A monopolist is deciding how to allocate output between two regions, 1 and 2. Th

Question

  A monopolist is deciding how to allocate output between two regions, 1 and 2. The two regions are sufficiently far apart to make price arbitrage prohibitively costly. The demand in each region is given by:

Region 1:                                Region 2:

Q1 = 15 – P1                            Q2 = 12.5 – 0.5P2

The firm’s total cost is given by: TC = 5 + 3Q, where Q is the quantity produced (which equals the sum of Q1 and Q2). If this firm price discriminates, determine the price and quantity the firm should set in each market. What is the value of the firm’s profit? Verify the region being charged the higher price faces the less elastic demand (i.e., price elasticity of demand is smaller in absolute value).

Explanation / Answer

(a) TC = 5 + 3(Q1 + Q2) = 5 + 3Q1 + 3Q2

With price discrimination, profit is maximized when MR1 = MC1 and MR2 = MC2.

In region 1,

Q1 = 15 - P1

P1 = 15 - Q1

Total revenue, TR1 = P1 x Q1 = 15Q1 - Q12

Marginal revenue, MR1 = dTR1 / dQ1 = 15 - 2Q1

MC1 = dTC / dQ1 = 3

So,

15 - 2Q1 = 3

2Q1 = 12

Q1 = 6

P1 = 15 - 6 = 9

In region 2,

Q2 = 12.5 - 0.5P2

0.5P2 = 12.5 - Q2

P2 = 25 - 2Q2

TR2 = P2 x Q2 = 25Q2 - 2Q22

MR2 = dTR2 / dQ2 = 25 - 4Q2

MC2 = dTC / dQ2 = 3

So,

25 - 4Q2 = 3

4Q2 = 22

Q2 = 5.5

P2 = 25 - (2 x 5.5) = 25 - 11 = 14

(b)

Firm's total revenue (TR) = (P1 x Q1) + (P2 x Q2) = (6 x 9) + (14 x 5.5) = 54 + 77 = 131

TC = 5 + 3 x (6 + 5.5) = 5 + 3 x 11.5 = 5 + 34.5 = 39.5

Profit = TR - TC = 131 - 39.5 = 91.5

(c)

Price elasticity of demand, eP = (dQ / dP) x (P / Q)

In Region 1, eP1 = (dQ1 / dP1) x (P1 / Q1) = - 1 x (9/6) = - 1.5

In region 2, eP2 = (dQ2 / dP2) x (P2 / Q2) = - 0.5 x (14 / 5.5) = - 1.27

So, elasticity is lower in region 2 where a higher price is charged (14 > 9).

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.