微观经济学成本函数

Chapter 8
第八章
COST FUNCTIONS
成本函数
Copyright ©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 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
Two 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
Economic Profits经济利润•Total costs for the firm are given by厂商的总成本为
total costs = C= w l+ vk
•Total revenue for the firm is given by厂商的总收入为
total revenue = pq= pf(k,l)•Economic profits (π) are equal to经济利润(π)为
π= total revenue -total cost
π= pq-w l-vk
π= pf(k,l) -w l-vk
Economic 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 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
) ，并且想最小化其成本
选定其产出水平(q
Cost-Minimizing 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 process 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
Cost-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= w l+ vk+ λ[q0-f(k,l)]
•First order conditions are一阶条件为
∂L/∂l= w-λ(∂f/∂l) = 0
∂L/∂k= v-λ(∂f/∂k) = 0
∂L/∂λ= q0-f(k,l) = 0
Cost-Minimizing Input Choices
成本最小化投入选择
q 0
Given output q 0, we wish to find the least costly point on the isoquant 给定产出q 0，我们希望在等产量线上，找到成本最少的点
C 1
C 2C 3
Costs 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 period
k per period
C 1< C 2< C 3
Cost-Minimizing Input Choices
成本最小化投入选择
C 1C 2
C 3
q 0The minimum cost of producing q 0 is C 2生产q 0的最小成本为C 2
l per period
k per period
k*
l *
The optimal choice is l *, k * 最优选择为l *, k *
This occurs at the tangency between the isoquant and the total cost curve 在等产量线和总成本曲线的切点处取得
Contingent Demand for Inputs
条件要素需求
•In Chapter 4, we considered an individual’s expenditure-minimization problem在第四章中，我们考虑了个人支出最小化问题
–we used this technique to develop the compensated demand for a good我们使用这一方法提出商品的补偿需求
•Can we develop a firm’s demand for an input in the same way?
我们可以用同样的方法提出厂商对一种投入的需求吗？•In the present case, cost minimization leads to a demand for capital and labor that is contingent on the level of output
being produced在当前情况下，成本最小化所引起的资本和劳动的需求是视生产的产出水平而定的
The Firm’s Expansion Path
厂商的扩展线
•The firm can determine the cost-minimizing combinations of k and l for every level of output厂商可以对任意产出水平，决定成本最小的k和l的组合
•If input costs （w，v) remain constant for all amounts the firm may demand, we can trace the locus of cost-
minimizing choices如果厂商在任何产量下都有唯一确定的成本组合（w，v)，那么我们可以找到成本最小化选择的轨迹
–called the firm’s expansion path叫做厂商的扩展线
The Firm’s Expansion Path
厂商的扩展线
l per period
k per period q 00The expansion path is the locus of cost-minimizing tangencies 扩展线是成本最小化切点的轨迹
q 0
q 1
E The curve shows how inputs increase as
output increases
曲线显示了随着产出增加，投入如何增加
The Firm’s Expansion Path
厂商的扩展线
•The expansion path does not have to be a straight line 扩展线并不一定是直线
–the use of some inputs may increase faster than others as output expands随着产出扩张，某些投入的使用可能会比其他投入增加
快
•depends on the shape of the isoquants取决于等产量线的形状•The expansion path does not have to be upward sloping 扩展线并不一定是向上倾斜的
–if the use of an input falls as output expands, that input is an inferior input如果随着产出扩张某种投入的使用减少，该投入为劣等投
入要素
Cost Minimization成本最小化•Suppose that the production function is Cobb-Douglas假设生产函数为柯布Ÿ道格拉
斯
q= kαlβ
•The Lagrangian expression for cost minimization of producing q0is 生产q0的成本最小化的拉格朗日表示为
L= vk+ w l+ λ(q0-kαlβ)
Cost Minimization成本最小化•The first-order conditions for a minimum are 最小值的一阶条件为
∂L/∂k= v-λαkα-1lβ= 0
∂L/∂l= w-λβkαlβ-1= 0
∂L/∂λ= q0-kαlβ= 0
Cost Minimization成本最小化•Suppose that the production function is CES 假设生产函数为CES：
q= (kρ+lρ)γ/ρ
•The Lagrangian expression for cost minimization of producing q0is生产q0的成本最小化的拉格朗日表示为
L= vk+ w l+ λ[q0-(kρ+lρ)γ/ρ]
Cost Minimization成本最小化•The first-order conditions for a minimum are 最小值的一阶条件为
∂L/∂k= v-λ(γ/ρ)(kρ+ lρ)(γ-ρ)/ρ(ρ)kρ-1= 0
∂L/∂l= w-λ(γ/ρ)(kρ+ lρ)(γ-ρ)/ρ(ρ)lρ-1= 0
∂L/∂λ= q0-(kρ+lρ)γ/ρ= 0
Total Cost Function总成本函数•The total cost function shows that for any set of input costs and for any output level, the minimum cost incurred by the firm is总成本函数表示对于任意一组投入要素价格和任意产出水平，厂商的最小总成本为
C= C(v,w,q)
•As output (q) increases, total costs increase 随着产出(q)增加，总成本增加
Graphical Analysis of Total Costs
总成本的图形分析
•Suppose that k1units of capital and l1units of labor input are required to produce one unit of output假设生产一单位产出
需要k
1单位资本和l
1
单位劳动
C(q=1) = vk1+ w l1
•To produce m units of output (assuming constant returns to scale)为了生产m单位产出（假定规模报酬不变）
C(q=m) = vmk1+ wm l1= m(vk1+ w l1)
C(q=m) = m⋅C(q=1)
Graphical Analysis of Total Costs 总成本的图形分析Output
Total
costs C With constant returns to scale, total costs are proportional to output
由于规模报酬不变，总成本和产出是成比例的
AC = MC
Both AC and MC will be constant
AC 和MC 都将是常数
Graphical Analysis of Total Costs
总成本的图形分析
•Suppose instead that total costs start out as concave and then becomes convex as output increases假如总成本开始为凹的，然后随着产出增加变为凸的
–one possible explanation for this is that there is a third factor of production that is fixed as capital and labor usage expands这种情况的一个可能的解释是，存在第三种不随资本和劳动使用增
加而变动的其他生产要素（如企业家才能）
–total costs begin rising rapidly after diminishing returns set in 总成本在报酬递减开始作用后，增加地很快
Graphical Analysis of Total Costs 总成本的图形分析Output
Total
costs C
Total costs rise dramatically as output increases after diminishing returns set in
随着产出增加，总成本在报酬递减开始作用后，急剧增加
Graphical Analysis of Total Costs 总成本的图形分析Output
Average and
marginal costs MC MC is the slope of the C curve MC 是曲线C 的斜率AC If AC > 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 AC
Shifts in Cost Curves
成本曲线的移动
•The cost curves are 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
Some Illustrative Cost 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
β
+αββ+ααβ+α=+=///1),,(w Bv q
w vk q w v C l where 等式中β+αβ−β+αα−β
αβ+α=//)(B which is a constant that involves only the parameters αand β这个值是常数，仅涉及参量αand β
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 ρ+l ρ)γ/ρ
•
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
ρ
−ρ−ρρ−ρργ+=+=/)1(1/1//1)(),,(w v q w vk q w v C l σ
−σ−σ−γ+=1/111/1)(),,(w v q q w v C
Properties of Cost Functions
成本函数的属性•Homogeneity齐次性
–cost functions are all homogeneous of degree one in the input prices成本函数对所有投入价格是一次齐次的
•cost minimization requires that the ratio of input
prices be set equal to RTS, a doubling of all input
prices will not change the levels of inputs purchased
成本最小化要求投入价格的比值等于RTS，所有成
本价格翻倍不会影响所购买的投入组合
•pure, uniform inflation will not change a firm’s input
decisions but will shift the cost curves up纯粹、均匀
的通货膨胀不会改变厂商的投入组合决策，但会
使成本曲线向上移动
Properties of Cost Functions
成本函数的属性•Nondecreasing in q, v, and w （成本函数）对q, v和w单调非递减
–cost functions are derived from a cost-
minimization process成本函数是由成本最小
化过程推导出来的
•any decline in costs from an increase in one of the
function’s 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
会导致矛盾
Properties of Cost Functions
成本函数的属性•Concave in input prices对投入价格为凹的–costs will be lower when a firm faces input
prices that fluctuate around a given level than
when they remain constant at that level当企业面临围绕一定水平波动的投入价格时，成本
会比投入价格固定时低
•the firm can adapt its input mix to take advantage
of such fluctuations厂商可以改变其投入组合，以
利用这种波动的优势
Concavity of Cost Function 成本函数的凹性C(v,w,q 1)
Since the firm’s input mix will likely
change, actual costs will be less than C pseudo such as C (v,w,q 1)
因为厂商的投入组合可能会改变，实际的成本会比C pseudo 低，比如C (v,w,q 1)C pseudo
If the firm continues to buy the same input mix as w changes, its cost function would be C pseudo 如果随着w 变化，厂商继续购买同样的投入组合，其成本函数将是C pseudo w
Costs At w 1, the firm’s costs are C(v,w 1,q 1)在w 1，厂商的成本为C(v,w 1,q 1)C (v ,w 1,q 1)w 1
Properties of Cost Functions
成本函数的属性
•Some of these properties carry over to average and marginal costs某些属性对平均和边际成本也适用
–Homogeneity齐次性
–effects of v, w, and q are ambiguous
v, w和q的影响是不确定的
Size of Shifts in Costs Curves
成本曲线移动的大小
•The increase in costs will be largely influenced by the relative significance of the input in the production process 成本增加很大程度上，受到投入在生产过程中相对重要程度的影响
•If firms can easily substitute another input for the one that has risen in price, there may be little increase in costs 如果厂商可以轻易地用其他投入替代价格增加的投入，成本增加就会较小
Technical Progress技术进步•Improvements in technology also lower cost curves技术进步可以降低成本曲线•Suppose that the production function (with constant returns to scale) is假设生产函数（规模报酬不变）为：
q= A(t)f(k,l), while A(0)=1•Total costs are总成本为
C0= C0(q,v,w) = qC0(v,w,1)
Technical Progress技术进步•Because the same inputs that produced one unit of output in period zero will produce A(t) units in period t 因为在时期0生产一单位产出的投入，在时期t可以生产A(t)单位
C t(v,w,A(t)) = A(t)C t(v,w,1)= C0(v,w,1)
•Total costs are given by总成本为
C t(v,w,q)= qC t(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
Contingent Demand for Inputs
条件要素需求
•Contingent demand functions for all of the firms inputs 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
–Shephard’s lemma谢泼德定理
•the contingent demand function for any input is given by the
partial derivative of the total-cost function with respect to that
input’s price 任意条件要素需求由总成本函数对要素价格的
偏微分得到。