Hierarchies in Goods-Characteristics Analysis

Kelvin Lancaster, Columbia University
ABSTRACT - Two major problems in demand theory are those of justifying the analysis of demand for goods within a group without reference to goods outside the group, and constructing a model of consumer behavior that conforms to recognized limits on human information-processing abilities. This paper investigates the use of hierarchical structures in solving these problems.
[ to cite ]:
Kelvin Lancaster (1976) ,"Hierarchies in Goods-Characteristics Analysis", in NA - Advances in Consumer Research Volume 03, eds. Beverlee B. Anderson, Cincinnati, OH : Association for Consumer Research, Pages: 348-352.

Advances in Consumer Research Volume 3, 1976      Pages 348-352

HIERARCHIES IN GOODS-CHARACTERISTICS ANALYSIS

Kelvin Lancaster, Columbia University

[The author wishes to acknowledge the assistance of the National Science Foundation (SOC=7514252).]

ABSTRACT -

Two major problems in demand theory are those of justifying the analysis of demand for goods within a group without reference to goods outside the group, and constructing a model of consumer behavior that conforms to recognized limits on human information-processing abilities. This paper investigates the use of hierarchical structures in solving these problems.

INTRODUCTION

The problem with which this paper is concerned is that of narrowing down the set of all characteristics possessed by goods within a group to a smaller subset of relevant characteristics, in terms of which the behavior of consumers with respect to choices over the group can be fully explained. The notion of relevance in this sense was introduced in Lancaster (1971) and is related to, but not identical with, the notion of "salience" in Fishbein (1971). Relevance is of particular importance in the economist's version of the characteristics model, in which the characteristics are, so to speak, looked at from the goods end rather than the consumer's end and thus consist potentially of all the physical, chemical, biological and other objective properties of the goods. This view was stressed in the original model of Lancaster (1966) and again, more recently, in Ratchford (1975). Behavioral models, looking at characteristics from the consumer end, have generally pre-selected the apparently relevant characteristics, but may ignore taken-for-granted properties.

It is worth emphasizing at this point that the economist has always taken a holistic view of the consumer as a decision-maker with a unified overall objective (maximization of utility, or attainment of a preferred state), so that all his decisions should be considered to be interconnected. Both this view and the most common behavioral view of the consumer as fragmented into a complex of more or less independent decision-making particles are obvious over-simplifications, but they are oversimplified from quite different directions. Thus the economist does not take it for granted that a particular group of goods can be separated off and consumer behavior with respect to this group be analyzed without reference to events outside the group, but considers it necessary to specify what special structural properties enable this separation to be made.

Goods may be separated into distinct groups on the basis of horizontal properties. Goods in two groups may, for example, possess the same essential characteristics but may be clustered so that the relative contents of different characteristics are comparatively close within each group, but widely different between groups (sports cars and station wagons, for example). Our main interest here, however, is in vertical separation. This does not mean that the groups themselves are conceived of as standing in a vertical or hierarchical relationship, but that the separation between the groups can be established prior to any detailed investigation of the properties of the group.

THE UTILITY TREE

Leontief (1947) introduced the concept of a separable function, where F(x1,. . .,xn) is separable into s groups if the n variables can be partitioned into s groups and functions fr (xr) exist such that F can be written in the form [f1(x1),. . .,fr(xr),. . . ,fs(xs)], where xr is the collection of variables assigned to the rth group. Each fr is called a "branch" of the function F.

The importance of separability for a utility function, should it have that property, is that maximization of utility can be regarded as a consistent two-stage process of the kind first described by Strotz (1957), in which the consumer first makes an optimal allocation of his budget among the groups, then spends that allocation optimally within the group. After some debate in the literature, it has been established that the weak form of separability given above is sufficient for consistency of the two-stage process, due to the work of Blackorby et. al. (1970). If the utility function is separable, it is possible that one or more of the branch utility functions fr may themselves be separable into sub-branch utility functions, and that some of these may in turn be further separable (giving us a real "tree"), implying the existence of a consistent multi-stage optimization.

If utility maximization can be regarded as a multistage process, the final stage will consist of choice among a restricted group of goods, subject only to the group budget. Effects arising from, say, price changes within another group will be very indirect and will be apparent only insofar as they affect the budget allocation of the group. Thus separability of the utility function would go a long way towards enabling us to analyze the demand for a group of goods in comparative isolation from events outside the group.

The traditional discussions of separability have been in terms of a utility function on goods, with goods as the arguments of both the overall utility function and the branch functions. In the characteristics approach, the utility function is conceived of as a function of characteristics quantities. Now there is no reason why the utility function on characteristics should not be separable, the groups now being groups of characteristics rather than of goods, but the multi-stage optimization property (which is what we are really interested in) no longer follows from separable utility alone.

The problem is that, although the utility function is defined on characteristics, the budget constraint is in terms of goods and characteristics cannot be purchased separately but only by buying the goods which possess them. Characteristics which may be separable in the utility function may not be separable in the budget because they are possessed by the same good. Suppose, for example, that there are three characteristics and the utility function is separable into three branches, one for each characteristic. If all goods possess all three characteristics, no division of the optimization process is possible and the separability of the utility function has no operational significance.

It is obviously sufficient for multi-stage optimization in a characteristics model that (a) the utility function on characteristics be separable into groups of characteristics and (b) that the technological relationship between goods and characteristics be such that for every group of characteristics there be a group of goods which possess no characteristics other than those of the characteristics group. This requires not only that the consumption technology can be partitioned into groups of goods having characteristics exclusive to that group, but that this partitioning corresponds to the groups of characteristics into which the utility function is separable.

This double requirement on the partitions may seem to be a very strong one, but it is, of course, satisfied implicitly if the utility function on goods can in fact be shown to be separable, since the structure of the utility function on goods is a compound of the structure of the utility function on characteristics and the structure of the goods-characteristics technology. Stating the properties in terms of characteristics merely reveals how strong a property is that of separability in terms of goods, it does not of itself create the difficulties.

Separability is, of course, a question of fact. Unfortunately there is little in the corpus of empirical work on demand that provides any guidance as to whether individuals act as if their utility functions are separable or not. The two most used models in econometric demand studies, the "Rotterdam Model" and the "Linear Expenditure Model", both can be shown to implicitly assume strong separability of the utility function, along with a variety of other more or less heroic assumptions, and provide no test of any separability hypothesis. The most that can be said is that the assumption of separability (of goods into relatively broad groups) has not been shown to introduce any identifiable problems. For further details on the various econometric studies, the reader is referred to Brown and Deaton (1972) or Philips (1974).

In spite of the lack of any explicit confirmation from broad econometric studies, some kind of separability assumption is implicit in all attempts to study the demand for closely related goods, whether by economists or by others concerned with consumer research. It is simply taken for granted that we can sensibly study choices over a properly defined group of goods without reference to the details of choice over other groups.

THE GROUP

Following through the discussion of separability in the characteristics model, we can see that the separation of certain goods into a group that can be studied in isolation requires an appropriate coincidence between the way in which the utility function can be separated and the way in which the technology of goods-characteristics relationships can be partitioned. If the partition boundaries coincide exactly, there is no ambiguity about the division into groups. If the partitions coincide only in some cases, then the groups are defined by a sort of highest common factor logic. That is, if C1 and C2 are groups of characteristics which are separable in the utility function, but characteristics from both groups are possessed by all goods in some group G1, then we obtain only a single separable group (provided no goods outside G1 possess these characteristics), not two. Conversely if goods groups G1 and G2 possess no common characteristics, but the characteristics from the groups are not separable in the utility function, then the true separable group is that of the goods in G1 and G2 taken together.

Consider, for example, two food characteristics, "flavor'' (taken to be one-dimensional for the sake of illustration only) and "calorie content". It seems reasonable to suppose that flavor, a direct effect on the senses, and calorie content, related to health and personal appearance, impinge on different branches of separable utility function. Because of their technological jointness in food, however, the separability of the utility function is not the determining influence in the grouping. Or consider a weight-watcher dinner and an exercise-parlor session, two goods with no objective characteristics in common but which possess characteristics likely to appear in the same branch of the utility function, and thus not in totally separate groups.

In the worst-case situation, the utility function may be separable into many branches and the consumption technology be partitionable into many groups of goods with their own sets of characteristics, but there may be no coincidences between the divisions and no groups of goods that can be fully separated from each other.

Due to the role played by the structure of the consumption technology in determining groups, grouping may be changed as a result of technological innovation, either embodied in new goods or arising from newly discovered information about the properties of existing goods. Air travel, for example, has consolidated the formerly distinct groups of goods (using the term in the general sense) "visits to Europe" and "annual vacation activities'' into a single group, by removing the disparity in time required for the two sets of activities. Discovery of the carcinogenic properties of cigarette smoking was equivalent to a change in technology (even though cigarettes had always been considered vaguely unhealthy) and certainly changed the structure of the group by making the tar-nicotine content a highly relevant characteristic, although it may not have caused a regrouping. Technological change need not consolidate groups (as in the air travel example) but may also divide them. Modern audio technology permits separation of the activities "listing to a symphony" and "going to a concert", which were formerly always in combination.

CONSUMER DIVERSITY

The preceding discussion has been confined to the analysis of how separability of an individual utility function, combined with an appropriate structure of the technology, enables that individual to optimize in more than one stage and thus enables him to consider groups in relative isolation. Obviously, the concept of the group is of no real use in the analysis of market behavior unless the separation into groups has essentially the same pattern over many, if not all, individuals.

The two-part structure of the characteristics model provides a much firmer basis for the existence of groupings which are common to many consumers than does the traditional economic model which sets up utility as a direct and individual function of goods. In the characteristics model, groups can only be separated along partition lines in the technology, and only then of course, if utility is separable in a corresponding way. Since the technology is common to all consumers, the grouping for any consumer is built out of the same basic blocks as those for any other consumer. This does not guarantee the existence of any groups which are common to all consumers since the technology may, for example, be partitioned into three distinct blocks of goods (no two blocks possessing the same characteristics) while some consumers have utility functions separable in such a way as to make the first two blocks into one group, the third block into another, and other consumers group the second and third blocks together, with the first separate. In this case, all consumers see a pattern of two groups, but the groups are not the same for all. In spite of examples of this kind, it is .apparent that the .. existence of common building blocks containing many goods makes the existence of common groupings a reasonable hypothesis, and makes it a certainty if we can assume that some of the separability properties of utility are common to all persons.

Virtually all econometric demand studies (perhaps all) that are based on some underlying model of consumer behavior and are not a set of ad hoc equations, implicitly assume that the aggregate demand can be treated as if it resulted from the decisions of a single representative consumer. Such models have no place for consumer diversity, which would nullify some of the properties assigned to the aggregate demand functions. Taking the goods groupings to be common to all consumers is then a straightforward consequence of non-diversity, providing no test of commonality.

We cannot, of course, analyze the fine structure of demand within a group without supposing that consumers are diverse. The hypothetical representative consumer in the demand studies is someone who brushes his teeth with every brand of toothpaste, in the proportions in which they are sold on the market, so that this approach has nothing to say about the choice among kinds of toothpaste. Insofar as economist's traditional demand studies are useful at all, it is in the light they can throw on very broad properties of demand - in which "goods" are aggregates, even complete groups, rather than individual products. Although these studies are commonly considered "micro" rather than "macro" economics, they are really a kind of half-way "macro-micro".

"Micro-micro" demand theory, that can handle product differentiates and not just aggregates like "automobiles", depends on taking account of the real diversity that exists between individuals. The more micro the analysis, however, the greater the dependence on the ability to separate individual groups of goods. If this cannot be done, we are forced to contemplate a model in which a consumer is making a single-stage choice over tens of thousands of different products.

If we put all this together, we see that a feasible analysis of demand at the most micro level really requires

(1) that all (or most) individual consumers share a common structure of preferences with utility functions that are separable in a universal way, so that there exist goods groups that are the same for different persons

(2) but that the individual preferences are diverse within this common structure, so that different individuals make different choices within the groups and allocate different proportions of their total budget to the various groups.

Although the assumption of diversity constrained by a universal structure is presupposing a great deal about the human psyche, that is what is implicit in all fine-structure analysis of demand over relatively small groups of products. Since analysis at this level has been carried out by those in market research for many years, we can presume that the underlying assumption is not unacceptable.

INTRAGROUP ANALYSIS

We have now established the conditions which enable us to single out a group of goods and analyze the demand for goods within the group without explicit reference to goods in other groups, confident that this corresponds to the decision structure common to all individuals in the market yet leaves scope for diversity of choice within the group. Analysis of demand over a relatively small group of goods is clearly a much simpler problem than that of analysis over the set of all possible goods, but remains complex. We remain strongly interested in additional structural or other properties than permit further reduction of the problem.

If we take any set of goods likely to emerge as a group for decision-making purposes, and take the characteristics to be the set of all objective properties possessed by goods within the group, it is obvious enough that potential number of characteristics can be enormous. The number will vary a great deal - if the group is "automobiles" (or any subset of automobiles), the num-bet of discernible properties is very large, while if the group is "soft drinks", the number will be much smaller. Nevertheless, analysis in terms of characteristics is potentially useful only if the number of characteristics actually entering the decision process can be greatly restricted. We shall refer to these as relevant characteristics.

Lancaster (1971) explored some of the grounds on which characteristics might he rejected as irrelevant on a priori criteria. These included universal characteristics (possessed by all goods, and possessed by goods within the group to approximately the same extent per dollar), invariant characteristics, characteristics made irrelevant through certain types of satiation, redundant characteristics (in fixed proportion to other characteristics). In most cases, however, we can expect that there remain a very large number of potentially relevant characteristics even after the maximum possible use of the preceding criteria.

Now consider the group from the other end, from the viewpoint of the consumer rather than the technology. We all know that we, as human decision-makers, are very limited as to the number of different variables that we can handle at one time. The work of Miller (1956) and others has validated this introspection experimentally, and come up with five to nine as the maximum number of variables. Fishbein (1971) takes the number of salient outcomes considered by a consumer to lie in this same range, presumably a number not chosen at random. Let us accept the magic number seven as expressing the limits of hi. an decision-making powers, then we must conclude either that goods do not ever possess more than seven potentially relevant characteristics or that decision-making, even over the single group, occurs in such a way that not more than seven characteristics are being considered at any one time.

There are, of course, many studies which have investigated the number of relevant characteristics in particular markets at particular times, using a variety of methods to achieve results, including asking the consumers themselves in one way or another, using perceptual maps or principal component analysis, or analyzing brand-switching data. Our interest here is not in the actual characteristics chosen as relevant in some context, but in the process by which the decision of relevance or nonrelevance is reached.

The simplest hypothesis is that of random selection -consumer simply becomes aware of certain characteristics in a sequential and random way and closes his mind when he reaches his information-processing limit. This does not require diversity of preferences in order to generate different choices and would imply that the role of advertising was to make sure that the characteristic thought to be a good selling point was made prominent so as to get in before the door closed. Acceptance of such a model might be no big thing in marketing research, but would have profound consequences for economics because the foundation of all microeconomic policy is that consumer choice implies attainment of a preferred or constrained optimum position. The random model could lead to the conclusion that an expert might make a better choice for the individual that he could himself. This is certainly conceivable, but if true the market economy, even in the ideal perfect state so often modeled by economists, would lose its ultimate best claim to superiority over planning.

As an economist, the author is therefore biased in favor of some model that involves purposive choice at all stages, and thus towards the type of model in which there is a deliberate sequential choice process, each step of the sequence involving a consideration of an appropriately small number of characteristics.

HIERARCHICAL STRUCTURES

A decision process in which a choice involving a restricted number of parameters is made, after which a further choice is made from another restricted set of parameters, and so on down the sequence, is necessarily hierarchical unless it is purely random. The ordering of the hierarchy determines which set of parameters is considered first, second, and on through the sequence.

More than a century ago, Earl Menger discussed a model of the consumer in terms of a hierarchy of wants. He was concerned with a sequence of wants of diminishing importance, each satisfied by successive allocations of the same good rather than by different goods, and used the model primarily to establish the notion of marginal utility. This is not the kind of hierarchy which we need in the present context, but comes much closer to it if the successive "wants" are thought of as being satisfied by different characteristics of the same good.

A more recent hierarchical model, and one that is directly comparable with the characteristic model, is that given by Ironmonger (1972). It should be noted that the Ironmonger work, although published much later than the original Lancaster (1966) paper, was developed independently and much had been embodied in an unpublished 1961 dissertation. The Ironmonger analysis is cast in terms of wants rather than characteristics, but the effective variable in his utility function is the "number of units of satisfaction of want i" which is simply the view from the consumer's end of the number of units of characteristic i obtained from consumption. Ironmonger's utility function is-a strictly hierarchical one, with wants completely ordered (a lexicographic preference order with satiation) so that the consumer's decision at any one stage is confined to that of satiating the next unsatisfied want in the ordering. Interesting results follow from the analysis solely because goods possess multiple characteristics and thus concentration on a single want at a particular stage results in other wants being partly satisfied. The multiple characteristics properties of goods generate many results similar to those of Lancaster {1966, 1971), in spite of the great differences in the assumed structure of the preferences. (The standard Lancaster analysis assumes characteristics act simultaneously, not sequentially, on utility).

Two aspects of the Ironmonger structure seem to rule it out as far too extreme for our present purposes. A fully hierarchical preference ordering would imply that a consumer would never think about clothiers until after he had eaten (assuming the ordering was in that direction), while separable but nonhierarchical preferences would merely imply that the consumer could think about clothing without thinking about eating and vice versa. The latter seems surely to conform more accurately to reality. The other doubtful property is that of having only one characteristic at a time enter in to the decision process.

The utility model that fits best the reality of limited information processing ability and the various criteria for micro analysis of demand, together with the property that choice reveals preference, is one with a mixed structure. It should be separable, with the same general lines of separation for all individuals, but without hierarchical relations between the groups as a general pattern. Within the groups, there should be some hierarchical structure so that the final decisions can be made in several stages, in each of which only a few (but typically more than one) characteristics are taken into account.

Consider a potential scenario for choosing an automobile, assuming that the separability properties have already enabled this choice to be made without reference to goods other than automobiles. Stage I might consist of assessing the characteristics associated with whether the car actually runs or not - does it have an engine? Wheels? does the engine turn over? This is a simple blocking stage, and cars which are not accepted here will not be chosen whatever their other characteristics. (We assume repair is not possible). Stage II might consist of considering those characteristics that make the car suitable for family use - how many will it seat? how much baggage will it hold? Finally, Stage III might consist of comparing details in aesthetics, convenience, comfort, mechanical operation and so on.

This scenario illustrates the main features that would have to be taken into account in devising a multi-stage decision model, which we can summarize as follows:

(1) How many stages are there, and is some decision about the number of stages itself a first stage of the process?

(2) What determines the ordering of the stages, and is the determination of this ordering part of the decision process?

(3) Finally, what kind of decision is made at each stage? Simple accept/not accept, for example, or some assignment of an actual utility number?

Once again, the special interest of the economist in the consumer as a whole needs to be stressed. An ad hoc model of a limited decision process that implied something about the consumer's overall preference structure or utility function that was very different from that usually assumed would find little acceptance among economists, unless it could be shown compellingly that the truth and generality of the model justified scrapping accepted economic theorems.

We should distinguish between the technique and the substance of the decision-making, since it is only the latter that affects the fundamental structure of the implied utility function. The number of stages, for example, is not important, but the kind of decision made at each stage can have a fundamental bearing on the implied preference ordering.

Consider the pass/fail kind of decision implied in Stage I of the suggested scenario given above, in which a subset of characteristics is checked against a minimum acceptable level and the good is not considered further unless it passes on all counts. If this is followed by an optimizing choice over those goods which have passed, the implied preference structure is that of an incomplete ordering which is, however, well-be-hayed over those collections actually ordered. The ordering is incomplete because goods which fail the screening stage are not further considered and thus not" ranked with respect to each other. But the final choice can be considered to represent a properly preferred position because it is the preferred choice from the final set, and all goods (or goods collections) in the final set can be taken to be preferred to all goods which failed the screening. Thus a simple pass/fail stage in the decision process does not cause any fundamental upset in the basis of preference.

Stage II in the example given might be taken as a more advanced type of pass/fail decision. Instead of a series of characteristics which all must be satisfied in order to pass the screening, the various characteristics are assumed to contribute jointly to some index of suitability as a family transport, as characteristics contribute jointly to utility in a utility function. A car passes this stage if the index has at least some minimum value. Again, if this stage is followed by a pure maximization stage, the final choice is a true preferred position. A screening stage of this kind does, however, introduce problems of both incompleteness and continuity into the preference structure.

Sequential sorting processes have been suggested as appropriate models for market research purposes - the Hendry model, for example (see Herniter (1974)), and so have various models based on threshold effects. For market research it is, of course, of the highest importance to determine which particular characteristics are taken into account at which stage. For our more general approach, it is important only that an individual be consistent in the structure of his decisions. If the group is well-defined, the number and specification of decision stages can be taken as individual, contributing to the assumed diversity of preferences over the group. We do not need a general and universal theory of the order of sorting.

In the pure screening model, only the last decision stage is a true maximizing stage. In this final stage, presumably, the consumer chooses his preferred collection of those characteristics left to this stage from among those goods whose other characteristics have been sufficient to pass the previous screening. The process is that of maximizing a (branch) utility function of certain characteristics, subject to obtaining minimum levels of other characteristics or functions of these.

Although the screening model has many attractions, has been used as the basis for market behavior models, and can be made to fit in reasonably well with the most important assumptions on preferences usually made by economists, it is not perfect. One is uncomfortable, in particular, with the status of above-minimum characteristics levels at the screening stages, which fail to count in the final decision. If a car is especially suitable for family use) why should not that factor count at the final stage, even to the extent of resulting in the choice of that car over another that actually has more final stage characteristics?

One potential modification of the pure screening model is to introduce some concept of "carry-over", where properties above the minima and one stage are carried forward to be weighed together with the next stage characteristics. If above-threshold quantities of individual characteristics are carried forward, we lose the property of limiting the number of variables per stage, which was the object of introducing multi-stage analysis. However, we could carry forward the above-threshold level of some index (like family suitability in the example given) and still achieve reduction of variables. Why not carry forward the whole value of the early stage indices, so that the final decision was a maximization of a function of indices, subject to minimum levels of specific indices? This could be done on the basis of a kind of separability of the utility function alone, and would preserve most of the properties liked by economists. But, although it reduces the number of characteristics considered at any one time (since the making up of each index is a separate operation), it does not reduce the number of goods - all goods in the group would be considered together in the final judgment Thus it seems desirable to retain the essentials of goods-screening process.

Such a model, if it could be developed into an applicable form, would seem to come close to satisfying both the economists and the behaviorists. Until this, or some other model with equivalent properties, is fully worked out, economists and those concerned with detailed market behavior are likely to have somewhat different views on the use of characteristics analysis. Economists, interested in the broader analysis of market structures - as typified, for example, in Lancaster (1975) - will continue to look at characteristics from the objective, goods point of view. Market behaviorists will continue to look at characteristics from the subjective, consumer's end, as in Howard and Sheth (1969), Wilkie and Pessemier (1973), Fishbein (1971).

REFERENCES

Blackorby, C, Lady, G, Nissen, D and Russell, R (1970), "Homothetic separability and consumer budgeting", Econometrica, 38 (1970), 468-472.

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Fishbein, M (1971), "Some comments on the use of 'models' in advertising research". In European Society of Market Research, Proceedings: Seminar on Translating Advertising Theories into Research Reality, 1971, 297-318.

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