Question

1. Consider two firms facing the demand curve P=50-5Q where Q = Q1 + Q2. The firms’ cost functions are C1(Q1) = 20+10Q1 and C2(Q2) = 10+12Q2.
Suppose both firms entered the industry.
a) What is the joint profit-maximizing level of output?
b) What is total production (Q1+Q2) at the joint profit-maximizing level?
c) What is firm 1's output if they behave non-cooperatively (Cournet Model)?
d) What is firm 2's output if they behave non-cooperatively (Cournet Model)?
e) How much should firm 1 be willing to pay Firm 2 if collusion is illegal but a takeover is not?

Answer 1

a) If both firms enter the market, and they collude, they will face a marginal revenue curve with twice the slope of the demand curve:

MR= 50-10Q

Setting marginal revenue equal to marginal cost (the marginal cost of Firm 1, since it is lower than that of Firm 2) to determine the profit-maximizing quantity,

Q:50-10Q= 10, or Q= 40/10 = 4

b) Since the two firms have different cost functions, it will not be optimal for them to split the output evenly between them.

c), d) :

In the Cournot model, Firm 1 takes Firm 2’s output as given and maximizes profits.

The profit function derived becomes p1= (50 -5Q1-5Q2 )Q1-(20 + 10Q1 ), or

p=40Q1-5Q1 square -5Q1Q2-20

Setting the derivative of the profit function with respect to Q1to zero, we find Firm 1’s reaction function:

dp/d1Q =40-10Q1-5Q2=0,or

Q1=4-(Q2/2 )

Similarly, Firm 2’s reaction function is

Q2=3.8-(Q1/2)

To find the Cournot equilibrium, we substitute Firm 2’s reaction function into Firm 1’s reaction function:

Q1=4-(1/2)3.8-(Q1 /2) or

Q1=2.8

Substituting this value for Q1 into the reaction function for Firm 2, we find Q2= 2.4.