1. 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

2. 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

3. 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

4. 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

5. 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

6. 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*

7. 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

8. 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

9. 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

10. 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)

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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   = -

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

20. 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 < /

21. 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

22. 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

23. 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

24. 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