Reference no: EM133079155

Suppose that a market is composed of two firms that simultaneously choose quantities. Firm 1 has22 a cost function C1(q1) = q1^2. Firm 2 has a cost function C2(q2) = (2q2)^2. The inverse demand equation is

p(Q) = 100 - Q

Suppose first that firms are competing according to the Cournot model.

(a) Find the best response equation q1(q2) for firm 1.

(b) Find the best response equation q2(q1) for firm 2.

(c) Use your best response equations to mathematically solve for the equilibrium quantities q1∗, q2∗, Q∗, pi1*, pi2* equilibrium price p , and profit for each firm π1 , π2 . You may need to use the calculator for this part. Write the answers either as fractions or as numbers round to the second decimal place.

Now assume the two firms are colluding.

(d) Setup the joint profit maximization for the colluding firms (also known as the joint monopoly or cartel problem).

(e) Solve the collusion profit maximization problem for equilibrium quantities q1∗, q2∗, Q∗, equilibrium price p , and profit for each firm π1 , π2 . Suppose the firms split their joint profits in the following way: if π is joint profits, firm 1 gets 0.65π, while firm 2 gets 0.35π.

Suppose for simplicity that each firm can choose among only two levels of output: the Cournot level found in part (c), or the collusion level found in part (e).

The objective is to write down a payoff matrix that represents the single period game in which each firm chooses the Cournot or the collusion quantity. To do that, you first need to:

(f) Find each firm's profits when firm 1 chooses the Cournot quantity while firm 2 chooses the collusion quantity. (Hint: you need to first find aggregate quantity and price in order to derive profits).

(g) Find each firm's profits when firm 2 chooses the Cournot quantity while firm 1 chooses the collusion quantity.

(h) Using the profits found in parts (c), (e), (f), and (g) as payoffs, write down a payoff matrix. Round profits to the closest integer. There should be two players, firm 1 and firm 2, and two possible actions for each player, namely producing the Cournot quantity and producing the collusion quantity. Solve for the Nash equilibrium in pure strategies.

(i) Now assume that the firms play the previous game for infinitely many periods. In each period, they simultaneously choose quantities. They both discount future profits with a discount factor βwhere0≤β<1 so that total profit are π1+βπ2+β^2π3 = ...
Suppose that each firm follows the "grim trigger" strategy, that is:

(i) Each firm produces the collusion quantity as long as the other firm does the same.

(ii) Each firm produces the Cournot quantity in period t + 1 and thereafter if the other firm produces the Cournot output at time t. What is the value of β such that collusion forever is an outcome of the repeated game?