(a) Suppose we have the following production function: Q = K^3/4L^1/4_. Confirm the technology is constant returns to scale (CRS). Show your work and explain what it means.

(b) Now suppose K is fixed in the short-run at 21.544 (so that K^3/4 = 10). Let r = $10 and w = $20. What is the firm's cost minimization problem? Explain.

(c) Derive the short-run cost function. Show that short-run costs are increasing in Q and that average variable and marginal costs also rise as Q rises. You can use math or a table/figure.

(d) Suppose P = $100. What is the firm's profit maximization problem?

(e) Show that optimal Q = 23.20 in the short-run. You can use math or a spreadsheet. Also show costs, revenues, and profits. Should the firm simple choose Q = 0 and shut-down?

(f) Now let both K and L be variable. What is the firm's long-run cost minimization problem now? What two conditions must be met to ensure the firm is minimizing costs. Explain and use a diagram.

(g) We can show that the MRTS = -3 L/K. What is the MRTS measuring and why does it change along the isoquant?

(h) Let r = $10 and w = $20. Find the long-run cost function (C = wL + rK). Show that average and marginal costs are constant (eg the same value for any Q). Why is this the case?