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QUESTION:

Assume there are two types of products, labelled $l$ and $n$. Firms compete in the market by choosing which product to sell and then choosing the quantities. Let $Q_n$ and $Q_l$ denote the total demand of product $n$ and $l$, respectively. Let the inverse demand functions 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 $P_l$ and $P_n$ denote the prices of product $l$ and $n$, respectively, and $a$, $\gamma$, $\delta$ are all constants greater than zero. Let $q_l^i$ and $q_n^i$ denote the $i$th firm's output of product $l$ and $n$, respectively. Let $X_l^i$ and $X_n^i$ denote the output of the other firms producing product $l$ and $n$, respectively. Let $N_n$ and $N_l$ denote the number of firms selling product $n$ and $l$, 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 sub-game perfect Nash equilibria in this game.

My working so far:

I have almost solved the question but I am stuck towards the end of my working. What I've done so far is as follows. First, fix the number of firms selling each product and solve for the equilibrium quantity choices. Then, we can solve for the equilibrium number of firms making each product.

A firm choosing to sell the $l$ product earns profits: $$\pi_l = (P_l - c_n - c_p)q_l^i \ \cdots \ (1) $$ while a firm choosing to sell the $n$ product earns profits: $$\pi_n = (P_n - c_n )q_n^i \ \cdots \ (2) $$ Noting that $Q_l = q_l^i + X_l^i$ and $Q_n = q_n^i + X_n^i$ and substituting into the above and then taking first-order conditions with respect to $q_l^i$ (for $(1)$) and $q_n^i$ (for $(2)$), respectively, yields: \begin{align*} & (a+\gamma) - (1+\delta)X_l^i - Q_n - (c_n + c_p) - 2(1+\delta)q_l^i = 0 \ \cdots \ (1') \\ & a - X_n^i - Q_l - c_n - 2q_n^i =0 \ \cdots \ (2') \end{align*} From $(1')$, the best response function of a firm choosing 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 noting that $X_l^i = Q_l - q_l^i$, we have $$q_l^i = \frac{(a+\gamma) - (1+\delta)Q_l - Q_n - (c_n + c_p)}{1+\delta} \ \cdots \ (3) $$. From $(2')$, the best response function of a firm choosing 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 noting that $X_n^i = Q_n - q_n^i$, we have $$q_n^i = a-Q_n-Q_l-c_n \ \cdots \ (4)$$. Since the right-hand sides of $(3)$ and $(4)$ are constants, the first-order conditions imply that firms making the same product produce the same quantity in equilibrium. Since there are $N_n$ firms making $n$ and $N_l$ firms making $l$, therefore: \begin{align*} & Q_l = N_lq_l^i \\ & Q_n = N_nq_n^i. \end{align*} Substituting in $(3)$ and $(4)$ 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*} Solving $(5)$ and $(6)$ simultaneously for $Q_l$ and $Q_n$, we obtain the total sales of each product (with each firm selling a given 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 quantities chosen by firms selling $l$ and $n$ 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 sub-game perfect Nash equilibrium, we need an additional property, that is, no firm can have an incentive to switch and produce the other product. The profits of firms producing $l$ and $n$, 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 $N_l$ and $\pi_n^i(N_l, N_n)$ is decreasing in $N_n$. Let $N = N_l + N_n$ denote the total firms in the market, then two types of equilibria can be summarized as follows:

  1. If $\pi_l(1, N-1) < \pi_n(0, N)$, each of the $N$ firms sells $q_n^* = Q_n(0,N)/N$ of product $n$ where $Q_n$ satisfies $(8)$ and no firms sell product $l$.
  2. If $\pi_n(N-1, 1) < \pi_l(N, 0)$, each of the $N$ firms sells $q_l^* = Q_l(N,0)/N$ of product $l$ where $Q_l$ satisfies $(7)$ and no firms sell product $n$.

The intuition behind equilibrium listed in 1. above is simple to see. If $\pi_l(1, N-1) < \pi_n(0, N)$, 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 equilibrium, any firm that is producing $l$ are strictly better off by deviating to producing $n$, so every firm will produce $n$ in equilibrium. The intuition for 2. is similar.

Where I am stuck:

I am told that there is another equilibrium which is characterized as:

If the number of firms in the market and the parameter values are such that the monopoly profits from selling one product exceed the Cournot profits if all firms sell the other product, then, ignoring integer problems, equilibrium is found by setting the profits from selling the two products equal and so $N_l^*$ and $N_n^*$ satisfy $$ (1+\delta)(N_l^* + 1)(a - c_n)^2\left[(1+\delta)(N_l^*+1)(N-N_l^* + 1) - (N-N_l^*)^2 \right] = (N-N_l^* +1)(a+\gamma - c_n - c_p)^2\left[(1+\delta)(N-N_l^*+1)(N_l^*+1) - (N_l^*)^2 \right] \ \cdots \ (9) $$ $$ N_n^* = N - N_l^* \ \cdots \ (10) $$

If $\pi_l(1, N-1) \ge \pi_n(0, N)$ and $\pi_n(N-1, 1) \ge \pi_l(N, 0)$, then $N_l^*$ firms sell $q_l^* = Q_l(N_l^*, N_n^*)/N_l^*$ of $l$; $N_n^*$ sell $q_n^* = Q_n(N_l^*, N_n^*)/N_n^*$ of $n$ when equations (7), (8), (9), and (10) are satisfied.

What is the reasoning behind this equilibrium? Specifically, how do Equations (9) and (10) come about? And what exactly are $N_n^*$ and $N_l^*$ and how do they come about in the construction of the equilibrium?

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