Measurement of Consumption Efficiency in Price-Quantity Space: A Distance Function Approach

Measurement of Consumption Efficiency in Price-Quantity Space: A Distance Function Approach

In standard consumer demand analysis, it is implicitly assumed that consumers behave optimally and, thus, efficiently. However, optimality is a restrictive assumption to make for consumers’ actual behaviour. This study moves away from this restrictive assumption and develops a theoretical model for the analysis of consumer’s inefficiency in price-quantity space.

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Standard consumer demand analysis assumes a priori that consumers always behave optimally, that is, they do succeed in obtaining maximum utility from given purchased commodities, or they do succeed in choosing the minimum quantities required for the achievement of a utility level. However, optimality is a restrictive assumption to make for consumers’ actual behaviour. As Afriat (1988) points out, “The ordinary theory of the consumer is based on utility – and unquestioned efficiency. Even when the utility is granted, perfect efficiency seems an extravagant requirement. The familiar volatilities of real consumers make such intolerance unsuitable.” (p. 252). It is then more reasonable to assume that consumers may not behave optimally and employ theoretical and empirical models that accommodate any departure for optimality, and, hence, inefficiency, and allow it to be measured.

The importance of studying inefficiency in consumption lies not only on the fact that optimal behaviour, and hence, efficiency, is a restrictive assumption to make for consumers’ actual behaviour. It also lies on the fact that consumer’s non-optimal behaviour has a negative impact on welfare levels. In particular, it has a negative impact on consumer’s welfare levels in terms of budget that was wasted and which could have been allocated to the satisfaction of other wants. In addition, over- consumption leads to increased and more industrialised production, which itself fuels over-consumption, through, say, advertising. This circle implies excessive use of natural resources and/or wrong allocation of them in the production of commodities, increased waste from both consumption and production, and a negative impact on social welfare.

The assumption of consumer’s non-optimal behaviour can be accommodated in the case of commodities, such as highly perishable foods, meat, fish and agricultural products. In such cases, consumers may be inefficient because they are making rough estimates of the volume of the commodities and the quantity combination of them that are enough for the achievement of some desired utility level: when consumers choose a commodity bundle, they choose it on the basis of their estimates of what commodity combination is the suitable one for their wants. Consumers may also be inefficient because they cannot predict the future exactly: since individuals’ every day lives cannot be programmed to the detail, it is not unexpected that a portion of the purchased quantities of the commodities are not consumed but – in the case of highly perishable foods that cannot be stored – are disposed of instead. Or it could be lack of information, awareness and responsibility from the part of consumers with respect to the full social costs of their consumption decisions that lead to excess purchases and spending, and consumption inefficiency. Thus, consumers may purchase a commodity bundle which is non-optimal: they could have bought less of all the commodity quantities (commodity inefficiency), thus reducing expenditures, and/or they could have re-allocated their expenditures by choosing a different quantity mix (allocative inefficiency), thus reducing expenditures even more.

In this context, the aim of this paper is to propose a theoretical framework for the analysis of consumer’s efficiency in price-quantity space. The theoretical model which is developed is based on the simple observation that consumer preferences are commonly defined over the consumption levels and no distinction is being made between the quantities of the commodities purchased and the consumption levels themselves, that is, it is implicitly assumed that the purchased quantities and the consumed quantities are the same. However, if it is assumed that consumers are free to dispose of any unwanted quantities of the commodities they have purchased, then it becomes possible to define a measure of efficiency of the consumers in their effort to mimimise expenditure for commodities. Past attempts to study consumption efficiency in price-quantity space have been based on revealed preference relations or money-metric utility functions in order to construct non-parametric or parametric efficiency indices (Afriat, 1967, 1988; Varian 1982, 1983, 1985, 1990).1 The focus of these studies, however, is on the examination of the goodness-of-fit of optimising models to actual data by measuring the departure from optimisation. Moreover, what is implied by these efficiency measures is that inefficiency occurs because a portion of the consumers’ budget is wasted, and not a portion of the purchased quantities. However, it is this latter assumption that allows the construction of a measure of what we define here as commodity efficiency. Finally, since these models do not allow for the possibility that an observed commodity bundle may also be commodity inefficient, no distinction is being made between what we define as allocative efficiency and expenditure (or overall) efficiency. As a result, the efficiency score that these models assign to consumers may be higher than it should.

Our analysis is carried out under the consumer’s expenditure-minimisation framework, and the starting point is the assumption that the consumer’s objective is to choose a feasible commodity vector in order to achieve a desirable utility level. Assuming also that the consumer need not make use of all the quantities of the purchased commodities and may dispose of any unwanted quantities of them, the quantities of the purchased commodities may well be higher than the ones required to just attain the desirable utility level, and the consumer may well have chosen an inefficient way of attaining this utility level. This type of efficiency is what we are going to define as commodity efficiency. Another type of efficiency is what we call expenditure, or overall, efficiency, and which we describe as the consumer’s ability to avoid wasting expenditures, by minimising the cost of purchased commodities in the achievement of a utility level. A third type of efficiency is allocative efficiency: allocative efficiency is concerned with how close an observed commodity vector is to the expenditure-minimising commodity vector on the same indifference curve. Finally, we show the relation between the three types of efficiency, i.e., the decomposition of expenditure efficiency into commodity efficiency and allocative efficiency.

Econometric estimation of our theoretical model calls for establishment of an appropriate empirical framework which will accommodate consumers’ non-optimal behaviour. In particular, the index which is proposed for the measurement of commodity efficiency is based on a distance function representation of consumer preferences. Computation of the commodity efficiency index requires knowledge of the value of the distance function, which can be acquired though econometric estimation of the latter. However, the difficulty in estimation of a distance function representation of consumer preferences lies on that it is a function, not only of observed commodity quantities, but also of consumer’s utility level which is unobserved. In order to illustrate how this knowledge can be acquired, we estimated a translog distance function with a panel data set of British household purchases of milk & yoghurt, fruits, and vegetables. The methodology that is adopted for estimation of the translog distance function lies on treating consumer’s unobserved utility level as a random error term. Specifically, treatment of the terms associated with the utility level and the distance as one-sided positive error terms gives rise to a density for the composite error term which resembles the two-tiered frontier estimation framework by Polachek and Yoon (1987, 1996). The estimated distance function can then be used as an index to measure commodity inefficiency.3 As far as calculation of the measure of allocative efficiency is concerned, knowledge of either the expenditure function or the expenditure-minimising commodity vector is required; standard procedures employed in production efficiency analysis for computing the index of allocative efficiency are employed for computation of the measure of allocative efficiency in consumption. Finally, the measure of expenditure efficiency can be computed with the use of the proposed relation for the decomposition of expenditure efficiency into commodity and allocative efficiency. 

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