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In the context of Intermediate Microeconomics II, the Cournot duopoly model describes a market with two firms producing a homogeneous good, deciding simultaneously and independently how much quantity to produce. Each firm chooses its output level assuming the output of the other firm is fixed. The price in the market is then determined by the total output produced by both firms, through the market demand function. In this problem, we are given that the second firm has an identical cost function as the first firm. We are to determine the Cournot equilibrium output for each firm. Let’s proceed step by step to determine the Cournot equilibrium: ### Assumptions * There are two firms: Firm 1 and Firm 2. * The product is homogeneous. * Firms have identical cost functions: Suppose the cost function for both firms is: $C(q_i) = c \cdot q_i$, where $c$ is the constant marginal cost and $i = 1, 2$. * The inverse demand function in the market is: $P(Q) = a - bQ$, where $Q = q_1 + q_2$, $a > 0$, $b > 0$. * Each firm maximizes its own profit, taking the other firm’s output as given. ### Step 1: Profit Function of Each Firm The revenue of firm $i$ is: $$ R_i = P(Q) \cdot q_i = (a - b(q_1 + q_2)) \cdot q_i $$ The cost function of firm $i$: $$ C(q_i) ________ _________ ____ _________ ______ ____ __________ ________ _____ __________.
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