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

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Unformatted text preview: Economics 3010 Fall 2009 Professor Daniel Benjamin Cornell University Problem Set 6 Solutions 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) = A K2/3L1/3 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, where α = w K −2 and F = R K are positive constants. The short run cost minimization problem is: Min L wL + R K s.t. y= f(K, L) = K 2/3L1/3 Note: this short‐run cost‐minimizing problem has only one variable factor (labor). The optimal amount of labor can be obtained from the production function constraint for L. From y = K 2/3L1/3, y3 = K 2L . Hence L* = K ‐2 y3 Therefore, c (w, R, K , y) = wL + R K = w K ‐2 y3 + R K , or SRTC(y) = α y3 + F where α = w K ‐2 and F = R K are positive constants. 1 (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. (2) with K =2 (w = 1 and R = 1) (1) with K =1 (w = 1 and R = 1) 3 −2 3 SRTC(y) = w K y + R K = y + 1 SRTC(y) = w K −2 y3 + R K = ¼ y3 + 2 2 SRAVC(y) = y SRAVC(y) = ¼ y2 2 SRAC (y) = y + 1/y SRAC (y) = ¼y2 + 2/y 2 SRMC (y) = 3y SRMC (y) = ¾ y2 $ $ SRAC SRMC SRAC SRMC SRAVC SRAVC y y (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, 1/3 −2/3 where β =2 +2 is a positive constant. The long run cost minimization problem is: Min wL + RK L,K s.t. y= f(K, L) = K2/3L1/3 Lagrangian function is L = wL + RK – λ(K2/3L1/3– y) F.O.Cs: ∂L /∂L = w – (1/3) λ K2/3L‐2/3 = 0 ‐1/3 1/3 ∂L /∂K = R – (2/3) λ K L = 0 2/3 1/3 ∂L /∂λ = K L – y = 0 Multiply (1) by L and (2) by K to get wL = (1/3) λ K2/3L‐2/3L = λ (1/3) y RK = (2/3) λ K‐1/3 L1/3K = λ (2/3) y (1) (2) (3) Hence, L = λ (1/3) (y/w) Hence, K =λ (2/3) (y/R) 2 (4) (5) From (3), y = K2/3L1/3, and using (4) and (5), y = [λ (2/3) (y/R)] 2/3 [λ (1/3) (y/w)] 1/3 We can solve for λ and can get the conditional factor demand functions for L and K: K(w,R,y) = 21/3R‐1/3w1/3y L(w,R,y) = 2‐2/3R2/3w‐2/3y Finally, we can come up with a cost function: c(w,R,y) = wL+rK = w{2‐2/3R2/3w‐2/3y} + R{21/3R‐1/3w1/3y}= [2‐2/3 +21/3] w1/3 R2/3 y In our example, w = 1 and R = 1. Hence, by re‐writing above: c(w,R,y) =[2‐2/3 +21/3] y = β y where β =21/3+2−2/3 (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. From LRTC(y) = ( 21 / 3 + 2 −2 / 3 )y LRAC = ( 21 / 3 + 2 −2 / 3 ) ≈ 1.89 LRMC = ( 21 / 3 + 2 −2 / 3 ) ≈ 1.89 Both LRAC and LRMC are constant at about 1.89. We can also predict the shape of these curves from the fact that the production function has constant returns to scale. First, let’s prove that production function of constant returns to scale have constant identical LRAC and LRMC. Suppose Y=F(K, L), LRAC Cobb‐Douglas production function of y = Ax1ax2b has constant returns to scale when a + b = 1 , which is the case for the given production function, y= f(K, L) = K2/3L1/3. Therefore, we can predict the constant average and marginal cost. (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. 3 SRAC w/ K =1 $ SRAC w/ K=2 LRAC y1 y2 In the long run all inputs are flexible, while in the short run some inputs are fixed. Hence, short‐ run cost is always greater than or equal to long‐run cost and they are tangent at a single point. At this tangency point, the level of fixed input specified in short‐run production is actually the optimal level in the long run at the corresponding unit production. In our example, in (c), we’ve derived that the cost minimization solution in the long run is, K*(w,R,y) = 21/3R‐1/3w1/3y L*(w,R,y) = 2‐2/3R2/3w‐2/3y When, K is fixed at 1 (with R=1 w=1), the corresponding y = 2‐1/3. This point is shown as y1 in the graph. Recall that the short‐run cost function is the lowest‐cost way of producing y with given fixed output. If K = 1 is the optimal solution to produce y1, the corresponding L* should be the same both in both the short‐run and long‐run. Hence, we have the same average cost at y1 in both the short run and the long run. 4 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: % of seats filled Revenue per flight 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.) Definitions used: Marginal Cost: the change in total cost for a given change in output. Marginal Revenue: the change in total revenue for a given change in output. Average Total Cost: total cost divided by output. Whether or not profit increases from an additional flight depends on whether or not marginal revenue is above marginal cost. When Continental Airlines lets a plane take off with 40% capacity, they get $200,000. Although this is less than the average total cost of $300,000 per flight, profits will increase if the marginal cost of the flight is below $200,000. The majority of costs in the airline industry are fixed costs: leasing the planes, leasing the landing rights, and salaries for the pilots and flight attendants. The main variable cost is fuel. It was profitable to fly a plane at 40% capacity because the marginal revenue exceeded the variable cost (despite being less than the average cost). (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‐ 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? Even though average revenue per hour is less than the average cost per hour at night, it’s possible that profits could increase by staying open. Profit will increase as long as the marginal revenue from staying open an additional hour exceeds the marginal cost. Marginal costs are actually not that high: the variable costs of workers’ hourly wages and electricity. The major component of a convenience store’s 5 average costs is the fixed costs of renting the space, stocking the store, installing equipment to prevent theft by customers (and employees). But these fixed costs are irrelevant for whether it is profit‐ maximizing to stay open 24 hours. Twenty‐four hour stores are less common in small towns because the marginal revenue from staying open an additional hour is much smaller; there are far fewer customers at night than in a bigger city. Even though the variable costs of staying open are the same whether the store is in Ithaca or in a bigger city, the marginal revenue is large enough to justify staying open only in a bigger city. 3. (Firm Supply: Shut‐down condition) [Note: This question is basedon 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 the 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). 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. TR TC TFC TVC π if stay open π if shut down US Firm $200 $250 $25 $225 Japanese Firm $200 $250 $75 $175 (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. If the US Firm stays open, they get Profit = TR – TC = 200 – 250 = ‐50 If the US Firm closes, they get Profit = ‐TFC = ‐25 If the Japanese Firm stays open, they get Profit = TR – TC = 200 – 250 = ‐50 If the Japanese Firm closes, they get Profit = ‐TFC = ‐75 Recall that the short‐run shutdown condition says that the firm should shut down if average revenue is less than average variable cost (for the case of constant price, this is simply saying that price is less than average variable cost). Here, the Japanese Firm’s revenue exceeds total variable costs, so they continue to operate in the short‐run and incur 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.) If the price of steel were to remain low, the firms could get rid of their fixed factors and exit the market. The long‐run exit condition says that a firm should leave if price (or average revenue) is below average total cost. In this example, both firms would exit the market because average revenue is below average total cost. 6 ...
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