ECON6203Lecture 1

8/8/2019 ECON6203Lecture 1

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Lecture 1

Utility Theory I


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Ordinal Utility ( )HH ,Property of the consumption Set X:

X is not empty

X is closedX is convex

Properties ofPreferences:Description of consumer preferenceConsumers have preferences over consumption bundles forall possible pairsofalternatives

| H HNot [ X X] and not [X X]X is indifferent to X: X~ XH | HX Not [X X]X is at least as good as X: X

H XX is preferred to X : X

X

RX n

0

}


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HH

H

H

Only one of the three possibilities is true forany pair ofalternatives.

ordering is complete forany two bundles X, X either

X

X

Consumer is suppose to have preference over the consumption

bundles that satisfy the following set of axioms.

Axiom 1. Completeness

The

X

OrX

Orboth

Remark if X X

HX X

Then X~ X


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The preference relation is Asymmetric if X X than Not [X X].H H H

Axiom 2. Transitivity

The preference relation and the indifference relation are transitive.

If X X And X X

Then X X

H

H

H

If X ~ X And X ~ X

Then X ~ X

H Weak preference

H Strict preference

~ indifference

X Y X Then X XH H

H

H


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Axiom 3. Continuity

Forall Y in X the sets

{ X: X Y}a

nd {X: X Y}a

re closed sets.

Thus {X: X Y} and {X:X Y} are open sets.

H R

The reason for this assumption to rule out any discontinuous behavior.In other words, it rules out the sudden reversal in preferences.

Sequence

A sequence (Xn) is determined by a rule which assigns to eachnatural number n a unique real number Xn. (Xn) the sequence wherewith term is Xn =-1, +1, -1, +1.

n1

H


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Convergence

A sequence is said to converge to the limit iff the following criterion is

satisfied

nX l

Given >0 we can find an N such that, forany n>N.I

Iln

g"

!

n

llim

1 2 76543 111098

X1l

I

Il

I

I


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Observe that the definition of convergence begins given any > 0. It is clearthat the smaller the value of > 0 the bigger must be the corresponding N.

I

I

If (Xi) is a sequence of consumption bundles that are all at least as good as abundle y, If this sequence convergence to some bundle Xt. Then Xt is at leastas good as y.

This property

if Y>Zif X is close enough to Ythen X>Z


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Continuity

Given a condition A1 we can consider the set ofall combinations at least as

well liked as A1 . A set of combination not more liked than A1 .

These two sets are closed if you select a infinite sequence of combinationswhich converge to some limiting combination A0 . and if each member ofthe sequence were at least as will liked as A1 , then A0 is at least as wellliked as A1 . N0 jumps.

Example violating continuity

A1=( )1

1q 1

2q A2=( )

2

1q 2

2q

A1 is preferred to A2 if > or = and >

Lexicographic ordering A=( ) and

1

1q 1

1q2

1q 2

1q 1

2q 2

2q

0

1q 0

2q

1q(

2q(

The combination ),(2

0

21

0

1qqqq (( ),)

2

1((

2

0

21

0

1qqqqA ii ((!

AqqqAbutAA ii

i RH ),(lim 20

2

0

1(!

gp


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At least as much of everything is at least as good

If X Y then X Yu H

And more of some

If X Y then X Y" HAxiom 4. Local nonsatiation

Given any x X and any > 0 then there is some bundle y X withsuch that Y X

Rules out thick indifference curves

I

IYX H

X2

X1

One can always do a bit better even

with small change in the

consumption bundle.

Axiom 4 imply 4 but not the vice versa.

Axiom 4. Strong Monotonocity


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Axiom 5. Convexity

Given X, Y,

and Z X such th

at X Z

and Y Z then

H H

tX + (1-t) Y Z forall 0 t 1H ee

Axiom 5. Strict Convexity

Given X Y and Z X if X Z and Y Z

Then tX + (1-t) Y Z forall 0 < t < 1

{ H H

H

The set of all consumption bundles that are indifferent to each

other is called the indifference curve.


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X Y Cartesian product

Given any two sets X and Y an ordered (X, Y) where x X andy Y the collection ofall these.

Metric Space: Given an arbitrary set X ifa function d from X X intoR is defined and if d satisfies

d(X, Y)=0 iff X=Yd(X,Y)+d(Y,Z) d(Z,X)

Then X is metric space. (X,d)

Ex with Euclidian distancenR

!

!

n

i

ii YXYXd

1

2)(),(


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Considera point in a metric space (X, d) say, , and define a set0Xn

}),(,:{)( 00 rXXdXxxXBr

Where r is some positive real number.

The set is called the open ball about with radius r.0

X)( 0Xr

1X

2X

r

0X


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Given an open ball in quick any point X in . Thenwe can always find another open ball about this point X.

)( 0XBrnR )( 0XBr

Which is contained in open set . Let S be a subset in a metricspace (X,d), This set S is called an open set if forany X in S there exists apositive real number r such that

)( 0Xr

SXr )(

0X

0X

Closed setis the complementofan open set

SXorSC \


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Given a metric space (X,d), a point in X, and a

subset S of X, a sequence in S is said to convergeto if forany real number >0 , there exists apositive integer such that implies

. Is called the limit of , and

such a sequence is called a convergentsequence in S if S.

0X

}{q

X0X I

q

u qq

I),( qo XXd0X

}{ qX

}{ qX

0X


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Euclidian Distance

!

!

n

i

ii YXYXd1

2)(),(

X(X1, X2)

Y (Y1, Y2)

2

22

2

11)()(),( YXYXYXd !

d(X,Y)=0 iff X=Y

Triangular inequality d(X,Y)+d(Y,Z)>d(Z,X)

d(X,Y) 0 forall X and Y

Symmetry d(X,Y)=d(Y,X)

u


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Let

U(X)= where e~ X

U(X)= where e~ Y

Then if then e e and transitivity

imply X~ e e ~ y

xtxt

xt

xtR

xt yt

ytyt

yt

ytR

R

Reflexive forall x X x X

Utility function:

Such that X Y ifand only if U(X) > U(Y)

RXu p:

H

H


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If preference ordering is complete, reflexive, transitive and continuous

then it can be represented by a continuous utility function.

If u(X) represents some preferences is a monotonicfunction. Then f(u(X)) represent the preferences.

RRf p:H

f is strictly increasing on the set Sx S and y Sx < y f(x) < f(y)or f(x) > f(Y) if x>y

Properties of the utility function:

1. u(X)is strictly increasing ifand only if is strictly monotonic.

2. u(X) is quasi concave ifand only if is convex.

3. u(X) is strictly quasi concave ifand only if is strictly convex.

H

H

H


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Let X be the consumption set, the satisfaction from consumptioncan be expressed by an index which is a real number. This u utility

index which isa

function from X into R. This function denotedb

yu(X)

Preference is represent able by natural order of real number

X Y iff u(X) u(Y)

X Y iff u(X) u(Y)

X ~ Y iff u(X) = u(Y)

u

H H

Any monotone increasing function can replace it.

H

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ECON6203Lecture 1