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1、Chapter 7 第七章 PRODUCTION FUNCTIONS 生产函数 1 Production Function 生产函数 The firms production function for a particular good (q) shows the maximum amount of the good that can be produced using alternative combinations of capital (k) and labor (l) 企业的特定商品的生产函 数，表示了使用不同的资本(k)和劳动(l)组 合，所能生产的商品的最大量 q = f(k,l)
2、 2 Marginal Physical Product 边际实物产量 To study variation in a single input, we define marginal physical product as the additional output that can be produced by employing one more unit of that input while holding other inputs constant为了研究单个投入的变化，我 们定义边际实物产量为：当保持其他投入不变时，多使用 一单位的该投入所能额外生产的产出 3 kkfkqMP c
3、apital of product physical marginallllfqMP labor of product physical marginalDiminishing Marginal Productivity 边际生产力递减 The marginal physical product of an input depends on how much of that input is used一种投入的边际产量，依赖于所 使用的该投入的数量 In general, we assume diminishing marginal productivity一般 我们假定边际生产力递减 4 0
4、1122 ffkf kMPkkk02222 fffMPlll llDiminishing Marginal Productivity 边际生产力递减 Because of diminishing marginal productivity, 19th century economist Thomas Malthus worried about the effect of population growth on labor productivity 因为边际生产力递减， 19世纪经济学家， 托马斯马尔萨斯担心人口增长对劳动 生产力的影响 But changes in the marginal
5、productivity of labor over time also depend on changes in other inputs such as capital 但是劳动的边际生产力随时间的变化也取决于其他的投入， 比如资本 we need to consider flk which is often 0 我们需要考虑flk （通常 0） 5 Average Physical Product 平均实物产量 Labor productivity is often measured by average productivity 劳动生产力通常以平均生产力衡量 6 ll ll),( i
6、nput laboroutputkfqAP Note that APl also depends on the amount of capital employed注意，APl也依赖于所使用的 资本的数量 A Two-Input Production Function 两种投入的生产函数 Suppose the production function for flyswatters can be represented by 假设苍蝇拍的生产函数可以表示为 q = f(k,l) = 600k 2l2 - k 3l3 To construct MPl and APl, we must assum
7、e a value for k 为了得到MPl和APl，我们必须假定k值 let k = 10 令k = 10 The production function becomes 生产函数变为 q = 60,000l2 - 1000l3 7 A Two-Input Production Function 两种投入的生产函数 The marginal productivity function is 边际生产力函数为 MPl = q/l = 120,000l - 3000l2 which diminishes as l increases (assume l20) 上式随着l增加递减(假 定l20）
8、,因为fll = 120,000 - 6000l This implies that q has a maximum value这意味着，q存在 最大值： 120,000l - 3000l2 = 0 40l = l2 l = 40 Labor input beyond l = 40 reduces output 超过l = 40的劳动 投入会减少产出 8 A Two-Input Production Function 两种投入的生产函数 To find average productivity, we hold k=10 and solve 为了得到平均生产力，我们令k=10并求解 APl =
9、 q/l = 60,000l - 1000l2 APl reaches its maximum where APl达到其最大值， 当 APl/l = 60,000 - 2000l = 0 l = 30 9 A Two-Input Production Function 两种投入的生产函数 In fact, when l = 30, both APl and MPl are equal to 900,000 事实上，当l = 30，APl和 MPl都等于900,000 Thus, when APl is at its maximum, APl and MPl are equal 所以，当APl达
10、到其最大值时， APl和 MPl相等 10 Isoquant Maps 等产量图 To illustrate the possible substitution of one input for another, we use an isoquant map为了描述投入间可能的 替代，我们使用等产量图 An isoquant shows those combinations of k and l that can produce a given level of output (q0) 等产量线，表示了 可以生产给定产出水平的k 和l的组合 f(k,l) = q0 11 Isoquant Ma
11、p 等产量线 12 l per period k per period Each isoquant represents a different level of output 每条等产量线，代表了一个不同的产出水平 output rises as we move northeast 随着我们向东北方向移动， 产出增加 q = 30 q = 20 Marginal Rate of Technical Substitution (RTS) 边际技术替代率 13 l per period k per period q = 20 -slope = marginal rate of technical
12、 substitution (RTS) -斜率 = 边际技术替代率(RTS) The slope of an isoquant shows the rate at which l can be substituted for k 等产量线的斜率，表示了l可以被k替 代的比率 lA kA kB lB A B RTS 0 and is diminishing for increasing inputs of labor RTS 0并且随着劳动投入的增加而 递减 Marginal Rate of Technical Substitution (RTS) 边际技术替代率 The marginal ra
13、te of technical substitution (RTS) shows the rate at which labor can be substituted for capital while holding output constant along an isoquant 边际技术替代率(RTS) ，表示了保持产出在同一等产量 线上不变，劳动可以被资本替代的比值 14 0) for ( qqddkkRTSllRTS and Marginal Productivities RTS和边际生产率 Take the total differential of the production
14、 function: 对生产函数求全微分 15 dkMPdMPdkkfdfdqkllll Along an isoquant dq = 0, so 沿同一等产量线dq = 0 ，所以 dkMPdMPk llkqqMPMP ddkkRTSl ll0) for ( Hence the RTS is given by the ratio of the inputs marginal productivities. 所以RTS等于投入的边际生产力之比 RTS and Marginal Productivities RTS和边际生产率 Because MPl and MPk will both be n
15、onnegative, RTS will be positive (or zero) 因为MPl和MPk都是非负的，RTS将是正的 （或者0） However, it is generally not possible to derive a diminishing RTS from the assumption of diminishing marginal productivity alone 然而，单独从边际生产力递减的假设推出RTS递减， 一般是不可能的 16 RTS and Marginal Productivities RTS和边际生产率 To show that isoquants are convex, we would like to show that d(RTS)/dl 0, the denominator is positive 因为我们已假设fk 0，所以分母为正 Because fll and fkk are both assumed to be negative, the ratio will be negative if fkl is po
