中级微观经济学(英语) 20161010

Inferior goods:

Exercise:
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2016-2017学年上学期第四次课
INCOME CHANGES

How does the value of x1*(p1,p2,y) change as y changes, holding both p1 and p2 constant?
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m
(a b) p1 * m x1 a
Engel curve for good 1
m
x 1*
( a b) p2 * m x2 b
x 2*
Engel curve for good 2
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2016-2017学年上学期第四次课
INCOME CHANGES

In every example so far the Engel curves have all been straight lines? Q: Is this true in general? A: No. Engel curves are straight lines if the consumer’s preferences are homothetic.


homogeneity: A real-valued function f(x) is called homogeneous of degree k if f(tx)=t^{k}*f(x) for all t > 0.
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HOMOTHETICITY
Exercises:
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INCOME EFFECTS -- A NONHOMOTHETIC EXAMPLE

Quasilinear preferences are not homothetic.
U( x1 , x 2 ) f ( x1 ) x 2 .

For example,
Rearranged to isolate y, these are:
m ( p1 p2 ) x m ( p1 p2 ) x
* 1 * 2
Engel curve for good 1 Engel curve for good 2
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2016-2017学年上学期第四次课
Income Changes
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2016-2017学年上学期第四次课
HOMOTHETICITY
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2016-2017学年上学期第四次课
HOMOTHETICITY
Extension:

Homotheticity VS homogeneity
homotheticity:F(x) is a homothetic function if it can be written as F(x)=f(g(x)), where f is strictly increasing and g is homogeneous of degree one.
The demand equations are

U ( x1 , x2 ) x1 x2
0,if p1 p2 x ( p1 , p2 , m) m / p1,if p1 p2
* 1
0,if p1 p2 x ( p1 , p2 , m) m / p2 ,if p1 p2 .
2016-2017学年上学期第四次课
HOMOTHETICITY

A consumer’s preferences are homothetic if and only if for every t > 0.
(x1,x2)
p (y ,y )
1 2
(tx1,tx2)
p(ty ,ty )
1 2

That is, the consumer’s MRS is the same anywhere on a straight line drawn from the origin.

U ( x1 , x2 ) minx1 , x2
m x x . p1 p2
* 1 * 2
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2016-2017学年上学期第四次课
INCOME CHANGES AND PERFECTLYCOMPLEMENTARY PREFERENCES
m x x . p1 p2
* 1 * 2
m
x2*
x1’ x1’’
x1’’’
x1
x1’ x1’’
x1’’’
x1*
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2016-2017学年上学期第四次课
INCOME CHANGES AND PERFECTLYSUBSTITUTABLE PREFERENCES

An example of computing the equations of Engel curves; the perfectly-substitution case.
* 1
Rearranged to isolate m, these are:
(a b) p1 * m x1 a ( a b) p2 * m x2 b
Engel curve for good 1
Engel curve for good 2
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2016-2017学年上学期第四次课
INCOME CHANGES AND COBB-DOUGLAS PREFERENCES

a b U( x1 , x 2 ) x1 x 2 .
am bm * x ;x 2 . (a b) p1 ( a b) p2
* 1
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2016-2017学年上学期第四次课
INCOME CHANGES AND COBB-DOUGLAS PREFERENCES
am bm * x ;x 2 . (a b) p1 ( a b) p2
2016-2017学年上学期第四次课
INTERMEDIATE MICROECONOMICS
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授课教师 : 刘 璐 / Lu Liu
2016-2017学年上学期第四次课
CHAPTER SIX
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Demand
2016-2017学年上学期第四次课
OUTLINE
Properties of Demand Functions
2016-2017学年上学期第四次课
Income Changes
Fixed p1 and p2.
x2 m’ < m’’ < m’’’ Income offer curve x2’’’ x2’’ x 2’
m m’’’ m’’ m’ x 2’ x2’’ m’’’ m’’ m’ Engel curve; good 1 x2’’’ Engel curve; good 2

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2016-2017学年上学期第四次课
NORMAL AND INFERIOR GOODS

Normal goods:
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2016-2017学年上学期第四次课
NORMAL AND INFERIOR GOODS

Inferior goods:
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2016-2017学年上学期第四次课
NORMAL AND INFERIOR GOODS
Each curve intersects both axes.
~ x1
x1
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Income Changes; Quasilinear Utility
Baidu Nhomakorabea
x* 2 0.
x1* Engel curve for good 1
0 Engel curve for good 2
x2 *
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2016-2017学年上学期第四次课
INCOME CHANGES AND PERFECTLYSUBSTITUTABLE PREFERENCES
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* 2
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2016-2017学年上学期第四次课
INCOME CHANGES AND PERFECTLYSUBSTITUTABLE PREFERENCES
0,if p1 p2 x ( p1 , p2 , m) m / p1,if p1 p2
* 1
0,if p1 p2 x ( p1 , p2 , m) m / p2 ,if p1 p2 .


Exceptions:

Luxury good: demand goes up by a greater proportion than income. Necessary good: demand goes up by a lesser proportion than income.
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x1’ x1’’
x1’’’
x1
x1’
x1’’’ x1’’
x1*
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2016-2017学年上学期第四次课
Income Changes
Fixed p1 and p2.
m m’’’ m’’ m’ Engel curve; good 2
m ( p1 p2 ) x
* 2
m’’’
m
x 2’ x2’’
x2’’’
INCOME CHANGES AND PERFECTLYCOMPLEMENTARY PREFERENCES

Another example of computing the equations of Engel curves; the perfectly-complementary case. The demand equations are
Fixed p1 and p2.
x2
m’ < m’’ < m’’’ Income offer curve x2’’’ x2’’ x2’
m m’’’ m’’ m’ x 2’ x2’’ Engel curve; good 1 x2’’’ x2* Engel curve; good 2
m m’’’ m’’ m’
* 2
Suppose p1 < p2. Then
m x p1
* 1
and
x 0
* 2
m px
* 1 1
and
x 0
* 2
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2016-2017学年上学期第四次课
INCOME CHANGES AND PERFECTLYSUBSTITUTABLE PREFERENCES
m m
* y p1x1

Income Changes Own-Price Changes Cross-Price Changes


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2016-2017学年上学期第四次课
PROPERTIES OF DEMAND FUNCTIONS

The consumer’s demand functions: x1*(p1,p2,m) x2*(p1,p2,m) key: they give the optimal amounts of each of the goods as a function of the prices and income faced by the consumer. Comparative statics analysis of demand functions -the study of how demands x1*(p1,p2,m) and x2*(p1,p2,m) change as prices p1, p2 and income m change. key: before and after the change in the economic environment; not concerned with any adjustment process; only examine the equilibrium choice
x2*
m ( p1 p2 ) x
* 1
m’’ m’ x1’ x1’’’ x1’’
Engel curve; good 1
x1*
2016-2017学年上学期第四次课
INCOME CHANGES AND COBB-DOUGLAS PREFERENCES

An example of computing the equations of Engel curves; the Cobb-Douglas case. The demand equations are
U ( x1 , x2 ) ln x1 x2 .

“zero income effect” for good 1
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2016-2017学年上学期第四次课
Income Changes; Quasilinear Utility
x2 Each curve is a vertically shifted copy of the others.