Production

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5 Productioin

OVERVIEW

This chapter reviews the general problem of transforming productive resources in goods and services for 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 production is defined as that period of time during which at least one factor of production held fixed. fixed. The long run in production is defined as that period of time during which all factors of  production produ ction are are variable variable.. In the short short run, the firm is subject subject to the law law of  diminishing returns (sometimes referred to as the law of diminishing marginal product), which states that as additional units of a variable input input are combined with one or more fixed inputs, 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 of labor usage usage where the average product of labor (APL) is increasing increasing.. Over this this range of output, output, the marginal product of labor (MPL) is greater than the average product of  labor. Stage I ends and stage II begins begins where the average product product of labor is maxim maximize ized, d, i.e 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. positive. In other words,, stage II of production words production begins begins where APL is maximi maximized zed,, and ends ends wither MPL = 0.

Managerial Economics: Theory and Practice

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Stage III of production is the range of product where the marginal product of labor is negative. In stage II and stage III of production, APL > MPL.According to economic theory, production in the short run for a “rational” firm takes place in stage II of production. If we assume two factors of production, 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. If capital and labor are substitutable, the marginal rate of technical substitution is defined as the ratio of the marginal product of labor to the marginal product of  capital, i.e., MPL/MPK. Returns to scale refer to the proportional increase in output given an equal proportional increase in all inputs. Since all inputs are variable, returns to scale is a long-run production phenomenon. Increasing returns to scale (IRTS) occurs when a proportional increase in all inputs results in a more than proportional increase in output. Constant returns to scale (CRTS) occurs when a proportional increase in all inputs results in the same proportional increase in output. Decreasing returns to scale (DRTS) occurs when a proportional increase in all inputs results in a less than proportional increase in output. Another way to measure returns to scale is the coefficient of output elasticity (eQ), which is defined as the percentage increase (decrease) in output with respect to a percentage increase (decrease) in all inputs. The coefficient of output elasticity is equal to the sum of the output elasticity of labor (eL) and the output elasticity of capital (eK), i.e., eQ = eL + eK. IRTS occurs when eQ > 1. CRTS occurs when eQ = 1. DRTS occurs when eQ < 1. The Cobb-Douglas production function is the most popular specification in empirical research. Its appeal is largely due to the fact that it exhibits several desirable mathematical properties, including substitutability between and among inputs, the law of diminishing returns to a variable input, and returns to scale. The Cobb-Douglas production function has several shortcomings, however, including an inability to show marginal product in stages I and III of production. Most empirical studies of cost functions use time series accounting data, which present a number of problems. Accounting data, for example, tend to ignore opportunity costs, the effects of changes in inflation, tax rates, social security contributions, labor insurance costs, accounting practices, etc. There are also other problems associated with the use of accounting data including output heterogeneity and asynchronous timing of costs. Economic theory suggests that short-run total cost as a function of  output first increases at an increasing rate, then increases at a decreasing rate. Cubic cost functions exhibit this theoretical relationship, as well as the expected “U-shaped” average total, average variable and marginal cost curves.

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Multiple Choice Questions

MULTIPLE CHOICE QUESTIONS

5.1 Production function represent: A. The maximum amounts of inputs required to product a given amount of output. B. The maximum amount of output that can be obtained from a given set of inputs. C. The minimum amount of inputs required to produce a given amount of output. D. Both B and C are correct. 5.2 Production in the short run is: A. The period of time during which at least one factor of  production is fixed. B. The period of time during which all factors of production are variable. C. The period of time during which all factors of production are fixed. D. Less than one year. 5.3 Production in the long run is: A. The period of time during which at least one factor of  production is fixed. B. The period of time during which all factors of production are variable. C. The period of time during which all factors of production are fixed. D. Greater than one year. 5.4 The firm operates in the  ______, but plans in the  ______. A. long-run; short-run B. long-run; long-run C. short-run; long-run D. short-run; short-run 5.5 The production function: A. Is only applicable in the short run. B. Cannot be empirically estimated. C. Summarizes the relationship between output and factors of  production. D. Summarizes the least-cost combination of inputs to produce a given output.

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5.6 Consider the production function Q =  f (K, L, M),   where K  is capital, L is labor, and M  is land. The Cobb-Douglas production: A. Is Q = AK a Lb M g , where A is a constant and 0 ≤ (a  , b  , g ) ≤ 1. B. Illustrates the substitutability of the factors of production. C. Can be used to determine returns to scale. D. Is consistent with the law of diminishing marginal product. E. All of the above. 5.7 Consider the production function Q =  f (K, L), where K  is capital and L is labor. The marginal product of labor is: A. The change in total output resulting from a change in labor input. B. Total output per unit of labor input. C. The change in labor output resulting from a change in total output. D. The contribution to total output from the last unit of labor employed. E. Both A and D are correct. 5.8 Suppose that Q =  f (K, L), where K  is capital and L is labor. Diminishing marginal product of labor implies that: XIII. The marginal product of labor is less than the average product of labor. XIV. The marginal product of labor is falling. XV. The total product of labor is increasing at a decreasing rate. Which of the following is correct? A. I only. B. II only. C. I and II only. D. II, and III only. 5.9 The average product of capital increases when: A. MP L > MP K.  B. MP K > AP K.  C.  AP L > MP L. D. MP K > MP L. E. MP L = MP K.  5.10 In response to an increase in demand, a microchip manufacturer increased its labor force by  5 percent. This action resulted in an increase in the total product of labor. For this to happen:

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Multiple Choice Questions

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I. The average product of labor must have increased. II. Additional units of capital must have been employed. III. The marginal product of labor must have increased. Which of the following is correct? A. I only. B. II only. C. III only. D. I and III only. E. None of the above. 5.11 When MP L < AP L, then AP L must be: A. Rising. B. Falling. C. Equal to MP L. D. Greater than AP K.  5.12 Suppose that the firm’s production function is Q =  25K 0.5L0.5. If  K  = 121 and L =  36, then total output is: A. 1,250. B. 1,500. C. 1,650. D. 1,750. 5.13 Suppose that the firm’s production function is Q =  25K 0.5L0.5. If  K  = 100 and L =  25, then the marginal product of labor is: A.  2.5. B. 6.25. C. 10. D.  25. 5.14 Suppose that the firm’s production function is Q =  25K 0.5L0.5. If  K  =  225 and L =  36, then the marginal product of capital is: A.  2. B. 4. C.  5. D. 6. 5.15 Decreasing returns to scale occurs when: A. Output decreases following a proportional increase in all inputs. B. Output increases at a decreasing rate following a proportional increase in all inputs. C. Output increases less than proportionally to a proportional increase in all inputs. D. Input usage falls as the rate of output increases.

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5.16 If the output remains unchanged when all inputs are doubled, then the production function exhibits: A. Increasing returns to scale. B. Decreasing returns to scale. C. Constant returns to scale. D. Zero returns to scale. 5.17 Consider Figure 1, 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). I. A movement from A to B exhibits decreasing returns to scale. II. A movement from A to C exhibits increasing returns to scale. III. A movement from A to D exhibits decreasing returns to scale. Which of the following is correct? A. I only. B. II only. C. III only. D. I and II only. E. II and III only.

FIGURE 1

5.18 Consider the production function Q =  f (K, L), where K  is capital and L is labor. Diminishing returns to labor occurs when: A. MP L declines as more labor is added to a fixed amount of  capital. B. MP L declines as more labor is added to an increased amount of  capital. C. MP L declines because of increased specialization. D. MP L declines because the ratio of capital to labor is increasing. 5.19 The law of diminishing returns: C. Implies that productive resources are not efficiently employed. D. Is a long-run phenomenon. E. Can only occur when all inputs are increased proportionally. F. Is a short-run phenomenon.

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Multiple Choice Questions

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5.20 Which of the following production functions exhibits the law of  diminishing returns as soon as production begins? A. 100 + 5L B. 100 + 5L + 0.025L2 C. 100 + 5L - 0.025L2 D. Cannot be determined on the basis of the information provided. 5.21 Consider the production function Q =  f (K, L), where K  is capital and L is labor. If MP L < 0, then the firm must be operating in: A. Stage I of production. B. Stage II of production. C. Stage III of production. D. Stage IV of production. 5.22 In Stage II of production: A. TP L must be increasing at an increasing rate. B. MP L must be greater than  AP L. C.  AP L must be greater than MP L. D.  AP L is increasing. E. MP L must be increasing. 5.23 Stage I of production ends where: A.  AP L = MP L. B.  AP L is maximized. B.  AP L = 0. C. MP L is maximized. D. Both A and B are correct. 5.24 Stage II of production ends where: A. MP L is maximized. B.  AP L is maximized. C. TP L is maximized. D. MP L = 0. E. Both C and D are correct. 5.25 Consider the production function Q =  f (K, L), where K  is capital and L is labor. An isoquant: A. Summarizes all the combinations of K  and L necessary to produce a given level of output. B. Summarizes all the combinations of K  and L that can be efficiently produced with a given production technology. C. Summarizes all the combinations of two outputs that can be produced with a given combination of  K  and L. D. Illustrates the least-cost combination K  and L necessary to produced a given level of output.

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5.26 Consider the production function Q =  f (K, L), where K  is capital and L is labor. The slope of an isoquant is: I. The capital-labor ratio. II. The negative of the ratio of marginal products of the inputs. III. The rate at which one input can be substituted for another input. Which of the following is correct? A. I only. B. II only. C. III only. D. I and II only. E. II and III only. 5.27 Consider the production function Q =  f (K, L), where K  is capital and L is labor. A convex isoquant indicates that: A. K  and L are perfectly substitutable. B. K  and L must be used in fixed proportions. C. K  and L are imperfectly substitutable. D. As more of K  is used, increasingly smaller amounts of L must be substituted for it in order to produce a given level of output. 5.28 Consider the production function Q =  f (K, L), where K  is capital and L is labor. The marginal rate of technical substitution may be defined as: I. The rate at which K  and L may be substituted for each other while total output remains constant. II. Equal to the slope of the isoquant. III. The rate at which all combinations of K  and L equal total cost. Which of the following is correct? A. I only. B. II only. C. I and II only. D. II and III only. 5.29 Consider the production function Q =  f (K, L), where K  is capital and L is labor. An “L-shaped” isoquant indicates that: A. K  and L are perfect substitutes for one another. B. K  and L must be used in fixed proportions. C. K  and L are imperfect substitutes for one another. D. None of the above statements are true.

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5.30 Consider the production function Q =  f (K, L), where K  is capital and L is labor. A linear isoquant indicates that: A. K  and L are perfect substitutes for one another. B. K  and L must be used in fixed proportions. C. K  and L are imperfect substitutes for one another. D. None of the above statements are true. 5.31 Suppose that the firm’s production function is Q =  2L0.25K 0.5, where K  is capital and L is labor. With K  on the vertical axis and L on the horizontal axis, the marginal rate of technical substitution is: A. - 2K/L. B. -K/L. C. -K/2L. D. -L/K . 5.32 The output elasticity of capital is: A. ∂ AP K /K  .   B. MP K /AP    K.  C. ∂MP K /K  .   D. K/L. 5.33 Consider the Cobb-Douglas production function Q =  33K 0.33L0.66, where K  is capital and L is labor. This production function exhibits: A. Increasing returns to scale. B. Constant returns to scale. C. Decreasing returns to scale. D. Zero returns to scale. E. Cannot be determined from the information provided. 5.34 The production function is Q = KL-1 exhibits: A. Increasing returns to scale. B. Decreasing returns to scale. C. Constant returns to scale. D. Zero returns to scale. E. None of the above. 5.35 Consider the Cobb-Douglas production function Q = 47K 0.45L0.55, where K  is capital and L is labor. The coefficient of output elasticity is: A. 0.45. B. 0.55. C. 1. D. Cannot be determined from the information provided.

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SHORTER PROBLEMS

5.1 A firm’s production function is: Q = 250K0.7L0.3 Determine the marginal product of capital and the marginal product of labor when K  =  50 and L = 150. 5.2 Suppose that a firm’s production function is Q = 7K0.5L0.5 where Q is units of output, K  is machine hours, and L is labor hours. Suppose that the amount of  K  available to the firm is fixed at 144 machine hours. A. What is the firm’s total product of labor equation? B. What is the firm’s marginal product of labor equation? C. What is the firm’s average product of labor equation? 5.3 A firm’s production function is: Q = 250K0.7L0.3 Verify that  AP L = MP L when AP L is maximized. 5.4 A firm’s production function is: Q = 200KL - 10KL2 Verify that  AP L = MP L when AP L is maximized. 5.5 A firm’s production function is: Q = 250K0.7L0.3 Verify that this expression exhibits the law of diminishing marginal product with respect to variable labor input. 5.6 A firm’s production function is: Q = 250K0.7L0.3 Suppose that Q = 1,000. A. What is the equation of the corresponding isoquant in terms of  L? B. Demonstrate that this isoquant is convex with respect to the origin.

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Longer Problems

LONGER PROBLEMS

5.1 Suppose that the average product of labor is given by the equation APL = 100 + 500L - 5L2 A. What is the equation for the total product of labor (TPL)? B. What is the equation for the marginal product of labor (MPL)? C. At what level of labor usage is AP L = MP L? 5.2 The research department of Merriweather International has estimated the following production function Q = 50K0.5L0.4F0.4 where Q represents of units of output, K  represents units of capital, L represents labor hours, and F  represents thousands of square feet of factory floor space. A. Does this production function exhibit increasing, decreasing or constant returns to scale? B. Estimate the coefficient of output elasticity. C. Determine the marginal product equations. D. Suppose that K  = 625, L =  20, and F  =  50. Estimate total output. 5.3 Consider the following production functions: 1) 2) 3) 4)

Q = 200K0.5L0.5 Q = 2K + 2L + 5KL Q = 50 + 5K + 5L Q = 3K/3L

where Q represents of units of output, K  represents units of capital, and L represents labor hours. A. Suppose that K  = L = 1. Calculate total output for each production function. B. Suppose that K  = L =  2. Calculate total output for each production function. C. Based upon your answers to parts A. and B., do these production functions exhibit increasing, decreasing, or constant returns to scale? 5.4 Consider the following production function: Q = 300KL + 18L2 + 3K2 - 0.5L3 - 0.25K3 where Q is units of output per month, L is the number of workers, and K is units of capital. Assume that L =  20 and K  = 10.

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A. Calculate total output per month. B. Calculate MP L and MP K.  C. Calculate AP L and  AP K.  5.5 Consider the following production function: Q = 225L + 100K - 7.5L2 - 5K2 + KL where Q represents units of output per month, L is the number of  workers, and K  is units of capital. A. How much K  and L will maximize Q? B. What is the maximum output per month?

ANSWERS TO MULTIPLE CHOICE QUESTIONS

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18

D. A. B. C. C. E. A. D. B. E. B. C. D. C. C. B. C. A.

5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 5.30 5.31 5.32 5.33 5.34 5.35

D. C. C. C. D. E. A. E. C. C. B. A. C. B. C. B. C.

SOLUTIONS TO SHORTER PROBLEMS

5.1 MPK = ∂QK/∂K = 0.7(250)K-0.3L0.3 = 175K-0.3L0.3 = 175(L/K)0.3 = 243.32 MPL = ∂QL/∂L = 0.3(250)K0.7L0.3-1 = 75K0.7L-0.7 = 75(K/L)0.7 = 34.76 5.2 A. TPL = 7(144)0.5L0.5 = 84L0.5 B. MPL = dTPL/dL = 0.5(7)(144)0.5L-0.5 = 42L-0.5 C. APL = TPL/L = 84L0.5/L = 84L-0.5

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Solutions to Shorter Problems

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5.3 MPL = 0.3(250K0.7L-0.7) APL = Q/L = (250K0.7L0.3)/L ∂APL/∂L = [0.3(250K0.7L-0.7)L - 250K0.7L0.3]/L2 = 0, i.e., the first-order condition for  AP L maximization. Since L > 0, this implies that 0.3(250K0.7L-0.7)L - 250K0.7L0.3 = 0 0.3(250K0.5L-0.7) = 250K0.7L0.3/L MPL = APL 5.4 MPL = 200K - 20KL APL = (200KL - 10KL2)/L ∂APL/∂L = [(200K - 20KL)L - (200KL - 10KL2)]/L2 = 0, i.e., the first-order condition for  AP L maximization. L > 0 implies that (200K - 20KL)L - (200KL - 10KL2) = 0 (200K - 20KL)L = 200KL - 10KL2 200K - 20KL = (200KL - 10KL2)/L MPL = APL 5.5 MPL = ∂QL/∂L = 0.3(250)K0.7L-0.7 = 75K0.7L-0.7 > 0 since L and K  are positive. The second partial derivative of the production function is ∂MPK/∂K = ∂2QK/∂K2 = -0.7(75)K0.7L-1.7 < 0 which is clearly negative since L-1.7  = 1/L1.7 > 0. 5.6 A. 1,000 = 250K0.7L0.3 Solving this equation for K  in terms of  L yields K0.7 = 1,000(250L0.3)-1 = 1,000(250-1L-0.3) = 4L-0.3 (K0.7)1/0.7 = (4L-0.3)(1/0.7) K = 41/0.7 L-0.3/0.7 B. Taking the first and second derivatives of this expression yields dK/dL = (-0.3/0.7)(41/0.7L-1.3/0.7 ) < 0 Since L-1.3/0.7 = 1/L1.3/0.7 > 0 d2K/dL2 = (-1.3/0.7)(-0.3/0.7)(41/0.7L-2.3/0.7 ) = (0.39/0.7)(41/0.7L-2.3/0.7 ) > 0 Since L-2.3/0.7 = 1/L2.3/0.7 > 0 Since the first derivative is negative and the second derivative is positive then the isoquant is convex with respect to the origin.

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SOLUTIONS TO LONGER PROBLEMS

5.1 A. TPL = (APL)L = (100 + 500L - 5L2)L = 100L + 500L2 - 5L3 B. MPL = dTPL/dL = 100 + 1,000L - 15L2 C. APL = MPL 100 + 500L - 5L2 = 100 + 1,000L - 15L2 10L2 = 500L 10L = 500 L = 50 Alternatively, dAPL/dL = 500 - 10L = 0, i.e., the first-order condition for AP L maximization. L = 50 d2APL/dL2 = -10 < 0, i.e., the second-order condition for  AP L maximization is satisfied. 5.2 A. Let l be some factor of proportionality. E¢ = 50(l K)0.5(l L)0.4(l F)0.4 = 50l 0.5K0.5l 0.4L0.4l 0.4F0.4 = l 1.350K0.5L0.4F0.4 = l 1.3Q If all inputs are increased by the scalar l , enrollments will increase by l 1.3. Thus, this production function exhibits increasing returns to scale. Alternatively, the above production function exhibits increasing returns to scale because 0.5 + 0.4 + 0.4 = 1.3 > 0. B. eQ = eK + eL + eF = 0.5 + 0.4 + 0.4 = 1.3 C. MPK = 0.5(50)K-0.5L0.4F0.4 = 25K-0.5L0.4F0.4 MPL = 0.4(50)K0.5L-0.6F0.4 = 20K0.5L-0.6F0.4 MPF = 0.4(50)K0.5L0.3F-0.6 = 20K0.5L0.4F-0.6 D. Q = 50K0.5L0.4F0.4 = 50(625)0.5(20)0.4(50)0.4 = 50(25)(3.31)(4.78) = 19,777.25 5.3 A. 1) 2) 3) 4) B. 1) 2) 3) 4)

Q = 200(1)0.5(1)0.5 = 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)0.5(2)0.5 = 400 Q = 2(2) + 2(2) + 5(2)(2) = 28 Q = 50 + 5(2) + 5(2) = 70 Q = 3(2)/3(2) = 1

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C. 1) Since output doubled as K  and L were doubled, then this production function exhibits constant returns to scale. Alternatively, for Cobb-Douglas production functions, returns to scale may be determined by adding the values of the exponents. Since 0.5 + 0.5 = 1, then this production function exhibits constant returns to scale. 2) Since output more than doubled as K  and L were doubled, then this production function exhibits increasing returns to scale. 3) Since output less than doubled as K  and L were doubled, then this production function exhibits decreasing returns to scale. 4) Since output less than doubled (remained unchanged) as K  and L were doubled, then this production function exhibits decreasing returns to scale. 5.4 A. Q = 300(10)(20) + 18(20)2 + 3(10)2 - 0.5(20)3 - 0.25(10)3 = 60,000 + 7,200 + 300 - 4,000 - 250 = 63,250 B. MPL = ∂Q/∂L = 300K + 36L - 3L2 = 300(10) + 36(20) - 3(20)2 = 3,000 + 720 - 1,200 = 2,520 MPK = ∂Q/∂K = 300L + 6K - 0.75K2 = 300(20) + 6(10) - 0.75(10)2 = 6,000 + 600 - 75 = 6,525 C. APL = Q/L = (300KL + 18L2 + 3K2 - 0.5L3 - 0.25K3)/L = 300K + 18L + 3K2/L - 0.5L2 - 0.25K3/L = 300(10) + 18(20) + 3(10)2/20 - 0.5(20)2 -0.25(10)3/10 = 3,000 + 360 + 15 - 200 - 25 = 3,150 APK = Q/K = (300KL + 18L2 + 3K2 - 0.5L3 - 0.25K3)/K = 300L + 18L2/K + 3K - 0.5L3/K - 0.25K2 = 300(20) + 18(20)2/10 + 3(10) - 0.5(20)3/10 - 0.25(10)2 = 6,000 + 720 + 30 - 400 - 25 = 6,325

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5.5 A. ∂Q/∂L = 225 - 15L + K = 0, i.e., the first-order condition for Q maximization. ∂Q/∂K = 100 - 10K + L = 0, i.e., the first-order condition for Q maximization. Solving the first-order conditions simultaneously yields L* = 15.77 K* = 11.77 B. Q* = 225(15.77) + 100(11.74) - 7.5(15.77)2 - 5(11.74)2 + (11.74)(15.77) = 3,548.66 + 1,174.50 - 1,865.63 - 670.16 + 185.24 = 2,372.61