Answer to Question #318333 in Microeconomics for hasthea

Question

1.    A homogeneous products duopoly (from firm 1 and firm 2) faces an inverse market demand function given by P = 300 - 3Q, where Q = q₁ +q₂. The cost functions for Firm 1 and Firm 2 are given as C1(q1) =100q1 and C2(q2) = 100q2.

a)    If the firms compete under Cournot assumptions; 

                i.         What is Firm 1’s profit-maximizing quantity, given that Firm 2 produces q₂ output per year?

              ii.         What is Firm 2’s profit-maximizing quantity, given that Firm 1 produces q₁ output per year?

            iii.         Sketch the equation of each firm’s reaction curve (Firm 1 and Firm 2)

             iv.         Find the Cournot equilibrium quantity per firm and price in this market.

b)    If the firms collude and agree to share total profits equally in the industry, what will be the price?

c)    Now suppose that firm 1 is a Stackelberg leader while firm 2 is a follower. Calculate the price in the industry and output of the leader.

Answer 1

What is Firm 1’s profit-maximizing quantity, given that Firm 2 produces q₂ output per year?

$P =300 -3Q$

$P= 300- 3(Q_{1}+Q_{2})$

$PQ = (300-3Q_{1}-3Q_{2})Q$

$300Q_{1}-3Q_{1}^2-3Q_{1}Q_{2}$

$MR_{1} = 300-6Q_{1}- 3Q_{2}$

To find Q1 we use the reaction function of firm 1

we get the reaction function by equating the marginal revenue equation to the marginal cost equation and making Q1 the subject.

therefore

$Q_{1} = \frac{200-6Q_{2}}{3}$

What is Firm 2’s profit-maximizing quantity, given that Firm 1 produces q₁ output per year?

Applying the same approach used in finding Q1 we can obtain the value of Q2;

Q2 is therefore;

$Q_{2} =\frac{200-3Q_{1}}{6}$

b)    If the firms collude and agree to share total profits equally in the industry, what will be the price?

If the firms decide to share profits equally, then they produce the same output

Revenue $= P =(300 -3Q)Q = 300Q-3Q^2$

$MR = 300-6Q$

$MR =MC$

$300-6Q = 100$

$6Q=200$

$Q = 33.33$

The price can be computed by keying the value of Q in the inverse demand function.

$300-3(33.33)= 200.01$

P= 2000.01