EC2310
1. A distributor (Zorro, Ltd.) supplies soft drinks to two retail shops. The distributor can
buy the drinks at a cost-price of 𝑐 𝑍 per case and also faces fixed costs of 𝐹 𝑍 . There are
two retailers 𝑖 = 1,2, with fixed costs of 𝐹 𝑖 . Zorro, Ltd. supplies them with drinks at a
wholesale price of 𝑝𝑊 per case; the retail price 𝑝𝑅 can be determined from the (local)
market demand curve 𝑄 = 𝑞1 + 𝑞 2 = 𝐴 − 𝐵𝑝𝑅 . Retailer 𝑖 chooses how many cases 𝑞 i
to sell as a function of 𝑞 j , 𝑝W and 𝐹 i. Suppose 𝐹 𝑍 = 10; 𝐹1 = 100; 𝐹 2 = 50; 𝑐 𝑍 =
4; 𝐴 = 50; 𝐵 = 0.5.
(a) First, assume that 𝑝𝑊 is set at a level that permits both retailers to operate
profitably. Find the profit-maximising wholesale price, equilibrium retail price,
corresponding retailer quantities and profits for Zorro and the two retailers.
Hint: Find each retailer’s best reply quantity 𝑞𝑖∗ (𝑞 𝑗 |𝑝𝑊 , 𝐹 𝑖 ) to the other retailer’s
quantity. Find the retailers’ equilibrium quantities as functions of 𝑝𝑊 in the duopoly
case where both retailers operate. (9 marks)
Answer: Profit of retailer 𝑖 as a function of both retail quantities and the wholesale
price is
𝑊
(𝐴 − 𝐵𝑝𝑊 − 𝑞𝑗 − 𝑞𝑖 )𝑞𝑖
𝜋𝑖 (𝑞𝑖 , 𝑞𝑗 , 𝑝 , 𝐹𝑖 ) = − 𝐹𝑖
𝐵
𝐴 1
For simplicity, 𝛼 = 𝐵 and β= 𝐵; 𝜋𝑖 (𝑞𝑖 , 𝑞𝑗 , 𝑝𝑊 , 𝐹𝑖 ) = (𝛼 − 𝛽(𝑞𝑗 + 𝑞𝑖 ) − 𝑝𝑊 )𝑞𝑖 − 𝐹𝑖
This lets us find the best reply
𝛼 − 𝑝𝑊 − 𝛽𝑞𝑗
𝑞𝑖𝐵𝑅 (𝑞𝑗 , 𝑝𝑊 , 𝐹𝑖 ) = { if 𝜋𝑖 (𝑞𝑖𝐵𝑅 , 𝑞𝑗 , 𝑝𝑊 , 𝐹𝑖 ) ≥ 0
2𝛽
0 otherwise
From this, assuming both firms operate, we get the equilibrium retailer quantities
and retail price:
𝛼 − 𝑝𝑊
𝑞𝑖𝐶𝑁 (𝑝𝑊 ) =
3𝛽
𝑅 (𝑝 𝑊 )
𝛼 + 2𝑝𝑊
𝑝̂ =
3
Comment: this ignores second-order conditions. Note that the Cournot profit for
firm 𝑖 is
(𝛼 − 𝑝𝑊 )2
− 𝐹𝑖
9𝛽
We can add this as a constraint; if 𝐹𝑚 ≡ max{𝐹1 , 𝐹2 }, letting us write the
wholesaler’s profit maximisation problem as
𝑊
2(𝛼 − 𝑝𝑊 )
max (𝑝 − 𝑐 𝑍 ) − 𝐹𝑍 subject to
𝑝𝑊 3𝛽
𝑝𝑊 ≤ 𝛼 − 3√𝛽𝐹𝑀
The first-order condition is
𝑑𝜋𝑍 (𝑝𝑊 ) ∝ +𝑐𝑍
𝑊
= 0 ⇒ 𝑝𝑊 =
𝑑𝑝 2
Checking against the constraint, this applies if
𝛼 − 𝑐𝑍
≥ 3√𝛽𝐹𝑀
2
Or, using the supplied parameter values (so 𝛼 = 100 and β=2)
100 − 4
= 58 ≥ 3√200 = 13.41
2
1
, EC2310
The requested answers are:
𝑝𝑊 = £52 𝜋𝑍 = £758
𝑝𝑅 = £68 𝜋1 = £28
𝑞1 = 𝑞2 = 8 𝜋2 = £78
(b) At what wholesale price would Retailer 1 (the high-cost firm) just be priced out of
the market? At that price, what would be the retail quantity and price, and the
profits of Zorro and the retailers? (5 marks)
Answer: Using the constraint from part a, we get the critical wholesale price as
𝑝𝑊 = 𝛼 − 3√𝛽𝐹𝑀 = 57.574
Using the supplied parameter values gives (after rounding)
𝑝𝑊 = £57.6 𝜋𝑍 = £748
𝑝𝑅 = £71.7 𝜋1 = £0
𝑞1 = 𝑞2 = 7.1 𝜋2 = £50
For the remaining parts, assume Firm 1 is out of business, so only Firm 2 exists.
(c) What are the profit-maximising wholesale and retail prices and the corresponding
profits for Zorro and Firm 2? (5 marks)
Answer: In this case, Firm 2’s profit-maximising price is the monopoly price for a retailer
facing a marginal cost of 𝑝𝑊 (or the best reply to a rival producing 0 using the best reply
from part a.)
𝐵𝑅 (𝑝 𝑊 )
𝛼 − 𝑝𝑊 𝑊) 𝑊
𝛼 − 𝑝𝑊
𝑞2 = 𝜋𝑍 (𝑝 = (𝑝 − 𝑐𝑍 ) − 𝐹𝑍
2𝛽 2𝛽
𝑅 (𝑝 𝑊 )
𝛼 + 𝑝𝑊 𝛼 + 𝑐𝑍
𝑝 = 𝑝𝑊 =
2 2
Using the supplied parameters, this gives:
𝑝𝑊 = £52 𝜋𝑍 = £566
𝑅
𝑝 = £76 𝜋1 = £0
𝑞1 = 0; 𝑞2 = 12 𝜋2 = £238
Students might note that this would require an exclusive dealing arrangement to keep
firm 1 out, and would not be in the interest of Zorro or all firms together, due to double
marginalisation.
(d) If Zorro acquired Firm 2 (vertical integration), what would be the optimal retail price,
quantity and overall profit? (9 marks)
Answer: In this case, the profit function can be written in terms of the retail price alone:
𝜋(𝑝𝑅 ) = (𝑝𝑊 − 𝑐𝑍 )(𝐴 − 𝐵𝑝𝑅 ) − 𝐹𝑍 − 𝐹2
𝛼+𝑐𝑍
From this, we get 𝑝𝑅 = , as in part c. Using the parameters, this gives
2
𝑝𝑅 = £52 𝑄 = 24 𝜋 = £1092
(e) How could Zorro get the profit level in part d without vertical integration? (2 marks)
Answer: A vertical restraint in which Zorro paid Firm 2’s fixed cost, required Firm 2 to
charge its customers £52 and set a wholesale price of £52.
2. Assume a market where firms locate on a circle of unit circumference. The firms’ costs
include a marginal cost per unit of £3 and a fixed cost per firm of £8. There are 420
2
1. A distributor (Zorro, Ltd.) supplies soft drinks to two retail shops. The distributor can
buy the drinks at a cost-price of 𝑐 𝑍 per case and also faces fixed costs of 𝐹 𝑍 . There are
two retailers 𝑖 = 1,2, with fixed costs of 𝐹 𝑖 . Zorro, Ltd. supplies them with drinks at a
wholesale price of 𝑝𝑊 per case; the retail price 𝑝𝑅 can be determined from the (local)
market demand curve 𝑄 = 𝑞1 + 𝑞 2 = 𝐴 − 𝐵𝑝𝑅 . Retailer 𝑖 chooses how many cases 𝑞 i
to sell as a function of 𝑞 j , 𝑝W and 𝐹 i. Suppose 𝐹 𝑍 = 10; 𝐹1 = 100; 𝐹 2 = 50; 𝑐 𝑍 =
4; 𝐴 = 50; 𝐵 = 0.5.
(a) First, assume that 𝑝𝑊 is set at a level that permits both retailers to operate
profitably. Find the profit-maximising wholesale price, equilibrium retail price,
corresponding retailer quantities and profits for Zorro and the two retailers.
Hint: Find each retailer’s best reply quantity 𝑞𝑖∗ (𝑞 𝑗 |𝑝𝑊 , 𝐹 𝑖 ) to the other retailer’s
quantity. Find the retailers’ equilibrium quantities as functions of 𝑝𝑊 in the duopoly
case where both retailers operate. (9 marks)
Answer: Profit of retailer 𝑖 as a function of both retail quantities and the wholesale
price is
𝑊
(𝐴 − 𝐵𝑝𝑊 − 𝑞𝑗 − 𝑞𝑖 )𝑞𝑖
𝜋𝑖 (𝑞𝑖 , 𝑞𝑗 , 𝑝 , 𝐹𝑖 ) = − 𝐹𝑖
𝐵
𝐴 1
For simplicity, 𝛼 = 𝐵 and β= 𝐵; 𝜋𝑖 (𝑞𝑖 , 𝑞𝑗 , 𝑝𝑊 , 𝐹𝑖 ) = (𝛼 − 𝛽(𝑞𝑗 + 𝑞𝑖 ) − 𝑝𝑊 )𝑞𝑖 − 𝐹𝑖
This lets us find the best reply
𝛼 − 𝑝𝑊 − 𝛽𝑞𝑗
𝑞𝑖𝐵𝑅 (𝑞𝑗 , 𝑝𝑊 , 𝐹𝑖 ) = { if 𝜋𝑖 (𝑞𝑖𝐵𝑅 , 𝑞𝑗 , 𝑝𝑊 , 𝐹𝑖 ) ≥ 0
2𝛽
0 otherwise
From this, assuming both firms operate, we get the equilibrium retailer quantities
and retail price:
𝛼 − 𝑝𝑊
𝑞𝑖𝐶𝑁 (𝑝𝑊 ) =
3𝛽
𝑅 (𝑝 𝑊 )
𝛼 + 2𝑝𝑊
𝑝̂ =
3
Comment: this ignores second-order conditions. Note that the Cournot profit for
firm 𝑖 is
(𝛼 − 𝑝𝑊 )2
− 𝐹𝑖
9𝛽
We can add this as a constraint; if 𝐹𝑚 ≡ max{𝐹1 , 𝐹2 }, letting us write the
wholesaler’s profit maximisation problem as
𝑊
2(𝛼 − 𝑝𝑊 )
max (𝑝 − 𝑐 𝑍 ) − 𝐹𝑍 subject to
𝑝𝑊 3𝛽
𝑝𝑊 ≤ 𝛼 − 3√𝛽𝐹𝑀
The first-order condition is
𝑑𝜋𝑍 (𝑝𝑊 ) ∝ +𝑐𝑍
𝑊
= 0 ⇒ 𝑝𝑊 =
𝑑𝑝 2
Checking against the constraint, this applies if
𝛼 − 𝑐𝑍
≥ 3√𝛽𝐹𝑀
2
Or, using the supplied parameter values (so 𝛼 = 100 and β=2)
100 − 4
= 58 ≥ 3√200 = 13.41
2
1
, EC2310
The requested answers are:
𝑝𝑊 = £52 𝜋𝑍 = £758
𝑝𝑅 = £68 𝜋1 = £28
𝑞1 = 𝑞2 = 8 𝜋2 = £78
(b) At what wholesale price would Retailer 1 (the high-cost firm) just be priced out of
the market? At that price, what would be the retail quantity and price, and the
profits of Zorro and the retailers? (5 marks)
Answer: Using the constraint from part a, we get the critical wholesale price as
𝑝𝑊 = 𝛼 − 3√𝛽𝐹𝑀 = 57.574
Using the supplied parameter values gives (after rounding)
𝑝𝑊 = £57.6 𝜋𝑍 = £748
𝑝𝑅 = £71.7 𝜋1 = £0
𝑞1 = 𝑞2 = 7.1 𝜋2 = £50
For the remaining parts, assume Firm 1 is out of business, so only Firm 2 exists.
(c) What are the profit-maximising wholesale and retail prices and the corresponding
profits for Zorro and Firm 2? (5 marks)
Answer: In this case, Firm 2’s profit-maximising price is the monopoly price for a retailer
facing a marginal cost of 𝑝𝑊 (or the best reply to a rival producing 0 using the best reply
from part a.)
𝐵𝑅 (𝑝 𝑊 )
𝛼 − 𝑝𝑊 𝑊) 𝑊
𝛼 − 𝑝𝑊
𝑞2 = 𝜋𝑍 (𝑝 = (𝑝 − 𝑐𝑍 ) − 𝐹𝑍
2𝛽 2𝛽
𝑅 (𝑝 𝑊 )
𝛼 + 𝑝𝑊 𝛼 + 𝑐𝑍
𝑝 = 𝑝𝑊 =
2 2
Using the supplied parameters, this gives:
𝑝𝑊 = £52 𝜋𝑍 = £566
𝑅
𝑝 = £76 𝜋1 = £0
𝑞1 = 0; 𝑞2 = 12 𝜋2 = £238
Students might note that this would require an exclusive dealing arrangement to keep
firm 1 out, and would not be in the interest of Zorro or all firms together, due to double
marginalisation.
(d) If Zorro acquired Firm 2 (vertical integration), what would be the optimal retail price,
quantity and overall profit? (9 marks)
Answer: In this case, the profit function can be written in terms of the retail price alone:
𝜋(𝑝𝑅 ) = (𝑝𝑊 − 𝑐𝑍 )(𝐴 − 𝐵𝑝𝑅 ) − 𝐹𝑍 − 𝐹2
𝛼+𝑐𝑍
From this, we get 𝑝𝑅 = , as in part c. Using the parameters, this gives
2
𝑝𝑅 = £52 𝑄 = 24 𝜋 = £1092
(e) How could Zorro get the profit level in part d without vertical integration? (2 marks)
Answer: A vertical restraint in which Zorro paid Firm 2’s fixed cost, required Firm 2 to
charge its customers £52 and set a wholesale price of £52.
2. Assume a market where firms locate on a circle of unit circumference. The firms’ costs
include a marginal cost per unit of £3 and a fixed cost per firm of £8. There are 420
2
