In this exercise, we will explore how changes in output

In this exercise, we will explore how changes in output and input prices affect output supply and input demand curves.
A: Suppose your firm has a production technology with diminishing marginal product throughout.
(a) With labor on the horizontal axis and output on the vertical, illustrate what your production frontier looks like.
(b) On your graph, illustrate your optimal production plan for a given p and w. True or False: As long as there is a production plan at which an isoprofit curve is tangent, it is profit maximizing to produce this plan rather than shut down.
(c) Illustrate what your output supply curve looks like in this case.
(d) What happens to your supply curve if w increases? What happens if w falls?
(e) Illustrate what your marginal product of labor curve looks like and derive the labor demand curve.
(f) What happens to your labor demand curve when p increases? What happens when p decreases?
B: Suppose that your production process is characterized by the production function x = f (ℓ) = 100ln (ℓ+1). For purposes of this problem, assume w > 1 and p > 0.01.
(a) Set up your profit maximization problem.
(b) Derive the labor demand function.
(c) The labor demand curve is the inverse of the labor demand function with p held fixed. Can you demonstrate what happens to this labor demand curve when p changes?
(d) Derive the output supply function.
(e) The supply curve is the inverse of the supply function with w held fixed. What happens to this supply curve as w changes? (Recall that ln x = y implies ey = x, where e is the base of the natural log.)
(f) Suppose p = 2 and w = 10. What is your profit maximizing production plan and how much profit will you make?