A lumber company uses labor (L) and capital (K) to produce joint products, hardwood (H) and plywood (P). These items can be produced by one of two processes:
Process 1: 1 unit of L and 2 units of K to yield 2 units of H and 1 unit of P or
Process 2: 2 units of L and 2 units of K to yield 2 units of H and 4 units of P
Profit contribution is $2 per unit of H and $1 per unit of P. The firm has 110 units of L and 160 units of K available.
a. Formulate and solve the firm's profit-maximization problem. (Hint: Don't be distracted by the fact that the processes produce joint products. The correct decision variables are the levels of each process.)
b. Find the shadow price of labor.
c. Answer part (a), assuming the contribution of P rises to $3 per unit.