 = 225