# LX20201204 742

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1. Let c1 represent the marginal cost for Firm 1, and Let c2 represent the marginal cost
for Firm 2. Here c1 = c2 = R10.
A Bertrand competitor maximizes profits by choosing the optimal price, taking into
account its competitor’s price. For Firm 1, this means:
π
1
= p1q1(p1, p2) c1(q1) = (p1 c1) (100 2p1 + p2)
2. We can use the same approachfind both best-response functions (profit-maximizing
price as a function of the other firm’s price), and then find the set of prices that satisfy
both equations simultaneously:
π
1
= p1q1(p1, p2) c1(q1)
= p1(100 2p1 + p2) 30(100 2p1 + p2)
∂π/∂p1 = 100 – 4p1 + p2 +60 = 0
→ p1 = 40 +p2/40
π
2
= p2q2(p1, p2) c2(q2)
= p2 (100 2p2 + p1) 10 (100 2p2 + p1)
∂π2/∂p2 = 100 – 4p2 + p1 +20 = 0
→ p1 = 30 +p1/4
So, p1 = 152/3 and p2 = 128/3
Note the firm 1 charges a higher price than firm 2. The higher marginal cost shifts its
best-response function out relative to the other firm.
3a. Find each firm’s reaction function in terms of the other firm’s price:
π
H
= pH(50 0.01pH + 0.005pN) 2000 (50 0.01pH + 0.005pN) 20,000
∂πH/∂pH = 50 – 0.02pH + 0.005pN + 20 = 0
pH = 3500 + pN/4
π
N
= pN(50 0.01pN + 0.005pH) 2000 (50 0.01pN + 0.005pH) 20,000
∂πN/∂pN = 500 – 0.02pN + 0.005pH + 20 = 0

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pN = 26,000 + pH/4
Find the Nash equilibrium as the intersection of reaction functions:
pN = 26,000 + 875 + pN/16 → pN = 16/15 (26,875) ≈ \$28,666.67
pH = 3500 + 28666.67/4 ≈ \$10,666.67
Notice that without at least three appendectomies, H would lose money and would
rather shut down (due to fixed costs).
3b. After the merger the hospitals choose prices at each hospital to maximize joint
profits:
Π = πH + πN
∂π/∂pH = 50 – 0.02pH + 0.005pN + 20 + 0.005pN 10 = 0
pH = 3000 + pN/2
∂π/∂pN = 500 – 0.02pN + 0.005pH + 20 + 0.005pH 10 = 0
pN = 25,500 + pH/2
Solve the system of equations:
pN = 25,500 + 1,500 + pN/4
pN = \$36,000
pH = \$21,000
3c. The merger raised prices at both hospitals. Before the merger, H had a quantity of
50 0.01 (10,666.67) + 0.005 (28,666.67) ≈ 87 while N had a quantity of 500 0.01
(28,666.67) + 0.005 (10,666.67) ≈ 267 so the average price was
87 × 10,666.67 +267 × 28,666.67/87 + 267 ≈ \$24,243
After the merger H has a quantity of 50 0.01 (21,000) + 0.005 (36,000) = 20 while
N has the quantity of 500 0.01 (36,000) + 0.005 (21,000) = 245 so the average price
becomes
20 × 21,000 +245 × 36,000/20 + 267 ≈ \$34,868

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1. Let c1 represent the marginal cost for Firm 1, and Let c2 represent the marginal cost for Firm 2. Here c1 = c2 = R10. A Bertrand competitor maximizes profits by choosing the optimal price, taking into account its competitor’s price. For Firm 1, this means: π1 = p1q1(p1, p2) – c1(q1) = (p1 – c1) (100 – 2p1 + p2) 2. We can use the same approach—find both best-response functions (profit-maximizing price as a function of the other firm’s price), and then find the set of prices that satisfy both equations simultaneously: π1 = p1q1(p1, p2) – c1(q1) = p1(100 – 2p1 + p2) – 30(100 – 2p1 + p2) ∂π/∂p1 = 100 – 4p1 + p2 +60 = 0 → p1 = 40 +p2/40 π2 = p2q2(p1, p2) – c2(q2) = p2 (100 – 2p2 + p1) – 10 (100 – 2p2 + p1) ∂π2/∂p2 = 100 – 4p2 + p1 +20 = 0 → p1 = 30 +p1/4 So, p1 = 152/3 and p2 = 128/3 Note the firm 1 charges a higher price than firm 2. The higher marginal cost shifts its best-response function out relative to the other firm. 3a. Find each firm’s reaction function in terms of the other firm’s price: πH = pH(50 – 0.01pH + 0.005pN) – 2000 (50 – 0.01pH + 0.005pN) – 20,000 ∂πH/∂pH = 50 – 0.02pH + 0.005pN + 20 = 0 pH = 3500 + pN/4 πN = pN(50 – 0.01pN + 0.005pH) – 2000 (50 – 0.01pN + 0.005pH) – 20,000 ∂πN/∂pN = 500 – 0.02pN + 0.005pH + 20 = 0 pN = 26,000 + pH/4 Find the Nash equilibrium as the intersection of reaction functions: pN = 26,000 + 875 + pN/16 → pN = 16/15 (26,875) ≈ \$28,666.67 pH = 35 ...
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