Final answer:
To solve for the firms' reaction functions, we use the market demand function and firms' cost functions to derive their respective profit functions. By differentiating these with respect to each firm's output and setting it equal to zero, we find the optimal output levels that are functions of the other firm's output. This results in the reaction functions for both firms.
Step-by-step explanation:
To find the reaction functions for the two firms producing an identical good, we need to consider how each firm's output decision affects the overall market quantity and thus the price. We know that market demand is given by the inverse demand function p(Y)=14−Y, where Y=y₁+y₂ is the total market quantity and p is the price. The cost function for both firms is given by Cᵢ(yᵢ) = c yᵢ, where c₁=c₂=2. The profit for firm i is the revenue minus the cost, which can be expressed as πᵢ(yᵢ, y₂) = p(Y)yᵢ - Cᵢ(yᵢ). Substituting the demand function and the cost given, this becomes πᵢ(yᵢ, y₂) = (14 - Y)yᵢ - 2yᵢ.To derive each firm's reaction function, we need to calculate the derivative of the profit function with respect to the firm's output and set it to zero to find the profit-maximizing output level, given the output level of the other firm. For firm 1, we differentiate the profit function with respect to y₁ and set the result equal to zero, solving for y₁ in terms of y₂. This will give us firm 1's reaction function. Similarly, we do the same for firm 2. The process involves some algebra and results in the reaction functions, where each firm's optimal output depends on the output of the competitor.Conclusion Upon solving for optimal outputs, we obtain the reaction functions for both firms, which describe how each firm's output level is a function of its competitor's output level. These reaction functions are crucial in predicting behavior in markets where firms are interdependent and strategically interact with one another.