Returns to scale, average cost and the run Consider the Cobb-Douglas production function f(L, K) = L^ 2/ 3 K 1/ 3. The price of labor is w = 2. The price of capital is r = 1. Suppose that in the short run, capital is fixed at K¯ = 100.
1. What are the returns to scale in the short run?
2. Derive the short-run average cost function and, using the software available at https://www. flashandmath.com/mathlets/calc/fungraph/index.html, plot it in a graph with price on the vertical axis.
3. In the long run where capital can be adjusted freely by the firm, what are the returns to scale? What is the shape of the average cost curve? the marginal cost curve? You do not need to justify your answer.
4. Derive the long-run average cost of the firm using the method of your choice, and plot it in your previous graph.
Now assume that in the long run, capital can only be varied in discrete amounts. To simplify, assume that capital can only take on the two values K = 100 and K = 400.
5. Derive the short-run average cost function when K¯ = 400. Plot it in your previous graph.
6. Show the new long-run average cost curve on your graph. (You can draw by hand on the graph using the mouse.)
7. Comment on the shapes of the two long-run average cost curves that you derived.