35
Problems in this chapter focus on comparative statics analyses of income and own-price
changes. Many of the problems are fairly easy so that students can approach the ideas
involved in shifting budget constraints in simplified settings. Theoretical material is
confined mainly to the analytical problems that stress various elasticity measures and
introduce the almost ideal demand system.
Comments on Problems
5.1 This problem is an example of perfect substitutes. Solving this problem is easy
edge” nature of demand with perfect substitutes.
5.2 This problem is a fixed-proportions example. The problem illustrates how the
single good looking at utility-maximizing choices.
5.3 An exploration of the notion of homothetic functions. This problem shows that
Giffen’s paradox cannot occur with homothetic functions.
5.4 This problem asks students to pursue the analysis of Example 5.1 to obtain
compensated demand functions. The analysis essentially duplicates Examples
5.3 and 5.4.
5.5 This problem is another utility-maximization example. In this case, utility is not
separable and cross-price effects are important.
5.6 This problem is in revealed preference theory. The bundles here violate the
strong axiom.
5.7 This problem is an example with no substitution effects. It shows how price
income shares.
5.8 This problem shows the convenient result that budget shares can be computed
Analytical Problems
CHAPTER 5:
Income and Substitution Effects
Chapter 5: Income and Substitution Effects
36
5.9 Share elasticities. This problem shows that many conventional elasticity
measures can be derived from “share elasticities.” This is useful because many
budgetary studies proceed mainly by focusing on expenditure shares.
5.10 More on elasticities. This problem shows how the elasticity of substitution
5.11 Aggregation of elasticities for many goods. This problem shows how the
aggregation relationships introduced in Chapter 5 for two goods can be
generalized to any number of goods.
5.12 Quasi-linear utility (revisited). This problem extends Problem 3.13 to consider
of the expenditure function that is widely used in empirical studies of demand.
The connections between this problem and Problem 5.9 are quite important in
the interpretation of many empirical studies.
5.14 Price indifference curves. This problem introduces a graphical concept that is
sometimes used to illustrate theoretical points.
5.15 The multiself model. This behavioral economics problem illustrates how a
model in which the individual has two different utility functions can be used to
examine: (1) situations where the utility function used to make decisions differs
from the true function and (2) situations where the person does not know what
his or her precise preferences are.
Solutions
if
3 8,
xy
<
pp
then
*x
x = I p
and
*0.y
If
3 8,
xy
pp
then
*0x =
and
*.
y
y I p
c.
d. Increases in
I
shift demand for
x
outward. Reductions in
y
p
do not
x
x
is a vertical line so
long as
3 8.
xy
pp
If the person buys only
,x
holding utility constant
requires that
0.75xU
no matter what the price of
x
is.
b
stand
for peanut butter. Utility maximization requires
2.bj
The budget
constraint is
0.05 0.1 3. bj
Substitution gives
*30b
and
*15.j
b. If
$0.15,
j
p
substitution now yields
*12j
and
*24.b
c. To continue buying
*15,j
*30.b
David would need to buy 3 more
ounces of jelly and 6 more ounces of peanut butter. This would require
an increase in income of
3(0.15) 6(0.05) 0.75.
d.
fixed proportions), and because bread is free, it is just as though he buys
the good “sandwiches,” where
2.
s b j
p p p
In part (a),
0.20
s
p
and
15.
s
q
Chapter 5: Income and Substitution Effects
36
2
() 1.
x x x
p x I Ip x I p x
II
1
j
jj
pI
Multiplication by
1i
x
yields the desired result.
n
Chapter 5: Income and Substitution Effects
36
1 2 1 2
2
0 1 1 1 2 2 2 11
22
11 1 11 1 22 22 2
22
22 2 12 12 1 12 2
()
12 1 2 0 1 2
1
ln ln ln ln (ln )
2
11
ln ln (ln ) (ln ) ln ln
22
1(ln ) (ln ) ln ln ln ln
2
ln ln .
a k p k p k
k p p k k p
p k k p k p
p p k u p p
Chapter 5: Income and Substitution Effects
36
ii. With perfect knowledge of preferences, however,
12
8 2ln2 9.39, 7 3ln3 10.30UU
, so in each case there is a
utility loss of about 0.05.
iii. As a result of part (ii), this person would pay up to about 0.05 to learn