Exercise Set 9
Cost Minimization

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

Given: F(K,L) = 10K^3/4L^1/4, w = 50, r = 30.

The firm wishes to produce 1300 units of output at the lowest possible cost.

1. How much labor and capital should it use?
2. What is the firm's least possible cost of production?
3. Sketch the firm's isoqant and isocost curves. Indicate the optimal combination of inputs on the graph.

Problem, stated formally:

min     50L + 30K

s.t.     10K^3/4L^1/4 = 1300

Lagrangian function:

LAGR = 50L + 30K + λ(1300 - 10K^3/4L^1/4)

Use FOC and SOC to find the solution:

• The firm will use 38.9 units of labor and 194.4 units of capital.
• The minimum cost of production is 7775.81.

2. Suppose the price of labor rises. Select a new value for w. How will this affect the firm's optimal choice of inputs and its total cost? Provide a sketch.

3. Suppose the price of capital falls. Select a new value for r. How will this affect the firm's optimal choice of inputs and its total cost? Provide a sketch.

4. Suppose the firm wishes to produce more output. Select a new value for Q. How will this affect the firm's optimal choice of inputs and its total cost? Provide a sketch.

5. In Question 1, what was the value of the Lagrange multiplier? What is its interpretation?