Margarita Robotics has a daily production function given by $Q=K^{0.5} L^{0.5},$ where $K$ is the monthly number of hours of use for a precision lathe (capital) and $L$ is the monthly number of machinist hours (labor). Suppose that each unit of capital costs $\$ 40,$ and each unit of labor costs $\$ 10$.
a. In the short run, $\bar{K}$ is fixed at 16,000 hours. What is the short-run demand for labor?
b. Given that $\bar{K}$ is fixed at 16,000 hours, what are total cost, average total cost, average variable cost, and marginal cost in the short run?
c. What are the long-run demands for capital and labor?
d. Derive total cost, average cost, and marginal cost in the long run.
e. How do Margarita Robotics' marginal and average costs change with increases in output? Explain.