微观经济学第20章(范里安) 上财

y’ output units?
x2* = y
min{4x1,x2} y’
x1*
x1
= y/4
A Perfect Complements Example of Cost
Minimization The firm’s production function is
y min{4x1, x2}
and the conditional input demands are
For the production function
y f (x1, x2 ) x11/ 3x22 / 3
the cheapest input bundle yielding y output
units is
x*1(w1, w2, y), x*2(w1, w2, y)
w2 2w1
2/ 3
1/
3
y
12
2/ 3
w11/ 3 w
2/ 2
3y
21/ 3
w11/ 3 w
2/ 2
3y
3
w1w 4
2 2
1/ 3
y.
六、A Perfect Complements Example of Cost Minimization
The firm’s production function is
y min{4x1, x2}.
y,
2w1 w2
1/3 y
.
So the firm’s total cost function is
c(w1, w2, y) w1x*1(w1, w2, y) w2x*2(w1, w2, y)
So the firm’s total cost function is
c( w1, w 2, y) w1x*1( w1, w2, y) w 2x*2( w1, w2, y)
2/3
x*1.
(a) y (x*1)1/3(x*2 )2/3
From (b),
x*2
2w1 w2
x*1.
(b)
w1 w2
x*2 2x*1
.
Now substitute into (a) to get
y
( x*1 )1/ 3
2w1 w2
x*1
2/3
2ww21
2/3
x*1.
So
x*12ww21 x2 f(x1,x2)=y
x1
2.等成本线 c=w1x1+w2x2
x2
c=w1x1+w2x2
x2＝－w1/w2×x1＋c/w2
x1
Iso-cost Lines
x2
Slopes = -w1/w2.
c” w1x1+w2x2
c’ w1x1+w2x2
c’ < c”
x1
x2
等成本线的移动
x1
Input prices w1 and w2 are given. What are the firm’s conditional
demands for inputs 1 and 2? What is the firm’s total cost function?
A Perfect Complements Exx2 ampl4ex1o=fxC2 ost Minimization
x2*
f(x1,x2) y’
x1*
x1
（三）均衡条件：
TRS＝ =-w1/w2 或TRS12＝ =-w1/w2 或者-MP1/MP2 =-w1/w2
即： MP1/w1＝MP2/w2
四、关于既定产量条件下的成本最小化的 数学证明
在f（x1，x2）＝y条件下，已知w1、w2，求x1 、x2使w1x1＋w2x2具有最小值？
A Cobb-Douglas Example of
Cost Minimization
At the input bundle (x1*,x2*) which minimizes the cost of producing y output units:
(a)
y (x*1)1/3(x*2 )2/3
and
min{4x1,x2} y’ x1
A Perfect Complements Exx2 ampl4ex1o=fxC2 ost Minimization
min{4x1,x2} y’ x1
A Perfect Complements Exx2 ampl4ex1o=fxC2 WoshterMe iisntihmeilzeaastitocnostly
The (smallest possible) total cost for producing y output units is therefore
c(w1, w2, y) w1x*1(w1, w2, y)
w2x*2(w1, w2, y).
三、图形分析法
（一）等产量线与等成本线 1.等产量线 f(x1,x2)=y
f(x1,x2) y’ x1
At an interior cost-min input bundle: x2 (a) f (x*1, x*2 ) y and
(b) slope of isocost = slope of
isoquant; i.e.
w1 TRS MP1
w2
MP2
at
(x*1, x*2 ).
A firm’s Cobb-Douglas production
function is
y
f
(
x1,
x
2
)
x11/
3x
2/ 2
3
.
Input prices are w1 and w2.
What are the firm’s conditional input demand functions（有条件的要素需求 函数）?
（二）成本最小化的图形分析
x2 c1=w1x1+w2x2
f(x1,x2)=y c3=w1x1+w2x2 c2=w1x1+w2x2
x1
The Cost-Minimization
x2
Problem
All input bundles yielding y’ units
of output. Which is the cheapest?
L＝ w1x1＋w2x2 － [y－f（x1，x2）] L/x1＝w1－×f/x1＝0 L/x2＝r－×f/x2＝0
即 MP1/w1＝MP2/w2 f（x1，x2）＝y 使L取min即 w1x1＋w2x2取得min。
由此我们可以得到:
有条件的要素需求函数（对应于某个产 量的要素需求）
w1
2ww21
2/ 3
y
w
2
2w1 w2
1/ 3
y
So the firm’s total cost function is
c( w1, w 2, y) w1x*1( w1, w 2, y) w 2x*2( w1, w 2, y)
w1
w2 2w1
2/ 3
y
w
2
2w1 w2
1/ 3
y
The Cost-Minimization Problem
Consider a firm using two inputs to make one output.
The production function is y = f(x1,x2).
Take the output level y 0 as given.
12
2/ 3
w11/
3w
2/ 2
3y
21/
3
w11/
3w
22/ 3 y
So the firm’s total cost function is
c( w1, w 2, y) w1x*1( w1, w 2, y) w 2x*2( w1, w 2, y)
w1
w2 2w1
2/ 3
y
w
2
2ww21
So the cheapest input bundle yielding y output units is
x*1(w1, w2, y), x*2(w1, w2, y)
w2 2w1
2/ 3
y,
2w1 w2
1/3 y
.
A Cobb-Douglas Example of
Cost Minimization
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 firm’s costminimization problem is to solve
x1=x1(w1,w2,y) 和x2=x2(w1,w2,y) 也可以得到成本函数
c= w1x1＋w2x2= c (w1,w2,y) 它表示的是当要素价格为(w1,w2) 时，生
产y单位产量的最小成本。
五、A Cobb-Douglas Example of Cost Minimization
w1 w2
x*2 2x*1
.
(a) y (x*1)1/3(x*2 )2/3
From (b),
x*2
2w1 w2
x*1.
(b)
w1 w2
x*2 2x*1
.
(a) y (x*1)1/3(x*2 )2/3
From (b),
x*2
2w1 w2
x*1.
(b)
w1 w2
x*2 2x*1
.
Now substitute into (a) to get
c(y) denotes the firm’s smallest possible total cost for producing y units of output.
c(y) is the firm’s total cost function.
Cost Minimization
When the firm faces given input prices w = (w1,w2,…,wn) the total cost function will be written as c(w1,…,wn,y).
第20章 成本最小化
一、成本最小化与利润最大化 如果产量是既定的，价格是常数
min w1x1+w2x2 与max pf(x1,x2)-w1x1-w2x2完全相同的 即就是说，既定产量条件下的成本最小
与利润最大化是一致的。
二、Cost Minimization
A firm is a cost-minimizer if it produces any given output level y 0 at smallest possible total cost.
y
(x*1
)1/ 3
2w1 w2
x*1
2/ 3
(a) y (x*1)1/3(x*2 )2/3
From (b),
x*2
2w1 w2
x*1.
(b)
w1 w2
x*2 2x*1
.
Now substitute into (a) to get
y
( x*1 )1/ 3
2w1 w2
x*1
2/3
2ww21
min w1x1 w 2x2
x1,x2 0
subject to f (x1,x2 ) y.
我们可以求最优的x1和x2 进而求出成本函数。
The levels x1*(w1,w2,y) and x1*(w1,w2,y) in the least-costly input bundle are the firm’s conditional demands for inputs 1 and 2.
(b)
w1 w2
y/ y/
x1 x2
(1 (2
/ /
3)(x*1 )2/3(x*2 )2/ 3)(x*1 )1/3(x*2 )1/
3 3
x*2 2x*1
.
A Cobb-Douglas Example of Cost Minimization
(a) y (x*1)1/3(x*2 )2/3
(b)
input bundle yielding y’ output units?
min{4x1,x2} y’
x1
A Perfect Complements Example of Cost
Minimization
x2
4x1 = x2 Where is the least costly
input bundle yielding
x*1 ( w1 ,
w2, y)
y 4
and x*2(w1, w2, y) y.
A Perfect Complements Example of Cost
2/
3
y
is the firm’s conditional demand for input 1.
Since
x*2
2w1 w2
x*1
and
x*1
w2 2w1
2/3
y
x*2
2w1 w2
2ww21
2/ 3
y
2w1 w2
1/ 3
y
is the firm’s conditional demand for input 2.