Question

Consider a firm that produces a single output with a single input, labor, using 2 different plants. Denote by L1 the assignment of labor input into plant 1 and by L2 the assignment of labor input
into plant 2. Plant 1’s production function is F1 (L1) = 4√L1, for L1 ≥ 0. Plant 2’s production
function is F2(L2) = 8√L2, for L2 ≥ 0.
1. State the average product function of each plant as a function of the labor assignment. Denote
them by AP1(L1) and AP2(L2).
2. State the marginal product function of each plant as a function of the labor assignment.
Denote them by MP1(L1) and MP2(L2).
3. Define total quantity produced for a given labor assignment by Q(L1, L2) = F1(L1) + F2(L2). Suppose the firm has a total of 100 units of labor available, L = 100. It can freely assign them across the two plants subject to L1 + L2 = L. In a graph show total output produced for different choices of L1 ∈ [0, 100] where L2 = L − L1.
4. For L = 100, find the input assignment, (L1*, L2*), that maximizes total output, Use the insight that MP1(L1*) = MP2(L2*).
5. We want to derive the firm’s efficient production function frontier for any total labor input L ≥ 0. Call it F (L). It is the greatest output that can be produced with L units of workers.
(a) For a given L, find the input assignment (L1*(L),L2*(L) ) that maximizes total output. Verify that (L1*(100),L2*(100) matches your answer to question 4.
(b) Define F (L) = F1(L1∗(L)) + F2(L2∗(L)). Show that it can be written in the form F (L) = A√L. What is the value of A?

Answer 1