Question

Suppose 2 firms compete in a market for widgets. Each of them produces identical widgets. Each firm incurs a cost of $10 per widget produced. The two firms simultaneously (and independently) decide how many widgets to produce. The inverse demand for widgets is given by P=100−3Q, where Q=q1+q2, where q1denotes the output of firm 1 and q2 denotes the output of firm 2.

What is firm 1's best response to q2=2?

What is firm 1's best response to q2=20?

What is firm 2's output in the Nash equilibrium?

Answer 1

In this case, firms' decisions are not independent of their rivals' actions. This is a case of a Cournot simultaneous move game where firms face a common aggregate demand curve in the market, and they simultaneously select their outputs, given their marginal cost functions, as the best response corresponding to the opponents' strategy.

Here, we have the given inverse demand function in the market as P = 100 - 3Q. We can rewrite Q = q₁ + q₂, where q₁ represents firm 1's output, and q₂ represents firm 2's output.

Hence, P = 100 - 3(q₁ + q₂)

Since, the firms are perfectly symmetric to each other, with the same MC function, we need only find one firm's optimal output. Hence, for firm 1's optimal output:

P = (100 - 3q₂) - 3q₁

Remember, a firm's total revenue, which we denote by R, is calculated by multiplying price with the output produced. Therefore,

R = P.q₁ = (100 - 3q₂)q₁ - 3q₁²

or R = 100q₁ - 3q₂q₁ - 3q₁² --------------- [1]

We can calculate the marginal revenue of the firm by differentiating [1] with respect to q₁

MR = 100 - 3q₂ - 6q₁

In equilibrium, we have Marginal cost (given as $10) equated with Marginal Revenue, and thus, we get

100 - 3q₂ - 6q₁ = 10

or 90 - 6q₁ - 3q₂ = 0

or 30 - 2q₁ - q₂ = 0

Therefore, q₁ = 15 - 0.5q₂ is the best response function of firm 1 in response to firm 2 in the market. Since we noted earlier that firm 1 and firm 2 are perfectly symmetric firms, we can easily infer the following:

q₂ = 15 - 0.5q₁ ----------- [2]

(this can be verified easily by calculating in a similar fashion the MR of firm 2, and equating with MC =10)

(a) Firm-1's best response to q₂ = 2 is q₁ = 15 - 0.5(2) = 15 -1 = 14 units

(b) Firm-1's best response to q₂ = 20 is q₁ = 15 - 0.5(20) = 15 - 10 = 5 units

(c) To calculate Nash equilibrium, we must first understand what a Nash equilibrium is. A Nash equilibrium is an n-tuple strategy profile (a₁*,......, a_n*) where a_i* for all i = 1,..., n is the best response strategy. That is, in a Nash equilibrium, all players are playing their best response strategies.

To compute the Nash equilibrium outcome, we substitute the value of q₁ in [2] and compute the value of q₂ in the best response function, which is the Nash equilibrium output:

q₂ = 15 - 0.5(15 - 0.5q₂)

q₂ = 15 - 7.5 + 0.25q₂

0.75q₂ = 7.5

q₂ = 10 = q₁ in Nash equilibrium