1. PREFERENCES AND UTILITY
Chapter 3
PREFERENCES AND UTILITY
MICROECONOMIC THEORY
BASIC PRINCIPLES AND EXTENSIONS
EIGHTH EDITION
WALTER NICHOLSON
Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.

2. 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

3. 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

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

5. 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
If A is preferred to B, then the utility assigned to A exceeds the utility assigned to B
U(A) > U(B)

6. Utility Utility rankings are ordinal in nature
They record the relative desirability of commodity bundles
Because utility measures are nonunique, 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

7. 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

8. Utility
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)
Keep in mind that everything is being held constant except X1, X2,…, Xn

9. 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*

10. Combinations (X1, Y1) and (X2, Y2) provide the same level of utility
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

11. 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

12. 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

13. Indifference Curve Map
Each point must have an indifference curve through it
Quantity of Y
Increasing utility
U1 < U2 < U3 U3 U2 U1
Quantity of X

14. because B contains more
Transitivity
Can 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

15. 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 X* and Y* form a convex set Y* U1
Quantity of X X*

16. 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

17. 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

18. Utility and the MRS MRS = -dY/dX = 100/X2
Note that as X rises, MRS falls
When X = 5, MRS = 4
When X = 20, MRS = 0.25

19. marginal utility of X1 = MUX1 = U/X1
Suppose that an individual has a utility function of the form
utility = U(X1, X2,…, Xn)
We can define the marginal utility of good X1 by
marginal utility of X1 = MUX1 = U/X1
The marginal utility is the extra utility obtained from slightly more X1 (all else constant)

20. Marginal Utility The total differential of U is
The extra utility obtainable from slightly more X1, X2,…, Xn is the sum of the additional utility provided by each of these increments

21. Deriving the MRS
Suppose we change X and Y but keep utility constant (dU = 0)
dU = 0 = MUXdX + MUYdY
Rearranging, we get:
MRS is the ratio of the marginal utility of X to the marginal utility of Y

22. 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

23. Marginal Utility and the MRS
Again, we will use the utility function
The marginal utility of a soft drink is
marginal utility = MUX = U/X = 0.5X-0.5Y0.5
The marginal utility of a hamburger is
marginal utility = MUY = U/Y = 0.5X0.5Y-0.5

24. 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

25. 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

26. 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

27. Examples of Utility Functions
CES Utility (Constant elasticity of substitution)
utility = U(X,Y) = X/ + Y/
when   0 and
utility = U(X,Y) = ln X + ln Y
when  = 0
Perfect substitutes   = 1
Cobb-Douglas   = 0
Perfect complements   = -

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

29. CES Utility Function
One of the assumptions of the Ordinal Theory is the assumption of continuity or substitutability. How easy it is to substitute one good for another has an impact on the shape of the utility function, thus, the indifference curve.

30. CES Utility Function
Along an indifference curve the MRS decreases as the Y/X ratio decreases. we wish to define some parameter that measures this degree of responsiveness. If the MRS does not change at all as the ratio of Y/X changes, we might say that substitution is easy—perfect substitutes where MRS = α / β and σ = ∞.

31. CES Utility Function
Alternatively, if the MRS changes rapidly for small changes in Y/X ratio, we would say the substitution is difficult, because minor variations in the Y/X ratio will have a substantial effect on the good’s relative utility, thus, the shape of the indifference curve.

32. CES Utility Function
A scale-free measure of this responsiveness is provided by the elasticity of substitution. Elasticity of substitution (σ) is:

33. CES Utility Function
For Cobb-Douglass Utility Function: δ = 0  σ = 1 For Perfect Substitutes: δ = 1  σ = ∞ For Perfect Complements: δ = -∞  σ = 0 σ is always positive because Y/X ratio and MRS change in the same direction.

34. 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 < /

35. Nonhomothetic Preferences
Some utility functions do not exhibit homothetic preferences
utility = U(X,Y) = X + ln Y
MUY = U/Y = 1/Y
MUX = U/X = 1
MRS = MUX / MUY = Y
Because the MRS depends on the amount of Y consumed, the utility function is not homothetic

36. Monotonic Transformation
A monotonic transformation is a way of transforming a set of numbers to another set of numbers in a way that preserves the order of the numbers. A monotonic transformation of a utility function is a utility function that represents the same preferences as the original utility function.

37. 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

38. Important Points to Note:
The negative of the slope of the indifference curve measures the marginal rate of substitution (MRS)
This shows 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
This is consistent with the notion that individuals prefer some balance in their consumption choices

39. 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