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23
4.8
a. If
( , ) min( , )U x y x y
, utility maximization requires
xy
. Substitution into
the budget constraint yields
()
xy
x I p p y
. Hence,
( , . ) ,
( , , ) ( ) .
xy
xy
x y x y
I
V p p I pp
E p p V p p V
( , , ) min( , ) .
x y x y
E p p V p p V
b. It is interesting that the discontinuous utility function has continuous indirect
4.9 Given
( , ) .U x y ax by
There are two cases to consider. First, assume
,
xy
ab
pp
implying
(1) .
x
y
pa
pb
Then
*x
x I p
and
*0.y
Second, assume
(2) .
x
y
pa
pb
Then
*y
y I p
and
*0.x
Hence,
x
E p U a
for condition (1) and
y
E p U b
for condition (2). For the knife-edged case of equality,
(3) ,
x
y
pa
pb
we have
.
xy
E p U a p U b
Analytical Problems:
4.10 Cobb–Douglas utility
a. The demand functions in this case are
b. Interchanging
I
and
V
yields
11
( , , ) .
x y x y
E p p V B p p V
c. As for all exponential equations, the exponent
gives the elasticity of
expenditures with respect to
.
x
p
That is, the more important
x
is in the utility
a. For utility maximization,
1
x
y
p
U x x
MRS U y y p
.
Hence,
1
1,
xx
yy
pp
x
y p p
where
1.
1
b. If
0,
,
y
x
p
x
yp
implying
.
xy
p x p y
c. Part (a) shows
1
.
xx
yy
p x p
p y p
Chapter 4: Utility Maximization and Choice
24
d. The algebra is a bit tricky here, but worth doing once. Let’s solve for indirect
Chapter 4: Utility Maximization and Choice
24
rr
1
2r
rr
( 1)(1 ) 0,rk
where
1.
r
x
k p K
4.14 Altruism
a. When , , so Michele is completely self-interested. When ,
so she cares only about others, not herself. Definitions of a “perfect
altruist” may vary. According to the “Golden Rule” standard (“Regard others
as you would have them regard you”), Michele would have a symmetric
having solutions
proportional to her altruism, .
c. A proportional income tax just reduces her net income from to .
and
. Allowing a charitable deduction reduces
the relative “price” of Sofia’s consumption: is still 1 but falls to .
(1 )(1 ) (1 )(1 )
tt
Charitable contributions still fall compared to the no-tax case because of the
income effect, but they rise relative to Michele’s own consumption because of
d(1). Substituting Sofia’s utility function into Michele’s and solving for yields
1 (1 ) (1 )
1 1 2 1 2
( , ) .U c c c c
Solving the utility-maximization problem yields
**
1,.
11
c I c I
For a given , Michele reduces her charitable contributions compared to part
(b) because she takes into account Sofia’s benefit from Michele’s
consumption, leading Michele to keep her own consumption higher.
d(2). Substituting Sofia’s utility into Michele’s and solving for gives the same
function as in part (b).
Chapter 4: Utility Maximization and Choice
24
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