x* 1
y x 1 p1’
Own-Price Changes
Fixed p2 and y. x2
p1
p1 = p1’’ = p2
p2 = p1’’ p1’
x* 2
y p2
0
x* 1
y p2
x 1*
x* 2
0
x* 1 0
x* 1 y p2
x1
Own-Price Changes
Fixed p2 and y. x2
x1
0
x2.
, if p 1 p2
y / p 1 , if p 1 p 2 0 , if p 1 p 2 y / p 2 , if p 1 p 2 .
Own-Price Changes
Fixed p2 and y. x2
p1
p1 = p1’ < p2
p1’
x* 1
y x* 1 p1’
x* 2
0
x2.
The ordinary demand equations are
Income Changes and PerfectlySubstitutable Preferences
* x 1 ( p1 , p 2 , y )
* x 2 ( p1 , p 2 , y )
Suppose p1 < p2. Then
rectangular hyperbola 双曲线.
Own-Price Changes
Fixed p2 and y. x2
x* 2 by ( a b)p2
p1
Ordinary demand curve for commodity 1 is ay * x1 ( a b )p1
x 1*
x1*(p1’’’)
x 1*
x1
Own-Price Changes
What does a p1 price-offer curve look like for a perfect-substitutes utility function?
U(x1, x 2 )
* x 1 ( p1 , p 2 , y )
* x 2 ( p1 , p 2 , y )
U( x1 , x 2 )
* x1 * x2
m in x 1 , x 2 .
y p1 p2 .
The ordinary demand equations are
y y
( p1 ( p1
* p 2 ) x1 * p 2 )x 2
Engel curve for good 1 Engel curve for good 2
Own-Price Changes
* x 1 ( p1 , p 2 , y )
and
a
a b a
b b
* x 2 ( p1 , p 2 , y )
y p1 y . p2
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
Fixed p1 and p2. y’ < y’’ < y’’’
y’’’ y’’ y’ y y’’’ y’’ y’
Engel curve; good 2
Income offer curve
x2’’’ x2’’ x 2’
x2’ x2’’’ x2’’
x 2*
Engel curve; good 1 x1’ x1’’’ x1’’
p1
p1’’’
Ordinary demand curve for commodity 1
x* 1
p2 = p1’’
y p1
y p2
p1 price offer curve
p1’
0
x* 1
y p2
x 1*
x1
Own-Price Changes
p1 Given p1’, what quantity is demanded for commodity 1? Answer: x1’ units. The inverse question is: Given x1’ units are demanded, what is the price of commodity 1? x1* Answer: p1’
Own-Price Changes
x2 Fixed p2 and y. p1x1 + p2x2 = y p1 = p1’
p1= p1’’’
p1= p1’’ x1
Own-Price Changes
Fixed p2 and y.
p1
p1’’’ p1’’ p1’
Ordinary demand curve for commodity 1
x* 1 ay ; ( a b )p 1 x* 2 by . ( a b )p 2
y y
( a b )p1 * x1 a ( a b )p 2 * x2 b
Engel curve for good 1 Engel curve for good 2
Income Changes and CobbDouglas Preferences
0
0
, if p 1
, if p 1
p2 p2
p2
0
0.
y / p 1 , if p 1
y / p 2 , if p 1 p 2 . y * * and x 2 x1 p1
y
* and * p 1x 1 x2
* b )x1
is the ordinary demand function and
p1
is the inverse demand function.
Own-Price Changes
A perfect-complements example:
* x1
y p1
y
* x1
p2
p2
is the ordinary demand function and
Own-Price Changes
What does a p1 price-offer curve look like for Cobb-Douglas preferences? Take a b U( x 1 , x 2 ) x 1 x 2 . Then the ordinary demand functions for commodities 1 and 2 are
北京大学国家发展研究院2014年秋季学期 本科双学位课程 中级微观经济学(2班)
第四讲： 需求
任课教师：李力行
第六章-需求
Comparative statics analysis of demand function How ordinary demands x1*(p1,p2,y) and x2*(p1,p2,y) change as price and income change
x1’
x1’’’ x1’’
x1
x1’ x1’’’ x1’’
x1*
Income Changes
Fixed p1 and p2.
y
y
(p1
* p 2 )x 2
y’’’ y’’ y’ y
Engel curve; good 2
y
(p1
* p 2 )x1
y’’’ y’’ y’
x2’ x2’’’ x2’’
x 2*
p1
is the inverse demand function.
Income Changes
How does the value of x1*(p1,p2,y) change as y changes, holding both p1 and p2 constant?
Income Changes
Fixed p1 and p2. y’ < y’’ < y’’’
Income offer curve 收入提供曲线
x2’’’ x2’’ x 2’
x1’ x1’’’ x1’’
A plot of quantity demanded against income is called an Engel curve 恩格尔曲线 y
x1*(p1’’’)
x1*(p1’)
x 1*
x1*(p1’’) x1*(p1’’’) x1*(p1’)
x1*(p1’’)
Own-Price Changes
Fixed p2 and y.
p1
p1’’’ p1’’ p1’
Ordinary demand curve for commodity 1
p1 price offer curve
x1*(p1’’’)
x1*(p1’)
x 1*
x1*(p1’’) x1*(p1’’’) x1*(p1’)
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 价格提供曲线.
x1’ x1’’’ x1’’
x1*
Income Changes and CobbDouglas Preferences
An example of computing the Engel curves: the Cobb-Douglas case. a b U( x 1 , x 2 ) x 1 x 2 . The ordinary demand equations are