ASSIGNMENT 10
Optimal use of inputs

A firm uses labor and energy to produce output. The firm's production function is given by Q = F(L,E). The cost of a unit of labor is denoted by w, that of a unit of energy by r.

Given: F(K,L) = 60L^3/4E^1/4, w = 10, r = 8.

1. Suppose the firm wishes to produce 1200 units of output at the lowest possible cost.

1. How much labor and energy should it use?
2. What is the value of the Lagrange multiplier?
3. What is the firm's least possible cost of production?
4. Using the bordered Hessian, show that the second-order sufficient condition for a minimum is met.
5. Plot the isoquant and isocost line. Indicate the optimal point.

2. Interpretation of the Lagrange multiplier
Suppose the firm increases its output by 1 unit. Obtain the firm's optimal values of L and E. By how much will the firm's cost of production rise? How is this linked to the Lagrange multiplier computed in Question 1?

3. Effect of input price changes
Suppose, as in Question 1, the firm wishes to produce 1200 units of output, but that the price of energy rises.

1. Use the envelope theorem to obtain the change in the firm's cost. Show steps involved.
2. Increase r by 1. Will the firm use more, or less of labor? Of energy? Compute the total cost. Is the change in cost consistent with your result in (a)?
3. Plot the isoquant and the new isocost line. Indicate the new optimal point.