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Two types of demand relationships are stressed in the problems to Chapter 6: cross-price effects
and composite commodity results. The general goal of these problems is to illustrate how the
demand for one particular good is affected by economic changes that directly affect some other
portion of the budget constraint. Several examples are introduced to show situations in which the
analysis of such cross-effects is manageable.
Comments on Problems
6.1 Another use of the Cobb–Douglas utility function that shows that cross-price effects are
6.2 Shows how some information about cross-price effects can be derived from studying
6.3 A simple case of how goods consumed in fixed proportion can be treated as a single
commodity (buttered toast).
6.4 An illustration of the composite commodity theorem. Use of the Cobb–Douglas utility
produces quite simple results.
6.5 An examination of how the composite commodity theorem can be used to study the
6.6 Illustrations of some of the applications of the results of Problem 6.5. More extensive
6.7 This problem demonstrates a special case in which uncompensated cross-price effects are
symmetric.
6.8 This problem looks at cross-substitution effects in a three-good CES function.
Analytical Problems
6.9 Consumer surplus with many goods. This illustrates how expenditure functions can
help to clarify consumer surplus ideas when several prices change.
CHAPTER 6:
Demand Relationships among Goods
6.10 Separable utility. This problem shows that many of the complications in a many good
utility function can be greatly simplified if utility is assumed to be separable.
6.11 Graphing complements. The problem draws on Samuelson’s famous paper on
6.12 Shipping the good apples out. This repeats the analysis in the Borcherding–Silberberg
6.13 Proof of the composite commodity theorem. This problem outlines two general
6.14 Spurious product differentiation. This behavioral problem shows how firms may be
able to receive higher prices for their products if they can convince (spuriously)
consumers that they are better.
6.1 a. As for all Cobb–Douglas applications, first-order conditions show
that
0.5
ms
p m p s I
. Hence,
0.5 s
s I p
and
0.
m
sp
b. Because indifference curves are rectangular hyperboles (ms = constant)
m
are precisely balanced.
c. We have the two conditions
0
mm
U
s s s
m
p p I
ss
U
m m m
p p I
But
.
ms
UU
sm
pp
So
.
sm
ms
II
d. From part (a),
0.5 0.5 0.5 .
s m m
sm
m m m s s
I p p m s p I
r
r
r
jr
must fall,
j
will fall. Hence,
0.
r
jp
bt b t
b. Since
0.5 ,
cc = I
p
0.
bt
c p =
,
t
b t b b
b
p
p b p t p b t p g
p
where
.
t
b
p
g b t
p
3
2
33
.
kp t
pt
= =
p t p t
The relative price is less than 1 for
0. t
The relative price
1
as
.t
d. Rise in
t
should reduce relative spending on
2
x
more than on
3
x
since this raises
its relative price. However, see the Borcherding and Silberberg analysis.
6.6 a. Transport charges make low-quality produce relatively more expensive at distant
locations. Hence, buyers will have a preference for high quality.
b. Increase in baby-sitting expenses raise the price cheap meals relative to expensive
ones.
c. High-wage individuals have higher value of time and hence a lower relative price
of Concorde flights.
d. Increasing search costs lower the relative price of expensive items.
ii
jj
.
j
i
j i j i
x
x
x = a a I = x
II
So income effects (in addition to substitution effects) are symmetric.
.
x x y x z
I
x = p p p p p
Clearly,
, 0.
yz
xx
pp
So these are gross complements.
b. The Slutsky equation shows
x x x
1 2 3 1 2 3
nn
b.
Notice that the rise in
1
p
shifts the compensated demand curve for
2.x
c. Symmetry of compensated cross-price effects implies that order of calculation is
irrelevant.
d. The figure in part (a) suggests that compensation should be smaller for net
complements than for net substitutes.
0.
xy
U
That is, the marginal utility of
x
does not
depend on the amount of y consumed. Though unlikely in a strict sense, this
independence might hold for large consumption aggregates such as “food” and
“housing.”
b. Because utility maximization requires
,
x x y y
MU p MU p
an increase in
income with no change in
x
p
or
y
p
must cause both
x
and
y
to increase to
x x y y
x
x
d. If
,U = y
x
then
1.
x
MU x y
But,
ln = ln + ln ;U x y
and so
x
MU = x .
Hence, the first case is not
