Question

Problem 1 Consider the following Cournot's duopoly, where two identical firms compete by setting quantities. Suppose the Market demand is P = 80 - 2Q and firm's cost function is TC = 20qi, where i = 1,2 (a) Determine each firm's equilibrium quantity, profit and the market equilibrium price. Explain (b) Suppose firms decide to form a cartel. In the static (one-period) case, will they be able to sustain the cartel? Explain using the appropriate pay-off matrix (c) How will your answer to part (b) change if you consider the infinitely-repeated game? Show all your work! (Hint: you will need to find the range of values for the discount factor)

Answer 1

Solution:

Market demand: P = 80 - 2Q = 80 - 2(q1 + q2)

a) Profit function for firm i, Wi = P*qi - TCi

Wi = (80 - 2qi - 2qj)*qi - 20qi ; where j is the other firm

Wi = (60 - 2qj)*qi - 2qi²

Then, finding the best response function of firm i (using the first order condition of partial derivative being equal to 0, that is, $\partial Wi/\partial qi$ = 0)

$\partial Wi/\partial qi$ = 60 - 2qj - 2*2qi

$\partial Wi/\partial qi$ = 0 then gives us, 4qi + 2qj = 60

qi*(qj*) = (60 - 2qj*)/4. This is the best response function of firm i to optimal quantity production by firm j.

So, firm 1's best response function becomes: q1* = (60 - 2q2*)/4

And firm 2's best response function becomes: q2* = (60 - 2q1*)/4

Solving the two equations simultaneously, we get the equilibrium quantity levels as:

q1* = q2* = 10

So, equilibrium total quantity in the market, Q* = 10 + 10 = 20 units, and equilibrium price, P = 80 - 2(20) = $40

Since the cost and quantity are same for both firms for a price level, their profits will also be same. So, profits are

W1 = W2 = 40*10 - 20*10 = $200

(b) If the firms decide to form a cartel, they will act as a monopoly. Then the profit of cartel firm is

W = total revenue - total cost

W = P*(q1 + q2) - (TC1 + TC2)

Or (with similar cost it becomes), W = P*Q - 20*Q

W = (80 - 2Q)*Q - 20*Q

W = 60Q - 2Q²

Again using the first order condition: $\partial W/\partial Q$ = 0

So, $\partial W/\partial Q$ = 60 - 2*2Q

60 - 4Q = 0 gives us, Q = 60/4 = 15 units (total quantity).

q1 = q2 = Q/2 = 15/2 = 7.5 units (by each firm)

So, we have found that in cartel both firms produce 7.5 units each and in cournot, both produce 10 units each. We can easily form the required payoff matrix.