ECON6021 Microeconomic Analysis

Consumption Theory II

Topics covered

1. Price Change

2. Price Elasticities

3. Income Elasticities

4. Market Demand

y

AB

xP

I

xP

I'

yP

IPrice consumption curve (PCC)Or Price expansion path (PEP)

x

A B

),,( IPPxx yx

Ordinary (Marshallian)Demand function

Price effect

Px

x

AB

S

X

Y

yP

I

yP

TI

xP

TI

xP

I'xP

Ix0 xsx1

J KM

Q

Price Effects

• Initial consumption: A

• Price decreases from Px to Px’

• Real income—Hick’s definition: an initial level of utility

• x0 to xs (or A to S) is the sub. effect

• xs to x1 (or S to B) is the income effect

Price Effects

• Price Effects= substitution effect

+ Income effect

• Substitution Effect a.k.a (also known as) pure price effect: a change in relative price while keeping utility constant

For income effects, S is the reference point.

M: no income effect

M-Q: X is normal

J-M: X is inferior

A is the reference point for the analysisof combined effect of income and substitution effect.

K-Q:

J-K: Giffen gd.

Giffen gd inferior gd.

0I

X

0I

X

0I

X

0xP

X

0xP

X

Price Elasticities

/

/x

xxx x x

x P dx xe

P x dP P

Own Price Elasticity

1

1

1

xx

xx

xx

e

e

e Elastic demand

Unitary demand

Inelastic demand

Price and Expenditure Elasticities

( ),

( )

( ) 1

1

1 1 1

x x

x xp x p

x x

x

x

xx

x x

xxx xx

x

P x Pe

P P x

P x

P x

P xx P

x P P

x Pe e

P x

Price Elasticity of Expenditure

>1 Elastic

<1 Inelastic

=1

Unitary

No change

No change

xxe( ),

( 1 )x xp x p

xx

e

e xP xP

0

0

0

xPx

xPx

xPx

xPx

1

11

p A Bx

A Px

B Bdx

dP Bdx P A Bx A

dP x B x Bx

0

1 1 / 2

0 /xx

if xA

e if x A BBx

if x A B

An Example: Linear demand

An Example: Linear Demand

BxAdx

xPdMR

BxAxxP

2)(

2

,

,

,

100 (demand)

or 100 (inverse demand)

( 1)100

when P 100

1 when P 50100

0 when P 0

decreases

x

x

x

x P

x P

x P

Q P

P Q

Q P P Pe

P Q Q P

Pe

P

e

2

,

from to 0 as P decreases from 100 to 0.

* (100- )* -[ 100 ] ( 50) 2500

100 2 0 when 50.

TR reaches a max when 1xx P

TR Q P Q Q Q Q Q

dTRQ Q

dQ

e

Q

1, xPx

e

P

Q

TR

Review: Linear Demand

IEP

X

AOG AOG

X

IEP (Income ExpansionPath)

x

is normal

0 (meaning that , fixed)

where ( , , )

x y

y

x

xP P

Ix P P I

x has no income effect

0x

I

Income Change

IEP

0x

inferior isx

I

x

Px

),,( IPPx yx

variable fixed

Demand

I

x ),,( IPPx yx

fixed

variable

Engel Curve

Income Elasticities

/

/xI

x I x xe

I x I I

1 superior good (luxury)

0 1 normal, necessity

0 no income effect

0 inferior good

xI

xI

xI

xI

e

e

e

e

Income Elasticity

x

expenditure on x

budget share for x

x

x

P x

P xs

I

( ), ,

2

,

2

2

( )

/ /

/

/ /1

1

x

x

xp x I x x I

x x

xxP x

Ix xI

xI

P x I x Ie P e

I P x I P x

P x I x II Ie P

I P x I I P x

I x I x I I I x I

I x I x

e

0

0

0

1, xIIS eex

if exI>1

if exI=1

If exI<1

1

x y

x y

x y

x y

yx

x xI y yI

I P x P y

dI P dx P dy

dx dyP PdI dI

dx x I dy y IP PdI x I dI y I

P yP x dx I dy I

I dI x I dI y

s e s e

Aggregate Income elasticity=1

Engel Aggregation (Adding-up condition)

Y

X

A

B

C

D

E

C’

I0

I1

xS

From C' C

budget share of x does not change,

e 0 1 0 1I xI xIe e

A-BBB-CCC-DDD-E

X YInferior superiorNo income eff superiorNormal only superiorNormal only normal onlySuperior normal onlySuperior no income effectSuperior inferior

Consider an income change…

,

, 2

,

,

, ,

,

max

subject to

, .2 2

12

0

1.

1 0.

/ 2 1

2

0.

x

y

x

x

x y

x y

x y

x x xx P

x x x

yx P

y

x I

S I x I

xx

xS I

x

U xy

P x P y I

I Ix y

P P

x P I P I Pe

P x P x P I

Pxe

P I

x Ie

I xe e Check

P x IS

I IS I

eI S

Cobb-Douglas Utility: U=xy

Homogenous function

• Homogenous function of degree k– If there exists a constant k so that for all m>0 and for

all a, b

Then, we say F(.) is homogenous of degree k.

(1) ),(),( baFmmbmaF k

Euler Theorem

• Euler Theorem– If F(a,b) is homogenous of degree k, then we have

• Proof of Euler Theorem.• Differentiate equation (1) with respect to m & then set

m=1

kFbb

Fa

a

F

0

0

0

xIxyxx

F

I

I

F

F

P

P

F

F

P

P

F

II

FP

P

FP

P

F

eee

y

y

x

x

yy

xx

Since demand = ( , , ) is homo. of degree 0,x yx F P P I

Corollary of Euler Theorem

xP

I0

S

A

B

yP

I

0y

1y

2y

0x1x 2xx

AOG

1110

0

00

0

B,At

)(

levied ison x t valorem)(ad tax excisean

, hence

,, :conditions Initial

yPxPtxI

yPxtPI

yx

PPI

yx

yx

yx

Lump Sum Principle

1

0

1

a value

Lump-sum tax: T dollars

so that T tx

Hence,

x y

x y x y

I T P x P y

P x P y P x P y

Chosen dependent on IC

Note that the new consumption at (S) is in a higher IC. In order to get a fixed amount of taxation, lump-sum tax is less harmless to consumers/citizens.

Lump Sum Principle

The amount of A is a free gift from government. A sum of money equivalent to the value of gift is even better.

AOG

X

0I

A0

Lump Sum Principle

Market Demand

Individual demand ),,( IPPxx yx

Assume 2 agents (1 and 2)

xxx

yx

xx

P

II

P

I

P

Ixx

x

IP

P

Ix

x

IP

P

Ix

222x

demand inverse 2

2

demand inverse 2

2

212121market

2

22

1

111

Market Demand

100

12.5

50

100 112.5

112.5 5 / 4 if 50

100 if 50 100

0 o.w.A B

P P

x x x P P

Market Demand

o.w. 0

100P if 100100

PxxP AA

12.5 if p 5050 4 4

0 o.w.B B

PP x x

The End