Consumer Economics Research Paper

Two basic ingredients are involved in this selection: limited means and consumer preferences. The first one, limitation of the budget, expresses the fundamental scarcity that is the raison d’etre of modern microeconomics. The second one, the ordering of the available alternatives according to preferences, reflects the role played by personal attitudes, biological needs, social aspirations, cultural trends and so on in deter-mining how the scarce resources are being allocated. In the marketing approach to consumer decision making the emphasis is mostly on what shapes preferences. In the theory of consumption and saving, the focus is on the lifetime wealth constraint and to a lesser degree on the explanation of (time) preference. On the level of generality used here both ingredients will be discussed in an even-handed way.

Neither the budget nor the preferences are fixed or necessarily exogenous to the economic system. In the present context, however, the budget or total available means will be taken as given for the consumer, while also the preferences are not an object of choice for her or him.

Consumer theory is about individual behavior. Empirical data frequently refer to an aggregate of consumers, for example, a nation. Without some additional assumptions the theoretical results do not carry over to such an aggregate and the empirical relevance of the theory is minimal. The additional assumptions imply the existence of the representative consumer, who can be treated as if he or she were the individual consumer of the theory.

Empirical applications are abundant. Davenant (1699) is the earliest known example. It explains the price of corn as a function of the amount harvested. For a long time empirical work and theory were virtually unrelated. It was only just before World War II that the applications were first based on theory. Still, one usually focused attention on the demand for a single good. In more recent times attention has also been directed at the explanation of demand for all goods simultaneously, for which there is an explicit theory that offers the appropriate framework.

2. The Budget

The total means available, the budget, are used to pay for the quantities of the various goods and services, goods for short, that the consumer wants to acquire. It is assumed that the number of different goods is finite and fixed. The set of quantities is called a bundle. The budget is then the total amount to be paid for a bundle. Theory assumes the budget to be given for the consumer. Not quite correctly, ‘income’ is frequently used as a synonym for total means.

The price is the amount of money to be paid for a unit of a particular good. The prices are assumed to be positive, with negative and zero prices not admitted. Quantities are most of the time taken to be positive as well. The framework can be extended to also cover negative demands: ‘bads,’ or the supply of home-produced goods, for example. Here the focus is on positive demands. With positive prices and positive demands the budget must be positive.

Theory assumes that the goods are fully divisible. This is a realistic assumption in the case of fluids like gasoline, but not empirically valid for many types of household durables. Theoretically, one can replace the durables themselves by the services these render, which are continuously divisible, but empirically the stream of services is not directly observable. It has been attempted to reformulate consumer theory in terms of integer quantities. This development has not entered the mainstream of consumer analysis.

Most of the time the quantities are taken to be the objects of choice and the prices given for the consumer, at least in the short run. There are cases where the quantities are fixed and the prices are endogenous. The Davenant study just mentioned is an example where the prices are being explained and the harvests of corn being fixed. This reversal of causality is specifically of relevance for nonstorable agricultural produce. If for all goods the quantities are the variables explained one speaks of regular demands. The case of endogenous prices is one of in verse demands. When for some goods the quantities are endogenous and for other goods the prices are endogenous, one speaks of mixed demands. Although inverse and mixed demands are easily handled by the theory, the mainstream is concerned with regular demands.

3. Preferences

According to the basic postulate of consumer theory, the consumer selects that bundle among the available alternatives that gives the most satisfaction. Assumptions are made that guarantee the existence of a complete, reflexive, and transitive preference order defined on the set of bundles. The requirement of completeness, that is, the ability to order all possible alternatives, is somewhat stronger than needed. It is enough if the consumer is able to order the bundles in the neighborhood of the set of bundles that fit in the budget. The assumptions of reflexivity and transitivity are of a logical nature.

The assumptions about the preference order do not exclude the possibility that bundles occupy the same rank in that order. All bundles with the same order form an indifference class. In the case of the choice between only two goods the graphical representation of the indifference class is the indifference cur e. The set of all indifference classes is named the indifference map. Theory specifies a continuity postulate for this map. This amounts to the property that a drop in the quantity of one good of a bundle can always be compensated for by a change in the quantities of one or more other goods to stay within the same indifference class. This assumption rules out lexicographic preferences, where the preference for one good absolutely dominates the desire for other goods.

The preference order is equipped with two further properties. One is that of desirability. More of any good is preferred, keeping the quantities of the other goods the same. There is no satiation. Moreover, it excludes the existence of intersecting indifference classes. It also implies that the budget is always used fully. The other property is that of con exity of pBibliography: a linear convex combination of two bundles is preferred to the least preferred one of the two bundles themselves. This property is of special importance to guarantee a unique solution to the consumer’s choice problem.

4. Utility

Under the assumptions made about the preference order there exists what is known as the utility function. It assigns a (real) value, utility, to bundles in such a way that for any pair of bundles the one that is preferred to the other obtains a higher value. In the case that the consumer is indifferent between the two bundles they obtain the same value.

Given the desirability property of preferences the utility function is monotonously increasing in the quantities of the goods. Convexity of preferences implies that the utility of a convex combination of two bundles obtains more utility than the one with the least utility. It is known as the property of quasiconcavity.

The name ‘utility function’ is slightly misleading. It is a preference indicator. Utility is not ‘produced’ by a bundle. The utility function associates with a bundle a certain value that reflects the rank of its indifference class. There is also no uniqueness of the index. Adding to all rank values the same value yields another utility function that corresponds with the same preference order as the original one. In more general terms: given a particular utility function for a preference order any order preserving or monotone increasing transformation of it will yield an equally valid utility function. As far as consumer demand is concerned only ordinal properties of a utility function matter. Cardinal properties that are not invariant under order preserving transformations have no effect on choice behavior.

In fact, consumer theory does not need a utility function at all. However, a utility function appears to be a convenient tool for describing the consumer’s choice. No damage is done using a particular utility function, as long as it correctly reflects the preference order.

A related concept is that of marginal utility. Mathematically, it is simply the first derivative of the utility function with respect to the quantity of a certain good. It represents the increment in utility for an infinitesimal increase in that quantity. In view of the desirability of preferences it is always positive.

5. Demands

Taking the case of exogenous prices, the consumer selects that bundle among the bundles that can be acquired with the given budget that he or she prefers most. Under the assumption made for the preference order the selected bundle is unique. It can be written as a set of functions, one for each good, of the budget and of the set of prices. As the selected bundle fits in the budget also the demand functions will fit in it. This is the adding-up property of demand. The composition of the bundles that satisfy the budget will remain the same if all prices and the budget change by the same positive factor. This then also holds for the set of demand functions. These are homogeneous of degree zero in the budget and the prices.

One of the implications of the assumptions made is that for two bundles in the same indifference class the amount demanded of a certain good is less if its price is higher. This is the Law of Demand. Note that it concerns bundles belonging to the same indifference class. It is the most important qualitative result of consumer theory. The more general version stating that demand for a good is a negative function of its price is not true with the same degree of theoretical validity. This law was put forward for the first time by Davenant (1699).

Observe that the adding-up, homogeneity, and negativity conditions have been formulated without the use of the concept of utility. For the derivation of further properties of demand functions the utility function proves to be convenient.

6. Consumer Equilibrium

With the utility function as a tool the consumer’s choice problem can be described as maximizing utility with respect to the quantities of the various goods under the condition that the bundle chosen fits in the budget. The first-order condition for this constrained optimum is the proportionality of the respective marginal utilities with their prices. This result is known as the Second Law of Gossen. It characterizes the consumer’s equilibrium. It implies that for all goods the marginal utilities divided by the prices are the same.

One interpretation of this outcome is that an extra unit of currency is allocated over the various goods in such a way that it yields the same increase in utility per unit of currency for all goods. The marginal utility of a unit of money spent on each good is the same.

The marginal utilities are cardinal concepts, but their ratio is ordinal. There is a tendency to work with this ratio that is known as the marginal rate of substitution. In equilibrium the marginal rate of substitution between two goods is then equal to the price ratio of these goods.

Proportionality of marginal utilities and prices is not only the equilibrium condition in the case that the quantities are endogenous and the prices exogenous, but also when the prices and quantities reverse roles in this respect. In fact, this condition also holds when for some goods the prices are exogenous and the quantities endogenous, while for other goods the opposite is true.

Under the usual assumptions made, the equilibrium conditions can be solved for a unique set of values for the endogenous variables given the values for the exogenous variables and the shape of the utility function. In the case of endogenous goods the solution can be expressed as a set of demand functions in terms of total means available and the prices. These demand functions have a number of interesting properties, which play a role in empirical analysis.

7. Properties Of Demand Functions

Under rather general conditions the demand functions are differentiable with respect to income or total means and prices. The derivatives satisfy certain conditions. Frequently one expresses those conditions in terms of elasticities, defined as the proportional change in demand as a consequence of a (small) proportional change in means (income elasticities) or in prices (price elasticities), respectively. The elasticities can be easily derived from the derivatives of the demand functions. Note that neither derivatives nor elasticities are necessarily constants according to the theory. The choice of constant coefficients is a matter of empirical expedience. Income elasticities can be negative. In this case, an increase in income reduces demand for the good. This usually happens if there exists a similar product that is much more attractive. If the income elasticities are negative one speaks of inferior goods. Goods with positive elasticities are superior or normal goods. Income elasticities can be larger than one. Then as income increases, the budget share, being the share of expenditure on the good in question in total expenditure, will increase. A good with such a property is named a luxury. A necessity is a good with an income elasticity less than unity.

In Sect. 5 some properties were mentioned. The adding-up property reflects that the demands exhaust the budget. The Engel adding-up condition states that the sum of the income elasticities multiplied by the respective budget shares adds up to one. The Cournot adding-up condition refers to the price elasticities. The sum of the price elasticities for the prices of the various goods with respect to the price of one good, say good i, multiplied by the budget shares, equals minus the budget share of the ith good. These properties imply that if one knows all but one of, say, the income elasticities, one can reconstruct the remaining one as a residual.

The homogeneity condition entails that the sum of the income and price elasticities equals zero. The same proportional increase in income and all prices will have no effect on the quantity demanded. Thus, knowing all but one of those elasticities enables one to find the remaining one.

The price effects can be split up in a part that represents the move from one indifference class to the other and in a part that reflects substitutions within an indifference class. The first effect can be neutralized by an adequate change in means, such that one stays in the original indifference class. It is named the income effect of price changes. The other part of the effect of a price change is known as the substitution effect of price changes or as the compensated effect of price changes. The income effect of price changes can be combined with the effect of an income change to form the effect of a real income change. This represents the move from one indifference class to the other.

The substitution effects can be collected to form a matrix, the Slutsky matrix. On its diagonal one finds the substitution effect on the demand for a good by its own price. On the basis of the assumptions made it should be negative, see Sect. 5. For superior goods the income effect of the own price change is also negative. Then the two effects work in the same, negative, direction. This cannot be generalized. If the good is an inferior commodity, the income effect will be positive and may, in absolute value, be larger than the substitution effect. The total own price effect is then positive. Such a good is known as a Giffen good. No empirical cases are on record, but they cannot be ruled out on theoretical grounds. For practical purposes the total own price effect may be taken to be negative.

A further property of the Slutsky matrix is its symmetry, first stated by Slutsky (1915). It is a mathematical property of great empirical con-sequence, but with hardly any intuitive appeal. The empirical importance can be illustrated for the case that there are 20 goods in the budget. The number of Slutsky cross-price effects is then 20*20–20 = 380. The property that these are pairwise equal reduces the empirical problem by 190 fewer coefficients to estimate separately.

The elements of the Slutsky matrix contain a considerable amount of information about the preferences of the consumer. Two instances of this will be taken up in the following sections.

8. Complementarity, Substitution, And Independence

Preferences are specifically of interest when they refer to interactions between goods. If two goods reinforce their desirability, one speaks of complementary goods. Examples are wine and cheese. Two goods are substitutes if one can replace the other to a certain degree; beef and pork, for instance. In the case of independence there is no specific relation between the two goods. How are these fundamental relations between goods reflected in empirical demand behavior?

The early marginalists started off from the marginal utility of a good. If it goes up if one has more of another good, the two are complements. If it goes down, the pair are substitutes. Unfortunately, this rather straightforward definition is not an ordinal one. Depending on the type of utility indicator corresponding to the same preference order, two goods may be complements rather than substitutes. The original meaning of the terms ‘complements’ and ‘substitutes’ clearly refers to the preference order and not to a particular representation of it.

It has been attempted to associate complementarity and substitution with the sign of the marginal rate of substitution, an ordinal concept. This definition is remote from intuition and the consequence for the signs of the compensated price elasticities are not easy to trace. Hicks (1939) cut that Gordian knot by using the sign of the elements of the Slutsky matrix. A negative sign means complementarity and a positive one substitution. However, there are problems with this definition. One of these is that positive off-diagonal elements of the Slutsky matrix tend to be larger or more numerous than negative elements, reflecting the tightness of the budget, which, as such, has nothing to do with preferences. The approach of Allais (1943) appears to meet most of the difficulties, but is not that simple to apply.

In the early days of empirical application of full-sized demand systems the assumption of complete independence was used frequently. Under this assumption there exists a utility function that is additive in (functions of ) the various goods. The best known example is the Samuelson–Klein–Rubin utility function. It is a weighted sum of the logarithms of the quantities of the goods as far as they exceed some minimum value. The implied demand functions are correspondingly also more simple than in the case of real preference interaction between goods. The impact of price changes other than that of the own price takes the form of a change in a price index where the identity of the other price change does not matter. A con-sequence of complete independence is the limited number of coefficients to estimate. Given the paucity of data, typical for the early years of empirical demand analysis, a small number of unknown parameters was an attractive feature, even if the underlying assumption was not quite realistic.

9. Separability Of Preferences

Terms like ‘wine’ and ‘cheese’ actually refer to conglomerates of different types of wine and different types of cheese. If the preferences for the various types of wine are independent of the amounts of the diverse cheeses consumed, one speaks of separability of preferences. This does not mean that the total amount spent on wines as a group does not depend on the total amount of cheese consumed. For the preference interaction between wines and cheeses the wine nature of the wines and the cheese nature of the cheeses dominate the relation and the specific brands do not matter. As a consequence the consumer’s behavior can be modeled as a stepwise procedure. First, the consumer decides on the total amounts of wine and cheese he or she wants to consume. Next, given those amounts and the prices, he or she determines how much of each type of wine and of each type of cheese he or she will purchase. For the first level the consumer needs to know total available (real) income and all the prices. However, those prices can be represented by a price index for each group. The consequence is that the impact of a change in the price of, say, Gouda cheese on the demand for, say, a high-class Burgundy wine is only by way of the effect of a change in the price index of cheese. The origin of the price change does not need to be identified.

It is a simple matter to extend this framework to more than two levels. For example, one can think of a hierarchy of the type: total consumption, food, meat, beef, with beef taken as the elementary good.

For the full set of demand relations separability means a rather simple constraint on the Slutsky coefficients for interactions between goods belonging to different groups.

Separability is very convenient for empirical work. One can estimate the first level model separately from the lower level models and in this way control the dimensionality of the estimation problem. In fact, the data usually are only available in already aggregated form. The assumption of separability justifies working with those aggregates as if they were elementary goods.

A special case of separability is strong separability. There the aggregates have no special type of inter-action like that of substitution or complementarity discussed in the previous section. They are all in-dependent vis-a-vis each other. The Slutsky matrix then becomes even more constrained than for the case of ordinary or weak separability.

Under separability the utility function can be written as a function of a set of subutility functions, one for each group. Under strong separability the preference order can be represented by a utility function that is additive in those subutility functions.

10. Indirect Utility And Expenditure Functions

The utility function is a function of the quantities of the various goods. One can replace these quantities by the optimal quantities, the demand functions. One obtains the indirect utility function, an expression in terms of total means and all prices. It indicates the maximum attainable utility for the given total means and these prices. Its derivatives are the marginal utility of total means and the marginal utilities of the various prices. The former shows how much utility can be increased if total means are slightly increased. The marginal utility of a price is the reduction in utility when the price goes up. These properties of the indirect utility function are based on those of the demand functions. Another property is its homogeneity of degree zero in means and prices. Otherwise said: an increase of means and prices by the same factor leaves utility unchanged.

The indirect utility function is a cardinal concept like the original utility function. The properties just mentioned, together with a curvature assumption, are ordinal, however. The Identity of Roy can be used to derive the demand functions from a well-specified indirect utility function. To obtain the demand function for good i one simply takes the negative of the ratio of the marginal utility of the price of good i and that of total means. It is a convenient way to arrive at a proper functionally specified set of demand functions. The alternative to deriving the demand functions as the optimal values for the direct utility function under the budget turns out to be practical only for very simple functional specifications of that function. The Roy identity can be seen as a convenient short cut.

A concept similar to the indirect utility function is the expenditure function, also known as cost function. It yields the minimum outlay or means to obtain for a given set of prices a given level of utility. It can be obtained from the indirect utility function by fixing the level of utility and solving for the means. It has, mutatis mutandis, similar properties to the indirect utility function. It reacts positively to both an increase in the level of utility to be reached and to price increases. It also has to satisfy a certain curvature condition. Corresponding to the identity of Roy there is Shephard’s lemma. By differentiating the expenditure function with respect to the price of good i, keeping utility constant, one obtains the optimal demands as a function of the prices and the level of utility.

To distinguish these demand functions from the ones that are expressed in prices and means, they are named Hicksian demand functions. The former ones are then the Marshallian demand functions. One can shift easily from the Hicksian to the Marshallian demand functions by replacing the level of utility by the expenditure function. Likewise, one may change from the Marshallian demand functions to the Hicksian ones by substituting the total means variable by the expenditure function. For empirical purposes the Hicksian demand functions are not so useful because one would need a value for the utility level, which would be rather arbitrary. However, starting from a well-specified expenditure function one can derive the Hicksian demand functions in which one replaces the utility level by the indirect utility obtained from the expenditure function.

From a theoretical point of view, the Hicksian demand functions are of considerable interest. Their price derivatives represent the effect of price changes with utility kept constant. They are precisely the Slutsky coefficients introduced in Sect. 7. The Slutsky matrix turns out to be the matrix of second-order price derivatives of the expenditure function.

Another interesting feature of the expenditure function itself is its role in the derivation of the constant utility price index. Consider the case of a consumer who is in equilibrium. The expenditure function gives the minimal amount of expenditure to reach a given level of utility with the current prices. Take another set of prices but keep utility constant. The expenditure function will now show the minimum outlay required to reach the original utility level with the new prices. The ratio of the latter and former amounts can be seen as a price index. It indicates the factor by which one has to multiply the total means to keep the same level of welfare. The factor could be more or less than one. In the case of a general price increase it will be larger than one. The practical use of the constant utility price index meets with the difficulty of specification of the expenditure function. However, its theoretical appeal cannot be questioned.

11. Empirical Validity

Do data on consumer behavior reflect the properties that theory postulates? In the case of demand functions for a single good the properties of interest are the negativity in the response to a change in the own price for the good in question and the homogeneity co-dition, implying no response to an equiproportional change in income (budget) and the prices. Usually negativity does not create serious problems and the demand equations in estimated form readily display this property.

Statistical tests tend to be less lenient for the homogeneity condition. This is somewhat puzzling because of the plausibility of this property, which is sometimes interpreted as the absence of a monetary veil. Homogeneity justifies the proposition that al-location of means is based on relative prices rather than on absolute ones, which is a central issue in microeconomics. Many reasons have been advanced to explain away this dilemma such as inappropriate data, specification errors (absence of explicit dynamics, for instance), and the use of incorrect test statistics. Correction of some of these shortcomings appears to have achieved some success.

In the case of systems of demand equations explain-ing demand for a set of goods simultaneously, negativity does also not create a serious validity problem, but homogeneity is less easily accepted. Additional properties of empirical interest here are Slutsky symmetry and separability. Symmetry appears to agree with the data. This may be more a matter of the wide confidence intervals of the estimates of the estimated Slutsky coefficients than of the truth of the symmetry hypothesis itself. Separability of preferences has such attractive properties for applied work that it is frequently used in spite of its demonstrated lack of empirical validity.

Also in consumer analysis, theory and application are being handled in a balanced way, reflecting doubts about both theory and facts and about the procedures to reconcile them.

Bibliography:

1. Allais M 1943 Les donnees general de l’economie pure. Traite d’economie pure. Imprimerie Nationale, Paris
2. Barten A P, Bohm V 1982 Consumer theory. In: Arrow K J, Intriligator M D (eds.) Handbook of Mathematical Economics, II. North Holland, Amsterdam
3. Davenant C 1699 An Essay Upon the Probable Methods of Making a People Gainers in the Ballance (sic) of Trade. J. Klapton, London
4. Deaton A, Muellbauer J 1980 Economics and Consumer Behavior. Cambridge University Press, Cambridge, UK
5. Hicks J R 1939 Value and Capital. An Enquiry into Some Fundamental Principles of Economic Theory. Clarendon Press, Oxford, UK
6. Samuelson P A 1947 Foundation of Economic Analysis. Harvard University Press, Cambridge, MA
7. Slutsky E 1915 Sulla teoria del bilancio del consomatore. Giornale degli economisti 51: 1–26
8. Theil H 1975 and 1976 Theory and Measurement of Consumer Demand. North-Holland, Amsterdam