Lecture 4 - Demand

4

(Demand)

x1*(p1,p2,m)

x2*(p1,p2,m)

p1, p2m

x*=(x1*(p1,p2,m) , x2*(p1,p2,m))

1.

x1*(p1,p2,y)-

m-

p1 and p2 ?

x2

x1

p1 and p2

m < m < m

x2

x1x1x1

x1

x2x2x2

p1 and p2

m < m < m

x2

x1x1x1

x1

x2x2x2

p1 and p2

m < m < m (Income offer

curve)

(Income ExpansionPath) .

(Engelcurve) .

x2

x1x1x1

x1

x2x2x2

p1 and p2


curve)

x2

x1x1x1

x1

x2x2x2

x1*x1x1

x1

mmm

p1 and p2

m < m < m

m

(Income offer

curve)

x2

x1x1x1

x1

x2x2x2

x1*x1x1

x1

Engel

curve;

good 1

p1 and p2

m < m < m

mmm

m

(Income offer

curve)

x2

x1x1x1

x1

x2x2x2

x2*

m

x2x2

x2

mmm

p1 and p2


curve)

x2

x1x1x1

x1

x2x2x2

x2*x2x2

x2

Engel

curve;

good 2

p1 and p2

m < m < m

m

mmm

(Income offer

curve)

x2

x1x1x1

x1

x2x2x2

x1*

x2*

x1x1

x1

x2x2

x2

Engel

curve;

good 2

Engel

curve;

good 1

p1 and p2

m < m < m

mmm

m

m

mmm

(Income offer

curve)

(normal) .

.

(inferior) .

.

x2

x1

Goods 1 & 2 Normal

x1x1

x1

x2x2x2

x1*

x2*

x1x1

x1

x2x2

x2

Engel

curve;

good 2

Engel

curve;

good 1mmm

m

m

mmm

(Income offer

curve)

Good 2 Is Normal, Good 1 Becomes

Income Inferior

x2

x1

x2

x1


Income Inferior

x2

x1

Income

offer curve


Income Inferior

x2

x1 x1*

mEngel curve

for good 1


Income Inferior

x2

x1 x1*

x2*

m

Engel curve

for good 2

Engel curve

for good 1


Income Inferior

m

Engel curves

:

1 2 1 2U(x , x ) = x + x

* 1 2

1 1 2

1 1 2

0 , if p > px (p , p ,m) =

m / p , if p p .

Suppose p1 < p2. Then

* 1 2

1 1 2

1 1 2

0 , if p > px (p , p ,m) =

m / p , if p p .

Suppose p1 < p2. Then and

* 1 2

1 1 2

1 1 2

0 , if p > px (p , p ,m) =

m / p , if p p .

*

1

1

mx =

p

*

2x = 0

Suppose p1 < p2. Then and

and

*

2x = 0

* 1 2

1 1 2

1 1 2

0 , if p > px (p , p ,m) =

m / p , if p p .

*

2x = 0*

1 1m = p x

*

1

1

mx =

p

x2 0*

.y p x 1 1

*

m m

x1* x2*0

Engel curve

for good 1

Engel curve

for good 2

Slope is p1

1 2 1 2U(x , x ) = min x , x .

Engel curves

:

m- :

Engel curve for good 1


* *

1 2

1 2

mx = x =

p + p

*

1 2 1

*

1 2 2

m = (p + p )x

m = (p + p )x

p1 and p2

x1

x2

x1

x2m < m < m

p1 and p2

x1

x2

x1x1

x2x2x2

x1

m < m < m

p1 and p2

x1

x2

x1x1

x2x2x2

x1 x1*

m

mmm

Engel

curve;

good 1

x1x1

x1

m < m < m

p1 and p2

x1

x2

x1x1

x2x2x2

x1

x2*

m

x2x2

x2

mm

m

Engel

curve;

good 2

m < m < m

p1 and p2

x1

x2

x1x1

x2x2x2

x1 x1*

x2*x2x2

x2

Engel

curve;

good 2

Engel

curve;

good 1

x1x1

x1

m

mm

m

m < m < m

p1 and p2

m

mmm

x1*

x2*x2x2

x2

x1x1

x1

Engel

curve;

good 2

Engel

curve;

good 1

p1 and p2 m

mm

m

m

mmm

*

1 2 2m = (p + p )x

*

1 2 1m = (p + p )x

-

Engel curves

:

a b

1 2 1 2U(x ,x ) = x x

* *

1 2

1 2

am bmx = ; x = .

(a + b)p (a + b)p



* *

1 2

1 2

am bmx = ; x = .

(a + b)p (a + b)p

*11

*22

(a + b)pm = x

a(a + b)p

m = xb

m- :

mmx1*

x2*

*11

(a + b)pm = x

aEngel curve

for good 1

*22

(a + b)pm = x

bEngel curve

for good 2

? ?

: . (homothetic) .

Homotheticity

(homothetic) k>0 :

MRS .

1 2 1 2 1 2 1 2x , x y , y kx ,kx ky ,ky

A Non-homothetic Example

- .

,

1 2 1 2U(x , x ) = f(x ) + x

1 2 1 2U(x , x ) = x + x

-

x2

x1

.

2 .

-

x2

x1x1~

x2

x1x1~

x1*

m

x1~

Engel

curve

for

good 1

x2

x1x1~

x2*

m Engel

curve

for

good 2

x2

x1x1~

x1*

x2*

x1~

Engel

curve

for

good 2

Engel

curve

for

good 1

m

m

2.

x1*(p1,p2,m)

p1-

p2 and m ?

p1 p1 p1 , p1 .

x1

x2

p1 = p1

p2 and m

p1x1 + p2x2 = m

x1

x2

p1= p1

p1 = p1

p1x1 + p2x2 = m

p2 and m

x1

x2

p1= p1p1=

p1

p1 = p1

p1x1 + p2x2 = m

p2 and m

x2

x1

p1 = p1p1x1 + p2x2 = m

p2 and m

x2

x1x1*(p1)

p1 = p1

p2 and m

x2

x1x1*(p1)

p1

x1*(p1)

p1

x1*

p1 = p1

p2 and m

x2

x1x1*(p1)

p1

x1*(p1)

p1

p1 = p1

x1*

p2 and m

x2

x1x1*(p1)

x1*(p1)

p1

x1*(p1)

p1

p1 = p1

x1*

p2 and m

x2

x1x1*(p1)

x1*(p1)

p1

x1*(p1)

x1*(p1)

p1

p1

x1*

p2 and m

x2

x1x1*(p1)

x1*(p1)

p1

x1*(p1)

x1*(p1)

p1

p1

p1 = p1

x1*

p2 and m

x2

x1x1*(p1) x1*(p1)

x1*(p1)

p1

x1*(p1)

x1*(p1)

p1

p1

p1 = p1

x1*

p2 and m

x2

x1x1*(p1) x1*(p1)

x1*(p1)

p1

x1*(p1)x1*(p1)

x1*(p1)

p1

p1

p1

x1*

p2 and m

x2

x1x1*(p1) x1*(p1)

x1*(p1)

p1

x1*(p1)x1*(p1)

x1*(p1)

p1

p1

p1

x1*

1-

p2 and m

x2

x1x1*(p1) x1*(p1)

x1*(p1)

p1

x1*(p1)x1*(p1)

x1*(p1)

p1

p1

p1

x1*

p1

p2 and m 1-

p1- (price offer curve) (price expansion path) .

(x1, p1) 1- (ordinary demand curve) .

x1

x2

p2 and m

, (ordinary).

x1

x2

p1 price

offer

curve

p2 and m

x1

x2

p1 price

offer

curve

x1*

1- (ordinary)

p1

p2 and m

x1

x2

p2 and m

(Giffen) .

x1

x2 p1 price offer

curve

p2 and m

x1

x2 p1 price offer

curve

x1*

1- Giffen

p1

p2 and m

p1 ?

1 2- .

1 2 1 2U(x , x ) = x + x

* 1 2

1 1 2

1 1 2

0 , if p > px (p , p ,m) =

m / p , if p p

and

x2

x1

*

2x = 0

,

1

*

1

mx =

p

p1 = p1 < p2

p2 and m

x2

x1

p1

x1*

p1

p1 = p1 < p2

,

1

*

1

mx =

p

*

2x = 0

,

1

*

1

mx =

p

p2 and m

x2

x1

p1

x1*

p1

p1 = p1 = p2

p2 and m

x2

x1

p1

x1*

*

2x = 0

,,

1

*

1

mx =

p

p1

p1 = p1 = p2

*

1x = 0

*

2

2

mx =

p

p2 and m

x2

x1

p1

x1*

p1

p1 = p1 = p2

*1

2

m0 x

p

p2 = p1

*

2x = 0

,,

1

*

1

mx =

p

*

1x = 0

*

2

2

mx =

p

p2 and m

x2

x1

p1

x1*

*

2

2

mx =

p

p1

p1

*

1x = 0

p2 = p1

*

1x = 0

p2 and m

p1 = p1 > p2

x2

x1

p1

x1*

p1

p2 = p1

p1

*

1

1

mx =

p

2

m

p

p1 price

offer

curve

1-

p2 and m

*1

2

m0 x

p

p1 ?

1 2- .

1 2 1 2U(x , x ) = min x , x

* *

1 1 2 2 1 2

1 2

mx (p , p ,m) = x (p , p ,m) =

p + p

p2 m , p1 x1* x2*- .

* *

1 1 2 2 1 2

1 2

mx (p , p ,m) = x (p , p ,m) =

p + p

* *1 1 2

2

mp 0, x = x

p

p2 m , p1 x1* x2*- .

* *

1 1 2 2 1 2

1 2

mx (p , p ,m) = x (p , p ,m) = .

p + p

* *1 1 2

p , x = x 0

* *1 1 2

2

mp 0, x = x

p

p2 m , p1 x1* x2*- .

* *

1 1 2 2 1 2

1 2

mx (p , p ,m) = x (p , p ,m) = .

p + p

x1

x2

p2 and m

p1

x1*,

1

*

2

2

mx =

p + p

*

1 ,

1 2

mx =

p + px1

x2

p1

,

1

*

1

2

mx =

p + p

p1 = p1

y/p2

p2 and m

p1

x1*

x1

x2

p1

p1

p1 = p1

y/p2

,,

1

*

2

2

mx =

p + p

,

*

1 ,

1 2

mx =

p + p

,,

1

*

1

2

mx =

p + p

p2 and m

p1

x1*

x1

x2

p1

p1

p1

p1 = p1

y/p2

p2 and m

,,,

1

*

2

2

mx =

p + p

,,

*

1 ,

1 2

mx =

p + p

,,,

1

*

1

2

mx =

p + p

p1

x1*

1-

*

2

1 2

mx =

p + p

*

1

1 2

mx =

p + p

xy

p p1

1 2

*.

x1

x2

p1

p1

p1

2

m

p

y/p2

p2 and m

p1 ?

1 2- .

a b

1 2 1 2U(x ,x ) = x x .

-

*1 1 2

1

a mx (p , p ,m) =

a + b p

*

2 1 2

2

b mx (p , p ,m) =

a + b p

and

x2* p1 p1

*1 1 2

1

a mx (p , p ,m) =

a + b p

*

2 1 2

2

b mx (p , p ,m) =

a + b p

and

x2* p1 p1 1-

*1 1 2

1

a mx (p , p ,m) =

a + b p

*

2 1 2

2

b mx (p , p ,m) =

a + b p

and

x2* p1 p1 1- .

x1*(p1) x1*(p1)

x1*(p1)

x2

x1

x

by

a b p

2

2

*

( )

xay

a b p1

1

*

( )

p2 and m

x1*(p1) x1*(p1)

x1*(p1)

x2

x1

p1

x1*

1-

x

by

a b p

2

2

*

( )

xay

a b p1

1

*

( )

xay

a b p1

1

*

( )

p2 and m

3.

(Cross-Price Effects)

p2-

1- 1- 2- (gross substitute) .

1- 1- 2- (gross complement) .

:

*

1

1 2

mx =

p + p

*

1

22 1 2

x m= < 0.

p p + p

2- 1- .

- :

*

2

2

bmx =

(a + b)p

*

2

1

x= 0

p

1 2- , .

4.

? .

? .

p1

x1*

p1

p1 ?

p1

x1*

p1

x1

p1 ?: x1 .

p1

x1*x1

: x1 ?

p1 ?: x1 .

p1

x1*

p1

x1

p1 ?: x1 .

: x1 ?

: p1

(inverse demand function) .

:

-:

*

1

1

amx =

(a + b)p

1 *

1

amp =

(a + b)x

:

*1

1 2

mx =

p + p

1 2*

1

mp = - p

x

:

:

11 2

2

pMRS = p = MRS p

p

2 1

p = 1 p = MRS

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Lecture 4 - Demand