Question

2004 Suppose that two firms, 1 and 2, compete in a marketwith market demand being P-A-Q.where O is total output of the two firms, and A is the realization of a random variable 10 with probability 8 with probability 1-a Both firms have marginal cost that is equal to 0. Each rm simutanoously chooses its own outpot to maximize profit. Before production strts bowever, the realized value of A is privately learned by firm 1, bus this value remains unknown to fim 2. The game is common know ledge. Write down each firms, optimization oblet i) Derive the firms" reaction functions Bi) Find the Bayesian Nash Equilibeium(BNE) this game

Answer 1

P = A-Q

So Total Revenue will be Q(A-Q) = AQ- Q^2

Marginal Revenue will be A- 2Q

Since probability of A being 10 is $\alpha$ and A being 8 is 1- $\alpha$

So expected marginal revenue will  $\alpha$(10-2Q) + (1-$\alpha$) (8-2Q) = 10$\alpha$- 2Q$\alpha$ + 8- 2Q -8$\alpha$+2Q$\alpha$ = 2$\alpha$ -2Q +8

So Marginal Revenue function is 2$\alpha$ -2Q +8

So optimization problem for each firm will be profit maximization which MR = MC

So 2$\alpha$ -2Q +8 = 0 will be optimization problem.

ii) First firm will produce quantity Q1 and second firm will produce quantity Q2

So for first firm , Total Revenue = Q1(A- Q1-Q2)

A can be 10 for $\alpha$ probability and can be 8 for 1-$\alpha$ probability

so Total Revenue = Q1(10$\alpha$ +8(1-$\alpha$) -Q1-Q2)

Or 10$\alpha$Q1+ 8Q1 -8$\alpha$Q1 -Q1^2 - Q1Q2 Or 2$\alpha$Q1 - Q1^2 -Q1Q2

So MR will be its derivative which will be 2$\alpha$- 2Q1 -Q2 which shoul be 0 for profit maximization as MC is 0

So Q2 = 2$\alpha$- 2Q1 (1)

Same way for second company MR will be 2$\alpha$- 2Q2 -Q1 which should be 0 for profit maximization

Replacing Q2 by Q1 from using (1) equation

2$\alpha$ -2(2$\alpha$-2Q1) - Q1 = 0

OR 2$\alpha$- 4$\alpha$+ 4Q1- Q1 = 0

Or 3Q1= 2$\alpha$

Or Q1 = 2$\alpha$/3

Hence by putting value of Q1 in (1) equation

Q2 = 2$\alpha$- 2Q1 or 2$\alpha$- 2(2$\alpha$/3) Or 2$\alpha$- 4$\alpha$/3 = 2$\alpha$/3

iii Bayesian Nash Equilibrium will set when both firms will produce 2$\alpha$/3 units

IF A = 10, Total Payoff of each firm will 2$\alpha$/3 (10 - 4$\alpha$/3) = 20$\alpha$/3 - 8($\alpha$^2)/9

For A = 8, toal payoff of each firm will be 16$\alpha$/3 - 8($\alpha$^2)/9

Thanks