1. 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?

2. 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.

3. Assumptions about Preferences
1. Complete - either or
or both, i.e
2. Reflexive -

4. Transitive - if
and
then transitivity =>

5. 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.

6. 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

7. 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
The boundary of the set is the indifference curve x1

8. y x y x y x y x

9. 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

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

11. Assumption 5. Monotonocity
Gives us downward sloping indifference curves
Are we out of the woods yet
NO!

12. All these satisfy properties 1-5x y x y x y x

13. Convexity
A function is convex if

14. And along an indifference curve this special property holds:
When U(x1,y1) = U(x2,y2)

15. Convexity So a function is convex if
But we don’t have to use the fraction 1/2

16. Convexity More generally a function is convex if
Where l lies between 0 and 1

17. y
But this indifference curve is convex y1 y3 y2 x x3 x2 x1

18. 6. Strict Convexity So we need Strict convexity
And it is STRICTLY convex if
Where l lies between 0 and 1

19. y y1
Strictly Convex y3 y2 x x3 x2 x1

20. 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

21. Axioms of Consumer Theory
1.Completeness
2. Reflexivity
3. Transitivity
4. Continuity
5. Monotonicity
6. Convexity
7. Differentiability

22. Conditions 1-5 allow us to write a utility function: u = u(x,y).
E.g. u=x1/2y1/2
Formally, if
then u is a mathematical function such that

23. 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

24. 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
We say that a utility function is unique up to any positive monotonic transformation

25. Table showing PMT of Utility in column A and non-PMT in Column E

26. 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)]