A Theory of Demand with Variable Consumer Preferences
R. L. Basmann
Econometrica, Vol. 24, No. 1. (Jan., 1956), pp. 47-58.
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A THEORY OF DEMAND WITH VARIABLE
CONSUMER PREFERENCES1
Existing theory of consumer demand does not contain a body of theorems purporting t o explain the consumption behavior of individuals when their preferences
are changed, either autonomously or by advertising and other forms of selling
effort. The objective of this paper is t o present a theory of consumer demand with
variable preferences. The assumption that the individual consumer has a unique
ordinal utility index function is replaced by the assumption that he has a family
of ordinal utility functions; advertising expenditures by the sellers of commodities
are assumed t o determine which one of these ordinal utility functions is t o be maximized. From these assumptions are derived a number of theoretical relations which
measurements defining advertising elasticities of demand must satisfy. The relations involving shifts in demand and advertising elasticities of demand are shown
to be analogues of the theorems of consumer demand under fixed preferences. For
example, a theorem corresponding t o the well-known Leontief-Hicks theorem on
aggregate commodities has been worked out, thus showing that, under certain
advertising conditions, a group of commodities may be treated as a single good.
1.
INTRODUCTION
THE MODERN THEORY of consumer demand as formulated by E d g e ~ o r t h , ~
Antonelli14and Pareto,6 and worked out by Slutsky16Hicks and Allen,' and
Hicks18is based on the assumption that the individual consumer allocates expenditures on commodities as if he had a fixed, ordered set of preferences described by an indifference map or by an ordinal utility function which he maximizes subject to restraints imposed by the money income he receives and the
prices he must pay. From the point of view of the econometrician, this theory
has served well by indicating criteria to govern the construction of aggregate
commodity and consumer price indexeslgby providing certain plausible a priori
constraints to be imposed on estimates of the parameters of empirical demand
Journal Paper No. 5-2772 of the Iowa Agricultural Experiment Station, Ames, Iowa.
Project No. 1200.
T h e author is indebted to Professors Gerhard Tintner, John A. Nordin, and Donald
R. Kaldor and t o Mr. George Elsasser of Iowa State College and t o Professor Leonid Hurwicz of the University of Minnesota for their advice and criticism. Responsibility for any
errors found in this article belongs to the author alone.
3 F. Y. Edgeworth, Mathematical Psychics, London: Kegan Paul, 1881.
G. B. Antonelli, Sulla teoria matematica della economia pura, Pisa: Folchetto, 1886.
Reprinted in Giornale degli Economisti, N. S., Vol. 10, 1951, pp. 233-263.
6 V. Pareto. Manuel d'lhonomie Politique, 2d ed., Paris: Giard, 1927.
E . Slutsky, "Sulla teoria del bilancio del consumatore," Giornale degli Economisti,
Vol. 51, 1915, pp. 1-26.
J. R. Hicks and R. G. D. Allen, "A Reconsideration of the Theory of Value," Economica, N. S., Vol. 1, 1934, pp. 52-73, 196-219.
J. R. Hicks. Value and Capital, Oxford: Clarendon Press, 1939.
Cf. E. von Hofsten, Price Indexes and Quality Changes, London: Allen and Unwin,
1952, passim.
48
R. L. BASMANN
function^,^^ and by suggesting several hypotheses to be subjected to statistical
test. Consumer demand theory, however, not having taken variable preferences
explicitly into account until very recently, has not asserted any hypothetical
laws governing relations among the shifts in demand functions which a change
in preference orderings should cause. It is desirable that such hypothetical laws
be derived, and that econometricians bring them under empirical test. The
logical consequences of the assumption of fixed preferences differs markedly
from experience in the modern economy, for in the latter it is commonly observed
that the consumption behavior of real individuals and households is changed
more or less systematically by advertising and other forms of selling effort,
and by changes in social and technological factors exogenous with respect to
the consumer economy. In econometric demand analysis, the introduction of
time as an independent trend variable to "explain" the effects of changes in
taste is a t best an expedient it would be better to avoid, if possible, since trend
parameters are not capable of causal or legal interpretation.
In recent years two economists have cleared the way for the development of
a theory of consumer demand with variable preferences. In two short papers
growing out of the Lange-Robertson-Hicks debate over the nature of related
goods," Ichimura12and Tintner13 defined a change in preferences by a change
in the form of the ordinal utility function or indifference map and derived, for
shifts in demand, algebraic expressions which are linear combinations of the
Slutsky-Hicks substitution terms14 which play a central role in existing consumer demand theory. Straightforward linear transformations of the expressions derived by Tintner can be shown to follow the same mathematical rules
as the corresponding linear transformations of substitution terms. Economic
interpretations of the latter transformations and the rules which govern them
constitute all but one of the major analytical laws of existing consumer demand
theory; thus, in general, the same mathematical rules that govern the reactions
of consumers' expenditure and consumption to income-compensated price
changes govern the reactions of consumers' expenditure and consumption to
changes in the variables that determine the form of their preference orderings.
In this article the laws of consumer demand with variable preferences are
worked out and interpreted. Throughout the theoretical analysis, it is assumed
that the form of a representative individual's utility function depends on the
l o Cf. H . Wold. Demand Analysis: A Study in Econometrics, N e w Y o r k : John W i l e y .
1953. p. 47; and T . C . Koopmans and W . C . Hood, " T h e Estimation o f Simultaneous Linear
Economic Relationships," i n W . C . Hood and T . C . Koopmans, eds., Studies i n Econometric
Method, N e w Y o r k : John W i l e y , 1953, pp. 120-122.
l1 Cf. 0 . Lange, "Complementarity and Interrelations o f S h i f t s i n Demand," Review
of Economic Studies, V o l . 8 , 1940, pp. 58-63; and D . H . Robertson, J . R. Hicks, and 0.
Lange, " T h e Interrelations o f Shifts i n Demand," Review of Economic Studies, V o l . 12,
1944-45, pp. 71-78.
l2 S. Ichimura, "A Critical Note on t h e ~ k f i n i t i o n
of Related Goods," Review of Economic Studies, V o l . 18, 1950-51, pp. 179-183.
l3 G . Tintner, "Complementarity and Shifts i n Demand," Metroeconomica, V o l . 4 , 1952,
pp. 1-4.
l4 C f . J. R. Hicks, Value and Capital, 2d ed., Oxford: Clarendon Press, 1946, pp. 309-311.
49
VARIABLE CONSUMER PREFERENCES
expenditures which sellers make for advertising goods and services. The analysis
is entirely in terms of comparative statics; it provides theoretical explanations
of the way a hypothetical consumer, who always maximizes his utility function,
reallocates a fixed total expenditure to purchases of goods and services when
his preference structure (ordinal utility function) is changed by advertising in
certain defined ways. From the point of view of the econometrician engaged in
empirical demand analysis existing consumer demand theory has shown itself
to be essentially adequate for the analysis of effects of prices and income on
demand, and it seems plausible that the theory of consumer demand with
variable preferences will be equally useful in the empirical analysis of the effects
of advertising on demand functions.
2. DEFINITIONS AND ASSUMPTIONS
The point of view adopted here is that advertising costs are incurred by a seller
in an effort to secure a favorable change in consumers' subjective evaluations of
the commodity he offers for sale; in terms of economic theory, the seller seeks
by advertising his goods and services to increase their marginal utilities to consumers with respect to the marginal utilities of the commodities offered by other
sellers. This point of view conforms to that expressed in the institutional studies
of advertising by Bordenl5 and Lever.16
In order to express this concept of advertising in terms of ordinal utility theory,
it is first assumed that an individual's preferences can be represented by the
utility function, u(xl , . . . , xn ; 81, . - . , en), where the xi denote the quantities
of distinct goods and services he consumes during a given period of time during
which he receives money income, M, and pays prices, pl , . . . , p, , which he
cannot influence. The ei denote parameters which describe the form of the ordinal
utility function. They are assumed to depend on the variables a i l where a j
denotes the money expenditure by the seller of xi in advertising that commodity
to the consumer. It is also assumed that the form of the utility function is
uniquely determined by the advertising expenditures, i.e., that
Oj = &(al , .
(1)
, a,/xl, .. . , xn)
( j = 1, -
. , n),
are single-valued functions. The utility function is presumed to possess first-order
partial derivatives, ul , . - , un , with respect to the xi , and these denote the
marginal utilities of the commodities, xl , . . . , x, The marginal utilities are
presumed to possess in turn the first-order partial derivatives, u i j , and uie,.
It is assumed, moreover, that theinverse of transformation (1) exists, so that the
second-order partial derivatives, uiai and uiek, i, j, k = 1, . . . , n, are related
according to
.
'6
N. H. Borden, T h e Economic Eflects of Advertising, Chicago: Richard D. Irwin, 1947,
pp. 162-163.
l6 E. A. Lever, Advertising and Economic Theory, London: Oxford University Press,
1947, p. 28.
50
R. L. BASMANN
3.
DERIVATION O F THE TINTNER-ICHIMURA RELATION
The derivation of expressions for shifts in demand functions with respect to
small changes in advertising expenditures begins here with the assumption that
the usual first and second order conditions for individual consumer equilibrium
are fulfilled, i.e., that
and the matrix
(i, j = 1, ... , n),
(5)
is negative definite. The Lagrangean multiplier, A, denotes the marginal utility
of money income, M, and is positive. Income, M, and all prices, pi , are assumed
to remain constant.
An important invariant derived from the equilibrium conditions is the SlutskyHicks17 substitution term, sij . It is obtained by differentiating the first order
conditions (4), along with the side condition
u(xl,
(6)
, xn ; 01, . - , On)= constant,
partially with respect to pj and solving the resulting system of equations for
Sii =
I Uij (
---
X
IUI
'
where ( U i j ( is the cofactor of uij in the matrix U . The substitution term, sij ,
denotes the rate of change in demand for xi with respect to a small change in
the price of xj , all other prices remaining constant, but with income adjusted
to maintain a constant level of utility. The mathematical laws governingrelations
among substitution terms play a central role in the theory of consumer demand
with variable preferences, and so are summarized here, though without complete
proofs being furnished as these are available elsewhere.'S The four major theorems
are :
< 0,
(8)
~ i i
(9)
where not all x i
sij = sji
=
(i,j
=
1,
, n),
0.
J . R. Hicks, Value and Capital, 2d ed., p. 309.
J . L. Mosak, General Equilibrium Theory in International Trade, Bloomington, Ind.:
Principia Press, 1944, pp. 9-30.
17
'8
51
VARIABLE CONSUMER PREFERENCES
Expressions for shifts in demand with respect to advertising are obtained from
the equilibrium conditions (3) and (4) by differentiating them partially with
respect to a j and solving the resulting system of equations. Denote the shift in
demand for xi and xiai , where xi,, = axi/daj ; then
(i,j
=
1,
, n),
where Rhaj = aRn/aaj, Rh = uh/u,, and p, = 1, x, being the numeraire.
The relation (12) was first derived by Tintner;lS and a special case of (12) was
assumption was that there is a solitary
derived earlier by I ~ h i m u r aIchimura's
.~~
increase in the marginal rate of substitution of (say) XI for one of the other
commodities, all other marginal rates of substitution remaining constant; thus,
the expression he derived is equivalent t o (13) when Rjai > 0, and Riai = 0,
for all i # j.21
The expressions (12) and (13), which are called the Tintner-Ichimura Relations, satisfy the criterion of invariance; for if the equally valid utility function,
v = +(u), where +'(u) > 0, is substituted for u, v h = +'(u)u~,v h a i = +'(u)uhaj
+'I(u)uhuai , and the marginal utility of money income, M, is +'(u)X. SlutskyThe TintHicks substitution terms are invariant against this transf~rmation.~~
ner-Ichimura Relation, (12), after elementary simplification becomes
+
According to the first order conditions (4), uh/X = ph , and so it follows from the
linear dependence relation (10) that the second right-hand term of (14) is equal
to zero; xi,, is therefore invariant against the transformation +(u).
It follows from the initial assumption that the individual consumer spends his
entire money income, M, that the expressions for shifts in demand are related
according to the linear dependence
C pix*,
=
0.
i-1 That the Tintner-Ichimura relations satisfy this linear dependence can easily be
verified by straightforward substitution of (12) into (15), having regard to the
linear dependence rule for Slutsky-Hicks substitution terms.
'9 G . Tintner, "Complementarity and S h i f t s i n Demand," Metroeconomica, V o l . 4 , 1952,
pp. 1-4.
S. Ichimura, " A Critical Note o n t h e Definition o f Related Goods," Review of Economic Studies, V o l . 18, 1950-51, pp. 179-183.
2 1 Cf. R. L. Basmann, "A Note on M r . Ichimura's Definition of Related Goods," Review
of Economic Studies, V o l . 22, 1954-55, pp. 67-69.
Za Cf. J. R. Hicks, op. k t . , pp. 306-307.
52 R. L. BASMANN
The Tintner-Ichimura market relation is obtained by summing over all
(say, N) individuals in the market; that is,
(The notation, X i ,will henceforth denote market demand.) Since each individual
is presumed to be charged the same prices for his purchases of commodities, i t
follows from (15) that the Tintner-Ichimura market relations (16) satisfy the
linear dependence
4. INTERRELATIONS
The transformations
(18) OF ELASTICITIES O F DEMAND
eij
=
Qi
xei
(i,j = 1,
, n),
Xi
define the empirically useful concepts of individual and market advertising
elasticities of demand.23e i j is a dimensionless number representing the percentage
change in demand for x i with respect to a one per cent change in advertising
expenditure, aj ; E i j represents the corresponding concept for market demand
functions. These concepts are closely related to that of the elasticity of substitution sometimes employed in existing consumer demand theory. We define
(20) 2..
23
-
-8..
23 p j / x i
(i,j
=
1,
, n).
z i j represents the percentage change in the consumer's demand for x i with respect
to a one per cent income-compensated change in the price of x j If bhiRh is substituted for Rhaj in (13), then, after simplification, the expression for e i j becomes
.
where w h j = aj b h i is the elasticity of the marginal rate of substitution of x, for
with respect to the advertising expenditure, a j , and represents the percentage change in the marginal rate of substitution with respect to a one per
cent change in expenditure on advertising x i . Formula (21) shows that advertising elasticities of demand are linear combinations of substitution elasticities
with advertising elasticities of the marginal rates of substitution acting asweights.
Since, in empirical demand analysis, it is sometimes possible to obtain inde-
xh
23 Cf. R. Dorfman and P. 0. Steiner, "Optimal Advertising and Optimal Quality,"
American Economic Review, Vol. 44, 1954, pp. 826-836.
53
VARIABLE CONSUMER PREFERENCES
pendent estimates of substitution elasticities and advertising elasticities of demand, it is also possible to obtain estimates of the advertising elasticities of
marginal rates of substitution. Empirical verification of several hypotheses concerning the whj might then throw some light on problems encountered in the
theory of selling costs.24Some possibilities will be mentioned below.
Using relations (15) and (17), one can easily verify that advertising elasticities
of demand are linearly dependent according to
( j = 1,
- - , n),
(j
... , n),
for individual consumers, and
n
C
(piXi)Eij = 0
i=l
=
1,
for the market demand functions.
The linear dependence relations impose several restrictions on the signs and
magnitudes of advertising elasticities of demand. Since the coefficients, pix(
and p i x i , are nonnegative, it follows from (22) and (23) that the elasticity of
demand with respect to a j must be positive for at least one i, and negative for
at. least one other, except in the logically (but not empirically) trivial case where
all advertising elasticities are zero.
Another restriction imposed by the linear dependence relations limits the
number of elasticities, eij and Eij which can be constant. Not all the eij (or Eij)
can be constant unless all are (trivially) zero, for if they were all constant, then
expenditures on the several commodities would be determined according to the
system of equations
(24)
[eji]m = 0
(j, i
=
1,
, n),
where m is the column vector (ml , ... , m,), mi = pixi, and the rank of the
matrix [eji]is n - 1. It follows from (24) that the ratios of expenditures, mi/m,
are constant; hence
and this implies
a j am,
mi aaj
a j am,
m, aaj
'
24 Cf.N. S. Buchanan, "Advertising Expenditures: A Suggested Treatment," Journal
of Political Economy,Vol. 20, 1942, pp. 537-557.
54
R. L. BASMANN
Thus, if all the elasticities of demand with respect to a j are constant, and if this is
true for all j = 1, . . . ,n, then the elasticities of all the demand functions with
respect to each a, are equal; but the only magnitude of the eij which is consistent with this result is zero.
Some interesting results are obtained when it is assumed that advertising
expenditure, a i , affects only the marginal rate of substitution of x, for xi leaving
all other marginal utilities ~ n c h a n g e d .Let
~ ~ Ria, = bi,Ri > 0, Rjai = 0, for
all j # i. In this case the Tintner-Ichimura relation (13) becomes, after some
simplification,
and
(29)
e . . = w..z..
1 %
t%
I$
( j = 1, . . . , n).
From (28) and (8) it follows that xi,, > 0; and from (16) that Xi,, > 0. According to the definition of related goods given by
xi is a substitute for x j if
sij > 0; xi is a complement of xi if sij < 0; and xi and x j are independent
commodities if sij = 0. Accordingly, if xi is a substitute for x i , then by (28)
xi,, < 0; if xi is a complement of x j , then xiai > 0; that is, if there is an increase
in advertising expenditure, a ; , there results an increase in the demand for x i , a
decrease in the demand for commodities which are substitutes for xi , an increase
in the demand for commodities which are complements of x i , and no change in the
demand for commodities which are independent of x i . From the definition of
zji (formula (20)), the algebraic sign of zji is opposite to that of sji . Moreover,
the elasticity, wii , of the marginal rate of substitution of x, for xi with respect
to advertising expenditure, a i l is a positive number. I t follows, therefore,
from the definition of related goods, that the advertising elasticity of demand
for xi with respect to a; is greater than zero, and that the advertising elasticity
of demand for xj with respect to a i is less than zero if xi is a substitute for xj ,
is greater than zero if xi is a complement of x i , and is zero if xi is independent
of x j .
In this case where the advertising expenditure, a i l affects only the marginal
rate of substitution of x, for xi , theadvertising elasticities of demand are directly
proportional to the elasticities of substitution. Indeed, the advertising elasticity
of demand for xj with respect to a <is simply the product of the elasticity of
the marginal rate of substitution of x, for xi with respect to a i and the elasticity
of substitution of xi for x j . If Ri is relatively inelastic, i.e., if wi; < 1, then
formula (29) indicates that the consumer's demand for xj responds relatively less
to a one per cent increase in advertising expenditure, a i l than to a one per cent
income-compensated decrease in the price of xi ; if R; is relatively elastic, i.e.,
if wii > 1, then (29) indicates that the consumer's demand for xi responds relatively more to a one per cent increase in advertising expenditure, a i l than to a
one per cent income-compensated decrease in the price of xi . This points to the
Cf. G. Tintner, o p . cit., p. 3.
J. R. Hicks, o p . cit., pp. 309-311. Also see S. Ichimura, o p . cdt., p. 181 and R. L. Basmann, op. cit., p. 69.
26
VARIABLE CONSUMER PREFERENCES
55
possible convenience of classifying commodities as "necessities" or "luxuries"
according as the elasticities, wii , are either less than or equal to or else greater
than unity. Possibly such a classification would agree closely with a corresponding
one based on the magnitude of price e l a ~ t i c i t i e s . ~ ~
5. SOME THEOREMS ON COMPOSITE COMMODITIES
From the point of view of empirical demand analysis, one of the most important propositions of existing consumer demand theory is the well-known
Leontief-Hicks theorem,28which asserts that if the prices of a group of goods all
change in equal proportion, then that group of goods can be treated logically
as if it were but a single commodity. It is frequently the experience with market
data that the prices of individual goods in a general class of commodity undergo
approximately uniformly proportionate changes. By using the Leontief-Hicks
theorem, one is then able to define the concepts of related commodities as applied
to composite goods in exactly the same way as one defines complementarity,
substitutability, and independence of single, well-defined goods, and to define
the concepts of price elasticity, income elasticity, and substitution elasticity of
demand for composite commodities, and to verify that these measures satisfy
exactly the same laws as do the corresponding measures defined for single goods.
There is an analogous theorem in the theory of consumer demand with variable
preferences, but here it is the marginal rates of substitution of x, for each good
in a group of goods which are assumed to change in equal proportion. If X I ,
. . . , x, , are all sub-commodities of a general class, it may sometimes be plausible to assume that expenditure devoted to advertising the class of commodity
will affect the marginal utilities of the sub-commodities more or less uniformly.
The theorem asserts that if the marginal rates of substitution are all increased
in the same proportion, then the group of goods can be treated logically as a
single commodity; specifically, an aggregate quantity, called the composite
demand for the group, which is a weighted sum of the demands for individual
commodities within the group, can be defined such that this quantity behaves
exactly as do the xjai in formula ( 2 8 ) ; and elasticities of composite demand
can be defined such that they behave exactly as the eji in formula (29).The proof
of the theorem is essentially equivalent to one of the steps in the proof of the
Leontief-Hicks theorem.
Before the proof is undertaken, it is necessary t o define the concept of related
composite commodities. Suppose that the prices of xl , . . . , x, all increase in
the same proportion, and denote this aggregate by vl ; suppose that all other
prices remain constant, but partition the remaining commodities into two aggregates, X,+I , . . . , x, , denoted by u2 , and x,+~, . . . , x, , denoted by v 3 . According
to the Leontief-Hicks theorem, each of these groups of goods, ul , uz , and u3,
may be treated as if it were a single commodity. In addition, suppose that the
equal proportionate changes in the prices, pl , . . . , p, , are income-compensated.
Cf. H. Wold, o p . cit., pp. 114-115.
Cf. W. Leontief, "Composite Commodities and the Problem of Index Numbers,"
Econometrica, Vol. 4, 1936, pp. 39-59; J. R. Hicks, o p . cit., pp. 312-313; and H. Wold, o p .
cit., pp. 108-109.
l7
28
56
R. L. BASMANN
Thus, the change in expenditure on x i , j = 1, . . . , n, due to the proportionate
change in the price of xi , i = 1, . ,m, is equal to sji p j pi . If x j is a substitute
for xi , then i t follows from the definition of related goods that sjipjpi > 0; if
x j is a complement of x i , it follows that sjipjpi < 0; and, in particular,
..
Summing over the commodities making up the aggregate, v l , one obtains the
result that the change in aggregate expenditure on v l with respect to the proportionate increase or decrease in prices, p ~ . ,. . , p, , obeys the same law of
sign as s i i p i ;for, according to the theorem ( l l ) , i t follows that
where the left-hand member is the rate of change in expenditure on the group v l
with respect to the equal proportionate change in prices.
is the rate of change in expenditure on the aggregate v 2 , and
is the rate of change in expenditure on the aggregate v8 with respect to the proportionate change in prices, p ~ . , . , p, If &: > 0, the commodities in the
aggregate vz are said to be predominantly substitutable for the commodities in
the aggregate v l ; if Q; < 0, the commodities in the aggregate v2 are said to be
predominantly complementary with the commodities in vl . It follows from the
linear dependence rule (10) that if Q; < 0, Q; > 0, i.e., if vz is predominantly
complementary with v l , then v3 is predominantly substitutable for vl . This
last result is derived ultimately from the assumption that the consumer spends
his entire income, M
Suppose that there is advertising which increases all the marginal rates of
substitution, R1, . . , Rm , in equal proportion, and denote the proportional
increase by b. Denote by a the total advertising on v l ; denote by rl the total
expenditure on v l , by rz the total expenditure on vs , and by r3 the total expenditure on u3 ; i.e.,
.
.
.
(34)
From (13) the Tintner-Ichimura relation for the single good, x i , is given by
m
(37)
~ j =
a
-bC
sjipi
i=l
( j = 1, . . . , n).
VARIABLE CONSUMER PREFERENCES
The rates of change of expenditures, rl , r 2 , and ra , are given by
> 0. From the definitions of related
aggregate commodities given early in this section it follows that dr2/da < 0,
if vz is predominantly substitutable for vl , and that arz/da > 0, if v z is predominantly complementary with vl . Comparing (34) with (38), one sees that drl/aa
obeys the same law of sign as pixiai ; and comparing (34) with (39), one sees
that ar2/da and dr3/aa obey the same law of sign as pjxjai, where j # i.
From the linear dependence rule (15) i t follows that
It follows once more from (11) that arl/da
from which i t follows that
which is the linear dependence rule for advertising elasticities of composite
commodities. Or, in a more condensed notation,
where
is the advertising elasticity of demand for the composite commodity, vi , i =
1, 2, 3, with respect to the aggregate advertising expenditure on vl . (43) is
seen to have exactly the same form as (22). This completes the proof of the
theorem that a group of goods, all of whose marginal rates of substitution increase
or decrease in equal proportion, can be treated as a single commodity.
6.
FURTHER INTERRELATIONS
OF ELASTICITIES OF DEMAND: COMPOSITE
GOODS
I n formulas (21) and (29) the connections between advertising elasticities of
demand and substitution elasticities were pointed out. Similar connections
exist between advertising elasticities of demand for composite goods and the
advertising elasticities of demand for the sub-commodities that make up the
aggregates, which elasticities are in turn related to elasticities of substitution
between the sub-commodities, I n order to show these relations, the concept of
elasticity of substitution between two aggregate commodities is first defined.
58
R . L. BASMANN
This is done in a fashion similar to that in which the concept of related composite
commodities was defined above. From formula (20) one has
Analogously, in terms of the composite commodities, one may define the elasticity
.
and similarly for v s l and v l l
It follows from the definitions of elasticities, e,,,
formulas (38), (39), and (40) that
,i
=
1, 2, 3, and from the
where w,, = ab defines the elasticity of the marginal rate of substitution of x,
for the composite commodity 8 1 with respect to aggregate advertising expenditure, a. That the relations (46) have the same form as (29) shows that there are
the same relations between aggregate elasticities of demand with respect to
advertising expenditure and the income-compensated price elasticities of one
aggregate commodity for another as there are between elasticities of demand
with respect to advertising which affects the marginal rate of substitution of
x, for only one commodity and the income-compensated price elasticities of
that one commodity.
7. CONCLUSION
By summing over all individual consumers in a given market, or within a
market stratum, market formulas equivalent to (29) through (46) can easily be
derived and shown to have the same algebraic form. In empirical demand
analysis it is frequently convenient to postulate that aggregate market behavior
of individuals can be represented as the market behavior of a conceptual average
individual; the formulas then purport to describe the consumption behavior of
this statistical individual.
The linear dependence rules (22) and (42) may be employed in empirical
demand analysis in essentially two different ways. First, the linear dependence
may be imposed a priori, i.e., as side conditions which estimated advertising
elasticities of demand are forced to satisfy. One might do this when, for example,
one is chiefly interested in testing the hypothesis that an increase in expenditure
on advertising a commodity, xi , affects only the marginal rate of substitution of
x, for x i . Second, el~sticitiesestimated unconditionally may be "tested" by
substitution ill the appropriate linear dependence relation.
If independent estimates of elasticities of substitution and advertising elasticities can be obtained, it is possible, using the relations (21), for one to estimate
advertising elasticities of marginal rates of substitution and to test hypotheses
such as those represented by formula (29).
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A Theory of Demand with Variable Consumer Preferences
R. L. Basmann
Econometrica, Vol. 24, No. 1. (Jan., 1956), pp. 47-58.
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[Footnotes]
7
A Reconsideration of the Theory of Value. Part I
J. R. Hicks; R. G. D. Allen
Economica, New Series, Vol. 1, No. 1. (Feb., 1934), pp. 52-76.
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11
Complementarity and Interrelations of Shifts in Demand
Oscar Lange
The Review of Economic Studies, Vol. 8, No. 1. (Oct., 1940), pp. 58-63.
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12
A Critical Note on the Definition of Related Goods
S. Ichimura
The Review of Economic Studies, Vol. 18, No. 3. (1950 - 1951), pp. 179-183.
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20
A Critical Note on the Definition of Related Goods
S. Ichimura
The Review of Economic Studies, Vol. 18, No. 3. (1950 - 1951), pp. 179-183.
Stable URL:
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21
A Note on Mr. Ichimura's Definition of Related Good
R. L. Basmann
The Review of Economic Studies, Vol. 22, No. 1. (1954 - 1955), pp. 67-69.
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23
Optimal Advertising and Optimal Quality
Robert Dorfman; Peter O. Steiner
The American Economic Review, Vol. 44, No. 5. (Dec., 1954), pp. 826-836.
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24
Advertising Expenditures: A Suggested Treatment
Norman S. Buchanan
The Journal of Political Economy, Vol. 50, No. 4. (Aug., 1942), pp. 537-557.
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26
A Critical Note on the Definition of Related Goods
S. Ichimura
The Review of Economic Studies, Vol. 18, No. 3. (1950 - 1951), pp. 179-183.
Stable URL:
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26
A Note on Mr. Ichimura's Definition of Related Good
R. L. Basmann
The Review of Economic Studies, Vol. 22, No. 1. (1954 - 1955), pp. 67-69.
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28
Composite Commodities and the Problem of Index Numbers
Wassily Leontief
Econometrica, Vol. 4, No. 1. (Jan., 1936), pp. 39-59.
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