1. PREFERENCES AND UTILITY
Chapter 3
PREFERENCES AND UTILITY
Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved.

2. Outline Preference Utility Axioms of rational choice
Completeness
Transitivity
Continuity
Utility
Marginal rate of substitution (MRS) and convexity
Graph
Mathematical derivation

3. Axioms of Rational Choice
Completeness
if A and B are any two situations, an individual can always specify exactly one of these possibilities:
A is preferred to B
B is preferred to A
A and B are equally attractive

4. Axioms of Rational Choice
Transitivity
if A is preferred to B, and B is preferred to C, then A is preferred to C
assumes that the individual’s choices are internally consistent

5. Axioms of Rational Choice
Continuity
if A is preferred to B, then situations suitably “close to” A must also be preferred to B
used to analyze individuals’ responses to relatively small changes in income and prices

6. Utility
Given these assumptions, it is possible to show that people are able to rank in order all possible situations from least desirable to most
Economists call this ranking utility (introduced by Jeremy Bentham 19 century)
if A is preferred to B, then the utility assigned to A exceeds the utility assigned to B
U(A) > U(B)

7. Utility Utility rankings are ordinal in nature
they record the relative desirability of commodity bundles
Any set of numbers that accurately reflects a person’s preference ordering will do.
Because utility measures are not unique, it makes no sense to consider how much more utility is gained from A than from B
It is also impossible to compare utilities between people (may use very different scale)

8. Utility
Utility is affected by the consumption of physical commodities, psychological attitudes, peer group pressures, personal experiences, and the general cultural environment
Economists generally devote attention to quantifiable options while holding constant the other things that affect utility
ceteris paribus assumption

9. utility = U(x1, x2,…, xn; other things)
Assume that an individual must choose among consumption goods x1, x2,…, xn
The individual’s rankings can be shown by a utility function of the form:
utility = U(x1, x2,…, xn; other things)
x’s refer to the quantities of the goods that might be chosen
this function is unique up to an order-preserving transformation

10. utility Quite often to write as utility = U(x1, x2,…, xn)
If only two goods
utility = U(x, y)
Everything else is being held constant

11. Arguments of utility functions
The utility functionis used to represent how an individual ranks certain bundles of goods that might be purchased at one point in time. On occasion we will use other arguments in the utility function
utility = U(w): wealth
utility = U(c,h): consumption and nonwork time
utility = U(c1, c2): consumption in period 1 and 2

12. Economic Goods
In the utility function, the x’s are assumed to be “goods”
more is preferred to less
Quantity of y
Preferred to x*, y* ? y*
Worse
than
x*, y*
Quantity of x x*

13. Trade and substitution
Most economic activity involves trading between individuals.
When someone buys a loaf of bread, he or she is voluntarily giving up one thing (money)

14. Indifference Curves
An indifference curve shows a set of consumption bundles among which the individual is indifferent
Quantity of y
Combinations (x1, y1) and (x2, y2)
provide the same level of utility y1 y2 U1
Quantity of x x1 x2

15. Indifference curve
The slope of the indifference curve is negative, showing that if the individual is forced to give up some y, he or she must be compensated by an additional amount of x

16. Marginal Rate of Substitution
The negative of the slope of the indifference curve at any point is called the marginal rate of substitution (MRS)
Quantity of y y1 y2 U1
Quantity of x x1 x2

17. Marginal Rate of Substitution
MRS changes as x and y change
reflects the individual’s willingness to trade y for x
Quantity of y
At (x1, y1), the indifference curve is steeper.
The person would be willing to give up more
y to gain additional units of x
At (x2, y2), the indifference curve
is flatter. The person would be
willing to give up less y to gain
additional units of x y1 y2 U1
Quantity of x x1 x2

18. MRS
The slope of U1 and the MRS tell us something about the trade this person will voluntarily make.
MRS diminishes between x1,y1 and x2,y2

19. Indifference Curve Map
Each point must have an indifference curve through it
Quantity of y U1 U2 U3
Increasing utility
U1 < U2 < U3
Quantity of x

20. Indifference curve map
Indifference curves are similar to contour lines on a map in that they represent lines of equal “altitude”

21. Transitivity
Can any two of an individual’s indifference curves intersect?
The individual is indifferent between A and C.
The individual is indifferent between B and C.
Transitivity suggests that the individual
should be indifferent between A and B
Quantity of y
But B is preferred to A
because B contains more
x and y than A C B U2 A U1
Quantity of x

22. Convexity of indifference curve
An alternative way of stating the principle of a diminishing marginal rate of substitution (MRS) uses the mathematical notion of a convex set

23. Convexity
A set of points is convex if any two points can be joined by a straight line that is contained completely within the set
Quantity of y
The assumption of a diminishing MRS is
equivalent to the assumption that all
combinations of x and y which are
preferred to or x* and y* form a convex set y* U1
Quantity of x x*

24. Convexity
If the indifference curve is convex, then the combination (x1 + x2)/2, (y1 + y2)/2 will be preferred to either (x1,y1) or (x2,y2)
Quantity of y
This implies that “well-balanced” bundles are preferred
to bundles that are heavily weighted toward one
commodity y1
(y1 + y2)/2 y2 U1
Quantity of x x1
(x1 + x2)/2 x2

25. Convexity and balance in consumption
Individual prefer some balance in their consumption
Any proportional combination of two indifferent bundles of goods will be preferred to the initial
The assumption of convexity and diminishing MRS rule out the possibility of an indifference curve being straight over any portion of its length

26. Utility and the MRS
Suppose an individual’s preferences for hamburgers (y) and soft drinks (x) can be represented by
Solving for y, we get
y = 100/x
Solving for MRS = -dy/dx:
MRS = -dy/dx = 100/x2

27. Utility and the MRS Note that as x rises, MRS falls
MRS = -dy/dx = 100/x2
Note that as x rises, MRS falls
when x = 5, MRS = 4 (the person is willing to give up 4 hamburger for another soft drink)
when x = 20, MRS = 0.25
Point C is midway between points A and B and has 12.5 hamburgers and 12.5 soft drinks.
Utility =12.5

28. A mathematical derivation
A mathematical derivation of the MRS concept provided additional insights about the shape of indifference curves and the nature of preferences

29. Marginal Utility
Suppose that an individual has a utility function of the form
utility = U(x,y)
The total differential of U is
Along any indifference curve, utility is constant (dU = 0)

30. Deriving the MRS Therefore, we get:
MRS is the ratio of the marginal utility of x to the marginal utility of y

31. Diminishing Marginal Utility and the MRS
Intuitively, it seems that the assumption of decreasing marginal utility is related to the concept of a diminishing MRS
diminishing MRS requires that the utility function be quasi-concave
this is independent of how utility is measured
diminishing marginal utility depends on how utility is measured
Thus, these two concepts are different

32. Convexity of Indifference Curves
Suppose that the utility function is
We can simplify the algebra by taking the logarithm of this function
U*(x,y) = ln[U(x,y)] = 0.5 ln x ln y

33. Convexity of Indifference Curves
Thus,
MRS is diminishing as x increases and y decreases. The indifference curves are convex

34. Convexity of Indifference Curves
If the utility function is
U(x,y) = x + xy + y
There is no advantage to transforming this utility function, so

35. Convexity of Indifference Curves
Suppose that the utility function is
For this example, it is easier to use the transformation (order preserving)
U*(x,y) = [U(x,y)]2 = x2 + y2

36. Convexity of Indifference Curves
Thus,
as x increases and y increases, the MRS increases. The curve are concave, not convex, and clearly not a quasi-concave function

37. Utility functions for specific preferences
Individuals’ rankings of commodity bundles and the utility functions implied by this rankings are unobservable.
All we can learn about people’s preferences must come from the behavior we observe when they respond to changes in income, prices, and other factors
Examine a few of the forms of utility functions is useful , both because such an examination may offer some insights into observed behavior and understand the properties of such functions can be of some help in solving problems.

38. Examples of Utility Functions
Cobb-Douglas Utility
utility = U(x,y) = xy
where  and  are positive constants
The relative sizes of  and  indicate the relative importance of the goods
Since utility is unique only up to a monotonic transformation, it is often convenient to normalize these parameters so that α+β=1

39. Examples of Utility Functions
Perfect Substitutes
utility = U(x,y) = x + y
Quantity of y
The indifference curves will be linear.
The MRS will be constant along the
indifference curve. U1 U2 U3
Quantity of x

40. Perfect substitutions
utility = U(x,y) = x + y

41. Perfect substitutes
MRS is constant along the entire indifference curve. A person would be willing to give up the same amount of y to get one more x no matter how much x was being consumed.
A diminishing MRS do not apply
Might describe the relationship between different brands of the same product (e.g. willing to give up 10 gallons of Shell in exchange for 10 gallons of Exxon

42. Examples of Utility Functions
Perfect Complements
utility = U(x,y) = min (x, y)
Quantity of y
The indifference curves will be
L-shaped. Only by choosing more
of the two goods together can utility
be increased. U1 U2 U3
Quantity of x

43. utility = U(x,y) = min (x, y)
Perfect complements
utility = U(x,y) = min (x, y)
Neither of the two goods will be in excess only if x = y, that is y/x= / 
MRS=infinite for y/x> / 
=undefined for y/x= / 
= for y/x< / 

44. Perfect complements
These preferences would apply to goods that “ go together”– Coffee and cream, Peanut butter and jelly, and cream cheese and lox

45. Examples of Utility Functions
CES Utility (Constant elasticity of substitution)
utility = U(x,y) = x/ + y/
when   0 and  ≦ 1.
utility = U(x,y) = ln x + ln y
when  = 0
Perfect substitutes   = 1
Cobb-Douglas   = 0
Perfect complements   = - (less obvious-- using a limits argument)

46. Examples of Utility Functions
CES Utility (Constant elasticity of substitution)
The elasticity of substitution () is equal to 1/(1 - )
Perfect substitutes   = 
Fixed proportions   = 0

47. Homothetic Preferences
If the MRS depends only on the ratio of the amounts of the two goods, not on the quantities of the goods, the utility function is homothetic
Perfect substitutes  MRS is the same at every point
Perfect complements  MRS =  if y/x > /, undefined if y/x = /, and MRS = 0 if y/x < /

48. Homothetic Preferences
For the general Cobb-Douglas function, the MRS can be found as

49. Homothetic preference
The importance of homothetic functions is that one indifference curve is much like another. Slopes of the curves depend only on the ratio y/x, not on how far the curve is from the origin.
We can only look at one indifference curve or at a few nearby curves without fearing that our results would change dramatically at very different levels of utility.

50. Nonhomothetic Preferences
Some utility functions do not exhibit homothetic preferences
utility = U(x,y) = x + ln y
MRS diminishes as y decreases, but it is independent of the quantity of x consumed.

51. Nonhomothetic preference
Contrary to the homothetic case, a doubling of both x and y doubles the MRS in this case rather than leaving it unchanged.

52. The Many-Good Case Suppose utility is a function of n goods given by
utility = U(x1, x2,…, xn)
The total differential of U is

53. The Many-Good Case
We can find the MRS between any two goods by setting dU = 0
Rearranging, we get

54. Multigood Indifference Surfaces
We will define an indifference surface as being the set of points in n dimensions that satisfy the equation
U(x1,x2,…xn) = k
where k is any preassigned constant

55. Multigood Indifference Surfaces
If the utility function is quasi-concave, the set of points for which U  k will be convex
all of the points on a line joining any two points on the U = k indifference surface will also have U  k

56. Important Points to Note:
If individuals obey certain behavioral postulates, they will be able to rank all commodity bundles
the ranking can be represented by a utility function
in making choices, individuals will act as if they were maximizing this function
Utility functions for two goods can be illustrated by an indifference curve map

57. Important Points to Note:
The negative of the slope of the indifference curve measures the marginal rate of substitution (MRS)
the rate at which an individual would trade an amount of one good (y) for one more unit of another good (x)
MRS decreases as x is substituted for y
individuals prefer some balance in their consumption choices

58. Important Points to Note:
A few simple functional forms can capture important differences in individuals’ preferences for two (or more) goods
Cobb-Douglas function
linear function (perfect substitutes)
fixed proportions function (perfect complements)
CES function
includes the other three as special cases

59. Important Points to Note:
It is a simple matter to generalize from two-good examples to many goods
studying peoples’ choices among many goods can yield many insights
the mathematics of many goods is not especially intuitive, so we will rely on two-good cases to build intuition

60. The end