In our development of producer theory, we have found it convenient to assume that the production technology is homothetic.
A: In each of the following, assume that the production technology you face is indeed homothetic. Suppose further that you currently face input prices (wA,rA) and output price pA — and that, at these prices, your profit maximizing production plan is A = (ℓA,kA,xA).
(a) On a graph with ℓ on the horizontal and k on the vertical, illustrate an isoquant through the input bundle (ℓA,kA). Indicate where all cost minimizing input bundles lie given the input prices (wA,rA).
(b) Can you tell from what you know whether the shape of the production frontier exhibits increasing or decreasing returns to scale along the ray you indicated in (a)?
(c) Can you tell whether the production frontier has increasing or decreasing returns to scale around the production plan A = (ℓA,kA,xA)?
(d) Now suppose that wage increases to w′. Where will your new profit maximizing production plan lie relative to the ray you identified in (a)?
(e) In light of the fact that supply curves shift to the left as input prices increase, where will your new profit maximizing input bundle lie relative to the isoquant for xA?
(f) Combining your insights from (d) and (e), can you identify the region in which your new profit maximizing bundle will lie when wage increases to w′?
(g) How would your answer to (f) change if wage fell instead?
(h) Next, suppose that, instead of wage changing, the output price increases to p′. Where in your graph might your new profit maximizing production plan lie? What if p decreases?
(i) Can you identify the region in your graph where the new profit maximizing plan would lie if instead the rental rate r fell?
B: Consider the Cobb-Douglas production function f (ℓ,k) = Aℓαkβ with α,β > 0 and α + β<1.
(a) Derive the demand functions ℓ(w,r,p) and k(w,r,p) as well as the output supply function
x(w,r,p).
(b) Derive the conditional demand functions ℓ(w,r,x) and k(w,r,x).
(c) Given some initial prices (wA,rA,pA), verify that all cost minimizing bundles lie on the same ray from the origin in the isoquant graph.
(d) If w increases, what happens to the ray on which all cost minimizing bundles lie?
(e) What happens to the profit maximizing input bundles?
(f) How do your answers change if w instead decreases?
(g) If instead p increases, does the ray along which all cost minimizing bundles lie change?
(h) Where on that ray will the profit maximizing production plan lie?
(i) What happens to the ray on which all cost minimizing input bundles lie if r falls? What happens to the profit maximizing input bundle?