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1、Chapter 8 第八章COST FUNCTIONS 成本函数1Copyright 2005 by South-western, a division of Thomson learning. All rights reserved.Economic Cost经济成本 The economic cost of any input is the payment required to keep that input in its present employment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
2、receive in its best alternative employment该投入用于其他用途所能获得的 最大报酬 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 Assumptions 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 homogeneous capital (k), measured in machine-hours同质的资本(k)， 以
3、机器小时衡量 entrepreneurial costs are included in capital costs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 + vk Total revenue fo
4、r the firm is given by厂商的总收入为 total revenue = pq = pf(k,l) Economic 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 and labor employed经济利润是所使用的资本和劳动数量的函 数 we could examine how
5、 a firm would choose k and l to maximize profit我们可以检验厂商如何选择k和l来最大化其利润 “derived demand” theory of labor and capital inputs 劳动和资本投入 的“引致需求”理论 for now, we will assume that the firm has already chosen its output level (q0) and wants to minimize its costs现在，我们假定厂商已经 选定其产出水平(q0) ，并且想最小化其成本5Cost-Minimizing
6、 Input Choices 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 point on the isoquant at which the RTS 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/v it should equate the rate at which k can be traded for l in the productive proce
7、ss to the rate at which they can be traded in the marketplace k和l在生产过程的替代比率应该与它们在市场 中交易价格的比值相等6Cost-Minimizing Input Choices 成本最小化投入选择 Mathematically, we seek to minimize total costs given q = f(k,l) = q0 数学上，我们在q = f(k,l) = q0的条件下最小化 成本 Setting up the Lagrangian建立拉格朗日函数L = wl + vk + q0- f(k,l) Firs
8、t 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 Input Choices 成本最小化投入选择Dividing the first two conditions we get前两个一阶条件相除， 得到8) for ( /kRTSkff vwll= The cost-minimizing firm should equate the RTS for the two inputs to the ratio of their prices成本最小
9、化的 厂商会使得两种投入的RTS与其价格比值相等Cost-Minimizing Input Choices 成本最小化投入选择 Cross-multiplying, we get交叉相乘，我们得到9wf vfkl= For costs to be minimized, the marginal productivity per dollar spent should be the same for all inputs为了成本最小化，每美元的 边际生产力对于所有投入应该是相同的Cost-Minimizing Input Choices 成本最小化投入选择 Note that this equa
10、tions inverse is also of interest注意这 一等式的倒数也是有意义的10=kfv fwl The Lagrangian multiplier shows how much in extra costs would be incurred by increasing the output constraint slightly拉格朗日乘子表示如果稍微增加产出 约束时，会产生的额外成本Cost-Minimizing Input Choices 成本最小化投入选择11q0Given output q0, we wish to find the least costly
11、point on the isoquant给定产出q0，我们希望在等产量线上，找到成 本最少的点C1C2C3Costs are represented by parallel lines with a slope of -w/v 成本由斜率为-w/v的 平行线表示l per periodk per periodC1 MC, AC must be falling 如果AC MC，AC一 定是下降的If AC MC, AC must be rising 如果AC MC，AC 一定是上升的min ACShifts in Cost Curves 成本曲线的移动 The cost curves are
12、drawn under the assumption that input prices and the level of technology are held constant成本曲线是在投 入价格和技术水平保持不变的假设下，得 出的 any change in these factors will cause the cost curves to shift这些因素的任何变化都会引起成 本曲线的移动31Some Illustrative Cost Functions 成本函数的例证 Suppose we have a fixed proportions technology such t
13、hat 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-shaped isoquants (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)32+=bw avqqvwC),(Some Illustrative Cost Functions 成本函数的例证 Suppose we have a Cobb-Douglas technology such that 假设技术为柯布?
14、道格拉斯，那么 q = f(k,l) = kl Cost minimization requires that最小化成本要求33lk vw=l =vwkSome Illustrative Cost Functions 成本函数的例证 If we substitute into the production function and solve for l, we will get 如果我们代入生产函数并求解l，我们可以 得到34+ + =/ /1vwql A similar method will yield相似的方法可以得到+ + =/ /1vwqkSome Illustrative Cos
15、t Functions 成本函数的例证 Now we can derive total costs as现在我们可以得到总成 本为35+=+=/1),(wBvqwvkqwvClwhere 等式中+=/)(Bwhich is a constant that involves only the parameters and 这个值是常数，仅涉及参量 and Some Illustrative Cost Functions 成本函数的例证 Suppose we have a CES technology such that假设技术为 CES,那么 q = f(k,l) = (k+l)/ To derive the total cost, we would use the same method and eventually get为得到总成本，我们需要用同样的方 法，最终得到36+=+=/ )1(1/1/1)(),(wvqwvkqwvCl+=1/111/1)(),(wvqqwvCProperties of Cost Functions 成本函数的属性 Homogeneity齐次性 cost functions are all homogeneous of de
