**QUESTION:**

Suppose there are two types of products, labeled $ l $ Y $ n $. Companies compete in the market by choosing which product to sell and then choosing the quantities. Leave $ Q_n $ Y $ Q_l $ denote the total product demand $ n $ Y $ l $, respectively. Let the functions of reverse demand be given by:

begin {align *}

& P_l (Q_l, Q_n) = (a + gamma) – Q_n – (1+ delta) Q_l \

& P_n (Q_l, Q_n) = a – Q_n – Q_l

end {align *}

where $ P_l $ Y $ P_n $ denote product prices $ l $ Y $ n $, respectively, and $ a $, $ gamma $, $ delta $ All constants are greater than zero. Leave $ q_l ^ i $ Y $ q_n ^ i $ denote the $ i $The product output of the firm. $ l $ Y $ n $, respectively. Leave $ X_l ^ i $ Y $ X_n ^ i $ Denote the output of the other companies producing the product. $ l $ Y $ n $, respectively. Leave $ N_n $ Y $ N_l $ denote the number of companies that sell products $ n $ Y $ l $, respectively. Let the marginal cost of producing the $ l $ product be $ c_n + c_p $ and the marginal cost of producing the $ n $ product be $ c_n $. Find all the perfect subgame Nash equilibria in this game.

**My work so far:**

I have almost solved the question, but I am stuck towards the end of my work. What I have done so far is the following. First, set the number of companies that sell each product and solve the equilibrium amount options. Then, we can solve the equilibrium number of the companies that make each product.

A company that decides to sell the $ l $ Product earns profits:

$$ pi_l = (P_l – c_n – c_p) q_l ^ i cdots (1) $$

while a company decides to sell the $ n $ Product earns profits:

$$ pi_n = (P_n – c_n) q_n ^ i cdots (2) $$

Noting that $ Q_l = q_l ^ i + X_l ^ i $ Y $ Q_n = q_n ^ i + X_n ^ i $ and replacing the former and then taking first-order conditions with respect to $ q_l ^ i $ (for $ (1) $) Y $ q_n ^ i $ (for $ (2) $), respectively, produces:

begin {align *}

& (a + gamma) – (1+ delta) X_l ^ i – Q_n – (c_n + c_p) – 2 (1+ delta) q_l ^ i = 0 cdots (1 & # 39;)

& a – X_n ^ i – Q_l – c_n – 2q_n ^ i = 0 cdots (2 & # 39;)

end {align *}

Since $ (1 & # 39;) $, the best response function of a company that decides to sell. $ q_l ^ i $ of product $ l $ is given by

$$ q_l ^ i = frac {(a + gamma) – (1+ delta) X_l ^ i – Q_n – (c_n + c_p)} {2 (1+ delta)} $$

but noticing that $ X_l ^ i = Q_l – q_l ^ i $, we have

$$ q_l ^ i = frac {(a + gamma) – (1+ delta) Q_l – Q_n – (c_n + c_p)} {1+ delta} cdots (3) $$.

Since $ (2 & # 39;) $, the best response function of a company that decides to sell. $ q_n ^ i $ of product $ n $ is given by

$$ q_n ^ i = frac {a-X_n ^ i – Q_l – c_n} {2} $$

but noticing that $ X_n ^ i = Q_n – q_n ^ i $, we have

$$ q_n ^ i = a-Q_n-Q_l-c_n cdots (4) $$.

From the right sides of $ (3) $ Y $ (4) $ they are constant, first order conditions imply that companies that make the same product produce the same amount in equilibrium. Since there are $ N_n $ companies that do $ n $ Y $ N_l $ companies that do $ l $, Thus:

begin {align *}

& Q_l = N_lq_l ^ i \

& Q_n = N_nq_n ^ i.

end {align *}

Substituting in $ (3) $ Y $ (4) $ we have the following

begin {align *}

& Q_l = N_l left ( frac {(a + gamma) – (1+ delta) Q_l – Q_n – (c_n + c_p)} {1+ delta} right) cdots (5)

& Q_n = N_n left (a-Q_n-Q_l-c_n right) cdots (6)

end {align *}

Resolving $ (5) $ Y $ (6) $ simultaneously for $ Q_l $ Y $ Q_n $, we obtain the total sales of each product (with each company selling a certain product, selling the same amount):

begin {align *}

& Q_l (N_l, N_n) = lambda N_l left ((N_n + 1) (a + gamma – c_n – c_p) – N_n (a-c_n) right) cdots (7) \

& Q_n (N_l, N_n) = lambda N_n left ((1+ delta) (N_l + 1) (a-c_n) – N_l (a + gamma – c_n – c_p) right) cdots ( 8)

end {align *}

where

$$ lambda = frac {1} {(1+ delta) (N_l + 1) (N_n + 1) – N_lN_n} $$. Therefore, in equilibrium, the amounts chosen by the companies that sell $ l $ Y $ n $ are, respectively:

begin {align *}

& q_l (N_l, N_n) = frac {Q_l (N_l, N_n)} {N_l} \

& q_n (N_l, N_n) = frac {Q_n (N_l, N_n)} {N_n}

end {align *}

To find the perfect Nash equilibrium of the subgame, we need additional ownership, that is, no company can have an incentive to change and produce the other product. The benefits of the production companies. $ l $ Y $ n $, respectively, are given by

begin {align *}

& pi_l ^ i (N_l, N_n) = left[a+gamma – Q_n(N_l, N_n) – (1+delta)Q_l(N_l, N_n) – c_n – c_p right]q_l (N_l, N_n) \

& pi_n ^ i (N_l, N_n) = left[a – Q_n(N_l, N_n) – Q_l(N_l, N_n) – c_n right]q_n (N_l, N_n).

end {align *}

One can show that $ pi_l ^ i (N_l, N_n) $ is decreasing in $ N_l $ Y $ pi_n ^ i (N_l, N_n) $ is decreasing in $ N_n $. Leave $ N = N_l + N_n $ denotes the total number of companies in the market, then two types of equilibria can be summarized as follows:

- Yes $ pi_l (1, N-1) < pi_n (0, N) $, each of the $ N $ companies sell $ q_n ^ * = Q_n (0, N) / N $ of product $ n $ where $ Q_n $ satisfy $ (8) $ and no company sells product $ l $.
- Yes $ pi_n (N-1, 1) < pi_l (N, 0) $, each of the $ N $ companies sell $ q_l ^ * = Q_l (N, 0) / N $ of product $ l $ where $ Q_l $ satisfy $ (7) $ and no company sells product $ n $.

The intuition behind the balance that is listed in 1. is easy to see. Yes $ pi_l (1, N-1) < pi_n (0, N) $, then we have

$$ underbrace { pi_l (N, 0) < cdots < pi_l (1, N-1)} _ { text {Since} pi_l ^ i (N_l, N_n) text {is decreasing in } N_l} < underbrace { pi_n (0, N) < cdots < pi_n (N-1, 1)} _ { text {Since} pi_n ^ i (N_l, N_n) text { is decreasing in} N_n} $$

Therefore, in balance, any company that is producing. $ l $ they are strictly better by diverting to production $ n $, so that each firm produces $ n $ in equilibrium. Intuition for 2. is similar.

**Where I'm stuck**

They tell me that there is another balance that is characterized by:

*If the number of firms in the market and the values โโof the parameters are such that the profits of the monopoly of selling a product exceed the profits of Cournot if all the firms sell the other product, then, ignoring the whole problems, the balance is found when establishing the profits from the sale of the product. two equal products and so $ N_l ^ * $ Y $ N_n ^ * $ satisfy*

$$

(1+ delta) (N_l ^ * + 1) (a – c_n) ^ 2 left[(1+delta)(N_l^*+1)(N-N_l^* + 1) – (N-N_l^*)^2 right] = (N-N_l ^ * +1) (a + gamma – c_n – c_p) ^ 2 left[(1+delta)(N-N_l^*+1)(N_l^*+1) – (N_l^*)^2 right] cdots (9) $$

$$ N_n ^ * = N – N_l ^ * cdots (10) $$

*Yes $ pi_l (1, N-1) ge pi_n (0, N) $ Y $ pi_n (N-1, 1) ge pi_l (N, 0) $, so $ N_l ^ * $ companies sell $ q_l ^ * = Q_l (N_l ^ *, N_n ^ *) / N_l ^ * $ of $ l $; $ N_n ^ * $ to sell $ q_n ^ * = Q_n (N_l ^ *, N_n ^ *) / N_n ^ * $ of $ n $ when equations (7), (8), (9) and (10) are met.*

What is the reasoning behind this balance? Specifically, how are equations (9) and (10) produced? And what are they exactly $ N_n ^ * $ Y $ N_l ^ * $ And how are they produced in the construction of equilibrium?