Some Conceptual Issues in the Testing of the Economic Theory of Consumer Demand

Penny Baron, University of Iowa
Gerald Eskin, University of Iowa
Cliff Lloyd, Simon Fraser University
[ to cite ]:
Penny Baron, Gerald Eskin, and Cliff Lloyd (1975) ,"Some Conceptual Issues in the Testing of the Economic Theory of Consumer Demand", in NA - Advances in Consumer Research Volume 02, eds. Mary Jane Schlinger, Ann Abor, MI : Association for Consumer Research, Pages: 679-692.

Advances in Consumer Research Volume 2, 1975      Pages 679-692

SOME CONCEPTUAL ISSUES IN THE TESTING OF THE ECONOMIC THEORY OF CONSUMER DEMAND

Penny Baron, University of Iowa

Gerald Eskin, University of Iowa

Cliff Lloyd, Simon Fraser University

The economic theory of consumer demand attempts to explain the choice among goods available in the market place based on preferences, prices and the consumer's income. This paper reviews some problems encountered when attempting an empirical test of this theory and discusses some promising methodologies to alleviate some of these difficulties. Some feasible controlled market place experiments are suggested based on these methodological considerations. Specific research sites, treatments, methods of control and measurement techniques are discussed.

How do consumers choose among the goods available in the market place? Although there are many partial answers forthcoming from various theories, only one theory claims great generality and comprehensiveness. It is the economic theory of consumer choice. In this theory consumption occurs in such a way that the selected goods bundle maximizes the consumer's utility subject to prices and income. This theoretical structure has existed more or less in its current form since Slutsky (1915), yet definitive empirical tests have not been made.

Our purpose. here, is to review some problems encountered when attempting to construct an empirical test of the theory and to discuss some promising methodolOgies to alleviate some of these difficulties. Our approach utilizes the Revealed Preference formulation of the theory and certain treatment manipulations in the context of controlled market place experiments. The treatments discussed combine price changes with income compensation. Under these particular treatment manipulations, practical testing of the theory appears to be feasible.

I

THE TRADITIONAL THEORY OF CONSUMER BEHAVIOR

The basic idea of the economist's theory of consumer behavior is that the consumer chooses, from among those things that he can afford, the ones that he prefers. It is supposed that his preferences have a certain regularity of structure (i.e. that he prefers more to less, that if one bundle is preferred to another and the second is preferred to a third then the first is preferred to the third etc.). On the basis of these assumptions a complex and fairly rich characterization of the consumer's behavior may be derived.

Consider for example the case of an individual who is trying to decide how much he will buy of two commodities,the quantities of which we shall designate as X1 and X2. Suppose that the price of commodity 1 is P1 and that of commodity 2 is P2. Then if the consumer had Y dollars he could just afford any combination of one and two such that the equation was satisfied. This would be any point in or on the boundary of the triangle in Figure 1.

P1X1 + P2X2 < Y

FIGURE 1

The triangle is the set of combinations that the consumer can afford and its boundary, the line Y/P1 to Y/P2 is his budget restraint representing the equation

P1X1 + P2X2 = Y

The assumptions that are traditionally made regarding the consumer's preferences imply that these may be represented by a set of indifference curves as in Figure 2. Each such curve represents a set of combinations of X1 and X2 such that the consumer would be indifferent between them. Thus the consumer would be indifferent between points A and B. He would not prefer either of them over the other. Each such curve also divides the entire commodity space (the X1, X2 plane) into three regions. One such region is the set of points below the indifference curve. Every point in this region is inferior to every point on the curve. Thus in Figure 2 points A and B are both preferred to point D. Another region is the set of points above the curve, every one of which is preferred to every point on the curve. Accordingly C is preferred to both A and B in Figure 2. The third region is, of course, the curve itself.

FIGURE 2

The problem of finding the preferred combination of commodities consistent with a given budget is simply identical with that of finding the highest indifference curve that the budget restraint touches. Thus in Figure 3 the consumer will choose point A consuming X1 of commodity 1 and X of commodity 2.

FIGURE 3

II

EMPIRICAL IMPLICATIONS OF THE THEORY

To see what sort of implications this theory has, let us consider a consumer confronted with a price change. Suppose that for the price line 0 in Figure 4 the consumer chooses the point A. Now suppose that the price of commodity 1 falls so that the budget line becomes less steep like the price line 1. It can easily be seen that the consumer could buy more or less of either commodity after this price change than before it. He will choose some point between the two points "a" on line 1 but which such point our theory does not predict. Now suppose that simultaneous with the fall in the price we had "compensated" the individual for the price change by altering his purchasing power so that he could just buy after the price change what he did buy before it. This would yield the budget line 2 in Figure 4. The consumer confronted with this "compensated variation" in price would necessarily choose some point between point A and b on line 2 in Figure 4. But every such point is to the right of A. The consumer in this case would have to buy more of the commodity the price of which had fallen or our theory must be wrong.

FIGURE 4

The choice process described above is presumed to lead to a demand relation

(1) Xi = fi (P1,...,Pi,...Pn,Y).

The empirical content of the theory must concern the observable quantities (X, P, Y) and the function f which can in principle be determined by the observation set (X, P, Y).

It is often assumed that the preference ordering which resulted in the demand function (1) is describable by a utility indicator which is twice differentiable, and convex in all arguments. It can be shown that

(2)    EQUATION

where the expression EQUATION represents the ratio of differentials EQUATION for the situation: dPj 0, dPk = 0 for all k j and dY = dPJXJ. That is the expression measures the change in consumption of some good X. given a price change for good j under conditions of income compensation of the type illustrated in Figure 4. We shall subsequently write this expression

EQUATION

the conditionality dY = dPjXj being implicitly assumed.

It can be shown that certain restrictions on f and EQUATION are implied by the preference (utility maximation) model. For the two good case these restrictions are:

(3)    EQUATION

(this is the property being described in Figure 4)

(4), (5)  and (6) EQUATION

Results (3) through (6) and their generalization to n variables constitute potentially refutable implications of traditional demand theory. These results have been known, more-or-less since the work of Slutsky in 1915. Still they have not been conclusively tested. In the section that follows we consider some reasons for this state of affairs.

III

PROBLEMS IN TESTING CONSUMER THEORY

The empirical testing of the theory of consumer demand involves several substantive problems. We discuss below the most important ones for our purpose.

Aggregation

The empirical implications of the theory are about the behavior of the individual consumer confronted with price and income changes. The theory has little to say regarding the behavior of groups of consumers. This is not to say that one can not use it to understand the actions of groups, but rather that one can not test it by observations of the behavior of aggregates of people. The reason for this is simply that the income variables relevant to consumption purchases are individual incomes (Hicks, 1946; Lloyd, 1967) . It matters who has the money. This means that research based on aggregated data is not directly relevant. [But see Brown and Deaton (1972) for a review of the current state of this literature and its confusions.].

Koo (1963) and Koo and Hasenkamp (1972) have attempted to test the theory using diary records of all food items purchased by individual families in a consumer panel. This procedure avoids much of the aggregation problem but is subject to numerous kinds of reporting errors. Battalio, Kagel et. al. (1973) report a study in a controlled "token economy" which relies on the direct observation of all the purchases by each individual consumer.

Required Controls

In order to test the theory, it must be possible to observe the individual's reaction to single price changes for one commodity (or possibly a composite commodity). [A composite commodity is a group of commodities the prices of which always vary by the same percentage.] In a normal environment, without controls, numerous price changes over a variety of commodities will occur on any given day. In such a situation it would be impossible to attribute individual consumption adjustments to particular price changes as the theory requires. It is necessary to be able to control the goods and prices confronting the consumer. Whenever stocks run out,unmeasured price changes occur, or new products are offered for sale,the conditions underlying the predictions of the theory are not met. The consumer's choices in such situations can not be used to indicate whether he is behaving in a way that is consistent with the theory. These considerations suggest that one must either locate an actual economy which can be subject to controls, or build one's own temporary experimental economy. Battalio, Kagel et. al. (1973) chose to do the latter with notable success. The work by Koo (1963) and Koo and Hasenkamp (1972) suffers from a lack of controls over the actual prices and goods confronted bs their consumers.

The key consideration with respect to control is that the consumer be presented with a detectable opportunity to behave in a way that is unambiguously inconsistent with the theory. The situation in the usual market economy does not offer such an opportunity. Price changes are not only numerous, but small. Yet Battalio, Kagel, et. al. (1973) have shown that, if attention is given to errors in measuring the consumer's adjustment to these changes, only relatively large price changes produce reliable effects. A single relative price change which is also large, and which occurs in an economic environment where all of the other prices are constant, is not a very likely naturally occurring event. Deliberate experimental manipulation and control are necessary.

Requirement of Complete Observations

Some of the implications of the theory (in particular (5) above and the revealed preference notions to be discussed below) require that we observe all of the individual's purchases. This requirement is particularly difficult to meet if one must rely on records of consumption kept by consumers themselves rather than on direct observations, or if there are numerous places in which a consumer can choose to make his purchases. This consideration suggests that, if we wish to enumerate by consumer, all transactions at the point of sale, then a single purchase site is desirable. This requirement indicates that a very small and isolated economy is required.

Shifts in Preference

The consumer in a normal environment is frequently confronted with stimuli likely to alter his preferences. Advertisements bombard him constantly. Weather changes alter his clothing, his appetite and his use of appliances. New commodities are introduced into the set available to him. All of these forces are likely to alter his preferences, and results (2) through (6) hold only if preferences are constant. This problem can be mitigated by attempting to isolate the consumer from as many of these confounding influences as possible, and by employing experimental treatments which are robust enough to be little affected by small disturbances.

Measurement Error

Tests of consumer demand theory are very sensitive to possible measurement errors. Battalio, Kagel, et. al. (1973) found that, when allowance was made for measurement error, many cases which were apparently inconsistent with the theory, could not be distinguished from measurement error. Experimental procedures allow for the control of measurement errors, and the use of direct observations makes it possible to employ multiple measures on each transaction. Only multiple measurements allow for the assessment of magnitudes as well as direction of measurement errors.

Estimation and Approximation Errors

There are two basic approaches to testing the various implications of the theory (i.e. equations 3-6). The most frequently used method involves estimation of the parameters of some general demand function which has the property that for some values of its parameters equations (3) through (6) are satisfied while for others it is not. In this procedure, estimates are developed, then tested to see if they differ significantly from values that are consistent with the theory.

This approach assumes that a general class of demand functions is known; the problem is simply to discover which of two conditions (consistent vs. inconsistent) prevails. Unfortunately many of the functional forms used do not have sufficient generality. Hence, they impose undesired restrictions on preference and demand. Given these restrictions,when a null hypothesis concerning (3) through (6) is falsified there remains the possibility that it is the improper specification of the model within which estimation is conducted which has caused the result.

If the null is sustained, a specification error could also be responsible. Poorly fitting models tend to produce large error terms. When these estimated errors are used in the statistical tests. bias in favor "no difference" nulls result.

An alternate approach involves the calculation of ratios based on direct observations such as the ratio of demand changes forthcoming from a price change (EX/AP) or the demand change forthcoming from an income change (aX/LY). These ratios are considered estimates of the partial derivatives, UX/UP and UX/BY. Based on these estimates one can see if conditions (3) through (6) are satisfied.

It is well known that these ratios of discrete changes are not identical to the partial derivatives in most cases. An exception is the linear case, But unfortunately linear demand functions are not consistent with the utility maximization premise. There is then the possibility that nonsatisfaction of the various conditions in a particular data set could be due to approximation error.

As an example, consider the prospects for testing the negativity of the own substitution effect (equation 3)* In order to approximate (2) hence test (3) we would calculate the expression, EQUATION. If both (DX1/DP1) and (DX1/DI) are negative, our estimate of (3) will be negative and support the theory. If however opposite signs prevail, the total expression might turn out to be positive. This result might obtain in a situation where there was upward bias in the component with a positive sign or a bias towards zero in the component with the negative sign. An additional ambiguity concerns how to estimate the X term in this expression. It is not clear whether the estimate should be based on the quantity of good 1 before the price change or the quantity demanded after it. The theory does not tell us because it applies only to an infinitesimally small change in P and I, hence to small dX.

The above problems are inherent in a calculus treatment of a theory in which functional forms are not known. Next we consider a reformulation which does suffer from these difficulties.

IV

REVEALED PREFERENCE AND EXPERIMENTAL METHOD AS TESTING AIDS

Some of the above mentioned testing difficulties are alleviated by an alternative formulation of the theory, the revealed preference approach. Tests of revealed preference are based on the directly observable quantities (X,P,Y) as compared to the derivatives of the calculus approach. The usefulness of this"revealed preference" formulation is enhanced by the fact that it is implied by the classical approach. Indeed Uzawa [1960] has shown that satisfaction of the strong axiom of revealed preference is equivalent to the existence of a Slutsky-Hicks system of demand relations.

The basic idea of revealed preference is that if a consumer purchases a goods vector X1 when he could afford some vector X2, then it must be the case that he does not prefer x2 to X1. Also he will not buy x2 if he could afford X1. One might even be willing to assert that not only is it the case that x2 is not preferred, but that X1 is preferred to X2. This latter formulation is the basis of the strong-axiom of revealed preference. Assuming that X is bought at prices p1 and X at prices p2, a requirement of the axiom is:

(8)   (p1X1 > p1X2) -> (p2x1 > p2x2).

That is, if when the consumer buys X1 he could more than afford X2, then, when he buys x2 he could not afford X1.

Direct test of the theory is possible based on the revealed preference approach. For example, consider a consumer facing the budget line 1 in Figure 5 and who purchases the quantities designated by point A. Subsequently, if prices and income should shift in such a way that the new budget line is the one labeled 2 then, if he should select the point C, it could be concluded that his actions were inconsistent with the strong axiom of revealed preference. When faced with the original budget 1 point C was available, hence A was revealed preferred to C. Subsequent selection of C violates this preference ordering, hence falsifying the theory. If B had been selected, falsification could not have resulted. It would merely have established that B is a preferred point. Similarly, original selection of D and subsequent selection of C is consistent (because the consumer could not afford D when C was bought).

FIGURE 5

The above establishes that if a consumer was faced with income and prices resulting in budget sets 1 and 2 and if we could observe the consumption pattern under these two situations, then there exists the possibility of observing behavior inconsistent with the theory. Thus, conceptually, testing is possible. But just any arbitrary price or income variation will not necessarily provide data for testing. For example, consider a consumer faced with the budget restriction 1 in Figure 6. Now let there be a decrease in the price of X1 such that budget line 2 becomes the relevant restriction. Point B is revealed preferred to A by this manipulation but there does not exist a point on budget line 2 such that selection of that point is inconsistent with the revealed preference axiom. Price variations as in Figure 6 help establish the preference ordering but do not, alone,provide tests of the theory.

FIGURE 6

This suggests,that even abstracting from the numerous difficulties discussed above,actual market data may not always vary in ways that would allow proper testing. In general, the realism of the marketplace will involve relatively small variations in price of a capricious sort. However, when controlled manipulation of prices and income is possible, variables can be manipulated in ways that maximize the strength of the tests. Assuming this possibility exists, what testable propositions can be formulated using the revealed preference approach? The following are suggested:

1. Reversibilities--Consider some change in the budget constraint and corresponding shift from some point A to some point B. Next consider a reversion to the original budget line. Reversibility requires that the quantity consumed return to point A. This implication follows from the static nature of the theory. Although shifts in preference or lengthy adjustment processes could account for nonreversible behavior, the admission of such arguments renders the theory vacuous (i.e. nontestable). In the.research suggested below we seek conditions under which relative stability in preference should hold and argue that if nonreversibility is observed under these conditions, the theory has questionable utility as a guide to empirical analysis using time series data.

2. Homogeneity--If all prices and income are changed by the same proportion then no shift in consumption should result. This follows immediately from the observation that such manipulations do not alter the budget-restriction.

3. Negativity of the Compensated Price Effect--Consider a price change for some good Xi from p0i to P1i, all other prices unchanged. If an income compensation of the amount (P1i-P0i) X0i is given then the strong axiom of revealed preference implies that consumption of good Xi will fall.

That is X1i - X0i < 0.

The situation is as depicted in Figure 7 where a consumer originally at point A on budget line 1 must shift to the shaded area of budget line 2 when faced with the new budget situation, a shift to the unshaded area being inconsistent with the theory. Any such shift entails increased consumption of commodity 1. Any consumer, originally at point A on budget line 2 when subsequently faced with budget line 1 must shift to the shaded portion of that line, the unshaded area being inconsistent with the theory. These sorts of compensated price variations then have the following properties. For a compensated price decrease, each consumer must increase his consumption of the good in question. If he does not change the quantity consumed or if he decreases it, inconsistency with the theory is established. With such compensated price variations we are not faced with situations as in Figure 6 where all behavior patterns are consistent with the theory under some pairs of price and income configurations.

FIGURE 7

In a sense the compensated price variation maximizes the opportunities for the consumer to behave in an inconsistent manner if the consumer chooses to do so.

The exception to the above argument is the consumer who under the initial set of conditions purchases a zero quantity of the good in question. Here the situation is as in Figure 6. The value of income compensation is zero and inconsistency is not possible.

4. Acyclic Behavior-- The strong axiom of revealed preference requires that if C is revealed preferred to B and B is revealed preferred to A then it can not be the case that A is revealed preferred to C. Behavior satisfying this condition is said to be acyclic.

It is possible to construct a test using income compensation such that, under subsequent pairs of treatment, the negativity of (3.) could hold but acyclicity would be violated.

An example is given in Figure 8. Consider a consumer who purchases a positive quantity of both goods X1 and X2. Under a price increase for good X1 with appropriate income compensation, if this consumer satisfies (3.), observed consumption will be at a point such as B which will thus be revealed to be preferred to A. Next consider an increase in the price of X2, again income compensated, but now from the point B. If the price changes are such that the final budget line 3 intersects budget line 1 in the positive orthant, then three results are possible. Either condition (3.) is violated by the selection of a point to the left of point B or the selected point is revealed preferred to B. If the selected point is revealed preferred to B then it may be the case that it is a point such as C. That is, it is located in the shaded portion of budget line 3 In this case C is revealed preferred to A by the transitive nature of revealed preference. But it is also the case that A is preferred to C because C is in the interior of budget set 1 Experimental manipulations designed to produce budget sets as in Figure 7 and 8 could be employed in experimental tests of the theory. Two operational versions of these tests are as follows:

FIGURE 8

All commodities available to consumers at retail in an isolated local market are divided into 3 groups (C1, C2, C3). Compensated price variations are instituted in the manner described in Exhibit I.

Design I attempts to produce choice situations for consumers like those described in Figures 5 and 7. This design allows for two tests of the negativity of compensated price effect, comparison of purchase behavior under Treatment A with the behavior observed under treatment B and a similar comparison between treatments A and C. (See 3. above.) It allows for one test of reversibility, comparison of purchase behavior under treatment A during Period 1 with the behavior under treatment A during Period 3.

Design II is intended to produce situations like the one described by Figure 8. This design allows for two tests of the negativity effect (3.), AB and BC, and one test of reversibility, comparison of Periods 1 and 4 under treatment A.

EXHIBIT I

PRICE LEVELS BY COMMODITY GROUPINGS FOR EACH TREATMENT PERIOD

Additionally, this design allows one test for acyclic behavior, comparison of purchase patterns under treatments A, B, and C.

Design II seems somewhat more attractive on account of this additional testing possibility. However, its success depends on being able to actually produce a choice situation which looks like Figure 8 for many consumers. Since it requires very rapid adjustment to successive price changes, it is somewhat risky in the absence of information about adjustment lags. Design I is our preferred design for an initial investigation.

V

TESTING PROCEDURES

The above discussion has established the general requirements for a test of the theory. Proper manipulations of price coupled with appropriate income compensations have been shown to constitute appropriate test conditions for the theory within the context of a revealed preference approach. Designs for testing the propositions of the theory have been suggested (Section IV). It remains to be shown that the designs proposed are executable and that there exists a site which allows adequate experimental control and at the same time retains some external validity.

It is necessary to discover a "real world" site which allows for the necessary controls, where conditions are stable enough so that consumer preferences do not change, where the economy is closed, where it is possible to observe all market transactions for each consumer in the research population, and where all the relevant conditions of such transactions are controllable or measurable.

Procedures must exist to execute the required design. Adequate measures of dependent variables must exist which yield known and small measurement errors. It must also be possible to execute the required price treatments and income compensation over appropriate time horizons.

Below we briefly discuss some proposed research, currently being considered for funding, which attempts to meet these requirements.

Possible sites include the villages of Davis Inlet, Hopedale, Makkovik, and Postville, All are isolated villages along the northeast coast of Labrador, Newfoundland. Each village is served by a store run by the Newfoundland Government Department of Recreation and Rehabilitation. In each town no significant competition exists for the government store. Each of these stores receives a large shipment of goods by boat in early fall. Shortly thereafter very cold weather sets in and no significant shipments of goods can be sent in until the following summer. Prices are set on a cost plus basis and are constant through the winter.

Postville appears to be the preferred site for initial investigations. It has only the government store, with no competition of any sort. The population is estimated at 140. It is a permanent fishing village with a settled population of Scottish origin. The population earns its income by fishing during the period of open water and receives unemployment compensation during the long winter.

The community is extremely isolated and most villagers do not travel outside it during the winter. Accordingly, by controlling prices in the government store, all prices faced by the local consumers will be controlled. Permission has been obtained from a representative of the Newfoundland Government to use the government store as a test site.

Testing will be conducted during the period January 1 through April 31. This will avoid Christmas buying and allow completion of the study well ahead of the spring thaws. During this period the weather in the area is consistently very cold. Accordingly no taste changes due to seasonal variation should occur. There is no advertising done in the local market and the village is sufficiently isolated to be beyond the reach of external advertising. No shifts in consumer preferences will thus arise from advertising efforts. Obviously no new products will be introduced into the environment during its period of more-or-less complete isolation. Procedures are available for dealing with anticipated minor sources of external effect.

Treatments. The variable to be manipulated in this experiment is the price of certain goods with income compensation. Other relevant variables can either be controlled or measured.

Approximately 60 goods which are frequently purchased and which account for a reasonably large proportion of the consumers bi-monthly incomes will be selected for experimental manipulation. The selected goods will be divided in half in such a way that virtually everyone makes purchases within each grouping and that expenditures are approximately equal. Price manipulations will be made over each of these two product groupings. All the remaining goods will constitute a third grouping whose prices will remain constant throughout the course of the experiment.

Timing The treatment periods will be one month long. This will allow completion of the experiment in the time available. Each treatment period covers two bi-monthly pay periods which should allow some measurement of time effects.

Income Compensation. A continuous record of each consumers purchases will be maintained-via cash register tapes and records from tear off price tags. At the time of a price change each consumer's expenditure, during the previous period, for items whose prices are to be raised, will be determined exactly and the consumer will be given 15% or 25% of this amount depending on the treatment requirements. Consumers will be told at the start of this study that the general interest of the study is to gain a better understanding of how retail transactions are carried out and that some income adjustments to maintain consistent buying power may occur should prices go up.

Dependent Variable. The dependent variable is the purchase behavior of individual consumers under the various treatment conditions discussed above.

Measurement Error Direct observations of purchases will be used. Register receipts and inventory tags are the primary records. Errors should be minimal and will be calculated separately for each consumer. The size of measurement error can be determined by comparison of the two separate methods of measuring transactions. Since records are obtained directly during the transaction of a sale, many types of bias have no opportunity to arise. The records themselves should also suffer from minimal error since, in one case, the record is the actual cash register receipt which displays a completely itemized listing of each purchase, and, in the other case, it is the coded tags from each item purchased.

Data Analysis The analysis procedure is straightforward. The expenditure pattern of each consumer will be examined for changes in relative amounts spent for each commodity grouping across treatment periods. Compatibility of these expenditure patterns with revealed preference theory is verifiable using the obtained data on prices and quantity purchased. The number of consumers who reveal themselves to be consistent with respect to each proposition being tested will be determined.

REFERENCES

Battalio, Raymond C., Kagel, John H., Winkler, Robin G., Fisher, Edwin B. Jr., Bassman, Robert L., Krasner, Leonard. A test of consumer demand theory using observations of individual consumer purchases. Western Economic Journal, 1973.

Brown, J.A.C. and Deaton, A.S. Surveys in applied economics: Models of consumer behavior. Economic Journal, 1972, 82, 1145-1236.

Hicks, J. R. Value and capital. (2nd ed.) Oxford: Clarendon Press, 1946.

Kagel, J. H. Token economics and experimental economics. Journal of Political Economy, 1972, 80, 779-785.

Koo, Anthony Y. C. An empirical test of revealed preference theory. Econometrica, 1963, 31, 636-664.

Koo, Anthony Y. C., and Hasenkamp, George. Structure of revealed preference: Some preliminary evidence. Journal Political Economy, 1972, 80, 724-744.

Lloyd, Cliff. Microeconomic analysis. Homewood, Illinois: Richard D. Irwin, 1967.

Marimont. A new method of checking the consistency of preference matrices. Journal of the Association for Computing Machinery, 1959, 6, 67-71.

Neter, John. Measurement errors in reports of consumer expenditures. Journal Marketing Research, 1970, 7, 11-25.

Slutsky, E. E. Sculla teoria del bilancio del consumatore. Giornale degli Economistie Rivista di Statistica, 1915, 51, 1-26.

Uzawa, H. Preference and rational choice in the theory of consumption. In K. Arrow, S. Karlin and P. Suppes (Eds.), Mathematical Methods in the Social Sciences. Stanford: Stanford Press, 1960.

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