Q1: Consider an economy with 10 firms having identical marginal cost c = 1 engaging in Cournot

competition. The inverse demand curve is P = 10 − Q.

Part a. (3 marks)

State the formula of Herfendahl Index. Find out its value in this economy.

Part b. (2 marks)

Find the Cournot output of one firm in this economy.

Part c. (2 marks)

Find the Cournot price in this economy.

Part d. (1 mark)

Find the value of Lerner Index in this economy.

 

Q2: Suppose firm 1 and firm 2 each produce the same product and face a market demand curve

Q = 5000 − 200P. Firm 1 has a marginal cost c1 = 6 and Firm 2 has a marginal cost c2 = 10.

The smallest indivisible unit of currency is ε > 0 which can be assumed to be very small

but strictly positive.

Part a. (2 marks)

Write down a possible pair of Bertrand equilibrium prices p1, p2 (in terms of ε).

Part b. (2 marks)

In your proposed equilibrium in Part a, write down the quantity supplied q1, q2 (in terms of

ε).

Part c. (1 mark)

What are the profits of each firm in equilibrium? (in terms of ε)

Part d. (2 marks)

For Part a, b and c, we assumed that ε is sufficiently small to make our arguments work.

What is the critical value such that if ε is strictly greater than it, your arguments would fail?

 

P.S the attached screenshots are the exact same questions. Just for your reference if you need a clearer display

1. a. 1000

b. 0.81

c. 1.82

d. 0.45

 

2. a. Firm 1 charges 7.67 and firm 2 charges 3.67

Q = 2732

c. Profit of firm 1 is 4562.44 and firm 2 incurs loss of 17,293.

 

Q-1 (a) Herfendalh index is calculated as follows:

?HHI=∑s12?+s22?+...+S102??

As there are 10 firms and each firm have same marginal cost therefore each firm have 10% of the market share.

 

The HHI index is:

?HH1=(10)2+(10)2+(10)2+(10)2+(10)2+(10)2+(10)2+(10)2+(10)2+(10)2HHI=1000?

 

(b) Cournot output is the (n/n+1) of the perfectly competitive industry.

Perfectly competitive firm produces where P = MC

?P=MC10−Q=1Q=9?

 

Output produced in Cournot model:

?QCournot?=n+1n?×QPC?QCournot?=10+110?×9QCournot?=8.18?

 

Output produced by each firm is:

?q=10QCournot??q=108.18?q=0.81?

Output produced by each firm is 0.81.

 

c. Substitute the value of Q in demand function to determine price:

?P=10−8.18P=1.82?

 

The price is 1.82.

 

d. The Lerner price index is calculated as follows:

?L=PP−MC?L=1.821.82−1?L=0.45?

 

The Lerner index is 0.45

 

Q-2 (a) The demand function is:

?Q=5000−200P1?−200P2??

 

The profit function of firm 1:

?π=QP1?−MC(Q1?)π=(5000−200P1?−200P2?)P1?−6(5000−200P1?−200P2?)π=5000P1?−200P12?−200P2?P1?−30,000−1200P1?−1200P2?To maximize profit differenciate profit function with respect to price and equate it to zeroP1?dπ?=5000−400p1?−200p2?−1200=03800−400p1?=200p2?19−2p1?=p2? (1)?

 

The profit function of firm 2:

?π=QP2?−MC(Q2?)π=(5000−200P1?−200P2?)P2?−10(5000−200P1?−200P2?)π=5000P2?−200P1?P2?−200P22?−50,000−2000P1?−2000P2?To maximize profit differenciate profit function with respect to price and equate it to zeroP1?dπ?=5000−200p1?−400p2?−2000=03000−400p2?=200p1?15−2p2?=p1? (2)?

 

Substitute equation 2 in equation 1:

?19−2(15−2p2?)=p2?19−30+4p2?=p2?11=3p2?p2?=3.67?

 

Substitute value of p₂ in equation 2:

?15−2(3.67)=p1?p1?=7.67?

 

Price charged by firm 1 is 7.67 and price charged by firm 2 is 3.67.

 

(b) The quantity supplied by each firm is:

?Q=5000−200(7.67+3.67)Q=2732?

 

Output produced by each firm is 2732

 

(c) Profit of each firm is:

?π1?=QP1?−MC(Q1?)π1?=2732(7.67)−6(2732)π1?=4562.44?

 

?π2?=QP2?−MC(Q2?)π2?=2732(3.67)−10(2732)π2?=−17,293.56?

 

Profit of firm 1 is 4562.44 and firm 2 incurs loss of 17,293.