Transcript Axioms of Choice (Preference Ordering)

ECON6021 Microeconomic
Analysis
Consumption Theory I
1
Topics covered
1.
2.
3.
4.
Budget Constraint
Axioms of Choice & Indifference Curve
Utility Function
Consumer Optimum
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Bundle of goods
• A is a bundle of goods
consisting of XA units
of good X (say food)
and YA units of good Y
(say clothing).
Y
YA
YB
A
B
• A is also represented
by (XA,YA)
XA XB
X
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Convex Combination
( xc , yc )  (txA  (1  t ) xB , ty A  (1  t ) yB )
y
(xA, YA)
A
C
(xA, YB)
B
x
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Convex Combination
y A  yB
(ty A  (1  t ) yB )  yB ty A  tyB


 Slope of AB
Slope of CB 
tx A  txB
x A  xB
(tx A  (1  t ) xB )  xB
 C is on the st. line linking A & B
Conversely, any point on AB can be written as
(txA  (1  t ) xB , ty A  (1  t ) yB ) where t [0,1]
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Slope of budget line
Px
dy

dx
Py
Unit:
(market rate of substitution)
$ per jar
 loavesper jar (independent of $)
$ per loaf
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I0
Py
Example:
jar of beer
loaf of bread
|Slope|=
Px=$4
Py=$2
Px
$4 per jar

 2 loavesper jar
Py $2 per loaf
Px
Py
feasible
consumption set
I0
Px
Both Px and Py double,
P' x
$8 per jar

 2 loavesper jar
'
P y $4 per loaf
No change in market rate of
substitution
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Tax: a $2 levy per unit is imposed for each good
Px  t $4  $2 3


Py  t $2  $2 2
 Slope of budget line changes
y
I0
2
after levy is imposed
I0
4
After doubling
the prices
I0
8
I0
6
I0
4
x
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Axioms of Choice
& Indifference Curve
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Axioms of Choice
• Nomenclature:
– : “is preferred to”
– : “is strictly preferred to”
– : “is indifferent to”
• Completeness (Comparison)
– Any two bundles can be compared and one of the
following holds: AB, B A, or both ( A~B)
• Transitivity (Consistency)
– If A, B, C are 3 alternatives and AB, B C, then A
C;
– Also If AB, BC, then A C.
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Axioms of choice
•
Continuity
–
•
Strong Monotonicity (more is better)
–
•
AB and B is sufficiently close to C, then A C.
A=(XA , YA), B=(XB , YB) and XA≥XB, YA≥YB with at
least one is strict, then A>B.
Convexity
–
–
–
If AB, then any convex combination of A& B is
preferred to A and to B, that is, for all 0 t <1,
(t XA+(1-t)XB, tYA+(1-t)YB)  (Xi , Yi), i=A or B.
If the inequality is always strict, we have strict
convexity.
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Indifference Curve
• When goods are divisible and there are only
two types of goods, an individual’s
preferences can be conveniently represented
using indifference curve map.
• An indifference curve for the individual
passing through bundle A connects all
bundles so that the individual is indifferent
between A and these bundles.
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Properties of Indifference Curves
Y
• Negative slopes
• ICs farther away from
origin means higher
satisfaction
Preferred bundles
I
A
II
X
Not preferred bundles
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Properties of Indifference Curves
• Non-intersection
– Two indifference
curves cannot
intersect
• Coverage
– For any bundle, there
is an indifference
curve passing
through it.
Y
A
Q
P
X
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Properties of Indifference Curves
• Bending towards
Origin
– It arises from
convexity
axiom
– The right-handside IC is not
allowed
Y
X
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Utility Function
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Utility Function
• Level of satisfaction depends on the amount
consumed: U=U(x,y)
• U0 =U(x,y)
– All the combination of x & y that yield U0 (all
the alternatives along an indifference curve)
• y=V(x,U0), an indifference curve
• U(x,y)/x, marginal utility respect to x,
written as MUx.
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Y
YA
A
B
YB
XA XB
dU 
U0
U ( x, y )
U ( x, y )
dx 
dy
x
y
X
U ( x, y )
U ( x, y )
dU 0 
dx 
dy  0
x
y
dy
Slope:
dx
U U
(by construction)
(if strong monotonicity
U / x
MU x


0
U / y
MU y
holds)
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Y
MRS  MRS
A
A
xy
B
B
xy
U0
X
The MRS is the max amount of good y a consumer
would willingly forgo for one more unit of x,
holding utility constant (relative value of x
expressed in unit of y)
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• Marginal rate of substitution
dy
dy U / x MU x
MRSxy 
 

0
dx U U0
dx U / y MU y
DMRS:
dMRS
dx
U 0  constant
0
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Measurability of Utility
V=200
V=2001
V=100
U=30
U=20
U=10
An order-preserving re-labeling of ICs does not alter the preference
ordering.
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Positive monotonic (orderpreserving) transformation
• They are called positive monotonic
transformation
U  xy
U '  U 2  x2 y 2
U ''  U  6  xy  6
U '''  2U  2 xy
U '' ''  U 2  1  x 2 y 2  1
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Positive Monotonic Transformation
What is the MRS of U at (x,y)?
U
y
x
U
x
y

U / x y
MRS 

U / y x
How about U’?
U '
 2 xy 2
x
U '
 2x2 y
y

U '/ x 2 xy 2 y
MRS' 
 2 
U '/ y 2 x y x
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Positive Monotonic Transformation
• IC’s of order-preserving transformation U’
overlap those of U.
• However, we have to make sure that the
numbering of the IC must be in same order
before & after the transformation.
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Positive Monotonic Transformation
• Theorem: Let U=U(X,Y) be any utility
function. Let V=F(U(X,Y)) be an orderpreserving transformation, i.e., F(.) is a
strictly increasing function, or dF/dU>0 for
all U. Then V and U represent the same
preferences.
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Proof
Consider any two bundles
B  ( xB , yB ).Then we have:
A
U
A  ( xA , y A )
and
B
 U ( x A , y A )  U ( xB , y B )
 F (U ( x A , y A ))  F (U ( xB , y B ))
 V ( x A , y A )  V ( xB , y B )
A
V
B
Q.E.D.
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Consumer Optimum
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Constrained Consumer Choice
Problem
• Preferences: represented by indifference curve
map, or utility function U(.)
• Constraint: budget constraint-fixed amount of
money to be used for purchase
• Assume there are two types of goods x and y, and
they are divisible
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Consumption problem
• Budget constraint
– I0= given money income in $
– Px= given price of good x
– Py= given price of good y
• Budget constraint: I0Pxx+Pyy
• Or,
I0= Pxx+Pyy (strong monotonicity)
dI0= Pxdx+Pydy=0 (by construction)
Pxdx=-Pydy
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yB  yD
MRSxy 
XA  XB
YB
YD
Psychic willingness
to substitute
B
A
D
At B, my MRS is very high for X. I’m
willing to substitute XA-XB for
YB-YD. But the market provides me more
X to point D!
Px
 MRSxy 
Py
C
XB XA
XD
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Consumer Optimum
• Normally, two conditions for consumer
optimum:
• MRSxy = Px/Py
(1)
• No budget left unused
(2)
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Y
Both A & C satisfy (1) and (2)
U0
U1
Problem: “bending toward origin”
does not hold.
A
C
X
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Special Cases
coffee
coffee
Generally low MRS
Generally high MRS
U2
U1
U0
tea
Px
MRSxy 
for all (x, y)
Py
tea
Px
MRSxy 
for all (x, y)
Py
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Quantity Control
• Max U=U(x,y)
Subject to
(i) I ≥Pxx+Pyy
(ii) R≥x
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y
(1)
(2)
(3)
(4)
(1)
(2)
Corner at x=0
Interior solution 0<x<R
“corner” at R
“corner” at R
(3)
(4)
x
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An Example: U(x,y)=xy
max U  xy
x, y
subject to I  Px x  Py y
(1)
MU x  y , MU y  x
MRS 
MU x y Px
 
 market rate of sub.
MU y x Py
(2)
or Py y  Px x
Hence x 
I
I
,y
Px
Py
(
(1))
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B
A satisfies (1) but not (2)
B, C satisfy (2) but not (1)
Only D satisfies both (1) &(2)
D
A
C
x I
1 I
1
I


1
I x 2 Px x 2 Px I / 2 Px
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Other Examples of Utility
Functions
U ( x, y)  x  y
U ( x, y)  min{x, y}
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An application: Intertemporal
Choice
• Our framework is flexible enough to deal
with questions such as savings decisions
and intertemporal choice.
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Intertemporal choice problem
Income in period 2
C2
u(c1,c2)=const
1600
500
Slope = -1.1
1000
C1
Income in period 2
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1000-C1=S
500+S(1+r)=C2
Substituting (1) into (2), we have
500+(1000-C1)(1+r)=C2
Rearranging, we have
1500+1000r-(1+r) C1=C2 > C
Using C1=C2=C, we finally have
C
(1)
(2)
1600 1500  1000r
500

 1000 
2.1
2r
2r
r   C  (S
)
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