# BNE: Incomplete Information Cournot

Consider two risk-neutral firms that compete in quantities (Cournot). The aggregate inverse demand is given by $$P(Q) = 3-Q$$. Each firm can only observe its own cost. Find a symmetric BNE.

The constant marginal costs are distributed as follows: My Solution:

There are two types: MC is 1 (low type) and MC is 2 (high type).

First, let's define types and quantities

1. When Firm 1 is high type, it knows $$q_1 = q_1^H$$, but it believes $$q_2 = 1/6q_2^L + 1/3 q_2^H$$

2. When Firm 1 is low type, it knows $$q_1 = q_1^L$$, but it believes $$q_2 = 1/6q_2^H + 1/3 q_2^L$$

3. When Firm 2 is high type, it knows $$q_2 = q_2^H$$, but it believes $$q_1 = 1/6q_1^L + 1/3 q_1^H$$

4. When Firm 2 is low type, it knows $$q_2 = q_2^L$$, but it believes $$q_1 = 1/3q_1^L + 1/6 q_1^H$$

Second, calculate the expected payoffs:

$$EU_1^H(q_1^H,q_2) = (3-q_1^H-q_2)q_1^H -2q_1^H$$

$$EU_1^H(q_1^H,q_2) = (3-q_1^H- [1/6q_2^L + 1/3 q_2^H])q_1^H -2q_1^H$$

$$EU_1^L(q_1^L,q_2) = (3-q_1^L-q_2)q_1^L -q_1^L$$

$$EU_1^L(q_1^L,q_2) = (3-q_1^L- [1/3q_2^L + 1/6 q_2^H])q_1^L -q_1^L$$

$$EU_2^H(q_1,q_2^H) = (3-q_1-q_2^H)q_2^H -2q_2^H$$

$$EU_2^H(q_1,q_2^H) = (3-[1/6q_1^L + 1/3 q_1^H ]-q_2^H)q_2^H -2q_2^H$$

$$EU_2^L(q_1,q_2^L) = (3-q_1-q_2^L)q_2^L -q_2^L$$

$$EU_2^L(q_1,q_2^L) = (3-[1/3q_1^L + 1/6 q_1^H ]-q_2^L)q_2^L -q_2^L$$

Third, Find FOCs:

$$q_1^H = \frac{1-1/6 q_2^L-1/3q_2^H}{2}$$ $$q_1^L = \frac{2-1/3 q_2^L-1/6q_2^H}{2}$$ $$q_2^H = \frac{1-1/6 q_1^L-1/3q_1^H}{2}$$ $$q_2^L = \frac{2-1/3 q_1^L-1/6q_1^H}{2}$$

Finally, at equilibrium, $$q_1^H = q_1^L = 2/5$$ and $$q_2^H = q_2^L = 2/5$$

HOWEVER, IN SOLUTION MANUAL, its solution is very different. WHAT is my mistake? Why is it different?

SOLUTION MANUAL says:

Expected utility of low and high type firms are follow:

$$EU_1^L (q_1, q_2')=q_1^L (2-q_1^L-q_2')q_1^L$$

$$EU_1^H (q_1, q_2^{''})=q_1^H (2-q_1^H-q_2^{''})q_1^H$$

$$EU_2^L (q_1', q_2)=q_2^L (2-q_1'-q_2^L)q_2^L$$

$$EU_2^L (q_1^{''}, q_2)=q_2^H (2-q_1^{''}-q_2^H)q_2^H$$

FOCs are

$$q_1^L = (2-q_2')/2$$

$$q_1^H = (1-q_2^{''})/2$$

$$q_2^L = (2-q_1')/2$$

$$q_2^H = (1-q_1^{''})/2$$

where consider type L of firm 1. It believes that probability of 2/3, firm 2 of low type and with probability 1/3 firm 2 is of high type.So,

$$q_2' = 2/3 q_2^L + 1/3 q_2^H$$

$$q_2^{''} = 1/3 q_2^L + 2/3 q_2^H$$

$$q_1' = 2/3 q_1^L + 1/3 q_1^H$$

$$q_1^{''} = 1/3 q_1^L + 2/3 q_1^H$$

At equilibrium, $$q_1^H = q_1^L = 5/7$$ and $$q_2^H = q_2^L = 2/7$$

In the solution manual, I especially don't understand the probability part and last part (the following part:)

where consider type L of firm 1. It believes that probability of 2/3, firm 2 of low type and with probability 1/3 firm 2 is of high type.So,

$$q_2' = 2/3 q_2^L + 1/3 q_2^H$$

$$q_2^{''} = 1/3 q_2^L + 2/3 q_2^H$$

$$q_1' = 2/3 q_1^L + 1/3 q_1^H$$

$$q_1^{''} = 1/3 q_1^L + 2/3 q_1^H$$

Please explain the solution manual's solution or please share your more understandable clear solution way with me. Thank you a lot.

Let $$f_i = L$$ be the event that firm $$i$$ is low and let $$f_i = H$$ be the event that firm $$i$$ is a high type.
Assume that firm 1 knows she herself is a high type then: \begin{align*} \Pr(f_2 = H|f_1 = H) &= \frac{\Pr(f_2 = H \text{ and } f_1 = H)}{\Pr(f_1 = H)},\\ &= \frac{1/3}{1/3 + 1/6},\\ &= \frac{1/3}{1/2} = \frac{2}{3} \end{align*} Then also $$\Pr(f_2 = L|f_1 = H) = 1- \dfrac{2}{3} = \dfrac{1}{3}$$.
The expected profits of the solution manual seems strange. They should not be pre-multipied by $$q_1^L$$ ($$q_1^H$$ etc.)