We will now re-consider the problem from exercise 12.5 but will focus on the two-step optimization method

Question:

We will now re-consider the problem from exercise 12.5 but will focus on the two-step optimization method that starts with cost minimization.
A: Suppose again that you face a production process such as the one depicted in Graph 12.9.

(a) What do the horizontal slices—the map of isoquants—of this production process look like? Does this map satisfy our usual notion of convexity as “averages better than extremes”?

(b) From this map of isoquants, how would you be able to infer the vertical shape of the production frontier? Do you think the producter choice set is convex?

(c) Suppose this production frontier is homothetic. For a given set of input prices (w,r), what can you conclude about how the cost minimizing input bundles in your isoquant map will be related to one another.

(d) What can you conclude about the shape of the cost curve for a given set of input prices?

(e) What will the average and marginal cost curves look like?

(f) Suppose again that A = (ℓ^A,k^A,x^A) is a profit maximizing production plan at the current prices (and suppose that A is not a corner solution). Illustrate the isoquant that represents the profit maximizing output quantity x A. Using the conditions that have to hold for this to be a profit maximum, can you demonstrate that these imply the producer is cost minimizing at A?

(g) Where else does the cost minimizing condition hold? Do the profit maximizing conditions hold there as well?

(h) What happens to output as p falls? What happens to the ratio of capital to labor in the production process (assuming the production process is indeed homothetic)?

B: Consider the same production function as the one introduced in part B of exercise 12.5.

(a) Write down the problem you would need to solve to determine the least cost input bundle to produce some output level x.

(b) Set up the Lagrange function and derive the first order conditions for this problem.

(c) To make the problem easier to solve, substitute y = e^{−(ℓ−β)} and z = e^−(k−γ) into the first order conditions and solve for y and z as functions of w, r , x (and α).

(d) Recognizing that y and z were placeholders for e^{−(ℓ−β)} and e^−(k−γ) , use your answers now to solve for the conditional input demands ℓ(w,r,x) and k(w,r,x).

(e) Derive from your answer the cost function for this firm — i.e. derive the function that tells you the least it will cost to produce any output quantity x for any set of input prices. Can you guess the shape of this function when α, β, γ, w and r are held fixed?

(f) Derive the marginal cost function. Can you guess its shape when α, w and r are held fixed?

(g) Use your expression of the marginal cost curve to derive the supply function. Can you picture what this looks like when it is inverted to yield a supply curve (with input prices held fixed)?

(h) In (g) of exercise 12.5, you were asked to calculate the profit maximizing output level when α= 100, β = γ = 5 = p and w = r = 20. You did so using the input demand functions calculated from the profit maximization problem. You can now use the supply function derived from the cost minimization problem to verify your answer (which should have been 91.23 units of output.) Then verify that your answer is also the same as it was before (93.59) when r falls to 10.