Post on 06-Jan-2016
description
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