Problem%20Set%206 - Economics 3010 Fall 2009 Professor...

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Unformatted text preview: Economics 3010 Fall 2009 Professor Daniel Benjamin Cornell University Problem Set 6 This problem set is due in section on Friday, November 6. Please collect your answers into a stapled packet, and write your own name and your TA’s name on the front of each packet, as well as your netID and the time of the section you usually attend. 0. (Aplia) Please do the assigned problems on the Aplia website. These problems are due at 11:45pm on Thursday, October 5. 1. (Cost curves: The Cobb‐Douglas production function) Consider a firm that produces widgets. Suppose that the number of widgets produced (y) depends on the amount of capital (K), the amount of labor (L), and the level of technology (A) according to the following Cobb‐Douglas production function (f): y= f(K, L) = K2/3L1/3. (Recall from Problem Set 5 that f(K, L) = AK1/3L2/3 is a reasonable approximations for a representative firm in the U.S. economy, where L is a combination of skilled and unskilled labor, and K is physical capital, and A represents the level of technology. The specification f(K, L) = AK2/3L1/3 is in fact a reasonable approximation if L is unskilled labor only, and K represents a combination of physical and human capital. As is common in microeconomic analysis, we normalize the level of technology, A, to equal 1 throughout this problem.) In the short run, capital is a fixed factor, K = K , while labor, L, is free to vary. Let w denote the wage rate, let R denote the rental rate on capital, and suppose there are many firms demanding labor and capital, so the firm takes w = 1 and R = 1 as given. (a) Write down the firm’s short‐run cost‐minimization problem. Solve the problem, and show that the firm’s short‐run total cost function can be written as SRTC(y) = α y3 + F, −2 where α = wK and F = RK are positive constants. (b) On a single graph, draw the short‐run average variable cost (SRAVC), short‐run average cost (SRAC), and short‐run marginal cost (SRMC) curves when K = 1 . On a separate graph, draw the SRAVC, SRAC, and SRMC curves when K = 2 . In both cases, show that the SRMC curve goes through the minimum point of the SRAC curve. 1 (c) Write down the firm’s long‐run cost‐minimization problem. You do not need to solve the problem yourself, but follow along the derivation in the appendix to chapter 20, and write down the key steps of the derivation required to show that LRTC(y) = β y, where β = 21/ 3 + 2−2 / 3 is a positive constant. (d) On a single graph, draw the long‐run average cost (LRAC) and long‐run marginal cost (LRMC) curves. Explain why you could have predicted the shape of these curves from the fact that the production function has constant returns to scale. (e) Draw a new graph showing the LRAC curve, the SRAC curve when K = 1 , and the SRAC curve when K = 2 . Explain intuitively why each SRAC curve has a single point that intersects the LRAC curve. 2. (Firm Supply: Average Costs vs. Marginal Costs) [Note: This question is based on true events, but the numbers in part (a) are completely made up.] The accountants for the major airlines reported that the total cost per flight (calculated as Total Costs divided by number of flights) was $300,000. The revenue per flight depended on the number of passengers according to the following table: Revenue per flight % of seats filled 40% $200,000 50% $250,000 60% $300,000 (a) For years, airlines only scheduled a flight if they expected the flight to have at least 60% of its seats filled. As a result, at times of the week with high demand, all the company’s airplanes were in use, but at times of the week with low demand, many of the planes remained idle. The airlines believed that running additional flights would reduce their profits. Suddenly in the 1960s, Continental Airlines started running any flight that was expected to be at least 40% full – and their profit increased! How was that possible? (Note: There is not enough information here for you to know the answer for sure, but using what you have learned, make an educated guess.) (Within a few years, all the other airlines followed Continental’s lead, and these days, many flights run with even fewer seats filled.) (b) Prior to the 1970s, all convenience stores closed at night, during hours when the expected revenue per hour was less than the average cost per hour (total cost divided by number of hours open) of keeping the store open. However, in the 1970s, 24‐hour convenience stores burst onto the scene and earned greater profit as a result of staying open during late‐night hours, even though average revenue per hour during those hours was indeed lower than average cost per hour. How is it possible that 24‐ 2 hour convenience stores earned greater profit? Why do you think 24‐hour stores are far less common in small towns like Ithaca than they are in bigger cities? 3. (Firm Supply: Shut‐down condition) [Note: This question is based on true events, but the numbers are completely made up.] There are both American and Japanese firms that sell steel in the U.S. Periodically, the price of steel falls low enough that steel‐operating firms become unprofitable. When this happens, the American firms typically stop operating their steel factories, but Japanese firms continue to produce and sell steel in the U.S. at a loss. American steel firms often accuse the Japanese firms of “dumping” – an illegal act according to U.S. trade law, in which a foreign company sells a product in the U.S. at “less than fair value,” e.g., at a price that causes the company to lose money. Let’s apply what you’ve been learning to figure out what is going on. For a typical American firm and a typical Japanese firm, the following table shows (in $millions) annual levels of Total Revenue (TR), Total Cost (TC), Total Fixed Cost (TFC), and Total Variable Cost (TVC). TR TC TFC TVC π if stay open π if shut down US Firm $200 $250 $25 $225 Japanese Firm $200 $250 $75 $175 For the American firm, labor is a variable cost. But for the Japanese firm, labor is a much more of a fixed cost because it is very difficult to fire workers once they are hired. That is why relatively more of the Japanese firm’s costs are fixed. (a) Fill in the last two columns of the table, profit if the firm stays open and profit if the firm shuts down. Explain in words why the Japanese firm continues to produce at a loss. (b) What would happen to these firms if the price of steel remained low for a long time? (Hint: Think about the short run versus the long run.) 3 ...
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