1. What properties would we 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 ruled out?
What else do we need to assume about properties to generate the nice indifference curves required to produce sensible demand curves?

2. Axioms of Consumer Theory
This presentation covers the lectures on the Axioms of Consumer preference. There are some changes compared to the coverage in the lectures. Continuity has been moved up the order to a more appropriate and I have simplified the coverage on Convexity.

3. 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) (x2, y2) .
the individual is indifferent between the two bundles.

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

5. Transitive - if
and
then transitivity =>

6. 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.

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 x1

8. 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

9. Could indifference curves ever cross?
Units of good Y u1
Units of good X

10. Proof by Contradictionu2
Could we get this?
Ans: No
Proof by Contradiction
Units of good Y u1
Units of good X

11. Consider points a,b and c And assume a is preferred to c
Units of good Y a b u1 c
Units of good X

12. Claim: a is preferred to c u2
Indifference curve u1 implies that Point a is indifferent to b
And u2=> point b is indifferent to c
Units of good Y a
=> a is indifferent to c b
Contradicts Claim u1 c
Units of good X

13. u2
Crossing Indifference curves lead to a logical contradiction, so indifference curves can never cross
Units of good Y a b u1 c
Units of good X

14. So what other properties must Indifference curves have?
Well thus far - none.
Almost anything goes.

15. y x y x
(a)
(b) y x y x
(c)
(d)

16. Or even these cases here:y x y x
(e)
(f)

17. What does case (a) mean? y x u2 u1 u2 u1 u0 u0
(a)

18. 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

19. Effectively means that points on the indifference curve are ‘close’ to one another and that there are no gaps. y1 x1

20. 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

21. 5. Monotonicity
The next assumption we need is called the assumption of monotonicity or non-satiation. It says that if

22. What does this mean? y1 x1

23. Shaded area is the set of points where motonocity holds andx1

24. So up here is definitely better than
Down Here x1

25. So up here is definitely better than
Down Here
So the indifference curve cannot go through either of these areas x1

26. So up here is definitely better than
Down Here u1
So it must slope down x1

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

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

29. Lets take the third case firsty1 u1 x1

30. Why is this case a problem?u3 u2 u1 u0
Why is this case a problem?

31. Let’s draw in a budget constrainty1 u3 u2 u1 u0
Let’s draw in a budget constraint

32. y1 u3 u2 u1 u0
So now we have two equilibria or two demands for x at this set of prices

33. The Demand Curve will look like thispx
The Demand Curve will look like this x

34. Why would that be a problem?px
Why would that be a problem? x

35. Why would that be a problem?px
Why would that be a problem?
Suppose that we observed this data set x

36. If we assumed it was a straight line demand curve we would getpx
If we assumed it was a straight line demand curve we would get x

37. x px But the true demand curve is:
And if you told the boss it’s the red one, your fired x

38. Since determining equilibrium is a problem and supply curve shifts even more sopx S1 S0 x

39. So how do we rule this case out?y1 u1
So how do we rule this case out?

40. This is known as a convex curvey1
We want the indifference curve to look like this

41. Concavity and (Quasi-) Convexity
In the lectures I tried to simplify this topic compared with previous years and feel in the circumstances I may have made things worse. I have confined the material here to the essential concept of what is called quasi-convexity. More detailed coverage of the topic is contained in a separate presentation Concavity and Quasi-convexity

42. If we look at a map of the utility mountain then a nice, well behaved indifference curve should look like this.
We say it is Quasi-convex because the cross-sections look convex looking from the x,y origin y x

43. What does Quasi-convex mean?
Suppose we take a weighted average of two bundles on the same indifference curve and compare the utility we get from this new bundle compared with the utility we got from the originals.
If it is higher we say that the function is quasi-convex.

44. Formerly we can state this as: A function is quasi-convex iff
Where U(x1,y1) = U(x2,y2)

45. y
U(x1,y1) = U(x2,y2) y1
U(x2,y2) y2 x x2 x1

46. U(x1,y1) U(x2,y2) Consider a new bundle: (x3, y3) where
x3= half of x1 and x2 and
y3= half of y1 and y2 y
U(x1,y1) y1 y3
U(x2,y2) y2 x x3 x2 x1

47. U(x1,y1) U(x2,y2) What is the utility associated with this new bundle?

48. y y1
U(x3,y3) y3 y2 x x3 x2 x1

49. Then we say the indifference curve is quasi-convex
U(x3,y3) y2 x x3 x2 x1

50. y y1 y3 y2 x x3 x2 x1

51. y y1 y3 y2 x x3 x2 x1

52. Note The bundle need not be x3, y3, but any point on the red line
Note The bundle need not be x3, y3, but any point on the red line. That is, we could use any fraction l instead of 1/2. If the indifference curve is quasi-convex the condition
would still hold y y1 y3 y2 x x3 x2 x1

53. But this indifference curve is convex, sincey
But this indifference curve is convex, since y1 y3 y2 x x3 x2 x1

54. But not Strictly convex
U(x3,y3) y3
But not Strictly convex y2 x x3 x2 x1

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

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

57. Strict Convexity rules out every case here except case (b)
(d)

58. Why is case (b) troublesome?
Because it has a pointy bit where there can be an equilibrium for more than one price: y x u1

59. Differentiability
To rule out case (b) we assume that the indifference curve is differentiable everywhere. That is, the function is smooth and has no corners.
=> y x y x

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

61. 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

62. 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

63. Since utility is ordinal I can change the function as long as it does not change the ordering of the the bundle:
So I can change u(x2, y2) to 2u(x2, y2) so that
u(x2, y2)=4 and u(x1, y1)=2 becomes
2u(x2, y2)=8 and 2u(x1, y1)=4
But the initial condition still holds:
We say that a utility function is unique up to any positive monotonic transformation

64. PMT Positive Monotonic Transformation
If u(x2,y2) > u(x1,y1), then any Positive Monotonic Transformation of u, say f (u), implies that
f [u(x2,y2)] > f [u(x1,y1)]

65. Consider utility from each bundle below

66. Eg of PMT : B = A +3

67. Eg2 of PMT: C=2.5A

68. Eg 3 of PMT: D=2 A+4

69. Eg of NOT a PMT: E=100/A