Answer to Question #221308 in Microeconomics for sauni

Question

Suppose the inverse demand function for two Cournot dupolists is given by P= 10 – (Q1+Q2) and their cost are zero.

a) What is each firm marginal revenue?

b) What are the reaction function for the two firms

c) What are the Cournot equilibrium output

d) What is the equilibrium price?

Answer 1

The solution

$P= 10- (Q1+Q2)$

$P= 10- Q1-Q2$

The total revenue for firm 1 is obtained by multiplying the price by quantity Q.

$TR1= [10-Q1-Q2]Q1$

$= 10Q1-Q1^2 - Q2Q1$

The marginal revenue is obtained by computing the first derivative of TR₁

with respect to Q

Marginal revenue is;

$\frac{\delta TR1}{\delta Q1}$ $= 10- 2Q1- Q2$

To calculate the reaction function for firm 1, we make Q1 the subject of the formula

The reaction function is therefore;

$Q1= 5- 0.5Q2$

For firm 2

$TR2= [10-(Q1+Q2)]Q$$2$

$= 10Q2-Q1Q2- Q2^2$

Marginal revenue will be computed as by computing the first derivative of the total revenue with respect to the Q

$\frac{\delta TR2}{\delta Q2}$=$10-Q1-2Q2$

The reaction function of firm 2 is;

$Q2= 5- 0.5Q1$

Calculating the equilibrium output

Replace Q2 into the reaction function of firm one and vice versa

$Q1= 5- 0.5(5-0.5Q1)$

$Q1= 5- 2.5+ 0.25Q1$

$Q1-0.25Q1=2.5$

$0.75Q1 = 2.5$

$Q1 = \frac{2.5}{0.75}$ =$3.33333$

$Q2= 5- 0.5(3.3333)= 3.33335$

Equilibrium Price

$P = 10-( 3.3333+3.33335)$

$P =3.33335$