4 No production in the short-run and positive production in the long-run Suppose that the marginal cost curves in the short-run and in the long-run are identical for a certain firm, and that the short-run average cost curve is below the long-run average cost curve. Then for a certain price, we can show that the firm wants to produce in the short-run, but not in the long-run. In this question, we will investigate whether a reversed situation is possible. Consider the following production function: Q = L³K². The wage for labor (L) is w = 1. The cost of using capital (K) has two parts: the first is the rent that the firm needs to pay capitalists (r = 2), and the second is the maintenance cost (m = 2). The firm can change labor and capital in the long-run, but it can change only labor in the short-run. Thus, the long-run economic cost with (L,K) is wL+ (r+m)K, while the short-run economic cost of running the firm is wL + mko for given capital Ko. (Here we are implicitly assuming the following. If the firm decides to produce positive amount, it needs to pay the maintenance cost for all Ko. On the other hand, if it decides not to produce at all, it pays zero maintenance cost. In short, mKo is NOT a sunk cost even in the short-run.) (a) Find the long-run cost function, the long-run marginal cost function, and the long-run average cost functions (as functions of Q). (b) Draw the long-run marginal cost and the long-run average cost functions in one graph carefully, and show that the firm wants to produce for any market price P. (c) For given Ko = 1, find the short-run cost function, the short-run marginal cost function, and the short-run average cost function (as functions of Q). Note again that the short-run cost is wL + mK. (d) Draw (i) the long-run marginal cost, (ii) the long-run average cost, (iii) the short-run marginal cost, (iv) the short-run average cost functions in one graph carefully. Find the range of prices under which the firm wants to produce in the long-run, but not in the short-run.
4 No production in the short-run and positive production in the long-run Suppose that the marginal cost curves in the short-run and in the long-run are identical for a certain firm, and that the short-run average cost curve is below the long-run average cost curve. Then for a certain price, we can show that the firm wants to produce in the short-run, but not in the long-run. In this question, we will investigate whether a reversed situation is possible. Consider the following production function: Q = L³K². The wage for labor (L) is w = 1. The cost of using capital (K) has two parts: the first is the rent that the firm needs to pay capitalists (r = 2), and the second is the maintenance cost (m = 2). The firm can change labor and capital in the long-run, but it can change only labor in the short-run. Thus, the long-run economic cost with (L,K) is wL+ (r+m)K, while the short-run economic cost of running the firm is wL + mko for given capital Ko. (Here we are implicitly assuming the following. If the firm decides to produce positive amount, it needs to pay the maintenance cost for all Ko. On the other hand, if it decides not to produce at all, it pays zero maintenance cost. In short, mKo is NOT a sunk cost even in the short-run.) (a) Find the long-run cost function, the long-run marginal cost function, and the long-run average cost functions (as functions of Q). (b) Draw the long-run marginal cost and the long-run average cost functions in one graph carefully, and show that the firm wants to produce for any market price P. (c) For given Ko = 1, find the short-run cost function, the short-run marginal cost function, and the short-run average cost function (as functions of Q). Note again that the short-run cost is wL + mK. (d) Draw (i) the long-run marginal cost, (ii) the long-run average cost, (iii) the short-run marginal cost, (iv) the short-run average cost functions in one graph carefully. Find the range of prices under which the firm wants to produce in the long-run, but not in the short-run.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:4 No production in the short-run and positive production in the
long-run
Suppose that the marginal cost curves in the short-run and in the long-run are identical for a
certain firm, and that the short-run average cost curve is below the long-run average cost curve.
Then for a certain price, we can show that the firm wants to produce in the short-run, but not
in the long-run. In this question, we will investigate whether a reversed situation is possible.
Consider the following production function:
Q = L³K².
The wage for labor (L) is w = 1. The cost of using capital (K) has two parts: the first is
the rent that the firm needs to pay capitalists (r = 2), and the second is the maintenance cost
(m = 2). The firm can change labor and capital in the long-run, but it can change only labor
in the short-run. Thus, the long-run economic cost with (L,K) is wL+ (r+m)K, while the
short-run economic cost of running the firm is wL + mko for given capital Ko. (Here we are
implicitly assuming the following. If the firm decides to produce positive amount, it needs to
pay the maintenance cost for all Ko. On the other hand, if it decides not to produce at all, it
pays zero maintenance cost. In short, mKo is NOT a sunk cost even in the short-run.)
(a) Find the long-run cost function, the long-run marginal cost function, and the long-run
average cost functions (as functions of Q).
(b) Draw the long-run marginal cost and the long-run average cost functions in one graph
carefully, and show that the firm wants to produce for any market price P.
(c) For given Ko = 1, find the short-run cost function, the short-run marginal cost function, and
the short-run average cost function (as functions of Q). Note again that the short-run cost is
wL + mK.
(d) Draw (i) the long-run marginal cost, (ii) the long-run average cost, (iii) the short-run
marginal cost, (iv) the short-run average cost functions in one graph carefully. Find the range
of prices under which the firm wants to produce in the long-run, but not in the short-run.
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