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
with intuition, but students should not try to use calculus because of the “knife–
edge” nature of demand with perfect substitutes.
5.2 This problem is a fixed-proportions example. The problem illustrates how the
goods used in fixed proportions (peanut butter and jelly) can be treated as a
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
elasticities are determined only by income effects, which, in turn, depend on
income shares.
5.8 This problem shows the convenient result that budget shares can be computed
from expenditure functions through logarithmic differentiation.
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
affects the sizes of price elasticities.
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
the special form of the Slutsky equation for the quasi-linear function.
5.13 The almost ideal demand system. This problem introduces a parametrization
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
c.
Chapter 5: Income and Substitution Effects
37
d. Increases in
I
shift demand for
x
outward. Reductions in
y
p
x
x
do not
5.2 a. To avoid confusing goods’ names with prices, let
b
stand
j
d.
e. Because David uses only peanut butter and jelly to make sandwiches (in
Chapter 5: Income and Substitution Effects
38
5.3 a. As income increases, the ratio
xy
pp
stays constant, and the utility-
5.4 a. Since
c. It is easiest to show the Slutsky equation in elasticities by just reading
5.5 a. The Lagrange method yields
b. The indirect utility function is
Chapter 5: Income and Substitution Effects
39
c. Clearly, the compensated demand function for
x
depends on
,
y
p
5.6 Year 2’s bundle is revealed preferred to year 1’s since both cost the same in
5.7 a. Because of the fixed proportions between
h
and
,c
we know that the
b. With fixed proportions, there are no substitution effects. Here, the
compensated price elasticities are zero, so the Slutsky equation shows
d. If this person consumes only ham and cheese sandwiches, the price elasticity of
Analytical Problems
5.9 Share elasticities
c. Because
I
may be cancelled out of the derivation in part (b),
5.10 More on elasticities
5.11 Aggregation of elasticities for many goods
Chapter 5: Income and Substitution Effects
41
a. Because the demand for any good is homogeneous of degree zero,
Euler’s theorem states
b. Parts (b) and (c) are based on the budget constraint:
1.
n
ii
ip x I
==
5.12 Quasi-linear utility (revisited)
a. First, we need to find the demand functions for both the goods. This is
done by straightforward application of the Lagrange method to give:
b. Now, we need to find the compensated demand functions for both the
goods by first finding the indirect utility function:
Chapter 5: Income and Substitution Effects
42
c. For the Slutsky equation, the own-price effects of the demand functions
are also needed:
Chapter 5: Income and Substitution Effects
43
For the elasticity version of the equation, we need the own-price
elasticity of each good:
5.13 The almost ideal demand system
a.
b.
ln ( , , )
E kp kp u
Chapter 5: Income and Substitution Effects
44
Next, we gather together the terms with
k
in them. From the constraints,
we can see that
This simplifies the mess to a great extent to give:
1 2 1 2
homogeneous to degree 1 in prices.
c. From Problem 5.8,
ln ln .
xx
s E p=
Hence,
5.14 Price indifference curves
b. The slope shows the rate at which the prices should move relative to one
another for the utility to remain constant, given I:
Chapter 5: Income and Substitution Effects
45
5.15 The multiself model
a. Decision utility
iii. This problem illustrates that with two targets (a utility of 10.30 and
iv. Utility could also be raised to 10.30 (from the 10.08 calculated in part i)
b. Preference uncertainty
Chapter 5: Income and Substitution Effects
46
ii. With perfect knowledge of preferences, however,