Suppose that two manufacturers produce identical fireproof safes at a constant marginal cost of $90. The market inverse demand curve for fireproof safes is P = 450 - 2Q, where Q is the total output of fireproof safes produced by the two manufacturers, q₁ + q₂. The firms compete by simultaneously choosing their quantity to produce. At Nash equilibrium, what is the market price of a fireproof safe?

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To find the Nash equilibrium in a Cournot competition, we first determine the reaction functions of the two firms. Given the inverse demand function P = 450 - 2Q and the constant marginal cost of $90 for both firms, we can set up the profit function for each firm, which is $\pi = (P - MC)q$, where $P$ is the price from the inverse demand function, $MC$ is the marginal cost, and $q$ is the quantity produced by the firm.The total quantity $Q$ is the sum of quantities produced by both firms, $Q = q_1 + q_2$. Substituting the inverse demand function, we get $P = 450 - 2(q_1 + q_2)$.Each firm maximizes its profit by taking the derivative of its profit function with respect to its own quantity and setting it to zero. This process yields the reaction functions for both firms, which are functions of the quantity produced by the other firm.Solving the system of equations formed by the reaction functions of both firms gives us the equilibrium quantities $q_1^*$ and $q_2^*$. The total quantity $Q^*$ at equilibrium is the sum of these quantities.Substituting $Q^*$ back into the inverse demand function gives us the equilibrium price.For Cournot duopoly, the equilibrium quantities can be found using the formula $q_i^* = \frac{a - c}{3b}$ for linear demand curves of the form $P = a - bQ$, where $a = 450$, $b = 2$, and $c = 90$. This yields $q_i^* = \frac{450 - 90}{3*2} = 60$ for each firm, so $Q^* = q_1^* + q_2^* = 120$.Substituting $Q^* = 120$ into the inverse demand function, $P = 450 - 2*120 = 450 - 240 = 210$, gives us the equilibrium market price.

Suppose That Two Manufacturers Produce Identical Fireproof Safes at a Constant