Question

1. Acme Inc. produces widgets. Its production function is ? = ? 1/3 (? − 1) 1/3 , for ? ≥ 0 and ? ≥ 1, where ? denotes units of capital input, ? denotes units of labor input, and ? denotes units of output. (Note that ? = 0 for all ? < 1.) The price of a unit of capital input is ? and the price of a unit of labor input is ?.

a. Find Acme’s ??? and ??? functions.

b. Find Acme’s ??? function.

c. Suppose that ? = 8 in the short run. Find Acme’s short run (contingent) demand function for labor input. (Hint: This will be a function of the target output level, ?.)

d. Suppose that ? = 8 in the short run. Find Acme’s short run total cost function. (Hint: This will be a function of the target output level, ?, and the input prices, ? and ?.)

e. Find Acme’s long run (contingent) demand functions for labor input and capital input. (Hint: These will be functions of the target output level, ?, and the input prices, ? and ?.)

f. Find Acme’s long run total cost function. (Hint: This will be a function of the target output level, ?, and the input prices, ? and ?.)

2. Apex Inc. produces gadgets. Its production function is ? = min(4?, ? − 1), for ? ≥ 0 and ? ≥ 1, where ? denotes units of capital input, ? denotes units of labor input, and ? denotes units of output. (Note that ? = 0 for all ? < 1.) The price of a unit of capital input is ? and the price of a unit of labor input is ?.

a. Find Apex’s ??? and ??? functions.

b. Find Apex’s ??? function.

c. Suppose that ? = 8 in the short run. Find Apex’s short run (contingent) demand function for labor input. (Hint: This will be a function of the target output level, ?.)

d. Suppose that ? = 8 in the short run. Find Apex’s short run total cost function. (Hint: This will be a function of the target output level, ?, and the input prices, ? and ?.)

e. Find Apex’s long run (contingent) demand functions for labor input and capital input. (Hint: These will be functions of the target output level, ?, but not the input prices, ? and ?. Why?)

f. Find Apex’s long run total cost function. (Hint: This will be a function of the target output level, ?, and the input prices, ? and ?.)

3. Pinnacle Inc. produces industrial pins. Its production function is ? = ?? 1/2 ? 1/2 , for ? ≥ 0 and ? ≥ 0, where ? denotes units of capital input, ? denotes units of labor input, and ? denotes units of output. The price of a unit of capital input is ? and the price of a unit of labor input is ?.

a. Find Pinnacle’s ??? and ??? functions.

b. Find Pinnacle’s ??? function.

c. Find Pinnacle’s long run (contingent) demand functions for labor input and capital input. (Hint: These will be functions of the target output level, ?, and the input prices, ? and ?.)

d. Find Pinnacle’s long run total cost function. (Hint: This will be a function of the target output level, ?, and the input prices, ? and ?.)

e. Pinnacle currently produces in the U.S., where ? = 2, ? = 100, and ? = 25. What is Pinnacle’s per-unit cost of a pin?

f. Suppose that Pinnacle has an opportunity to move production to Mexico, where ? = 1 and ? = 100. For which values of ? in Mexico would moving production to Mexico reduce the per-unit cost of pin production?

Answer 1

Answering only first Ques