A New Approach to Consumer Theory Author(S): Kelvin J

A New Approach to Consumer Theory Author(s): Kelvin J. Lancaster Source: The Journal of Political Economy, Vol. 74, No. 2 (Apr., 1966), pp. 132-157 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/1828835 Accessed: 09/12/2010 04:36

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http://www.jstor.org A NEW APPROACH TO CONwXSUMERTHEORY*

KELVIN J. LANCASTER Johns Hopkins University

I. THE CURRENT STATUS state of the theory of consumer behav- OF CONSUMER THEORY ior. All intrinsic properties of particular goods, those properties that make a dia- TIHEm theory of consumer behavior in deterministic situations as set out mond quite obviously something dif- from a by, say, Debreu (1959, 1960) or ferent loaf of bread, have been Uzawa (1960) is a thing of great aesthetic omitted from the theory, so that a con- beauty, a jewel set in a glass case. The sumer who consumes diamonds alone is product of a long process of refinement as rational as a consumer who consumes from the nineteenth-century utility theo- bread alone, but one who sometimes con- rists through Slutsky and Hicks-Allen to sumes bread, sometimes diamonds (cete- the economists of the last twenty-five ris paribus, of course), is irrational. Thus, years,1 it has been shorn of all irrelevant the only property which the theory can postulates so that it now stands as an build on is the property shared by all example of how to extract the minimum goods, which is simply that they are of results from the minimum of assump- goods. tions. Indeed, we can continue the argument To the process of slicing away with further, since goods are simply what con- Occam's razor, the author made a small sumers would like more of; and we must contribution (I 957). This brought forth a be neutral with respect to differences in reply by Johnson (1958) which suggest- consumer tastes (some consumers might ed, somewhat tongue-in-cheek, that the like more of something that other con- determinateness of the sign of the substi- sumers do not want), that the ultimate proposition is that goods are what are tution effect (the only substantive result thought of as goods. of the theory of consumer behavior) In spite of the denial of the relevance could be derived from the proposition of intrinsic properties to the pure theory, that goods are goods. there has always been a subversive un- Johnson's comment, on reflection, dercurrent suggesting that economists would seem to be almost the best sum- continue to take account of these prop- mary that can be given of the current erties. Elementary textbooks bristle with * The author wishes to acknowledge helpful substitution examples about butter and comments from various sources, including Gary margarine, rather than about shoes and Becker, Harry Johnson, and colleagues and students at Johns Hopkins University, especially Carl Christ, ships, as though the authors believed F. T. Sparrow, William Poole, C. Blackorby, T. that there was something intrinsic to Amemiya, and T. Tsushima. butter and margarine that made them 1 The American Economic Association Index of good substitutes and about automobiles Economic Journals lists 151 entries under category 2.111 (utility, demand, theory of the household) and gasoline that made them somehow over the period 1940-63. intrinsically complementary. Market re- 132 NEW APPROACHTO CONSUMERTHEORY 133 searchers, advertisers, and manufactur- tempts to deal with complementarity, ers also act as though they believe that substitution, independence, and neutral knowledge of (or belief in) the intrinsic want associationswithin the convention- properties of goods is relevant to the way al framework. consumers will react toward them. The clearest case of conflict between H. A NEW APPROACH a belief that goods do have intrinsic prop- Like many new approaches, the one erties relevant to consumer theory but set out in this paper draws upon several that they are not taken into account has elements that have been utilized else- been the long search for a definition of where. The chief technical novelty lies "intrinsic complementarity." The search in breaking away from the traditional was successful only where Morishima approach that goods are the direct ob- (1959) turned from traditional theory to jects of utility and, instead, supposing an approach somewhat similar to that of that it is the propertiesor characteristics the present paper. of the goods from which utility is derived. Perhaps the most important aspects We assume that consumptionis an ac- of consumer behavior relevant to an tivity in which goods, singly or in com- economy as complex as that of the United bination, are inputs and in which the States are those of consumer reactions to output is a collection of characteristics. new commodities and to quality varia- Utility or preference orderings are as- tions. Traditional theory has nothing to sumed to rank collections of characteris- say on these. In the case of new com- tics and only to rank collections of goods modities, the theory is particularly help- indirectly through the characteristics less. We have to expand from a commod- that they possess. A meal (treated as a ity space of dimension n to one of dimen- single good) possessesnutritional charac- sion n + 1, replacing the old utility teristics but it also possesses aesthetic function by a completely new one, and characteristics, and different meals will even a complete map of the consumer's possess these characteristicsin different preferences among the n goods provides relative proportions.Furthermore, a din- absolutely no information about the new ner party, a combination of two goods, preference map. A theory which can a meal and a social setting, may possess make no use of so much information is a nutritional, aesthetic, and perhaps in- remarkably empty one. Even the tech- tellectual characteristics different from nique of supposing the existence of a the combination obtainable from a meal utility function for all possible goods, in- and a social gathering consumed sepa- cluding those not yet invented, and re- rately. garding the prices of nonexistent goods In general-and the richnessof the ap- as infinite-an incredible stretching of proach springs more from this than from the consumers' powers of imagination- anything else-even a single good will has no predictive value. possess more than one characteristic,so Finally we can note the unsuitability that the simplest consumption activity of traditional theory for dealing with will be characterized by joint outputs. many of the manifestly -important aspects Furthermore, the same characteristic of actual relationships between goods (for example, aesthetic properties) may and consumers in I. F. Pearce's (1964) be included among the joint outputs of recent heroic but rather unsuccessful at- many consumption activities so that 134 KELVIN J. LANCASTER goods which are apparently unrelated in of utility. The theory set out here goes certain of their characteristics may be much further than these ideas. Multiple related in others. characteristics, structurally similar to We shall assume that the structure we those of the present paper but confined have interposed between the goods them- to a particular problem and a point util- selves and the consumer's preferences is, ity function, are implicit in the classical in principle, at least, of an objective kind. "diet problem" of Stigler (1945), and That is, the characteristics possessed by multidimensioned utilities have been a good or a combination of goods are the used by workers in other fields, for ex- same for all consumers and, given units ample, Thrall (1954). The third assump- of measurement, are in the same quanti- tion, of activities involving complemen- ties,2 so that the personal element in con- tary collections of goods, has been made sumer choice arises in the choice between by Morishima (1959) but in the context collections of characteristics only, not in of single-dimensioned utility. the allocation of characteristics to the A variety of other approaches with goods. The objective nature of the goods- similarities to that of the present paper characteristics relationship plays a cru- occur scattered through the literature, cial role in the analysis and enables us to for example, in Quandt (1956), or in distinguish between objective and pri- Becker (1965), or in various discussions vate reactions to such things as changes of investment-portfolio problems. These in relative prices. are typically set out as ad hoc approaches The essence of the new approach can to particular problems. Perhaps the most be summarized as follows, each assump- important aspect of this paper is that the tion representing a break with tradition: model is set out as a general replacement 1. The good, per se, does not give util- of the traditional analysis (which remains ity to the consumer; it possesses charac- as a special case), rather than as a special teristics, and these characteristics give solution to a special problem. rise to utility. It is clear that only by moving to mul- 2. In general, a good will possess more tiple characteristics can we incorporate than one characteristic, and many char- many of the intrinsic qualities of indi- acteristics will be shared by more than vidual goods. Consider the choice be- one good. tween a gray Chevrolet and a red Chev- 3. Goods in combination may possess rolet. On ordinary theory these are either characteristics different from those per- the same commodity (ignoring what may taining to the goods separately. be a relevant aspect of the choice situa- A move in the direction of the first tion) or different commodities (in which assumption has already been made by case there is no a priori presumption that various workers including Strotz (1957, they are close substitutes). Here we re- 1959) and Gorman (1959), with the gard them as goods associated with satis- "utility tree" and other ideas associating faction vectors which differ in only one a particular good with a particular type component, and we can proceed to look in much the same way 2 Since the units in which the characteristics are at the situation measured are arbitrary, the objectivity criterion as the consumer-or even the economist, relating goods and characteristics reduces to the in private life-would look at it. requirement that the relative quantities of a par- is forever being ticular characteristic between unit quantities of any Traditional theory pair of goods should be the same for all consumers. forced to interpret quite common real-life NEW APPROACH TO CONSUMER THEORY 135 happenings, such as the effects of adver- hold for all individuals, the coefficients tising in terms of "change of taste," an ajk being determined by the intrinsic entirely non-operational concept since properties of the goods themselves and there is no way of predicting the relation- possibly the context of technological ship between preference before and after knowledge in the society. the change. The theory outlined here, al- 2. More heroically, we shall assume though extremely rich in useful ways of that each consumption activity produces thinking about consumer behavior, may a fixed vector of characteristics3 and that also be thought to run the danger of add- the relationship is again linear, so that, ing to the economist's extensive collec- if zi is the amount of the ith characteris- tion of non-operational concepts. If this tic were true, it need not, of course, inhibit Z = bikyk, (3) the heuristic application of the theory. k Even better, however, the theory implies or predictions that differ from those of tra- z= By. (4) ditional theory, and the predictions of Again, we shall assume that the coeffi- the new approach seem to fit better the cients bik are objectively determined-in realities of consumer behavior. principle, at least-for some arbitrary III. A MODEL OF CONSUMER BEHAVIOR choice of the units of zi. 3. We shall assume that the individual To obtain a working model from the possesses an ordinal utility function on ideas outlined above, we shall make some characteristics U(z) and that he will assumptions which are, on balance, nei- choose a situation which maximizes U(z). ther more nor less heroic than those made U(z) is provisionally assumed to possess elsewhere in our present economic theo- the ordinary convexity properties of a rizing and which are intended to be no standard utility function. more and no less permanent parts of the The chief purpose of making the as- theory. sumption of linearity is to simplify the 1. We shall regard an individual good problem. A viable model could certainly or a collection of goods as a consumption be produced under the more general set activity and associate a scalar (the level of relationships of the activity) with it. We shall assume that the relationship between the level Fk(151X)= 01 k = ... m (5) of activity k, and the goods consumed Yk, The model could be analyzed in a similar in that activity to be both linear and ob- way to that used by Samuelson (1953b) jective, so that, if Xjis the jth commodity and others in analyzing production, al- we have though the existence of much jointness Xj= aikyk, (1) among outputs in the present model pre- k sents difficulties. and the vector of total goods required IThe assumption that the consumption tech- for a given activity vector is given by nology A, B is fixed is a convenience for discussing those aspects of the model (primarily static) that x=Ay. (2) are the chief concern of this paper. The consequences of relaxing this particular assumption is only one of many possible extensions and expansions of the Since the relationships are assumed ideas presented and are discussed by the author objective, the equations are assumed to elsewhere (Lancaster, 1966). 136 KELVIN J. LANCASTER In this model, the relationshipbetween question, does he prefer characteristics the collections of characteristics avail- collection z1 or Z2, does have such an able to the consumer-the vectors z- answer. which are the direct ingredients of his If we take the standard choice situa- preferencesand his welfare, and the col- tion facing the consumer in a free mar- lections of goods available to him-the ket, with a linear budget constraint, this vectors x-which represent his relation- situation, in our model, becomes: ship with the rest of the economy, is not direct and one-to-one, as in the tradi- Maximize U(z) tional model, but indirect, through the subject to px < k activity vector y. with z= By Consider the relationships which link z and x. These are the equation systems: x= Ay x = Ay (2) and z = By (4). Supposethat x, y, z > 0. there are r characteristics, m activities, and n goods. Only if r = m = n will This is a non-linear program of an in- there be a one-to-one relationship be- tractable kind. The problem of solution tween z and x. In this case both the B need not worry us here, since we are in- and A matrixes are square (the number terested only in the properties of the so- of variables equals the number of equa- lution. tions in both sets of equations) and we IV. THE SIMPLIFIED MODEL can solve for y in terms of x, y =A-x, giving z = BA-'x. U(z) can be written We shall simplify the model in the directly and unambiguouslyas a function initial stages by supposing that there is u(x). Otherwisethe relationsare between a one-to-one correspondence between vectors in spaces of different dimensions. goods and activities so that we can write Consider some x* in the case in which the consumer-choice program in the sim- m > n: equation (2) places only n re- pler form strictions on the m-vector y, so that y Maximize U(z) can still be chosen with m - it degrees subject to px ? k of freedom.If r < m, then there are m - r degrees of freedom in choosing y, given with z=Bx some z, but whether the ultimate rela- z, x ! 0. tionship gives several choices of z for a given x, or several x for a given z, and This is still, of course, a non-linear pro- whether all vectors z are attainable, de- gram, but we now have a single step be- pends on the relationshipsbetween r, m, tween goods and characteristics. and n and the structures of the matrixes The model consists of four parts. There A, B. In general, we will expect that the is a maximand U(z) operating on charac- consumermay face a choice among many teristics, that is, U is defined on charac- paths linkinggoods collectionswith char- teristics-space (C-space). The budget acteristics collections. The simple ques- constraint px ? k is defined on goods- tion asked (in principle) in the tradition- space (G-space). The equation system al analysis-does a particular consumer z = Bx represents a transformation be- prefer collection x1 or collection x2-no tween G-space and C-space. Finally, longer has a direct answer, although the there are non-negativity constraints z, NEW APPROACH TO CONSUMER THEORY 137 x > 0 which we shall assume to hold ini- economy, and the behavior of consumers. tially, although in some applications and Certain properties of the transforma- with some sign conventions they may not tions between G- and C-space follow always form part of the model. immediately from the fact that B is a In traditional consumer analysis, both matrix of constants, and the transforma- the budget constraint and the utility tion z = Bx is linear. These can be stated function are defined on G-space, and we as follows, proof being obvious. can immediately relate the two as in the a) A convex set in G-space will transform ordinary textbook indifference-curve dia- into a convex set in C-space, so that the budget gram. Here we can only relate the utility constraint px < k, x _ 0 will become a convex function to the budget constraint after constraint on the z's. inverse transformation will not neces- defined on the same b) An both have been sarily exist, so that an arbitrary vector z in space. We have two choices: (1) We can C-space may have no vector x in G-space cor- transform the utility function into G- responding to it. space and relate it directly to the budget c) Where an inverse transformation does constraint; (2) we can transform the exist from C-space into G-space, it will trans- sets into convex sets so that, for and relate form convex budget constraint into C-space any set of z's which do have images in G-space, it directly to the utility function U(z). the convexity of the U function on the z's will Each of these techniques is useful in be preserved in relation to the x's. different circumstances. In the case of The properties are sufficient to imply the first, we can immediately write U(z) that utility maximization subject to con- = U (Bx) = u(x), so we have a new straint will lead to determinate solutions utility function directly in terms of for consumer behavior. goods, but the properties of the function u(x) depend crucially on the structure of V. THE STRUCTURE OF CONSUMPTION the matrix B and this, together with the TECHNOLOGY constraints x >? and z = Bx > 0 give a The consumption technology, which is situation much more complex than that important a determinant of consumer of conventional utility maximization. as of The second technique again depends cru- behavior as the particular shape the cially on the structure of B and again utility function, is described fully only by will generally lead to a constraint of a the A and B matrixes together, but cer- more complex kind than in conventional tain types of behavior can be related to analysis. more generalized descriptions of the tech- The central role in the model is, of nology. We shall distinguish broadly be- course, played by tche transformation tween structural properties of the tech- equation z = Bx and the structure and nology, such as the relationship between qualitative4 properties of the matrix B. the number of rows and columns of B Most of the remainder of the paper will and/or A and whether A, B are decom- be concerned with the relationship be- posable, and qualitative properties, such tween the properties of B, which we can as the signs of the elements of A and B. call the consumption technology5 of the The leading structural property of the 4 "Qualitative" is used here in a somewhat more 5 If the relationship between goods and activities general sense than in the author's work on the prop- is not one-to-one, the consumption technology con- -erties of qualitatively defined systems for which see sists of the two matrixes B, A, as in the technology Lancaster (1962, 1965). of the Von Neumann growth model. 138 KELVIN J. LANCASTER consumption technology is the relation- r - n characteristics being determined ship between the number of characteris- from the remaining r - n equations and tics (r) and the number of activities (m), the goods vector x corresponding to z. In that is, between the number of rows and this case, it is generally most useful to columns of B. It will be assumed that B analyze consumer behavior by transform- contains no linear dependence, so that its ing the utility function into G-space, rank is the number of rows or columns, rather than the budget constraint into whichever is less. We shall assume, unless C-space. What does the transformed util- otherwise stated, a one-to-one relation- ity function look like? ship between goods and activities. As shown in the Appendix, the utility 1. The number of characteristics is function transformed into G-space re- equal to the number of goods. In this tains its essential convexity. An intuitive case, there is a one-to-one relationship way of looking at the situation is to note between activities vectors and character- that all characteristics collections which istics vectors. We have z = Bx, x = are actually available are contained in an B-1z. If B is a permutation of a diagonal n-dimensional slice through the r-dimen- matrix then there is a one-to-one rela- sional utility function, and that all slices tionship between each component of z through a convex function are themselves and each component of y, and the model convex. The transformation of this n-di- becomes, by suitable choice of units, ex- mensional slice into G-space preserves actly the same as the traditional model. this convexity. If B is not a diagonal permutation, the For investigation of most aspects of objects of utility are composite goods consumer behavior, the case in which the rather than individual goods, and the number of characteristics exceeds the model has some important differences number of goods-a case we may often from the conventional analysis. Note wish to associate with simple societies- how specialized is the traditional case in can be treated along with the very spe- relation to our general model. cial case (of which conventional analysis If B is a diagonal permutation but is a special subcase) in which the number of there is not a one-to-one relationship be- characteristics and goods is equal. In tween activities and goods so that A is other words, given the consumption tech- not a diagonal permutation, we have a nology, we concern ourselves only with model similar to that of Morishima the particular n-dimensional slice of the r-dimensional utility function implied by (1959). 2. The number of characteristics is that technology6 and, since the slice of the has greater than the number of goods. In utility function the same general properties as any n-dimensional this case, the relationships Bx = z con- utility function, we can proceed as if the utility tain more equations than variables xi so function was defined on only n charac- that we cannot, in general, find a goods teristics. vector x which gives rise to an arbitrarily specified characteristics vector z. We can 6 Assuming no decomposability or singularities in take a basis of any arbitrarily chosen n the consumption technology matrix B, then, if z,, is the vector of any n components of z and Bn, the cor- characteristics and consider the reduced responding square submatrix of B, the subspace of n X n system 11= a, which gives a one- C-space to which the consumer is confined, is that to-one relationship between n character- defined by zr-n = Br-nBn7 Zn, where zr-n, Br-n are the vector and corresponding submatrix of B con- istics and the n goods, with the remaining sisting of the components not included in zn, Bn- NEW APPROACH TO CONSUMER THEORY 139

3. In the third case, in which the num- constraint px = k, can determine a char- ber of goods exceeds the number of characteristics frontier consisting of all z such acteristics, a situation probably descrip- that the value of the above program is tive of a complex economy such as that just equal to k. There will be a determi- of the United States, there are properties nate goods vector associated with each of the situation that are different from point of the characteristics frontier. those of the two previous cases and from As in the previous case, it is easy to the conventional analysis. show that the set of characteristics vec- Here, the consumption technology, tors in C-space that are preferred or in- z = Bx, has fewer equations than vari- different to z transforms into a convex ables so that, for every characteristics set in G-space if it is a convex set in vector there is more than one goods vec- C-space; it is also easy to show that the tor. For every point in his characteristics- set of z's that can be obtained from the space, the consumer has a choice between set of x's satisfying the convex constraint different goods vectors. Given a price px ? k is also a convex set. The charac- vector, this choice is a pure efficiency teristics frontier is, therefore, concave to

I ;

zip ' ~~~~~~zF 1 'I :z I .1~~~~~~~~~~

FIG. I

choice, so that for every characteristics the origin, like a transformation curve. vector the consumer will choose the most For a consumption technology with four efficient combination of goods to achieve goods and two characteristics, the fron- that collection of characteristics, and the tier could have any of the three shapes efficiency criterion will be minimum cost. shown in Figure 1. Note that, in general, The efficiency choice for a characteris- if B is a positive matrix, the positive tics vector z* will be the solution of the orthant in G-space transforms into a canonical linear program cone which lies in the interior of the positive orthant in C-space, a point illus- Minimize px trated in the diagrams. subject to Bx = A* A consumer's complete choice subject x> 0. to a budget constraint px ? k can be considered as consisting of two parts: Since this is a linear program, once we a) An efficiency choice, determining the have the solution x* for some z*, with characteristics frontier and the associat- value k*, we can apply a scalar multiple ed efficient goods collections. to fit the solution to any budget value k b) A private choice, determining which and characteristics vector (k/k*)z*. By point on the characteristics frontier is varying z*, the consumer, given a budget preferred by him. 140 KELVIN J. LANCASTER The efficiency choice is an objective tive goods prices may leave activity not a subjective choice. On the assump- prices, and the consumer's choice situation that the consumption technology is tion, unchanged. objective, the characteristics frontier is In most of the succeeding analysis, we also objective, and it is the same for all will be concerned with the B matrix and consumers facing the same budget con- the relationship between activities and straint. Furthermore the characteristics characteristics, since this represents the frontier is expanded or contracted linear- most distinctive part of the theory. ly and proportionally to an increase or decrease in income, so that the frontier VI. THE EFFICIENCY SUBSTITUTION EF- has the same shape for all consumers FECT AND REVEALED PREFERENCE facing the same prices, income differences At this stage, it is desirable to examine simply being reflected in homogeneous the nature of the efficiency choice so that expansion or contraction. we can appreciate the role it plays in the We should note that, if the consump- consumer behavior implied by our model. tion technology matrix has certain spe- Consider a case in which there are two cial structural properties, we may obtain characteristics, a case that can be illus- a mixture of the above cases. For exam- trated diagrammatically, and, say, four ple, a matrix with the structure activities. The activities-characteristics portion [OB2] of the consumption technology is defined by the two equations where B1 is an (s X k) matrix and B2 is an (r - s) X (n - k) matrix, partitions the 1- bily, + b12Y2 + bl3Y3 + b14y4 technology into two disconnected parts, (6.1) + + b23y,+ b24y4. one relating s of the characteristics to k of z2 b2ly1 b22Y2 the goods, the other separately relating With activity 1 only, the characteris- r -s of the characteristics to n - k of the tics will be obtained in proportion, 1ll/b2l goods. We can have s > k and r-s < (the ray labeled 1 in Fig. 2). Similarly n-k giving a mixed case. with activities 2, 3, 4, one at a time, Dropping the assumption of a one-to- characteristics will be obtained in pro- one relationship between goods and activities does not add greatly to the diffi- portions b12/b22, b13/b23, b14/b24, respec- culties of the analysis. We have, as part tively, corresponding to the rays 2, 3, 4 of the technology, x Ay, so that the in the diagram. budget constraint px ? k can be written We are given a budget constraint in immediately as pAy ? k. The goods goods space of the form ipixi_ k. If prices transform directly into implicit there is a one-to-one correspondence be- activity prices q = pA. Interesting cases tween goods and activities, the prices of arise, of course. If the number of goods the activities are given by pi. If there is is less than the number of activities, then not a one-to-one relationship, but a not all q's are attainable from the set of goods-activities portion of the consump- p's; and if the number of goods exceeds tion technology the number of activities, different p vec- = ai2y2 ai4y4 tors will correspond to the same q vector. xi ailyi + + ai3y3 + (6.2) This implies that certain changes in rela- i = 1 .. e NEW APPROACH TO CONSUMER THEORY 141 then the budget constraint can be trans- are efficient, giving the characteristics formed immediately into characteristics frontier, while combinations 1 and 3, 2 space and 4, or 1 and 4 are inefficient. Suppose that the consumer chooses (E piaii) yi+( IPiai2) y2 characteristics in the combination represented by the ray z*, giving a point E* (6.3) on the frontier. Now suppose that rela- +( -(v piai3) Y3+( E Pai4)Y4? k tive prices change: in particular, that the price of activity 2 rises so that, with in- wherethe compositeprices qj= =ipja, come still at k, the point E2 moves in- j= 1 . . 4 representthe prices of each ward on ray 2. If the movement is small

Li)

E l

II 2

E&3) C~~~~~~~~~~~~~~~~~

FIG. 2

activity. The number of goods in relation enough, the characteristics frontier con- to the number of activities is irrelevant tinues to have a corner at E2, and the at this stage, since each activity has a consumer will continue to obtain charac- unique and completely determined price teristics in proportion z* by a combina- qj, given the prices of the goods. tion of activities 1 and 2. If income is Given q1,q2, q3, q4, and k, the maximum adjusted so that the new frontier goes attainable level of each activity in isola- through E*, the consumer will use the tion can be written down (corresponding same activities in the same proportions to the points E1, E9, E3, E4 in Fig. 2,) and as before. the lines joining these points represent If the price of activity 2 rises suffi- combinations attainable subject to the ciently, however, the point E2 will move budget constraint. In the diagram it has inward past the line joining E1 and E3 been assumed that prices are such that to E2'. Combinations of 1 and 2 and of combinations of 1 and 2, 2 and 3, 3 and 4 2 and 3 are now inefficient combinations 142 KELVIN J. LANCASTER of activities, their place on the efficiency ing of activities will not, in general, frontier being taken by a combination of occur7) but does not require that the 1 and 3. The consumer will switch from number of goods exceed the number of a combination of activities 1 and 2 to a characteristics. In fact, with two goods, combination of 1 and 3. two characteristics, and three activities, Thus there is an efficiency substitution the effect may occur. With two goods, effect which is essentially a switching two characteristics and one hundred ac- effect. If price changes are too small to tivities (well spread over the spectrum), cause a switch, there is no efficiency sub- an almost smooth efficiency substitution stitution effect: If they are large enough, effect would occur. the effect comes from a complete switch Since the efficiency substitution effect from one activity to another. implies that consumers may change goods The manifestation of the efficiency collections as a result of compensated substitution effect in goods space de- relative price changes, simply in order to pends on the structure of the A (goods- obtain the same characteristics collection activities) matrix. There are two polar in the most efficient manner, it is obvious cases: that the existence of substitution does a) If there is a one-to-one relationship not of itself either require or imply con- between goods and activities, the effi- vexity of the preference function on char- ciency substitution effect will result in a acteristics. In other words, the axiom of complete switch from consumption of revealed preference may be satisfied even one good to consumption of another. if the consumer always consumes charac- This might be regarded as typical of sit- teristics in fixed proportions (and possi- uations involving similar but differen- bly even if the consumers had concave tiated products, where a sufficiently preferences), so that the "revelation" large price change in one of the products may be simply of efficient choice rather will result in widespread switching to, or than convexity. A formal proof is given away from, the product. in the Appendix. b) If there is not a one-to-one relationship between goods and activities and, in VI]. OBJECTIVE AND SUBJECTIVE particular, if all goods are used in all CHOICE AND DEMAND THEORY activities, the efficiency substitution ef- In an economy or subeconomy with a fect will simply result in less consump- complex consumption technology (many tion of a good whose price rises, not a goods relative to characteristics), we complete disappearance of that good have seen that there are two types of from consumption. If all cakes require substitution effect: eggs but in different proportions, a rise 1. Changes in relative prices may rein the price of eggs will cause a switch sult in goods bundle I becoming an in- from egg-intensive cakes to others, with a decline in the consumption of eggs, but 7This is a somewhat imprecise statement in that, if the B matrix is partitionable into disconnected not to zero. subtechnologies, for some of which the number of The existence of an efficiency substitu- activities exceeds the number of characteristics and tion effect depends, of course, on the for others the reverse, an efficiency-substitution effect may exist over certain groups of activities, number of activities exceeding the num- although the number of activities is less than the ber of characteristics (otherwise switch- number of characteristics over-all. NEW APPROACH TO CONSUMER THEORY 143 efficient method of attaining a given bun- dependent-either may occur without dle of characteristics and being replaced the other in certain circumstances but by goods bundle II even when the char- in general we will expect them both to acteristics bundle is unchanged. take place and hence that their effects 2. Changes in relative prices, with or will be reinforcing, if we are concerned without causing efficiency substitutions with a complex economy. Thus, the con- as in type 1, may alter the slope of the sumer model presented here, in the con- characteristics frontier in a segment rele- text of an advanced economy, has, in a vant to a consumer's characteristics sense, more substitution than the tradi- choice. The change in the slope of the tional model. Furthermore, since part of frontier is analogous to the change in the the total substitution effect arises from budget line slope in the traditional case objective, predictable, and income-dis- and, with a convex preference function, tribution-free efficiency considerations, will result in a substitution of one char- our confidence in the downward slope of acteristics bundle for another and, hence, demand curves is increased even when of one goods bundle for another. Note income redistribution takes place. that, even with smoothly convex prefer- Since it is well known that satisfaction ences, this effect may not occur, since of the revealed preference axioms in the the consumer may be on a corner of the aggregate (never guaranteed by tradition- polyhedral characteristics frontier, and al theory) leads to global stability in thus his characteristics choice could be multimarket models (see, for example, insensitive to a certain range of slope Karlin, 1959), the efficiency substitution changes on the facets. effect increases confidence in this sta- The first effect, the efficiency substitu- bility. tion effect, is universal and objective. In a simple economy, with few goods Subject to consumer ignorance or ineffi- or activities relative to characteristics, ciency,' this substitution effect is inde- the efficiency substitution effect will be pendent of the shapes of individual con- generally absent. Without this reinforce- sumers' preference functions and hence ment of the private substitution effect, we of the effects of income distribution. would have some presumption that per- The second effect, the private substi- verse consumer effects ("Giffen goods," tution effect, has the same properties, in backward-bending supply curves) and general, as the substitution effect in tra- lower elasticities of demand would char- ditional theory. In particular, an aggre- acterize simple economies as compared gately compensated relative price change with complex economies. This seems to combined with a redistribution of income be in accord with at least the mythology may result in no substitution effect in of the subject, but it is certainly empiri- the aggregate, or a perverse one. cally verifiable. On this model, consump- These two substitution effects are in- tion technology as well as income levels 8 One of the properties of this model is that it differentiate consumers in different so- gives scope for the consumer to be more or less effi- cieties, and we would not necessarily cient in achieving his desired characteristics bundle, although we will usually assume he is completely expect a poor urban American to behave efficient. This adds a realistic dimension to consumer in his consumption like a person at the behavior (traditional theory never permits him to level in a econ- be out of equilibrium) and gives a rationale for the same real-income simple Consumers' Union and similar institutions. omy. 144 KELVIN J. LANCASTER

VIII. COMMODITY GROUPS, SUBSTITUTES, erty that efficiency substitution effects COMPLEMENTS will occur only for relative price changes In a complex economy, with a large within the group and will be unaffected number of activities and goods as well as by changes in the prices of other goods. characteristics, and with a two-matrix If the utility function on characteristics (A, B) consumption technology, it is ob- has the conventional properties, there vious that taxonomy could be carried out may, of course, be private substitution almost without limit, an expression of effects for goods within the group when the richness of the present approach. Al- the prices of other goods changes. For an though an elaborate taxonomy is not intrinsic commodity group, the whole of very useful, discussion of a few selected the objective analysis can be carried out types of relationships between goods can without reference to goods outside the be of use. One of the important features group. of this model is that we can discuss rela- Goods from different intrinsic com- tionships between goods, as revealed in modity groups can be regarded as intrin- the structure of the technology. In the sically unrelated, goods from the same conventional approach, there are, of group as intrinsically related. course, no relationships between goods If, within a group, there are two activ- as such, only properties of individual's ities, each in a one-to-one relationship preferences. with a different good, and if the bundles The simplest taxonomy is that based of characteristics derived from the two on the zero entries in the technology ma- goods differ only in a scalar (that is, have we can regard the trixes. It may be that both matrixes A, B identical proportions), are almost "solid," in which case there two goods in question as intrinsic perfect is little to be gained from a taxonomic substitutes. If the associated characteris- are are approach. If, however, the B matrix tics bundles similar, the goods close substitutes. We can give formal re- contains sufficient zeros to be decompos- spectability to that traditional butter- able as follows, margarine example of our texts by con- =B10[ 1. sidering them as two goods giving very LOB2J (7.1) similar combinations of characteristics. a so that there is some set of characteris- On the other hand, if certain activity requires more than one good and if these tics and some set of activities such that goods are used in no other activity we these characteristics are derived only can consider them as intrinsic total com- from these activities and these activities plements and they will always be con- give rise to no other characteristics, then sumed in fixed proportions, if at all. we can separate that set of characteris- Many goods within a commodity tics and activities from the remainder of group will have relationships to each the technology. If, further, the activities other which are partly complementary in question require a particular set of and partly substitution. This will be true goods which are used in no other activi- if two goods, for example, are used in ties (implying a decomposition of the A different combinations in each of several matrix), then we can regard the goods as activities, each activity giving rise to a forming an intrinsic commodity group. similar combination of characteristics. Goods within the group have the prop- The goods are complements within each NEW APPROACH TO CONSUMER THEORY 145 activity, but the activities are substi- the consumption good, the characteris- tutes. tics resulting from the work-consumption activity are given by a linear combina- IX. LABOR, LEISURE, AND OCCU- tion of the characteristics from work and PATIONAL CHOICE consumption separately, the weights in Within the structure of the present the combination being given by the wage theory, we can regard labor as a reversed rate. activity, using characteristics as inputs Add another activity, leisure, which and producing commodities or a com- gives rise to the two characteristics, and modity as output. This is similar to the the constraint that the weighted sum of standard approach of generalized con- the levels of activity labor and activity ventional theory, as in Debreu (1959). leisure is a constant. We can add to this approach in an im- The model is illustrated in Figure 3. portant way within the context of the W represents a work-consumption activ- present model by noting that a work ac- ity giving positive levels of both charac- tivity may produce characteristics, as teristics, I represents a leisure activity, well as the commodity labor, as outputs. also giving positive levels of both charac- This is structurally equivalent to per- teristics. The constraint on total time mitting some of the columns of the B (so that a linear combination of w and I matrix to have both negative and posi- is a constant) is represented by some tive elements, corresponding to activities line joining w, 1. that "use up" some characteristics (or If the constraint line has, like AB in produce them in negative quantities) and the diagram, a negative slope, then indi- produce others. In a work activity, the vidual consumers' utility functions will corresponding column of the A matrix be tangent to the constraint at different will contain a single negative coefficient points (like in, in') and we will have a for the commodity labor, or, more differ- neoclassical type of labor-leisure choice entiated, for one or more types of labor. in which the proportions depend on in- If a work activity corresponds to a col- dividual preferences. Some consumers' umn of mixed signs in the B matrix, it is preferences may be such that they will a recognition of the obvious truth that choose A (maximum work) or B (maxi- some work activities give rise to valued mum leisure), but it is a private choice. characteristics directly from the work it- In this model, however, for a certain self. level of the wage, given the coefficients Consider a very simple model of two of the technology, the constraint may characteristics with two commodities, have a positive slope as in A'B, or AB'. labor and consumption goods. Both labor If the constraint is A'B (corresponding, and consumption goods correspond to ceteris paribus, to a sufficiently low real separate activities giving rise to the two wage), all individuals will choose B, the characteristics in different proportions- only efficient point on the constraint set perhaps negative in the case of labor. OA'B. At a sufficiently high wage, giving With no income other than labor, and constraint set OAB', A, the maximum only one good available to exchange for labor choice, is the only efficient choice labor, we can collapse work and consump- and will be chosen by all individuals. tion into a single work-consumption ac- The above effect, in which for some tivity. Given the wage rate in terms of wage range there is a private labor-leisure 146 KELViN J. LANCASTER choice between efficient points while out- below which no peasant will offer himself side the range all individuals will take as paid labor and that this is an efficiency maximum work or maximum leisure, can choice and not a private choice. only occur if both the work-consumption We can use the same type of model and leisure activities give both charac- also to analyze occupational choice. Sup- teristics in positive amounts. If the using pose that we have two types of work up of characteristic 2 in labor exceeded (occupations) but otherwise the condi- the amount of that characteristic gained tions are as above. If and only if the by consumption, then the work-consump- characteristics arising from the work it- tion activity might lie outside the posi- self are different in the two occupations, tive quadrant, like w'. In this case, a the two work-consumption activities will constraint like A'B can exist, but not one give rise to activities in different com-

C- A

11~~~~~~~

II z2,

FI.I

like AB'. Furthermore, if the consumer binations. If the work characteristics are will choose only positive characteristics in the same proportion, the characteris- vectors, no consumer will choose maxi- tics of the work-consumption activity mum work. will be in the same proportions and one This model of the labor-leisure choice, or the other occupation will be the only which provides for objective and univer- efficient way to achieve this characteris- sal efficiency choices as well as private tics bundle. choices, may be the basis for a useful Figure 4 illustrates one possible set of working model for an underdeveloped relationships for such a model. In the area. If the "leisure" be defined as diagram, w(, w2 represent the characteris- "working one's own field," the work-con- tics combinations from work-consump- sumption activity as entering the market tion activities in occupations 1 and 2, 1 economy, we see that there will be wages the characteristics combinations from NEW APPROACH TO CONSUMER THEORY 147 leisure. The frontier consists of the lines have the very interesting effect, where AC (combinations of wl and leisure) and those who choose occupation 1 will work AB (combinations of w2 and leisure). We very hard at it; leisure-lovers will choose shall impose the realistic restriction that private combinations of occupation 2 and asnindividual can have only a single occu- leisure surely a good description of pation so that AB is not a possible com- effects actually observed. bination of activities. The loss to certain individuals from The choice of occupation, given the confinement to a single occupation is ob- relationships in the figure, depends on vious. Could he choose a combination of personal preferences, being M1 (combi- occupations 1 and 2, the individual at nation of w2 and leisure) for an individual M2 would do so and be better off than

z / r i 2 \ t

FIG. 4 with preferences skewed towards Z2 and with a combination of occupation 1 and M2 for an individual with preferences leisure. In a two-characteristic, three- skewed towards z1. But note a special activity model, of course, two activities effect. For some individuals whose in- will be chosen at most, so that leisure difference curves cannot touch BC but plus both occupations will not appear. can touch AC, the efficient choice will be The configuration in the diagram (Fig. the corner solution M3 (= B). There is, 4) represents the situation for some set in fact, a segment of AC to the left of of technical coefficients and specific wages w2 (the part of AC to the right of w2 is in the two occupations. A large number dominated by BC), lying below the hori- of other configurations is possible. In zontal through B which is inefficient rela- particular, if the wage rate in occupation tive to B and will never be chosen. 2 fell sufficiently, BC would lie inside A C In a configuration like the above we and occupation 2 would cease to be chosen 148 KELVIN J. LANCASTER by any individual. All individuals, in this characteristics related to "transporta- case, would choose their various personal tion" may remain with constant coeffi- combinations of occupation 1 and leisure. cients over several seasons. Confinement to a single occupation Elementary textbooks stress the multi- need not result in a welfare loss, even dimensional characteristics of money and when neither occupation dominates the other assets. The present model enables other in an efficiency sense. If the tech- this multidimensionality to be appropri- nical coefficients were different, so that ately incorporated. "Safety," "liquidity," the characteristics vectors representing and so forth become workable concepts occupation 2 and leisure changed places, that can be related to characteristics. We then the work-leisure combinations would can use analysis similar to that of the be given by AB and BC, both efficient preceding sections to show why efficiency relative to any combination of occupa- effects will cause the universal disappear- tions 1 and 2. In this case, all individuals ance of some assets (as in Gresham's would optimize by some combination of Law) while other assets will be held in leisure and any one of the occupations. combinations determined by personal Approaches similar to those outlined preferences. It would seem that develop- above seem to provide a better basis for ment along these lines, coupled with de- analysis of occupational choice than the velopment of some of the recent ap- traditional, non-operational, catch-all proaches to consumer preferences over "'non-monetary advantages." time as in Koopmans (1960), Lancaster (1963), or Koopmans, Diamond, and X. CONSUMER DURABLES, ASSETS, Williamson (1964) might eventually lead AND MONEY to a full-blooded theory of consumer be- Within the framework of the model, havior with respect to assets-saving and we have a scheme for dealing with du- money-which we do not have at present. rable goods and assets. A durable good In situations involving risk, we can use can be regarded simply as giving rise to multiple characteristics better to analyze an activity in which the output consists individual behavior. For example, we of dated characteristics, the characteris- might consider a gamble to be an activity tics of different dates being regarded as giving rise to three characteristics-a different characteristics. mathematical expectation, a maximum Given characteristics as joint outputs gain, and a maximum loss. One consum- and two types of dimension in charac- er's utility function may be such that he teristics space-cross-section and time- gives more weight to the maximum gain any asset or durable good can be regard- than to the maximum loss or the expected as producing a combination of several ed value, another's utility function may characteristics at any one time, and that be biased in the opposite direction. All combination need not be regarded as kinds of models caD be developed along continuing unchanged through time. In these lines, and they are surely more the decision to buy a new automobile, realistic than the models (Von Neu- for example, the characteristic related to mann and Morgenstern, 1944; Friedman "fashion" or "style" may be present in and Savage, 1952) in which the expected relative strength in the first season, rela- value, alone, appears in the utility-maxi- tively less in later seasons, although the mizing decisions. NEW APPROACH TO CONSUMER THEORY 149

XI. NEW COMMODITIES, DIFFERENTIATED Given the technology (or the relevant GOODS, AND ADVERTISING portion of it) and given the intrinsic Perhaps the most difficult thing to do characteristic of the activity associated with traditional consumer theory is to with the new good, we simply insert it introduce a new commodity-an event in the appropriateplace in the technol- that occurs thousands of times in the ogy, and we can predictthe consequences. U.S. economy, even over a generation, If a new good possesses characteristics without any real consumers being unduly in the same proportionsas some existing disturbed. In the theory of production, good, it will simply fail to sell to anyone where activity-analysis methods have be- if its price is too high, or will completely

91 9

I,,~~~~

g.3

FIG. 5

come widely used, a new process or prod- replace the old good if its price is suffi- uct can be fitted in well enough; but in ciently low. consumer theory we have traditionally More usually, we can expect a new had to throw away our n-dimensional good to possess characteristics in some- preference functions and replace them by what different proportions to an existing totally new (n + 1) dimensional func- good. If its price is too high, it may be tions, with no predictable consequences. dominated by some combination of exist- In this model, the whole process is ex- ing goods and will fail to sell. If its price traordinarily simple. A new product sim- is sufficiently low, it will result in adding ply means addition of one or more ac- a new point to the efficiency frontier. In tivities to the consumption technology. Figure 5, ABC represents the old effi- 150 KELVIN J. LANCASTER ciency frontier, on which some individ- an enables the consumer more efficiently uals will consume combinations of goods to reach his preferred combination of g1 and g2 in various proportions, some characteristics. combinations of g2 and g3. If the price Many economists take a puritanical of the new good, g4, is such that it repre- view of commodity differentiation since sents a point, D, on the old efficiency their theory has induced them to believe frontier, some persons (those using com- that it is some single characteristic of a binations of gi and g2) will be indifferent commodity that is relevant to consumer between their old combinations and com- decisions (that is, automobiles are only binations of either gi and g4 or g2 and g4. for transportation), so that commodity If the price of g4 is a little lower, it will variants are regarded as wicked tricks to push the efficiency frontier out to D'. trap the uninitiated into buying unwant- Individuals will now replace combina- ed trimmings. This is not, of course, a tions of gi and g2 with combinations of gl correct deduction even from the conven- and g4 or g2 and g4, depending on their tional analysis, properly used, but is preferences. The new good will have manifestly incorrect when account is taken away some of the sales from both gi taken of multiple characteristics. and g2, but completely replaced neither. A rather similar puritanism has also If the price of g4 were lower, giving been apparent in the economist's ap- point D", then combinationsof g4 and g3 proach to advertising. In the neoclassical would dominate g2, and g2 would be re- analysis, advertising, if it does not rep- placed. At an even lower price, like D"', resent simple information (and little in- combinations of g4 and g3 would domi- formation is called for in an analysis in nate g2, and the corner solution g4 only which a good is simply a good), is an would dominate all combinations of g1 attempt to "change tastes" in the con- and g4 (since AD"' has a positive slope), sumer. Since "tastes" are the ultimate so that g4 would now replace both gi datum in welfare judgments, the idea of and g2 changing them makes economists uncom- Differentiation of goods has presented fortable. almost as much of a problem to tradi- On the analysis presented here, there tional theory as new commodities. In the is much wider scope for informational present analysis, the difference is really advertising, especially as new goods ap- one of degree only. We can regard a dif- pear constantly. Since the consumption ferentiated good typically as a new good technology of a modern economy is clear- within an existing intrinsic commodity ly very complex, consumers require a group, and within that group analyze it great deal of information concerning that as a new commodity. Sometimes there technology. When a new version of a appear new commodities of a more fun- dishwashing detergent is produced which damental kind whose characteristics cut contains hand lotion, we have a product across those of existing groups. with characteristics different from those We may note that differentiation of of the old. The consumption technology goods, if successful (that is, if the differ- is changed, and consumers are willing to entiated goods are actually sold) repre- pay to be told of the change. Whether sents a welfare improvement since it the new product pushes out the efficiency pushes the efficiency frontier outward frontier (compared, say, with a combina- NEW APPROACH TO CONSUMER THEORY 151 tion of dishwasher and hand lotion con- must have a price low enough relative to sumed separately) is, of course, another the prices of other commodities to be matter. represented on the efficiency frontier, In any case, advertising, product de- otherwise it will be purchased by no one sign, and marketing specialists, who have and will not appear in the economy. This a heavy commitment to understanding implies that if there are n viable com- how consumers actually do behave, them- modities in a group, each in a one-to-one selves act as though consumers regard a relation to an activity, the equilibrium commodity as having multiple charac- prices will be such that the efficiency teristics and as though consumers weigh frontier has n-1 facets in the two-charac- the various combinations of characteris- teristic case. In Figure 6, for example, tics contained in different commodities where the price of commodity 3 brings

I .2

3 FIG.~~~~~~~

FIG. 6 in reaching their decisions. At this pre- it to point A on the efficiency frontier, liminary stage of presenting the model that price could not be allowed to rise to set out here, this is strong evidence in a level bringing it inside point B, or it its favor. would disappear from the market; and if its price fell below a level correspond- XII. GENERAL EQUILIBRIUM, WELFARE, ing to C, commodities 2 and 4 would dis- AND OTHER MATTERS appear from the market. Thus the limits Since the demand for goods depends on prices necessary for the existence of on objective and universal efficiency ef- all commodities within a group can be fects as well as on private choices, we can established (in principle) from objective draw some inferences relative to equilib- data. Only the demand within that price rium in the economy. range depends on consumer preferences. A commodity, especially a commodity With a large number of activities rela- within an intrinsic commodity group, tive to characteristics, equilibrium prices 152 KELVIN J. LANCASTER would give a many-faceted efficiency vant, while there are other individuals frontier that would be approximated by for whom the Cadillac/Continental ratio a smooth curve having the general shape is the relevant datum. If the A matrix of a production possibility curve. For is strongly connected, however, the im- many purposes it may be mathematically plicit price ratios between different ac- simple to analyze the situation in terms tivities can correspond to price ratios of a smooth efficiency frontier. We can between the same sets of goods, and the then draw on some of the analysis that Paretian conditions may be relevant. exists, relating factor inputs to outputs Finally, we may note that the shape of goods, as in Samuelson (1953b). Goods of the equilibrium efficiency frontier and in our model correspond to factors in the the existence of the efficiency substitu- production model, and characteristics in tion effect can result in demand condi- our model to commodities in the produc- tions with the traditionally assumed prop- tion model. erties, even if the traditional, smooth, The welfare implications of the model convex utility function does not exist. In set out here are quite complex and de- particular, a simple utility function in serve a separate treatment. We might which characteristics are consumed in note several important aspects of the constant proportions-the proportions welfare problem, however, which arise perhaps changing with income-can be directly from a many-faceted, many- substituted for the conventional utility cornered efficiency frontier: function. 1. Consumers whose choices represent a corner on the efficiency frontier are not, XIII. OPERATIONAL AND PREDICTIVE in general, equating marginal rates of sub- CHARACTERISTICS OF THE MODEL stitution between characteristics to the In principle, the model set out here ratio of any parameters of the situation can be made operational (that is, em- or to marginal rates of substitution of pirical coefficients can be assigned to the other consumers. technology). In practice, the task will be 2. Consumers whose choices represent more difficult than the equivalent task points on different facets of the efficiency of determining the actual production frontier are equating their marginal rates technology of an economy. of substitution between characteristics To emphasize that the model is not to different implicit price ratios between simply heuristic, we can examine a sim- characteristics. If there is a one-to-one ple scheme for sketching out the effi- relationship between goods and activi- ciency frontier for some commodity ties, the consumers are reacting to rela- group. We shall assume that there is a tive prices between different sets of one-to-one relationship between activi- goods. The traditional marginal condi- ties and goods, that at least one charac- tions for Paretian exchange optimum do teristic shared by the commodities is not hold because the price ratio relevant capable of independent determination, to one consumer's decisions differs from and that a great quantity of suitable the price ratio relevant to another's. In market data is available. common-sense terms, the price ratio be- In practice, we will attempt to operate tween a Cadillac and a Continental is with the minimum number of character- irrelevant to my decisions, but the price istics that give sufficient explanatory ratio between two compact cars is rele- power. These may be combinations of NEW APPROACH TO CONSUMER THEORY 153 fundamental characteristics (a factor- efficiency frontier. But with a third char- analysis situation) or fundamental characteristic, detergent 3 could be adjacent acteristics themselves. to detergents 2 and 1 in an extra dimen- Consider some commodity group such sion, and we could build up an efficiency as household detergents. We have a pri- frontier in three characteristics. mary objective characteristic, cleaning Other evidence could, of course, be power, measured in some chosen way. used to determine the efficiency frontier We wish to test whether one or more for a given market situation. Among this other characteristics are necessary to evidence is that arising from ordinary describe the consumer-choice situation. activity-analysis theory, that, with r We take a two-dimensional diagram characteristics we would expect to find with characteristic "cleaning power" some consumers who used r commodities along one axis. Along the axis we mark at the same time, unless all consumers the cleaning power per dollar outlay of were on corners or edges of the efficiency all detergents observed to be sold at the frontier. same time. If this is the same for all de- Last, but possibly not least, simply tergents, this single characteristic de- asking consumers about the characteris- scribes the situation, and we do not seek tics associated with various commodities further. However, we shall assume this is may be much more productive than at- not so. From our observed market data, tempts to extract information concerning we obtain cross-price elasticities between preferences within the context of conven- all detergents, taken two at a time. From tional theory. the model, we know that cross-price elas- In general, if consumer preferences are ticities will be highest between deter- well dispersed (so that all facets of the gents with adjacent characteristics vec- efficiency frontier are represented in some tors, so that the order of the characteris- consumer's choice pattern), a combina- tics vectors as we rotate from one axis to tion of information concerning interper- the other in the positive quadrant can be sonal variances in the collections of goods established. chosen and of the effects of price changes The ordering of "cleaning power per on both aggregate and individual choices dollar" along one axis can be compared can, in principle, be used to ferret out with the ordering of the characteristics the nature of the consumption technol- vectors. If the orderings are the same, an ogy. Some of the problems that arise are equilibrium efficiency frontier can be similar to those met by psychologists in built up with two characteristics as in measuring intelligence, personality, and Figure 7a. The slopes of the facets can other multidimensional traits, so that be determined within limits by the limit- techniques similar to those used in psy- ing prices at which the various detergents chology, such as factor analysis, might go off the market. If the ordering in prove useful. terms of cleaning power does not agree Even without specification of the con- with the ordering in terms of cross-elas- sumption technology, the present theory ticity, as in Figure 7b, two characteris- makes many predictions of a structural tics do not describe the market appro- kind which may be contrasted with the priately, since detergent with cleaning predictions of conventional theory. Some power 3 in the figure cannot be on the of these are set out in Chart 1. 154 KELVIN J. LANCASTER

XIV. CONCLUSION characteristics in fixed proportions and not goods In this model we have extended into that it is these characteristics, consumption theory activity analysis, themselves, on which the consumer's which has proved so penetrating in its preferences are exercised. application to production theory. The The result, as this brief survey of the crucial assumption in making this appli- possibilities has shown, is a model very cation has been the assumption that many times richer in heuristic explana- goods possess, or give rise to, multiple tory and predictive power than the con-

FiG. 7a

4 3 z I j 3

FIG. 7b

4 4-z

I / / f ':I of L

4 3 2a Z

FiG. 7b NEW APPROACH TO CONSUMER THEORY 155 ventional model of consumer behavior great number of possible ways in which and one that deals easily with those many the model can be used. It is hoped that common-sense characteristics of actual a door has been opened to a new, rich behavior that have found no place in treasure house of ideas for the future de- traditional exposition. velopment of the most refined and least This paper is nothing more than a powerful branch of economic theory, the condensed presentation of some of the theory of the consumer himself.

CHART 1 THIS THEORY CONVENTIONAL THEORY Wood will not be a close substitute for bread, No reason except "tastes" why they should not since characteristics are dissimilar be close substitutes A red Buick will be a close substitute for a gray No reason why they should be any closer sub- Buick stitutes than wood and bread Substitution (for example, butter and marga- No reason why close substitutes in one context rine) is frequently intrinsic and objective, should be close substitutes in another will be observed in many societies under many market conditions A good may be displaced from the market by No presumption that goods will be completely new goods or by price changes displaced The labor-leisure choice may have a marked Labor-leisure choice determined solely by indi- occupational pattern vidual preferences; no pattern, other than between individuals, would be predicted (Gresham's Law) A monetary asset may cease No ex ante presumption that any good or asset to be on the efficiency frontier, and will dis- will disappear from the economy appear from the economy An individual is completely unaffected by price An individual is affected by changes in all changes that leave unchanged the portion of prices the efficiency frontier on which his choice rests Some commodity groups may be intrinsic, and No presumption that commodities forming a universally so group (defined by a break in spectrum of cross-elasticities) in one context will form a group in another context

APPENDIX I. TRANSFORMATION OF THE UTILITY set S, is preferredor indifferent to x*. If we FUNCTION INTO G-SPACE take some other z' in P, every x in S' such that Bx ? z' is also preferred or indifferent Consider some characteristics vector z* to x'*. Similarly for z" in P and S" such that which does have an image x* in G-space, that Bx > z", and so on. From the theory and consider the set P of all vectors z pre- of inequalities, the sets S, S', S" . . . are all ferred or indifferent to z*. If U has the tra- convex, and since P is convex, a linear com- ditional properties, the set P is convex with bination of z', z" is in P, so that a linear com- an inner boundary which is the indifference bination of x's in S', S" is also preferred or surface through z*. Now z ? z* implies z is indifferent to x*. Hence the set P of all x in P so that every x such that Bx > z*, a preferred or indifferent to x* is the linear 156 KELVIN J. LANCASTER combination of all the sets S. S', S", . .. and the set of attainable z's and the set of z's so is convex. preferred or indifferent to z*. Thus the utility function transformed For the same satisfactions vector Z* and into G-space retains its essential convexity. a new price vector p** the efficiency choice A more intuitive way of looking at the situ- will be the solution y** (giving x**), ation is to note that all characteristics col- V*, of lections which are actually available are = > contained in an n-dimensional slice through Min p**Ay:By z* y (8.2) the r-dimensional utility function and that Max va*:vb < p**A . all slices through a convex function are themselves convex. The transformation of Since z* is the same in (8.1) and (8.2), this n-dimensional slice into G-space pre- y** is a feasible solution of (8.1) and y* of serves this convexity. (8.2). From the fundamental theorem of linear programing we have II. "REVEALED PREFERENCE " IN = A COMPLEX ECONOMY p**Ay* > V**x* p**Ay**, (8.3) We shall use the structural properties of p*Ay** > V*z* - p*Ay*. (8.4) the consumption technology A, B (dropping the assumption of a one-to-one relationship A program identical with (8.2) except between goods and activities) to show that that z* is replacedby hz* will have a solution in a complex economy with more activities hy**, v**. Choose h so that hp**Ay** = than characteristics the efficiency choice al- p**Ay*. From (8.3) h > 1. From (8.4), ways satisfies the weak axiom of revealed > preference and will satisfy the strong axiom hp*Ay** p*Ay** > p*Ay*. (8.5) for sufficiently large price changes, so that If we now write p for p*, p' for p**; satisfaction of even the strong axiom does x = Ay*, x' = hAy** we have not "reveal" convexity of the preference function itself. p'x' = p'x implies px' > px, (8.6) Consideran economy with a consumption technology defined by satisfying the weak axiom of revealedprefer- ence. z = By, The equality will occur on the right in (8.6) only if equalities hold in both(8.3) and x = Ay, (8.4), and these will hold only if y** is optimal as well as feasible in (8.1), and y* is and a consumer subject to a budget con- optimal as well as feasible in (8.2). In general, straint of the form p*x < k who has chosen if the number of activities exceeds the num- goods x* for activities y", giving character- ber of characteristics,we can always find two istics ?. prices p*, p** so related that neither of the We know that if the consumer has made solutions y**, y* is optimal in the other's an efficient choice, y* is the solution of the program. program (the value of which is k). Hence, if the number of activities exceeds the number of characteristics (representing Minimize p*Ay (= p*x): a) the number of primary constraints in the program), we can find prices so related that By= z* , y 0, the strong axiom of revealed preference is even the has ob- which has a dual satisfied, though consumer (solution v*). tained characteristics in unchanged propor- Maximize vz*:vB < p*A . (8.1b) tions (z*, hz*) and has revealed nothing of his preference map. The dual variables v can be interpreted as The above effect represents an efficiency the implicit prices of the characteristics substitutioneffect which would occur even if themselves. From the Kuh-Tucker Theo- characteristics were consumed in absolutely rein, we can associate the vector v with the fixed proportions. If the consumer substi- slope of the separating hyperplane between tutes between different satisfactions bundles NEW APPROACH TO CONSUMER THEORY 157 when his budget constraint changes, this compensation" in the conventional analysis private substitution effect is additional to (see Samuelson 1948, 1953a), so the efficien- the efficiency substitution effect. cy effect leads to "external overcompensa- Just as the conceptual experiment im- tion" additional to private overcompensa- plicit in rcecalcd preference implies "over- tion.

REFERENCES Becker, Gary S. "A Theory of the Allocation nology of Consumption," A.E.R., Papers and of Time," Econ. J., September, 1965. Proceedings, May, 1966 (to be published). Debreu, Gerald. Theory of Value. Cowles Foun- Morishima, M. "The Problem of Intrinsic Coin- dation Monograph 17, 1959. plementarity and Separability of Goods," . "Topological Methods in Cardinal Metroeconomica,Vol. XI (December, 1959). Utility Theory," in K. J. Arrow, S. Karlin, Pearce, I. F. A Contributionto Demand Analysis. and P. Suppes (eds.). Mathematical Methods New York: Oxford Univ. Press, 1964. in the Social Sciences, 1959. Stanford, Calif.: Quandt, R. E. "A Probabilistic Theory of Con- Stanford Univ. Press, 1960. sumer Behaviour," Q.J.E., Vol. LXX (No- Friedman, Milton, and Savage, L. J. "The Ex- vember, 1956). pected-Utility Hypothesis and the Measura- Samuelson, P. A. "Consumption Theory in bility of Utility," J.P.E., Vol. LX (Decem- Terms of Revealed Preference," Economica, ber, 1952). N.S. 15 (November, 1948). Gorman, W. M. "Separable Utility and Aggre- . "Consumption Theorems in Terms of gation," Econometrica, Vol. XXVII (July, Over-Compensation Rather than Indiffer- 1959). ence Comparisons," ibid., N.S. 20 (Febru- Johnson, Harry G. "Demand Theory Further ary, 1953). (a) Revised or Goods Are Goods," Economica, . "Prices of Factors and Goods in General N.S. 25 (May, 1958). Equilibrium," Rev. Econ. Studies, Vol. XXI Karlin, S. Mathematical Methods and Theory in (1953). (b) Games, Programming and Economics. New Stigler, G. J. "The Cost of Subsistence," J. York: Pergamon Press, 1959. Farm Econ., Vol. XXVII (1945). Koopmans, T. C. "Stationary Ordinal Utili- Strotz, Robert, "The Empirical Implications of ty and Impatience," Econometrica, Vol. a Utility Tree," Econometrica, Vol. XXV XXIII (April, 1960). (April, 1957). Koopmans, T. C., Diamond, P. A., and Wil- . "The Utility Tree: A Correction and liamson, R. E. "Stationary Utility and Time Further Appraisal," ibid., Vol. XXVII Perspective," ibid., Vol. XXXII (January- (July, 1959). April, 1964). Thrall, Robert M., Coombs, C., and Davis, Lancaster, Kelvin J. "Revising Demand Theo- R. L. Decision Processes. New York: Wiley ry," Economica, N.S. 24 (November, 1957). & Sons, 1954. . "The Scope of Qualitative Economics," Uzawa, H. "Preference and Rational Choice in Rev. Econ. Studies, Vol. XXIX (1962). the Theory of Consumption," in K. J. . "An Axiomatic Theory of Consumer Arrow, S. Karlin, and P. Suppes (eds.). Time Preference," Internat. Econ. Rev., Vol. Mathematical Methods in the Social Sciences, IV (May, 1963). 1959. Stanford, Calif.: Stanford Univ. Press, . "The Theory of Qualitative Linear 1960. Systems," Econometrica, Vol. XXXIII Von Neumann, J., and Morgenstern, 0. Theory (April, 1965). of Gamesand Economic Behaviour.Princeton, . "Change and Innovation in the Tech- N.J.: Princeton Univ. Press, 1944.