经济学应用 1

1、1,Chapter 2,The Theory of the Firm,2,Part I Production Function,厂商理论,厂商是生产商品的技术单位。企业家决定生产多少和怎样生产，企业因其决策而获得利润或遭受损失。 企业家在由其生产函数确定的技术法则的约束下，将投入品转化为产品。生产函数是企业主使用的投入品数量与产品数量之间关系的数学表述。,厂商理论,对厂商的正规分析与对消费者的正规分析在许多方面是相似的。 消费者购买商品以“生产”满足，企业家购买投入品以生产产品。消费者有一个效用函数，厂商有一个生产函数。消费者的预算方程是他购买的商品数量的线性函数，竞争性厂商的成本方程是它购买

2、的投入品数量的线性函数。,厂商理论,厂商分析和消费者分析当然是存在差异的，如效用函数是主观的，效用不具有确切的基数测量单位；生产函数是客观的，厂商的产出很容易测量。单个厂商生产的产品可能不止一种，企业主的极大化过程会超出消费者的极大化过程。 理性消费者使既定收入的效用极大化，企业家的类似行为是使既定成本水平的产量极大化，但他经常会把成本看作是可变的，他可能希望使生产既定产量的成本极小化，或者使生产和销售某种商品所得到的利润极大化。,生产函数,考察某个简单的生产过程： 企业家利用两种可变的投入(X1和X2)，一种或多种固定投入，生产某一种产品(Q)。其生产函数把产量(q)作为可变投入量(x1和x

3、2)的函数： (2-1) 假定(2-1)是具有连续的一阶和二阶偏导数的单值连续函数。生产函数只对非负的投入和产出水平有定义，小于零的数值在这里是没有意义的。,生产函数,一般假定，生产函数在定义域内是递增的，即fi0。当产出达到极大或成本达到极小时，生产函数被假定为正则严格拟凹函数，当利润达到极大时，生产函数被假定是严格凹函数。 技术是所有的关于生产产品所必需的投入组合的技术信息。生产函数与技术的不同之处在于，生产函数预先假定技术的效率，并说明每种可能的投入组合所能达到的最大产出。任何特定的投入组合的最优利用是一个技术问题，而不是经济问题。为生产某一特定产量而进行的最优投入组合的选择，决定于投入

4、品和产品的价格，是经济分析的对象。,生产函数,短期生产函数,10,短期生产函数,As the use of an input increases in equal increments, a point will be reached at which the resulting additions to output decreases (i.e. MP declines).,The Law of Diminishing Marginal Returns,When the input of i is small, MP increases due to specialization. Whe

5、n the input of i is large, MP decreases due to inefficiencies.,The Law of Diminishing Marginal Returns,短期生产函数,X1的产出弹性用 表示，定义为Q变化率对X1的变化率的比率： (2-2) 假定生产函数是 ，求X1的产出弹性。,What does a technology look like when there is more than one input? The two input case: Input levels are x1 and x2. Output level is y.

6、 Suppose the production function is,长期生产函数,E.g. the maximal output level possible from the input bundle(x1, x2) = (1, 8) is And the maximal output level possible from (x1,x2) = (8,8) is,长期生产函数,1. 等产量线 An isoquant (等产量线) is the set of all possible combinations of inputs 1 and 2 that are just sufficie

7、nt to produce a given amount of output. 等产量曲线是能够带来相同产量水平的两种要素不同组合比例的轨迹,长期生产函数,12040556575 24060758590 3557590100105 46585100110115 57590105115120,12345,劳 动,资 本,Isoquants,1,2,3,4,1,2,3,4,5,5,q1 = 55,A,D,B,q2 = 75,q3 = 90,C,E,K,L,Isoquants,The complete collection of isoquants is the isoquant map. The

8、isoquant map is equivalent to the production function - each is the other. E.g.,长期生产函数,y 8,y 4,x1,x2,y 6,y 2,Cobb-Douglas Technologies,A Cobb-Douglas production function is of the form E.g.with,x2,Cobb-Douglas Technologies,x1,All isoquants are hyperbolic (双曲线), asymptoting (渐进) to, but never touchin

9、g any axis.,Cobb-Douglas Technologies,x2,x1,All isoquants are hyperbolic,asymptoting to, but nevertouching any axis.,Fixed-Proportions Technologies,x2,x1,minx1,2x2 = 14,4,8,14,2,4,7,minx1,2x2 = 8,minx1,2x2 = 4,x1 = 2x2,y = minx1,2x2,Perfect-Substitution Technologies,y=x1 + 3x2,Marginal (Physical) Pr

10、oducts,The marginal product (边际产量) of input i is the rate-of-change of the output level as the level of input i changes, holding all other input levels fixed. That is,Marginal (Physical) Products,E.g. if,then the marginal product of input 1 is,and the marginal product of input 2 is,Marginal (Physica

11、l) Products,Typically the marginal product of one input depends upon the amount used of other inputs. E.g. if,then,and if x2 = 27 then,if x2 = 8,Marginal (Physical) Products,The marginal product of input i is diminishing if it becomes smaller as the level of input i increases. That is, if,Marginal (

12、Physical) Products,and,so,and,Both marginal products are diminishing.,E.g. if,then,At what rate can a firm substitute one input for another without changing its output level?,Marginal Rate of Technical Substitution (边际技术替代率),Marginal Rate of Technical Substitution (边际技术替代率),Marginal Rate of Technica

13、l Substitution (边际技术替代率),Marginal Rate of Technical Substitution (边际技术替代率),How is a marginal rate of technical substitution computed? The production function is A small change (dx1, dx2) in the input bundle causes a change to the output level of,But dy = 0 since there is to be no changeto the output

14、 level, so the changes dx1and dx2 to the input levels must satisfy,Marginal Rate of Technical Substitution (边际技术替代率),rearranges to,so,Marginal Rate of Technical Substitution (边际技术替代率),is the rate at which input 2 must be givenup as input 1 increases so as to keepthe output level constant. It is the

15、slopeof the isoquant.,Marginal Rate of Technical Substitution (边际技术替代率),so,and,The MRTS is,MRTS; A Cobb-Douglas Example,x2,MRTS; A Cobb-Douglas Example,x1,x2,x1,8,4,MRTS; A Cobb-Douglas Example,x2,x1,6,12,MRTS; A Cobb-Douglas Example,2.脊线和生产区域 尽管生产要素之间具有替代性，但这种替代性有一个限度，当替代到一定程度时，会发生技术上的困难。 假定生产中使用两种

16、要素：劳动和资本，劳动增加，相应资本减少，总产量不变，那么可否完全用劳动来替代资本进行生产呢？,长期生产函数,2.脊线和生产区域 答案当然是否定的，因为光靠一种要素的投入进行生产是不可能的。 从下图来看，投入K1 ,L1 , K2 , L2 , K3入等这些不同的组合生产相同数量的产出y，随着资本投入量的下降，劳动投入量不断增加。但当K减少到B点时(这时等产量线斜率为零)不能再减少了，B点的纵坐标是生产y单位产出所需要的最低量的。,长期生产函数,2.脊线和生产区域 同样，A点(这时等产量线的斜率为无穷大)是生产y单位产出所需要的最低量的L。 A点、 B点之外的等产量线的斜率为正，这表明两种要素

17、必须同时增加才能维持总产量不变。 比较B点和D点，不仅使用的劳动量增加了，使用的资本也增加了。,长期生产函数,2.脊线和生产区域 如果仅仅劳动增加，资本不变则有C点，在C点上产量反而小于B点，这表明在B点以右的等产量线上，劳动的边际产量为负。 同样，可以说明A点以外的等产量线上，资本的边际产出也为负值。,长期生产函数,O,L,K,E,y,C,D,B,A,F,要素的投入区域,K2,K3,K1,L1,L2,L3,长期生产函数,2.脊线和生产区域 从等产量线的性质可以知道，一个等产量线的坐标平面上有无数条等产量线，每条等产量线又类似于上图，都有类似的A点和B点。我们把所有等产量线上切线斜率为零和无穷

18、大的点与原点一起连接起来，如下图，形成的两条线称为“脊线”(ridge line)。,长期生产函数,2.脊线和生产区域 超过脊线范围之外，必须同时增加两种投入要素的数量才能使总产量不变。 脊线表明生产要素替代的有效范围，厂商只能在脊线范围之内从事生产，实现不同要素组合，因此，这一脊线围成的区域是“生产区域”。,长期生产函数,长期生产函数,长期生产函数,3.替代弹性 如果生产函数有凸的等产量线，那么X1对X2的技术替代率和投入品比率x2/x1，都随着X1沿着等产量线替代X2而下降。替代弹性 是测量替代比率的一个纯数，它定义为投入品比率变化的比例与技术替代率变化率的比值：,长期生产函数,3.替代弹

19、性 通过计算得 其中 ，根据严格拟凹性的假定，D为正数，所以替代弹性是正数。 假定生产函数为 ，求其替代弹性。,4. 规模报酬 厂商规模变化指厂商所有的要素投入按同一比例增加或减少 产量变化与规模变化有区别：短期内，规模不变，通过可变投入的调整变动产量，是规模不变条件下的产量变化；长期内，通过所有投入的调整变动产量，是规模变化条件下的产量变化,长期生产函数,Returns-to-Scale（规模报酬）,Marginal products describe the change in output level as a single input level changes. Returns-to

20、-scale describes how the output level changes as all input levels change in direct proportion(正比例) (e.g. all input levels doubled, or halved).,Returns-to-Scale,If, for any input bundle (x1,xn),then the technology described by theproduction function f exhibits constantreturns-to-scale（规模报酬不变）.E.g. (k

21、 = 2) doubling all input levelsdoubles the output level.,Returns-to-Scale,One input, one output,Returns-to-Scale,If, for any input bundle (x1,xn),then the technology exhibits diminishingreturns-to-scale （规模报酬递减）.E.g. (k = 2) doubling all input levels less than doubles the output level.,Returns-to-Sc

22、ale,One input, one output,Returns-to-Scale,If, for any input bundle (x1,xn),then the technology exhibits increasingreturns-to-scale （规模报酬递增）.E.g. (k = 2) doubling all input levelsmore than doubles the output level.,Returns-to-Scale,One input, one output,Returns-to-Scale,A single technology can local

23、ly exhibit different returns-to-scale.,Returns-to-Scale,One input, one output,Examples of Returns-to-Scale,The perfect-substitutes productionfunction is,Expand all input levels proportionatelyby k. The output level becomes,The perfect-substitutes productionfunction exhibits constant returns-to-scale

24、.,Examples of Returns-to-Scale,The perfect-complements productionfunction is,Expand all input levels proportionatelyby k. The output level becomes,The perfect-complements productionfunction exhibits constant returns-to-scale.,Examples of Returns-to-Scale,The Cobb-Douglas production function is,Expan

25、d all input levels proportionatelyby k. The output level becomes,Examples of Returns-to-Scale,The Cobb-Douglas production function is,The Cobb-Douglas technologys returns-to-scale isconstant if a1+ + an = 1increasing if a1+ + an 1decreasing if a1+ + an 1.,Returns-to-Scale,Q: Can a technology exhibit

26、 increasing returns-to-scale even though all of its marginal products are diminishing?,Returns-to-Scale,Q: Can a technology exhibit increasing returns-to-scale even if all of its marginal products are diminishing? A: Yes. E.g.,Returns-to-Scale,so this technology exhibitsincreasing returns-to-scale.,

27、But,diminishes as x1,increases and,diminishes as x2,increases.,Returns-to-Scale,A marginal product is the rate-of-change of output as one input level increases, holding all other input levels fixed. Marginal product diminishes because the other input levels are fixed, so the increasing inputs units

28、have each less and less of other inputs with which to work.,Returns-to-Scale,When all input levels are increased proportionately, there need be no diminution(减少) of marginal products since each input will always have the same amount of other inputs with which to work. Input productivities need not f

29、all and so returns-to-scale can be constant or increasing.,Part II Cost Function,Cost Minimization,A firm is a cost-minimizer if it produces any given output level y 0 at smallest possible total cost. c(y) denotes the firms smallest possible total cost for producing y units of output. c(y) is the fi

30、rms total cost function （总成本函数）.,Cost Minimization,When the firm faces given input prices w = (w1,w2,wn) the total cost function will be written asc(w1,wn,y).,The Cost-Minimization Problem,Consider a firm using two inputs to make one output. The production function isy = f(x1,x2). Take the output le

31、vel y 0 as given. Given the input prices w1 and w2, the cost of an input bundle (x1,x2) is w1x1 + w2x2.,The Cost-Minimization Problem,For given w1, w2 and y, the firms cost-minimization problem is to solve,subject to,The Cost-Minimization Problem,The levels x1*(w1,w2,y) and x1*(w1,w2,y) in the least

32、-costly input bundle are the firms conditional demands for inputs 1 and 2 (条件要素需求). The (smallest possible) total cost for producing y output units is therefore,Conditional Input Demands,Given w1, w2 and y, how is the least costly input bundle located? And how is the total cost function (成本函数)comput

33、ed?,Iso-cost Lines (等成本线),等成本线：是生产要素价格一定时，花费一定总成本(C) 所能购买的生产要素所有可能组合的轨迹. A curve that contains all of the input bundles that cost the same amount is an iso-cost curve. E.g., given w1 and w2, the $100 iso-cost line has the equation,Iso-cost Lines,Generally, given w1 and w2, the equation of the $c iso

34、-cost line isi.e. Slope is - w1/w2.,Iso-cost Lines,The y-Output Unit Isoquant,The Cost-Minimization Problem,The Cost-Minimization Problem,The Cost-Minimization Problem,x1,x2,f(x1,x2) y,x1*,x2*,At an interior cost-min input bundle:(a) and(b) slope of isocost = slope of isoquant; i.e.,The Cost-Minimiz

35、ation Problem,A Cobb-Douglas Example of Cost Minimization,A firms Cobb-Douglas production function is Input prices are w1 and w2. What are the firms conditional input demand functions?,A Cobb-Douglas Example of Cost Minimization,A Cobb-Douglas Example of Cost Minimization,A Cobb-Douglas Example of C

36、ost Minimization,A Cobb-Douglas Example of Cost Minimization,So the cheapest input bundle yielding y output units is,Fixed w1 and w2.,Conditional Input Demand Curves,x,2,x,1,Fixed w1 and w2.,Conditional Input Demand Curves,x,2,x,1,Fixed w1 and w2.,Conditional Input Demand Curves,x,2,x,1,Fixed w1 and

37、 w2.,Conditional Input Demand Curves,x,2,x,1,Fixed w1 and w2.,Conditional Input Demand Curves,outputexpansionpath,Cond. demand for input 2,Cond.demandfor input 1,x,2,x,1,A Cobb-Douglas Example of Cost Minimization,For the production function the cheapest input bundle yielding y output units is,A Cob

38、b-Douglas Example of Cost Minimization,So the firms total cost function is,A Perfect Complements Example of Cost Minimization,The firms production function is Input prices w1 and w2 are given. What are the firms conditional demands for inputs 1 and 2? What is the firms total cost function?,A Perfect

39、 Complements Example of Cost Minimization,A Perfect Complements Example of Cost Minimization,Average Total Production Costs,For positive output levels y, a firms average total cost of producing y units is,Returns-to-Scale and Av. Total Costs,The returns-to-scale properties of a firms technology dete

40、rmine how average production costs change with output level. Our firm is presently producing y output units. How does the firms average production cost change if it instead produces 2y units of output?,Constant Returns-to-Scale and Average Total Costs,If a firms technology exhibits constant returns-

41、to-scale then doubling its output level from y to 2y requires doubling all input levels. Total production cost doubles. Average production cost does not change.,Decreasing Returns-to-Scale and Average Total Costs,If a firms technology exhibits decreasing returns-to-scale then doubling its output lev

42、el from y to 2y requires more than doubling all input levels. Total production cost more than doubles. Average production cost increases.,Increasing Returns-to-Scale and Average Total Costs,If a firms technology exhibits increasing returns-to-scale then doubling its output level from y to 2y require

43、s less than doubling all input levels. Total production cost less than doubles. Average production cost decreases.,Returns-to-Scale and Av. Total Costs,Returns-to-Scale and Total Costs,What does this imply for the shapes of total cost functions?,Returns-to-Scale and Total Costs,y,$,c(y),y,2y,c(y),c(

44、2y),Slope = c(2y)/2y = AC(2y).,Slope = c(y)/y = AC(y).,Av. cost increases with y if the firmstechnology exhibits decreasing r.t.s.,Returns-to-Scale and Total Costs,Returns-to-Scale and Total Costs,Short-Run e.g. the number of dollars of profit earned per hour.,Opportunity Costs (机会成本),All inputs mus

45、t be valued at their market value. Labor Capital,Economic Profit,How do we value a firm? Suppose the firms stream of periodic economic profits is P0, P1, P2, and r is the rate of interest. Then the present-value of the firms economic profit stream is,Profit Maximization,A competitive firm seeks to m

46、aximize its present-value. How?,Suppose the firm is in a short-run circumstance in which Its short-run production function is The firms fixed cost isand its profit function is,Short-Run Profit Maximization,Short-Run Iso-Profit Lines,A $P iso-profit line (等利润线) contains all the production plans that

47、yield a profit level of $P . The equation of a $P iso-profit line is I.e.,Short-Run Iso-Profit Lines,Short-Run Iso-Profit Lines,Short-Run Profit-Maximization,The firms problem is to locate the production plan that attains the highest possible iso-profit line, given the firms constraint on choices of

48、 production plans. Q: What is this constraint? A: The production function.,Short-Run Profit-Maximization,x1,Technicallyinefficientplans,y,The short-run production function andtechnology set for,Short-Run Profit-Maximization,x1,Increasing profit,y,Short-Run Profit-Maximization,x1,y,Short-Run Profit-M

49、aximization,x1,y,Given p, w1 and the short-runprofit-maximizing plan is,Short-Run Profit-Maximization,x1,y,Given p, w1 and the short-runprofit-maximizing plan is And the maximumpossible profitis,Short-Run Profit-Maximization,x1,y,At the short-run profit-maximizing plan, the slopes of the short-run p

50、roduction function and the maximaliso-profit line areequal.,Short-Run Profit-Maximization,x1,y,At the short-run profit-maximizing plan, the slopes of the short-run production function and the maximaliso-profit line areequal.,Short-Run Profit-Maximization,is the marginal revenue product(边际 收益产量) of i

51、nput 1. If then profit increases with x1. If then profit decreases with x1.,Short-Run Profit-Maximization; A Cobb-Douglas Example,Short-Run Profit-Maximization; A Cobb-Douglas Example,Short-Run Profit-Maximization; A Cobb-Douglas Example,Short-Run Profit-Maximization; A Cobb-Douglas Example,Short-Ru

52、n Profit-Maximization; A Cobb-Douglas Example,Short-Run Profit-Maximization; A Cobb-Douglas Example,Comparative Statics of Short-Run Profit-Maximization,What happens to the short-run profit-maximizing production plan as the output price p changes?,Comparative Statics of Short-Run Profit-Maximization

53、,x1,y,Comparative Statics of Short-Run Profit-Maximization,Comparative Statics of Short-Run Profit-Maximization,Comparative Statics of Short-Run Profit-Maximization,An increase in p, the price of the firms output, causes an increase in the firms output level (the firms supply curve slopes upward), a

54、nd an increase in the level of the firms variable input (the firms demand curve for its variable input shifts outward).,Comparative Statics of Short-Run Profit-Maximization,Comparative Statics of Short-Run Profit-Maximization,Comparative Statics of Short-Run Profit-Maximization,Comparative Statics o

55、f Short-Run Profit-Maximization,What happens to the short-run profit-maximizing production plan as the variable input price w1 changes?,Comparative Statics of Short-Run Profit-Maximization,Comparative Statics of Short-Run Profit-Maximization,x1,y,Comparative Statics of Short-Run Profit-Maximization,

56、x1,y,Comparative Statics of Short-Run Profit-Maximization,x1,y,Comparative Statics of Short-Run Profit-Maximization,An increase in w1, the price of the firms variable input, causes a decrease in the firms output level (the firms supply curve shifts inward), and a decrease in the level of the firms v

57、ariable input (the firms demand curve for its variable input slopes downward).,Comparative Statics of Short-Run Profit-Maximization,Comparative Statics of Short-Run Profit-Maximization,Comparative Statics of Short-Run Profit-Maximization,Comparative Statics of Short-Run Profit-Maximization,Long-Run

58、Profit-Maximization,Now allow the firm to vary both input levels, i.e., both x1 and x2 are variable. Since no input level is fixed, there are no fixed costs.,The profit-maximization problem is FOCs are:,Long-Run Profit-Maximization,Factor Demand Functions(要素需求函数),Demand for inputs 1 and 2 can be solved as,Inverse Factor Demand Functions,For a given optimal demand for x2, inverse demand functio