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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:

= 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:

= p1q1(p1, p2) – c1(q1)

= p1(100 – 2p1 + p2) – 30(100 – 2p1 + p2)

∂π/∂p1 = 100 – 4p1 + p2 +60 = 0

→ p1 = 40 +p2/40

= 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:

= 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

= 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 = 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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