Preferences

What properties would expect preferences to exhibit?

Which of these properties allow us to derive an indifference curve?

What will that indifference curve look like?

More importantly, what is not rules out?

What else do we need to assume about properties to generate the nice indifference curves required

to produce sensible demand curves?

Terminology

The bundle containing x1 and y1 is strictly preferred to (better than) the bundle containing x2 and y2.

(x1, y1) is weakly preferred to (at least as good as) (x, y) .

the individual is indifferent between the two bundles.

),(),( 2211 yxyx

),(~),( 2211 yxyx

),(),( 2211 yxyx

Assumptions about Preferences

1. Complete - either

or

or both, i.e

2. Reflexive -

),(),( 2211 yxyx

),(),( 1122 yxyx

),(~),( 2211 yxyx

),(),( 1111 yxyx

3. Transitive - if

and

then transitivity =>

),(),( 2211 yxyx

),(),( 3322 yxyx

),(),( 3322 yxyx

Fundamental Axioms of Consumer Theory

1.Completeness

2. Reflexivity

3. Transitivity

Known as the Three Fundamental Axioms of consumer theory.

These allow consumers to arrange bundles in order of preference.

If we have the three axioms above we can rank all bundles in the x,y space we have drawn below.

For example, if we take any bundle (x1, y1) then we can establish the bundles which satisfy the two relationships:

y1

x1

),(),( 2211 yxyx

),(),( 2211 yxyx

If we have the three axioms above we can rank all bundles in the x,y space we have drawn below.

For example, if we take any bundle (x1, y1) then we can establish the bundles which satisfy the two relationships:

y1

x1

),(),( 2211 yxyx

),(),( 2211 yxyx

The boundary of the set is the indifference curve

y

x

y

x

y

x

y

x

4. Continuity

• For any given bundle the set of bundles which are weakly preferred to it, and the set of bundles to which it is weakly preferred, are closed sets (that is, they contain their own boundary).

• Closed set: Football Pitch, Tennis Court

• Open set: Rugby Pitch, Cricket

5. Monotonicity

• So the next assumption we need is called the assumption of monotonicity or non-satiation. It say that if

),(),( 1122

1212

1212

yxyxthen

yyorxx

eitherand

yyandxx

Assumption 5.

• Monotonocity

• Gives us downward sloping indifference curves

• Are we out of the woods yet

• NO!

y

x

y

x

y

x

y

x

All these satisfy properties 1-5

Convexity

• A function is convex if

)y,U(x2

1)y,U(x

2

1

)y5.0y5.0,x5.0U(0.5x)y,U(x

2211

212133

And along an indifference curve this special property holds:

)y,U(x)y,U(x2

1)y,U(x

2

1

)y5.0y5.0,x5.0U(0.5x)y,U(x

112211

212133

When U(x1,y1) = U(x2,y2)

Convexity

• So a function is convex if

• But we don’t have to use the fraction 1/2

)y,U(x2

1)y,U(x

2

1)y,U(x 221133

Convexity

• More generally a function is convex if

)y,U(x)y,U(x

))y-(1y,)x-(1xU(

2211

2121

Where lies between 0 and 1

y

xx1

y1

y3

x2x3

y2

But this indifference curve is convex

6. Strict Convexity

• So we need Strict convexity

• And it is STRICTLY convex if

)y,U(x)y,U(x

))y-(1y,)x-(1xU(

2211

2121

Where lies between 0 and 1

y

xx1

y1

y3

x2x3

y2

)y,U(x2

1)y,U(x

2

1)y,U(x 221133

Strictly Convex

7. Differentiability

• To rule out case (b) above we assume that the conference is differentiable everywhere. That is, the function is smooth and has no corners.

• =>

y

x

y

x

Axioms of Consumer Theory

1.Completeness

2. Reflexivity

3. Transitivity

4. Continuity

5. Monotonicity

6. Convexity

7. Differentiability

• Conditions 1-5 allow us to write a utility function: u = u(x,y).

• E.g. u=x1/2y1/2

• Formally, if

),(),( 1122 yxyx

then u is a mathematical function such that

),(),( 1122 yxuyxu

• Utility is ordinal, I.e.the function merely orders bundles the actual number associated with u is irrelevant.

• E.g. If u(x2,y2)=4 and u(x1,y1)=2, then the x2, y2

bundle is preferred to the x1, y1 bundle, but we can’t say it is twice as good.

• That is utility is not Cardinal

• Since utility is ordinal I can change the function as long as it does not change the ordering of the the bundle:

• So if I change u(x2, y2) to 2u(x2, y2) then

• u(x2, y2)=4 and u(x1, y1)=2 becomes

• 2u(x2, y2)=8 and 2u(x1, y1)=4.

• However, this conveys the same essential information that

),(),( 1122 yxyx

We say that a utility function is unique up to any positive monotonic transformation

Table showing PMT of Utility in column A and non-PMT in

Column E

A=U(x,y) B=A+3 C=2.5A D=2A+4 E=100/ABundle 1 15 18 37.5 34 6.67Bundle 2 16 19 40 36 6.25Bundle 3 20 23 50 44 5Bundle 4 40 43 100 84 2.5

PMT Positive Monotonic

Transformation• If u(x2,y2) > u(x1,y1), then any PMT of u, for

example, f (u), implies that

• f [u(x2,y2)] > f [u(x1,y1)]