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This point is recorded as point x, on the 0x ray in Figure 6.2. Similarly, the points y, and z,
represent the combinations of a1 and a2 that would be obtained if all income were spent on
good y or good z, respectively.
Bundles of a1 and a2 that are obtainable by purchasing both x and y (with a fixed
budget) are represented by the line joining x, and y, in Figure 6.2.8 Similarly, the line x,z,
represents the combinations of a1 and a2 available from x and z, and the line y,z, shows
combinations available from mixing y and z. All possible combinations from mixing the
three market goods are represented by the shaded triangular area x,y,z,.
Corner solutions
One fact is immediately apparent from Figure 6.2: A utility-maximizing individual would
never consume positive quantities of all three of these goods. Only the northeast perime-
ter of the x,y,z, triangle represents the maximal amounts of a1 and a2 available to this
person given his or her income and the prices of the market goods. Individuals with a
preference toward a1 will have indifference curves similar to U0 and will maximize utility
by choosing a point such as E. The combination of a1 and a2 specified by that point can
be obtained by consuming only goods y and z. Similarly, a person with preferences
The points x,, y,, and z, show the amounts of attributes a1 and a2 that can be purchased by buying only
x, y, or z, respectively. The shaded area shows all combinations that can be bought with mixed bundles.
Some individuals may maximize utility at E, others at E0 .
a*2
a2
a1a*10
U ′0
U0
x*
y*
z*
E′
zE
y
x
8Mathematically, suppose a fraction a of the budget is spent on x and (1 – a) on y ; then
a1 ¼ aa1xx
, þ ð1% aÞa1yy
, ,
a2 ¼ aa2xx
, þ ð1% aÞa2yy
, .
The line x,y, is traced out by allowing a to vary between 0 and 1. The lines x,z, and y,z, are traced out in a similar way, as is
the triangular area x,y,z,.
FIGURE 6.2
Utility Maximization in
the Attributes Model
Chapter 6: Demand Relationships among Goods 199
represented by the indifference curve U0
0 will choose point E0 and consume only goods x
and y. Therefore, the attributes model predicts that corner solutions at which individuals
consume zero amounts of some commodities will be relatively common, especially in
cases where individuals attach value to fewer attributes (here, two) than there are market
goods to choose from (three). If income, prices, or preferences change, then consumption
patterns may also change abruptly. Goods that were previously consumed may cease to
be bought and goods previously neglected may experience a significant increase in pur-
chases. This is a direct result of the linear assumptions inherent in the production func-
tions assumed here. In household production models with greater substitutability
assumptions, such discontinuous reactions are less likely.
SUMMARY
In this chapter, we used the utility-maximizing model of
choice to examine relationships among consumer goods.
Although these relationships may be complex, the analysis
presented here provided a number of ways of categorizing
and simplifying them.
• When there are only two goods, the income and substitu-
tion effects from the change in the price of one good
(say, py) on the demand for another good (x) usually
work in opposite directions. Therefore, the sign of @x/@py
is ambiguous: Its substitution effect is positive but its
income effect is negative.
• In cases of more than two goods, demand relationships
can be specified in two ways. Two goods (xi and xj) are
‘‘gross substitutes’’ if @xi/@pj > 0 and ‘‘gross comple-
ments’’ if @xi/@pj < 0. Unfortunately, because these price
effects include income effects, they need not be symmet-
ric. That is, @xi/@pj does not necessarily equal @xj/@pi.
• Focusing only on the substitution effects from price
changes eliminates this ambiguity because substitution
effects are symmetric; that is, @xci=@pj ¼ @xcj =@pi. Now
two goods are defined as net (or Hicksian) substitutes if
@xci =@pj > 0 and net complements if @xci =@pj < 0.
Hicks’ ‘‘second law of demand’’ shows that net substi-
tutes are more prevalent.
• If a group of goods has prices that always move in uni-
son, then expenditures on these goods can be treated as
a ‘‘composite commodity’’ whose ‘‘price’’ is given by
the size of the proportional change in the composite
goods’ prices.
• An alternative way to develop the theory of choice
among market goods is to focus on the ways in which
market goods are used in household production to yield
utility-providing attributes. This may provide addi-
tional insights into relationships among goods.
PROBLEMS
6.1
Heidi receives utility from two goods, goat’s milk (m) and strudel (s), according to the utility function
Uðm, sÞ ¼ m & s:
a. Show that increases in the price of goat’s milk will not affect the quantity of strudel Heidi buys; that is, show that @s/@pm ¼ 0.
b. Show also that @m/@ps ¼ 0.
c. Use the Slutsky equation and the symmetry of net substitution effects to prove that the income effects involved with the
derivatives in parts (a) and (b) are identical.
d. Prove part (c) explicitly using the Marshallian demand functions for m and s.
6.2
Hard Times Burt buys only rotgut whiskey and jelly donuts to sustain him. For Burt, rotgut whiskey is an inferior good that
exhibits Giffen’s paradox, although rotgut whiskey and jelly donuts are Hicksian substitutes in the customary sense. Develop an
intuitive explanation to suggest why an increase in the price of rotgut whiskey must cause fewer jelly donuts to be bought. That
is, the goods must also be gross complements.
200 Part 2: Choice and Demand
6.3
Donald, a frugal graduate student, consumes only coffee (c) and buttered toast (bt). He buys these items at the university
cafeteria and always uses two pats of butter for each piece of toast. Donald spends exactly half of his meager stipend on coffee
and the other half on buttered toast.
a. In this problem, buttered toast can be treated as a composite commodity. What is its price in terms of the prices of butter
(pb) and toast (pt)?
b. Explain why @c/@pbt ¼ 0.
c. Is it also true here that @c/@pb and @c/@pt are equal to 0?
6.4
Ms. Sarah Traveler does not own a car and travels only by bus, train, or plane. Her utility function is given by
utility = b & t & p,
where each letter stands for miles traveled by a specific mode. Suppose that the ratio of the price of train travel to that of bus
travel (pt/pb) never changes.
a. How might one define a composite commodity for ground transportation?
b. Phrase Sarah’s optimization problem as one of choosing between ground (g) and air (p) transportation.
c. What are Sarah’s demand functions for g and p?
d. Once Sarah decides how much to spend on g, how will she allocate those expenditures between b and t?
6.5
Suppose that an individual consumes three goods, x1, x2, and x3, and that x2 and x3 are similar commodities (i.e., cheap
and expensive restaurant meals) with p2 ¼ kp3, where k < 1—that is, the goods’ prices have a constant relationship to one
another.
a. Show that x2 and x3 can be treated as a composite commodity.
b. Suppose both x2 and x3 are subject to a transaction cost of t per unit (for some examples, see Problem 6.6). How will this
transaction cost affect the price of x2 relative to that of x3? How will this effect vary with the value of t?
c. Can you predict how an income-compensated increase in t will affect expenditures on the composite commodity x2 and x3?
Does the composite commodity theorem strictly apply to this case?
d. How will an income-compensated increase in t affect how total spending on the composite commodity is allocated between
x2 and x3?
6.6
Apply the results of Problem 6.5 to explain the following observations:
a. It is difficult to find high-quality apples to buy in Washington State or good fresh oranges in Florida.
b. People with significant babysitting expenses are more likely to have meals out at expensive (rather than cheap) restaurants
than are those withoutsuch expenses.
c. Individuals with a high value of time are more likely to fly the Concorde than those with a lower value of time.
d. Individuals are more likely to search for bargains for expensive items than for cheap ones. Note: Observations (b) and
(d) form the bases for perhaps the only two murder mysteries in which an economist solves the crime; see Marshall Jevons,
Murder at the Margin and The Fatal Equilibrium.
6.7
In general, uncompensated cross-price effects are not equal. That is,
@xi
@pj
6¼
@xj
@pi
:
Use the Slutsky equation to show that these effects are equal if the individual spends a constant fraction of income on each good
regardless of relative prices. (This is a generalization of Problem 6.1.)
Chapter 6: Demand Relationships among Goods 201
6.8
Example 6.3 computes the demand functions implied by the three-good CES utility function
Uðx, y, zÞ ¼ % 1
x
% 1
y
% 1
z
:
a. Use the demand function for x in Equation 6.32 to determine whether x and y or x and z are gross substitutes or gross
complements.
b. How would you determine whether x and y or x and z are net substitutes or net complements?
Analytical Problems
6.9 Consumer surplus with many goods
In Chapter 5, we showed how the welfare costs of changes in a single price can be measured using expenditure functions and
compensated demand curves. This problem asks you to generalize this to price changes in two (or many) goods.
a. Suppose that an individual consumes n goods and that the prices of two of those goods (say, p1 and p2) increase. How
would you use the expenditure function to measure the compensating variation (CV) for this person of such a price
increase?
b. A way to show these welfare costs graphically would be to use the compensated demand curves for goods x1 and x2 by
assuming that one price increased before the other. Illustrate this approach.
c. In your answer to part (b), would it matter in which order you considered the price changes? Explain.
d. In general, would you think that the CV for a price increase of these two goods would be greater if the goods were net sub-
stitutes or net complements? Or would the relationship between the goods have no bearing on the welfare costs?
6.10 Separable utility
A utility function is called separable if it can be written as
Uðx, yÞ ¼ U1ðxÞ þ U2ð yÞ,
where Ui
0 > 0, Ui
00 < 0, and U1, U2 need not be the same function.
a. What does separability assume about the cross-partial derivative Uxy? Give an intuitive discussion of what word this
condition means and in what situations it might be plausible.
b. Show that if utility is separable then neither good can be inferior.
c. Does the assumption of separability allow you to conclude definitively whether x and y are gross substitutes or gross
complements? Explain.
d. Use the Cobb–Douglas utility function to show that separability is not invariant with respect to monotonic
transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter.
6.11 Graphing complements
Graphing complements is complicated because a complementary relationship between goods (under Hicks’ definition) cannot
occur with only two goods. Rather, complementarity necessarily involves the demand relationships among three (or more)
goods. In his review of complementarity, Samuelson provides a way of illustrating the concept with a two-dimensional
indifference curve diagram (see the Suggested Readings). To examine this construction, assume there are three goods that a
consumer might choose. The quantities of these are denoted by x1, x2, and x3. Now proceed as follows.
a. Draw an indifference curve for x2 and x3, holding the quantity of x1 constant at x01. This indifference curve will have the
customary convex shape.
b. Now draw a second (higher) indifference curve for x2, x3, holding x1 constant at x01 % h. For this new indifference curve,
show the amount of extra x2 that would compensate this person for the loss of x1; call this amount j. Similarly, show that
amount of extra x3 that would compensate for the loss of x1 and call this amount k.
c. Suppose now that an individual is given both amounts j and k, thereby permitting him or her to move to an even higher
x2, x3 indifference curve. Show this move on your graph, and draw this new indifference curve.
d. Samuelson now suggests the following definitions:
• If the new indifference curve corresponds to the indifference curve when x1 ¼ x01 % 2h, goods 2 and 3 are independent.
• If the new indifference curve provides more utility than when x1 ¼ x01 % 2h, goods 2 and 3 are complements.
202 Part 2: Choice and Demand
• If the new indifference curve provides less utility than when x1¼ x01 % 2h, goods 2 and 3 are substitutes. Show that these
graphical definitions are symmetric.
e. Discuss how these graphical definitions correspond to Hicks’ more mathematical definitions given in the text.
f. Looking at your final graph, do you think that this approach fully explains the types of relationships that might exist
between x2 and x3?
6.12 Shipping the good apples out
Details of the analysis suggested in Problems 6.5 and 6.6 were originally worked out by Borcherding and Silberberg (see the
Suggested Readings) based on a supposition first proposed by Alchian and Allen. These authors look at how a transaction
charge affects the relative demand for two closely substitutable items. Assume that goods x2 and x3 are close substitutes and are
subject to a transaction charge of t per unit. Suppose also that good 2 is the more expensive of the two goods (i.e., ‘‘good
apples’’ as opposed to ‘‘cooking apples’’). Hence the transaction charge lowers the relative price of the more expensive good
[i.e., (p2 þ t)/(p3 + t) decreases as t increases]. This will increase the relative demand for the expensive good if @ðxc2=xc3Þ=@t > 0
(where we use compensated demand functions to eliminate pesky income effects). Borcherding and Silberberg show this result
will probably hold using the following steps.
a. Use the derivative of a quotient rule to expand @ðxc2=xc3Þ=@t.
b. Use your result from part (a) together with the fact that, in this problem, @xci=@t ¼ @xci=@p2 þ @xci =@p3 for i = 2, 3, to show
that the derivative we seek can be written as
@ðxc2=xc3Þ
@t
¼ xc2
xc3
s22
x2
þ s23
x2
% s32
x3
% s33
x3
# $
,
where sij ¼ @xci =@pj.
c. Rewrite the result from part (b) in terms of compensated price elasticities:
ecij ¼
@xci
@pj
&
pj
xci
:
d. Use Hicks’ third law (Equation 6.26) to show that the term in brackets in parts (b) and (c) can now be written as
[(e22 – e23)(1/p2 – 1/p3) + (e21 – e31)/p3].
e. Develop an intuitive argument about why the expression in part (d) is likely to be positive under the conditions of this
problem. Hints: Why is the first product in the brackets positive? Why is the second term in brackets likely to be small?
f. Return to Problem 6.6 and provide more complete explanations for these various findings.
SUGGESTIONS FOR FURTHER READING
Borcherding, T. E., and E. Silberberg. ‘‘Shipping the Good
Apples Out—The Alchian-Allen Theorem Reconsidered.’’
Journal of Political Economy (February 1978): 131–38.
Good discussion of the relationships among three goods in
demand theory. See also Problems 6.5 and 6.6.
Hicks, J. R. Value and Capital, 2nd ed. Oxford, UK: Oxford
University Press, 1946. See Chapters I–III and related
appendices.
Proof of the composite commodity theorem. Also has one of the
first treatments of net substitutes and complements.
Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeco-
nomic Theory. New York: Oxford University Press, 1995.
Explores the consequences of the symmetry of compensated
cross-price effects for various aspects of demand theory.
Rosen, S. ‘‘Hedonic Prices and Implicit Markets.’’ Journal of
Political Economy (January/February 1974): 34–55.
Nice graphical and mathematical treatment of the attribute
approach to consumer theory and of the concept of ‘‘markets’’
for attributes.
Samuelson, P. A. ‘‘Complementarity—AnEssay on the 40th
Anniversary of the Hicks-Allen Revolution in Demand
Theory.’’ Journal of Economic Literature (December 1977):
1255–89.
Reviews a number of definitions of complementarity and shows
the connections among them. Contains an intuitive, graphical
discussion and a detailed mathematical appendix.
Silberberg, E., and W. Suen. The Structure of Economics: A
Mathematical Analysis, 3rd ed. Boston: Irwin/McGraw-Hill,
2001.
Good discussion of expenditure functions and the use of indirect
utility functions to illustrate the composite commodity theorem
and other results.
Chapter 6: Demand Relationships among Goods 203
EXTENSIONS
SIMPLIFYING DEMAND AND
TWO-STAGE BUDGETING
In Chapter 6 we saw that the theory of utility maximization in
its full generality imposes rather few restrictions on what
might happen. Other than the fact that net cross-substitution
effects are symmetric, practically any type of relationship
among goods is consistent with the underlying theory. This
situation poses problems for economists who wish to study
consumption behavior in the real world—theory just does not
provide much guidance when there are many thousands of
goods potentially available for study.
There are two general ways in which simplifications are
made. The first uses the composite commodity theorem
from Chapter 6 to aggregate goods into categories within
which relative prices move together. For situations where
economists are specifically interested in changes in relative
prices within a category of spending (such as changes in the
relative prices of various forms of energy), however, this
process will not do. An alternative is to assume that con-
sumers engage in a two-stage process in their consumption
decisions. First they allocate income to various broad group-
ings of goods (e.g., food, clothing) and then, given these
expenditure constraints, they maximize utility within each
of the subcategories of goods using only information about
those goods’ relative prices. In that way, decisions can be
studied in a simplified setting by looking only at one cate-
gory at a time. This process is called two-stage budgeting. In
these Extensions, we first look at the general theory of two-
stage budgeting and then turn to examine some empirical
examples.
E6.1 Theory of two-stage budgeting
The issue that arises in two-stage budgeting can be stated suc-
cinctly: Does there exist a partition of goods into m nonover-
lapping groups (denoted by r ¼ 1, m) and a separate budget
(lr) devoted to each category such that the demand functions
for the goods within any one category depend only on the
prices of goods within the category and on the category’s
budget allocation? That is, can we partition goods so that
demand is given by
xið p1, . . . , pn, IÞ ¼ xi2rð pi2r , IrÞ (i)
for r ¼ 1, m? That it might be possible to do this is suggested
by comparing the following two-stage maximization problem,
V,ð p1, . . . , pn, I1, . . . , ImÞ
¼ max
x1, ... , xn
Uðx1, . . . , xnÞ s.t.
X
i2r
pixi ) Ir , r ¼ 1,m
" #
(ii)
and
max
I1, ... , Im
V, s.t.
XM
r¼1
Ir¼ I,
to the utility-maximization problem we have been studying,
max
xi
Uðx1, . . . , xnÞ s.t.
Xn
i¼1
pixi ) I: (iii)
Without any further restrictions, these two maximization
processes will yield the same result; that is, Equation ii is just
a more complicated way of stating Equation iii. Thus, some
restrictions have to be placed on the utility function to ensure
that the demand functions that result from solving the two-
stage process will be of the form specified in Equation i. Intui-
tively, it seems that such a categorization of goods should
work providing that changes in the price of a good in one cat-
egory do not affect the allocation of spending for goods in any
category other than its own. In Problem 6.9 we showed a case
where this is true for an ‘‘additively separable’’ utility function.
Unfortunately, this proves to be a special case. The more gen-
eral mathematical restrictions that must be placed on the util-
ity function to justify two-stage budgeting have been derived
(see Blackorby, Primont, and Russell, 1978), but these are not
especially intuitive. Of course, economists who wish to study
decentralized decisions by consumers (or, perhaps more
importantly, by firms that operate many divisions) must do
something to simplify matters. Now we look at a few applied
examples.
E6.2 Relation to the composition
commodity theorem
Unfortunately, neither of the two available theoretical
approaches to demand simplification is completely satisfying.
The composite commodity theorem requires that the relative
prices for goods within one group remain constant over time,
an assumption that has been rejected during many different
historical periods.
On the other hand, the kind of separability and two-stage
budgeting indicated by the utility function in Equation i also
requires strong assumptions about how changes in prices for
a good in one group affect spending on goods in any other
group. These assumptions appear to be rejected by the data
(see Diewert and Wales, 1995).
Economists have tried to devise even more elaborate,
hybrid methods of aggregation among goods. For example,
Lewbel (1996) shows how the composite commodity theorem
might be generalized to cases where within-group relative
prices exhibit considerable variability. He uses this generaliza-
tion for aggregating U.S. consumer expenditures into six large
groups (i.e., food, clothing, household operation, medical care,
transportation, and recreation). Using these aggregates, he con-
cludes that his procedure is much more accurate than assuming
two-stage budgeting among these expenditure categories.
E6.3 Homothetic functions and
energy demand
One way to simplify the study of demand when there are
many commodities is to assume that utility for certain subcat-
egories of goods is homothetic and may be separated from the
demand for other commodities. This procedure was followed
by Jorgenson, Slesnick, and Stoker (1997) in their study of
energy demand by U.S. consumers. By assuming that demand
functions for specific types of energy are proportional to total
spending on energy, the authors were able to concentrate
their empirical study on the topic that is of most interest to
them: estimating the price elasticities of demand for various
types of energy. They conclude that most types of energy (i.e.,
electricity, natural gas, gasoline) have fairly elastic demand
functions. Demand appears to be most responsive to price for
electricity.
References
Blackorby, Charles, Daniel Primont, and R. Robert Russell.
Duality, Separability and Functional Structure: Theory
and Economic Applications. New York: North Holland,
1978.
Diewert, W. Erwin, and Terrence J. Wales. ‘‘Flexible Func-
tional Forms and Tests of Homogeneous Separability.’’
Journal of Econometrics (June 1995): 259–302.
Jorgenson, Dale W., Daniel T. Slesnick, and Thomas M.
Stoker. ‘‘Two-Stage Budgeting and Consumer Demand
for Energy.’’ In Dale W. Jorgenson, Ed., Welfare, vol. 1:
Aggregate Consumer Behavior, pp. 475–510. Cambridge,
MA: MIT Press, 1997.
Lewbel, Arthur. ‘‘Aggregation without Separability: A Standard-
ized Composite Commodity Theorem.’’ American Eco-
nomic Review (June 1996): 524–43.
Chapter 6: Demand Relationships among Goods 205
consuming more a1, say, requires the use of more of the ‘‘ingredients’’ x, y, and z, this activ-
ity obviously has an opportunity cost in terms of the quantity of a2 that can be produced.
To produce more bread, say, a person must not only divert some flour, milk, and eggs from
using them to make cupcakes but may also have to alter the relative quantities of these
goods purchased because he or she is bound by an overall budget constraint. Hence bread
will have an implicit price in terms of the number of cupcakes that must be forgone to be
able to consume one more loaf. That implicit price will reflect not only the market prices of
bread ingredients but also the available household production technology and,in more
complex models, the relative time inputs required to produce the two goods. As a starting
point, however, the notion of implicit prices can be best illustrated with a simple model.
The linear attributes model
A particularly simple form of the household production model was first developed by
K. J. Lancaster to examine the underlying ‘‘attributes’’ of goods.7 In this model, it is the
attributes of goods that provide utility to individuals, and each specific good contains a
fixed set of attributes. If, for example, we focus only on the calories (a1) and vitamins (a2)
that various foods provide, Lancaster’s model assumes that utility is a function of these
attributes and that individuals purchase various foods only for the purpose of obtaining
the calories and vitamins they offer. In mathematical terms, the model assumes that the
‘‘production’’ equations have the simple form
a1 ¼ a1xx þ a1yy þ a1zz,
a2 ¼ a2xx þ a2yy þ a2zz,
(6:46)
where a1x represents the number of calories per unit of food x, a2x represents the number
of vitamins per unit of food x, and so forth. In this form of the model, there is no actual
‘‘production’’ in the home. Rather, the decision problem is how to choose a diet that pro-
vides the optimal mix of calories and vitamins given the available food budget.
Illustrating the budget constraints
To begin our examination of the theory of choice under the attributes model, we first
illustrate the budget constraint. In Figure 6.2, the ray 0x records the various combinations
of a1 and a2 available from successively larger amounts of good x. Because of the linear
production technology assumed in the attributes model, these combinations of a1 and a2
lie along such a straight line, although in more complex models of home production that
might not be the case. Similarly, rays of 0y and 0z show the quantities of the attributes a1
and a2 provided by various amounts of goods y and z that might be purchased.
If this person spends all his or her income on good x, then the budget constraint
(Equation 6.45) allows the purchase of
x, ¼ I
px
, (6:47)
and that will yield
a,1 ¼ a1xx
, ¼ a1xI
px
,
a,2 ¼ a2xx
, ¼ a2xI
px
:
(6:48)
7See K. J. Lancaster, ‘‘A New Approach to Consumer Theory,’’ Journal of Political Economy 74 (April 1966): 132–57.
198 Part 2: Choice and Demand
	PART TWO: Choice and Demand
	CHAPTER 6 Demand Relationships among Goods
	Summary��������������
	Problems���������������
	PART TWO: Choice and Demand
	CHAPTER 6 Demand Relationships among Goods
	Suggestions for Further Reading��������������������������������������
	PART TWO: Choice and Demand
	CHAPTER 6 Demand Relationships among Goods
	Extensions: Simplifying Demand and Two-Stage Budgeting�������������������������������������������������������������