1. Subject:Economics Class:1st Semester
Topic:
Consumers Preferences & Indifference Curve Analysis

2. Outline Indifference Curves (Properties)
The Marginal Rate of Substitution
The Utility Function
Marginal Utility
Some Special Functional Forms

3. Indifference Curves
Definition: An Indifference Curve is the set of all baskets for which the consumer is indifferent
Definition: An Indifference Map illustrates the set of indifference curves for a particular consumer

4. Properties of Indifference Maps:
Completeness
Each basket lies on only one indifference curve
Monotonicity
Indifference curves have negative slope
Indifference curves are not “thick”

5. y
Monotonicity • A x

6. y
Monotonicity
Preferred to A • A x

7. y
Monotonicity
Preferred to A • A
Less
preferred x

8. y
Monotonicity
Preferred to A • A
Less
preferred
IC1 x

9. Indifference Curves are NOT Thicky
Indifference Curves are NOT Thick B • • A
IC1 x

10. Properties of Indifference Maps:
3. Transitivity
Indifference curves do not cross
4. Averages preferred to extremes
Indifference curves are bowed toward the origin (convex to the origin).

11. Indifference Curves Cannot Crossy
Indifference Curves Cannot Cross
Suppose a consumer is indifferent between A and C
Suppose that B preferred to A.
IC1 • B • A C • x

12. Indifference Curves Cannot Crossy
Indifference Curves Cannot Cross
It cannot be the case that an IC contains both B and C
Why? because, by definition of IC the consumer is:
Indifferent between A & C
Indifferent between B & C
Hence he should be indifferent between A & B (by transitivity).
=> Contradiction.
IC2
IC1 • B • A C • x

13. Averages Preferred to Extremesy
Averages Preferred to Extremes A • •
IC1 B x

14. Averages Preferred to Extremesy
Averages Preferred to Extremes A •
(.5A, .5B) • •
IC1 B x

15. Averages Preferred to Extremesy
Averages Preferred to Extremes A •
(.5A, .5B) •
IC2 •
IC1 B x

16. Marginal Rate Of Substitution
There are several ways to define the Marginal Rate of Substitution
Definition 1: It is the maximum rate at which the consumer would be willing to substitute a little more of good x for a little less of good y in order to leave the consumer just indifferent between consuming the old basket or the new basket

17. Marginal Rate Of Substitution
Definition 2: It is the negative of the slope of the indifference curve:
MRSx,y = — dy (for a constant level of
dx preference)

18. Diminishing Marginal Rate Of Substitution
An indifference curve exhibits a diminishing marginal rate of substitution:
The more of good x you have, the more you are willing to give up to get a little of good y.
The indifference curves
Get flatter as we move out along the horizontal axis
Get steeper as we move up along the vertical axis.

19. Example: The Diminishing Marginal Rate of Substitution

20. The Utility Function
Definition: The utility function measures the level of satisfaction that a consumer receives from any basket of goods.

21. The Utility Function
The utility function assigns a number to each basket
More preferred baskets get a higher number than less preferred baskets.
Utility is an ordinal concept
The precise magnitude of the number that the function assigns has no significance.

22. Ordinal and Cardinal Ranking
Ordinal ranking gives information about the order in which a consumer ranks baskets
E.g. a consumer may prefer A to B, but we cannot know how much more she likes A to B
Cardinal ranking gives information about the intensity of a consumer’s preferences.
We can measure the strength of a consumer’s preference for A over B.

23. Example: Consider the result of an exam
An ordinal ranking lists the students in order of their performance
E.g., Harry did best, Sean did second best, Betty did third best, and so on.
A cardinal ranking gives the marks of the exam, based on an absolute marking standard
E.g. Harry got 90, Sean got 85, Betty got 80, and so on.

24. The Utility Function Implications of an ordinal utility function:
Difference in magnitudes of utility have no interpretation per se
Utility is not comparable across individuals
Any transformation of a utility function that preserves the original ranking of bundles is an equally good representation of preferences.
eg. U = xy U = xy + 2 U = 2xy
all represent the same preferences.

25. y
Example: Utility and a single indifference curve 5 2
10 = xy 2 5 x

26. y
Example: Utility and a single indifference curve 5
Preference direction
20 = xy 2
10 = xy 2 5 x

27. Marginal Utility
Definition: The marginal utility of good x is the additional utility that the consumer gets from consuming a little more of x
MUx = dU dx
It is is the slope of the utility function with respect to x.
It assumes that the consumption of all other goods in consumer’s basket remain constant.

28. Diminishing Marginal Utility
Definition: The principle of diminishing marginal utility states that the marginal utility of a good falls as consumption of that good increases.
Note: A positive marginal utility implies monotonicity.

29. Example: Relative Income and Life Satisfaction (within nations)
Relative Income Percent > “Satisfied”
Lowest quartile 70
Second quartile 78
Third quartile 82
Highest quartile 85
Source: Hirshleifer, Jack and D. Hirshleifer, Price Theory and Applications. Sixth Edition. Prentice Hall: Upper Saddle River, New Jersey

30. Marginal Utility and the Marginal Rate of Substitution
We can express the MRS for any basket as a ratio of the marginal utilities of the goods in that basket
Suppose the consumer changes the level of consumption of x and y. Using differentials:
dU = MUx . dx + MUy . dy
Along a particular indifference curve, dU = 0, so:
0 = MUx . dx + MUy . dy

31. Marginal Utility and the Marginal Rate of Substitution
Solving for dy/dx:
dy = _ MUx
dx MUy
By definition, MRSx,y is the negative of the slope of the indifference curve:
MRSx,y = MUx
MUy

32. Marginal Utility and the Marginal Rate of Substitution
Diminishing marginal utility implies the indifference curves are convex to the origin (implies averages preferred to extremes)

33. Example:
U= (xy)0.5
MUx=y0.5/2x0.5
MUy=x0.5/2y0.5
Marginal utility is positive for both goods:
=> Monotonicity satisfied
Diminishing marginal utility for both goods
=> Averages preferred to extremes
Marginal rate of substitution:
MRSx,y = MUx = y
MUy x
Indifference curves do not intersect the axes

34. y
Example: Graphing Indifference Curves
IC1 x

35. y
Example: Graphing Indifference Curves
Preference direction
IC2
IC1 x

36. Special Functional Forms
Cobb-Douglas (“Standard case”)
U = Axy
where:  +  = 1; A, , positive constants
Properties:
MUx = Ax-1y
MUy = Axy-1
MRSx,y = y x

37. y
Example: Cobb-Douglas
IC1 x

38. y
Example: Cobb-Douglas
Preference direction
IC2
IC1 x

39. Special Functional Forms
2. Perfect Substitutes:
U = Ax + By
where: A,B are positive constants
Properties:
MUx = A
MUy = B
MRSx,y = A (constant MRS) B

40. Example: Perfect Substitutes (butter and margarine)y
IC1 x

41. Example: Perfect Substitutes (butter and margarine)y
IC1
IC2 x

42. Example: Perfect Substitutes (butter and margarine)y
Slope = -A/B
IC1
IC2
IC3 x

43. Special Functional Forms
3. Perfect Complements:
U = min {Ax,By}
where: A,B are positive constants
Properties:
MUx = A or 0
MUy = B or 0
MRSx,y = 0 or  or undefined

44. y
Example: Perfect Complements (nuts and bolts)
IC1 x

45. y
Example: Perfect Complements (nuts and bolts)
IC2
IC1 x

46. Special Functional Forms
4. Quasi-Linear Utility Functions:
U = v(x) + Ay
where: A is a positive constant, and v(0) = 0
Properties:
MUx = v’(x)
MUy = A
MRSx,y = v’(x) (constant for any x) A

47. y
Example: Quasi-linear Preferences (consumption of beverages)
IC1 • x

48. • • y IC2 IC’s have same slopes on any IC1 vertical line x
Example: Quasi-linear Preferences (consumption of beverages)
IC2
IC’s have same slopes on any
vertical line
IC1 • • x

49. Summary
Characterization of consumer preferences without any restrictions imposed by budget
Minimal assumptions on preferences to get interesting conclusions on demand…seem to be satisfied for most people. (ordinal utility function)