You are the manager of a firm that produces output in two plants.The demand for your firm's product is P = 96 − 15Q,where Q = Q₁ + Q₂.The marginal costs associated with producing in the two plants are MC₁ = 6Q₁ and MC₂ = 3Q₂.How much output should be produced in plant 2 in order to maximize profits? A) 1 B) 2 C) 3 D) 4

Question

Quizplus · Accepted Answer

To maximize profits, we need to find the level of output that equates marginal revenue with marginal cost in each plant. First, we can rewrite the demand equation as follows: Q = (96 - P)/15 Substituting this into the expression for Q, we get: Q = (96 - P)/15 = Q₁ + Q₂ Simplifying, we get: Q₂ = (96 - P)/15 - Q₁ Now we can express profits as: Profit = (P - MC₁)Q₁ + (P - MC₂)Q₂ Substituting the expressions for P and MC₂, we get: Profit = (96 - 15Q - 6Q₁)Q₁ + (96 - 15Q - 3Q₂)Q₂ Taking the derivative of profit with respect to Q₁ and Q₂ and setting them equal to zero, we get: dProfit/dQ₁ = 96 - 15Q - 12Q₁ + 3Q₂ = 0 dProfit/dQ₂ = 96 - 15Q - 6Q₂ + 2Q₁ = 0 Solving these two equations simultaneously, we get Q₁ = 2 and Q₂ = 4. Therefore, the optimal level of output for plant 2 is 4 units.

You Are the Manager of a Firm That Produces Output