Exercise Set 7
Profit Maximization by a Multi-Product Firm

1. Consider a firm that produces two goods. The demand for each good, and the firm's cost function are given:

Demand for Good 1: Q₁ = 40 - 2P₁ - P₂

Demand for Good 2: Q₂ = 35 - P₁ - P₂

Cost function: C(Q₁,Q₂) = Q₁² + 2Q₂² + 10

The firm wishes to maximize profits by choosing the optimal values of Q₁ and Q₂.

1. Rewrite the demand functions as P₁ = f(Q₁,Q₂) and P₂ = g(Q₁,Q₂).
2. Obtain the revenue functions R₁(Q₁,Q₂) and R₂(Q₁,Q₂).
3. Obtain the profit function.
4. Obtain the first-order and second-order derivatives of the profit function.
5. Write down the FOCs.
6. Solve for the optimal values of the output levels.
7. Check the SOC. [Use the Hessian.]
8. Obtain the maximum value of the profit function.

2. Suppose, ceteris paribus, the demand for Good 1 changes to: Q₁ = 50 - 2P₁ - P_2.

1. Obtain the firm's optimal values of output and price for each good.
2. What is the firm's maximum profit?