Allocate production among two firms

Question

1. Beta Inc. has two production facilities in two different towns that use labor to produce its final product. The marginal product of labor at plant 1 is MPL1 = 100/L1 and MPL2 = 50/L2.

A. Beta Inc. wants to find the best assignment of 90 workers between the two plants. What is it?

B. What if the second facility is upgraded with state-of-the-art technology? The marginal product of labor at the second facility after the upgrade is MPL2= 200/L2. Assuming Beta keeps its total payroll at 90 workers, how will this firm’s workforce be re-allocated across the two facilities and, thus, the two communities?

Answer 1

To maximize production set marginal utilities equal to each other subject to the labor budget L1 + L2 = 90.

MPL1 = 100/L1 = MPL2 = 50/L2

100L2 = 50L1

L1 = 2L2

L1 + L2 = 90 = 2L2 + L2 = 3L2

L2 = 30

L1 = 60

MPL1 = MPL2 = 5/3

Part b is solved in the same way.