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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:

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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.

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The probabilities are obtained using Bayes updating.

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 other probabilities can be computed in a similar way.

Also notice that for the expected profits that in general, maximising expected profits is not the same as maximising the profits of playing against a firm with the expected output. In case of linear inverse demand, it probably does not make a difference, but it might make a difference in general (e.g. if the inverse demand is non-linear). It seems like the solution manual makes somewhat of an error here.

The expected profits of the solution manual seems strange. They should not be pre-multipied by $q_1^L$ ($q_1^H$ etc.)

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