中级微观经济学中微课件

1、Chapter 8第八章COST FUNCTIONS成本函数Copyright 2005 by South-western, a diviof Thomson learning.s.1Economic Cost经济成本The economic cost of any input is the paymentrequired to keept input in its presentemployment任何投入的经济成本是保持该投入处于现行使用状态所需支付的费用the remuneration the input would receive in its best alternative emp

2、loyment该投入用于其他用途所能获得的最大注意与会计成本的区别区别主要在资本成本、企业家才能的成本方面在劳动力成本方面很一致2Two Simplifying Amptions两个简化假设There are only two inputs仅有两种要素投入homogeneous labor (l), measured in labor-hours同质的劳动(l)，以劳动小时衡量achine-hours同质的资本(k)，homogeneous capital (k), measured以机器小时衡量entrepreneurial costs are included in capital cos

3、ts企业家的成本包括在资本成本内Inputs are hired in perfectly competitive markets投入来源于完全竞争市场 firms are price takers in input markets厂商在投入市场上，为价格接受者3Economic Profits经济利润Total costs for the firm are given by厂商的总成本为total costs = C = wl + vkTotal revenue for the firm is given by厂商的总收入为 total revenue = pq = pf(k,l)Econo

4、mic profits () are equal to经济利润()为 = total revenue - total cost= pq - wl vk= pf(k,l) - wl vk4Economic Profits经济利润Economic profits are a function of the amount of capital andlabor employed经济利润是所使用的资本和劳动量的函数we could examine how a firm would choose k and l toprofit可以检验厂商如何选择k和l来最大化其利润“derived demand” t

5、heory of labor and capital inputs 劳动和资本投入的“引致需求”理论izefor now, we will amet the firm has already chosen its outputlevel (q0) and wants to minimize its costs现在假定厂商已经选定其产出水平(q0) ，并且想最小化其成本5Cost-Minimizing Inphoi成本最小化投入选择To minimize the cost of producing a given level of output,a firm should choose a po

6、on the isoquanwhich theRTS is equal to the ratio w/v在给定产出水平下，为了最小化其成本，厂商需要选择等产量线上的一点，在该点RTS等于比值w/vit should equate the rate at which k can be traded for lheproductive promarketplaces to the rate at which they can be tradedhek和l在生产过程的替代比率应该与它们在市场易价格的比值相等6Cost-Minimizing Inphoi成本最小化投入选择Mathematically,

7、 we seek to minimize total costs given q =f(k,l) = q0 数学上，成本在q = f(k,l) = q0的条件下最小化Setting up the Lagrangian建立日函数L = wl + vk + q0 - f(k,l) order conditions are一阶条件为L/l = w - (f/l) = 0L/k = v - (f/k) = 0L/ = q0 - f(k,l) = 07Cost-Minimizing Inphoi成本最小化投入选择two conditions we get前两个一阶条件相除，Dividing the得到f

8、 / lw RTS (l for k )f / kvThe cost-minimizing firm should equate the RTS forthe two inputs to the ratio of their pri成本最小化的厂商会使得两种投入的RTS与其价格比值相等8Cost-Minimizing Inphoi成本最小化投入选择Cross-multiplying, we get交叉相乘，得到fkflvwFor costs to be minimized, the marginal productivity per dollar spent should be the sam

9、e for all inputs为了成本最小化，每边际生产力对于所有投入应该是相同的的9Cost-Minimizing Inphoi成本最小化投入选择Notet this equations inverse is also oferest注意这一等式的倒数也是有意义的w v flThe Lagrangian multipfkr shows how much in extracosts would be incurred by increasing the outputconstraslightly日乘子表示如果稍微增加产出约束时，会产生的额外成本10Cost-Minimizing Inphoi

10、成本最小化投入选择Given output q0, we wish to find the least costly poonthe isoquant给定产出q0，本最少的点希望在等产量线上，找到成k periodCosts are represented by parallel linesC1wislope of -w/v 成本由斜率为-w/v的C3平行线表示C2C1 C2 MC, AC must be falling如果AC MC，AC一定是下降的ACIf AC MC, ACmust be rising如果AC MC，AC一定是上升的min ACOutput30Shifts in Cost

11、 Curves成本曲线的移动The cost curves are drawn under theamptiont input priand the level oftechnology are held constant成本曲线是在要素价格和技术水平保持不变的假设下，得出的 any changehese factors will cause the costcurves to shift这些本曲线的移动的任何变化都会引起成31Some Illustrative Cost Functions成本函数的例证Supe we have a fixed proportions technology s

12、ucht假定技术为固定比例，那么q = f(k,l) = min(ak,bl)Production will occur at the vertex of the L-shdisoquants (q = ak = bl)生产在L形等产量线的顶点处发生(q = ak = bl)C(w,v,q) = vk + wl = v(q/a) + w(q/b) v w ab C(w, v,32Some Illustrative Cost Functions成本函数的例证Supe we have a Cobb-Douglas technology sucht假设技术为，那么q = f(k,l) = k l t

13、最小化成本要求Cost minimization requireswklvwk v l33Some Illustrative Cost Functions成本函数的例证If we substitutel, we will get 如果得到o the production function and solve for代入生产函数并求解l，可以 / l q1/ w / v / A similar method will yield相似的方法可以得到 / k q1/ w / v / 34Some Illustrative Cost Functions成本函数的例证Now we can derive

14、total costs as现在本为可以得到总成C(v,w,q ) vk wl q1/ Bv / w / where 等式中B ( ) / / which is a constantt involves only the parameters and 这个值是常数，仅涉及参量 和 35Some Illustrative Cost Functions成本函数的例证t假设技术为Supe we have atechnology such,那么q = f(k,l) = (k + l )/To derive the total cost, we would use the same methodand

15、eventually get为得到总成本，法，最终得到C(v,w,q ) vk wl q1/ (v / 1需要用同样的方 w / 1 )(1) / 1/ (v 1 w 1 )1/ 1C(v,w,36Properties of Cost Functions成本函数的属性性Homogeneitycost functions are all homogeneous of degree onehe成本函数对所有要素价格是一次的input pricost minimization requirest the ratio of inputpripribe set equal to RTS, a doubl

16、ing of all inputwill not change the levels of inputs purchased成本最小化要求要素价格的比值等于RTS，所有要素价格翻倍不会影响所的要素组合pure, uniform inflation will not change a firms inputs but will shift the cost curves up纯粹、均匀deci的通货膨胀不会改变厂商的要素组合决策，但会使成本曲线向上移动37Properties of Cost Functions成本函数的属性Nondecreasing in q, v, and w （成本函数）对

17、q, v和w单调非递减 cost functions are derived from a cost-s成本函数是由成本最小minimization pro化过程推导出来的any decline in costs from an increase in one of the functions arguments would lead to a contradiction由于函数中一个参量增加而引起的任何成本减少会导致38Properties of Cost Functions成本函数的属性对要素价格为凹的Concave in input pri costs will be lower whe

18、n a firm fainputprit fluctuate around a given levelnwhen they remain constant level当企业围绕一定水平波动的要素价格时，成本会比要素价格固定时低the firm can adapt its input mix to take advantageof such fluctuations厂商可以改变其要素组合，以利用这种波动的优势39Conof Cost Function成本函数的凹性At w1, the firms costs are C(v,w1,q1)在w1，厂商的成本为C(v,w1,q1)If the fir

19、m continues to buy the same input mix as w changes, its cost function would be Cpseudo如果随着w变化，厂商继续购买同样的要素组合，其成本函数将是CpseudoCostsCpseudoC(v,w,q1)C(v,w1,q1)Since the firms input mix will likelychange, actual costs will be lessnCpseudosuch as C(v,w,q1)因为厂商的要素组合可能会改变，实际的成本会比Cpseudo低，比如C(v,w,q1)ww140Prope

20、rties of Cost Functions成本函数的属性Some of these properties carry over to averageand marginal costs某些属性对平均和边际成本也适用 Homogeneity性 effects of v, w, and q are ambiguousv, w和 q的影响是不确定的41Input Substitution要素替代A changehe price of an input will cause the firm to alterits input mix一种投入价格的变化会使厂商改变其投入组合We wish to s

21、ee how k/l changes in response to a change in想看看当q保持不变时，w/v, while holding q constant随着w/v的变化k/l是如何变化 k l w v 42Input Substitution要素替代Putting this in proportional terms as写作比例项(k / l) w / v ln( k / l )s (w / v )k / l ln(w / v )givesternative definition of the elasticity ofsubstitution提供了替代弹性的另一种定义he

22、 two-inpase, s must be nonnegative在两种要素的情况中，s一定是非负的 large values of s indicate mix significantly if input prit firms change their input change较大的s值，意味着如果要素价格变化，厂商会较大地改变其要素组合43Size of Shifts in Costs Curves成本曲线移动的大小The increase in costs will be largely influenced by therelative significance of the in

23、puthe production pros成本增加很大程度上，受到要素在生产过程中相对重要程度的影响If firms can easily substitute another input for the onehas risen in price, there may be little increase in costst如果厂商可以轻易地用其他要素替代价格增加的要素，成本增加就会较小44Technical Progress技术进步Improvementsechnology also lowercost curves技术进步可以降低成本曲线Supet the production func

24、tion (withconstant returns to scale) is 假设生产函数（规模不变）为：q = A(t)f(k,l),while A(0)=1Total costs are总成本为C0 = C0(q,v,w) = qC0(v,w,1)45Technical Progress技术进步Because the same inputst produced one unit of output inperiod zero will produce A(t) units in period t 因为在时期0生产一产出的要素，在时期t可以生产A(t)Ct(v,w,A(t) = C0(v,

25、w,1)Note又因为So所以，Ct(v,w,A(t) = A(t)Ct(v,w,1)C0(v,w,1) = A(t)Ct(v,w,1)Total costs are given by总成本为Ct(v,w,q) = qCt(v,w,1) = qC0(v,w,1)/A(t) = C0(v,w,q)/A(t)So，total costs fall over time at the rate of technical change 所以总成本以技术变化率随时间降低46Contingent Demand for Inputs条件要素需求Contingent demand functions for a

26、ll of the firmsinputs can be derived from the cost function 所有条件要素需求函数可以从成本函数中推得定理Shephards lemmathe contingent demand function for any input is given by thepartial derivative of the total-cost function with respect tot任意条件要素需求由总成本函数对要素价格的inputs price偏微分得到47Contingent Demand for Inputs条件要素需求Supe we

27、have a fixed proportionstechnology假设技术为固定比例The cost function is 成本函数为 v w ab C(w, v,48Contingent Demand for Inputs条件要素需求For this cost function, contingent demand functionsare quite simple对这一成本函数，条件要素需求非常简单k c (v,w,q ) C(v,w,q ) qval c (v,w,q) C(v,w,q) qwbhis case, the contingent demands for inputs d

28、o notdepend on inputs pri决于投入价格在此情况下，条件要素需求不取49Contingent Demand for Inputs条件要素需求Supe we have a Cobb-Douglastechnology假设技术为式的形The cost function is成本函数为C(v,w,q ) vk wl q1/ Bv / w / 50Contingent Demand for Inputs条件要素需求For this cost function, the derivation is messier对于这一成本函数，导数较为复杂kc (v,w,q) C q1/ Bv

29、/ w / v w / q1/ B v 51Contingent Demand for Inputs条件要素需求lc (v,w,q) C q1/ Bv / w / w w / q1/ B v The contingent demands for inputs depend on条件要素需求取决于两种投both inputs pri入的价格52Short-Run, Long-Run Distinction短期与长期的区别he short run, economic actors have only limited flexibilityheir actions在短期，经济主体的行为灵活性有限Am

30、et the capital input is held constank1 and thefirm is free to vary only its labor input假定资本投入保持不变为k1 ，厂商可以任意改变其劳动投入es生产函数为The production functionq = f(k1,l)53Short-Run Total Costs短期总成本Short-run total cost for the firm is厂商的短期总成本为SC = vk1 + wlThere are two types of short-run costs有两种短期成本short-run fix

31、ed costs are costs assoted with fixed inputs (vk1)短期固定成本是与固定投入相关的成本(vk1)short-run variable costs are costs assoted with variable inputs (wl)短期可变成本是与可变投入相关的成本(wl)54Short-Run Total Costs短期总成本Short-run costs are not minimal costs for producing thevarious output levels短期成本不是生产不同产出水平的最小成本the firm does no

32、ve the flexibility of inphoice厂商并没有投入选择的灵活性to vary its outputhe short run, the firm must use nonoptimalombinations为改变其短期内的产出，厂商必须使用非最inp优的投入组合the RTS will not be equal to the ratio of input priRTS可能不等于投入价格之比55Short-Run Total Costs短期总成本eriodk pxed at k1, theRTS with the厂商不能确保比k1l periodl1l2l356Becaus

33、e capital is fi firm cannot equate ratio of input pri因为资本固定为k1，RTS等于投入价格之q2q1q0Short-Run Marginal and Average Costs短期边际和平均成本age total cost (SAC) function is 短期平均成本The short-ru(SAC)函数为SAC = total costs/total output = SC/qThe short-run marginal cost (SMC) function is短期边际成本(SMC)函数为SMC = change in SC/ch

34、ange in output = SC/q57Relationship betn Short-Run andLong-Run Costs短期和长期成本的关系SC (k2)TotalSC (k1)costsCSC (k0)The long-run C curve can be derived by varying the level of k长期的C曲线可以由改变k的水平得到Outputq0q1q258Relationship betn Short-Run andLong-Run Costs 短期和长期成本的关系CostsThe geometric relationshipMCSMC (k0)S

35、AC (k0)ACbetn short-runSMC (k1)SAC (k1)and long-run ACand MC c be shownso短期与长期AC和MC的几何关系Outputq0q159Relationship betn Short-Run andLong-Run Costs 短期和长期成本的关系of the AC curve在AC曲线的最低点AC = MC = SAC = SMCAt the minimum pothe MC curve crosses the AC curveMC曲线穿过AC曲线在该点，MC = ACMC = AC at this poSAC曲线与AC曲线相切

36、the SAC curve is tangent to the AC curveSAC (for this level of k) is minimized at the same level of output as ACSAC（对于这一k水平）与AC在同一产出水平上得以最小化SMC和SAC也在该点相交SMCersects SAC also at this po60Important Pos to Note需注意的要点：t wishes to minimize the economic costs ofA firmproducing a particular level of output

37、should chooset inpombination for which the rate of technicalsubstitution (RTS) is equal to the ratio of the inputsrental pri希望在生产特定产出水平上，最小化其经济成本的企业，应该选择技术替代率(RTS)等于投入租用价格之比的投入组合61Important Pos to Note:需注意的要点:Repeated application of this minimization procedurepath 重复这一最小化过程，yields the firms expan可以得

38、到厂商的扩展线the expanpath shows how input usage expands with thelevel of output扩展线表示投入的使用如何随着产出水平变化it also shows the relationship betn output level and total cost也表示产出水平和总成本之间的关系this relationship immarized by the total cost function, C(v,w,q)这一关系可以用总成本函数C(v,w,q)总结62Important Pos to Note:需注意的要点：The firms average cost (AC = C/q) and marginal cost(MC = C/q) can be derived direc