Production Notes

TEACHING NOTES This chapter represents a turning point away from the theory of demand and toward a theory of supply. This is a good time to review the derivation of demand and present an overview of the theory of competitive supply, particularly because many ideas that were presented in Chapters 3 and 4 will now be used in a slightly different context. Point out that while we could not measure utility, we can measure output, and thus develop a richer theory for production than we could for consumption. In presenting the production function, emphasize the assumption of technical efficiency: the production function defines the highest level of output for every combination of inputs. Also stress that throughout this chapter we focus on real quantities; there are no prices, which will be introduced in Chapter 7. Exercises (3) through (5) deal more with intuitive concepts, while Exercises (6) through (8) are more mathematical. After introducing production, we hold capital fixed and examine the productivity of labor. Emphasize the time dimension of fixed vs. variable: in the short run, some forms of capital are fixed and some forms of labor are variable. Build on students’ appreciation of Examples (1) and (2) in the text. Because of the national attention given to the competitiveness of U.S. industry, students appreciate the discussion of labor productivity in developing countries; see Table 6.4. In this context, you can add a discussion of the relationship between total, average, and marginal products. If you have not discussed marginal utility, now is the time to make sure that they know the difference between average and marginal. An example that captures students’ attention is the relationship between average and marginal test scores. If their latest mid-term grade is greater than their average grade to date, this will increase their average. The three stages of production help students remember the relationship between average and marginal products: Stage 1 MP > AP AP rising MP positive Stage 2 MP < AP AP falling MP positive Stage 3 MP < AP AP falling MP negative Exercises (1), (2), and (5) rely on the relationship between AP and MP. This chapter also focuses on the use of isoquants to represent the relationships between inputs (the marginal rate of technical substitution) and between inputs and output (returns to scale). Rely on the students’ understanding of indifference curves when discussing isoquants. As with indifference curves, isoquants are a two-dimensional representation of a three-dimensional production function. Use Examples 6.3 and 6.4 to give concrete meaning to MRTS and returns to scale. Exercise (4) discusses MRTS. Exercise (6) discusses returns to scale. REVIEW QUESTIONS 1. What is a production function? How does a long-run production function differ from a short-run production function? A production function represents how inputs are transformed into outputs by a firm. We focus on the firm with one output and aggregate all inputs or factors of production into one of several categories, such as labor, capital, and materials. In the short run, one or more factors of production cannot be changed. As time goes by, the firm has the opportunity to change the levels of all inputs. In the long-run production function, all inputs are variable. 2. Why is the marginal product of labor likely to increase and then decline in the short run? 59 Chapter 6: Production When additional units of labor are added to a fixed quantity of capital, we see the marginal product of labor rise, reach a maximum, and then decline. The marginal product of labor increases because, as the first workers are hired, they may specialize in those tasks in which they have the greatest ability. Eventually, with the quantity of capital fixed, the workplace becomes congested and the productivity of additional workers declines. 3. Diminishing returns to a single factor of production and constant returns to scale are not inconsistent. Discuss. Diminishing returns to a single factor are observable in all production processes at some level of inputs. This fact is so pervasive that economists have named it the “law of diminishing marginal productivity.” If there is diminishing marginal productivity, at least one factor of production is held constant. For example, when holding the level of capital constant, each additional unit of labor has less capital to work with. Unlike the returns to a single factor, returns to scale are proportional increases in all inputs. While each factor by itself exhibits diminishing returns, output increases more than proportionately (increasing returns to scale), proportionately (constant returns to scale), or proportionally less (decreasing returns to scale). 4. You are an employer seeking to fill a vacant position on an assembly line. Are you more concerned with the average product of labor or the marginal product of labor for the last person hired? If you observe that your average product is just beginning to decline, should you hire any more workers? What does this situation imply about the marginal product of your last worker hired? In filling a vacant position, you should be concerned with the marginal product of the last worker hired because the last worker hired will influence total product. The point at which the average product begins to decline is the point where average product is equal to marginal product. Although adding more workers results in a further decline in average product, total product continues to increase. Only when total product declines do we necessarily stop hiring. When average product declines, the marginal product of the last worker hired is lower than the average product of previously hired workers. 5. Faced with constantly changing conditions, why would a firm ever keep any factors fixed? What determines whether a factor is fixed or variable? Whether a factor is fixed or variable depends on the time horizon in consideration: all factors are fixed in the very short run; all factors are variable in the long run. Therefore, the firm must, by definition, have one or more factors fixed in the short run. As stated in the text: “All fixed inputs in the short run represent outcomes of previous long-run decisions based on firms’ estimates of what they could profitably produce and sell.” 6. How does the curvature of an isoquant relate to the marginal rate of technical substitution? The isoquant represents the trade-off between inputs in production while holding output constant. The curvature of the isoquant represents the change in this tradeoff as inputs are added or subtracted. The marginal rate of technical substitution is the slope of the isoquant. One can show that this slope is equal to the ratio of the marginal rates of productivity for the respective inputs. The marginal rate of technical substitution changes with different levels of each input because of declining marginal productivity. The marginal rate of technical substitution changes at different input levels, thus forming the isoquant in its typical curved shape. If inputs are perfect substitutes, the ratio of marginal productivities is constant and the isoquant is linear. If the inputs are perfect complements, an addition unit of one 60 Chapter 6: Production input does not increase output, unless there is a complementary addition of the other input. This case leads to an L-shaped isoquant in which the slope of the horizontal segment is zero and the slope of the vertical segment is undefined. 7. Can a firm have a production function that exhibits increasing returns to scale, constant returns to scale, and decreasing returns to scale as output increases? Discuss. Most firms have production functions that exhibit increasing, constant, and decreasing returns to scale. At low levels of output, proportional increases in all inputs may lead to larger-than-proportional increases in output due to increased opportunities for specialized factors of production. As the firm grows, it may duplicate its level of operation such that output increases at the same rate as the increased use of all inputs. At some scale, the firm experiences an inability to increase output at the same rate as increases in input, a situation that can arise from management diseconomies. 8. Give an example of a production process in which the short run involves a day or a week and the long run any period longer than a week. Any small business where one input requires more than a week to change would be an example. The process of hiring more labor, which requires announcing the position, interviewing applicants, and negotiating terms of employment, can take a day, if done through a temporary employment agency. Usually, however, the process takes a week or more. Expansion, requiring a larger location, will also take longer than a week. EXERCISES 1. Suppose a chair manufacturer is producing in the short run when equipment is fixed. The manufacturer knows that as the number of laborers used in the production process increases from 1 to 7, the number of chairs produced changes as follows: 10, 17, 22, 25, 26, 25, 23. a. Calculate the marginal and average product of labor for this production function. The average product of labor, APL, is equal to MPL, is equal to Q . The marginal product of labor, L Q , the change in output divided by the change in labor input. L For this production process we have: b. L Q APL MPL 0 0 __ __ 1 10 10 10 2 17 8 1/2 7 3 22 7 1/3 5 4 25 6 1/4 3 5 26 5 1/5 1 6 25 4 1/6 -1 7 23 3 2/7 -2 Does this production function exhibit diminishing returns to labor? Explain. 61 Chapter 6: Production This production process exhibits diminishing returns to labor characteristic of all production functions with one fixed input. The first worker has a marginal product of 10; from the second to the fifth workers, the marginal product is declining, but still positive. The sixth and seventh workers have negative marginal products. c. Explain intuitively what might cause the marginal product of labor to become negative. Labor’s negative marginal product for L > 5 may arise from congestion in the chair manufacturer’s factory. Since more laborers are using the same, fixed amount of capital, it is possible that they could get in each other’s way, decreasing efficiency and the amount of output. 2. Fill in the gaps in the table below. Quantity of Variable Input Total Output 0 0 1 150 Marginal Product of Variable Input Average Product of Variable Input ___ ___ 2 200 3 4 200 760 5 150 6 Quantity of Variable Input 150 Total Output Marginal Product of Variable Input Average Product of Variable Input 0 0 ___ ___ 1 150 150 150 2 400 250 200 3 600 200 200 4 760 160 190 5 910 150 182 6 900 -10 150 3. A political campaign manager has to decide whether to emphasize television advertisements or letters to potential voters in a reelection campaign. Describe the production function for campaign votes. How might information about this function (such as the shape of the isoquants) help the campaign manager to plan strategy? The output of concern to the campaign manager is the number of votes. The production function uses two inputs, television advertising and direct mail. The use of these inputs requires knowledge of the substitution possibilities between them. If the inputs are perfect substitutes, the resultant isoquants are line segments, and the campaign manager will use only one input based on the relative prices. If the inputs are not perfect substitutes, the isoquants will have a convex shape. The campaign manager will then use a combination of the two inputs. 62 Chapter 6: Production 4. A firm has a production process in which the inputs to production are perfectly substitutable in the long run. Can you tell whether the marginal rate of technical substitution is high or low, or is further information necessary? Discuss. The marginal rate of technical substitution, MRTS, is the absolute value of the slope of an isoquant. If the inputs are perfect substitutes, the isoquants will be linear. To calculate the slope of the isoquant, and hence the MRTS, we need to know the rate at which one input may be substituted for the other. 5. The marginal product of labor is known to be greater than the average product of labor at a given level of employment. Is the average product increasing or decreasing? Explain. If the marginal product of labor, MPL, is greater than the average product of labor, APL, then each additional unit of labor is more productive than the average of the previous units. Therefore, by adding the last unit, the overall average increases. If MPL is greater than APL, then APL is increasing. If the MPL is lower than the APL, then the last unit reduces the average. The APL is at a maximum when the productivity of the last unit is equal to the average of the previous units ( i.e., when MPL = APL). 6. The marginal product of labor in the production of computer chips is 50 chips per hour. The marginal rate of technical substitution of hours of labor for hours of machinecapital is 1/4. What is the marginal product of capital? We know that factors of production are utilized efficiently by a firm at the point where the production process equates the ratio of the marginal product to the marginal rate of technical substitution. Here, we are given the marginal product of labor and the marginal rate of technical substitution. To determine the marginal product of capital, substitute the given values for the marginal product of labor and the marginal rate of technical substitution into the following formula: MPL MPK  MRTS, or 50 1  , or MPK 4 MPK = 200 computer chips per hour. 7. Do the following production functions exhibit decreasing, constant or increasing returns to scale? a. Q = 0.5KL Returns to scale refers to the relationship between output and proportional increases in all inputs. This concept may be represented in the following manner, where  represents a proportional increase in inputs: F(K, L) > F(K, L) implies increasing returns to scale; F(K, L) = F(K, L) implies constant returns to scale; and F(K, L) < F(K, L) implies decreasing returns to scale. Therefore, we can substitute K for K and L for L, and check the result against an equal increase in Q. Q* = 0.5(K)(L) = (0.5KL)2 = Q2 > Q This production function exhibits increasing returns to scale. b. Q = 2K + 3L Q* = 2(K) + 3(L) = (2K + 3L) = Q = Q. This production function exhibits constant returns to scale. 63 Chapter 6: Production 8. The production function for the personal computers of DISK, Inc., is given by Q = 10K0.5L0.5, where Q is the number of computers produced per day, K is hours of machine time, and L is hours of labor input. DISK’s competitor, FLOPPY, Inc., is using the production function Q = 10K0.6L0.4. a. If both companies use the same amounts of capital and labor, which will generate more output? Let Q be the output of DISK, Inc., Q2, be the output of FLOPPY, Inc., and X be the same equal amounts of capital and labor for the two firms. Then, according to their production functions, Q = 10X0.5X0.5 = 10X(0.5 + 0.5) = 10X and Q2 = 10X0.6X0.4 = 10X(0.6 + 0.4) = 10X. Because Q = Q2, both firms generate the same output with the same inputs. b. Assume that capital is limited to 9 machine hours but labor is unlimited in supply. In which company is the marginal product of labor greater? Explain. With capital limited to 9 machine units, the production functions become Q  30L0.5 and Q2  37.37L0.4. To determine the production function with the highest marginal productivity of labor, consider the following table: L Q Firm 1 MPL Firm 1 Q Firm 2 MPL Firm 2 0 0.0 ___ 0.00 ___ 1 30.00 30.00 37.37 37.37 2 42.43 12.43 49.31 11.94 3 51.96 9.53 58.00 8.69 4 60.00 8.04 65.07 7.07 For each unit of labor above 1, the marginal productivity of labor is greater for the first firm, DISK, Inc. 9. In Example 6.3, wheat is produced according to the production function Q = 100(K0.8L0.2 ). a. Beginning with a capital input of 4 and a labor input of 49, show that the marginal product of labor and the marginal product of capital are both decreasing. For fixed labor and variable capital: K = 4  Q = (100)(40.8 )(490.2 ) = 660.21 K = 5  Q = (100)(50.8 )(490.2 ) = 789.25  MPK = 129.04 K = 6  Q = (100)(60.8 )(490.2 ) = 913.19  MPK = 123.94 K = 7  Q = (100)(70.8 )(490.2 ) = 1,033.04  MPK = 119.85. For fixed capital and variable labor: L = 49  Q = (100)(40.8 )(490.2 ) = 660.21 L = 50  Q = (100)(40.8 )(500.2 ) = 662.89  MPL = 2.68 L = 51  Q = (100)(40.8 )(510.2 ) = 665.52  MPL = 2.63 64 Chapter 6: Production L = 52  Q = (100)(40.8 )(520.2 ) = 668.11  MPL = 2.59. Notice that the marginal products of both capital and labor are decreasing as the variable input increases. b. Does this production function exhibit increasing, decreasing, or constant returns to scale? Constant (increasing, decreasing) returns to scale imply that proportionate increases in inputs lead to the same (more than, less than) proportionate increases in output. If we were to increase labor and capital by the same proportionate amount (l) in this production function, output would change by the same proportionate amount: Q = 100(K)0.8 (L)0.2, or Q = 100K0.8 L0.2 (0.8 + 0.2) = Q Therefore, this production function exhibits constant returns to scale. 65