Question

Consider a Cournot type \( n \)-firm natural spring oligopoly where the market demand for natural spring water is given by \( P(Q) = a - Q, a>0 \). The \( n \) firms are symmetric. Each firm incurs a bottling cost of \( C_i = c q_i, c>0 \) and \( a>c \). The equilibrium market price will be:

• A. \( \frac{(n + 1)}{a} + \frac{n(n + 1)}{c} \)
• B. \( \frac{a}{(n + 1)} + \frac{nc}{(n + 1)} \)
• C. \( \frac{na}{(n + 1)} + \frac{nc}{(n + 1)} \)
• D. \( \frac{(a - c)}{(n + 1)} + \frac{c}{n(n + 1)} \)

Hint:

In a Cournot oligopoly, the equilibrium price is determined by the total quantity produced by all firms, which depends on each firm's marginal cost and the number of firms in the market.

Answer 1

The Correct Option is B

This is a Cournot oligopoly problem where each firm chooses its quantity of natural spring water, assuming the other firms' quantities are fixed. The market demand is given by \( P(Q) = a - Q \), where \( Q \) is the total quantity produced by all firms, and \( Q = \sum_{i=1}^n q_i \). Each firm \( i \) has a cost function of the form \( C_i = c q_i \), where \( c \) is the constant marginal cost of production. Step 1: Reaction Function of Each Firm To find the equilibrium market price, we first derive the reaction function of each firm. The profit function for firm \( i \) is: \[ \pi_i = P(Q) q_i - C_i = (a - Q) q_i - c q_i = (a - \sum_{i=1}^n q_i) q_i - c q_i \] Firm \( i \) maximizes its profit by setting its derivative with respect to \( q_i \) equal to zero: \[ \frac{d\pi_i}{dq_i} = a - Q - q_i - c = 0 \] Simplifying: \[ a - \sum_{i=1}^n q_i - q_i - c = 0 \] \[ a - q_i - c = \sum_{i \neq j} q_j \] Step 2: Symmetric Equilibrium Since the firms are symmetric, each firm will choose the same quantity \( q_i = q \). Therefore, the total quantity produced is \( Q = nq \), and the reaction function becomes: \[ a - q - c = (n - 1)q \] Solving for \( q \): \[ q = \frac{a - c}{n + 1} \] Step 3: Equilibrium Market Price The total quantity in the market is \( Q = nq \), so the equilibrium price is: \[ P(Q) = a - Q = a - n \left( \frac{a - c}{n + 1} \right) \] Simplifying: \[ P(Q) = a - \frac{n(a - c)}{n + 1} \] Thus, the equilibrium market price is: \[ P(Q) = \frac{a}{n + 1} + \frac{nc}{n + 1} \] This matches option (B).
Final Answer: \boxed{(B) \text{ \( \frac{a}{(n + 1)} + \frac{nc}{(n + 1)} \) }}