1 WSG6 7/7/03 4:36 PM Page 79 6 Cost OVERVIEW The previous chapter reviewed the theoretical implications of the technological process whereby factors of production are efficiently transformed into goods and services for sale in the market. The production function defines the maximum rate of output per unit of time obtainable from a given set of productive inputs. The production function, however, was presented as a purely technological relationship devoid of any behavioral assertions underlying motives of management. The purpose of this chapter is to bridge the gap between production as a purely technological relationship and the cost of producing a level of output to achieve a well-defined organizational objective. The cost function of a profit-maximizing firm shows the minimum cost of producing various output levels given market-determined factor prices, and the firm s budget constraint. Although largely the domain of accountants, the concept of cost to an economist carries a somewhat different connotation. Economists are concerned with any and all costs that are relevant to the managerial decision-making process. Relevant costs not only include direct, explicit, or out-of-pocket costs associated with the day-to-day operations of a firm, but also implicit (indirect) costs. The relevant cost concept is economic cost, which includes all opportunity costs, including explicit and implicit costs. Implicit costs may be made explicit, such as when the computer programmer pays himself or herself a salary equivalent to the foregone salary. In this case, implicit opportunity costs have been made explicit. Managerial Economics: Theory and Practice 79 Copyright 2003 by Academic Press. All rights of reproduction in any form reserved.
2 WSG6 7/7/03 4:36 PM Page Cost An analysis of the firm s short-run cost function follows directly from an analysis of the firm s short-run production function. Assuming constant factor prices, increasing marginal cost is a direct consequence of the law of diminishing returns, which in turn affects the pattern of behavior of the firm s average total cost and average variable cost. Short-run production functions assume that at least one factor of production is held fixed. The cost associated with these fixed inputs is called total fixed cost. Long-run production functions assume that all inputs are variable, which implies that there are no fixed costs. Moreover, the law of diminishing returns is no longer operable. The long-run cost concept of economies of scale follows directly from the long-run production concept of increasing returns to scale. Economies of scale occur when long-run average total costs fall as the firm increases its capacity, and is related to the production concept of increasing returns to scale. Diseconomies of scale occur when long-run average total cost increases, as when the firm increases its capacity, and is related to the long-run production concept of decreasing returns to scale. It can be argued that the firm s optimal scale of operation occurs when long-run average total cost is minimized, i.e., when the firm experiences constant returns to scale. Other important cost concepts are the learning curve effect, cost complementarities and economies of scope. The learning curve effect is the reduction in per unit cost resulting from increased worker productivity due to increased worker experience. Economies of scope refers to the reduction in per unit cost resulting from the joint production of two or more goods or services, such as the production of automobiles and trucks, or beef and leather, by the same producer. Cost complementarities exist when the marginal cost of producing one good is reduced by increasing the production of another good. MULTIPLE CHOICE QUESTIONS 6.1 Accounting profit is greater than economic profit because: A. Total economic costs include total explicit cost but does not include total implicit cost. B. Total economic costs include both total explicit and total implicit cost. C. Total economic profit does not include a normal rate of return. D. Total accounting profit includes a normal rate of return.
3 WSG6 7/7/03 4:36 PM Page 81 Multiple Choice Questions Costs that are not recoverable once incurred are called: A. Out-of-pocket expenses. B. Direct costs. C. Opportunity costs. D. Depreciation. E. Sunk costs. 6.3 Total fixed costs include: A. Only implicit costs because they do not change as output changes. B. Only implicit costs because they do not change as output changes. C. The costs of factors of production that do not vary with changes in output. D. The costs of plant and equipment. E. The costs of factors of production that last less than one year. 6.4 The change in total cost resulting from the implementation of a specific management decision is called: A. Managerial cost. B. Incremental cost. C. Marginal cost. D. Sunk cost. 6.5 In the short run: A. All costs are fixed. B. All costs are variable. C. All costs are incremental. D. Some costs are fixed and some costs are variable. 6.6 Tilly s Cellies manufactures designer cellular telephones. Tilly s total cost equation is TC = 5, Q Q 2, where Q represents the number of cellular telephones produced. The average total cost of producing 10 cellular telephones is: A. $48. B. $101. C. $752. D. $ Average total cost (ATC) falls when ATC is marginal cost (MC): A. greater than B. less than C. equal to D. Cannot be answered without knowledge of factor prices and the production function.
4 WSG6 7/7/03 4:36 PM Page Cost 6.8 Suppose that a firm s TFC is $15,000.If ATC = $5 and AVC = $4, then the total output of the firm is: A. 1,000 units. B. 5,000 units. C. 10,000 units. D. 15,000 units. 6.9 MC = AVC at an output level where: I. ATC is minimized. II. Where the MC curve intersects the AVC curve. III. Where AVC is minimized. IV. AVC is equal to the price of the product. Which of the following is correct? A. I only. B. II only. C. III only. D. I and IV only. E. II and III only Incremental cost is similar to: A. Marginal cost. B. Variable cost. C. Sunk cost. D. Operating cost The MC curve intersects the ATC curve: A. At an output level greater than where the MC cost curve intersects the AVC curve. B. At the same output level where the MC curve intersects the AVC curve. C. At a lower output level than where the MC cost curve intersects the AVC curve. D. Where MC = AFC If ATC falls when output increases then: I. MC must be less than ATC. II. MC must be less than AVC. III. TC must be increasing at a decreasing rate. Which of the following is correct? A. I only. B. II only. C. III only. D. I and III only.
5 WSG6 7/7/03 4:36 PM Page 83 Multiple Choice Questions Suppose that a firm s total cost is given by the equation TC = Q - 5Q 2 + 2Q 3, where Q represents total output. If Q = 10, then average fixed cost is: A. $10. B. $165. C. $175. D. $1, Suppose that a firm s total cost is given by the equation TC = Q - 5Q 2 + 2Q 3, where Q represents total output. If Q = 10, then average variable cost is: A. $10. B. $165. C. $175. D. $1, A firm s short-run average total cost is given by the equation ATC = Q + Q 2, where Q represents units of output. Short-run average total cost is minimized when output equals: A. 7 units. B. 14 units. C. 21 units. D. 28 units Suppose that a firm s production function is Q = f(l, K), where L is the variable input and K is the fixed input. An fall in the rental price of labor will cause the firm s marginal cost curve to: A. Shift to the right. B. Shift to the left. C. Shift up. D. Shift down Suppose that the marginal cost function is a linear function of output. This implies that: A. The AVC function is linear. B. The ATC function is linear. C. The MC curve intersects the AVC curve at its minimum point. D. The AVC curve is U shaped Assume that a firm s production function is Q = f(l, K) where L is variable and K is fixed. I. When the MP L starts to rise, then MC starts to fall. II. When MC starts to rise, then ATC starts to rise. III. When MC starts to rise, then AVC starts to rise. IV. When MP L starts to fall, then MC starts to rise.
6 WSG6 7/7/03 4:36 PM Page Cost Which of the following statements is correct? A. I only. B. II only. C. I and IV only. D. II and III only Suppose that a firm s total cost is given by the equation TC = Q + 0.5Q 2, where Q represents total output. This firm s production function exhibits: I. Increasing marginal product. II. Constant marginal product. III. Diminishing marginal product. Which of the following is correct? A. I only. B. II only. C. III only. D. I and II only. E. I, II, and III Which of the following suggests a linear short-run total cost curve? A. MC = AFC for all output levels. B. MC = ATC for all output levels. C. MC = AVC for all output levels. D. MC = ATC + AVC for all output levels Assume that a firm s production function is Q = f(l, K), where L and K represent labor and capital input respectively. Increasing marginal product of capital implies: A. dmc/dq > 0. B. datc/dq > 0. C. dmc/dq < 0. D. datc/dq < 0. E. Both C and D Suppose that a firm s total cost is given by the equation TC = Q - 44Q 2 + 2Q 3, where Q represents total output. AVC is minimized at what output level? A. 6.5 units. B. 11 units. C. 15 units. D units.
7 WSG6 7/7/03 4:36 PM Page 85 Multiple Choice Questions The learning curve effect indicates that: A. Labor productivity improves with an improvement in labor skills from on-the-job experience. B. The firm is experiencing economies of scale. C. The firm is experiencing diseconomies of scale. D. Labor productivity increases with the adoption of improved production technology, organizational and managerial techniques The experience curve effect indicates that: A. Labor productivity improves with an improvement in labor skills from on-the-job experience. B. The firm is experiencing economies of scale. C. The firm is experiencing diseconomies of scale. D. Labor productivity increases with the adoption of improved production technology, organizational and managerial techniques Diseconomies of scale is most closely associated with: A. Increasing returns to scale. B. Decreasing returns to scale. C. Constant returns to scale. D. Decreasing returns to scale. E. The experience curve effect The long-run ATC curve is downward sloping because: A. The law of diminishing marginal product. B. Economies of scope. C. Economies of scale. D. Diseconomies of scale The long-run ATC curve us upward sloping because: A. The law of diminishing marginal product. B. Economies of scope. C. Economies of scale. D. Diseconomies of scale Suppose that a firm s production function is Q = f(l, K), where L is the variable input and K is the fixed input. The long-run average total cost curve: I. Is the envelope of the short-run average-total-cost curves. II. Represents optimal levels output for alternative values of K. III. Is tangent to the minimum point on the short-run marginal cost curves.
8 WSG6 7/7/03 4:36 PM Page Cost Which of the following is correct? A. I only. B. II only. C. I and II only. D. II and III only Long-run ATC equals short-run ATC when: A. Long-run MC equals short-run MC. B. Long-run MC equals long-run ATC. C. Short-run MC is minimized. D. Short-run AVC equals long-run AVC. E. Long-run ATC equals long-run AFC Suppose that the firm s production function is Q = 200L 0.25 K 0.5, where K is capital and L is labor. This firm experiences: A. The law of diminishing marginal product. B. Economies of scope. C. Economies of scale. D. Diseconomies of scale. E. Both A and D are correct The minimum efficient scale of production: A. Is the output level corresponding to minimum long-run ATC. B. Is the output level corresponding to minimum long-run MC. C. Is the output level where long-run MC equals short-run ATC. D. Is the output level where short-run MC equals long-run ATC Economies of scale may result because: A. Of increased specialization in the application of factors of production. B. Some types of machinery are more efficient for large production runs. C. Large companies may be able to extract more favorable financing terms from its creditors than small companies. D. All of the above Economies of scope represents: A. Cost savings when firms specialize in the production of a single product. B. Cost savings arising from specialization of the factors of production. C. Cost savings arising from the joint production of two or more goods. D. Cost savings from economies of scale.
9 WSG6 7/7/03 4:36 PM Page 87 Shorter Problems Which of the following is most likely to result in economies of scope? A. The production of soft drinks and soda bottles. B. The production of cars and trucks. C. The production of computer hardware and computer software. D. Producing the same brand of wine from two different vineyards Cost complementarities exist when: A. The marginal cost of producing one good is reduced by increasing the production of another good. B. The total cost of producing two or more goods is lower when using the same production facilities. C. Average total cost of production falls using separate production facilities. D. Labor productivity increases with the adoption of more efficient capital. SHORTER PROBLEMS 6.1 A firm s total cost equation is: TC = Q - 2.5Q 2 + (1/3)Q 3 where Q represents units of output. Suppose that the firm produces 15 units of output. Calculate total fixed cost (TFC), total variable cost (TVC), average total cost (ATC), average fixed cost (AFC), average variable cost (AVC), and marginal cost (MC). 6.2 Suppose that a firm s total variable cost equation is: TVC = 180Q - 22Q 2 + Q 3 where Q represents units of output. Below what price should the firm shut down its operations? 6.3 A firm s average total cost equation is: ATC = 100Q Q where Q represents units of output. A. Determine the TC equation. B. Estimate MC when Q = 20.
10 WSG6 7/7/03 4:36 PM Page Cost 6.4 A firm s total cost equation is: TC = Q + 3.5Q 2 where Q represents units of output. A. Determine the output level that minimizes average total cost (ATC). B. Verify that marginal cost (MC) equals ATC at the output level that minimizes ATC. 6.5 A firm s long-run total cost (LRTC) equation is given by the expression LRTC = 7,396Q - 15Q 2 + (3/2)Q 3 A. What is the firm s long-run average-cost equation? B. What is the firm s minimum efficient scale (MES) of production? 6.6 The long-run average total cost (LRATC) equation of a perfectlycompetitive firm is: LRATC = Q Q 2 where Q represents units of output. A. Determine the minimum efficient scale of production. B. Calculate total cost at the minimum efficient scale of production. C. If the total level of output in the industry is 150,000 units, then how many firms can profitably operate in this industry? 6.7 Suppose that the per-unit labor cost of producing a single unit of output is $1,525. A. If the learning factor is 0.75 and the factor of proportionality is 2.5, estimate the per-unit labor cost of producing 50 units of output. B. If the wage rate is constant at $10.50 per hour, how many labor hours are required to produce the first unit of output? How many labor hours are required per unit of output when 50 units are produced? 6.8 Suppose that the total cost (TC) and demand equations for a monopolist is given by the equations: TC = Q 2 P = Q Determine the profit-maximizing price and output level.
11 WSG6 7/7/03 4:36 PM Page 89 Longer Problems 89 LONGER PROBLEMS 6.1 Loki Products, LLC., produces a line of home smoke alarms. Loki s total cost equation is: TC = 1, Q 2 A. Determine the output level that minimizes average total cost (ATC). B. At the output level that minimizes ATC, calculate ATC, TC, and MC. C. Verify that at the output level that minimizes ATC that the MC curve intersects the ATC curve from below. D. Determine the output level that minimizes average variable cost (AVC). E. At the output level that minimizes AVC, calculate AVC, TC, and MC. 6.2 The operations research department of Thunder Pomegranates, Inc. has estimated the following average variable cost equation for the firm s new brand of weed whackers: AVC = Q + Q 2 where Q represents the number of weed whackers produced. Suppose that Thunder is operating in the short run, and that capital is the fixed input. A. If Thunder employs 50 units of capital at $20 per unit, what is Thunder s total cost (TC) equation? B. Calculate the output level at which MC = AVC. C. Determine the output level that minimizes MC. 6.3 The total cost equation for a firm producing two products is TC(Q 1,Q 2 ) = Q Q Q 1 Q 2 A. Do cost complementarities exist for this firm? B. Under what circumstances do economies of scope exist for this firm? C. Suppose that Q 1 = Q 2 = 15. Do cost economies of scope exist? D. Suppose that Q 1 = Q 2 = 25. Do cost economies of scope exist? E. Suppose that the firm is currently producing 15 units of Q 1 and 25 units of Q 2. What is the firm s total cost of production? F. Suppose that the firm divests itself of the division selling Q 1 to a competitor. How much will it cost to continue producing 25 units of Q 2? What is the total cost of producing both Q 1 and Q 2 if the firm producing Q 1 produces 15 units?
12 WSG6 7/7/03 4:36 PM Page Cost 6.1 B. 6.2 E. 6.3 C. 6.4 B. 6.5 D. 6.6 C. 6.7 B. 6.8 D. 6.9 E A A A A B A D A C C C E B A D B C D C A E A D C B A. ANSWERS TO MULTIPLE CHOICE QUESTIONS
13 WSG6 7/7/03 4:36 PM Page 91 Solutions to Shorter Problems 91 SOLUTIONS TO SHORTER PROBLEMS 6.1 TFC = $300 TVC = 25Q - 2.5Q 2 + (1/3)Q 3 = 25(15) - 2.5(15) 2 + (1/3)(15) 3 = $ ATC = TC/Q = [ Q - 2.5Q 2 + (1/3)Q 3 ]/Q = 300Q Q + (1/3)Q 2 = 300(15) (15) + (1/3)(15) 2 = $82.50 AFC = TFC/Q = 300/15 = $20 AVC = TVC/Q = [25Q - 2.5Q 2 + (1/3)Q 3 ]/Q = Q + (1/3)Q 2 = (15) + (1/3)(15) 2 = $62.50 MC = dtc/dq = 25-5Q + Q 2 = 25-5(15) + (15) 2 = $ AVC = TVC/Q = Q + Q 2 davc/dq = Q = 0, i.e., the first-order condition for AVC minimization. d 2 AVC/dQ 2 = 2 > 0, i.e., the second-order condition for AVC minimization is satisfied. Solving the first-order condition for Q we obtain: Q* = 11 Since the firm operates at the point where P = MC, we substitute this result into the marginal cost equation to obtain P = MC = dtvc/dq = Q + 3Q 2 = (11) + 3(121) = $59 Thus, if the price falls below $59 per unit then the firm will shut down. 6.3 A. TC = ATC Q = (100Q Q)Q = Q Q 2 B. datc/dq = Q = (20) = $ A. ATC = TC/Q = 504Q Q datc/dq = -504Q = 0, i.e., the first-order condition for ATC minimization. Solving the first-order condition for Q we obtain 3.5Q 2 = 504 Q 2 = 144 Q* = 12 d 2 ATC/dQ = 1,008Q -3 = 1,009(12) -3 > 0, i.e., the second-order condition for ATC minimization is satisfied. B. ATC* = 504(12) (12) = $119 MC = dmc/dq = Q = (12) = $119
14 WSG6 7/7/03 4:36 PM Page Cost 6.5 A. LRATC = LRTC/Q = (7,396Q - 15Q 2 + (3/2)Q 3 )/Q = 7,396-15Q + (3/2)Q 2 B. dlratc/dq = Q = 0, i.e., the first-order condition for LRTC minimization. Solving the first-order condition we obtain Q* = 5 units d 2 LRATC/dQ 2 = 3 > 0, i.e., the second-order condition for LRTC minimization is satisfied. Thus, the minimum efficient scale of production is 5 units. 6.6 A. dlratc/dq = Q = Q = 3 Q = 1,500 B. LRATC = (1,500) (1,500) 2 = $7,177 TC = LRATC Q = 7,177(1,500) = $10,765,500 C. Number of firms in the industry = 150,000/1,500 = A. Per-unit labor cost = jq b = 1,525(50) ln0.75/ln2.5 = 1,525(50) = 1,525(0.293) = $ per unit B. The first unit of output will require $1,525/$10.50 = labor hours. 50 units of output will require $446.47/$10.50 = labor hours per unit of output. 6.8 TR = PQ = (500-25Q)Q = 500Q - 25Q 2 p=tr - TC = (500Q - 25Q 2 ) - ( Q 2 ) = Q - 50Q 2 dp/dq = Q = 0, i.e., the first-order condition for p maximization. d 2 p/dq 2 =-100 < 0, i.e., the second-order condition for p maximization is satisfied. Solving the first-order condition for Q we obtain Q* = 5 P* = (5) = $375
15 WSG6 7/7/03 4:36 PM Page 93 Solutions to Longer Problems 93 SOLUTIONS TO LONGER PROBLEMS 6.1 A. ATC = TC/Q = (1, Q 2 )/Q = 1,296Q Q datc/dq = -1,296Q = 0, i.e., the first-order condition for minimizing ATC. Solving the first-order condition we obtain Q 2 = 144 Q* = 12 d 2 ATC/dQ 2 = 2(1,296)Q -3 = 2(1,296)(12) -3 = 1.5 > 0, i.e., the second-order condition for minimizing ATC is satisfied. B. ATC = 1,296(12) (12) = $216 TC = 1, (12) 2 = $2,592 MC = dtc/dq = 18Q = 18(12) = $216 C. For MC to intersect ATC from below, then it must be the case that at Q* = 12, then the slope of the marginal cost curve must be positive. Taking the first derivative of the marginal cost equation with respect to Q we obtain dmc/dq = 18 > 0 D. TVC = 9Q 2 AVC = TVC/Q = 9Q davc/dq = 9 The AVC curve is linear and has no mathematical minimum. On the other hand, if we require that output be nonnegative (Q 0), then AVC is minimized where Q = 0. E. AVC = 9(0) = $0 TC = 1, (0) 2 = $1,291 = TFC MC = 18(0) = AVC 6.2 A. TC = TFC + TVC TFC = P K K 0 = $20(50) = $1,000 TVC = AVC Q = ( Q + Q 2 )Q = 200Q - 2.5Q 2 + Q 3 TC = 1, Q - 2.5Q 2 + Q 3 MC = dtc/dq = 200-5Q + 3Q 2 B. MC = AVC 200-5Q + 3Q 2 = Q + Q 2 2Q 2-2.5Q = 0 Q(2Q - 2.5) = 0 Q* = 1.25 units C. dmc/dq = Q = 0, i.e., the first-order condition for MC minimization. Solving the first-order condition for Q we obtain 6Q = 5 Q* = units d 2 MC/dQ 2 = 6 > 0, i.e., the second-order condition for MC minimization is satisfied.
16 WSG6 7/7/03 4:36 PM Page Cost 6.3 A. In the case of multiproduct cost functions, cost complementarities exist if the cross-second partial derivative of the multiproduct cost function is negative, i.e., MC 1 (Q 1,Q 2 )/ Q 2 = 2 TC/ Q 1 Q 2 < 0 MC 2 (Q 1,Q 2 )/ Q 1 = 2 TC/ Q 2 Q 1 < 0 MC 1 = TC/ Q 1 = 4Q 1 + Q 2 MC 1 / Q 2 = 2 TC/ Q 1 Q 2 = 1 > 0 MC 2 = TC/ Q 2 = 5Q 2 + Q 1 MC 2 / Q 1 = 2 TC/ Q 2 Q 1 = 1 > 0 Since the cross-second partial derivatives are positive, then cost complementarities do not exist. B. For economies of scope to exist, then TC(Q 1, 0) + TC(0, Q 2 ) > TC(Q 1,Q 2 ) TC(Q 1,Q 2 ) = Q Q 22 + Q 1 Q 2 TC(Q 1, 0) = Q 1 2 TC(0, Q 2 ) = Q 2 2 TC(Q 1, 0) + TC(0, Q 2 ) = ( Q 12 ) + ( Q 22 ) = 1, Q Q 2 2 1, Q Q 22 > Q Q 22 + Q 1 Q > Q 1 Q 2 For economies of scope to exist, then Q 1 Q 2 must be less than 500. C. Since 15(15) = 225 < 500, then economies of scope exist. D. Since 25(25) = 625 > 500, then economies of scope do not exist. E. TC(Q 1,Q 2 ) = Q Q 22 + Q 1 Q 2 = (15) (25) 2 + (15)(25) = , = $2, F. TC(0, Q 2 ) = (0) (25) 2 + (0)(25) = ,562.5 = $2, TC(Q 1, 0) = (15) (0) 2 + (15)(0) = = $950 The cost of producing Q 1 and Q 2 jointly is $2, The cost of producing Q 1 and Q 2 separately is $2, $950 = $3, Since TC(Q 1,Q 2 ) < TC(Q 1,0) + TC(0, Q 2 ) then economies of scope exit at these output levels.
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