EECE 340 Problems Book, Spring 2020

Chapter 7 Costs Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-2 Topics 1. The Nature of Costs. 2. Short-Run Costs. 3. Long-Run Costs. 4. Lower Costs in the Long Run. 5. Cost of Producing Multiple Goods. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-3 The Nature of Costs • Economists measure all relevant costs. – Explicit costs – direct, out-of-pocket payments for inputs to its production process within a given time period – Implicit costs – reflect only a forgone opportunity rather than an explicit, current expenditure. • Accountants measure costs in ways that are more consistent with tax laws and other laws. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-4 Opportunity Costs • Economic cost or opportunity cost - the value of the best alternative use of a resource. • The economic or opportunity cost includes both explicit and implicit costs. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-5 Solved Problem 7.1 • Meredith’s firm sends her to a conference for managers and has paid her registration fee. Included in the registration fee is free admission to a class on how to price derivative securities such as options. She is considering attending, but her most attractive alternative opportunity is to attend a talk by Warren Buffett about his investment strategies, which is scheduled at the same time. Although she would be willing to pay $100 to hear his talk, the cost of a ticket is only $40. Given that there are no other costs involved in attending either event, what is Meredith’s opportunity cost of attending the derivatives talk? Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-6 Solved Problem 7.1: Answer • To calculate her opportunity cost, determine the benefit that Meredith would forgo by attending the derivatives class. • Because she incurs no additional fee to attend the derivatives talk, Meredith’s opportunity cost is the foregone benefit of hearing the Buffett speech. Because she values hearing the Buffett speech at $100, but only has to pay $40, her net benefit from hearing that talk is $60 (= $100 – $40). Thus, her opportunity cost of attending the derivatives talk is $60. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-7 Costs of Durable Inputs • Durable good – a product that is usable for years. • Two issues may arise in measuring the cost of durable goods: – How to allocate the initial purchase cost over time. – What to do if the value of the capital changes over time. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-8 Sunk Costs • Sunk cost – a past expenditure that cannot be recovered. • Sunk cost is not relevant to a manager when deciding how much to produce now. • If an expenditure is sunk, it is not an opportunity cost. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-9 Short-Run Costs • Fixed cost (F) – a production expense that does not vary with output. • Variable cost (VC) – a production expense that changes with the quantity of output produced. • Cost (total cost, C) – the sum of a firm’s variable cost and fixed cost: C = VC + F Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-10 Marginal Cost • Marginal cost (MC) – the amount by which a firm’s cost changes if the firm produces one more unit of output. – And since only variable cost changes with output: q C MC    q VC MC   Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-11 Average Costs • Average fixed cost (AFC) – the fixed cost divided by the units of output produced: AFC = F/q. • Average variable cost (AVC) – the variable cost divided by the units of output produced: AVC = VC/q. • Average cost (AC) – the total cost divided by the units of output produced: AC = C/q AC = AFC + AVC. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-12 Table 7.1 Variation of Short-Run Cost with Output Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-13 C Figure 7.1 Short-Run Cost Curves 120 216 400 48 0 6 10 10 4 2 8 Quantity, q, Units per day Quantity, q, Units per day 6 A 4 2 8 F 1 27 1 20 VC MC AC AVC AFC Cost, $ Cost per unit, $ (a) (b) 60 28 27 20 8 0 B b a Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-14 10 Quantity, q, Units per day 6 4 2 8 AC Cost per unit, $ 60 28 27 20 8 0 When MC is lower than AVC, AVC is decreasing… and when MC is larger than AVC, AVC is increases …so MC = AVC, at the lowest point of the AVC curve! When MC is lower than AC, AC is decreasing… and when MC is larger than AC, AC is increases …so MC = AC, at the lowest point of the AC curve! AVC MC b a Relationship Between Average and Marginal Cost Curves Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-15 Figure 7.2 Variable Cost and Total Product of Labor 46 460 20 200 5 50 77 770 Total product, Variable cost 0 5 1 10 13 L, Hours of labor per day VC = wL, Variable cost, $ c a b d Quantity, q, Units per day Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-16 Shape of the Marginal Cost Curve MC = DVC/Dq. • But in the short run, DVC = wDL – Therefore, MC = wDL/Dq • The additional output created by every additional unit of labor is: Dq/ DL = MPL – Therefore, MC = w/ MPL Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-17 Shape of the Average Cost Curves AVC = VC/q. – But in the short-run, with only labor as an input: AVC = VC/q = wL/q – And since q/L = APL, then AVC = VC/q = w/APL Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-18 Application: Short-Run Cost Curves for a Beer Manufacturer Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-19 Effects of Taxes on Costs • Taxes applied to a firm shift some or all of the marginal and average cost curves. • For example, suppose that the government collects a specific tax of $10 per unit of output from the firm. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-20 Figure 7.3 Effect of a Specific Tax on Cost Curves Costs per unit, $ 15 5 8 10 0 q, Units per day 80 37 27 $10 $10 ACa = ACb + 10 ACb MCb MCa = MCb + 10 A $10.00 tax shifts both the AC and MC by exactly $10… Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-21 Solved Problem 7.2 • What is the effect of a lump-sum franchise tax on the quantity at which a firm’s after tax average cost curve reaches its minimum? (Assume that the firm’s before-tax average cost curve is U-shaped.) Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-22 Solved Problem 7.2: Answer Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-23 Long-Run Costs • Fixed costs are avoidable in the long run. – In the long-run, F = 0. – As a result, the long-run total cost equals: C = VC Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-24 Isocost Line • Isocost line - all the combinations of inputs that require the same (iso-) total expenditure (cost). • The firm’s total cost equation is: Labor Costs Capital Costs C = wL + rK. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-25 Isocost Line (cont.) • The firm’s total cost equation is: C = wL + rK. – We get the Isocost equation by setting the costs at a particular level: C = wL + rK. – And then solving for K (variable along y-axis): K = r - L w r C Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-26 Table 7.2 Bundles of Labor and Capital That Cost the Firm $200 Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-27 Figure 7.4 A Family of Isocost Lines K, Units of capital per year a d e $200 isocost L, Units of labor per year $200 $20 10 = $200 $10 = 20 Isocost Equation K = r - L w r C Initial Values C = $200 w = $10 r = $20 15 2.5 10 5 7.5 5 c b DL = 5 DK = 2.5 For each extra unit of capital it uses, the firm must use two fewer units of labor to hold its cost constant. Slope = -1/2 = w/r Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-28 Figure 7.4 A Family of Isocost Lines (cont.) K, Units of capital per year a e $300 isocost $200 isocost L, Units of labor per year $200 $20 15 = $200 $20 10 = $200 $10 = 20 $300 $10 = 30 Isocost Equation K = r - L w r C C = $300 w = $10 r = $20 An increase in C…. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-29 Figure 7.4 A Family of Isocost Lines (cont.) K, Units of capital per year a e $300 isocost $200 isocost $100 isocost L, Units of labor per year $200 $20 15 = $200 $20 10 = $100 $20 5 = $100 $10= 10 $200 $10 = 20 $300 $10 = 30 Isocost Equation K = r - L w r C C = $100 w = $10 r = $20 A decrease in C…. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-30 Combining Cost and Production Information • The firm can choose any of three equivalent approaches to minimize its cost: – Lowest-isocost rule - pick the bundle of inputs where the lowest isocost line touches the isoquant. – Tangency rule - pick the bundle of inputs where the isoquant is tangent to the isocost line. – Last-dollar rule - pick the bundle of inputs where the last dollar spent on one input gives as much extra output as the last dollar spent on any other input. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-31 Figure 7.5 Cost Minimization K, Units of capital per hour 50 0 L, Units of labor per hour $3,000 isocost $2,000 isocost $1,000 isocost Which of these three Isocost would allow the firm to produce the 100 units of output at the lowest possible cost? Isocost Equation K = r - L w r C Initial Values q = 100 w = $24 r = $8 Isoquant Slope MPL MPK = MRTS - Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-32 Figure 7.5 Cost Minimization K, Units of capital per hour 116 50 24 0 L, Units of labor per hour 100 303 28 q = 100 isoquant $3,000 isocost $2,000 isocost $1,000 isocost x Isocost Equation K = r - L w r C Initial Values q = 100 C = $2,000 w = $24 r = $8 Isoquant Slope MPL MPK = MRTS - y z Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-33 Cost Minimization • At the point of tangency, the slope of the isoquant equals the slope of the isocost. Therefore, r MP w MP r w MP MP MP MP MRTS r w MRTS K L K L K L      last-dollar rule: cost is minimized if inputs are chosen so that the last dollar spent on labor adds as much extra output as the last dollar spent on capital. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-34 Figure 7.5 Cost Minimization K, Units of capital per hour y x z 116 50 24 0 L, Units of labor per hour 100 303 28 q = 100 isoquant $3,000 isocost $2,000 isocost $1,000 isocost Spending one more dollar on labor at x gets the firm as much extra output as spending the same amount on capital. Initial Values q = 100 C = $2,000 w = $24 r = $8 w r MPL MPK = MPL = 0.6q/L MPK = 0.4q/K = 24 8 1.2 0.4 = = 0.05 Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-35 Figure 7.5 Cost Minimization K, Units of capital per hour y x z 116 50 24 0 L, Units of labor per hour 100 303 28 q = 100 isoquant $3,000 isocost $2,000 isocost $1,000 isocost if the firm shifts one dollar from capital to labor, output falls by 0.017 because there is less capital but also increases by 0.1 because there is more labor for a net gain of 0.083 more output at the same cost…. So …the firm should shift even more resources from capital to labor— which increases the marginal product of capital and decreases the marginal product of labor. Initial Values q = 100 C = $2,000 w = $24 r = $8 w r MPL MPK MPL = 0.6q/L MPK = 0.4q/K = 24 8 2.5 0.13 = = 0.1 = 0.017 Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-36 Solved Problem 7.3 • Use the tangency rule to determine the cost- minimizing bundles of labor and capital for a general Cobb-Douglas production function, q = ALαKβ, and for the specific beer production function that underlies Figure 7.5, q = 1.516L0.6K0.4, where w = 24 and r = 8. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-37 Solved Problem 7.3: Answer • Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-38 Figure 7.6 Change in Factor Price K, Units of capital per hour x 77 50 0 L, Workers per hour 100 52 q = 100 isoquant Original isocost, $2,000 New isocost, $1,032 v w r MPL MPK = Minimizing Cost Rule A decrease in w…. Initial Values q = 100 C = $2,000 w = $24 r = $8 w2 = $8 C2 = $1,032 Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-41 The Long-Run Expansion Path and the Long-Run Cost Function • Expansion path - the cost-minimizing combination of labor and capital for each output level. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-42 Figure 7.7(a) Expansion Path and Long-Run Cost Curve K, Units of capital per hour x 100 75 50 0 L,Workers per hour 150 200 100 Expansion path $3,000 isocost $2,000 isocost $4,000 isocost z y q = 100 Isoquant q = 150 Isoquant q = 200 Isoquant Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-43 Long-run cost curve Figure 7.7(b) Expansion Path and Long-Run Cost Curve X Y Z 0 q , Units per hour 4,000 3,000 2,000 200 100 150 C, Cost, $ Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-45 Figure 7.8 Long- Run Cost Curves • The shape of long run cost curves is determined by the production function relationship between output and inputs. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-46 Economies of Scale • Economies of scale - property of a cost function whereby the average cost of production falls as output expands. • Diseconomies of scale - property of a cost function whereby the average cost of production rises when output increases. Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-47 Table 7.3 Returns to Scale and Long- Run Costs Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-50 Figure 7.9 Long-Run Average Cost as the Envelope of Short-Run Average Cost Curves Average cost, $ SRAC 1 SRAC 2 SRAC 3 SRAC 3 LRAC q2 q1 q, Output per day 10 0 12 b c a d e Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-53 Figure 7.10 Long-Run and Short-Run Expansion Paths 100 x y z 50 159 Long-run expansion path 200 isoquant 100 isoquant Short-run expansion path $4,616 $4,000 $2,000 L, Workers per day 0 100 200 In the short run, the firm cannot vary its capital, so its short-run expansion path is horizontal at the fixed level of output. In the long run, the beer manufacturer increases its output by using more of both inputs, so its long-run expansion path is upward sloping. K, Capital per day Copyright ©2015 Pearson Education, Inc. All rights reserved. 7-57 Cost of Producing Multiple Goods • Economies of scope - situation in which it is less expensive to produce goods jointly than separately. • Production possibility frontier - the maximum amount of outputs that can be produced from a fixed amount of input.