Question

Exercise 5. Dayna's Doorstops, Inc. (DD) is a monopolist in the doorstop industry. Its cost is C = 100 –5Q+Q?, and inverse demand is P= 55 – 20. a. What price should DD set to maximize profit? What output does the firm produce? How much profit and consumer surplus does DD generate? b. What would output be if DD acted as a perfect competitor and set MC = P? What profit and consumer surplus would then be generated? c. What is the deadweight loss from monopoly power in part a.?

Answer 1

Answer : a) P = 55 - 2Q

TR (Total Revenue) = P*Q = (55 - 2Q) * Q

=> TR = 55Q - 2Q^2

MR (Marginal Revenue) = $\Delta$TR / $\Delta$Q

=> MR = 55 - 4Q

C = 100 - 5Q + Q^2

MC (Marginal Cost) = $\Delta$TC / $\Delta$Q  

=> MC = - 5 + 2Q

For monopolist the profit maximizing condition is, MR = MC. So,

55 - 4Q = - 5 + 2Q

=> 55 + 5 = 2Q + 4Q

=> 60 = 6Q

=> Q = 60 / 6

=> Q = 10

From demand function we get,

P = 55 - (2 * 10)

=> P = $35

Therefore, here the monopolist should charge $35 to maximize the profit level. The monopolist produces 10 units of output level.

TR = P * Q = 35 * 10

=> TR = $350

TC = 100 - (5 * 10) + (10)^2

=> TC = $150

Profit = TR - TC = 350 - 150

=> Profit = $200

Therefore, here the profit is $200.

To obtain the maximum willingness to pay we have to take Q = 0 for demand function. So,

P = 55 - (2 * 0)

=> P = $55

Consumer surplus (C.S.) = 0.5 * Height * Base

=> C.S. = 0.5 * (55 - 35) * 10

=> C.S. = $100

Therefore, here the consumer surplus is $100.

b) For DD as perfect competitor the profit-maximizing condition is, P = MC. So,

55 - 2Q = - 5 + 2Q

=> 55 + 5 = 2Q + 2Q
=> 60 = 4Q

=> Q = 60 / 4

=> Q = 15

From demand function we get,

P = 55 - (2 * 15)

=> P = $25

Therefore, as a perfect competitor DD will produce 15 units of output level.

TR = P * Q = 25 * 15 = $375

TC = 100 - (5 * 15) + (15)^2

=> TC = $250

Profit = TR - TC = 375 - 250

=> Profit = $125

Therefore, here the profit is $125.

Here the maximum willingness to pay is, P = $55.

C.S. = 0.5 * Height * Base = 0.5 * (55 - 25) * 15

=> C.S. = $225

Therefore, here the consumer surplus is $225.

c) Deadweight loss for monopolist = 0.5 * (Pm - Pc) * (Qc - Qm)

Here Pm = Monopoly price

Pc = Competitive price

Qc = Competitive quantity

Qm = Monopoly quantity

So,

Deadweight loss = 0.5 * (35 - 25) * (15 - 10)

=> Deadweight loss = $25

Therefore, here the deadweight loss for monopolist is $25.