(a) Suppose we have the following production function: Q = min[100 K, 2L]_. Confirm the technology is constant returns to scale (CRS). Show your work and explain what it means.

(b) Show the isoquants in a figure. What are the input requirements for Q = 100? Explain how this tells you the input requirements for Q = 1.

(c) Suppose K is fixed at K = 1 in the short-run but L is variable. Show that output rises proportionally with L but only up to the point that L = 50. Show that more L has no more impact on Q. Explain.

(d) Let r = $20 and w = $100. Suppose the market price is P = $100. What is optimal output Q and optimal input L given K = 1? Should the firm simply shut-down (Q = 0)?

(e) What is the shut-down price? That is, what is the minimum price that ensures the firm has positive output given r = $20 and w = $100? Explain.

(f) Now let K vary with L in the long-run. What is the firm's cost minimization problem now? Contrast with the short-run problem in (c) and (d).Show this in an isoquant/isocost figure.

(g) Find the firm's long-run cost function given r = $20 and w = $100. Confirm that average and marginal costs are constant. Why is this the case?

(h) What is the firm's exit price? That is, what is the minimum price that ensures the firm has positive output in the long-run? Explain.