Question

Suppose there are two firms operating in a market. The firms produce identical products, and the total cost for each firm is given by C = 10q_i, i = 1,2, where q_i is the quantity of output produced by firm i. Therefore the marginal cost for each firm is constant at MC = 10. Also, the market demand is given by P = 106 –2Q, where Q= q₁ + q₂ is the total industry output.

The following formulas will be useful:

If market demand is given by P = a –bQ, then

• MR₁ = a – 2bq₁ – bq₂
• MR₂ = a – bq₁ – 2bq₂

For parts a-c, assume the firms choose their quantities simultaneously.

a) What is firm 1’s reaction function? (Write the equation.)

b) Determine the Nash equilibrium in quantities; that is, how much output will each firm

produce in equilibrium?

c) What will be the market price?

For parts d-g, assume that the two firms decide to collude and act like a monopolist. According to their agreement, they will maximize their total profit and split it. Each firm will produce half of the total output.

d) How much output will each firm produce?

e) What will the market price be?

f) What is the deadweight loss?

g) What is the value of the Lerner Index in this situation?

For parts h-j, assume that firm 1 sticks to the agreement and produces the quantity determined in part d). However, Firm 2 decides to cheat on the agreement, and maximizes its profit given the quantity produced by firm 1.

h) What is the residual demand facing firm 2? (Write the equation.)

i) (4 points) How much output will firm 2 produce?

j) (2 points) What will the market price be?

(Stackelberg Duopoly) For parts k-m, assume there is no collusion, and that firm 1 produces its quantity, q₁, first. Firm 2 observes Firm 1’s choice of q₁, and then Firm 2 chooses their output, q₂. Assume each firm maximizes their own profit. For this section, the following formula will be useful:

• If TR = aq -bq², then MR = a – 2bq.

k) (4 points) In order to maximize profit, how much output should firm 1 produce?

l) (2 points) Given your answer to k), how much output should firm 2 produce in order to maximize its profit?

m) (2 points) What will the market price be?

Answer 1

A).

Here the market demand curve is “P=106 – 2*q”, => the “MR” for both firms are given by.

=> “MR1 = 106 - 2*2*q1 – 2*q2” and “MR2 = 106 – 2*2*q2 – 2*q1” respectively. So, the reaction function of “firm1” is given by.

=> MR1-MC1 = 0, => (106 - 2*2*q1 – 2*q2) – 10=0, => 96 – 4*q1 – 2*q2 = 0.

=> q1 = 96/4 – q2/2, => q1 = 24 – q2/2, be the reaction function of “firm1”. Similarly, the reaction function of firm 2 is given by.

=> MR2-MC2 = 0, => q2 = 24 – q1/2, be the reaction function of “firm1”.

B).

Now, if the two firm choose their output simultaneously, => the NE will be the intersection of the two reaction function, => if we simultaneously solve these reaction function then the optimum solution will be “q1=q2=16”.

So, here the NE is “q1=q2=16”.

C).

So, here the total quantity supplied is given by, “q=q1+q2=16+16=32”, => the market price is given by.

=> P = 106 – 2*q = 106 – 2*32 = 42, => the market equilibrium price is “P=42”.

D).

Now, if they collude and act as a monopolist, => they will choose their output by “MR=MC”.

=> P = 106 - 2*q, => MR = 106 – 2*2*q, => MR = 106 – 4*q and “MC=10”.

=> at the optimum “MR=MC”, => 106 – 4*q = 10, => q = 96/4 = 24, => q=24. SO, here the optimum production is “q=24”. So, here each firm will produce “q1=q2=24/2 =12 units”.

E).

So, here the equilibrium market price is given by, “P = 106 – 2*q = 106 – 2*24 = 58”.