Quản trị kinh doanh - Chapter 8: Costs curves

1Costs CurvesChapter 8Copyright (c)2014 John Wiley & Sons, Inc.2Chapter Eight OverviewIntroductionLong Run Cost FunctionsShiftsLong run average and marginal cost functionsEconomies of scaleDeadweight loss – "A Perfectly Competitive Market Without Intervention Maximizes Total Surplus"Short Run Cost FunctionsThe Relationship Between Long Run and Short Run Cost FunctionsChapter EightCopyright (c)2014 John Wiley & Sons, Inc.3Chapter EightLong Run Cost FunctionsDefinition: The long run total cost function relates minimized total cost to output, Q, and to the factor prices (w and r).TC(Q,w,r) = wL*(Q,w,r) + rK*(Q,w,r)Where: L* and K* are the long run input demand functionsCopyright (c)2014 John Wiley & Sons, Inc.4Chapter EightLong Run Cost FunctionsAs Quantity of output increases from 1 million to 2 million, with input prices(w, r) constant, cost minimizing input combination moves from TC1 to TC2 which gives the TC(Q) curve.Copyright (c)2014 John Wiley & Sons, Inc.5Chapter EightWhat is the long run total cost function for production function Q = 50L1/2K1/2?L*(Q,w,r) = (Q/50)(r/w)1/2K*(Q,w,r) = (Q/50)(w/r)1/2TC(Q,w,r) = w[(Q/50)(r/w)1/2]+r[(Q/50)(w/r)1/2]= (Q/50)(wr)1/2 + (Q/50)(wr)1/2= (Q/25)(wr)1/2What is the graph of the total cost curve when w = 25 and r = 100?TC(Q) = 2QLong Run Cost FunctionsExamplesCopyright (c)2014 John Wiley & Sons, Inc.6Q (units per year)TC ($ per year)TC(Q) = 2Q$4M.Chapter EightA Total Cost CurveCopyright (c)2014 John Wiley & Sons, Inc.71 M. $2M.Chapter EightTC ($ per year)Q (units per year)TC(Q) = 2QA Total Cost CurveCopyright (c)2014 John Wiley & Sons, Inc.81 M.2 M. $2M.$4M.Chapter EightA Total Cost CurveTC ($ per year)Q (units per year)TC(Q) = 2QCopyright (c)2014 John Wiley & Sons, Inc.9Chapter EightLong Run Total Cost CurveTracking MovementDefinition: The long run total cost curve shows minimized total cost as output varies, holding input prices constant.Graphically, what does the total cost curve look like if Q varies and w and r are fixed?Copyright (c)2014 John Wiley & Sons, Inc.10Chapter EightLong Run Total Cost CurveAn ExampleCopyright (c)2014 John Wiley & Sons, Inc.11Chapter EightLong Run Total Cost CurveCopyright (c)2014 John Wiley & Sons, Inc.12Chapter EightLong Run Total Cost CurveCopyright (c)2014 John Wiley & Sons, Inc.13Q (units per year)L (labor services per year)KTC ($/yr)00••L0L1K0K1Q0Q1TC = TC1TC = TC0Chapter EightLong Run Total Cost CurveCopyright (c)2014 John Wiley & Sons, Inc.14Q (units per year)L (labor services per year)KTC ($/yr)00LR Total Cost CurveQ0TC0 =wL0+rK0••L0L1K0K1Q0Q1TC = TC1TC = TC0Chapter EightLong Run Total Cost CurveCopyright (c)2014 John Wiley & Sons, Inc.15Q (units per year)L (labor services per year)KTC ($/yr)00LR Total Cost CurveQ0Q1TC0 =wL0+rK0••L0L1K0K1Q0Q1TC = TC1TC = TC0TC1=wL1+rK1Chapter EightLong Run Total Cost CurveCopyright (c)2014 John Wiley & Sons, Inc.16Chapter EightLong Run Total Cost CurveIdentifying ShiftsGraphically, how does the total cost curve shift if wages rise but the price of capital remains fixed?Copyright (c)2014 John Wiley & Sons, Inc.17LK0TC0/rChapter EightA Change in Input PricesCopyright (c)2014 John Wiley & Sons, Inc.18L0-w0/rTC0/rTC1/r-w1/rChapter EightKA Change in Input PricesCopyright (c)2014 John Wiley & Sons, Inc.19L••0AB-w0/rTC0/r-w1/rChapter EightTC1/rKA Change in Input PricesCopyright (c)2014 John Wiley & Sons, Inc.20LQ0••0A-w0/rTC0/r-w1/rChapter EightBTC1/rKA Change in Input PricesCopyright (c)2014 John Wiley & Sons, Inc.21Q (units/yr)TC ($/yr)TC(Q) postChapter EightA Shift in the Total Cost CurveCopyright (c)2014 John Wiley & Sons, Inc.22Q (units/yr)TC(Q) anteTC(Q) postChapter EightTC ($/yr)A Shift in the Total Cost CurveCopyright (c)2014 John Wiley & Sons, Inc.23Q (units/yr)TC(Q) anteTC(Q) postTC0Chapter EightTC ($/yr)A Shift in the Total Cost CurveCopyright (c)2014 John Wiley & Sons, Inc.24Q (units/yr)TC(Q) anteTC(Q) postQ0TC1TC0Chapter EightTC ($/yr)A Shift in the Total Cost CurveCopyright (c)2014 John Wiley & Sons, Inc.25Chapter EightHow does the total cost curve shift if all input prices rise (the same amount)? Input Price ChangesCopyright (c)2014 John Wiley & Sons, Inc.26Chapter EightAll Input Price ChangesPrice of input increases proportionately by 10%. Cost minimization input stays same, slope of isoquant is unchanged. TC curve shifts up by the same 10 percentCopyright (c)2014 John Wiley & Sons, Inc.27Chapter EightLong Run Average Cost FunctionDefinition: The long run average cost function is the long run total cost function divided by output, Q. That is, the LRAC function tells us the firm’s cost per unit of outputAC(Q,w,r) = TC(Q,w,r)/QCopyright (c)2014 John Wiley & Sons, Inc.28Chapter EightLong Run Marginal Cost FunctionMC(Q,w,r) = {TC(Q+Q,w,r) – TC(Q,w,r)}/Q= TC(Q,w,r)/Qwhere: w and r are constantDefinition: The long run marginal cost function measures the rate of change of total cost as output varies, holding constant input prices.Copyright (c)2014 John Wiley & Sons, Inc.29Chapter EightLong Run Marginal Cost FunctionRecall that, for the production function Q = 50L1/2K1/2, the total cost function was TC(Q,w,r) = (Q/25)(wr)1/2. If w = 25, and r = 100, TC(Q) = 2Q.ExampleCopyright (c)2014 John Wiley & Sons, Inc.30Chapter Eighta. What are the long run average and marginal cost functions for this production function?AC(Q,w,r) = (wr)1/2/25MC(Q,w,r) = (wr)1/2/25b. What are the long run average and marginal cost curves when w = 25 and r = 100?AC(Q) = 2Q/Q = 2.MC(Q) = (2Q)/Q = 2.Long Run Marginal Cost FunctionCopyright (c)2014 John Wiley & Sons, Inc.310AC, MC ($ per unit)Q (units/yr)AC(Q) =MC(Q) = 2$2Chapter EightAverage & Marginal Cost CurvesCopyright (c)2014 John Wiley & Sons, Inc.320AC(Q) =MC(Q) = 2$21MChapter EightAC, MC ($ per unit)Q (units/yr)Average & Marginal Cost CurvesCopyright (c)2014 John Wiley & Sons, Inc.330AC(Q) =MC(Q) = 2$21M 2MChapter EightAC, MC ($ per unit)Q (units/yr)Average & Marginal Cost CurvesCopyright (c)2014 John Wiley & Sons, Inc.34Chapter EightSuppose that w and r are fixed:When marginal cost is less than average cost, average cost is decreasing in quantity. That is, if MC(Q) AC(Q), AC(Q) increases in Q.When marginal cost equals average cost, average cost does not change with quantity. That is, if MC(Q) = AC(Q), AC(Q) is flat with respect to Q.Copyright (c)2014 John Wiley & Sons, Inc.36Chapter EightAverage & Marginal Cost CurvesCopyright (c)2014 John Wiley & Sons, Inc.37Chapter EightEconomies & Diseconomies of ScaleDefinition: If average cost decreases as output rises, all else equal, the cost function exhibits economies of scale.Similarly, if the average cost increases as output rises, all else equal, the cost function exhibits diseconomies of scale.Definition: The smallest quantity at which the long run average cost curve attains its minimum point is called the minimum efficient scale.Copyright (c)2014 John Wiley & Sons, Inc.380Q (units/yr)AC ($/yr)Q* = MESAC(Q)Chapter EightMinimum Efficiency Scale (MES)Copyright (c)2014 John Wiley & Sons, Inc.39When the production function exhibits increasing returns to scale, the long run average cost function exhibits economies of scale so that AC(Q) decreases with Q, all else equal.Chapter EightReturns to Scale & Economies of ScaleCopyright (c)2014 John Wiley & Sons, Inc.40Chapter EightReturns to Scale & Economies of Scale When the production function exhibits decreasing returns to scale, the long run average cost function exhibits diseconomies of scale so that AC(Q) increases with Q, all else equal. When the production function exhibits constant returns to scale, the long run average cost function is flat: it neither increases nor decreases with output.Copyright (c)2014 John Wiley & Sons, Inc.41Chapter Eight If TC,Q 1, MC > AC, so AC must be increasing in Q. Therefore, we have diseconomies of scale. If TC,Q = 1, MC = AC, so AC is just flat with respect to Q.Definition: The percentage change in total cost per one percent change in output is the output elasticity of total cost, TC,Q.TC,Q = (TC/TC)(Q /Q) = (TC/Q)/(TC/Q) = MC/ACOutput Elasticity of Total CostCopyright (c)2014 John Wiley & Sons, Inc.42Chapter EightShort Run & Total Variable Cost FunctionsDefinition: The short run total cost function tells us the minimized total cost of producing Q units of output, when (at least) one input is fixed at a particular level.Definition: The total variable cost function is the minimized sum of expenditures on variable inputs at the short run cost minimizing input combinations. Copyright (c)2014 John Wiley & Sons, Inc.43Chapter EightTotal Fixed Cost FunctionDefinition: The total fixed cost function is a constant equal to the cost of the fixed input(s).STC(Q,K0) = TVC(Q,K0) + TFC(Q,K0)Where: K0 is the fixed input and w and r are fixed (and suppressed as arguments)Copyright (c)2014 John Wiley & Sons, Inc.44Q (units/yr)TC ($/yr)TFCExample: Short Run Total Cost, Total Variable Cost and Total Fixed CostChapter EightKey Cost Functions InteractionsCopyright (c)2014 John Wiley & Sons, Inc.45TVC(Q, K0)TFCChapter EightQ (units/yr)TC ($/yr)Example: Short Run Total Cost, Total Variable Cost and Total Fixed CostKey Cost Functions InteractionsCopyright (c)2014 John Wiley & Sons, Inc.46TVC(Q, K0)TFCSTC(Q, K0)Chapter EightQ (units/yr)TC ($/yr)Example: Short Run Total Cost, Total Variable Cost and Total Fixed CostKey Cost Functions InteractionsCopyright (c)2014 John Wiley & Sons, Inc.47TVC(Q, K0)TFCrK0STC(Q, K0)rK0Chapter EightQ (units/yr)TC ($/yr)Example: Short Run Total Cost, Total Variable Cost and Total Fixed CostKey Cost Functions InteractionsCopyright (c)2014 John Wiley & Sons, Inc.48Chapter EightThe firm can minimize costs at least as well in the long run as in the short run because it is “less constrained”.Hence, the short run total cost curve lies everywhere above the long run total cost curve.Long and Short Run Total Cost FunctionsUnderstanding the RelationshipCopyright (c)2014 John Wiley & Sons, Inc.49Chapter EightLong and Short Run Total Cost FunctionsUnderstanding the RelationshipHowever, when the quantity is such that the amount of the fixed inputs just equals the optimal long run quantities of the inputs, the short run total cost curve and the long run total cost curve coincide.Copyright (c)2014 John Wiley & Sons, Inc.50LKTC0/wTC0/r0Chapter EightLong and Short Run Total Cost FunctionsCopyright (c)2014 John Wiley & Sons, Inc.51LTC0/w TC1/wTC1/rTC0/r•0BK0Chapter EightKLong and Short Run Total Cost FunctionsCopyright (c)2014 John Wiley & Sons, Inc.52LTC0/w TC1/w TC2/wTC2/rTC1/rTC0/r•••0ACBQ1K0Chapter EightKLong and Short Run Total Cost FunctionsCopyright (c)2014 John Wiley & Sons, Inc.53LTC0/w TC1/w TC2/wTC1/rTC0/rQ0•••Expansion Path0ACBQ1Q0K0Chapter EightTC2/rKLong and Short Run Total Cost FunctionsCopyright (c)2014 John Wiley & Sons, Inc.540Total Cost ($/yr)Q (units/yr)TC(Q)STC(Q,K0)Q0K0 is the LR cost-minimisingquantity of K for Q0Q1Chapter EightLong and Short Run Total Cost FunctionsCopyright (c)2014 John Wiley & Sons, Inc.550•Q0Q1ATC0Chapter EightTotal Cost ($/yr)Q (units/yr)TC(Q)STC(Q,K0)K0 is the LR cost-minimisingquantity of K for Q0Long and Short Run Total Cost FunctionsCopyright (c)2014 John Wiley & Sons, Inc.560•Q0Q1•ACTC0TC1Chapter EightTotal Cost ($/yr)Long and Short Run Total Cost FunctionsTC(Q)STC(Q,K0)Q (units/yr)K0 is the LR cost-minimisingquantity of K for Q0Copyright (c)2014 John Wiley & Sons, Inc.570•Q0Q1••ACBTC0TC1TC2Chapter EightTotal Cost ($/yr)Long and Short Run Total Cost FunctionsTC(Q)STC(Q,K0)Q (units/yr)K0 is the LR cost-minimisingquantity of K for Q0Copyright (c)2014 John Wiley & Sons, Inc.58Chapter EightShort Run Average Cost FunctionDefinition: The Short run average cost function is the short run total cost function divided by output, Q.That is, the SAC function tells us the firm’s short run cost per unit of output.SAC(Q,K0) = STC(Q,K0)/QWhere: w and r are held fixedCopyright (c)2014 John Wiley & Sons, Inc.59Chapter EightShort Run Marginal Cost FunctionDefinition: The short run marginal cost function measures the rate of change of short run total cost as output varies, holding constant input prices and fixed inputs.SMC(Q,K0)={STC(Q+Q,K0)–STC(Q,K0)}/Q = STC(Q,K0)/Qwhere: w,r, and K0 are constantCopyright (c)2014 John Wiley & Sons, Inc.60Chapter EightSummary Cost FunctionsNote: When STC = TC, SMC = MCSTC = TVC + TFCSAC = AVC + AFCWhere:SAC = STC/QAVC = TVC/Q (“average variable cost”)AFC = TFC/Q (“average fixed cost”)The SAC function is the VERTICAL sum of the AVC and AFC functionsCopyright (c)2014 John Wiley & Sons, Inc.61Q (units per year)$ Per Unit0AFCExample: Short Run Average Cost, Average Variable Cost and Average Fixed CostChapter EightSummary Cost FunctionsCopyright (c)2014 John Wiley & Sons, Inc.620AVCAFCChapter EightQ (units per year)$ Per UnitSummary Cost FunctionsExample: Short Run Average Cost, Average Variable Cost and Average Fixed CostCopyright (c)2014 John Wiley & Sons, Inc.630SACAVCAFCChapter EightQ (units per year)$ Per UnitSummary Cost FunctionsExample: Short Run Average Cost, Average Variable Cost and Average Fixed CostCopyright (c)2014 John Wiley & Sons, Inc.640SMCAVCAFCChapter EightSACQ (units per year)$ Per UnitSummary Cost FunctionsExample: Short Run Average Cost, Average Variable Cost and Average Fixed CostCopyright (c)2014 John Wiley & Sons, Inc.65$ per unit0•••AC(Q)SAC(Q,K3)Q1 Q2 Q3Chapter EightQ (units per year)Long Run Average Cost FunctionAs an Envelope CurveCopyright (c)2014 John Wiley & Sons, Inc.660•••AC(Q)SAC(Q,K1)Q1 Q2 Q3Chapter Eight$ per unitQ (units per year)Long Run Average Cost FunctionAs an Envelope CurveCopyright (c)2014 John Wiley & Sons, Inc.670•••AC(Q)SAC(Q,K1)SAC(Q,K2)Q1 Q2 Q3Chapter Eight$ per unitQ (units per year)Long Run Average Cost FunctionAs an Envelope CurveCopyright (c)2014 John Wiley & Sons, Inc.680•••AC(Q)SAC(Q,K1)SAC(Q,K2)SAC(Q,K3)Q1 Q2 Q3Chapter Eight$ per unitQ (units per year)Long Run Average Cost FunctionAs an Envelope CurveCopyright (c)2014 John Wiley & Sons, Inc.69Chapter EightLong Run Average Cost FunctionAs an Envelope CurveExample: Let Q = K1/2L1/4M1/4 and let w = 16, m = 1 and r = 2. For this production function and these input prices, the long run input demand curves are:Therefore, the long run total cost curve is:TC(Q) = 16(Q/8) + 1(2Q) + 2(2Q) = 8QThe long run average cost curve is:AC(Q) = TC(Q)/Q = 8Q/Q = 8L*(Q) = Q/8M*(Q) = 2QK*(Q) = 2QCopyright (c)2014 John Wiley & Sons, Inc.70Chapter EightRecall, too, that the short run total cost curve for fixed level of capital K0 is:STC(Q,K0) = (8Q2)/K0 + 2K0If the level of capital is fixed at K0 what is the short run average cost curve?SAC(Q,K0) = 8Q/K0 + 2K0/QShort Run Average Cost FunctionCopyright (c)2014 John Wiley & Sons, Inc.71Q (units per year)$ per unit0MC(Q)Chapter EightCost Function SummaryCopyright (c)2014 John Wiley & Sons, Inc.720AC(Q)Chapter EightQ (units per year)$ per unitMC(Q)Cost Function SummaryCopyright (c)2014 John Wiley & Sons, Inc.730••AC(Q)SAC(Q,K2)Q1 Q2 Q3SMC(Q,K1)Chapter EightQ (units per year)$ per unitMC(Q)Cost Function SummaryCopyright (c)2014 John Wiley & Sons, Inc.740•••AC(Q)SAC(Q,K1)SAC(Q,K2)SAC(Q,K3)Q1 Q2 Q3MC(Q)SMC(Q,K1)Chapter EightQ (units per year)$ per unitMC(Q)Cost Function SummaryCopyright (c)2014 John Wiley & Sons, Inc.750•••AC(Q)SAC(Q,K1)SAC(Q,K2)SAC(Q,K3)Q1 Q2 Q3SMC(Q,K1)Chapter EightQ (units per year)$ per unitMC(Q)Cost Function SummaryMC(Q)Copyright (c)2014 John Wiley & Sons, Inc.76Chapter EightEconomies of Scope – a production characteristic in which the total cost of producing given quantities of two goods in the same firm is less than the total cost of producing those quantities in two single-product firms.Mathematically, TC(Q1, Q2) 0 – constant representing AVC of first unit produced, -1 < B < 0 – experience elasticity (% change in AVC for every 1% increase in cumulative volume – slope of the experience curve tells us how much AVC goes down (as a % of initial level), when cumulative output doubles Economies of ExperienceCopyright (c)2014 John Wiley & Sons, Inc.78Chapter EightTotal Cost Function – a mathematical relationship that shows how total costs vary with factors that influence total costs, including the quantity of output and prices of inputs.Cost Driver – A factor that influences or “drives” total or average costs.Constant Elasticity Cost Function – A cost function that specifies constant elasticity of total cost with respect to output and input prices.Translog Cost Function – A cost function that postulates a quadratic relationship between the log of total cost and the logs of input prices and output. Estimating Cost FunctionsCopyright (c)2014 John Wiley & Sons, Inc.