Homework 5 Econ 361 Intermediate Microeconomic Analysis 30 points

Size: px
Start display at page:

Download "Homework 5 Econ 361 Intermediate Microeconomic Analysis 30 points"

Transcription

1 Homework 5 Econ 361 Intermediate Microeconomic Analysis 30 points 1. (3 points) Assume a firm uses two inputs, labor and capital. If the marginal product of capital divided by the user cost of capital is greater than the marginal product of labor divided by the wage, then to minimize costs should this firm use more labor and less capital? No, the opposite is true. A firm should use less labor and more capital. If, as described, MPK/r > MPL/w, then increasing K and decreasing L until the equality holds will increase output without increasing costs. In other words, if the output per dollar of a unit of an additional unit of capital is greater than the output per dollar of a unit of an additional unit of labor, then reallocating dollars from labor to capital will increase output without changing total costs. Alternatively, we can think about this as reducing costs for a given level of output. Either way, reducing L and increasing K is the correct strategy. If this firm s production function leads to a corner solution, then in fact all of the firm s inputs should be capital. 2. (3 points) If for a firm total costs less than double as output doubles, then is the firm is experiencing increasing returns to scale? Possibly, but not necessarily. This is the definition of economies of scale, so the question is asking whether economies of scale are always caused by increasing returns to scale. Recall that returns to scale deal with how proportional changes in inputs proportionally change output (the easiest way to think of this is whether doubling inputs more or less than double output). Increasing returns to scale could lead to economies of scale if increasing inputs had no effect on input prices (factor markets are perfectly competitive). But, this might not be the case, as factor prices could go up. More importantly, economies of scale might be caused by something entirely different. If a firm actually has constant or decreasing returns to scale, but as it purchases more inputs in factor markets it gains market power and can demand lower factor prices, then economies of scale can be achieved anyway. Returns to scale and economies of scale are different concepts, and while increasing returns to scale can be the cause of economies of scale, it often is not. 1

2 3. (3 points) A local business owner wonders if she can estimate her long run average costs by examining her short run costs. What would you tell her? First, recall that the definition of long run is time it takes for fixed costs, which means capital, to become variable. Average cost in the short run is calculated for a given, constant level of capital. In the long run, a firm s manager chooses the optimal level of capital as well as labor. This can lead to totally different conclusions regarding costs than in the short run. Figure 8.13 on page 293 illustrates this perfectly: average cost can be rising in the short run due to diminishing MPL but falling in the long run due to economies of scale. In other words, the trend in short run average cost is the opposite of the long run trend, making short run costs a terrible predictor of long run costs in this case. Why is this case? In the short run, increasing output is achieved by increasing labor. The marginal product of labor is likely diminishing, and thus marginal cost (w/mpl) is rising and average cost is also probably rising. Think of a crowded bakery with more and more bakers fighting over an oven. In the long-run, the firm expands the size of the bakery, purchasing more ovens to complement the additional bakers. Average costs fall, perhaps because of bulk pricing of wheat or even ovens, or perhaps due to specialization among bakers (Cookie bakers, cake bakers, bread bakers etc ). Regardless of the story, economies of scale are often present, while in the short run, the fixed nature of capital and the diminishing marginal product of labor can cause average cost to actually be rising. As such, short run average cost is likely a bad indicator of long run average cost. A better approach might be to consider reasons why her firm would face either economies or diseconomies of scale if output were changed (supplier X has offered a price reduction of Y if I increase purchases by Z, etc ). 4. (3 points) If a firm exhibits a constant marginal product of its variable input, will its marginal cost curve be horizontal? Yes, as long as we assume the wage rate is not a function of output (a result of uncompetitive factor markets) and that the firm is operating in the short run. Short run marginal cost is defined as w/mpl (assuming labor is the variable input). For a marginal cost curve to be horizontal, the expression w/mpl must evaluate to a constant. So, a constant MPL will only translate into constant marginal cost if the wage rate is also a constant. If, on the other hand, wages depend on the number of workers employed by the firm (perhaps labor is scarce), then the marginal cost curve will not be horizontal. In this case, as hiring more workers to increase output would raise wages, the marginal cost curve would be upward sloping. If the firm is operating in the long run, even if input prices do not change as q changes a constant MPL does not guarantee a flat MC, as long run MC is determined by what occurs to costs as firms alter scale, not what happens to costs as they change a variable input holding others fixe. 2

3 5. (3 points) Will a firm shut down if its total costs exceed total revenue? This is true in the long run, but not necessarily the case in the short run. In the long run, if a firm cannot attain a profit of 0 or greater, they will exit the industry. However, in the short run, certain costs are fixed (often capital investments) and certain costs are variable (the amount of worker-hours is adjustable). The key is that fixed costs are incurred whether the firm shuts down or not. If the firm shuts down, the total cost = the fixed cost = the loss incurred by the firm, as no revenue was generated. However, total revenue exceeds variable cost (TR > TVC), then by staying open, they will cover part of the fixed costs (TR TVC = portion of TFC covered). If TR < TVC, then each unit of output sold increase the loss, and the loss minimizing strategy is to shut down. To summarize: In the long run, if TR < TC, the firm will shut down and exit the industry. In the short run, if TR < TVC, the firm will shut down but remain in business. In all other cases, the firm will remain in business. 6. (7 points) A large corporate sugar beet farmer in western Minnesota hires you as a management consultant. He tells you that he uses two variable inputs in production, units of capital (K) and worker-hours of labor (L). Based on historical data from the industry, you determine that sugar beet production is characterized by the following production function: q = K 1/3 L 2/3 a. The farmer has day labor available to him, and wonders how much additional output he can get if he hires 5 workers for 8 hours each, holding capital constant. Provide him with the equation he needs to figure this out. Additional output from hiring 40 more workers-hours = output after hiring the extra workers output before hiring the extra workers. Mathematically, Δq = q2 q1 = K 1/3( L+ 40) 2/ 3 - K 1/3 L 2/3 The above equation will yield the additional output of adding 40 worker hours for any given level of labor (L) and a given level of capital (K). b. Now the farmer is interested in exploring increasing the scale of his operation. Show him mathematically what would happen to output if he were to double both of his inputs. q = K 1/3 L 2/3 q 2 = (2K) 1/3 (2L) 2/3 q 2 = (2 1/3 )*(2 2/3 )* K 1/3 L 2/3 q 2 = (2 1/3 )*(2 2/3 )*q (2 1/3 )*(2 2/3 ) = 2 q 2 = 2q 3

4 So, doubling inputs would scale output by or 2, implying that this production function is characterized by constant returns to scale. c. Last, the farmer would like use a spreadsheet to predict long run total costs as a function of the wage, the user cost of capital, and output, given the technology that currently characterizes his production. Derive the total cost equation C(r,w,q) that is associated with the production function written above. You must use Lagrangian optimization techniques in your derivation. The steps to answering this question are found in the appendix to chapter 7. Set up the Lagrangian: Φ = wl rk - λ[k 1/3 L 2/3 -q] Differentiate with respect to L, K, and λ: 1) dφ/dl = w - λ[2/3k 2/3 L -1/3 ] = 0 2) dφ/dk = r - λ[1/3k -2/3 L 2/3 ] = 0 3) dφ/dλ = K 1/3 L 2/3 q = 0 Solve 1) for w and 2) for r w = λ[2/3k 2/3 L -1/3 ] r = λ[1/3k -2/3 L 2/3 ] Solve for r/w: r/w =.5L/K Solve for L and K: L = 2rK/w K =.5Lw/r Substituting these back into 3) yields L and K as functions of q, r, and w: q = (.5Lw/r) 1/3 L 2/3 q = L*(.5w/r) 1/3 L = q/(.5w/r) 1/3 q = K 1/3 (2rK/w ) 2/3 q = K*(2r/w) 2/3 K = q/(2r/w) 2/3 4

5 Substitute back into the cost function to arrive at c(q,r,w): C = wl + rk C = w(q/(.5w/r) 1/3 ) + r(q/(2r/w) 2/3 ) C = wq/(.5w/r) 1/3 + rq/(2r/w) 2/3 ) C = q(w/(.5w/r) 1/3 + r/(2r/w) 2/3 ) 7. (8 points) Consider the following table, which provides the number of U.S. breweries (beer companies) by year, by their capacity (the number of barrels of beer they are capable of producing). Number of Breweries, by year, by capacity Capacity (Thousands of Barrels) , ,001-2, ,001-4, , Source: Brock, James. The Structure of American Industry (Upper Saddle River, NJ: Pearson, 2009), page 139. a. Based upon these numbers, what can you say about the optimal scale of a brewery? Use a graph or graphs with cost curves in your answer. As time passes, the number of firms in all but the category decreases, with more dramatic reductions the smaller the firm. For now, ignore the fact that the two smallest categories begin to grow again in 1989, as this is unrelated to cost. Based on these numbers, we can see that firms either expanded, consolidated to increase output or they exited the industry. One reason firms might try to grow output in the long run is economies of scale, as long run profit maximization requires that firms produce at the minimum of long run average cost. For example, this graph up to Q* is a long run average cost curve with economies of scale: 5

6 In the long run, if economies of scale indeed characterize these firms, then the optimal scale is large (4001+ group). b. What about the technology of brewing and the markets for beer might explain the patterns in the data that you see in the table? Economies of scale can be caused by increasing returns to scale. If a firm can double inputs and more than double outputs, the cost per unit of output decreases holding input prices constant. Breweries experience increasing returns to scale because doubling the amount of metal in the vats and pipes (capital) will more than double the volume that the vats and pipes can produce. At the simplest level, doubling the surface area of a vat will more than double the volume that vat can hold. This would explain why firms have gotten larger, but not why smaller firms have staged a comeback. Microbreweries that purport to focus on quality rather than quantity have successfully differentiated themselves in the eyes of consumers in recent decades. While they are still at a cost disadvantage, they do not compete directly with low cost giants like Miller and Molson Coors, because consumers view them as a higher quality good. This seems like a compelling reason why microbreweries have resurged despite a clear cost disadvantage. 6