Transcript Axioms of Choice (Preference Ordering)

ECON6021 Microeconomic
Analysis
Consumption Theory II
Topics covered
1.
2.
3.
4.
Price Change
Price Elasticities
Income Elasticities
Market Demand
Price effect
y
I
Py
Price consumption curve (PCC)
Or Price expansion path (PEP)
B
A
I
Px
Px
A
I
P'x
x
Ordinary (Marshallian)
Demand function
B
x  x(Px , Py , I )
x
Price Effects
Y
I
Py
I T
Py
J K
MB
A
S
Q
x0 xsx1
I
Px
I T
Px
I
Px'
X
• 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
 X


0


 I

 X

 0

 I

 X

 0

 I

A is the reference point for the analysis
of combined effect of income and
substitution effect.
K-Q:
 X


 0 
 Px

J-K: Giffen gd.
 X


 0 
 Px

Giffen gd  inferior gd.
Price Elasticities
Price and Expenditure Elasticities
Own Price Elasticity
x Px
dx / x
exx 

Px x dPx / Px
exx  1
Elastic demand
exx  1
Unitary demand
exx  1
Inelastic demand
Price Elasticity of Expenditure
e( px x ), px
 ( Px x ) Px

Px Px x
 ( Px x ) 1

Px x
1  Px
x 
 
x  Px
x  Px
Px 
x Px
 1
 1  exx  1  exx
Px x
e( px x ), px
exx
(  1  exx )
Px 
Px 
>1
Elastic
0
Px x 
Px x 
<1
Inelastic
0
Px x 
Px x 
=1
Unitary
0
No
change
No change
An Example: Linear demand
p  A  Bx
A P
x 
B B
dx
1

dP
B
dx P
1 A  Bx
A

 1
dP x
B x
Bx
  if
A 
exx  1 
  1 if
Bx 
 0 if
x0
x  A / 2B
x  A/ B
An Example: Linear Demand
xP  Ax  Bx
d ( xP)
MR 
 A  2 Bx
dx
2
P
Review: Linear Demand
Q  100  P
or P  100  Q
ex ,Px 
ex , Px
(demand)
(inverse demand)
ex , Px  1
Q P
P
P
 ( 1) 
P Q
Q 100  P

P


 1
100  P 
 0
when P  100
Q
when P  50
when P  0
ex , Px decreases from  to 0 as P decreases from 100 to 0.
TR
TR  Q * P  (100-Q )*Q  -[Q  100Q ]  (Q  50) 2  2500
dTR
 100  2Q  0 when Q  50.
dQ
TR reaches a max when ex ,Px  1
Q
Income Change
AOG
AOG
IEP (Income Expansion
Path)
IEP
X
X
x is normal
x
 0 (meaning that Px , Py fixed)
I
where x(Px , Py , I )
x has no income effect
x
0
I
Px
fixed
variable
x( Px , Py , I )
Demand
IEP
x
variable
x
x is inferior
x
0
I
x( Px , Py , I )
Engel Curve
fixed
I
Income Elasticities
Income Elasticity
x I x / x
exI 

I x I / I
exI  1
superior good (luxury)
0  exI  1
normal, necessity
exI  0
no income effect
exI  0
inferior good
e( px x ), I
e Px x 

 I
, I

Px x
expenditure on x
Px x
sx 
I
budget share for x
 ( Px x ) I
x I

 Px
 ex , I
I
Px x
I Px x
  Px x / I 
x / I  I2
I

 Px
I
Px x / I
I
Px x
2
 I  x / I  x  I / I  I
 x   I 


  1
2

I

 x
 I   x 
 exI  1
eS x , I
 0

 exI  1 0
 0

if exI>1
if exI=1
If exI<1
Engel Aggregation (Adding-up
condition)
I  Px x  Py y
dI  Px dx  Py dy
dx
dy
1  Px
 Py
dI
dI
dx  x   I 
dy  y   I 
 Px      Py    
dI  x   I 
dI  y   I 
Px x dx I Py y dy I


I dI x
I dI y
 sx exI  s y e yI
Aggregate Income elasticity=1
Consider an income change…
Y
X
A
A-B
B
B-C
C
C-D
D
D-E
B
C
D
C’
I1
I0
E
X
Inferior
No income eff
Normal only
Normal only
Superior
Superior
Superior
Y
superior
superior
superior
normal only
normal only
no income effect
inferior
From C'  C
budget share of x does not change,
eSx I  0  exI  1  0  exI  1
Cobb-Douglas Utility: U=xy
max U  xy
x, y
subject to Px x  Py y  I
 x
I
I
,y
.
2 Px
2 Py
ex , Px 
x Px
 I Px
 I Px


 1
2
Px x
2 Px x
Px I
ex , Py 
x Py
0
Py I
x I
 1.
I x
 ex , I  1  0.    Check
ex , I 
eS x , I
Px x
I /2
1


I
I
2
S x I

 0.
I S x
Sx 
eS x , I
Homogenous function
• Homogenous function of degree k
– If there exists a constant k so that for all m>0 and for all
a, b
F (ma, mb)  m F (a, b) (1)
k
Then, we say F(.) is homogenous of degree k.
Euler Theorem
• Euler Theorem
– If F(a,b) is homogenous of degree k, then we have
F
F
a
b  kF
a
b
• Proof of Euler Theorem.
• Differentiate equation (1) with respect to m & then set
m=1
Corollary of Euler Theorem
Since demand x=F ( Px , P y , I ) is homo. of degree 0,
F
F
F
Px 
Py 
I 0
Px
Py
I
F Px
F Py
F I


0
Px F
Py F
I F
exx  exy  exI  0
Lump Sum Principle
AOG
Init ialcondit ions: I 0 , Px , Py
I
Py
hence x0 , y0
an excise t ax (ad valorem)t on x is levied
I 0  ( Px  t ) x  Py y
A
y0
y1
y2
B
I 0  tx1  Px x1  Py y1
At B,
S
I0
Px
x1 x2 x0
x
Lump Sum Principle
Lump-sum tax: T dollars
so that T  tx1
I 0  T  Px x  Py y
Hence, Px x  Py y  Px x1  Py y
a value
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
AOG
I0
0
A
X
The amount of A is a free gift from government.
A sum of money equivalent to the value of gift is even better.
Market Demand
Market Demand
x  x(Px , Py , I )
Individual demand
Assume 2 agents (1 and 2)
I1
x1 
2 Px
I1
Px 
2 x1
inversedemand
I2
x2 
2 Px
I
Py 
2 x2
inversedemand
I1
I2
I1  I 2
x market  x1  x2 


2 Px 2 Px
2 Px
Market Demand
100 P
P  100 xA  xA  
0
100
P

12.5 
P  50  4 xB  xB  
4

 0
5
0
12.5
100 112.5
if P  100
o.w.
if p  50
o.w.
if P  50
112.5  5P / 4

x  x A  xB   100  P
if 50  P  100

0
o.w.

The End