Problems

1. For the following production functions,

   • Find the marginal product of each input.

   • Determine whether the production function exhibits diminishing marginal returns to each input.

   • Find the marginal rate of technical substitution and discuss how MRTS_LK changes as the firm uses more L, holding output constant.

   1. Q(K, L) = 3K + 2L

   2. Q(K, L) = 10K^0.5L^0.5

   3. Q(K, L) = K^0.25L^0.5

2. A more general form of the Cobb–Douglas production function is given by

   Q = AK^αL^β

   where A, α, and β are positive constants.

   1. Solve for the marginal products of capital and labor.

   2. For what values of α and β will the production function exhibit diminishing marginal returns to capital and labor?

   3. Solve for the marginal rate of technical substitution.

3. Catalina Films produces video shorts using digital editing equipment (K) and editors (L). The firm has the production function Q = 30K^0.67L^0.33, where Q is the hours of edited footage. The wage is $25, and the rental rate of capital is $50. The firm wants to produce 3,000 units of output at the lowest possible cost.

   1. Write out the firm’s constrained optimization problem.

   2. Write the cost-minimization problem as a Lagrangian.

   3. Use the Lagrangian to find the cost-minimizing quantities of capital and labor used to produce 3,000 units of output.

   4. What is the total cost of producing 3,000 units?

   5. How will total cost change if the firm produces an additional unit of output?

4. A firm has the production function Q = K^0.4L^0.6. The wage is $60, and the rental rate of capital is $20. Find the firm’s long-run expansion path.