Cost Minimization
LEARNING GOALS:
- IV.B.3: Solve the firm’s cost minimization function for conditional input demands, cost function, marginal cost function, average cost function, and supply function in both the short-run and long-run and describe how they differ.
- IV.B.4: Recognize that profit maximization implies cost minimization.
- IV.B.8: Recognize the relationship between short-run and long-run costs.
- IV.B.11: Interpret the Lagrange Multiplier as the shadow price (cost of producing an additional unit of the good while at the current optimum).
Suppose that a firm produces a good with the following production function: \(F(L,K)=5L^{0.7}K^{0.3}\)
Suppose also that a unit of labor costs $w=1$ and a unit of capital costs $r=2$.
Use the graph to aid in your responses to the following questions.
- Suppose that the firm wants to produce 100 units of output. Which curve on the graph represents the bundles of inputs that cost same amount (i.e., the isocost curve)? Which curve represents the bundles that produce the same quantity of output (i.e., the isoquant curve)?
- Use the graph to determine the bundle of inputs that produces 100 units and costs the least.
- Set up the Lagrangian for this cost minimization problem.
- Solve the firm’s cost minimization problem and show your firm’s optimal choices match what you found in (b).
- Solve for the Lagrange multiplier. What does it represent?
- Use the graph to calculate how much more it costs the firm to produce 110 units. Is this consistent with your interpretation of the Lagrange multiplier?
- Suppose that the firm’s marginal product of labor decreases. How might this be represented as a change in the parameters of the production function? Use the graph to see what happens to the firm’s optimal choice and explain.