Question

Suppose a software monopolist faces two markets for its software, students and professionals. The demand curve of professionals is given by QP = 200 − 2PP and the demand curve by students is given by QS = 150 − 3PS. The firm’s cost function is C(Q) = 400 + 5Q.

(a) If the firm can price discriminate, what price should it charge in each market to maximize profits? How much profit does it earn?

(b) If the firm cannot price discriminate, what price should it charge? Verify that it sells to both markets at this price. How much profit does it earn? Hint: If the firm cannot price discriminate, this means it must treat the two markets as a single combined market.

Answer 1

Solution:

Demand curve of professionals: QP = 200 - 2PP, so PP = 100 - 0.5QP

Demand curve of students: QS = 150 - 3PS, so PS = 50 - (1/3)*QS

Total cost, TC = 400 + 5Q

So, marginal cost, MC = = 5

a) Total revenue from professionals, TR = (100 - 0.5*QP)*QP = 100QP - 0.5*QP²

So, marginal revenue, MR = = 100 - QP

Profit maximizing under price discrimination occurs where the marginal cost equals the marginal revenue for each market.

Thus, 100 - QP = 5, QP = 95

Price charged by professionals is PP = 100 - 0.5*95 = $52.5

Total revenue for students is TR = (50 - (1/3)QS)*QS = 50QS - (1/3)*QS²

So, marginal revenue, MR = = 50 - (2/3)*QS

Profit maximizing under price discrimination occurs where the marginal cost equals the marginal revenue for each market.

Thus, 50 - (2/3)*QS = 5, QS = (50 - 5)*3/2 = 67.5

Price charged by students is PS = 50 - (1/3)*67.5 = $27.5

And total profit = total revenue - total cost

Profit = PP*QP + PS*QS - C(QP + QS)

Profit = 52.5*95 + 27.5*67.5 - (400 + 5*(95 + 67.5))

Profit = 4987.5 + 1856.25 - 1212.5 = $5,631.25

b) Two markets as single combined market:

Q = QS + QP

Q = (200 - 2P) + (150 - 3P) (P is the common price in both markets)

Q = 350 - 5P

So, P = (350 - Q)/5 = 70 - 0.2Q

So, total revenue now is TR = (70 - 0.2Q)Q

TR = 70Q - 0.2Q²

And marginal revenue, MR = = 70 - 0.4Q

MR = MC

70 - 0.4Q = 5

Q = (70 - 5)/0.4 = 162.5

Price charged is P = 70 - 0.2*162.5 = $37.5

From inverse demand functions obtained in the beginning, we see that the maximum willingness to pay for each group is $100 (professionals) and $50 (students), which both exceed $37.5, thus both markets will be served.

Profit earned = P*Q - TC(Q)

Profit = 37.5*162.5 - (400 + 5*162.5)

Profit = 6093.75 - 1212.5 = $4,881.25