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Eco 384 Intermediate Economics Problem 3

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Question 3

1) The demand is P = 500−20Q where Q = Q1 +Q2, Q1 is the number of bottle of wine produces and sold per month by Aubonvin, and Q2 is the number of bottles of wine produced and sold per month by Wineryplus.

MC = 20.

The program of firm 1 is max Q1 {Π1(Q1, Q2) = P(Q1, Q2)Q1 − 20Q1}

⇒ Max Q1 (500 − 20(Q1 + Q2))Q1 − 20Q1

The FOC ∂Π1 (Q1, Q2)/∂Q1 = 500 − 40Q1 − 20Q2 − 20 = 0

Solution is: Q1 = 12 − 1 2 Q2 (i)

The ideal response by firm 1 on quantity Q2. Due to the symmetric nature of the firms, same best response function for firm 2:

Q2 = 12 – ½ Q1 (ii)

Which gives Q2c=8

Indicating that Q1c=8

The price therefore is Pc = 500 − 20Q = 500 − 40 × 8 = 180

Each firm has a profit of Π1 (Qc 1, Qc 2) = Pc Qc 1 − 20Qc 1 = (180 − 20)8 = 1280

This is the same profit that firm 2 will be able to attain

2) Stackelberg

The response from firm 2 will be Q2 = 12 – ½ Q1

Firm 1’s profit Π1 (Q1, Q2) = (500 − 20Q1 − 20(2 – ½ Q1) − 20) Q1

Maximization for firm 1 Max Q1 Π1(Q1, Q2) = (240 − 10Q1)Q1

FOC 240 − 20Q1 = 0

QS 1 = 12 > Qc 1 = 8

And QS 2 = 12 – ½ 12 = 6 < Qc 2 = 8

The price is PS = 500 − 20 × 12 − 20 × 6 = 140 < 180

The profit of each firm is ΠS1 = (140 − 20)12 = 1440 > 1280

ΠS2 = (140 − 20)6 = 720 < 1280

3. Bertrand Paradox

P = 20

Quantities, 20 = 500 − 20Q

Q = 24

Q1 = Q2 = 24

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