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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,y
) if and only if the
consumer prefers (x, y)to(x,y
). If a consumer has the utility function
U(x1,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 (x1,x
2) is known as the 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(x1,x
2)=x1x2. 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 (x1,x
2) such that x1x2= 12. This last equation
Example: Arthur’s uncle, Basil, has the utility function U∗(x1,x
2)=
3x1x2−10. Notice that U∗(x1,x
2)=3U(x1,x
2)−10, where U(x1,x
2)is
Arthur’s utility function. Since U∗is a positive multiple of Uminus a con-
stant, it must be that any change in consumption that increases Uwill 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
34 UTILITY (Ch. 4)
When you have finished this workout, we hope that you will be able
to do the following:
•Calculate marginal utilities and marginal rates of substitution when
you know the utility function.
•Find utility functions that represent preferences when goods are per-
fect substitutes and when goods are perfect complements.
4.0 Warm Up Exercise. This is the first of several “warm up ex-
ercises” that you will find in Workouts. These are here to help you see
how to do calculations that are needed in later problems. The answers to
all warm up exercises are in your answer pages. If you find the warm up
exercises easy and boring, go ahead—skip them and get on to the main
problems. You can come back and look at them if you get stuck later.
This exercise asks you to calculate marginal utilities and marginal
rates of substitution for some common utility functions. These utility
functions will reappear in several chapters, so it is a good idea to get to
know them now. If you know calculus, you will find this to be a breeze.
NAME 35
u(x1,x
2)MU1(x1,x
2)MU2(x1,x
2)MRS(x1,x
2)
2x1+3x22 3 −2/3
4x1+6x24 6 −2/3
ax1+bx2a b −a/b
2√x1+x21
√x11−1
√x1
36 UTILITY (Ch. 4)
4.1 (0) Remember Charlie from Chapter 3? Charlie consumes apples and
bananas. We had a look at two of his indifference curves. In this problem
(a) Charlie has 40 apples and 5 bananas. Charlie’s utility for the bun-
0102030
40
10
20
30
Apples
Bananas
40
(b) Donna offers to give Charlie 15 bananas if he will give her 25 apples.
Would Charlie have a bundle that he likes better than (40,5) if he makes
Donna gives Charlie 15 bananas, he will have a total of 20 bananas. If he
has 20 bananas, how many apples does he need in order to be as well-off
as he would be without trade?)
4.2 (0) Ambrose, whom you met in the last chapter, continues to thrive
on nuts and berries. You saw two of his indifference curves. One indif-
ference curve had the equation x2=20−4√x1, and another indifference
curve had the equation x2=24−4√x1,wherex1is his consumption of
NAME 37
by the utility function U(x1,x
2)=4
(a) Ambrose originally consumed 9 units of nuts and 10 units of berries.
His consumption of nuts is reduced to 4 units, but he is given enough
(b) On the graph below, indicate Ambrose’s original consumption and
sketch an indifference curve passing through this point. As you can verify,
Ambrose is indifferent between the bundle (9,10) and the bundle (25,2).
If you doubled the amount of each good in each bundle, you would have
bundles (18,20) and (50,4). Are these two bundles on the same indiffer-
0 5 10 15 20
5
10
15
Nuts
Berries
20
(9,10)
(c) What is Ambrose’s marginal rate of substitution, MRS(x1,x
2), when
(d) We can write a general expression for Ambrose’s marginal rate of
substitution when he is consuming commodity bundle (x1,x
2). This is
38 UTILITY (Ch. 4)
(a) What is the slope of Burt’s indifference curve at the point where he
a line with this slope through the point (4,6). (Try to make this graph
fairly neat and precise, since details will matter.) The line you just drew
is the tangent line to the consumer’s indifference curve at the point (4,6).
(b) The indifference curve through the point (4,6) passes through the
04812
16
4
8
12
Cookies
Glasses of milk
16
a
b
Red Line
Black Line
Blue curve
(c) Burt currently has the bundle (4,6). Ernie offers to give Burt 9
glasses of milk if Burt will give Ernie 3 cookies. If Burt makes this trade,
(d) Ernie says to Burt, “Burt, your marginal rate of substitution is −2.
That means that an extra cookie is worth only twice as much to you as
an extra glass of milk. I offered to give you 3 glasses of milk for every
cookie you give me. If I offer to give you more than your marginal rate
of substitution, then you should want to trade with me.” Burt replies,
NAME 39
“Ernie, you are right that my marginal rate of substitution is −2. That
means that I am willing to make small trades where I get more than 2
glasses of milk for every cookie I give you, but 9 glasses of milk for 3
cookies is too big a trade. My indifference curves are not straight lines,
you see.” Would Burt be willing to give up 1 cookie for 3 glasses of milk?
(e) On your graph, use red ink to draw a line with slope −3 through the
point (4,6). This line shows all of the bundles that Burt can achieve by
trading cookies for milk (or milk for cookies) at the rate of 1 cookie for
every 3 glasses of milk. Only a segment of this line represents trades that
make Burt better off than he was without trade. Label this line segment
on your graph AB.
(a) On the graph below, use blue ink to draw and label the line whose
equation is x= 10. Also use blue ink to draw and label the line whose
equation is 2y= 10.
(c) Now use red ink to sketch in the indifference curve along which
0 5 10 15 20
5
10
15
x
y
20
Red
indifference
curve
Blue
lines
x=10
2y=10
