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82 CHAPTER 3 CONSUMER PREFERENCES AND THE CONCEPT OF UTILITY

basket A contains two units of food (x 2) and eight units of clothing ( y 8).
Thus, Brandon realizes a level of utility of U 1(2)(8) 4 with basket A. As the
graph indicates, Brandon can achieve the same level of utility by choosing other bas-
kets, such as basket B and basket C.
The concept of marginal utility is easily extended to the case of multiple goods.
The marginal utility of any one good is the rate at which total utility changes as the
level of consumption of that good rises, holding constant the levels of consumption of all
other goods. For example, in the case in which only two goods are consumed and the
utility function is U(x, y), the marginal utility of food (MU ) measures how the level
x
of satisfaction will change ( U ) in response to a change in the consumption of food
( x), holding the level of y constant:

¢U
MU 2 (3.2)
x
¢x y is held constant
) measures how the level of satis-
Similarly, the marginal utility of clothing (MU y
faction will change ( U ) in response to a small change in the consumption of clothing
( y), holding constant the level of food (x):

¢U
MU 2 (3.3)
y
¢y x is held constant
One could use equations (3.2) and (3.3) to derive the algebraic expressions for MU and
x
3
MU from U(x, y). When the total utility from consuming a bundle (x, y) is U 1xy,
y
the marginal utilities are MU 1y (21x) and MU 1x (21y). So, at basket A (with
x
y
x 2 and y 8), MU 18 (212) 1 and MU 12 (218) 1 4.
y
x
Learning-By-Doing Exercise 3.1 shows that the utility function U 1xy satisfies
the assumptions that more is better and that marginal utilities are diminishing. Because
these are widely regarded as reasonable characteristics of consumer preferences, we will
often use this utility function to illustrate concepts in the theory of consumer choice.
LEARNING-BY-DOING EXERCISE 3.1
S
E D
Marginal Utility
Let’s look at a utility function that satisfies Problem
the assumptions that more is better and that marginal
utilities are diminishing. Suppose a consumer’s prefer- (a) Show that a consumer with this utility function
ences between food and clothing can be represented by believes that more is better for each good.
the utility function U 1xy, where x measures the (b) Show that the marginal utility of food is diminishing
number of units of food and y the number of units of and that the marginal utility of clothing is diminishing.
clothing, and the marginal utilities for x and y are ex-
pressed by the following equations: MU x 1y (21x)
and MU y 1x (21y).

3 Learning-By-Doing Exercise A.7 in the Mathematical Appendix shows how to derive the equations of
MU x and MU y in this case.