A monopolist is deciding how to allocate output between two regions, 1 and 2. Th
ID: 1212043 • Letter: A
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).