Schweizerische Zeitschrift für Volkswirtschaft und Statistik Revue suisse d'Economie politique et de Statistique

Herausgegeben vonder Schweizerischen Gesellschaftför Statistik und Volkswirtschaft Publiée par la Société suisse de Statistique et d'Economie politique

Redaktion/Rédaction: Prof. Dr. H.G.Bieri

112. Jahrgang/112e année Heft/Fasc. 2 Juni/juin 1976

An Experimental Test of a Simple Theory of Aggregate Per-capita Demand Functions

By R.L. Basmann, R. C. Battalio, J.H.Kagel, College Station, Texas

1. Introduction. Hypothesis

This article reports the first experimental confirmation of an oftmentioned and important empirical hypothesis about per-capita demand functions that is sug­gested (but not deductively implied) by Hicks's famous mathematical theorem on individual consumer demand functions for composite commodities1. Moreover, this article presents a simple explanatory theory of individual consumer demand that accounts for the aforementioned experimental confirmation of the hypothesis

* The experimental study reported here was conducted in collaboration with E. Fisher, L. Krasner and R. Winkler. We should like to thank the staff and administration of the Central Islip State Hospital without whose assistance this research would not have been completed. We also thank J. Trimble for helpful comments. J. Knight and M. Buchanan performed the computations. This experiment and its analysis was partially supported by N.S.F. Grant GS32057, "Interpretation Systems for Empirical Economic Theories of Consumer Factor Demand," (Principal Investigators: R.L.Basmann, R.C. Bat­talio and J. H. Kagel)and by U. S. Public Health Service Grant # 11938 (Principal Investigator, Leonard Krasner).

Dr. René Kästli (Bern) has made several suggestions which have been incorporated into the final draft As a result, the reader will be better able to appraise the significance of the experimental data in relation to the underlying economic theory with which our experimental research is ultimately concerned.

1 See Table 1 for classification of the types of system of demand functions that are referred to in this article. Systems of demand functions referred to the "northwest box" of Table 1 characterize the demand behavior of individual consumers for individual commodities. We call such functions micro demand functions.

Systems of demand functions referred to in the "northeast box" of Table 1 are constructed from systems of micro demand functions in accordance with the antecedent condition clause of Hicks's theorem. Hicks's mathematical theorem characterizes the conditions under which a system ofn Slutsky-Hicks consumer demand functions can be reduced to a smaller system of q composite functions that satisfies the Slutsky-Hicks definition of a system of demand functions. The antecedent condition clause

Schweiz. Zeitschrift für Volkswirtschaft und Statistik, Heft 2/1976

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about per-capita demand functions. One practical use for the proffered explanatory theory is the design of sample surveys of consumer demand in nonexperimental demand analysis.

The strategy of theory formation followed in this study has been previously de­scribed in this Journal; cf. Basmann (1975). Methods employed and described in the present article serve to exemplify and elucidate principles for the reader of the previous paper, and citations will be made thereto where pertinent.

Hicks's strictly mathematical theorem asserts that if the prices of a group of goods change in the same proportion, then individual equilibrium expenditure ofthat group of goods behaves as if it were expenditure on a single commodity, cf. Hicks (1946), pp. 312-313. Although Hicks's theorem does not deductively imply that per-capita demand functions for such composite commodities satisfy the Slutsky-Hicks defi­nition of a system of demand functions2, the experimental data in Table 2 tend, not only to confirm the empirical hypothesis that per-capita equilibrium demand functions for composite commodities satisfy the Slutsky-Hicks criteria, but also that, within a very small allowance for effects of observation error, per-capita equi­librium demand functions for composite commodities characterized by the very simple mathematical form.

A i > 0 , i = l , 2 , . . . , q

f At = 1 (la-c) i - i

^ = A i M P r l

agree perfectly with the experimental data. ("M" designates per-capita total ex­penditure on commodities and "Pj" designates the price-index for the group of

specified constancy of the price-ratios within a group of individual commodities as the exactly one criterion for definition of a composite commodity as referred to by the theorem. Although the ante­cedent clause of Hicks's mathematical theorem on demand functions for composite commodities is stated in terms of summation over a collection of consumer demand functions for individual com­modities whose prices vary in exactly constant proportion, if a definite system of demand functions is specified, then that theorem can be reformulated to hold for approximate constancy of price ratios in the sense of mathematical (e, S) analysis.

2 As the antecedent condition clause of Hicks's theorem does not even mention summation of consumer demand functions over consumers, it cannot deductively imply any conclusion whatsoever in respect of properties of such aggregate demand functions. In particular, Hicks's theorem does not deductively imply that such quantitative relations among aggregate or per-capita quantities, per-capita income, and prices, satisfy the Slutsky-Hicks definition of a system of demand functions. However, the empirical hypothesis, namely, that summations or weighted averages of individual consumer Slutsky-Hicks demand functions for composite as well as individual commodities themselves compose a system of Slutsky-Hicks demand functions has been widely conjectured by demand analysts and used as an auxiliary maintained (nontested) hypothesis in market demand studies.

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commodities composing the composite commodity, i. %" designates per-capita equilibrium demand for the composite commodity, i, and "Ai" designates an un­known constant.)

Table 1 Classification of Systems of Equilibrium Demand

Functions Referred to in this Study

Individual Consumers, Individual Consumers, Individual Commodities Composite Commodities (Slutsky-Hicks) • (Slutsky-Hicks)

Per-capita, Per-capita, Individual Commodities • Composite Commodities

1"-»w indicates direction of deductive inference in Hicks's theorem; Hicks (1946), pp. 312-313.

Table 21

Experimental Data

Period

t =

1 2 3 4 5 6 7

Commodity Per-capita Group

Xl

19.359 38.919 22.722 8.514 8.677

13.265 13.844

Purchases

x2

10.103 5.973 8.194

13.568 14.649 10.735 8.813

28

12.103 11.135 11.500 10.514 10.676 10.618 11.031

Per-capita Expenditures

M

41.564 42.595 42.417 34.324 35.351 34.618 33.688

Commodity Group

Pi

1.000 0.5012

1.000 2.000 2.000 1.000 1.000

Indexes

P2

1.000 2.000 1.000 0.500 0.500 1.000 1.000

Price

P3

1.000 1.000 1.000 1.000 1.000 1.000 1.000

1 Data rounded in the 4th place. The primary data used in the computations are available from the authors. All computations of the estimates of the basic constants were performed (double precision) using an IBM 360 at Texas A & M University, October, 1974, with a WATFIV compiler.

2 A discussion of the experimental procedure accounting for this index is contained in Battalio, Kagel, Winkler et al, 1973.

The usual variant of the popular empirical hypothesis in respect of equilibrium demand functions is as follows: If (A) individual consumers' micro demand func­tions ("northwest" box of Table 1) are Slutsky-Hicks demand functions and the antecedent clause of Hicks's mathematical theorem on composite commodités holds, then (B) per-capita equilibrium expenditure on a composite commodity

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(group of goods) behaves as if it were expenditure on a single commodity. (Here the grouping of individual commodities to form composite commodities is under­stood to be in accordance with the antecedent clause of Hicks's theorem, and with maintenance of constant weights in group price-indexes Pi,..., Pq.) The antecedent condition statement of Hicks's mathematical theorem specified empirical circum­stances, which—if actually realized—assists in making the empirical analysis of individual consumer demand feasible in practice by reducing the number of in­dependent price magnitudes to be taken into account, cf. Hicks (1946), pp. 33-34. However, thedeductive consequences of Hicks's antecedent condition are not alone sufficient to render empirical analysis of per-capita demand feasible in practice. The additional empirical hypothesis that per-capita demand for composite com­modities satisfies Slutsky-Hicks criteria—cf., Hicks (1946), p. 55—when conjoined with the antecedent condition of Hicks's mathematical theorem on composite commodities to form (A) above, does assist empirical analysis of per-capita demand to the limited extent that it reduces the number of independent prices required in quantitative analysis, however, and its "plausibility" is conventionally appealed to in support of conclusions of empirical demand studies based on nonexperimental aggregate market data. Circumstances forcing demand analysts to make do with such data, it would be unjust to condemn the convention as such. Still, it would be fallacious to let this methodological convention become a stylized "fact", viz. to conclude that the foregoing hypothesis applies to other methods of grouping commodities that are commonly used in nonexperimental econometric demand analysis, e.g. such group price-indexes and expenditures indexes as are routinely employed and published by the U. S. Bureau of Labor Statistics. Those groupings do not satisfy the antecedent condition of Hicks theorem. Those groupings are constructed for purposes other than scientific demand analysis, i.e. administrative and political purposes, and (for that reason) are not to be faulted for their frequent lack of suitability for scientific consumer demand studies. Such data cannot pos­sibly contradict the empirical hypothesis commonly appealed to in market demand analysis; consequently apparent agreement between the hypothesis and such data is (strictly) without scientific significance, cf. Basmann (1975), esp. pp. 164-169, pp. 173-174. Nonetheless, the experimental confirmation reported here can strength­en the aforementioned conventional appeal in nonexperimental demand analy­sis, to a limited extent, at least, since the experimental data satisfying the ante­cedent condition of the hypothesis were ex ante capable of contradicting its con­clusion but in fact did not do so. The concept of equilibrium demand Ci is that of a limiting magnitude to which actual consumption £ti converges with time t as commodity prices remain constant following an initial price change. The theory of variation of preferences among individual consumers refers to equilibrium de­mands. As a practical matter of testing (1 a-c) experimentally, it is essential that commodity prices be varied significantly and that the analysis of the experimental

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data allow for deviations between actual consumption Çti and equilibrium demand Çti. Accordingly we introduce auxiliary dynamical hypotheses as described below.

The hypothetical dynamical per-capita composite demand functions adjusted to the experimental data (Table 2) are characterized by (1 a-c) and

S« = Ç« + Bi fi(APti, Pti, Pt-i) (2a^c)

where

APti = P u - Pt-i,i5

the hypothetical per capita dynamical adjustment functions fi satisfy the condition

fi(0,Pti,Pt-i,i) = 0,

and the B4 are unknown real constants. Two alternative forms of fi are used in this study, namely,

fi = AP« (3a-b)

fi = P t l-1-Pt-i ,r1 .

The selection of these hypothetical forms of per-capita dynamical adjustment functions is briefly explained at the end of Section 2. Briefly, according to the dynamical adjustment hypotheses (2a-c), there is a nonzero deviation between actual per-capita consumption £ti and equilibrium per-capita demand Çti only if Pt| i is different from Pt_i, i.

Equations (2 a-c) and (3) take account of actual observations, and a hypothesis put forward, by Allyon and Azrin (1968 a, b) on the basis of a series of experi­mental studies of individual consumer behavior in token economies. Their obser­vations suggest that gradualness of individual consumer response to price changes is attributable to persistance of previous patterns of consumption (called reinforcer sampling effects); cf. Battalio et al. (1973), pp.424—426. Directly examining the period-to-period variation in individual consumers' consumption patterns we con­cluded—on an empirical rather than intuitive basis —that individual consumers' consumption of composite commodities cannot be adequately accounted for by variation in equilibrium demand alone; this raised the question whether the re­inforcer sampling effects for individual consumers are transitory, e.g. persist only a few periods, or are long lasting. Cf. Battalio et al. (1973), p. 426.

Equations (2 a-c), which refer solely to per-capita consumption and per-capita equilibrium demand, allow for individual consumers' adjustment processes having durations of more than one period. They imply only that aggregate effects of in­dividual consumers' adjustment processes tend to average out after one period. The question whether individual adjustment processes tend to average out in the immediate period following a price change is an auxiliary question relative to the

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main hypothesis in respect of per-capita equilibrium demand. The selection of the mathematical forms (3a-b) of the adjustment components has not been guided by any formal economic theory respecting individual adjustment processes and their aggregation. This selection, based instead on considerations of precision and research economy, will be explained in Section 2.

Thus confirmed, the empirical hypothesis about equilibrium per-capita demand functions is justifiable (in terms of research economy) for use as an experimental "fact" (to use a metaphor) for theoretical explanation3. Although the simple ex­planatory theory presented in Section 4 was formulated twenty years ago, in view of the lack of empirical price data known to satisfy the antecedent statement of Hicks's theorem and exhibiting sufficiently large period-to-period variations to make statistical estimation and testing of its deductive conclusions more or less precise, elaborations of the explanatory theory as a purely mathematical theory (say) for design of consumer sample surveys has not been a practical consider­ation previous to the experimental confirmations reported here4.

2. Data and Antecedent Experimental Conditions

Basic experimental data, methods, and test procedures are described in Battalio et al. (1972). A detailed summary of test procedures is given in our first paper, Battalio et al. (1973), pp. 420-421. The collection of micro commodities was par­titioned into three groups. Within each group, ratios of micro commodity prices were held constant. Token money prices of commodities within one group were held constant, while very large variations (relative to those variations usually en­countered in market prices) were made in the prices of commodities in the other two groups. Price indexes Pi, P2, P3 corresponding to the three commodity groups are given in Table 2. Each price index is a weighted average of individual com­modity prices and, in order to avoid spurious contradictions in the subsequent analysis, constant weights were used in the construction of price-indexes. Token money wage rates already established in the token economy before the experiment began were held constant. On ethical as well as technical grounds, experimental controls were designed to ensure freedom of consumers to accept or reject jobs, or to remain unemployed, as desired. The technical grounds for instituting such controls will be described at the beginning of Section 4. Briefly, the aim is to allow the distribution of earnings to reflect the distribution of individual consumer pre-

3 The empirical hypothesis confirmed by experimental data in Table 2 is not a fact in the ordinary, or metaphysical, sense of that word; cf. Basmann (1975), p. 167. New experimental data are ex ante capable of dis-confirming it and when opportunity for conducting a new experiment arises we intend to test the empirical hypothesis again - of course ! Cf Basmann (1975), esp. pp. 160-164.

4 Professor J. B. McDonald of Brigham Young University has started this theoretical development recently.

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ferences already formed during the six months period previous to the commence­ment of the experiment. Earnings of individual consumers varied and their ranks in the distribution of earnings varied considerably from week to week according to jobs individuals sought and performed. On the other hand, the empirical distri­bution (Lorenz) of earnings by ranks remained approximately constant from period to period; the empirical Gini coefficients varied from 0.4 to 0.5. (The Gini coeffi­cient is used here solely in its only established referential meaning, namely, as a rough measure of the degree of numerical approximation between a given Lorenz distribution and the uniform distribution.) Comparison of these experimental Lorenz earnings distributions with corresponding nonexperimental Lorenz income distributions for various national economies indicated no significant differences either in their form or in magnitude of their respective Gini coefficients. Notice, however, that reliable estimates of effects of observational error in the Lorenz distri­butions for national economies is lacking.

Micro data from the first stage of our token economy experiment (Battalio et al. [1973]), in which consumers are individual human beings, reliably confirmed that the Slutsky-Hicks theory of consumer demand actually applied to the micro demand behavior of the subjects. Furthermore, since commodities were grouped according to constant price ratios, the composite commodity data (Table 2) ac­tually is characterized by the antecedent condition clause of Hicks's mathematical theorem on composite commodities (Hicks [1946], pp. 312-313; Battalio et al. [1973], p. 421).

In this experimental investigation, residual deviations between the xti data in Table 2 and estimates of the functions (2) and (3) are attributable to exactly two sources: (A) observational errors, eti,

eti = x t l-Çti, i = 1,2, 3, t = 1 7 (4)

which are committed by the experimenters in recording consumer purchases and in classifying and counting the individualized tokens used as money in the experi­ment; and (B) to potential errors of contradiction, or disagreement between hypo­thesis and "fact". No part of the residual deviation between data and estimated demand functions is attributed to suppositious randomness in the economic be­havior of the individual consumers.

The concept of an observational or measurement error deductively implies that such errors cannot be determined exactly. Were such not the case, we would have subtracted such errors from the data shown in Table 2, of course, in order that they not be conflated with errors of the type (B). Instead, we shall have to follow the practice of other experimental sciences and estimate bounds for observational or measurement errors. The reader can validly appraise our conclusions only by knowing how observational errors are treated in this study. As the treatment is not a traditional method in econometrics, we describe it briefly.

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The measurement errors eti, i = 1, 2, 3, which are committed by experiment­ers, are treated as random variables. In forming a hypothetical distribution func­tion of eti, i = 1,2, 3 we take account of some empirical regularities encountered in a previous direct study of measurement errors in our experiment. (Experimental procedures were designed to facilitate estimation of errors of measurement in earnings and expenditures independently of hypotheses about demand functions.) To begin with the proportionate error Wt in the measurement Mt of individual consumer total expenditure had an empirical distribution partially characterized by:

E[|Wt|;Mt] = constant

in which the mean absolute proportionate error Wt, given Mt, varied (over in­dividuals) directly with M^,/2 approximately. Accordingly, we have required that the hypothetical distribution function of eti, i = 1, 2, 3 deductively imply that the conditional mathematical expectation of | Wt|, given Mt, where

„7 Ptieu + Pt2et2 + Pt3et3 /c. W t = ™ > w

Mt

be directly proportional to M-1^2. The physical operations of classifying and counting tokens in the measurement

of total expenditure and independently conducted physical operations for deter­mining within-group expenditures are the same and subject to the same causes of inaccuracy. Errors in Pti can be neglected; see Table 2. xti is obtained by divid­ing group expenditure by the group price-index Pti. An auxiliary statistical hypo­thesis (the most economical to use at the outset) is that the errors of measurement, eti, and error in Mt, are independently distributed for i and t5.

We have examined and used three alternative forms of hypothetical joint distri­bution functions of eti, i = 1, 2, 3; Basmann et al (1974), pp.20-26. The most suitable form is the distribution function characterized by the density:

5 The sum of the errors in classifying and counting tokens within each groups is not identical to the error in classifying and counting tokens in order to determine total expenditures. Let et be the error in measuring total expenditure (by putting all tokens in a single bag, classifying them by de­nomination and counting within denominations). The numerator of the definiens of definition (5) is not identical to et, except fortuitously. Although this auxiliary hypothesis is suggested by the description of the operations of counting and classifying tokens no attempt to "justify" it on the grounds of its plausibility is intended. In principle the hypothesis is corrigible, of course, and estimates of hypothetical correlations among eti, i = 1, 2, 3 and of hypothetical serial correlations among eti, t — 1, 2, ..., i = 1, 2, 3 are not difficult to carry out. However, the practical significance of such estimates, parti­cularly estimates of series correlations is greatly diminished by their sensitivity to accumulated com­putational and observational errors.

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j0(eu, e«, et3) = ft (?«)-* Cg exp {-£ ^ - } ' (6a-b) 1=i" i-1 Z

Cti = K 2 P 2 1 M I 1 .

In other words, the quantities

M'1Zti, i = 1,2,3

where Zti is the error in measurement of the proportionate per-capita expenditure PtiÇti/Mt, are independently and identically normally distributed. From this it follows deductively that the conditional expectation of |Wt|, given Mt, is

EClW.hMJ-jf^ (7)

as required above.

Substituting (la-c), (2), and either of (3 a) or (3 b) for "Çti" in (4) one obtains from (6a-b) a definite likelihood function, from which estimates of Ai, Bi, i = 1,2,3 and K are computed from Table 2 by routine procedures6. Estimates are presented in Tables 3, 4 and 5.

The unknown constant K is not a constant of either individual or per-capita demand functions, of course; rather its magnitude is a property of the experiment­ers' observation and measurement procedures. However, its maximum likelihood estimate is used in conjunction with (7) and the independent estimate of average absolute proportionate error of measurement of total earnings, namely, 0.035, in the following way: If the maximum likelihood estimate of K results in a maximum likelihood estimate ofminimum E[| Wt|; M J that is greater than 0.035, then (per­haps) consideration of more complicated hypothetical forms of equilibrium de­mand functions and adjustment functions would not be unwarranted by the data. However, it the maximum likelihood estimate of minimum E [ | Wt| ; Mt] is smaller than 0.035, then (for example) to complicate the demand function hypotheses by (say) additional variables in order to obtain an even closer arithmetic "fit" to the numbers in Table 2 would be, at best, merely a numerical econometric exer­cise, and potentially misleading, cf. Basmann (1975), esp. pp.173-1747. The inde­pendent estimate of average absolute proportionate error of measurement of total

6 Maximum likelihood estimates of the equilibrium demand function constants Ai, i = 1,2,3 under the hypothesis that Bi = 0, i = 1,2,3 under the alternative hypotheses that eti, i = 1, 2 ,3 are distri­buted in accordance with (a) the double exponential frequency function and (b) the uniform frequency function, are presented in Basmann et al. (1974), pp. 20-28.

7 For a typical example consult Basmann et al. (1973).

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expenditures and earnings, 0.035, compares favorably with estimates of propor­tionate error in similar counting and classifying operations in other laboratory sciences.

Selection of forms (3 a) and (3 b) of hypothetical per-capita adjustment functions can now be explained briefly. Elsewhere in our sequence of experiments we are investigating the process of individual consumers' dynamical adjustment of com­modity purchases to equilibrium demand, and the forms (3 a) and (3 b) have orig­inated in the context of planning such special experiments. Of the forms con­sidered in that context, (3 a) and (3 b) are the most suitable for the immediate pur­pose here, namely, to make allowance for dynamical deviations of per-capita con­sumption Çti and per-capita demand Çti in the following sense: Given the statisti­cal hypothesis (6a-b) and the method of maximum likelihood, the greatest pre­cision of estimating (2a-b) is obtained in case the magnitudes Mt and Ptifi(APti, Pti, Pt-i, i) are "uncorrected", i.e., the sample vectors are orthongonal. The sim­plest forms (3 a) and (3 b) satisfy this condition to a close approximation. For i = 1, the "correlations" are -0.116 for (3a), and -0.0237 for (3b); for i = 2, the "cor­relations" are 0.185 and —0.0671. Consequently, this choice of hypothetical per-capita adjustment functions makes allowance for dynamical deviations between actual per-capita consumption £ti and per-capita equilibrium demand Çti at an essentially zero cost in terms of precision in estimating the per-capita equilibrium demand functions.

Since the adequacy of either of the auxiliary hypotheses (3 a) and (3 b) could be determined ex post, no effort was expended on their theoretical rationalization ex ante; cf. Basmann (1975), p. 160. Hopefully our current sequence of experiments specially designed for the investigation of dynamical adjustments to equilibrium of an explanatory theory for per-capita dynamical adjustments will correspond to that proffered in Section 4 in respect of per-capita equilibrium demand.

3. Test Results

Table 3 presents maximum likelihood estimates of the equilibrium demand function constants Aly i = 1, 2, 3 and K under the hypothesis that there is no reinforcer sampling effect, viz. Bi = 0,i = l , 2 , 3 . Call this hypothesis So- The interval estimates (a = 0.95) on Line 1(a) are exact, cf. Basmann (1974), pp. 210-211; they are free of so-called "nuisance parameters", computed from Student's t distribution, and (transparently) free of suppositious econometric "small sample bias". Point estimates on Line 1(b) satisfy the Slutsky-Hicks hypo­thetical restriction (1 b) exactly to within one unit in the seventh place of decimals ; this is computational accuracy only, and is without empirical significance since (lb) was imposed on the point estimates. The imputed estimates of the average

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Table 3 Maximum Likelihood Estimates of Constants of Per-capita Demand Functions

for Composite Commodities under the Hypothesis ôo (la-c)

Ka) (b)

II

III

Composite Commodity

1

0.427 g Ai ^0.503 0.465

2

' 0.204 g A2 = 0.279 0.242

3

0.255 £A3£ 0.33.1 0.293

K = 4.24

(min) 0.050 E(|Wt|;Mt)

(max) 0.056

absolute proportionate error of expenditure measurement, E [ | Wt | ; M J are shown on Line II of Table 3. Since the estimate of minimum E[| Wt|; M J is larger than our independent estimate, 0.035, a likelihood ratio test of the foregoing hypothesis So against either of the alternatives (3 a) or (3 b) is potentially able to disconfirm So without attributing an excessively small magnitude of E[|Wt|; MJ under the alternatives to the disconfirmation or confirmation of Ho.

Table 4 Maximum Likelihood Estimates of Demand Function Constants and Error

Distribution Constant under Hypothesis $i (la-c)-(2)-(3a)

I

II

III

Ai =0.465! A2 = 0.241 A3 = 0.293

Bi =0.06152

B2 = 0.0552

K =6.39

(min) 0.033 E(|W t |;M t):

(max) 0.037

1 Sum of unrounded estimates satisfies (1 b) to fourth place of decimals. Hypothetical restriction (1 b) was not imposed on these estimates.

2 Since Pt3 is constant for all t = 1, 2 , . . . , 7,- no estimate of B3 can be computed, nor is it needed, of course.

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Table 4 presents maximum likelihood estimates of the unknown demand func­tion constants Ai? i = 1,2, 3, Bi} i = 1, 2, 3 under (2) and (3a), and of the error distribution function constant K. Notice that differences between the estimates of Ai, i = 1, 2, 3 in Tables 3 and 4 are negligible as implied by remarks at the end of Section 2; however, we did not impose the Slutsky-Hicks hypothetical re­striction (lb) on the estimates of Ai shown in Table 4, so the agreement between the hypothesis (1 a-c)-(2)-(3a) and observation is significant in respect of (1 b).

Let Si designate the hypothesis characterized by (la-c), (2) and 3(a). We have performed a likelihood ratio test of So against Si.

The likelihood ratio for the test of So against Si is

LR = 1.27. (9)

Under the hypothesis So, the derived exact hypothetical probability of the event characterized by "LR=: 1.27" is very small, i.e.

Pr{LR=tl.27|S0}<0.001, (10)

Table 5 Maximum Likelihood Estimates of Demand Function

Constants and Error Distribution Function Constant under Hypotheses S2 (la-c), (2), (3c)

I

II

III

Ai = 0.465

A 2 = 0.241

A 3 = 0.293

Bi = 0.0516

B 2 = 0.0634

K = 6.28

E(|Wt|;Mt):

(min) 0.034

(max) 0.038

cf. Handbook of Tables for Probability and Statistics (1966), p. 246. Transparently, the foregoing test is free of econometric "small sample bias". Accordingly, we con­sider the hypothesis Si to be confirmed against So by the experimental data.

Table 5 presents estimates of the unknown demand function constants Ai, i = 1, 2, 3, Bi, i = 1, 2, 3 under the hypothesis S2 characterized by (la-c), (2) and (3 b), and the measurement error distribution function constant, K. Notice that the estimated demand function constants differ from Si only in respect of Bi, i = 1, 2, as is deductively implied by remarks on design at the end of Section 2.

The likelihood ratio for test of S2 against So is:

LR=1.19, (11)

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and sufficies to confirm S2 against So. Under So the derived exact hypothetical probability of the event characterized by "LR = 1.19", is very small, i.e.

Pr{LR=:1.19So}<0.001. (12)

Compare the imputed estimates of average absolute proportionate error E[|Wt|;Mt] of income measurement among So, Ôi and S2. Taken alone, the estimated minimum value 0.05 and maximum value 0.056 E [| Wt|; M J for So are not sufficiently greater than the independent estimate, 0.035, to warrant the attempt to improve the adjustment by adding additional variables to hypothetical per-capita equilibrium demand functions, or complicating the form of the latter. Those are still relatively small estimates for the counting operations involved. However, in addition to the fact that reinforcer sampling has been detected in the individual consumer data, neither of the dynamical adjustment hypotheses characterized by (2) and (3 a) or (3 b) to allow for dynamical adjustments introduces any significant degree of mathematical or syntactic complexity relative to the hypothetical per-capita equilibrium demand functions (la-b), nor does either of them proliferate suppositious variables that "plausibly might affect consumption". Moreover, in respect of economy in statistical analysis, the introduction of Si and S2 as alter­native hypotheses to So does not complicate matters by forcing the use of "large sample tests" based on conceptually infinite samples, nor do they significantly in­crease the cost of computing estimates.

So much for the ex ante justification for introducing Si and S2 as hypotheses alternative to S0. Ex post, their introduction is justified, partly by the outcomes of the respective likelihood ratio tests, and partly by the fact that neither of them requires, by deductive implication, the acceptance of an imputed estimate of aver­age absolute proportionate error containing a negative bias relative to the inde­pendent estimate. As shown in Table 4 and Table 5 the maximum likelihood esti­mates of minimum and maximum E[| Wt|; M J under the Si and S2 are in very close agreement with the independent estimate.

In view of the equally close agreements of Si and S2 in respect of derived maxi­mum likelihood estimates of E [| Wt|; M J, there is no ex ante justification for an attempt to test one against the other using the experimental data under consider­ation here. Nor is it feasible to attempt to do so. For the specific purpose of making precise tests of alternative dynamical adjustment hypothesis such as (2) and (3) design of a rather different sequence of price changes (from that displayed in Table 2) is required. What the results do support, however, is the hypothesis that dynamical adjustments of per-capita consumption of composite commodities to equilibrium demand tends to be rapid.

Referring to the question of agreement of hypothesis (la-c) with nonexperi­mental market data, we offer the following conjectures:

If it is assumed that nonexperimental market data commonly used in orthodox

166

econometrie demand analysis are beset by proportionate errors of measurement having like magnitudes to those encountered in controlled experiments, then the simple per-capita equilibrium composite demand functions (la-c) together with simple dynamical adjustment equations like those in (2 a-c) and (3a-b) can in many cases, at least, fit the market data with perfect significant agreement. Fur­ther, if allowance is made for the failure of composite market data to be grouped in accordance with the antecedent condition of Hicks's theorem on composite commodities —e.g. if an average absolute proportionate error of (say) 0.05-0.10 in measuring per-capita income or total expenditure is assumed —then the class of situations in which the simple per-capita composite demand hypotheses are in perfect significant agreement with available market data is greatly increased. Pre­cision of parameter estimation is likely to be poor, however, because period-to-period variation of market prices and group-indexes tends to be of the same magni­tude as errors of observation; cf Ciecka (1970), Chapter 3.

4. A Simple Explanatory Theory That Accounts for the Empirical Regularities

From the first part of our token economy experiment (Battalio et al. 1973) it was apparent by direct inspection of all micro data that, although individual con­sumer demand functions for individual commodities are Slutsky-Hicks demand functions, they require very complicated mathematical characterizations. Like­wise, the individual consumers' demand functions for composite commodities would require rather complicated mathematical characterizations. Although the per-capita composite demand functions are defined as weighted averages of the latter, an attempt to account (deductively) for the form (la-c), which the empirical per-capita equilibrium composite demand functions are found to possess, by direct aggregation is not at all promising from the point of view of research economy. Even if we were to estimate forms of unknown constants for each individual con­sumer's system of composite demand functions, direct aggregation would result in an explanatory theory far too unwieldy for practical use.

However, the fundamental explanatory paradigm used in statistical mechanics is perfectly suited to our present purpose8. We introduce a statistical hypothesis to characterize the variation of preferences among individual consumers. Differen­ces of preferences affect individual consumers' equilibrium micro demands (Xr

8 The empirical regularity (called Boyle's Law for gases), namely

(Pressure) x (Volume) = k (Temperature),

can be derived from Newtonian mechanics without resort to hypotheses about the statistical distri­bution of momenta of individual gas molecules. This nonstatistical derivation is sometimes presented in elementary textbooks of physics; cf. Riply (1964), pp.203-204. In our use of the paradigm, the ob­jective and the clues to the suitable form (14 a-c) parallel exactly those used in specifying the hypo­thetical (normal) distribution of the momenta of molecules of ideal gases; cf. Lindsay and Margenau (1957), pp.218-230, esp. pp.227-230.

167

X2,...,Xn)forn-commodities, and individual consumers' equilibrium total expen­diture9, m,

m = piXi + ... + pnXn. (13)

Consequently, we shall introduce a hypothetical relative frequency function g(Xi,..., Xn) with individual consumer equilibrium micro demands Xi, ..., Xn

as arguments, micro prices pi, ..., pn, and the maximum individual consumer equilibrium expenditure, y, as known parameters.

Transparendy, the hypothetical frequency function g(Xi,..., Xn) must satisfy the following restriction:

gPCl5...,Xn)>0, Xi^O,

0<piXi + ... + pnXn^y, (14a-b)

= 0, otherwise.

In our experiment prices Pi? i = 1,2,..., n, of micro commodities actually satis­fied the antecedent conditions of Hicks's theorem on composite commodities, viz.

Pj =XiPi fi = l ,2 f . . . fni

Pj = X.jP2,j = ni + 1,..., n2, (15a-c)

Pk = ^kP3,k = n2 + l , . . . ,n.

and magnitudes of corresponding composite commodities are defined by

Xi = A4X1 + ... + A,niXni

X2 = ^ m + i X n i + 1 + ... + A,n2Xn2 (16a-c)

X3 = /-n2 + lXn2 + l + ... + A,nXn;

where À,s, s = 1,2,..., n is fixed throughout the experiment, cf. Battalio et al. (1973), pp. 420-421. To successfully account for the form (la-b) of the experimental per-capita composite demand functions, the specified form of (14a-b) must deductively imply (only in the strict sense of modern logic, cf. Basmann (1975), pp. 157-158; also p. 154) that the conditional expectation ofXi given equilibrium total expenditure, m, be a linear function of m only. For reasons of research economy in practical applications we required that the derivation of the exact marginal frequency func­tions of the composite magnitudes Xi, X2, X3, be tractable cf. Basmann (1974), pp. 210-211, and amenable to numerical calculation of cumulated relative frequen-

9 In this connection we mention another experimental regularity in our data, namely, that the em­pirical distribution functions of individual consumers' total expenditures, m', can be closely approxi­mated by a hypothetical beta distribution function of the form (20a-c) below. However, due to the effect of dynamical adjustments, the parameters a, b vary from period-to-period, viz. a varies from 0.50 to 0.68, b varies from 1.3 to 2.2.

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cies. The class of frequency functions characterized by (14a-b) and the following density function of equilibrium quantities Xi,..., Xn demanded by individual con­sumers best satisfies the requirements states above10:

oti>0,b>o,i = l ,2, . . . ,n (17a-c)

Pl a lP2 a 2 . . .Pn a n g(X1,X2,...,Xn) =

(r«l+... + «n + b"1' y-i -r ... T - n T - B(ai,...fowb)

x(X1«1-1X2«,-1 . . .Xn«n-l)

x ( y - p i X i ~ . . . - p n X n ) b - 1 ,

0 < p i X i + ... + p n X < y ,

= 0, otherwise;

nf M r(a1)r(a2)...r(an)r(b) B ( a i , . . . , a n ; b ) = — — — • • — —

r(ai + ... + an-hb) and

oo

r(a) = Jer^u^du, a>0 . o

[For a convenient numerical table of the gamma function T(a) consult Selby (1974), p. 533.]

From (14a-b)-(17) it follows that the marginal frequency function of the indi­vidual consumers' equilibrium demands (16 a-c) for composite commodities is

ai>0, i = 1,2, 3 b>0 ( 1 8 a _ g )

g(Xi X2 X,) = P ? l P t 2 P 3 3 X?i-iXf2-iXg3-i B(ai, a2 , a3 , b) y»i + a2 + «3 + b-1

x(y-PiX1-P2X2-P3X3)b-1

Xi>0

0<PiXi + P2X2 + P3X3<y;

g(Xi,X2,X3) = 0, otherwise,

ai = <Xi +.. . + ocni

a2 = ani+i... + <*n2

<*3 = 0tn2 + l + ... + ttn.

10 Elsewhere we have mentioned use of the beta density function to characterize a stochastic theory of individual consumer behavior, cf Basmann et al. (1974), pp. 5-8. Transparently in the present case

169

The transformation of variables used in the derivation of (18a-g) from (14a-b) to (17) is presented in standard textboods of differential and integral calculus, e.g. Gillespie (1951), p.94.

From (15 a-c) it follows deductively that

E(Xt|m)= ^ i m P f , i = 1,2,3. (19) ai + a2 + a3

This conditional expectation of Xi, given m, defines the per-capita equilibrium composite demand for i taken over the subclass of every individual consumer whose equilibrium total expenditure is m. Consequently the hypothesis character­ized by (17a-c) and the antecedent condition of theorem on composite commod­ities deductively implies that per-capita equilibrium demand functions have the form (ld-c).

It follows deductively from (18 a-c) that individual consumers' total expendi­tures m, are marginally distributed according to the relative frequency function

a = ai + a2 + a3>0, b>0,

h(m>= p/ M JLi» 0<m<y> (20a-c) B(a,b)ya+b"1

= 0, otherwise.

The distribution function H(m*, a, b), i.e. m*

H(m*;a,b)= J h(m)dm (21) oo

designates that proportion of all individual consumers whose equilibrium total expenditures do not exceed m*.

The mean of the expenditure distribution H(m; a, b) is given by mH = ——. (22)

a + b Substituting mH for the placeholder m in (19),we obtain the theoretical per-capita equilibrium demand for the composite commodity i:

a ^ P f 1

EfXilmn) = (23) ai + a2 + a3

i = 1,2,3. (14a-b) and (17) do not deductively imply that individual consumer behavior is random, or impulsive. For the present it suffices to mention that we have eschewed (as scientifically premature) the "hypo­thesis" that individual consumer behavior is random, even in part. Such a "hypothesis" is, of course, neither verifiable nor falsifiable, but—due to its logical structure —essentially a definition, and must be treated deductively as such as spurious contradictions are to be avoided, cf. Basmann (1975), p. 169.

170

In this manner the theory of the distribution of individual consumer preferences (17a-c) explains the form of the empirical per-capita demand functions.

Table 6 presents some preliminary estimates of the unknown constants of (18a-g), which have been computed from experimental data on individual con­sumers' expenditures. (The estimates are preliminary in the sense that their con­sideration is limited to assisting plans for future experimentation.) (20 a-c) has been fitted to each of seven experimental expenditure distributions; estimates of a and b in Table 6 are weighted averages of the seven weekly estimates of (20a-c), the weights being proportional to weekly per-capita expenditures. Variation of the experimental expenditure distributions from week-to-week is proportionately very small, coefficients of variation being 9.6 and 6.9 for a and b respectively. Experimental correlation between week-to-week variations in estimates of a and b is small, i.e. 0.2. These results were to be expected in view of the purposive experimental design; cf. Section 2.

The estimates ai9 i = 1, 2, 3 shown in Table 6 are obtained by equating aia-1

to the estimates Ai of per-capita demand functions shown in Table 4. What is chiefly of interest here is the large variability of individual consumer preferences, as indicated by the coefficients of variation, vi5 i = 1, 2, 3, in Table 6, part II.

Table 6 Preliminary Estimates of the Distribution

of Individual Consumer Preferences (18a-g)

I

II

ai = 0.27,

a2 = 0.14,

a3 = 0.17

VÌ CTffiXi)

vi = 1.0

v2 = 0.7

v3 = 0.8

b = 1.7

These experimental results suggest that the conceptually "representative con­sumer" of conventional consumer demand theory is not significantly represen­tative at all11. The system of per-capita composite demand functions (19) derived

11 The term "representative consumer" has been widely used by demand theorists, but without an explication that is referentially adequate either for additional theory development or for practical use in economic and social policy. Its chief use has been (so far, at least) in stating a methodological con­vention that excludes from orthodox demand analysis the empirical study of systems of micro demand functions such as those listed in the "northwest box" of Table 1. For instance:

To assume that the representative consumer acts as an ideal consumer is a hypothesis worth testing;

171

from (18a-g) affords a definite scientific explication of the concept of "represen­tative consumer". The relatively small experimental coefficients of variation in Table 6 afford a partial measure of the degree to which the representative con­sumer is representative of actual individual consumers', cf Basmann et al. (1974), esp. pp. 7-8. These results afford an ex post justification of undertaking the system­atic study of variation of individual consumers' demand behavior; cf Basmann (1975), pp. 173-174. The ex ante justification for studying any experimental distri­bution qua distribution is that a single moment, such as a mean, is generally unrepresentative; as an elementary point of mathematics, the form as well as other properties of a distribution cannot be deduced from any finite sequence of its moments.

In closing, we note that the empirical success of the simple form (1 a-c) of per-capita demand functions against reliable, experimental data, alone provides an ex post confirmation of the importance of investigating individual consumers' de­mand behavior. Moreover, we call attention to the fact that the hypothetical dis­tribution of individual consumer demands (17a-c) does not deductively imply that individual consumer demand functions satisfy the Slutsky-Hicks criteria; nor does (17a-c) deductively imply that individual consumer demand functions do not satisfy those criteria, cf. Battalio et al. (1973), pp. 422-427. For several practical reasons our general theory of consumer preferences does not include a universal premise—cf. Basmann (1975), p. 167 —to the effect that individual consumer de­mand functions are of the Slutsky-Hicks type, although in some applications that state of affairs may be deduced as a conclusion. While a discussion of the afore­mentioned decision is beyond the scope of this paper, we note that it is based, in part, on previous experimental results in perception and learning.

to assume that an actual person, the Mr. Brown or Mr. Jones who lives around the corner, does in fact act in such a way does not deserve a moment's consideration. Hicks (1956), p. 55. For a recent methodological interpretation of this orthodox convention see Brown and Deaton

(1972), p. 1168. Presumably, Hicks and the authors just cited did not intend their statements to be taken as summaries of results of previous scientific investigations of individual human consumer be­havior, since few, if any such studies have been made, but merely as personal methodological resolu­tions concerning the limits of orthodox economic inquiry. Transparently, to know how representative the "representative consumer" actually is presupposes knowlege of a distribution function such as is exemplified by (14 a-c).

172

References

Ayllon, T., and N.H.Azrin: The Token Economy: A Motivational System for Therapy and Rehabilitation. New York: Appleton-Century-Crofts, 1968 (a).

- "Reinforcer Sampling: A Technique for Increasing the Behavior of Mental Patients," Journal of Ap­plied Behavioral Analysis, No. 1 (Spring, 1968), 13-20 (b).

Basmann, R.L.: "Exact Finite Sample Distribution for Some Econometric Estimators and Test Statis­tics: A Survey and Appraisal." In Intriligator and Kendrick (1974), 209-270.

- "Modern Logic and the Suppositious Weakness of the Empirical Foundations of Economic Science," Schweizerische Zeitschrift fur Volkswirtschaft und Statistik, Heft 2 (1975), 153-176.

Basmann, R.L., R.C.Battalio, J.H.Kagel: "Comment on R.P.Byron's The Restricted Aitken Estima­tion of Sets of Demand Relations'", Econometrica, Vol.41 (1973), 365-370.

- "Aggregate Per-capita Systems of Demand Functions," (mimeo) N.S.F. Project GS32057, Technical Report No. 17 (1974).

Battalio, R. C, J.H. Kagel, R. C. Winkler, E.B.Fisher, Jr., R.L.Basmann, and L.Krasner: "Central Islip Token Economy Experiment I. Sources and Methods of Data Collection and Processing," N. S. F. Project GS23057, Technical Report 1 (mimeographed), Texas A & M University, July, 1972.

- "A Test of Consumer Demand Theory Using Observations of Individual Consumer Purchases," Western Economic Journal, Vol. 2 (1973), 411-428.

Brown, A. and A.Deaton: "Surveys in Applied Economics: Models of Consumer Behavior," The Eco­nomic Journal, 82, No. 328 (December, 1972), 1145-1236.

Ciecka, J.E.: Ph.D. Dissertation. Purdue University, 1970. Gillespie, R.P.: Integration. London: Oliver and Boyd, 1951. Hicks, J.R.: Value and Capital, 2nd Ed., Oxford: Clarendon Press, 1946. - A Revision of Demand Theory, Oxford: Oxford University Press, 1956. Intriligator, M.D., and David Kendrick (eds.): New Frontiers in Quantitative Economics. Amsterdam:

North-Holland Publishing Company, 1974. Lindsay, R.B., and H.Margenau: Foundations of Physics, New York: Dover Publications, Inc., 1957. Ripley, Julien A.: The Elements and Structure of the Physical Sciences. New York: John Wiley and

Sons, 1964. Selby, Samuel M. (ed.): Standard Mathematical Tables, 19th Ed., Cleveland: Chemical Rubber Com­

pany, 1971.

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Zusammenfassung

Experimentelle Untersuchung einer einfachen Theorie über die Gesamtnachfragefunktion pro Kopf der Bevölkerung

Eine einfache Form der Gleichgewichtsnachfragefunktion pro Kopf der Bevölkerung stimmt mit experimentellen Daten völlig überein - innerhalb einer kleinen Toleranzzone für Beobachtungsfehler. Diese empirische Regelmässigkeit kann deduktiv durch eine einfache Theorie der Verteilung von indi­viduellen Verbraucherpräferenzen, welche ebenfalls deduktiv für die empirische Verteilung individuel­ler Gesamtausgaben massgeblich sind, erklärt werden.

Da es Ziel der Arbeit war, das Präferenzverhalten des einzelnen Verbrauchers caeteris paribus -das heisst: unabhängig von den Bemühungen des Produktionssektors, Verbraucherpräferenzen in Übereinstimmung mit Produktions- und Beschäftigungsplänen zu verändern - zu untersuchen, ist die experimentelle Volkswirtschaft durch eine beinahe vollständige Kontrolle des Verbrauchers über das Beschäftigungsniveau charakterisiert.

Entwurf und Methode des Experimentierens werden klar dargelegt

Résumé

Test expérimental d'une théorie simple concernant la fonction de demande globale per-capita

Une forme simple de la fonction de demande équilibrée per-capita est entièrement conforme aux données expérimentales - dans la marge d'une zone réduite de tolérance réservée aux erreurs d'obser­vation possibles. Une telle symétrie d'ordre empirique peut être mise en évidence par déduction en appliquant une théorie simple de la répartition de préférences individuelles des consommateurs, les­quelles s'avèrent - également par déduction - déterminantes en matière de répartition empirique des dépenses individuelles globales.

Du fait que l'objectif du travail a consisté à analyser caeteris paribus le comportement préférentiel du consommateur individuel - c'est-à-dire indépendamment des efforts déployés par le secteur de la production, visant à modifier les préférences du consommateur dans le sens d'une harmonisation avec les plans de production et d'emploi -, l'économie expérimentale se caractérise par l'existence d'un contrôle quasi total exercé par le consommateur sur le niveau de l'emploi.

Tant la méthode que le type d'expérimentation y sont clairement exposés.

Summary

An experimental Test of a simple theory of aggregate per-capita demand functions

A simple form of per-capita equilibrium demand function fits experimental data perfectly within a small allowance for observational error. This empirical regularity is deductively explained by a simple theory of the distribution of individual consumers' preferences which also deductively accounts for the empirical distribution of individual total expenditures.

The experimental economy is characterized by nearly complete consumer control over employment levels, since the objective was to investigate preference behavior of individual consumers caeteris paribus, i.e. independently of the production sector's efforts to alter demand preferences in conformity with production and employment plans.

Design and method of experimentation are fully disclosed.