bDepartment of Polymer Engineering Centerfor Polymer Engineering UniversityofAkron Akron, Ohio 44325 Weizmann Institute of Science Rehovot 76100, IsraeI
INTRODUCTION A most typical aspect of human behavior is undoubtedly consumer behavior, which is governed, as is any other side of conduct, by certain characteristics of the individ's personality structure. A study of consumer behavior is, therefore, an interesting test of the adequacy of the mathematical formalism used to represent individual behavior as a whole, and reveals those objective (numerical) personality characteristics that control the consumption of goods and services. At the same time, although economic science has long been studying individual consumption, attention has been focused on economic rather than behavioral aspects of the problem. So the theory has appeared to be an analytical tool for economic situations rather than personality structure and similar problems. Economic aspects of consumption are studied with the aid of various methods ranging from statistical analysis of the empirical data to the development of fairly general mathematical models that are, in fact, not aimed at observations of Deterministic approaches, which are rather popular in economics, are based upon either the preordering of preference^^.^ or utility function^.^-^ It has been shown in this way that many realistic features of consumption can be deduced under very general assumptions, say, of the utility function-the generality of consideration being dictated by the fact that the utility function should remain essentially unknown. Even when deduced from empirical data, the utility function may be known only incompletely as it allows a transformation via an arbitrarily increasing function.R A general theory of mass behavior of the human community elements has been outlined in our papersy--"where the foundation is formed by the maximum principle for a comfort function that covers all sides of human behavior and is, in this sense, a generalization of the utility function used in microeconomics to study consumption problems. Some aspects of group theory have been applied to this problem, seem277
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ingly for the first time,IOJ and they suggest that in reality the comfort function has a unique form that may b e represented as a quadratic-linear form of the extensive behavior variables. A consumer utility function of similar form has been considered in a frequently cited paperIz that includes a proof that the coefficients of the corresponding forms can be derived from the observation data. The present paper is concerned with a qualitative analysis of individual consumer behavior that follows from the general constructive theory of mass behavior.%" We shall not dwell on the results that are inferred from the convexity of the utility function in traditional economic theory, but shall concentrate our attention on those novel nonlocal effects that vividly manifest themselves under the specified comfort function. On the other hand, we will use the formalism developed to trace how personality structure and external conditions make u p the individ's behavior. Yet another important aim of this paper is to demonstrate that a fairly realistic representation of an individ's consumer behavior is achievable even with the simplest nontrivial comfort function. It is worth emphasizing that if such a fact were established in a traditional natural science, this would have been sufficient to fix the corresponding phenomenological function. Note that neither the nonlocal effects mentioned nor the good fit of the model to real human behavior has been discovered in the paper cited above1*; in this paper, attention was concentrated on economic rather than behavioral aspects. The utility function, in particular, was written in such a form that its parameters could not be interpreted as pictorial personality parameters. O n the other hand, the extremal problem was studied in local terms,I2 so that the global shape of the extremal (as a function of the income) remained, in fact, an open question. This uncertainty may account for the misapprehension, common in the literature: about the straightness of the extremal under the quadratic-linear utility function. As will b e shown below, the extremal actually constitutes a continuous polygonal line composed of straight segments. Such an extremal incorporates a fascinating peculiarity of consumer behavior: the dimensionality of the actual consumption space changes when the income is varying. W e will next study in this paper the essential problem of determining how the limitedness of the time resource of an individ (who spends h i d h e r time on the purchase and utilization of goods) affects his/her consumer behavior. This circumstance has also been taken into account in some recent economic-mathematical studies.I3 It has, however, been carried out from such a viewpoint, one drastically different from our own, that the consideration has been rather formal and concerned with whether a solution to the problem exists. Consequently, the theory has failed to reveal those novel qualitative features of consumer behavior that will be shown to arise from the limitedness of individual time resources and that are undoubtedly characteristic of human behavior.
CHARACTER OF CONSUMPTION AS A FUNCTION OF THE INCOME OF THE I N D M D
Let us apply the principle of maximum comfort" to study pure consumer behavior on the part of an individ. For this purpose we fix all nonconsumption
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variables to study what goods and what amounts of these goods are consumed by an individ depending on his income I , provided the preferences are given. When performing this analysis, we disconnect our scheme, which covers the whole society, and consider a single consumer. We neglect the fact that the accessible goods result from the production and consumption of the whole community. Production and consumption, therefore, are themselves certain solutions of the whole-community variational problem. Thus, the specific effects of interactions between the community elements and, in particular, the control effects will be left outside our consideration. We will utilize below the quadratic-linear representation for the individual comfort function,"'J1 which allows us to consider the goods consumed as certain independent sets of goods. We will also assume that among the goods for sale there are preciselyftypes of goods of interest for the consumer under consideration. Let the individ consuming good i in amount X, take the pleasure determined by the partial comfort function of the form
where P,stands for an amplitude of the pleasure (comfort) and X,, stands for the coordinate of the maximum comfort from good i (in fact its saturation limit). Then the consumption of allfgoods, each in amount X,,is associated with the total comfort function S, which, according to references 9-1 1 can be represented as 1
~ ( x=)
PrXr(xmr -
x,);
P,
>0
(1)
1=I
Function (1) characterizes certain aspects of the individ's personality that are associated with consumption. Therefore the quantities P, and X,, (i = 1, . . . ,f ) are the parameters of the individ's personality. We will not consider compelled purchases dictated by circumstances (or control), and will assume, therefore, that
X,; > 0
(i = 1 , 2 , . . . , f )
Besides, consumption vector X = ( X f ,. . . ,X,) should satisfy the obvious condition that the total cost of all the purchases does not exceed the income I of the given individ r
where P, = ( p : , . . . , p [ ) is the price vector with componentsp;. These components represent the price per unit of good i and are assumed to be positive: p ; > 0,
i = 1,2 , . . . f (
(4)
This condition causes no loss of generality because inequality (3) does not impose any constraints on the consumption of goods with zero prices, so that maximization of the corresponding partial comfort functions is trivial. As follows from the very
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sense of the consumption variables Xi, they must obey the conditions Xj20;
i=1,2, ...,f
(5)
Inequalities (3) and ( 5 ) determine a closed domain f+ (f-dimensional simplex) of admissible values of X
af(I)= ( X : P ; X
II,X, 2 O
(i = 1,2,,
. . . ,f))
(6)
and the extremal problem considered is thus the determination of the vector X E a, maximizing S(X)from (1). Let us rewrite expression (1) for S(X)in the form
c f
S(X)=
x f
PjXk; -
i= I
P;(Xm,- Xj)2
(7)
i= I
where the first member gives the value of the absolute maximum of comfort function S(X),and the second one
has a meaning of the discomfort function, which, obviously, minimizes when maximizing S(X)and vice versa. The value of discomfort function Df(X)may be considered as a measure (index) of the individ's displeasure. According to (7) and (8), the hypesurfaces S ( X ) = const or 0, (X)= const constitute a family of similar ellipsoids with the center of symmetry in point X,; = (X,], Xm2, . . . ,X,,,f ). Let us perform the following scale transformation of coordinates and prices: xi =
~;fi, xmi = x,, fi,
pi = P L J P ~ (i = I , 2, . . . , f )
(9)
Using vector notation, we will mark with the superscript k the vectors whose first k components have generally nonzero values while the rest, ( f - k ) components, are equal to zero. In particular, vectors (9) with componentsxi.x,i and P,can be written asxcf), xi I ( ] ), which is indicative of the consistency of inequalities (26). Thus, the whole admissible region of income is, according to inequalities (14) and (26), divided into adjacent segments, each corresponding to a specific dimensionality of the consumption vector. When changing income I from If to zero, the extremal point x(Z) describes a continuous polygonal line starting from the point x,, and terminating in the origin. The sequential segments of this line are the projections of straight line (15) on the hyperplanes [x : p ( k ). x = I,xl = 0 , j = k 1,. . . , f ]with sequentially decreasing dimensionality k = f,f - 1, . . . , 1 (FIG.1). The discomfort function on the extremal constitutes a piecewise quadratic function of income I . As regards the interpretation of the results thus obtained, we shall observe that inequalities (21) can be written via the starting parameters as follows:
+
These inequalities demonstrate that the individ under study denies himself first those goods that are most expensive (Pr,is large), less necessary (Xmiis small), and less pleasing (Pi is small). Such a trend seems to be realistic. The dimensionality variation of the consumption vector with varying income reflects an important and undoubtedly observable qualitative distinction between the expenditure structures of different population groups discriminated from each other only by the income. The decline in variety of consumed goods with decreasing income and the widening of the variety with growing income manifests a familiar tendency. INDMDUAL CONSUMPTION: ACCOUNTING FOR THE TIME-COST (TIME CAPACITY)
The reader is undoubtedly familiar with the sorrowful fact that in real life goods consumed are paid not only with money but also with time. Purchasing goods is time consuming. Time is spent traveling to a store, choosing goods, transporting them from the store, installing and assimilating them (if needed), and, quite often, standing in line or waiting for service. The utilization of goods by the consumer may also take considerable time. We will be interested in the routine or regular consumption that permits the specific time-cost ti required for purchasing and utilization per unit of good i to be introduced. Hence, if good i is consumed in xi amount, the total
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time-costs T z for all goods consumed is f
Tt = c t l X l
(34)
i= 1
It is clear that the value of T t should not exceed the physical maximum Tof the time that the individ under study has at his disposal; that is
T,
I
T
(35)
Because we shall not dwell in the present work upon the detailed time balance of the individ, that is, the distribution of his/her time over various activities, the value of T will b e presumed to be given. Introducing again the normalized quantities x, defined by (29) and setting T,
(36)
= tl/&
one can rewrite relations (34) and (35) in the form
which is similar to financial condition (10). The problem of consumer behavior consists now in minimizing the discomfort function Of (x) =
11 x - x,,,, 11
= min
(38)
under the condition o , p . x _ < I , ~ . xTI]
xEo,,={X:x2
(39)
Let us find out what novel features of the individ's behavior stem from the additional requirement (37). Since the elementary approach of the previous section now which allows us involves difficulties, we shall resort to the Kuhn-Tucker to introduce the Langrangian
L
= 0,-(X)
+ 2t
T 'X
+2
S
' p'X
(40)
and to write the necessary and sufficient conditions for the minimum of 0,-(x) under requirements (39) in the form
grudL - 2p = 0 It follows that X =X,
-
tT - Sp
-k p
(41)
where the Langrangian multipliers p,,t, and s satisfy the requirements p,xl = 0;
x, 2 0, pI2 0
(i = 1,2,.
s ( I - p . x ) = 0, s 2 0, I - p
t(T - 7 . x ) = 0 , t
2
0, T
..,f)
(42)
'X
2
0
(43)
- T ' X
2
0
(44)
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Let us denote byJ, the set of numbers i such thatx, > 0, and byJo the complementary set
J,
=
(i :xi > 01, J, = 1 j :x, = 0)
(45)
We will also set
which, according to relations (42), yields T
'X
=
(T
' X),
,p
'
X
= ( p ' X)+
(47)
Relations (42) show that p, = 0 for i E j + . O n the other hand, if j E Jo, one can assume, generally speaking, that p, > 0. Therefore, equations (41) fall into two groups: x, = x,, - t7, - sp,, i E J, I+ = '7,
+ sp, - xml,j
E J,
(48) (49)
It is clear that the structure of consumptione is determined primarily by the composition of the set J,. As follows from relations (43) and (44), there are the four following cases: (A)s=O,t=O (C) s = 0,t > 0
(B)s>O,t=O (D) s > 0,t > 0
(50)
We will consider them in succession. The simplest case (A) arises if (and only if) 1 2 1 m , ~ p . x , ; T 2T , , ' T . x ,
(51)
where I,, and T,, obviously stand for the monetary-costs and the time-costs required for maximal desired consumption, respectively. Relations (51) are indeed equivalent to conditions (A) because equation (49) at t = 1 and s = 0 is consistent with the requirements II, 2 0 and xml > 0 (see (2)) if (and only if) Jo = 0, that is, if eachx, > 0. In accordance with (42), it follows that p = 0, and therefore the last inequalities in (43) and (44) reduce to (51). Inequalities (51) show that x, E olTin case (A), which implies, in geometrical terms, that pointx,,, is sited below both the hyperplanes T . x = T a n d p ' x = I; that is, point x, lies o n the same side as the origin does. The four different possibilities of mutual arrangement of point x,,, and the hyperplanes T . x = T a n d p ' x = I are shown 2 at an arbitrary section (x,,~,,).The regions corresponding to cases (A), in FIGURE (B), (C), and (D) are marked in the figure by the same letters. Thus, in case (A), condition (39) presents, in fact, no restrictions, and the discomfort function reaches its absolute minimum 0,= 0 in the point x = x,. (See will thus speak of the structure of consumption, referring to the list of goods and services consumed by the individ in nonzero amounts. A more detailed description of consumption, that is, one representing the amounts of each good consumed, will be referred to as the composition of consumption.
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also FIGURE 3, where the subdivision of the first quadrant of the ( I , T ) plane into regions corresponding to the different cases (50) is shown.) In the nontrivial situation in which at least one of inequalities (51) is unfulfilled, point x, is situated outside region A = WIT (FIG. 2), and there arise three qualitatively different alternatives, cases (B), (C), and (D). Considering case (B), we, according to (48)-(50), have XI = x,,
-
PI = sp, - xm, (i € J + , j € J " )
sp,;
and the inequalitiesx, > 0 and k,
2
(52)
0 yield
where i E J , and j E 10.
FIGURE 2. The consumption space, broken into the domains A, B, C, and D,shown at a section by the arbitrary coordinate plane (x,, x , ) . Domain A corresponds to complete satisfaction or comfort. Domain B, where the I hyperplane is situated under the T hyperplane, corresponds the deficit of money. Domain C, where the T hyperplane is over the I hyperplane, corresponds to the time deficit. Domain D,which is tangential to the edge, is associated with the deficit of both money and time.
Thus, at t = 0, each value of s E [O, m) is associated with a unique subdivision of the index set J = (i : 1 I i I f ] into the subsets J , and JO;that is, each value of s is associated with a certain structure of consumption. Inequalities (53) also define an ordering of the goods with respect to the parameter xnzIlp,,which will be given again by relations (21) or (33). This ordering determines a sequence of the embedded subspaces Rlk) ( k = 1 , 2 , . . . , f ) , each R j k ) being a projection of R j k + ' )on the hyperplane xk+I = 0.
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According to (43), the value of parameters in (52) is determined by the relation k
The value of income I under which the consumption vector is k-dimensional is determined by falling of I into semiinterval k
where II = 0 and at k = f one should set x,,f+l = 0 in the right-hand side of (55). Inequalities (55) are exactly the same as inequalities (26), which had been obtained without allowance made for the time capacity. The solutions given by (52) and (53) coincide, of course, in case (B) under consideration, with solutions (27) and (28) of simplified extremal problem (12). It is clear that such a coincidence takes place only if the time resource T is sufficiently large, because in case (B) the last inequality from (44) must be fulfilled, which according to (52) and (54), reduces to
Condition (56) separates from the (I, T) plane a region B of the incomes and time resources that correspond to case (B). This region is situated immediately beyond the polygonal region D (FIG.3). In compliance with (56), this polygonal line is made up off segments of the followingfstraight lines: I) (k = 1 , 2 , . . .
(57)
It is a remarkable fact that segments k and (k + 1) intersect at a point with the Hence, the vertical lines drawn through the vertices of the abscissa equal to I(k+l). polygonal line stratify domain B into stripes (55), each corresponding to a h e d consumption structure and dimensionality (FIG.3). Inequality (56) can also be interpreted in terms of the arrangement of pointx, in the consumption space, which gives rise to a simple geometrical picture. Condition (56) may be readily shown to indicate that pointx, belongs to a region ofx (domain B in FIG.2) in which hyperplane T . x = T lies over the hyperplane p * x = I. (It is self-evident that the projections ofx, on the subspace (xi : i E J+ ] are implied here.) Thus, whenx E B or, equivalently, (I, T) E B, time restriction (37) turns out to be nonessential and the solution coincides with that found in the previous section. Such a situation is characteristic of the individs with sufficiently large time resources but not very high incomes. The important fact is that if for a given individ with a certain income I and time resource Twe havex, E B or (I,T) E b, the same situation (B) will take place when I decreases at T = const or T increases at I = const. The dimensionality of consumption may decrease, however, in the former case.
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In case (C), according to (48)-(SO), we have
and the inequalities
1.1-
T I
1.P
I
FIGURE 3. The (I, T ) plane, broken ipto the following domains: A , complete comfort; B, the money deficit; C , the time deficit; and D,the money and time deficit.
In much the same way as in (53),we determine for each t E (0, m> the setsJ; andJ6: J:
=
(i:t < x m , / T l t ;
J;, = [ j ) : t
> x,,/T,)
(59)
It follows that different goods are ordered, in case (C), in accordance with the values of the parameterx,,/.r,. so that
This ordering is quite similar to that in (21). but with the substitution of the
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time-prices for the monetary ones. The value of parameter t is now given by
The consumption vector (x;:i E J 't] defined by (58) has dimensionality k(xj, > 0, r = 1,2, . . . ,k)if T(k)< T
< T(k+l)
(62)
where similarly to (30)
The case (C) is thus perfectly analogous to case (B), but with time and money interchanged. The validity conditions for relations (57)-(62) are formed by the requirement that point x, should fall into the domain C, which is shown in a section in FIGURE 2. In this domain the hyperplane T . x = Tis situated below the hyperplane p ex = I, which corresponds to the inequality
In the (I,T ) plane case (C) corresponds to the situation in which point (I, T ) falls in the domain I? shown in FIGURE 3, that is, in the domain bounded by the polygonal line
and the straight lines T = T,,, and T = 0. This is stratified into the strips (62), each strip corresponding to a fixed structure of consumption. Thus, if income I is sufficiently high but time resource T is not very large, that is, if I and T obey conditions (64), the corresponding individ when consuming takes into consideration only the time-price of the good. Let us now turn our attention to the most interesting and profound case, that is, case (D), for whichs > 0 and t > 0. In compliance with relations (43) and (44), in this case the isocomfort sphere D ( x ) = const is tangent in the extremal point to the edge [x : TX = T, px = I ] of region wIT, as distinct from cases (B) and (C), in which the sphere is tangent smoothly to one of the hyperplanesp . x = I or T . x = T (FIG.2). Case (D) is evidently associated in the (I,T ) plane with the d o m a i n h (FIG. 3), which should now be stratified into subdomains with different structures of consumption. Because of relations (43) and (44), and because S > 0 and t > 0, we have T=(T'X)+,
r=(p'X),
(66)
that is, the individ spends for consumption all h i d h e r time and money. Inequalities
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> 0 and pi > 0 now yield
t’+s-
Pi
Xmi
Xmi
> I,
jEJ6”)
(67)
Inequalities (67) show that each good i is associated in the ( s , t ) plane with the straight line
FIGURE 4. The plane of the ancillary parameters (s, t ) , broken into subdomains according to a certain structure of consumption. In domain 1 (DMCEF), only the first good is consumed (XI > 0,x2 = 0,x3 = 0). In domain 2 (BMD), the first and third goods are consumed (x > 0,x3 > 0, but x2 = 0). In domain 3 (AMC), we have xs = 0, but XI > 0 and x2 > 0. And domain 4 (OAMB), the consumption of all the three goods is considered.
which cuts a triangular domain from the first octant such that x, > 0 when point (s, t ) is inside the triangle and xl = 0 when the point is outside the triangle (FIG.4). The union of all f of these triangular domains constitutes a set of those values of (s, t ) under which the set J\”) is not empty. The closure of the set just described will be referred to below as the consumption support. The collection o f f straight lines (68) divides the consumption support into polygonal domains, each corresponding to a certain structure of consumption, that is, a certain setJtD’. It can be readily shown that the number of these domains is equal to f + c, where c (c I f - 1) is the number of intersections between straight lines (68) that are sited in the first quadrant of the (s, t ) plane. Indeed, if a given point (s, t )
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lies below straight line z , which is a boundary of a polygonal, good r is consumed in a nonzero amount; otherwise, when the point (s, t ) is sited on or over the line r the consumption of good r is ruled out. It is apparent that each system of prices and personal values is associated with a certain consumption support in combination with its subdivision into polygons. Now let point (s, t ) be inside a certain polygon on the (s, t ) plane, so that the sets J+ and Jo are determined and given by relations (67). The solution of the extremal problem is then determined by formulas (48) with parameters (s, t ) and ( I , T ) related by conditions (66), which readily reduce to the form T = T ( + )- tT: - S ( 7 * p ) ,
I
=
P+)- t(7 . p ) + - sp2,
(69)
As long as point (s, t ) belongs to the fixed polygon, equations (69) determine a linear mapping of the polygon into the ( I , T ) plane. The mapping possesses the determinant
A+
= T:
.p: - (7.p)+
(70)
which is nonzero if vectors 7+ andp, are not colinear, particularly if consumption is not one-dimensional. Under these assumptions the image of the polygon considered is an analogous polygon (that is, a polygon having the same number of vertices) on the ( I , T ) plane. When point (s, t ) passes from the given polygon to an adjacent polygon and intersects the mutual boundary of the polygons determined by equation (68), the coefficients of mapping (69) have a jump. This discontinuity is compensated for by a simultaneous jump of the quantities T + and I + in such a way that mapping (69) remains continuous. To prove this fact it is sufficient to show that any straight line (68) separating two polygons on the (s, t ) plane is mapped into the same straight line on the ( I , 7')plane under both the mappings of the form (69) that are associated with the two adjacent polygons that are separated by the given straight line. Let mapping (69) correspond to the first of the two adjacent polygons, and let the similar mapping T = T(+)'- t ( 7 ; ) 2 - s(7'p') I = I ( + ) ' - t(T'P')+ - S ( p : ) 2
(71)
correspond to the second polygon in which the structure of consumption differs by a single good i. Let us assume, for certainty, that good i is added to the consumption when passing from the first to the second polygon. We then have T(')' = T ( + )+ T i X m i , (7;)2
= T':
f
T,';
( p ;)'
=p:
I(+)' = I ( ' ) +p.X . I mi
+ p;; (p'?'), = ( p 7 ) + +pi?;
(72)
Using relations (72), one can readily rewrite mapping (71) in the form
I = I(-+)- t(7 .p ) + - sp:
+ pixmi Xmi
Xmi
(73)
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From this it immediately follows that the mappings (69) and (71) on the straight line (68) are identical. Thus, equations (69) determine a continuous piecewise linear mapping of the consumption support in the ( I , T ) plane. This mapping, consistent with the subdivision of the consumption support into the f + c polygons on the ( s , r ) plane, determines on the ( I , T ) plane a set of generally polygonal domains, each associated with a certain structure of consumption support. It is important that the consumption on the (s, t ) plane is mapped exactly on the closure of domain b in the ( I , T ) plane (FIGS.3 & 4). Indeed, it can be easily
FIGURE 5. The images o n the ( I , T ) plane-of the conspmption support according to th-e sybdivision shown in FIGURE4.The domains A, B, C, and D and the subdivision of domains B, C, and D into the subdomains with the fixed structures of consumption are also shown. The 4 is mapped into the origin, and the whole domain B is mapped into straight line FE in FIGURE segment OD! There, X I arises (12 > 0) and, simultaneously, x j vanishes when one passes D ' M ' through the segment D ' M ' (from left to right).
checked that mapping (69) at s ' 0 constitutes a parametrical form of the polygonal line (57) that separates domains B and D. Mapping (69) at r = 0 is, in much the same way, equivalent to the set of equations (65) determining the boundary between regions C and D on the ( I , T ) plane. FIGURE 5 shows in the ( T , T ) plane the image of the consumption support and the subdivisions that were displayed in FIGURE4. The images of the corresponding points are marked with the same, but primed, letters. FIGURE 5 also depicts domains A, B, and k and the subdivisions of domains B and C into the strips with the fixed structure of consumption.
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Let us now analyze the change in the individ’s consumption with income I and time resource Tvarying. Let point (I, T ) be situated in a certain polygon belonging to domain D. Determinant (70) of mapping (69) is, under these assumptions, nonzero, which permits equations (69) to be explicitly solved for (s, t). In doing so, and by taking into account equations (58), we obtain the following expressions for nonzero components xi of the consumption vector ( i E J + ) and also for the Lagrange multipliers pi ( j E Jo):
where
AT = T ( + )- T,
AI = I ( + ) - I
(76)
As follows from relations (74) and (76), the trend in the changing of componentsx, (i E J + ) with variations of income I and time resource T is controlled by the signs of the following expressions:
When I is growing at T = const, xi increases if a, > 0 and decreases if a, < 0. Similarly, when T is growing at I = const, one can observe the increase ofxi if PI > 0, and the opposite trend if PI < 0. Good i will be referred to as being normal with respect to money if ai > 0. But if a, < 0, good i will be referred to as being strange with respect to money. In much the same fashion one can introduce the goods that are normal (PI > 0) and strange ( p i < 0) with respect to time. Taking into account that if a/P < a / b ,we find that the inequalities a
a+a
a
-
0, the discomfort diminishes while the comfort grows. Thus, there is nothing unusual in the strange behavior of the individ whose decreased consumption of a certain good results only from its high time-price ( p , / ~ , is relatively small). Such a line of conduct enables the individ to afford more expensive but less time-consuming goods (pleasures, entertainments). The individ finds these goals affordable because of the time saved and the larger sum of money available. It should be noted that such a trend is fairly characteristic of individs who are climbing up the ladder of prosperity. Point ( I , T ) , with a sufficient increase of income, crosses the boundary of the polygon and passes into either domain i. o r an adjacent polygon belonging to domain b. In the former case, the value of s vanishes so that the discomfort function and the consumption vector become independent of I. In the latter case, the passage through the boundary results in a change of the consumption structure. This change reflects the suspension of the consumption of a strange good (with respect to money) and/or the allocation of a part of consumption to a new, normal goodx, that could not have been afforded by the individ previously ( j E Jo in the starting polygon). At this point we should bear in mind that the images of straight lines (68) that divide domain D into polygons are given by an equation of the form x, = 0, when we consider a polygon in which i E J+,or by an equation of the form p, = 0, when a subsequent polygon in which i E Jo is considered. One can immediately find out which of the strange x, vanish and which of the normal x, are added to the consumption under sufficiently increasing income.
SOME REMARKS ON THE CHOICE OF THE QUALITY OF THE GOODS CONSUMED So far we have dealt with behavior variables xi, denoting the amount of good i consumed. These variables can be considered as extensive behavior variables. In actuality, the individ’s behavior consists also in choosing between goods of the same kind i but with a different quality qi.We shall consider q as a normalized parameter15 such that q E [0, 11. It is quite clear that the individual comfort must depend on the quality of the good consumed. This dependence stems from the individ’s tastes and preferences and is, thus, determined by the personality parameters of the individ. Therefore, the
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quality q may also play the part of a behavior variable, and one can introduce the partial comfort function S,(q) with the amplitude vanishing at x + 0 . To make a preliminary estimate on the influence of the goods on the individ’s consumer behavior, we will use below a somewhat simplified approach and assume that consumption of a certain good in amount X is controlled by the partial comfort function S,(X) with the parameters P and X, depending on the good’s quality: P = P ( q ) and X , = X,(q). That is, for a given good one can write S&X) = P ( q ) X ( x m ( q ) - X )
(4 E 10, 11)
(83)
The price of the good under consideration is apparently a function of q ; that is
Pr = Pr(q 1
(84)
The best choice, made from the individ’s viewpoint, of the amount and quality of the good consumed is, obviously, described by the following extremal problem
S = maxS,(X) x.4
(85)
under the condition that
It is safe to assume that
but the trend of X , as a function of q cannot be preassigned in the general case. There are situations when the need for the good falls off as the quality increases, for example, when the growth of quality results in an increase of the life expectancy of the good. Of course the opposite situation may also occur. Let us first consider the case when the individ processes sufficiently large means to purchase the good considered. Then maximization of the comfort function (84) with respect t o X and under fixed q yields
The equation
determines the value q = qmof the quality chosen by the individ for consumption. It is evident that the sufficiently large means mentioned should correspond to the income I that obeys the condition
for all qmsatisfying (89). If the solution qmof equation (89) is not unique, for example, when P ( q ) X i ( q ) = const, one should resort to a more detailed and realistic formulation of the problem,
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such as the one considered in the previous section. As has been shown in the previous section, the individ in a usual situation is lacking either money o r time or both time and money. Therefore, in the more realistic problem formulation, the individ would determine qmeither from the financial condition X m ( q m ) pr(qrn) = min 5 I
(91)
which can be treated as the behavior of an economical individ, or from a condition such as Xm(qm) . t(qrn)= min 5 T
(92)
in which t(q) is the time capacity per unit of the given good with qualityq. It is evident that condition (91) describes a very busy individ who suffers from the time deficit. The more detailed and realistic problem may eventually also give rise to the mixed financial-time criterion for the choice of the quality qm. If we let condition (90) remain unfulfilled, the extremum, according to the Kuhn-Tucker theorem, will be reached at pointX such that
x = I/Pr(q)
(93)
which permits (85) and (86) to be rewritten in the following form:
Let us discuss different situations that arise when the expression in (94) is maximized. For certainty, let Xrn(q)increase with q growing. Then s ( q )constitutes a ratio of two increasing functions, which may either increase or decrease or have a maximum at an internal point. If the increase o f p ( q ) andX,(q) is sufficiently rapid, the first case takes place in which one obviously has qm
=
1
(95)
Such a behavior can be considered as that of a gourmand who prefers goods of the highest quality, the amount of the goods consumed being adjusted to the means at the individ’s disposal. Thus, a gourmand’s behavior will result if the price grows slowly enough as the quality increases. If the growth of p ( q ) and Xrn(q)is sufficiently slow, then function s ( q ) can be readily shown to represent a decreasing function of q whose maximum value is reached at the least value of q: q m = qrnm = 0
(96)
The corresponding behavior is characteristic of a “glutton” who consumes the available maximum of the good and pays no attention to the quality. Finally, the general case in which the point qm of the comfort maximum is an internal point of the segment [0, 11 may be associated with behavior of a moderate individ. Further progress in solving the problem faces difficulties because the dependen-
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cies of the problem parameters on q have not yet been measured and remain unknown at present.
CONCLUSION We have thus shown that the quadratic-linear comfort function introduced in our paper’l and substantiated by some group-theory considerations proves sufficient to represent a wide variety of human behavior in the field of consumption. Although the formal statement of the problem may seem concrete and very simple, the theory produces nontrivial results and the solution can be obtained explicitly in each particular case (in contrast to the general schemes developed in mathematical economics). The results on the strategy and tactics of the consumer under financial and time restrictions, which have been obtained, to the best of our knowledge, for the first time, seem to be fairly realistic. The theory developed may also be helpful in formulating and solving a number of actual problems concerning optimal market policy, such as the problem of whether investments to improve service are reasonable. REFERENCES 1. WOLD,H. O., & L. JUREEN. 1953. Demand Analysis. Wiley. New York, NY. 2. TSUJIMURA, K. 1960. Family budget data and market analysis. Paper presented at the International Statistical Institute’s Tokyo conference. May 30-June 9. 3. SAMUELSON, P. A. 1948. Foundations of Economic Analysis. Cambridge, MA. 4. EKELAND, I. 1979. ElCments D’Economic MathCmatique. Hermann Collection. Paris. 5. ALLEN, R. G. D. 1960. Mathematical Economy. 2nd edit. Macmillan. London. 6. MALINVAUD, E. 1971. Lefons De Theorie Microeconomique. Dunod. Paris. 7. ASHMANOV, S. A. 1984. Introduction to Mathematical Economics (in Russian). Nauka. Moscow. 8. VOLKONSKII, V. A. 1963. On the objective mathematical characteristics of personal consumption. In Economic-Mathematical Methods (in Russian). Vol. I: 201-240. Izdatelstvo Akademii Nauk SSSR. Moscow. 9. CHERNYAK, Y. B., A. 1. LEONOV & A. Y. LERNER. 1985. On the theory of mass behavior in developed human communities. Ann. N.Y. Acad. Sci. 452: 44-53. 10. CHERNYAK, Y. B., A. I. LEONOV & A. Y. LERNER. 1990. On a theory of developed human communities: Individual behavior in the case of performing labor activity and consumption. J. Acad. Proc. Sov. Jewry 2: 37-103. 11. CHERNYAK, Y. B., A. I. LEONOV & A . Y. LERNER. 1992. Toward the constructive theory of human social behavior. I. General principles. Ann. N.Y. Acad. Sci. This volume. 12. SLUTSKY, E. 1915. Sulla teoria del bilancio del consumatore. G. Econ. Riv. Stat. No. 51: 1-26. 13. DANILOV, V. I. 1984. Non-Walras equilibrium and the generalized Gale lemma. In Mathematical Economics and Extremal Problems (in Russian): 15-23. Nauka. Moscow. 14. ANDRONOV, A. A., A. A. Vim & S. E. KHAIKIN.1981. Theory of Oscillations (in Russian). Nauka. Moscow. 15. AZDOLOV,G. P. & E. P. RAIKHMAN. 1973. On Qualimetry (in Russian). Izdatelstvo Standartov. Moscow.
