# A: Consider a firm whose technology has decreasing returns to

A: Consider a firm whose technology has decreasing returns to scale throughout and who faces a recurring fixed cost. Denote the level of capita chosen in the long run at the lowest point of the long run AC as k∗ .
(a) Replicate the short run MC and long run AC curves from Graph 13.3.Where in your graph does the long run MC curve lie?
(b) Draw a separate graph with the ACLR curve. Suppose that k < k∗ in the short run. Illustrate where the AEk must now lie.
(c) Next illustrate where the ACk and MCk curves lie. Is the long run MC curve now different than in part (a)?
(d) On a separate graph, repeat (b) and (c) for k′ > k∗.
(e) Illustrate the short run MC curves you drew in parts (c) and (d) in the graph you first drew in part (a). How is this graph similar to Graph 13.7 in the text?
(f) True or False: The MCk curve crosses the ACLR curve at the lowest point of the AEk curve only if k = k∗.
(g) How would your answer to (f) change if the sentence had started with the words “If the production technology has constant returns to scale and there are no fixed costs, ...”.
(h) True or False: Unless the production technology has constant returns to scale and no long run fixed costs, the short run AE curves are tangent at the lowest point of the long run AC curve only if k = k∗.
B: Suppose that a firm’s production function is x = f (ℓ,k) = Aℓαkβ with α,β > 0 and α+β < 1. Suppose further that the firm incurs a recurring (long run) fixed cost FC.
(a) In equation (13.45) from exercise 13.5, we already provided the long run cost function for such a firm in the absence of fixed costs. What are this firm’s long run marginal and average cost functions?
(b) Derive the output level x∗ at which the lowest point of the long run average cost curve occurs.
(c) From here on, suppose that α = 0.2, β = 0.6, A = 30, w = 20, r = 10 and FC = 1,000. Given these values, what is x∗? How much capital k∗ does the firm hire to produce x∗? (The conditional input demand functions for a Cobb-Douglas production process are given in equation (13.47) of exercise 13.7.)
(d) What is the long run marginal cost of production at x∗? What about the long run average
(e) For a fixed level of capital k, what are the short run MC, AC, and AE functions?
(f) What is the short run AE, AC and MC for x = x∗ when capital is fixed at k∗? How do these compare to long run AC and MC of producing x∗?
(g) Now suppose capital is fixed in the short run at k = 200. How does your answer to (f ) change? What if capital were instead fixed at k = 400? Interpret your answer.

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