6.a Indifference curves
Bundles
·A "bundle" consists of a combination of goods that a consumer can utilize. Assuming there are only two
goods, a bundle is comprised of certain amount of good X and a certain amount of good Y. For example,
there may be two combinations of X & Y termed bundle A and bundle B. Bundle A has X A number of good
X and YA number of good Y. Similarly, bundle B has X B number of good X and YB number of good Y.
Assumptions about consumer behavior
1. Completeness: A consumer can say that either A is preferred to B, B is preferred to A, or A and B are
equally preferred. Shorthand notation for this will be A > B, B > A, A = B, respectively.
2. Dominance (more is preferred to less): A > B if X A is greater than XB and YA is greater than YB, OR if
XA is greater than XB while YA equals YB.
3. Transitivity: if A > B and if B > C, then A > C.
Utility functions and Indifference curves
A utility function shows the level of utility a consumer derives from a particular bundle of X & Y. An
example of a utility function is the Cobb-Douglas utility function:
U = Xa·Yb
U stands for a numerical value of utility and X & Y stand for the number of the goods in the bundle. Say
U = X0.5·Y0.5 what combinations of X & Y (bundles) yield a utility level of 10?
Bundle
X
Y
A
1
100
B
2
50
C
3
33 1/3
D
4
25
All bundles satisfy the equation 10 = X0.5·Y0.5. Therefore, A=B=C=D, and the consumer is indifferent
between the bundles because they all yield the same level of utility.
Showing these bundles on a graph forms an indifference curve.
Y
Diagram 6.a.1
100
A
50
B
Diagram 6.a.1 shows bundles A,B,C
and D which all produce a utility level of
10. Since they all produce the same
utility, they are indifferent to the
consumer, and comprise an indifference
curve (IC). Bundle N has more of both
X and Y, so it must be on a higher IC
N
C
D
In difference curve U = 10
X
1
2
3
4
Bundle N, shown above, has more of good X than bundle B and is therefore preferred since more is
preferred to less. Since the consumer is indifferent between bundles A,B, C and D, bundle N is preferred to
A,C and D as well. Bundle N must lie on a higher indifference curve because it is preferred to A,B,C and
D.
Properties of IC's
1. Must have negative slope. If IC's had a positive slope, point B would have more of both X and Y than
point A, and thus be preferred by our assumption that more is preferred to less. Therefore, they could
not be on the same IC curve because the consumer would not be indifferent between the two.
2. IC's cannot intersect. If they were to intersect, we would have a bundle (at the intersection) that
produces two different levels of utility.
3. There are infinite number of IC's. We can always add one more of good X or Y infinitely and the
bundle would raise utility, moving to a higher IC curve.
Marginal Rate of Substitution (MRS): The maximum amount of Y a consumer is willing to give up to get
one more unit of X. The MRS shows the number of good Y the consumer would trade to get one more of
good X.
MRS = X / Y = slope of IC curve
A diminishing MRS implies that as X increases, the slope of the IC curve gets flatter. In other words, as
we get more and more of X, we are willing to trade less and less of good Y to get the next unit of X. This
means that IC curves (of the Cobb-Douglas form) are convex.
Marginal Utility(MU) & MRS
MUx is the marginal utility of good X; the utility gained from having one more unit of X. MU y is
the marginal utility of good Y; the utility gained from having one more unit of Y. To find the marginal
utility, take the partial derivative of the utility function with respect to X and Y.
MUx = total utility / X = U / X
MUy = total utility / Y = U / Y
The marginal rate of substitution (MRS) is the marginal utility of X divided by the marginal utility of Y.
MRS = MUx / MUy
Therefore, in order to find the marginal rate of substitution (MRS) we follow three steps. First, take the
partial derivative of the utility function with respect to X to get MU x. Then take the partial derivative of the
utility function with respect to Y to get MUy.
Specific Utility Functions
The first type of utility function is the Cobb-Douglas utility function. It has the form U = Xa·Yb. The
marginal rate of substitution for the Cobb-Dpuglas utility function is MRS = (a/b)(Y/X). For example, say
our function is U= X0.5Y0.5. First, we take the partial derivative of U with respect to X to get MU x.
U/X = 0.5X-0.5Y0.5.
Next, we take the partial derivative with respect to Y to get MU y.
U/Y = 0.5X0.5Y-0.5.
Dividing MUx by MUy we get
MRS =
0.5X-0.5Y0.5
0.5X0.5Y-0.5
=
Y
X
The next form of utility function is called the perfect substitute form. This is when the two goods X & Y
can be substituted interchangeably and would make no difference to the consumer. An example of perfect
substitutes is Mobil gasoline and Exxon gasoline. Presuming the prices are the same and there is no unusual
preference for gasoline name brand, one gallon of Mobil gas can be substituted for one gallon of Exxon gas
and the consumer would be completely indifferent. The mathematical form of perfect substitutes utility
function is
U = a·X + b·Y.
The marginal utility for X and Y are
MUx = a MUy = b.
The MRS = a/b which is a constant. Therefore, the indifference curve for perfect substitutes is a straight
line. The indifference curve for perfect substitutes is shown in Diagram 6.a.2.
Diagram 6.a.2
Y
Diagram 6.a.2 shows the IC
for perfectly substitutable
goods. The MRS for perfect
substitutes is MRS= a/b
which is a constant. Since
MRS = slope of IC, a
constant MRS means the IC
curve is a straight line
IC pe rfect su bsti tute s
X
The next form of utility function is for goods that are called perfect compliments. An example of two
goods that are perfect compliments is a right shoe and a left shoe. Having an equal number of each good
raises utility. If you have one right shoe and one left shoe, your utility would be the same as having one
right shoe and twenty left shoes, since those extra left shoes do not have a complimentary right shoe. The
utility function for perfect compliments is in the form of
U = min(X,Y).
This means our value of utility is the minimum value between X and Y. For example, say we had 2 X and
1 Y; the utility function would look like U = min(2,1) = 1 since one is the minimum value. Our utility
would not raise to two until we had two of both X and Y. The graph of the indifference curve for perfect
compliments is shown in diagram 6.a.3.
Y le ft s hoe
Diagram 6.a.3
Diagram 6.a.3 shows the IC for
perfectly complimentary goods.
Adding one more right shoe
without adding another left shoe
does not raise utility and is on
the same indifference curve
1
IC Pe rfect
comp lime nts
X rig ht s hoe
1
2