Question

An investor observes that there is no car polish service available in the town that he has recently moved to and decides to open a shop offering this service. The estimated market demand for the car polish service is given by P = 300 – 5Q. The total cost of the car polish service is TC = 5Q2 +100 and the marginal cost is MC = 10Q.

If the investor engages in perfect price discrimination, how many car polish services will he offer? What would be the deadweight loss?

What will be will be the profit maximizing price and quantity if the investor is not allowed to price discriminate, and is forced to charge a uniform price to all the customers? What will be the profit, deadweight loss and consumer surplus?

Answer 1

by P = 300 – 5Q.

TC = 5Q2 +100

MC = 10Q.

a. perfect price discrimination, how many car polish services will he offer? What would be the deadweight loss?

in perfect price discrimination, seller sells each unit at a different price as per the maximum willingness to pay of different customers. the seller extracts all the consumer surplus as its own surplus. thus price charged is different for each unit and each buyer. the total unit sold is the perfectly competitive number of units.

in Perfect competition, P = MC. from P = 300 – 5Q and MC = 10Q,

300 – 5Q = 10Q

300 = 15Q

Q = 300/15 = 20

therefore sells 20 units. there is no deadweight loss as all the cosnumer surplus is extracted by the seller.

(note that in graph, corresponding to Q = 20, price would p = mc = 10*Q = 10*20 = 200)

b. profit maximizing price and quantity if the investor is not allowed to price discriminate, and is forced to charge a uniform price to all the customers? What will be the profit, deadweight loss and consumer surplus?

if single price is to be charged then profit is maximsed by selling units such that MR = MC

P = 300 – 5Q

TR = p*Q = (300 – 5Q)Q

MR = dTR/dQ = 300 - 10Q

putting MR = MC

300-10Q = 10Q

300 = 20Q

Q = 300/20 = 15

therefore, Q' = 15

P' = 300 - 5*15 = 300 - 75 = 225

therefore profit maximising single price P' = 225 and Q' = 15

profit = TR - TC = P'*Q' - (5Q'^2 +100)

profit = 225*15 - 5(15)^2 - 100 = 2150

deadweight loss = area ABC = 1/2*(20-15)*(225-200) = 0.5*5*25 = 62.5

CONSUMER surplus = area AP'E

= 1/2*(300-225)*(15) = 562.5