ECON 412 - Penn State University Labor Economics and Labor Markets Homework 2 Solutions

ECON 412 - Penn State University Labor Economics and Labor Markets Homework 2 Ewout Verriest Fall 2020 Problem 1 Consider a firm which produces according to the following production function: q = A( √ E + √ K) where q denotes output, E denotes the number of workers hired by the firm, K denotes capital, and A 0 is a technology parameter. The marginal product of labor is therefore given by MPE = A 2 √ E . In the short run, the firm rents 100 units of capital, at a rental cost of r = 10 dollars per unit. The firm sells its output at a market price of 3 dollars, and has to pay each employee a wage of 15 dollars. The current factory setup implies that the technology parameter A = 20. 1.1. In the short run, how many employees should the firm hire? How many units of output does the firm produce, and how much profit does it make? Show your work and make sure to verify all conditions for optimality. Answer: • Write down the short-run profit function: π(E, K0) = pA( √ E + √ K) − wE − rK0 • First step is to derive the FOC and solve for E: V MPE = w ⇐⇒ p A 2 √ E = w ⇐⇒ E ∗ = A2p 2 4w2 Plugging in K0 = 100, w = 15, p = 3 and A = 20 gives us a solution E ∗ = 3600 900 = 4 employees. (Note: E ∗ does not depend on K0 given the function we’ve assumed, but this is not always the case.) • Second, we must verify the second order condition, by checking whether the V MPE (or MPE) is decreasing at E = 4. Note that MPE = A 2 √ E is everywhere decreasing in E (since √ E is in the denominator), so the SOC is satisfied. (Or you could show the first derivative of MPE with respect to