Question

A) Suppose a monopoly sells to two identifiably different types of customers, A and B, who are unable to practice arbitrage. The inverse demand curve for group A is PA = 29 - QA, and the inverse demand curve for group B is PB = 19 - 2QB. The monopolist is able to produce the good for either type of customer at a constant marginal cost of 3, and the monopolist has no fixed costs. If the monopolist practices group price discrimination, what are the profit maximizing prices and quantities for each type of customer?

B) Suppose a monopoly sells to two identifiably different types of customers, A and B, who are unable to practice
arbitrage. The inverse demand curve for group A is PA = 29 - QA, and the inverse demand curve for group B is PB = 19
- 2QB. The monopolist is able to produce the good for either type of customer at a constant marginal cost of 3, and the
monopolist has no fixed costs. If the monopolist is able to practice group price discrimination, what would be the values
of the price elasticities of the two groups at the profit maximizing prices?

C) The price elasticity of demand for hotdogs on a beach in a small east coast resort town is -4 in the month of May,
while in July the elasticity falls to -2. A single vendor supplies the hotdogs to the beachcombers. If the marginal cost
per hotdog is $0.60, how much should the vendor charge per hotdog in May and July?

Answer 1

A) For group A,

Inverse demand curve is P_A = 29-Q_A

Then, TR_A=P_A*Q_A = 29Q_A-Q²_A

and MR_A = dTR/dQ_A = 29 - 2Q_A

Now, MC=3

Then, for equilibrium, MR_A=MC

or, 29 - 2Q_A = 3

or, 2Q_A = 26

or, Q_A = 13

and P_A = 29 - Q_A = 29-13 = $16

Similarly, for group B,

Inverse demand curve is P_B = 19-2Q_B

Then, TR_B = P_B*Q_B = 19Q_B - 2Q_B²

and MR_B=dTR/dQ_B = 19-4Q_B

Now, MC=3

Then, for equilibrium, MR_B=MC

or, 19-4Q_B = 3

or, 4Q_B = 16

or, Q_B = 4

and P_B = 19-2Q_B = 19-2*4 = $11

B) For group A,

Q_A=29-P_A

Then, dQ_A/dP_A = -1

Now, price elasticity for group A= P_A/Q_A*dQ_A/dP_A = 16/13 *(-1) = -16/13 = -1.2

For group B,

Q_B= 19/2 - P_B/2

Then, dQ_B/dP_B = -1/2

Now, price elasticity  for group B = P_B/Q_B * dQ_B/dP_B = 11/4 * (-1/2) = -11/8 = -1.4

C) Let price be p and quantity be 1.

Given, MC=0.6

Then, for profit maximization MR=MC=0.6

where MR=dq/dp

Then, absolute price elasticity in the month of May = p/q*dq/dp

or, 4 = p/1 * MR

or, p = 4 / MR = 4/0.6 = $6.7 per hotdog

Similarly, absolute price elasticity for the month of July = p/q * dq/dp

or, 2 = p/1*MR

or, p = 2/MR = 2/0.6 = $3.33

Hence, price of a hotgog in May is $6.7 and price of a hotdog in July is $3.33.