Question

Consider a Cournot-competition under incomplete information. Two firms decide their quantity of production simultaneously. The market price P is determined by P = 100 − (q1 + q2). Assume that firm 1’s per-unit cost is commonly known at zero. On the other hand, firm 2’s per-unit cost is private information and is either at 0 or at 2. Suppose in the firm 1’s belief, the probability of c2 = 0 is 1 3 and the probability of c2 = 2 is 2 3 . Find the Bayesian Nash equilibrium in this game.

Answer 1

General method of solving for Bayesian Nash equillibrium in games of incomplete information :

inverse market demand: P=a-Q if a>Q, P=0 otherwise.

Each firm decides on the quantity to sell : q1 and q2 respectively

total market demand : Q= q1+q2

marginal cost for each firm : c1 and c2 respectively

c1 is common knowledge, but c2 is known only by firm 2

Firm 1 believes that c2 is “high” cH with probability p and “low” cL with probability (1-p)

Firm 1's belief about firm 2's cost is common knowledge

Both firms seek to maximize profits

Firm 2's profit function , if it produces q2 is:

π2 = revenue - cost

π2 = (P-c2)q2 = [a-(q1+ q2)]q2 - c2q2

First order conditions:

dπ2 / dq2= a - 2q2 - q1 - c2 = 0

q2 = (a-c2- q1)/2

If Firm 2's cost is high qH2 = (a-cH- q1)/2

If Firm 2's cost is low qL2 = (a-cL- q1)/2 {Best Response of Firm 1}

Firm 1's expected profits, if it produces q1 are:

π1 = expected revenue - cost

π1 = p[a-(q1+ qH2)]q1 + (1-p)[a-(q1+ qL2 )]q1 - c1q1

First order conditions:

dπ1 / dq1 = p(a - 2q1 - qH2 ) - (1-p)( a - 2q1 - qL2 ) - c1 = 0

q1= p(a - c1 - qH2 )/2 + (1-p)(a - c1 - qL2 )/2 {Best Response of Firm 1}

we know,

If Firm 2's type is “high”: qH2=(a-cH-q1)/2

If Firm 2's type is “low”: qL2=(a-cL-q1)/2

putting these in Best Response of Firm 1

q1 = (a - 2c1 + pcH + (1-p)cL)/3

now putting q1 back in Best Response of Firm 2

qH2 = (a-2cH+c1)/3 + (1-p)(cH- cL)/6

qL2 = (a-2cL+c1)/3 - p(cH- cL)/6

in the given question:

P = 100 - (q1+q2)

a = 100

c1 = 0

c2 = 2 with p = 2/3

c2 = 0 with 1-p =1/3

q1 = [100 - (2x0) + (2/3 x 2) + (1/3 x 0)] / 3 = 33.77

qH2 = (100 - (2x2) + 0)/3 + [1/3(2-0)]/6 = 32.11

qL2 = (100 - (2x0) + 0)/3 + [2/3(2-0)]/6 = 33.55

Bayesian Nash Equillibrium:

Firm 1: q1 = 33.77

Firm 2: qH2 = 32.11 and  qL2 = 33.55