Chapter 5
NAME
Choice
Introduction.
You have studied budgets, and you have studied prefer
ences. Now is the time to put these two ideas together and do something
with them. In this chapter you study the commodity bundle chosen by a
utilitymaximizing consumer from a given budget.
Given prices and income, you know how to graph a consumer’s bud
get. If you also know the consumer’s preferences, you can graph some of
his indiFerence curves. The consumer will choose the “best” indiFerence
curve that he can reach given his budget. But when you try to do this, you
have to ask yourself, “How do I ±nd the most desirable indiFerence curve
that the consumer can reach?” The answer to this question is “look in the
likely places.” Where are the likely places? As your textbook tells you,
there are three kinds of likely places. These are: (
i
) a tangency between
an indiFerence curve and the budget line; (
ii
) a kink in an indiFerence
curve; (
iii
) a “corner” where the consumer specializes in consuming just
one good.
Here is how you ±nd a point of tangency if we are told the consumer’s
utility function, the prices of both goods, and the consumer’s income. The
budget line and an indiFerence curve are tangent at a point (
x
1
,x
2
)ifthey
have the same slope at that point. Now the slope of an indiFerence curve
at (
x
1
2
)i
sthera
t
io
−
MU
1
(
x
1
2
)
/M U
2
(
x
1
2
). (This slope is also
known as the marginal rate of substitution.) The slope of the budget line
is
−
p
1
/p
2
. Therefore an indiFerence curve is tangent to the budget line
at the point (
x
1
2
)when
1
(
x
1
2
)
2
(
x
1
2
)=
p
1
/p
2
.Th
i
sg
iv
e
s
us one equation in the two unknowns,
x
1
and
x
2
. If we hope to solve
for the
x
’s, we need another equation. That other equation is the budget
equation
p
1
x
1
+
p
2
x
2
=
m
. With these two equations you can solve for
(
x
1
2
).
∗
Example:
A consumer has the utility function
U
(
x
1
2
x
2
1
x
2
.T
h
e
price of good 1 is
p
1
= 1, the price of good 2 is
p
2
= 3, and his income
is 180.
Then,
1
(
x
1
2
)=2
x
1
x
2
and
2
(
x
1
2
x
2
1
h
e
r
e

fore his marginal rate of substitution is
−
1
(
x
1
2
)
2
(
x
1
2
−
2
x
1
x
2
/x
2
1
=
−
2
x
2
/x
1
. This implies that his indiFerence curve will be
tangent to his budget line when
−
2
x
2
/x
1
=
−
p
1
/p
2
=
−
1
/
3. Simplifying
this expression, we have 6
x
2
=
x
1
. This is one of the two equations we
need to solve for the two unknowns,
x
1
and
x
2
. The other equation is
the budget equation. In this case the budget equation is
x
1
+3
x
2
= 180.
Solving these two equations in two unknowns, we ±nd
x
1
= 120 and
∗
Some people have trouble remembering whether the marginal rate
of substitution is
−
1
2
or
−
2
1
. It isn’t really crucial to
remember which way this goes as long as you remember that a tangency
happens when the marginal utilities of any two goods are in the same
proportion as their prices.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document50
CHOICE
(Ch. 5)
x
2
= 20.
Therefore we know that the consumer chooses the bundle
(
x
1
,x
2
) = (120
,
20).
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '09
 JOHNG.SESSIONS
 Microeconomics, Utility, Alice in Wonderland, Line segment

Click to edit the document details