Catalina Films produces video shorts using digital editing equipment (K) and editors (L). The firm has the production function Q = 30K0.67L0.33 , 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.