1. 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.

a. Write out the firm's constrained optimization problem.

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

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

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

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

2. 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.