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The plot of the x1-coordinate of the p1- price offer curve against p1 is the ordinary demand curve for commodity 1.
The Case of Cobb-Douglas Utility Function
x1
Own-Price Changes p1
x 2 Fixed p2 and y. p1’’’
Ordinary demand curve for commodity 1
p1’’
p1 priceoff来自rp1’curve
x1*(p1’’’)
x1*(p1’) x1*
x1*(p1’’)
x1*(p1’’’) x1*(p1’)
U ( x 1 ,x 2 ) x 1 x 2 .
Then the ordinary demand functions for commodities 1 and 2 are
x * 1(p 1,p 2,y ) y 0/p 1,,iiffp p 1 1 p p 2 2
and
x * 2(p 1,p 2,y ) y 0/p 2,,iiffp p 1 1 p p 2 2.
Chapter Six
Demand 需求函数的静态比较分析
What Do We Do in This Chapter?
We conduct comparative statics analysis of ordinary demand functions -- the study of how ordinary demands x1*(p1,p2,y) and x2*(p1,p2,y) change as prices p1, p2 and income y change.
U ( x 1 ,x 2 ) m i n x 1 ,x 2 .
Then the ordinary demand functions for commodities 1 and 2 are
The Case of Perfect-Complements Utility Function
x * 1 (p 1 ,p 2 ,y ) x * 2 (p 1 ,p 2 ,y ) p 1 y p 2 .
x1
x1*(p1’’)
Own-Price Changes
The curve containing all the utilitymaximizing bundles traced out as p1 changes, with p2 and y constant, is the p1- price offer curve.
Own-Price Changes p1
x 2 Fixed p2 and y. p1’’’
p1’’ p1’
Ordinary demand curve for commodity 1
x1*(p1’’’) x1*(p1’) x1*(p1’’)
x1*(p1’’’)
x1*(p1’) x1*
x1*(p1’’)
for commodity 1
is
x*1
ay (ab)p1
x1*
x1*(pxx1’*11’’*)(p1(’a’x)1*a(byp)1p’)1
x1
The Case of Perfect-Complements
Utility Function
What does a p1 price-offer curve look like for a perfect-complements utility function?
x 2 Fixed p2 and y.
x
* 2
by
(a b )p 2
x1*(pxx1’*11’’*)(p1(’a’x)1*a(byp)1p’)1
x1
Own-Price Changes p1
x 2 Fixed p2 and y.
x
* 2
by
(a b )p 2
Ordinary
demand curve
Own-Price Changes p1
Fixed p2 and y. p1’’’ x2
y/p2
p1’’
x
* 2
p1’
y
p1 p2
x*1
p1
y p2
x1
Ordinary
demand curve
for commodity 1
is
x*1
p1
y. p2
y
x1*
p2
The Case of Perfect-Substitutes Utility Function
Notice that x2* does not vary with p1 so the
p1 price offer curve is flat and the ordinary
demand curve for commodity 1 is a
rectangular hyperbola.
Own-Price Changes
Theoretically, nothing new.
Own-Price Changes
How does x1*(p1,p2,y) change as p1 changes, holding p2 and y constant?
Suppose only p1 increases, from p1’ to p1’’ and then to p1’’’.
Take
U (x 1,x2)x 1 a x b 2.
Then the ordinary demand functions for commodities 1 and 2 are
Own-Price Changes
x* 1(p1,p2,y)a abp y 1 and x* 2(p1,p2,y)ab bp y 2.
Own-Price Changes
x * 1 (p 1 ,p 2 ,y ) x * 2 (p 1 ,p 2 ,y ) p 1 y p 2 .
With p2 and y fixed, higher p1 causes smaller x1* and x2*.
As p1 0, x* 1x* 2 p y 2. As p 1 , x * 1 x * 2 0 .
