Production

WSG5 7/7/03 4:35 PM Page 635 Productioin OVERVIEW This chapter reviews the general problem of transforming productive resources in goods and services for sale in the market. A production function is the technological relationship between the maximum amount of output that a firm can produce with a given combination of inputs (factors of production). The short run in production is defined as that period of time during which at least one factor of production held fixed. The long run in production is defined as that period of time during which all factors of production are variable. In the short run, the firm is subject to the law of diminishing returns (sometimes referred to as the law of diminishing marginal product), which states that as additional units of a variable input are combined with one or more fixed inputs, at some point the additional output (marginal product) will start to diminish. The short-run production function is characterized by three stages of production. Assuming that output is a function of labor and a fixed amount of capital, stage I of production is the range of labor usage where the average product of labor (APL) is increasing. Over this range of output, the marginal product of labor (MPL) is greater than the average product of labor. Stage I ends and stage II begins where the average product of labor is maximized, i.e., APL = MPL. Stage II of production is the range of output where the average product of labor is declining and the marginal product of labor is positive. In other words, stage II of production begins where APL is maximized, and ends wither MPL = 0. Copyright © 2003 by Academic Press. All rights of reproduction in any form reserved. Managerial Economics: Theory and Practice 63 Its appeal is largely due to the fact that it exhibits several desirable mathematical properties. If capital and labor are substitutable. IRTS occurs when eQ > 1. Decreasing returns to scale (DRTS) occurs when a proportional increase in all inputs results in a less than proportional increase in output. The Cobb-Douglas production function is the most popular specification in empirical research.e. Accounting data. the marginal rate of technical substitution (MRTSKL) is the amount of a factor of production that must be added (subtracted) to compensate for a reduction (increase) in the amount of another input to maintain a given level of output. . i. Another way to measure returns to scale is the coefficient of output elasticity (eQ). Returns to scale refer to the proportional increase in output given an equal proportional increase in all inputs. MPL/MPK. Most empirical studies of cost functions use time series accounting data. the law of diminishing returns to a variable input. The coefficient of output elasticity is equal to the sum of the output elasticity of labor (eL) and the output elasticity of capital (eK). There are also other problems associated with the use of accounting data including output heterogeneity and asynchronous timing of costs. including substitutability between and among inputs. the effects of changes in inflation. including an inability to show marginal product in stages I and III of production. Since all inputs are variable. In stage II and stage III of production.WSG5 7/7/03 4:35 PM Page 64 64 Production Stage III of production is the range of product where the marginal product of labor is negative. accounting practices.According to economic theory.. The Cobb-Douglas production function has several shortcomings. average variable and marginal cost curves. etc. which present a number of problems. as well as the expected “U-shaped” average total. If we assume two factors of production. tend to ignore opportunity costs. and returns to scale. for example. eQ = eL + eK. tax rates.e. DRTS occurs when eQ < 1. i. Cubic cost functions exhibit this theoretical relationship. which is defined as the percentage increase (decrease) in output with respect to a percentage increase (decrease) in all inputs. CRTS occurs when eQ = 1. social security contributions. Economic theory suggests that short-run total cost as a function of output first increases at an increasing rate. returns to scale is a long-run production phenomenon. production in the short run for a “rational” firm takes place in stage II of production. APL > MPL. the marginal rate of technical substitution is defined as the ratio of the marginal product of labor to the marginal product of capital. then increases at a decreasing rate. Constant returns to scale (CRTS) occurs when a proportional increase in all inputs results in the same proportional increase in output. labor insurance costs.. however. Increasing returns to scale (IRTS) occurs when a proportional increase in all inputs results in a more than proportional increase in output. long-run D. long-run. short-run. but plans in the ______. D. The firm operates in the ______. The period of time during which at least one factor of production is fixed.4 5. long-run C. Greater than one year. Summarizes the relationship between output and factors of production. C. The period of time during which at least one factor of production is fixed. The period of time during which all factors of production are variable. B. A. long-run. short-run. The minimum amount of inputs required to produce a given amount of output. D. B. Production in the long run is: A. D. B. 5.1 Production function represent: A. Is only applicable in the short run. The period of time during which all factors of production are variable. short-run B. Less than one year. The maximum amounts of inputs required to product a given amount of output. C. D.2 5. C. Summarizes the least-cost combination of inputs to produce a given output. Both B and C are correct.WSG5 7/7/03 4:35 PM Page 65 Multiple Choice Questions 65 MULTIPLE CHOICE QUESTIONS 5. Cannot be empirically estimated. B. Production in the short run is: A. C. The period of time during which all factors of production are fixed.5 . The period of time during which all factors of production are fixed. The maximum amount of output that can be obtained from a given set of inputs.3 5. short-run The production function: A. The total product of labor is increasing at a decreasing rate. C. C.7 5. The change in total output resulting from a change in labor input. The Cobb-Douglas production: A. D. MPL = MPK. L). b. II. M). Both A and D are correct. C. Diminishing marginal product of labor implies that: XIII. MPL > MPK. Is Q = AKaLbMg. Can be used to determine returns to scale.WSG5 7/7/03 4:35 PM Page 66 66 5. Total output per unit of labor input. E. For this to happen: . Consider the production function Q = f(K. L is labor.6 Production Consider the production function Q = f(K. D. MPK > APK. II only. 5. This action resulted in an increase in the total product of labor. L. XIV. The marginal product of labor is: A.8 5. B. XV. Illustrates the substitutability of the factors of production. and M is land. and III only. where K is capital.10 In response to an increase in demand.9 The average product of capital increases when: A. I only. L). Which of the following is correct? A. C. B. where K is capital and L is labor. E. The marginal product of labor is falling. APL > MPL. a microchip manufacturer increased its labor force by 5 percent. MPK > MPL. g) ≤ 1. E. D. Suppose that Q = f(K. D. The change in labor output resulting from a change in total output. where A is a constant and 0 ≤ (a. The marginal product of labor is less than the average product of labor. where K is capital and L is labor. B. I and II only. All of the above. 5. The contribution to total output from the last unit of labor employed. Is consistent with the law of diminishing marginal product. B. 6. Which of the following is correct? A.650. Rising. B. The marginal product of labor must have increased. C.5L0.13 Suppose that the firm’s production function is Q = 25K0. then APL must be: A. Output increases less than proportionally to a proportional increase in all inputs. Output decreases following a proportional increase in all inputs. 1.11 When MPL < APL. D. 5. D.5L0.5. B. 1. C. 4.12 Suppose that the firm’s production function is Q = 25K0. 5. then the marginal product of labor is: A. D. None of the above.5. If K = 100 and L = 25. Falling. III only.25. II. Additional units of capital must have been employed. D. Output increases at a decreasing rate following a proportional increase in all inputs. 2. 1.5L0. Input usage falls as the rate of output increases. D. 5. 2.5. 6. B. III. B. B.WSG5 7/7/03 4:35 PM Page 67 Multiple Choice Questions 67 I. . 5. If K = 121 and L = 36. D.15 Decreasing returns to scale occurs when: A. C. then the marginal product of capital is: A. II only. E.250. Greater than APK.14 Suppose that the firm’s production function is Q = 25K0.500. 25. I only. C. C. B. then total output is: A. 10. The average product of labor must have increased.5. 1. Equal to MPL. 5. C. I and III only.750. If K = 225 and L = 36. 5. L). C. Is a short-run phenomenon. F. Diminishing returns to labor occurs when: A. II only. II and III only.16 If the output remains unchanged when all inputs are doubled. III. MPL declines as more labor is added to a fixed amount of capital. Zero returns to scale. .WSG5 7/7/03 4:35 PM Page 68 68 Production 5. Can only occur when all inputs are increased proportionally. which shows two total product curves for producing silicon chips using different amounts of a fixed factor (capital) and different amounts of a variable factor (labor). Decreasing returns to scale. D. E. Constant returns to scale. D. E. FIGURE 1 5. Which of the following is correct? A. MPL declines as more labor is added to an increased amount of capital. A movement from A to B exhibits decreasing returns to scale.17 Consider Figure 1. where K is capital and L is labor. Is a long-run phenomenon. C. D. A movement from A to C exhibits increasing returns to scale. A movement from A to D exhibits decreasing returns to scale. III only. MPL declines because of increased specialization. C. Increasing returns to scale. D. then the production function exhibits: A. MPL declines because the ratio of capital to labor is increasing. I only. Implies that productive resources are not efficiently employed.19 The law of diminishing returns: C. 5. I.18 Consider the production function Q = f(K. 5. II. I and II only. B. B. B. D. B. TPL is maximized. E. C. Summarizes all the combinations of K and L that can be efficiently produced with a given production technology. 100 + 5L + 0. D. where K is capital and L is labor. If MPL < 0. MPL is maximized. Summarizes all the combinations of two outputs that can be produced with a given combination of K and L. B. 5. Summarizes all the combinations of K and L necessary to produce a given level of output.22 In Stage II of production: A. C. . Both C and D are correct. An isoquant: A. B. Stage I of production. then the firm must be operating in: A. APL = 0.025L2 C. D. 5. Stage IV of production.23 Stage I of production ends where: A. APL is maximized. C.24 Stage II of production ends where: A. C. L). APL = MPL.WSG5 7/7/03 4:35 PM Page 69 Multiple Choice Questions 69 5.25 Consider the production function Q = f(K. MPL must be greater than APL. 100 + 5L . 5. D. Stage III of production. APL is maximized. Cannot be determined on the basis of the information provided. C. where K is capital and L is labor. Illustrates the least-cost combination K and L necessary to produced a given level of output. Both A and B are correct.0. MPL is maximized. 100 + 5L B. APL must be greater than MPL. 5. MPL = 0. MPL must be increasing. D. 5. L).025L2 D. APL is increasing. Stage II of production. TPL must be increasing at an increasing rate.21 Consider the production function Q = f(K.20 Which of the following production functions exhibits the law of diminishing returns as soon as production begins? A. B. E. B. B. The marginal rate of technical substitution may be defined as: I. C. I and II only.27 Consider the production function Q = f(K. As more of K is used. Equal to the slope of the isoquant. II only. increasingly smaller amounts of L must be substituted for it in order to produce a given level of output. E. I only. D. K and L must be used in fixed proportions. The slope of an isoquant is: I. A convex isoquant indicates that: A. III. where K is capital and L is labor. K and L are imperfectly substitutable. The capital-labor ratio. L). L). B. The negative of the ratio of marginal products of the inputs. I and II only. where K is capital and L is labor. K and L must be used in fixed proportions. Which of the following is correct? A. 5. II. II. D. where K is capital and L is labor. 5. Which of the following is correct? A.26 Consider the production function Q = f(K. L). K and L are imperfect substitutes for one another. K and L are perfectly substitutable. D. The rate at which K and L may be substituted for each other while total output remains constant. B. I only. . II and III only.WSG5 7/7/03 4:35 PM Page 70 70 Production 5. II only. L). 5. III. The rate at which all combinations of K and L equal total cost. None of the above statements are true. C. K and L are perfect substitutes for one another. D. B. An “L-shaped” isoquant indicates that: A. C. B. III only. C. where K is capital and L is labor. The rate at which one input can be substituted for another input.29 Consider the production function Q = f(K. II and III only.28 Consider the production function Q = f(K. 45L0. E. where K is capital and L is labor. This production function exhibits: A. C. B. where K is capital and L is labor.33L0. A linear isoquant indicates that: A.25K0. Decreasing returns to scale. where K is capital and L is labor.30 Consider the production function Q = f(K. 1. C. L).32 The output elasticity of capital is: A. where K is capital and L is labor. Zero returns to scale. -L/K. 5. 0. B. K and L must be used in fixed proportions.66. None of the above.5.33 Consider the Cobb-Douglas production function Q = 33K0. Decreasing returns to scale. E. B. Zero returns to scale.55.34 The production function is Q = KL-1 exhibits: A. Constant returns to scale.31 Suppose that the firm’s production function is Q = 2L0. 5. The coefficient of output elasticity is: A. Cannot be determined from the information provided. 5. C. . None of the above statements are true. C. Cannot be determined from the information provided. Increasing returns to scale.35 Consider the Cobb-Douglas production function Q = 47K0. -K/L. -K/2L. 0.55. B. K and L are imperfect substitutes for one another. With K on the vertical axis and L on the horizontal axis. Increasing returns to scale.WSG5 7/7/03 4:35 PM Page 71 Multiple Choice Questions 71 5. K and L are perfect substitutes for one another.45. K/L. MPK/APK. the marginal rate of technical substitution is: A. 5. D. D. B. B. 5. D. C. ∂APK/K. ∂MPK/K. C. D. Constant returns to scale. D. D. -2K/L. 10KL2 Verify that APL = MPL when APL is maximized.2 Suppose that a firm’s production function is Q = 7K0. 5.4 A firm’s production function is: Q = 200KL . and L is labor hours.6 A firm’s production function is: Q = 250K0. What is the firm’s marginal product of labor equation? C. What is the equation of the corresponding isoquant in terms of L? B. 5. What is the firm’s average product of labor equation? 5. Suppose that the amount of K available to the firm is fixed at 144 machine hours. .3 A firm’s production function is: Q = 250K0.5L0.7L0. K is machine hours.3 Suppose that Q = 1.5 where Q is units of output.000.7L0.3 Production Determine the marginal product of capital and the marginal product of labor when K = 50 and L = 150.3 Verify that this expression exhibits the law of diminishing marginal product with respect to variable labor input.5 A firm’s production function is: Q = 250K0. A.3 Verify that APL = MPL when APL is maximized.7L0. 5. Demonstrate that this isoquant is convex with respect to the origin.WSG5 7/7/03 4:35 PM Page 72 72 SHORTER PROBLEMS 5.7L0. What is the firm’s total product of labor equation? B.1 A firm’s production function is: Q = 250K0. 5. A. 0. Suppose that K = L = 2. do these production functions exhibit increasing. decreasing or constant returns to scale? B. and B. L represents labor hours.5L2 A. At what level of labor usage is APL = MPL? 5. What is the equation for the marginal product of labor (MPL)? C. or constant returns to scale? 5. What is the equation for the total product of labor (TPL)? B. and L represents labor hours. decreasing.5L0. and F = 50. D. C.4F0. Suppose that K = 625.WSG5 7/7/03 4:35 PM Page 73 Longer Problems 73 LONGER PROBLEMS 5. .25K3 where Q is units of output per month. and F represents thousands of square feet of factory floor space.0. Estimate the coefficient of output elasticity. C. Suppose that K = L = 1.5L0.4 where Q represents of units of output.3 Consider the following production functions: 1) 2) 3) 4) Q = 200K0.5L3 . Calculate total output for each production function. Determine the marginal product equations. Based upon your answers to parts A.5 Q = 2K + 2L + 5KL Q = 50 + 5K + 5L Q = 3K/3L where Q represents of units of output. A. Does this production function exhibit increasing.2 The research department of Merriweather International has estimated the following production function Q = 50K0.1 Suppose that the average product of labor is given by the equation APL = 100 + 500L . Estimate total output. Assume that L = 20 and K = 10. 5.4 Consider the following production function: Q = 300KL + 18L2 + 3K2 . Calculate total output for each production function. A. L = 20.. B. L is the number of workers. and K is units of capital. K represents units of capital. K represents units of capital. C. C. Calculate APL and APK.3 = 175K-0.25 5. C. B. B.7L0.3L0. L is the number of workers.32 MPL = ∂QL/∂L = 0.5 C. How much K and L will maximize Q? B. C.15 5.3 = 243. D. C.5(7)(144)0.7 = 34.14 5.7(250)K-0.19 5.5 B.20 5. Calculate MPL and MPK. and K is units of capital.9 5.6 5. C.18 D. C.WSG5 7/7/03 4:35 PM Page 74 74 A. Calculate total output per month.5 5.5 Consider the following production function: Q = 225L + 100K .26 5.17 5. MPL = dTPL/dL = 0. D. B. C.5 = 42L-0.3 5.30 5.2 5.31 5.33 5. C. A.5L0.7 5. A. E.5/L = 84L-0. APL = TPL/L = 84L0. SOLUTIONS TO SHORTER PROBLEMS 5.24 5.3 = 175(L/K)0. 5.3L0. C. B.5K2 + KL Production where Q represents units of output per month.7. E. C.13 5. D.5L2 . A.7 = 75(K/L)0.3(250)K0.23 5. A. E. B.27 5.22 5.11 5.10 5. TPL = 7(144)0.28 5.5 = 84L0.7L-0. What is the maximum output per month? ANSWERS TO MULTIPLE CHOICE QUESTIONS 5. C.2 . B. C. A.1 MPK = ∂QK/∂K = 0. 5. C. E.5 5.4 5.76 A.29 5. B. C.5L-0.35 D.8 5.34 5.32 5.3-1 = 75K0.1 5.12 5.21 5. B. A.16 5. 7L0.3/0.000(250L0..10KL2)/L MPL = APL MPL = ∂QL/∂L = 0.3(250)K0.7)(-0.10KL2 200K .39/0.7 > 0 d2K/dL2 = (-1. 5.20KL = (200KL .10KL2) = 0 (200K .7)(41/0.20KL)L .7L-1.3/0.7) APL = Q/L = (250K0. Since L > 0.3 MPL = 0.3/0.e. A. 1.7L-0.000(250-1L-0.3/0.7)(41/0.7 = 1/L1.7L-0.3/0.3/0.20KL)L = 200KL .3(250K0.10KL2)]/L2 = 0.7L0.250K0.7L-2. The second partial derivative of the production function is ∂MPK/∂K = ∂2QK/∂K2 = -0.7L0.3 = 0 0.3/0.WSG5 7/7/03 4:35 PM Page 75 Solutions to Shorter Problems 75 5.3 Solving this equation for K in terms of L yields K0.10KL2)/L ∂APL/∂L = [(200K .7> 0.7 < 0 which is clearly negative since L-1. i.7)1/0.3/0.7(75)K0.7 = 1.7L-0..7 ) = (0.7L-0.3) = 4L-0. L > 0 implies that (200K .3/0.4 5. the first-order condition for APL maximization.3)-1 = 1.7L-0.7L0.6 .7 Since L = 1/L2.7L-1.(200KL .7)L .7L0.7L-2.250K0.3/0.7 ) > 0 -2.3)/L ∂APL/∂L = [0.7 > 0 Since the first derivative is negative and the second derivative is positive then the isoquant is convex with respect to the origin.7 = 1/L1.7 = (4L-0.7 L-0.7 ) < 0 Since L-1. Taking the first and second derivatives of this expression yields dK/dL = (-0. this implies that 0.7)L .7 > 0 since L and K are positive.3/0.20KL APL = (200KL .3 (K0.7)(41/0.5L-0. the first-order condition for APL maximization.7 = 75K0. i.7) = 250K0.7) K = 41/0.(200KL .3/L MPL = APL MPL = 200K .000 = 250K0.3(250K0.3(250K0.3]/L2 = 0.e.7 B.3)(1/0.20KL)L .5 5.3(250K0. 5L0. L = 50 d2APL/dL2 = -10 < 0.4F0.6 D.4F0.5L0. 1) 2) 3) 4) B.5(2)0.5 + 0.5L0.2 5.5K0.1 Production A.5L2 = 100 + 1.4F0.4 = 25K-0.WSG5 7/7/03 4:35 PM Page 76 76 SOLUTIONS TO LONGER PROBLEMS 5.3Q If all inputs are increased by the scalar l. Alternatively.350K0.3F-0.4 + 0. the first-order condition for APL maximization.4F-0. APL = MPL 100 + 500L .5(lL)0.4F0.000L .3 .5L0.4l0.e. B.5L0.6F0.4 = 50(25)(3.5 = 400 Q = 2(2) + 2(2) + 5(2)(2) = 28 Q = 50 + 5(2) + 5(2) = 70 Q = 3(2)/3(2) = 1 5.15L2 C. enrollments will increase by l1.4F0.15L2 10L2 = 500L 10L = 500 L = 50 Alternatively.5(20)0.4 = 1.5 = 200 Q = 2(1) + 2(1) + 5(1)(1) = 9 Q = 50 + 5(1) + 5(1) = 60 Q = 3(1)/3(1) = 1 Q = 200(2)0.4(50)0.4 + 0. Q = 50K0.3 > 0.5 + 0. dAPL/dL = 500 .4L0. this production function exhibits increasing returns to scale.4(50)K0.5L3 B. the above production function exhibits increasing returns to scale because 0. 1) 2) 3) 4) Q = 200(1)0.25 A..3.5L0.000L .5L2)L = 100L + 500L2 . the second-order condition for APL maximization is satisfied.4(lF)0.6F0. i.5l0. Let l be some factor of proportionality. i.78) = 19.777.4 = 20K0.5(1)0.4 = l1. Thus. MPL = dTPL/dL = 100 + 1.4 = 50l0.4 MPF = 0.4(50)K0.31)(4.5L-0.5(50)K-0.3 C.4 = l1. A. E¢ = 50(lK)0. TPL = (APL)L = (100 + 500L .4 = 50(625)0.4 = 1.5L-0.6 = 20K0.e.4 MPL = 0. eQ = eK + eL + eF = 0.. MPK = 0.10L = 0. 000 + 600 .250 B. then this production function exhibits decreasing returns to scale.5 + 0.25(10)3/10 = 3.0.WSG5 7/7/03 4:35 PM Page 77 Solutions to Longer Problems 77 C.5L3/K . then this production function exhibits constant returns to scale.5L3 .0.0. then this production function exhibits increasing returns to scale.200 + 300 .200 = 2.0.5(20)3/10 .25K3)/L = 300K + 18L + 3K2/L .5L3 . 4) Since output less than doubled (remained unchanged) as K and L were doubled. Alternatively.3(20)2 = 3.5(20)2 -0.325 .3L2 = 300(10) + 36(20) .25(10)2 = 6.75 = 6.000 + 7.1.000 + 720 + 30 .000 .000 + 360 + 15 .4 A.25(10)3 = 60.250 = 63.5L2 . 3) Since output less than doubled as K and L were doubled.25 = 6.5 = 1.5(20)3 .520 MPK = ∂Q/∂K = 300L + 6K .25K3/L = 300(10) + 18(20) + 3(10)2/20 .0.0. Since 0.525 C. MPL = ∂Q/∂L = 300K + 36L .0.150 APK = Q/K = (300KL + 18L2 + 3K2 .200 .0.000 + 720 .0.0.25 = 3. then this production function exhibits constant returns to scale. 1) Since output doubled as K and L were doubled. Q = 300(10)(20) + 18(20)2 + 3(10)2 .0.25K3)/K = 300L + 18L2/K + 3K . APL = Q/L = (300KL + 18L2 + 3K2 .0.75(10)2 = 6. then this production function exhibits decreasing returns to scale.25K2 = 300(20) + 18(20)2/10 + 3(10) .75K2 = 300(20) + 6(10) .4. for Cobb-Douglas production functions.0. 5.400 . 2) Since output more than doubled as K and L were doubled.0.0. returns to scale may be determined by adding the values of the exponents. 77)2 .15L + K = 0.77) = 3.24 = 2.WSG5 7/7/03 4:35 PM Page 78 78 5. ∂Q/∂L = 225 .7.16 + 185.5(15.372. the first-order condition for Q maximization.74) .74)(15. the first-order condition for Q maximization.77) + 100(11.e. i. ∂Q/∂K = 100 ..61 .77 B.174.548.10K + L = 0.50 .e. i.865. Solving the first-order conditions simultaneously yields L* = 15.1..5 Production A.63 .66 + 1.77 K* = 11.74)2 + (11.5(11. Q* = 225(15.670.