Chapter 4
NAME
Utility
Introduction.
In the previous chapter, you learned about preferences
and indiﬀerence curves. Here we study another way of describing prefer
ences, the
utility function
. A utility function that represents a person’s
preferences is a function that assigns a utility number to each commodity
bundle. The numbers are assigned in such a way that commodity bundle
(
x, y
) gets a higher utility number than bundle (
x
0
,y
0
) if and only if the
consumer prefers (
x, y
)to(
x
0
0
). If a consumer has the utility function
U
(
x
1
,x
2
), then she will be indiﬀerent between two bundles if they are
assigned the same utility.
If you know a consumer’s utility function, then you can ﬁnd the
indiﬀerence curve passing through any commodity bundle. Recall from
the previous chapter that when good 1 is graphed on the horizontal axis
and good 2 on the vertical axis, the slope of the indiﬀerence curve passing
through a point (
x
1
2
)isknownasthe
marginal rate of substitution
.An
important and convenient fact is that
the slope of an indiﬀerence curve is
minus the ratio of the marginal utility of good 1 to the marginal utility of
good 2.
For those of you who know even a tiny bit of calculus, calculating
marginal utilities is easy. To ﬁnd the marginal utility of either good,
you just take the derivative of utility with respect to the amount of that
good, treating the amount of the other good as a constant. (If you don’t
know any calculus at all, you can calculate an approximation to marginal
utility by the method described in your textbook. Also, at the beginning
of this section of the workbook, we list the marginal utility functions for
commonly encountered utility functions. Even if you can’t compute these
yourself, you can refer to this list when later problems require you to use
marginal utilities.)
Example:
Arthur’s utility function is
U
(
x
1
2
)=
x
1
x
2
. Let us ﬁnd the
indiﬀerence curve for Arthur that passes through the point (3
,
4). First,
calculate
U
(3
,
4) = 3
×
4 = 12. The indiﬀerence curve through this
point consists of all (
x
1
2
) such that
x
1
x
2
= 12. This last equation
is equivalent to
x
2
=1
2
/x
1
. Therefore to draw Arthur’s indiﬀerence
curve through (3
,
4), just draw the curve with equation
x
2
=12
/x
1
.A
t
the point (
x
1
2
), the marginal utility of good 1 is
x
2
and the marginal
utility of good 2 is
x
1
. Therefore Arthur’s marginal rate of substitution
at the point (3
,
4) is

x
2
/x
1
=

4
/
3.
Example:
Arthur’s uncle, Basil, has the utility function
U
*
(
x
1
2
3
x
1
x
2

10. Notice that
U
*
(
x
1
2
)=3
U
(
x
1
2
)

10, where
U
(
x
1
2
)is
Arthur’s utility function. Since
U
*
is a positive multiple of
U
minus a con
stant, it must be that any change in consumption that increases
U
will also
increase
U
*
(and vice versa). Therefore we say that Basil’s utility function
is a
monotonic increasing transformation
of Arthur’s utility function. Let