2.1 Introduction

The notion of a hierarchy of needs or wants is an old one. Plato in his famous Republic already wroteFootnote ² that “the first and the greatest of our needs is the provision of food to support existence and life … The second the provision of a dwelling place, and the third of clothing … .” Much later on, as stressed by Reference DrakopoulosDrakopoulos (2004), several economists such as Smith, Say, Jevons, Menger, and Marshall, were also aware of the importance of making a distinction between basic and nonbasic needs. Some of them even proposed a system of choice based on this idea. However, the fact that incorporating hierarchy into the formalistic methodology of the marginalists was not an easy task, explains probably why such a hierarchical approach to decision making, did not become very popular.

The goal of this section is to take a closer look at this notion of hierarchy of needs and at its implications. After briefly reviewing the way some famous economists understood the idea of hierarchy of needs, we look at the notion of wants which for some economists is derived from that of needs while others do not really make a distinction between these two notions. Particular attention is then given to the writings of the quite famous French economist René Roy whose articles on the concept of hierarchy of needs are unfortunately not very well known because they were published in French. For Roy the notion of hierarchy of needs applies to broadly defined categories of goods while substitution between goods take place within each of these categories when relative prices change. The last sections (Sections 2.6, 2.7, and 2.8) are devoted to the implications of the concept of hierarchical choice for the shape of Engel curves, the distribution of wealth and poverty and the diversity of expenditures.

2.2 The Notion of Needs

The notion of need has existed for a long time. Smith, for example, explains in the Wealth of Nations (pp. 869–70) what he means by “necessaries”:

Almost one hundred years later, as mentioned by Reference BecchioBecchio (2014), in the first edition of his Principles of Economics (1871, 1981), Carl Menger stressed the fact that there are four prerequisites for a good to be an economic good: “the presence of a human need; some properties able to render a thing capable of being brought into a causal connection with the satisfaction of this need; the human knowledge of this causal connection; a command of the thing sufficient to direct it to the satisfaction of the need” (Reference BecchioBecchio, 2014). Menger’s views imply first that the existence of a human need is only a necessary condition but not a sufficient one for a thing to become a good. As stressed by Reference Ruprecht and BianchiRuprecht (2007) this suggests that the existence of an objective cause–effect relationship is a necessary condition for a thing to become a good. In addition, an individual consumer must be aware of this relationship. Ruprecht gives as illustration the discovery of the impact of citrus fruits on scurvy prevention. Citrus fruits had been known for a long time for their caloric content, their juiciness, their taste, but the connection with the prevention of scurvy was something new of which people became aware only around the middle of the seventeenth century. Clearly the objective impact of citrus fruit consumption on the prevention of scurvy does not imply that consumers are aware of this effect. Ruprecht, however, challenges Menger’s (1953) view, according to which the objective character of this relationship is a necessary condition for a thing to become a good. He cites Reference HayekHayek (1979), for whom subjective beliefs matter if we want to explain human behavior toward things. For Hayek what is relevant is not what can be found about things by the objective methods of science, but what a person thinks about them. “Knowledge, which we may happen to possess about the true nature of the material thing, but which the people whose action we want to explain do not possess,” is not relevant to explain the actions of individuals. Ruprecht then asks how people should take a decision when consumers face competing cause–effect hypotheses. Following Reference PopperPopper (1959) he argues that “non-falsification” may be a plausible criterion. But what happens when consumers need to choose between several hypotheses that have not yet been falsified but there is no way of testing them? In such a case consumers will have to decide which hypothesis they trust and whether there is a case strong enough for them to act.

As far as the fourth condition mentioned by Menger is concerned (“a command of the thing sufficient to direct it to the satisfaction of the need”), one could have thought that income is a precondition for getting “personal command” of something. But the concept of “positional goods,” introduced by Reference HirschHirsch (1976), shows that consumers acquire wants via their consumption experiences as well as when observing the consumption of other people. Ruprecht mentions then the concept of “emulative consumption” stressed, for example, by Reference BourdieuBourdieu (1984), a behavior often summarized by the expression “keeping-up-with-the-Joneses.” In short Ruprecht concludes that social imitation provides a strong argument against giving “personal command” the status of a necessary condition for a thing to become a good.

In fact, Reference BecchioBecchio (2014) has argued that in the second edition of Reference MengerMenger’s Principles (1923), Menger’s views were somewhat different as he then emphasized the notion of social needs and social goods. As mentioned by Reference YamamoriYamamori (2020), the second edition of Reference MengerMenger’s Grundsätze (1923) had a new first chapter whose title was “theory of needs.” In this chapter Menger wrote that “the starting point for any investigation of economic theory is needy human nature. Without needs, there would be no economy, no national economy, no economic science” (Reference MengerMenger, 1923, cited by Reference YamamoriYamamori, 2020). Yamamori also stressed that Menger makes a distinction between need (“Bedürfnis” in German) and “its satisfier-provisions” (“Bedarf” in German), which is defined as “the quantities of consumption goods a person must have to satisfy his needs.” The former notion is considered as subjective, the second as objective.

At the same time as Carl Menger published the first edition of his Principles, Stanley Reference JevonsJevons (1871) published his Theory of Political Economy. Both authors took a subjective approach to the theory of value, at the difference of the objective approach of classical economists. Jevons’s analysis is based on the concept of utility, while Menger uses the word “satisfaction.” As mentioned by Reference LagueuxLagueux (1997), for Jevons, utility is “the abstract quality whereby an object serves our purposes and becomes entitled to rank a commodity” (Jevons, p. 38), while for Menger, satisfaction describes a subjective state of mind. Menger refers “to degrees of satisfaction in relation to various needs which are considered and draws comparisons only between these different degrees of satisfaction” (Reference LagueuxLagueux, 1997).

Note that in his essay on “Economic possibilities for our grandchildren,” Reference KeynesKeynes (1933, pp. 833–41) considered that there are two types of needs: absolute and relative. While absolute needs are satiable, relative needs are not because “they satisfy the desire for superiority” (Reference KeynesKeynes, 1933).

2.3 From Needs to Wants

Needs have to be carefully distinguished from wants. As stressed by Reference Lutz and LuxLutz and Lux (1979) needs can be hierarchically classified and determine consumers’ behavior. Wants are derived from needs and may be substituted for each other. They reflect “the various preferences within a common category or level of need” (Reference Lutz and LuxLutz and Lux, 1979).

For Reference DrakopoulosDrakopoulos (1994) a need implies something which is universally necessary, while a want refers to a personal preference and the latter is a trait of the individual. This difference explains why there may be an important degree of substitution between wants, while there is likely to be only a weak degree of substitution between needs, which are assumed to be universal. Such a view echoes that of Reference Cosmides and ToobyCosmides and Tooby (1994), who argue that cultural differences are vastly overstated, “because beneath existing surface variability all humans share the same set of preference-generating and decision-making devices.” Such a statement is derived from an evolutionary psychology approach to human behavior. For Reference Cosmides and ToobyCosmides and Tooby (1994) “natural selection’s invisible hand created the structure of the human mind, and the interaction of these minds is what generates the invisible hand of economics.” These authors also stress that evolutionary psychology can provide a list of human universal preferences, and of the “procedures by which additional preferences are acquired or reordered.” Such an approach would widen the scope of preferences and not limit the latter to goods and services. Reference Earl and PottsEarl and Potts (2004) have thus proposed a distinction between high- and low-level preferences. High-level preferences are innate, while low-level preferences refer to specific preferences that are acquired by learning and specialization. Such a distinction between high- and low-level preferences is a consequence of human’s “bounded rationality.” Consumers “have a problem of knowing what they want and how to get it” (Reference Earl and PottsEarl and Potts, 2004). While high-level preferences define in a way the problems that must be solved, low-level preferences have to be acquired either by learning or via agents that Earl and Potts call “preference entrepreneurs,” who are able to adapt products to lifestyles.

Another interesting approach is that of Reference WittWitt (2001), who asked what wants people pursue in their daily economic activities, wants they satisfy by consuming resources as direct inputs or as tools for serving these wants. Witt’s main argument is that there is a biological foundation of certain “basic needs” (e.g. thirst, hunger, sex) and they are fixed and universally shared. Biological evolution requires organisms to consume certain things to survive and this explains why, among the very poor, everyone spends most of his/her money consuming food. Once the basic needs are satisfied, consumers move on to satisfy other needs that may be “learnt.” Here Witt makes a distinction between cognitive and noncognitive learning. The latter concerns acquired wants that emerge as consumers “form associations between these experiences and the material environment which surrounds them.”Footnote ⁴ Cognitive learning, on the contrary, refers to a kind of problem-solving sequence where consumers try to find new solutions to a given problem.

2.4 On the Notion of Hierarchical Choice

This notion of hierarchy has been stressed by Reference DrakopoulosDrakopoulos (1994), who presents a survey of hierarchical choice in economics. He starts by reminding us that this is an old idea since Plato, as indicated at the beginning of this section, had a hierarchical view of needs. Drakopoulos then emphasized the fact that the notion of hierarchical choice appears in a way or another in the works of Reference JevonsJevons (1871, Reference Jevons1957), Reference MarshallMarshall (1949), Reference MengerMenger (1871, Reference Menger1981) but also Reference FishburnFishburn (1974), Reference LittleLittle (1950) and many other economists. Here are some interesting citations.

Reference Georgescu-RoegenGeorgescu-Roegen (1966), for example, wrote:

This is the reason why

Reference RobinsonRobinson (1956, p. 354, cited by Reference LavoieLavoie, 1994) argues that, “generally speaking, wants stand in a hierarchy (though with considerable overlaps at each level), and an increment in a family’s real income is not devoted to buying a little more of everything at the same level but to stepping down the hierarchy.”

For Reference PasinettiPasinetti (1981, p. 73, cited by Reference LavoieLavoie, 1994), although possibilities of substitution among commodities are, of course, relevant at any given level of real income, there exists a hierarchy of needs. More precisely, there exists a very definite order of priority in consumers’ wants, and therefore among groups of goods and services, which manifests itself as real incomes increase.”

Reference EichnerEichner (1986, pp.159–60, cited by Reference LavoieLavoie, 1994) believes that “in an economy that is expanding over time, it is the income effect that will predominate over the relative price, or substitution effects … Substitution can take place only within fairly narrow subcategories. Consumers’ preferences are, in this sense, lexicographically ordered … .”

As emphasized by Reference LavoieLavoie (1994), for those economists, like the so-called “post-Keynesian economists,” for whom a hierarchy of needs exists, households whose incomes are relatively similar will satisfy their needs in more or less the same order. This is so because of convention. In other words, once physiological needs are fulfilled, there will still be a common hierarchy of needs for individuals who share a similar culture. Consumers tend to conform to the norms of reference groups and they tend to imitate the behavior of those who are higher in the hierarchy of society. This idea that the relative position of consumers in society matters very much is not new, as it was already raised by Reference DuesenberryDuesenberry (1949), Reference LeibensteinLeibenstein (1950) and Reference VeblenVeblen (1931).

2.5 Hierarchy of Needs and the Approach of René Roy

The French economist René Roy, famous for introducing what is known as Roy’s identity (Reference RoyRoy, 1947), introduced the notion of group of goods and assumed the existence of a hierarchy of goods (Reference RoyRoy, 1930, Reference Roy1931, Reference Roy1933, Reference Roy1943). More precisely he made a distinction between what he calls “inferior and superior groups.” In his wordsFootnote ⁵ (Reference RoyRoy, 1931):

As will be seen, Roy makes a link between the distribution of incomes and the general expression of the law of demand. As far as the allocation of one’s income between the various goods is concerned, Reference RoyRoy (1930, p. 115) writes:

Reference RoyRoy (1930) then derives the following result concerning the price elasticity of the demand for necessities.

Let P and Q refer to a price and quantity index of the goods of a given group and $p_i$ and $q_i$ to the price and quantity of good i which belongs to this group. We may therefore write that

$PQ=k{{\sum }_i{p}_{i}q}_i$

where k is a constant. Let $P_0$ and $Q_0$ refer to the value of the price and quantity indices at some original time 0.

Roy then defines as $r_0$ the lower limit of individual incomes. He also defines (Reference RoyRoy, 1930, pp. 121–2) as $q_l$ “the maximum value of the quantity index for each individual.” He then writes:

we have previously admitted that this limit ensuring satiety for the group of products studied was the same for each individual and that it did not depend in particular on the amount of individual income. This limit $q_l$ is such that if the price index corresponding to the group studied is equal to P, the individual income ensuring satiety is represented by the product $q_{l}P.$

To arrive at the law of demand for basic necessities, we will observe that the set of payments Q.P relating to this category of objects, can be divided into two groups:

1. a) The group of people whose individual consumption is imperfect and therefore remains below the $q_l$ limit.

2. b) The group of consumers whose individual income is sufficient to allow the purchase of the quantity limit $q_l$ .

For the first group, that is, for individuals whose income is between the lower limit $r_0$ and the income $P{q}_l$ enabling them to acquire the quantity-limit $q_l$ , the total income is allocated to the acquisition of objects belonging to the group studied, since they are, by hypothesis, objects of first necessity.

The set of payments for this group of goods will hence be expressed as ${\int }_{r_0}^{P{q}_l}rn\left(r\right)dr$ where r refers to income and $n\left(r\right)$ to the number of individuals with an income r.

For the second category of individuals, those whose income allows satiety, all the payments corresponding to basic necessities are equal to the product of individual income $P{q}_l$ by the number of people whose income is higher than this amount; the set of payments in question is therefore represented by the expression $P{q}_l{\int }_{P{q}_l}^{\infty }n\left(r\right)dr$ .

We may therefore write that

$PQ={\int }_{r_0}^{P{q}_l}rn\left(r\right)dr+P{q}_l{\int }_{P{q}_l}^{\infty }n\left(r\right)dr.$

Reference RoyRoy (1930, pp. 127–8) then derives $PQ$ with respect to the price, and this derivative, using the second member of $PQ$ , is equal to $q_l{\int }_{P{q}_l}^{\infty }n\left(r\right)dr$ = $Q_s$ where $Q_s$ refers to the total consumption of those people whose income is greater than $P{q}_l$ , that is, to the total consumption of those who are able to ensure satiety for the products considered.

Reference RoyRoy (1930) then concludes that

$d\left(QP\right)=PdQ+QdP=Q_{s}dP$

$\leftrightarrow P\frac{dQ}{Q}+\frac{QdP}{Q}=\frac{Q_s}{Q}dP\leftrightarrow \frac{dQ}{Q}+\frac{dP}{P}=\frac{Q_s}{Q}\frac{dP}{P}$

$\leftrightarrow \frac{dQ}{Q}=\frac{dP}{P}\left(\frac{Q_s}{Q}-1\right)\leftrightarrow \frac{dQ}{Q}=-\frac{dP}{P}\left(\frac{Q-Q_s}{Q}\right)$

$\leftrightarrow -\frac{\left(\frac{dQ}{Q}\right)}{\left(\frac{dP}{P}\right)}=-\lambda =\left(\frac{Q-Q_s}{Q}\right)$

$\lambda =\left(\frac{Q-Q_s}{Q}\right)$ may hence be considered as the price elasticity of the “necessities.” It is estimated as the ratio of the consumption of necessities of those who do not reach satiety over the total consumption of necessities in society. This price elasticity will hence be lower, the higher the percentage of individuals who reach satiety, as far as necessities are concerned.

It is interesting to note that Reference LavoieLavoie (1994), referring to the work of Reference Deaton and MuellbauerDeaton and Muellbauer (1980), concluded that “those who have attempted to estimate the importance of pure substitution effects on the general categories of consumption expenditures, after having taken into consideration the income effects through time, have discovered that these substitution effects, own-price elasticities and cross-elasticities, are quite negligible.” Reference LavoieLavoie (1994) added that “the findings of Reference Houthakker and TaylorHouthakker and Taylor (1970) show that even at a much more disaggregated level, that is, with over eighty categories of consumer goods, consumption expenditures are mainly determined by habits and income effects, while price substitution effects play a fairly modest role.”

In another paper Reference RoyRoy (1943) repeated his belief in the notion of hierarchy of needs. Here is what he wrote:

2.6 Implications for Engel Curves

Once choice is assumed to be hierarchical, there are, as stressed by Reference DrakopoulosDrakopoulos (1994), implications concerning the shape of demand and Engel curves. The notion that for each good there is some satiety level of consumption which is supposed not to be surpassed, whatever the individual’s income, has been stressed in several empirical analyses. Reference PraisPrais (1952), for example, suggested that when selecting a mathematical formula for the Engel curve, one should make sure that it has such an asymptotic property. An upper asymptote is certainly desirable for necessities but, as stressed by Reference Aitchison and BrownAitchison and Brown (1954), it is less obvious for luxury goods. They, however, argue that “many commodities begin life as luxuries and eventually become semi-luxuries or necessities as increasing incomes and falling prices bring consumers nearer to an ultimate saturation level.”

In what is considered as the first statistical analysis of budgets, Reference EngelEngel (1857, Reference Engel1895) proposed the following law of consumption (see Reference StiglerStigler, 1954): “The poorer a family, the greater the proportion of its total expenditure that must be devoted to the provision of food.” Reference WrightWright (1875) gave then what Reference StiglerStigler (1954) labeled an “excessive free translation of Engel’s law” since Wright mentioned four propositions:

Reference StiglerStigler (1954) mentions that Reference SchwabeSchwabe (1868, Reference Schwabe1966) added the following law: “The poorer anyone is, the greater the amount relative to his income that he must spend on housing.” Reference WrightWright (1875) also suggested the following generalization (see Reference StiglerStigler, 1954): “The higher the income, generally speaking, the greater the saving, actually and proportionately.”

Engel, in fact, seems to have borrowed a notion introduced by Smith and according to which “the ultimate measure of welfare is the degree to which individuals are able to satisfy what Engel called “Bedürdfnisse” and which Reference Chai and MonetaChai and Moneta (2010) translated as “wants.” More precisely Reference EngelEngel (1895, p. 1) wrote that “every individual directs (out of his own impulse) his highest interest to the continuous satisfaction of those wants that stem directly from his human nature, to the expansion of these wants, and also to the attainment of the necessary means to satisfy the higher, expanded wants.” As stressed by Reference Chai and MonetaChai and Moneta (2010), Engel “shifted the focus of research away from examining how expenditure is distributed across individual goods consumed, towards focusing on how it is distributed across wants which goods ultimately satisfy. Second, Engel made a clear break from the common tendency amongst classical economists to assume that some “basic” goods are inherently more important to human welfare than other ‘luxury’ goods.”

This emphasis on wants appears clearly in his empirical work since he did not really focus his attention on food expenditure but on “nourishment” which also included expenditure on alcohol and tobacco. According to Reference Chai and MonetaChai and Moneta (2010), Engel believed that his empirical analysis showed that there was a hierarchy among wants, the most important want being the want for nourishment. Then there was the want for clothing, accommodation, heating, and lighting. Engel also stressed that when a family is not able to satisfy all its wants, it is likely to decide not to satisfy “higher-order” wants so that it can satisfy its basic wants.

The most famous approach to the idea that there exists a hierarchy of needs is certainly that of Reference MaslowMaslow (1970). The first set of need that he identifies is that of basic needs, which include air, water, food, sleep, and sex. The second group of needs refers to safety needs, like feeling safe and secure and settled. These are more psychological than physiological needs. The third set of needs is “belonging needs,” which concern the desire to belong to families, clubs, etc. As stressed by Reference Clarke, McGillivray and ClarkeClarke (2005) “this level of needs incorporates the need to feel (nonsexual) love and acceptance of others.” The fourth level of needs covers self-esteem needs, that is, the need to be admired by those around you. The last level of needs is labeled “self-actualization,” summarized by Reference Clarke, McGillivray and ClarkeClarke (2005) as the “self-fulfillment of ones’ own potential,” although one may wonder whether such a want is universally shared.Footnote ⁶

A few papers have attempted to test this notion of hierarchy of needs. Thus Reference HagertyHagerty (1999) adopted Maslow’s notion of hierarchy of needs as well as his classification into five categories of needs: physiological, safety, belonging and love, esteem, and self-actualization. The indicators he used for physiological needs were the number of daily calories available per person and then GDP per capita. For safety needs he used safety from war, safety from murder, and high life expectancy. For belonging and love the indicators were low divorce rate and low child death rate. For esteem the measures were political rights and women’s participation in work for pay. Finally for self-actualization the indicators used were tertiary, secondary, and primary education enrollment.Footnote ⁷ Data were available for eighty-eight countries between 1960 and 1994. Reference HagertyHagerty (1999) tested four hypotheses to confirm Maslow’s theory. First, that nations increasingly fill their needs over time. Second, a nation’s trajectory of need fulfillment follows an S-shaped pattern. Third, nations’ sequence of need fulfillment follows Maslow’s sequence. Fourth, higher growth in one need area is correlated with lower growth in other need areas. The results indicated that the three first hypotheses were generally confirmed, but not the fourth one.

Reference Jackson and MarksJackson and Marks (1999) adopted Reference Max-Neef, Elizalde and HopenhaynMax-Neef’s (1989, Reference Max-Neef1991) characterization of human needs and applied it to an analysis of the patterns of consumer expenditure in the United Kingdom between 1954 and 1994. Both Reference MaslowMaslow (1970) and Reference Max-NeefMax-Neef (1991) characterize certain needs as material needs. Material needs are mainly subsistence and protection needs. More generally, satisfying these needs requires materials. Non-material needs are more about processes (personal, social, and cultural) than about objects. Thus “participation, affection, understanding, idleness, creation and freedom relate more to individual and social psychology than they do to material things” (Reference Jackson and MarksJackson and Marks, 1999). In their empirical analysis, Jackson and Marks observed that the increase between 1954 and 1994 in expenditures related to material needs satisfaction was 50 percent, while the increase related to non-material needs satisfaction was 160 percent. But this increase in nonmaterial needs appears to be dominated by expenditure on material goods (increase in goods and durable goods rather than in services) and therefore remains a material-intensive form of consumption.

Reference Clarke, McGillivray and ClarkeClarke (2005) adopted Maslow’s classification of needs, which makes a distinction between basic needs (BN), safety needs (SN), belonging needs (BIN), self-esteem needs (SEN), and self-actualization needs (SA), to estimate well-being. He assumed that well-being WB could be considered as self-actualization SA and expressed it as the following function:

$WB=SA\left(BN,SN,BIN,SEN\right)$

$BN$ was assumed to include two indicators: daily calories available per person and access to safe water. For $SN$ the indicators were infant mortality and life expectancy, for $BIN$ they were telephone mainlines and fertility rates, and for $SEN$ the indicators were adult illiteracy and unemployment. Clarke gave a higher weight to higher needs and decided to give to $BN,SN,BIN,SEN$ the weights 1, 2, 3, and 4, respectively. Clarke also used normalized indexes by dividing each year’s figure by the highest figure occurring throughout the time series he used.

Reference Chai and MonetaChai and Moneta (2012) attempted to test Engel’s approach to a hierarchy of needs. Engel had made a distinction between the following needs (“Bedürfnisse”): nourishment, clothing, housing, heating and lighting, tools for work, intellectual education, public safety, health and recreation, and personal service. Chai and Moneta used the UK Family expenditure surveys for the years 1961, 1970, 1980, 1990, and 2000 and allocated the various household expenditure data to the need categories classified by Engel and mentioned previously. They then observed that between 1960 and 1980 the expenditure shares of these various needs for the lowest-income decile did not vary very much. After 1980, expenditure on nourishment declined significantly while housing expenditure increased significantly. They concluded that their results were quite consistent with Engel’s observed patterns. In particular, the budget shares of household expenditure on needs that Engel considered as central to physical sustenance dominate low-income consumption patterns in all years examined. These authors also observed that Engel curves for lower-order goods have shapes that are more similar to each other than to the shapes of Engel curves for higher-order goods, while Engel curves for higher-order goods have shapes more similar to each other than to shapes of Engel curves for lower-order goods. Chai and Moneta concluded that “a hierarchy of needs appears to consist of two levels in that it is only the most important needs, the need for nourishment, that appears to dominate over other needs. There appears to exist no order between other lower order needs.”

2.7 Implications Concerning the Distribution of Wealth and Poverty

These implications have been analyzed in a very interesting way by the French engineer and economist Dupuit (1859). Here is what he wrote:Footnote ⁸

If, when the insufficiency of the harvest is a tenth, everyone reduced their consumption by a tenth, there would doubtless be deprivation; however […] the suffering would nowhere be very acute; but this is not the way of things: however expensive bread may be, it is still, for the rich and well-to-do classes, the most economical food, and for them the high price of bread is never a reason to consume less. The rich or well-to-do household that spent 400 francs on bread will spend eight if the price doubles, but the expenditure on other items in its budget is reduced by 400 francs […].

If half the population consumes, in times of scarcity, the same quantity of bread as in times of plenty, then all the deprivation must fall on the other half. It is then no longer a question of a tenth but of a fifth, and […] such a reduction in food […] is distributed in a very unequal way among the lower classes, according to the resources at their disposal, and afterwards this deprivation of bread entails many others; because, to satisfy this imperious need of hunger […] the poor man sells his clothes, his blankets, his mattresses, deprives himself of wood, of light, and thus suffering from hunger, cold and humidity, falls prey to disease, and finds only in death the end of his sufferings […].Others […] by dint of privations on secondary needs, can wait for better times. Finally, in the well-to-do classes, not only is there no deprivation, but there is an increase in consumption. For, by the very fact that the lower classes have been obliged to restrict their consumption relative to certain objects, these objects, the production of which has not diminished, are offered to the rich classes at a lower price

Let us not lose sight of the fact that in this phenomenon of the distribution of wealth, we must always arrive at the result that consumption is equal to production. Now we suppose that there is no decrease in production except on the article of cereals, and it is a fact […] that in times of scarcity the poor classes are obliged to renounce any other expense than that of their food; it is therefore necessary that in these circumstances the other classes consume what the poor classes have not been able to consume. […] which compensates for the excess expenditure which they have to do to get bread.

We will say: first, that scarcity is borne entirely by the poor classes of society, and even in a very unequal way between them; second, that for the poorest the deprivation of food is complicated by a host of other deprivations, […]; third that these privations are the only ones suffered by the slightly higher classes; fourth, that the wealthy classes, far from supporting it of any kind, can consume more. (Dupuit [1859], p. 163–4 3)

Reference Chai and MonetaChai and Moneta (2012) also analyzed the implications of a hierarchy of needs on the diversity of expenditures. They thus noted that households located at low- or medium-income deciles tend to diversify their consumption patterns.

2.8 Implications Concerning the Diversity of Expenditures

Somewhat similar conclusions were derived by Reference Chai, Rohde and SilberChai et al. (2015). Using data from the 2001 cross-section of the UK Household Expenditure Survey and gathering information on the expenditures of over 5000 households over twelve different categories, they measured diversity over different expenditure levels, using various diversity measures. They concluded that, at low-income levels, households have relatively concentrated spending patterns but tend to diversify their spending quickly as their income rises. At the same time, the level of heterogeneity in spending diversity also grows with income.

Reference Chai, Stepanova and MonetaChai et al. (2022) started from the idea that at very low-income levels household demand is mainly dedicated to food, while richer households consume a much wider variety of goods, higher-quality goods, and more services. Economic growth, however, is often accompanied by rising income inequality. Therefore, if one assumes hierarchical preferences, greater income inequality will raise the distance between consumers across the spending hierarchy and hence lower the homogeneity of consumption patterns. Another reason why the homogeneity of spending patterns is likely to decline as household incomes rise is that relatively affluent consumers may concentrate on niche luxuries that may be specific to each consumer (see Reference Neiman and VavraNeiman and Vavra, 2019). Reference Chai, Stepanova and MonetaChai et al. (2022) based their empirical analysis on the World Bank’s Global Consumption Database (GCD), which provides 2011 data covering ninety countries, mainly from the less developed countries. The GCD covers 107 expenditure categories: 32 are food and beverages, 41 are services, and 34 are goods. These 107 expenditure categories are also aggregated into 10 larger groups: food, clothing, health, electricity, passenger transport, housing, means of communication, education, recreation, personal transport. Interestingly the GCD also provides data on the percentage of households in each income segment who consume a particular good. These segments are uniform across countries and are based on the global distribution of per capita income. The lowest-income segment corresponds to the bottom half of the global distribution; the low-income segment to the 51th to 75th percentiles; the middle-income segment to the 76th to 90th percentiles, and the high-income segment to the 91st percentile and above.

As mentioned previously, the focus of the Reference Chai, Stepanova and MonetaChai et al. (2022) paper was on diversity. They concluded their empirical investigation by stressing that among low levels of per capita GDP, spending diversity grows via a diversification of the food diet of the consumers, while at higher levels of per capita GDP, spending diversification takes place via an increase in the range of services that are consumed. Reference Chai, Stepanova and MonetaChai et al. (2022) then argue that these observations tend to support the thesis that the rise in the service sector may well be the consequence of shifts in final demand related to the growth of the per capita GDP. They also concluded that higher income inequality tends to reduce the average share of households consuming a particular good and that, in more affluent countries, there was a negative correlation between the level of income inequality in a country and the number of varieties consumed. Among poor countries, however, it appears that income inequality has a positive impact on the number of varieties consumed, confirming previous findings by Reference Falkinger and ZweimüllerFalkinger and Zweimuller (1996). Reference Chai, Stepanova and MonetaChai et al. (2022) also found that expenditure hierarchies observed in the world have a common characteristic, namely that the income elasticity of food is always lower than that of other goods and services. However, among higher-order goods, expenditure hierarchies are more country-specific. As far as niche consumption is concerned, it appears that the diversification of consumption among poor consumers is quite homogenous, while rich people tend to concentrate spending into different areas so that there is a decrease in the homogeneity of demand.

In a recent paper on the International Comparison Program (ICP), purchasing power parity (PPP), and household expenditure patterns, Reference Clements, Lan, Liu and VoClements et al. (2022) initially concluded that in all the surveys they analyzed there is enough evidence confirming Engel’s law. But they also stressed that Engel’s law has implications concerning the structure of relative prices: given that the income elasticity of food is smaller than one, one can expect a decline in the relative price of food as income grows. Using the 2011 ICP data, Clements et al. also found a confirmation of an empirical regularity observed by Reference WorkingWorking (1943): the proportion of total expenditure devoted to food tends to decrease in arithmetic progression as total expenditure increases in geometric progression. An additional implication may be derived from what Clements et al. called “Working’s law,” namely that food will attract a smaller fraction of an increment of income among the rich than among the poor.

2.9 Concluding Comments

This section attempted to show that the concept of hierarchical choice is not a new one, as it appears implicitly or explicitly in the writings of famous economists. It was also shown that this notion has far-reaching implications, not only concerning the shape of Engel curves and the diversity of expenditures but also pertaining to the distribution of wealth and poverty. The focus of the next two sections will be on one of the implications of the notion of hierarchical choice, namely the idea that there is an order of acquisition of durable goods. Two ways of deriving such an order will be presented: Section 3 will first focus on the approach of Jacob Reference ParoushParoush (1963, Reference Paroush1965, p. 973) which is related to the notion of Guttman (194) scale; then it will present what is known as Item Response Theory.

3.1 Introduction

While the focus of Section 2 was on the notion of hierarchical choice in the history of economic thought, this section looks at one aspect of hierarchical choice, namely the idea that there is an order of acquisition of durable goods and assets. We will show that there are two main approaches to the derivation of such an order. One is inspired by Reference GuttmanGuttman’s (1944) work on the scaling of qualitative data and led Reference ParoushParoush (1963) to assume that there is an order of acquisition of durable goods and to propose an approach allowing one to discover such an order. The other approach is borrowed from the field of psychometrics and applies the technique known as Item Response Theory to the derivation of such an order of acquisition of durable goods and assets. The section ends with a short presentation of two other ways of looking at the order of acquisition of durable goods and assets, one which applies to this issue the concept of Borda rule and the other one whose emphasis is on the structure of asset ownership.

3.2 On the Guttman (1944) Scale

Assume a qualitative variable (also called “attribute”) which includes several categories (e.g. various religions) which do not have any intrinsic ordering. Assume that y is such an attribute, while $x$ is a quantitative variable which we can divide in a certain number of intervals which will have a one-to-one correspondence with the values of y. In such a case we can state that the attribute y is a simple function of x. Guttman gives the example of a variable x which can take ten values (0 to 9) and for which the following correspondence exists:

x	0	1	2	3	4	5	6	7	8	9
Y	$y_1$	$y_1$	$y_1$	$y_3$	$y_3$	$y_2$	$y_2$	$y_2$	$y_2$	$y_2$

Guttman then states that “for a given population of objects, the multivariate frequency of a universe of attributes will be called a scale if it is possible to derive from the distribution a quantitative variable with which to characterize the objects such that each attribute is a simple function of that quantitative variable. Such a quantitative variable is called a scale variable.”

Guttman then stresses that one should not expect to observe perfect scales and that the deviation from a perfect scale will be measured via what he calls a coefficient of reproducibility. This coefficient gives the relative frequency with which the values of the attributes correspond to the intervals of a quantitative variable. Guttman calls scale score a value of a scale variable and scale order the ordering of objects according to the numerical order of their scale scores.

He then considers the following illustration. Assume a mathematical test composed of the following three questions:

• - If r is the radius of a circle, what is the area of the circle?

• - Which values of x satisfy the equation $a{x}^2+bx+c=0?$

• - To what is $d{e}^x/dx$ equal?

The answer given to each question will be either right or wrong. Although a priori there could be eight (i.e. $2^3)$ patterns of answers given by the different individuals, one may expect that if the individuals are somewhat familiar with this kind of questions, only four types of answers would be observed. For example, a first category of individuals would give a correct answer to all three questions, a second category to the first two questions, a third category to the first question only, and a last category would have given a wrong answer to each of the three questions. If this is the case, the four type of individuals previously described would be respectively given the scores 3, 2, 1, 0. In other words, once we know the score of an individual, we also know to which question he/she gave a correct answer.

Guttman gives then a graphical illustration of this example. Assume, for example, that 80 percent of the individuals gave the correct answer to the first question, 40 percent to the second, and 10 percent to the third. The chart below summarizes these results:

Reference GuttmanGuttman (1944) then writes that the multivariate distribution for the three questions, given that they form a scale for the population, may be summarized on the same chart, since all those belonging to the group answering a more difficult question right belong also to the group who gave a correct answer to an easier question. Here is therefore the chart summarizing the results:

More generally the Guttman scale includes a list of statements that can be ranked from the least to the most important statement. This implies that if an individual agrees with the last statement on the list, he/she would have agreed with all the previous statements on the list.

The Guttman scale approach has been applied to various types of data, among which data allowing to discover the order of acquisition of assets, mainly of durable goods.

3.3 The Paroush Approach to Discovering the Order of Acquisition of Durable Goods

Sixty years ago, Reference ParoushParoush (1963) proposed to use information which is available on the order of acquisition of durable goods to derive estimates of the standard of living of households.Footnote ⁹ As a simple illustration, assume we have information on the ownership of three durable goods, a refrigerator, a television, and a car. An individual or a householdFootnote ¹⁰ can own one, two, three, or none of these durable goods. It is easy to derive that there are 2³ = 8 possible profiles of ownership of these three durable goods. Table 1 shows all the possible combinations of ownership of durable goods. A number 1 implies that the household owns the corresponding good, a zero that it does not.

Let us start by assuming that every household follows the following order: refrigerator, television, car. In other words we assume that the first durable good a household will acquire is a refrigerator. Then if this household desires to acquire a second durable good, it will buy a television. Obviously the third durable good the household will acquire, in our simple illustration, is a car. Note that if this is the order of acquisition of durable goods, there will be no household with the profiles 3, 4, 6, and 7 in Table 1. Profile 3, for example, assumes that a given household acquires only one durable good and that it is a television. But a refrigerator, and not a television, is supposed to be the first durable good acquired by a household.

We do not want to assume, however, that every household will buy first a refrigerator, then a television, and finally a car. There are clearly other possible sequences of acquisition. Moreover, even if we find out what the most common order of acquisition of durable goods is, there are always households that will deviate from this most common order of acquisition. This is why Reference ParoushParoush (1963, Reference Paroush1965, Reference Paroush1973) recommended computing the number of changes in numbers (from 0 to 1 or from 1 to 0), which will put a deviating household back to one of the profiles corresponding to a given order of acquisition of durable goods.

More generally, let us suppose that there are G durable goods. Then, for a given order of acquisition, as we have just seen in the simple illustration of Table 1, there will clearly be only (G + 1) possible profiles in the acquisition path. However, with G durable goods, there will be $2^G$ possible acquisition profiles, as shown in Table 1 with three durable goods. Call $P_j$ one of the ownership profiles that is allowed, under the order of acquisition of durables that has been selected. More precisely express $P_j$ as $P_j=$ $\left\{p_{j1},\dots ,p_{jg,}\dots ,p_{jG}\right\}$ where each element $p_{jg}$ is equal to 0 or 1 (as in Table 1). Now let $Y_i$ be the vector defining the order of acquisition of household i with $Y_i=\left\{y_{i1},\dots ,y_{ig},\dots ,y_{iG}\right\}$ . We can now compare this acquisition profile $Y_i$ of household i, with each of the (G + 1) acquisition profiles $P_j$ that are allowed, for the order of acquisition selected. Let $D_i$ be the smallest distance between the vector $Y_i$ and any of these (G + 1) acquisition profiles $P_j.$ In other words we write that

$D_i=Mi{n}_{forthe\left(G+1\right)allowedprofiles}\left\{\left|y_{i1}-p_{j1}\right|,\dots ,\left|y_{ig}-p_{jg}\right|,\dots ,\left|y_{iG}-p_{jG}\right|\right\}$(1)

Assume now that there are $N_i$ households with the same acquisition profile as household i. Reference ParoushParoush (1963, Reference Paroush1965, Reference Paroush1973) recommended computing what Reference GuttmanGuttman (1944) called the coefficient R of Reproducibility, which is defined as

$R=1-\left[\frac{\left({\sum }_i{N}_i{D}_i\right)}{\left({\sum }_i{N}_{i}G\right)}\right]$(2)

since G is the highest possible distance $D_i$ ( $D_i$ would be equal to G if, for example, all the elements of vector $Y_i$ are equal to 1 while all the elements of the vector $P_j$ are equal to 0).

Reference ParoushParoush (1963, Reference Paroush1965, Reference Paroush1973) indicated that one can prove that $\left(\frac{1}{2}\right)\le R\le 1.$ Paroush also wrote that “for most practical applications of the order of acquisition of durable goods a population is considered sufficiently ‘scalable’ if about ninety percent of its purchases are ‘reproducible’, provided the number of commodities is not very small.”

As a simple illustration, let us go back to Table 1 and suppose that the acquisition profile of individual i is profile 4, while the most common order of acquisition of durables is assumed to be as follows: a refrigerator, a television, a car. In such a case only profiles 1, 2, 5, and 8 in Table 1 are possible. We will now compute the distance of the acquisition profile of individual i to each of these four profiles 1, 2, 5, and 8.

The “distance” $D_i$ for an individual with profile 4 to profile 1 in Table 1 will then be expressed as |0–0| + |0–0| + |1–0| = 1.

The “distance” $D_i$ for an individual with profile 4 to profile 2 in Table 1 will then be expressed as |0–1| + |0–0| + |1–0| = 2.

The “distance” $D_i$ for an individual with profile 4 to profile 5 in Table 1 will then be expressed as |0–1| + |0–1| + |1–0| = 3.

Finally, the “distance” $D_i$ for an individual with profile 4 to profile 8 in Table 1 will then be expressed as |0–1| + |0–1| + |1–1| = 2.

The smallest distance of the ownership profile of individual i to the possible ownership profiles when the order of acquisition is a refrigerator, a television, a car is therefore equal to 1.

We can proceed similarly for every other individual. We may also define a “standardized distance” $D_{si}$ by dividing the observed distance $D_i$ by its maximal value G. If there are $N$ households in the whole population $(N={\sum }_i{N}_i)$ , the “average standardized distance” ${\overline{D}}_{si}$ in the population will be expressed as the weighted average of the “standardized distance” for the various households, that is as

${\overline{D}}_{si}={\sum }_i\left(\frac{N_i}{N}\right)\left(\frac{D_i}{G}\right)$(3)

so that the “proximity index” R turns out to be equal to the complement to 1 of ${\overline{D}}_{si}:$

$R=1-{\overline{D}}_{si}$(4)

Note, however, that we do not know what the most commonly order of acquisition of durable goods in the population is. To find out what it is, we need to compute the distances $D_i$ and the proximity index R for each possible order of acquisition of durable goods. It is easy to check that there are $G$ ! such possible orders. Let $D_i^h,{\overline{D}}_{si}^h$ , and $R^h$ be respectively the distance $D_i$ for individual i, the corresponding “average standardized distance” ${\overline{D}}_{si}$ in the population and the proximity index R when profile h is the one with which the profile of household is compared. The most common order of acquisition in the population will then be the one with the highest value of the proximity index $R^h$ .

Clearly detecting this most common order of acquisition of durable goods requires a very high number of computations. Assume, for example, that the data give information on ten durable goods. For each household in the survey under study, the determination of the minimum distance $D_i$ for this household will be based on eleven comparisons. Suppose that the survey under study includes 2,000 households. Then $(11\times 2,000)=20,000$ comparisons will be needed in order to determine the proximity index for a single order of acquisition. This procedure will have to be repeated 10! = 3,628,800 times. This is the total number of possible orders of acquisition resulting with ten durable goods. Therefore, 20,000 × 3,628,800 = 72576000000 will be the total number of computations necessary to find the order of acquisition with the highest index of proximity R. Given the power of contemporary laptops, having to implement such a high number of computations is not any more a problem.

Here is a simple illustration of the procedure that has just been described. To simplify we will assume that there are only two durable goods, a and b. There are hence four possible acquisition profiles, as shown in Table 2.

We will now assume that there are three individuals named x, y, and z. Table 3 indicates which durable good(s) each of these three individuals owns.

We can now compute the distance between the ownership profile of each of these three individuals and each potential ownership profile. Let us call $d\left(i;j\right)$ such a distance where $i=x,y,\text{or}z$ and $j=1,2,3,\text{or}4$ . It is then easy to find out that

$d\left(x;1\right)=1,d\left(x;2\right)=2,d\left(x;3\right)=0,d\left(x;4\right)=1$

$d\left(y;1\right)=1,d\left(y;2\right)=0,d\left(y;3\right)=2,d\left(y;4\right)=1$

$d\left(z;1\right)=1,d\left(z;2\right)=2,d\left(z;3\right)=0,d\left(z;4\right)=1$

Note that these distances are computed by comparing what is in the cell $\left(i;j\right)$ in Table 3 with what is in the cell $\left(k;j\right)$ in Table 2 where $k=1,2,3,\text{or}4$ . We then take the absolute value of the difference in each comparison.

The average distance $\overline{d\left(x,y,z;1\right)}$ to profile 1 is then expressed as $\overline{d\left(=x,y,z;1\right)}=\left(\frac{1}{3}\right)\left[d\left(x;1\right)+d\left(y;1\right)+d\left(z;1\right)\right]=\frac{1+1+1}{3}=1$ We can then similarly compute the other average distances and get

$\overline{d\left(=x,y,z;2\right)}=\left(\frac{1}{3}\right)\left[d\left(x;2\right)+d\left(y;2\right)+d\left(z;2\right)\right]=\frac{2+0+2}{3}=\frac{4}{3}$ .

$\overline{d\left(=x,y,z;3\right)}=\left(\frac{1}{3}\right)\left[d\left(x;3\right)+d\left(y;3\right)+d\left(z;3\right)\right]=\frac{0+2+0}{3}=\frac{2}{3}$ .

$\overline{d\left(=x,y,z;4\right)}=\left(\frac{1}{3}\right)\left[d\left(x;4\right)+d\left(y;4\right)+d\left(z;4\right)\right]=\frac{1+1+1}{3}=1$ .

We now will look at the two possible orders of acquisition. The first one implies that individuals first acquire good a, and then good b. The second one assumes that individuals first acquire good b, and then good a. In the first case, Table 2 indicates that only profiles 1, 2, and 4 are relevant. In the second case Table 2 not 3 shows that only profiles 1, 3, and 4 are possible.

Let us now examine the first case. We indicate here again the values of the distances $d\left(i;j\right)$ where $j=\text{only}1,2,\text{or}4.$ , We then have $d\left(x;1\right)=1,d\left(x;2\right)=2,d\left(x;4\right)=1$ so that the minimal distance for individual x is $Mind\left(x\right)=1$ . Similarly, $d\left(y;1\right)=1,d\left(y;2\right)=0,d\left(y;4\right)=1$ so that the minimal distance for individual y is 0 and we write: $Mind\left(y\right)=0$ .

Finally $d\left(z;1\right)=1,d\left(z;2\right)=2,d\left(z;4\right)=1$ so that the minimal distance for individual z is 1 and we write: $Mind\left(z\right)=1$ .

The average minimal distance of the individuals to the order $\left(a,b\right)$ is hence:

$\left(\frac{1}{3}\right)\left\{Mind\left(x\right)+Mind\left(y\right)+Mind\left(z\right)\right\}=\frac{1+0+1}{3}=\frac{2}{3}$

We now use the same procedure for the order $\left(b,a\right).$ The authorized profiles are here 1, 3, and 4. We then derive that

$d\left(x;1\right)=1,d\left(x;3\right)=0,d\left(x;4\right)=1$ so that the minimal distance for individual x is 0 and we write $Mind\left(x\right)=0$ .

Similarly, $d\left(y;1\right)=1,d\left(y;3\right)=2,d\left(y;4\right)=1$ so that the minimal distance for individual y is 1 and we write: $Mind\left(y\right)=1$ .

Finally $d\left(z;1\right)=1,d\left(z;3\right)=0,d\left(z;4\right)=1$ so that the minimal distance for individual z is 0 and we write: $Mind\left(z\right)=0$ .

The average minimal distance of the individuals to the order $\left(b,a\right)$ is hence:

$\left(\frac{1}{3}\right)\left\{Mind\left(x\right)+Mind\left(y\right)+Mind\left(z\right)\right\}=\frac{0+1+0}{3}=\frac{1}{3}$

We therefore conclude that on average the three individuals are closer to the order of acquisition $\left(b,a\right)$ than to the order $\left(a,b\right).$

3.4 Item Response Theory

Item Response Theory (IRT) is a technique that was originally introduced to help educational and psychological assessments.Footnote ¹¹ More precisely it was used to analyze the results of psychometric tests, the idea being that the probability of giving a correct answer to a question in the test was a function of the intelligence or ability of the examinee. However, intelligence (or ability) is not a characteristic of the individual that can be easily measured. It is, in fact, a latent trait. To measure it one relies on the answers given to a certain number of questions (items). If the answer to a given question is correct, the individual will receive a score of 1; if it is no correct, his/her score will be 0. Let A refer to the ability of an individual. Given the ability of the individual, the latter will have a probability $P\left(A\right)$ to give a correct answer to the item so that individuals with a low ability will have a small probability to give a correct answer, while those with high ability will have a high probability to correctly answer the item. It is assumed that the link between the probability P(A) and the ability A can be represented by a S-shaped curve. Such a curve carries the name of “item characteristic curve.” For estimation purpose the item characteristic curve will be represented by the Logistic Function (see Figure 1):

Figure 1 A typical Item Characteristic Curve

$P\left(A\right)=\frac{1}{1+e^{-A}}$(5)

While Item Response Theory has been mainly applied in psychology and in the field of education, it has been also used to analyze deprivation and poverty, more precisely multidimensional poverty. The general idea is that poverty has many facets and, like ability, cannot be really measured. Let now R (for richness) refer to the antonym of poverty. Like poverty, richness will be assumed to be multidimensional. It will not refer only to material wealth. It may also include other positive aspects of life such as good health or good relations with the family. In other words, richness is a latent variable that can be indirectly measured via the answers given to various questions focusing on diverse aspects of wealth in the large sense of this word. Let now P(R) be the probability that, for example, an individual owns a given durable good (item), such as a refrigerator or a car. If the individual owns a car, the individual will have a score of 1 for this item; if he/she does not own a car, his/her score will be 0. Similar assumptions will be made concerning the ownership of other durable goods (items) or the answers given to questions covering other aspects of richness such as those mentioned previously.

Here also it will be assumed that the link between P(R) and R will be represented by a Logistic Item Characteristic Curve that is similar to that given in Figure 1.

Two aspects describe such a curve. The first one, which in the educational literature is called the “difficulty” or “severity” of the item, focuses on the probability that the individual owns the durable good and is determined by the location of the curve. It indicates in fact along the richness scale (represented by the horizontal axis) the probability of owning a good. Therefore, for durable goods that are commonly found among poor individuals, at the left side of the horizontal axis, the characteristic curve will display a relative high probability P(R), and for durable goods that are usually found only among relatively rich individuals, the characteristic curve will display at the same horizontal location a relative low probability P(R).

The second aspect of the characteristic curve is called “discrimination” and focuses on the steepness of the curve. The steeper the curve, the better the durable good (item) under study discriminates between the probability of those who have and those who do not have the good.

Figure 2 displays the Item Characteristic Curve for two different goods. The first one, which refers to a cell phone, has a lower level of difficulty as the probability of owning a cell phone is higher as compared to that of owning a dryer (the second durable good) at all levels of richness R. An extreme case of discrimination would be that of perfect discrimination: those whose richness is smaller than some critical value $R_c$ do not own the durable good, while those whose wealth is above $R_c$ all own the good. Note that in such a case, no distinction is made between those whose richness level is lower than $R_c$ or among those whose richness level is higher than $R_c$ (Figure 3).

Figure 2 Tow typical Item Characteristic Curves

Figure 3 The case of perfect discrimination

The Parameters of the Item Characteristic Curve

The typical shape of an Item Characteristic Curve is that of a logistic curve with two parameters: d (discrimination parameter) and l (location or difficulty parameter). It is expressed as

$P\left(R\right)=\frac{1}{1+e^{-d\left(R-l\right)}}$(6)

The location parameter l is defined as the point on the “richness” axis at which the probability $P\left(R\right)$ of having a given durable good is equal to 0.5. In other words, we obtain $P\left(R\right)=0.5$ by substituting in equation (6) $l=R.$

The slope of the Characteristic Curve is given by:

$\frac{dP}{dR}=\frac{d}{{\left[1+e^{-d\left(R-l\right)}\right]}^2}=d\cdot P\left(1-P\right)$(7)

Therefore, the larger the discrimination parameter d, the steeper the Characteristic curve.

Estimating the Item Characteristic Curve

The estimation of the Logistic Characteristic Curve is based on the method of maximum likelihood. Each observation is assumed to be a single independent draw from a Bernoulli distribution. In a sample of n individuals for a given good, we define the variable y receiving the value $y_i=1$ ( $i=1,2,\dots ,n_1$ ) with probability $P_i$ for individuals owning the good and the value $y_i=0$ ( $i=n_1+1,n_1+2,\dots ,n$ ) with probability $1-P_i$ for individuals who do not have the good.

The join probability or likelihood function is:

$L={\prod }_{i=1}^{n_1}{P}_i{\prod }_{i=n_1+1}^n(1-P_i)$(8)

Taking logs we obtain:

$\ln L={\sum }_{i=1}^{n_1}\ln P_i+{\sum }_{i=n_1+1}^n\ln (1-P_i)$(9)

Since $P=\frac{1}{1+e^{-d\left(R-l\right)}}$ and $\left(1-P\right)=\frac{e^{-d\left(R-l\right)}}{1+e^{-d\left(R-l\right)}}$ we maximize the log likelihood function with respect to d and l:

$\underset{d,l}{\text{max}}\ln L=-{\sum }_{i=1}^{n_1}\ln \left[1+e^{-d\left(R-l\right)}\right]-{\sum }_{i=1}^{n_1}d\cdot \left(R-l\right)+\ln \left[1+e^{-d\left(R-l\right)}\right]$(10)

The parameters and their asymptotic variances are derived using the iterative Newton–Raphson method.

The Rasch model: A one parameter model

The Rasch model is based on three assumptions:

• - Uni-dimensionality: the latent trait is unique, and all the answers given for the different items depend on the same latent trait.

• - Conditional independence of the responses: for each individual, the answers given to the different questions are statistically independent, given the latent trait. In other words, the latent trait is the only factor that explains the difference between the response patterns provided by two individuals.

• - Monotonicity: the conditional probability for an individual of not having a car, for example, is a monotonic non decreasing function of the individual level of material deprivation.

Note that within the Rasch measurement approach the items in the scale have similar power in discriminating among respondents; that is, there is equal discrimination of the items. This characteristic differentiates the Rasch model from any other IRT model. This characteristic of equal discrimination is also known as parameter separability. In fact in the RaschFootnote ¹² (Reference Rasch1960) model it is assumed that the discrimination parameter d is equal to 1, whatever the item (e.g. durable good) under study. In such a case expression (6) will be written as

$P\left(R\right)=\frac{1}{1+e^{-1\left(R-l\right)}}$(11)

It is important to stress that the Rasch model is a confirmatory model. In other words, only when the data meet the Rasch model’s requirements can a satisfactory measurement tool be derived. It is therefore of utmost importance to check that the data fit the assumptions of the Rasch model.

Three Parameters Model

In the psychometric tests literature such a model takes into consideration the fact that individuals may select an answer by just guessing. A third parameter, g, is therefore introduced which is not a function of the ability of the individual. In our case such a parameter would simply indicate the lowest value of the probability of having a durable good and would correspond to the intercept of the item characteristic curve with the vertical axis.

The equation for such a model is then expressed as

$P\left(R\right)=g+\left(1-g\right)\left[\frac{1}{1+e^{-d\left(R-l\right)}}\right]$(12)

The location parameter is now defined as the point on the horizontal axis where the probability $P\left(R\right)$ is equal to $\left(1+g\right)/2$ and not any more to 0.5. Note that the discrimination parameter d is still proportional to the slope of the item characteristic curve at the point where $R=l.$

3.5 The Borda Approach

In an original paper Reference FineFine (1983) suggested deriving the order of acquisition of consumer durables from the Borda Rule which is named after the eighteenth-century French mathematician and naval engineer Jean-Charles de Borda. Borda seems to have devised this rule in 1770, but, according to Reference BrianBrian (2008), 1784 is the correct date of attribution. What is Borda’s rule? According to Borda’s method, voters are asked to rank the candidates in order of preference from the first to the last choice. The lowest ranking candidate is then given 1 point, the second lowest 2 points, etc., and the top candidate is attributed a number of points which is equal to the number of candidates. Then one adds the number of points given by all the voters to each candidate and the candidate who received the highest number of votes is the winner.

Reference FineFine (1983) started by stressing the parallelism that exists between the idea of an order of acquisition of durables and social choice theory. More precisely, individuals are assumed to have an order of priority in acquiring durable goods, and what we are looking for is to find an ordering that would represent the individual orderings in a consistent way. And this is clearly similar to what is done in social choice theory, where individuals are faced with social alternatives, and we are looking for a social ordering that would represent the individual orderings according to some well-defined criteria. The idea is to construct a “representative consumer” who is supposed to represent the preferences of society, but this “representative consumer” does not represent the average in terms of aggregate demand.

Assume n individuals with individual orderings $O{R}_1,O{R}_2,\dots ,O{R}_n.$

Here is the set of assumptions made by Reference FineFine (1983):

• - Symmetry: This is a standard assumption. It says that if we swap the alternative orders as well as the individuals, we will end up with the original order of preference of the “representative consumer.”

• - Monotonicity: If one or several individuals move a durable up in their order of preference, this should be reflected in the order of preference of the “representative consumer.”

• - Composition condition: If groups of consumers who agree with each other are brought together, then the order of preferences of the “representative consumer” should not be affected.

• - Labeling neutrality: If we relabel each durable good, the new names given to the durables should not affect the ordering of the “representative consumer.”

• - Independence condition: If a new durable good is introduced and either each individual buys it first or each individual buys it last, the order of preference for the original durables of the “representative consumer” should not be affected.

• - Inversion condition: If rather than ordering the durable goods by ownership, we rank them by “non-ownership” the order of preference of the “representative consumer” should be identical to the inverse of the order of preference of the “representative consumer” for the owned durables.

Reference FineFine (1983) then proves that the order of preference of the “representative consumer” that obeys these six conditions is the Borda rule.

3.6 Comparing the Order of Acquisition with the Structure of Ownership

The idea here is to use a count approach and simply to compute the percentage of households that have no durable, one durable good, two durable goods, and so on. One then classifies these percentages by decreasing values and checks how different the ranking obtained is from the order of acquisition obtained using one of the methodologies that have been previously discussed.

4.1 Introduction

This section is devoted to empirical studies aiming at discovering the order of acquisition of durable goods and assets. We first review the empirical literature. In Section 4.2 we focus our attention on papers that applied the approach of Paroush, while in Section 4.3 we review studies that used Item Response Theory. Then in Section 4.4 we present a new empirical study based on data from the 2019 Eurobarometer survey. We first derive the order of acquisition obtained with each approach and then compare the two sets of results. Finally, we check to what extent the order of acquisition of a country depends on its standard of living.

4.2 Empirical Illustrations of the Order of Acquisition of Durable Goods or Financial Assets: The Approach of Guttman and Paroush

The paper by Reference KayKay (1964) seems to have been the first one to link the possession of durable goods to a Guttman scale. Kay examined data covering a single neighborhood (Manuhoe) of Papeete, the capital of French Polynesia. There were seven kinds of durable goods: a primus stove, a bicycle, a radio, a two-wheeled motor vehicle, a kerosene or gas stove, a refrigerator, and an automobile. This study covered only forty households and the most common order of acquisition was that following the order of durable goods in the list previously mentioned.

Reference ParoushParoush (1965) gives an illustration based on the 1961 Survey of Urban Families Expenditures in Israel and considers three durable goods: gas cooker, refrigerator, and washing machine. It turns out that 13 percent of the families owned the three durable goods, 32 percent a gas cooker and a refrigerator, 16 percent only a gas cooker, and 26 percent did not own any of these three durable goods. Looking at the deviations (12 percent of the sample) from the order “first gas cooker, then refrigerator, finally washing machine,” Paroush observed that 7 percent owned only a refrigerator, 2 percent a refrigerator and a washing machine, 2 percent a gas cooker and a washing machine, and 1 percent only a washing machine. These results covered 99 percent of the sample so that, given that three durable gods were taken into account, the coefficient of Reproducibility R turns out to be expressed as

$R=1-\left[\left(\frac{1}{3}\right)\left(\frac{12}{99}\right)\right]=0.96$

Reference McFallMcFall (1969) also studied priority patterns in acquiring durable goods. He assumed that “consumers have sets of needs and values which are satisfied by sets of products, or, more correctly, by sets of product attributes.” Such attributes could refer, for example, to “clean,” “cook,” “entertainment,” “comfort,” and so on. Using data obtained from a sample of upper income consumers in San Diego county, McFall considered the following four durable goods: automatic washer, electric blanket, dishwasher, air conditioner, and he found that this was the most common order of acquisition with 40 families owning all the four durables, 110 owning an automatic dishwasher, an electric blanket, and a dishwasher, 132 families owning only an automatic dishwasher and an electric blanket, 46 families owning only an automatic dishwasher, and 8 families not owning any of these four durable goods. There were 63 families that did not fit this order of acquisition (out of a total of 3999 families).

Reference Katsulis, Lusch and StaffordKasulis et al. (1979) used data on consumer durables that were obtained for the years 1975 and 1976 from the Continuing Consumer Audit of the Distribution Research Program at the University of Oklahoma and the Oklahoma Publishing Company. Their analysis included 12 household durables and covered 1,747 respondents in 1975 and 2,025 in 1976. Using the Guttman scaling they concluded that in 1976 the order of acquisition was as follows: first TV, first vehicle, refrigerator, range, washer, stereo or tape player, dryer, second vehicle, second TV, dishwasher, freezer, and microwave oven. The order was almost the same in 1975. They also made a separate analysis for renters and owners, and it appears that the orders of acquisition were somewhat different in these two groups.

Reference Stafford, Kasulis and LuschStafford et al. (1982) used also the Guttman scaling to derive the order of acquisition by households of the following eight assets: corporate bonds, corporate stocks, checking accounts, husband’s life insurance, mutual funds, savings accounts, trusts, and wife’s life insurance. The data were also obtained for the years 1975 and 1976 from the Continuing Consumer Audit of the Distribution Research Program at the University of Oklahoma and the Oklahoma Publishing Company. For the full sample (about 3500 respondents) the order of acquisition obtained was as follows: Checking Account, Husband’s Life Insurance, Savings Account, Wife’s Life Insurance, Stocks, Bonds, Trust and Mutual Funds. The order obtained was quite similar when looking separately at three age groups: under thirty-five, thirty-five to forty-nine, and over fifty.

Reference Clarke and SoutarClarke and Soutar (1982) analyzed a sample of 540 heads of households in the Perth (Australia) metropolitan area. The questions covered a list of fifteen durable goods and the order of acquisition obtained for the whole sample was as follows: Refrigerator, Washing machine, Vacuum cleaner, Color television, Power drill, Lawn-mover, Hi-fi stereo system, Deep-freeze unit, Air conditioner, Food processor, Clothes dryer, Video cassette recorder, Built-in swimming pool, Electric dishwasher, Microwave oven. The order was slightly different for owners and renters but very similar to that of the whole sample.

Reference Dickson, Lusch and WilkieDickson et al. (1983) used data from a 1978 Home Testing Institute screening survey of a panel of 5,000 households in the United States. Households were asked about the ownership of the following twelve appliances, and this is the order of acquisition obtained, using the Guttman scaling: Refrigerator, Clothes washer, Color TV, Sewing machine, Kitchen oven, Clothes dryer, Stereo AM/FM radio, Separate freezer, Dishwasher, Room air-conditioner, Microwave, and Video Recorder. This order turns out to be identical to that obtained by classifying the appliances by the percentage of households that own them.

Reference Dickinson and KirznerDickinson and Kirzner (1986) focused their attention on the patterns of acquisition of financial assets. They used data from a survey carried out on behalf of a professional association in Canada. A questionnaire was sent out to each of the 36,044 members of this association and 14,569 questionnaires were returned. Ten financial assets were considered. Guttman’s scalogram analysis revealed the following acquisition pattern: Checking account, Savings account, Husband insurance, RRSP (registered retirement savings plan), Wife insurance, Savings bonds, GIC (guaranteed investment certificates), Corporate stock, Mutual fund, and Corporate bond.

Reference Deutsch, Silber, Kakwani and SilberDeutsch and Silber (2008) used as database the 1995 Census of the Israeli population. This Census provides quite detailed information on the ownership of durable goods. Here are the durable goods that were taken into account in their study: telephone, television, washing machine, apartment (or house), VCR, car, microwave oven, air conditioner, dishwasher, computer, and dryer. Applying the approach of Reference ParoushParoush (1963), they observed that the most common order of acquisition of durable goods was that given by the list previously mentioned.

Reference Bérenger, Deutsch and SilberBérenger et al. (2013) used data from the Demographic and Health Surveys (DHS) conducted in Egypt in 2005, in Morocco in 2003–2004 and in Turkey in 2003. The following durable goods and access to basic services were taken into account: ownership of dwelling, fuel for cooking, car, television, refrigerator, washing machine, video, phone, air conditioning, and type of toilet facilities. The orders of acquisition obtained, using the approach of Reference ParoushParoush (1963), were quite similar in the three countries. Thus in Egypt the order was: housing, cooking, toilet, TV, refrigerator, phone, washing machine, video, car, air conditioning. In Morocco the order was: housing, cooking, toilet, TV, phone, refrigerator, video, washing machine, car, and air conditioning. Finally in Turkey the order was: housing, TV, refrigerator, phone, toilet, washing machine, cooking, car, video, and air conditioning. Not surprisingly there were, however, differences between the orders observed in urban and rural areas.

Reference Bérenger, Deutsch and SilberBérenger et al. (2016), applying the approach of Reference ParoushParoush (1963), analyzed data from samples of the 2000 and 2010 Mexican Censuses and derived the most common order of acquisition of durable goods in the various states of Mexico. In 2000, in most of the states, the most common order of importance seems to be toilet, television, radio, refrigerator, and washing machine. Then, depending on the states, they found hot water heater, car or phone, and finally computer. Quite similar patterns were observed in 2010. Thus the list of the four first goods remains the same, but in some states access to toilet becomes more essential in 2010 than in 2000.

Reference Deutsch, Silber, Wan and ZhaoDeutsch et al. (2020b) examined data on household consumer durables obtained from the Asian Barometer Survey for six countries of Southeast Asia: Cambodia, Indonesia, Malaysia, the Philippines, Thailand, and Vietnam. They applied the approach of Paroush and had data on the following eight assets: Car, jeep, or van; Color or black-and-white television; mobile phone; Electric fan or cooler; Scooter, motorcycle or bicycle; Radio transistor; Pumping Set; and Refrigerator. For the Philippines the order of acquisition obtained was as follows: TV, Fan/cooler, Transistor, Mobile phone, Refrigerator, Motorcycle, Pumping Set, and Car. For Thailand the order was: Fan/cooler, TV, Mobile phone, Motorcycle, Refrigerator, Transistor, Car, and Pumping Set. For Indonesia the order was: TV, Motorcycle, Mobile phone, Fan/cooler, Pumping Set, Refrigerator, Transistor, and Car. For Vietnam the order was: Motorcycle, TV, Fan/cooler, Mobile phone, Pumping Set, Refrigerator, Transistor, and Car. For Cambodia the order was: Motorcycle, TV, Mobile phone, Transistor, Fan/cooler, Pumping Set, Car, and Refrigerator. Finally for Malaysia the order was: Mobile phone, TV, Fan/cooler, Refrigerator, Motorcycle, Car, Transistor, and Pumping Set. Although the orders of acquisition vary from one country to the other, in many cases the correlation between the orders obtained in two countries was higher than 0.5 and often 0.6. But clearly the order reflects preferences for different durables, and preferences depend, among other factors, on culture, geographical conditions, and level of development.

Reference Deutsch, Silber, Wan and XuDeutsch et al. (2020a) looked at data from the 2009 and 2013 Caucasus Barometer surveys and obtained information on the ownership of durable goods in Armenia, Azerbaijan, and Georgia. The following durable goods were taken account of: color TV, washing machine, refrigerator, air conditioner, car, cell phone, and personal computer. For 2013 the order of acquisition observed for Armenia was as follows: color TV, cell phone, refrigerator, washing machine, car, personal computer, and air conditioning. For Azerbaijan in 2013 the order was: cell phone, color TV, refrigerator, washing machine, car, air conditioning, and personal computer. Finally for Georgia the order in 2013 was: cell phone, color TV, refrigerator, washing machine, personal computer, car, and air conditioning. Note that the rank correlation between two countries was at least equal to 0.85 so that the orders of acquisition may be considered quite similar in the three countries examined.

Reference Deutsch, Silber and SilberDeutsch and Silber (2023) examined data obtained from the Eurobarometer for the year 2017. The data were available for thirty countries: France (1), Belgium (2), The Netherlands (3), West Germany (4), Italy (5), Luxembourg (6), Denmark (7), Ireland (8), Great Britain (9), Northern Ireland (10), Greece (11), Spain (12), Portugal (13), East Germany (14), Croatia (15), Finland (16), Sweden (17), Austria (18), Cyprus Republic (19), Czech Republic (20), Estonia (21), Hungary (22), Latvia (23), Lithuania (24), Malta (25), Poland (26), Slovakia (27), Slovenia (28), Bulgaria (29), and Romania (30). The following information was available concerning the durables and services available in the households: Phone (Good 1), Mobile Phone (Good 2), Television (Good 3), DVD (Good 4), CD player (Good 5), Computer (Good 6), Laptop (Good 7), Tablet (Good 8), Smartphone (Good 9), Internet (Good 10), Car (Good 11), and House which is already paid for (Good 12). We will not give the orders of acquisition obtained, using the Paroush approach, for each of these thirty countries. But it is interesting to note that Reference Deutsch, Silber and SilberDeutsch and Silber (2023) regressed the rank correlation between the orders of acquisition of two countries on the absolute value of the difference between the equalized median incomes of these two countries. They found that the coefficient of the absolute value of this difference was negative and significant and the R-squared of this regression was equal to 0.46.

4.3 Empirical Illustrations of the Order of Acquisition of Durable Goods or Financial Assets: Using Item Response Theory

Although Item Response Theory (IRT) was originally used in psychometrics, this technique has also been used to measure deprivation, the idea being that poverty, like intelligence, is really a latent variable that is difficult to measure.

Let us first assume that deprivation is unidimensional. Deprivation, however, cannot be observed as it is a latent variable. What we observe is a distribution of a certain number of binary deprivation indicators observed for, say, n households or individuals. A simple way of estimating overall deprivation would be to assume that it is equal to the unweighted or weighted sum of all the dichotomous indicators. In the latter case one could assume, for example, that a higher weight is given to an indicator, the lower its prevalence, the idea being that if not many households (individuals) do not have a given item, this item should not contribute much to the overall level of deprivation.

Assume we give the same weight to all the deprivation indicators. Then the value $IN{D}_{jh}$ taken by indicator h for individual j could be expressed as

$IN{D}_{jh}=L{D}_j+u_{jh}$

where $L{D}_j$ refers to the value of the latent deprivation for individual j, while $u_{jh}$ is an error term assumed to have zero mean and to be independent of $L{D}_j$ . These error terms are also assumed to be mutually independent. It is then easy to observe, if the number of deprivation indicators is large enough, that the average value of the deprivation indicators for individual j will be equal to the value of the latent deprivation for this individual.

The problem is that these deprivation indicators are dichotomous and not continuous variables so that this approach to measuring latent deprivation cannot be adopted, hence the idea to use Item Response Theory.

The first applications of Item Response Theory to the analysis of poverty seem to have been the studies of Reference DickesDickes (1983), Reference Dickes, Gailly, Hausman and SchaberDickes et al. (1984) and Reference Gailly, Hausman and SarpellonGailly and Hausman (1984). They used the Rasch model, estimating a unique parameter per item, the one which measures the “difficulty.” Therefore, all the items have the same slope as far as the relationship between poverty and the probability of having a disadvantage is concerned. Poverty is hence defined as an accumulation of disadvantages.

Reference Soutar and Cornish-WardSoutar and Cornish-Ward (1997) analyzed data on durable and financial asset ownership. These data were collected in 1989 in the area of Perth in Western Australia, by a commercial social research organization using telephone interviews. The authors applied the Rasch model using information on the ownership of twenty-five durable goods. The Rasch model ranked them in the following order: refrigerator, vacuum cleaner, washing machine, first color TV, toaster, first car, power tools, electric fan, Hi-Fi/video, video recorder, microwave oven, lawn lower, deep freezer, second car, food processor, clothes dryer, second color TV, electric shaver, small-food heater, large space heater, air conditioner, juicer, solar hot water heater, computer, and car phone.

For financial assets the Rasch model gave the following order: savings account, property home, checking account, superannuation, life insurance, shares, fixed deposits in bank, fixed deposits in credit society, other property, cash management trust, equity trust, collectables, insurance bonds, government bonds, and debentures.

Reference Cappellari, Jenkins, Jenkins and MicklewrightCapellari and Jenkins (2007) used data from wave 6 (survey year 1996) of the British Household Panel Survey (BHPS). Their focus was on “basic life-style” deprivation and they used seven binary indicator variables. The first six variables asked whether the households were able to keep their home adequately warm, eat meat, chicken, fish every second day, buy new, rather than second hand, clothes, have friends or family for a drink or meal at least once a month, replace worn-out furniture, and pay for a week’s annual holiday away from home. The seventh binary indicator variable asked whether the responding household had any difficulties paying for their accommodation in the last twelve months. When using a one-parameter model, the results were very similar to those obtained when looking at the frequency of each of the seven deprivation aspects analyzed, namely that having difficulties paying for the accommodation is the most common deprivation, followed by the difficulty of taking a week’s vacation, and so on. Reference Cappellari, Jenkins, Jenkins and MicklewrightCapellari and Jenkins (2007) found similar results when using a two-parameter IRT model.

Reference Dickes, Fusco, Kakwani and SilberDickes and Fusco (2008) also applied the Rasch model. Their empirical illustration was based on the EU-SILC data. Luxembourg was part of the countries that launched their survey in 2003, and the initial sample of PSELL-3 (Panel Socio-Economique “Liewen zu Lëtzebuerg”) consisted of 3,500 representative households (9500 individuals). Dickes and Fusco used the data relative to the second wave of PSELL-3, conducted in 2004. They selected a set of items referring to the absence of housing facilities, to problems with the accommodation, to problems with the environment or neighborhood, to the inability to afford most basic requirements, to the inability to meet payment schedules, and to the lack of durable goods. A list of twenty-nine dichotomous items was originally selected, and for every item, an estimation of severity parameter was obtained. Dickes and Fusco then focused their attention on the following nine durable goods: color TV, computer, washing machine, private car, camera, video player, CD player, DVD player, and audio tape player. They then concluded that the following items fit the Rasch model assumptions: the possession of a video player, a camera, a private car, a washing machine, and a color TV. This is the order given by the severity (location) parameter, implying that this is the sequence of acquisition of these five durable goods. Dickes and Fusco also analyzed other deprivation domains such as financial difficulties and problems with the environment.

Reference Raileanu Szeles and FuscoRaileanu Szeles and Fusco (2013) applied IRT models to measure deprivation and analyze its determinants in Luxembourg. Their analysis used data from the Socio-Economic Panel “Liewen zu Lëtzebuerg” (PSELL-3), which is the Luxembourgish component of the EU Community Statistics on Income and Living Conditions (EU-SILC). Nine dichotomous items pertaining to the enforced lack of durable goods, housing facilities, and the capacity to afford basic requirements were taken account of. The items are listed below according to their increasing deprivation rates in Luxembourg computed on 3,001 observations: cannot afford to have a washing machine (if wanted to), the dwelling has no bath, cannot afford keeping home adequately warm, cannot afford to have a car (if wanted to), cannot afford eating meat or equivalent every second day, cannot afford one-week annual holiday away from home, and cannot afford facing unexpected expenses. Raileanu Szeles and Fusco estimated a one- and a two-parameter IRT model and this allowed them to rank the items of deprivation according to their parameter of severity. The items “washing machine” and “bath” were the most severe in the one-parameter IRT, implying that the probability that an individual who has no bath (or no washing machine) will be deprived of the other items is higher than 0.5. When using the two-parameter IRT models, the item “washing machine” is again the most difficult item, whereas the rank of the “bath” was lower than in the one-parameter model. The ability to face unexpected expenses is the easiest item in both models. Note also that the one- and two-parameter models yield different rankings of deprivation items in terms of their difficulty, a result which is different from what Reference Cappellari, Jenkins, Jenkins and MicklewrightCappellari and Jenkins (2007) had found.

Reference Dholakia and BanerjeeDholakia and Banerjee (2013) used data provided by Pathfinders, a research group based in Mumbai, India, that regularly surveys households on various consumption related attitudes and behaviors. The datasets contained 2,864 respondents in 1996, and 2,686 respondents in 2002, drawn from the western part of India. The authors used the two-parameter Item Response Theory to detect the priority pattern in the acquisition of household durables. In 1996 the list of durables was as follows: radio, pressure cooker, ceiling fans, color TV, food processor, sewing machine, refrigerator, cassette player, Two in One, video, washing machine, stereo, Air conditioning, vacuum cleaner, and cooking range. The priority followed this order so that a radio had the highest priority and a cooking range the lowest.

In 2002 the priorities were as follows: pressure cooker, food processor, refrigerator, telephone, washing machine, Hi-Fi, mobile, video, air conditioner, computer, DVD/CD player, sewing machine, oven, radio, dishwasher, Two in One, and cooking range. The authors stressed the fact that by 2002, the hierarchy had become somewhat similar to the pattern of durable ownership observed in countries like Israel (see Reference ParoushParoush, 1965).

Reference Deutsch, Guio and PomatiDeutsch, Guio, and Pomati (2015) used Item Response Theory to estimate the extent of material deprivation in twenty-seven European Union countries, using the EU-SILC 2009 data. As mentioned previously, this database was also used to apply the Paroush approach. The focus here was on the order of curtailment of expenditures when households start facing poverty. The expenditures in this analysis were the following ones: holidays, unexpected expenses, furniture, pocket money, leisure, drink/meal out, clothes, meat/chicken/fish, keeping the home warm, arrears, car, computer/Internet, and shoes. IRT analysis showed that, for the European Union as a whole, the sequence of curtailment of expenditures followed the order of the items that was just mentioned. In most countries a one-week holiday away from home is always one of the first three types of expenditures to be curtailed, together with the ability to face unexpected expenses. On the other hand, two pairs of all-weather shoes are at least the eighth item to be given up and “computer/Internet” at least the ninth item. Finally note that the orders of expenditures curtailment are very similar to those obtained when using the Paroush approach.

Reference Bérenger, Deutsch and SilberBérenger, Deutsch, and Silber (2016) applied Item Response Theory to Mexican Census data for the years 2000 and 2010, as they did when applying the Paroush approach (see Section 4.2) to these data. They selected the same nine assets, namely a phone, a car, a hot water heater, a computer, a washing machine, a refrigerator, a television, a radio, and a toilet. These authors then found that the order derived from IRT was almost the same as that obtained with the Paroush approach, this being true for almost all the Mexican states. Thus, in 2000, in every state, the first items were access to toilet, television, radio, refrigerator, washing machine, while the last one was a computer. Small differences between the two approaches were observed mainly when looking at the ranking of the three last goods in the order of acquisition. This ranking similarity between the IRT and the Paroush approach was also observed when analyzing the 2010 Census.

Reference Deutsch, Silber, Wan and ZhaoDeutsch et al. (2020b) applied Item Response Theory to the database they had used when applying the Paroush approach, namely the 2014 and 2016 Asian Barometer Survey (ABS). Here also they focused their attention on six countries in Southeast Asia (Cambodia, Indonesia, Malaysia, the Philippines, Thailand, and Vietnam) and eight durable goods (car, jeep or van; color or black-and-white television; mobile phone; electric fan or cooler; scooter, motorcycle or bicycle; radio transistor; pumping set; and refrigerator). They then computed the rank correlations between the countries. It turned out that ten of the fifteen correlations were higher than 0.5 and 8 higher than 0.6. The highest correlation observed was between Vietnam and Indonesia. Thailand and Malaysia also exhibited similar orders of acquisition. But the correlation between the rankings of Cambodia and Malaysia was low (0.12). It is clear that the order of preferences depends on the level of development but also on culture, geographical conditions, and other factors.

Deutsch, Silber, Xu, and Wan (2020) applied Item Response Theory to the 2009 and 2013 Caucasus Barometer surveys which provided information on the ownership of durable goods in Armenia, Azerbaijan, and Georgia. These authors used the same list of durable goods they had selected when applying the Paroush approach (see Section 4.2), namely a color TV, a washing machine, a refrigerator, an air conditioner, a car, a cell phone, and a personal computer. For each of the three countries examined they found a very high correlation (above 0.85) between the order of acquisition obtained when adopting the Paroush approach and that derived when using Item Response Theory. The correlation between the orders obtained when comparing two countries was very high also (at least equal to 0.92).

Reference 86Farcomeni, Pittau, Viviani and ZelliFarcomeni et al. (2022), using the Rasch model analytical procedure, derived from the Survey on Income and Living Conditions (EU-SILC) pre-Covid data, a European measurement reference scale for material deprivation which ensures validity and comparability across different countries. Their analysis included originally twenty-three items collected at both the household and individual level. However, they ended up working with eight items to guarantee uni-dimensionality. Here is the list of the eight items ranked by increasing severity:

• - The household cannot afford to face unexpected expenses.

• - The person cannot afford to spend a small amount of money each week on himself/herself.

• - The household has arrears on mortgage, rent, utility bills or loans.

• - The person cannot afford to get-together with friends/family (relatives) for a drink/meal at least once a month.

• - The household cannot afford a meal with meat, chicken and fish (or vegetarian equivalent) every second day.

• - The person cannot afford to have Internet connection for personal use at home.

• - The person cannot afford medical examination or treatment.

• - The household has not a telephone (including mobile phone).

Reference Deutsch, Silber and SilberDeutsch and Silber (2023) applied Item Response Theory to the same Reference Deutsch, Silber and Wan2017 Eurobarometer database they had used to examine the Paroush approach to the order of acquisition of durable goods. Their analysis covered the same thirty countries and included the same twelve durable goods mentioned in Section 4.2. For each country, they therefore derived the ranking of the different goods and compared these rankings with that obtained when using the Paroush approach. More precisely they computed for each country the correlation coefficient between the set of rank correlations derived when using the Paroush approach and that obtained when implementing Item Response Theory and found that this coefficient of correlation was at least equal to 0.9, except for Spain, where it was equal to 0.87. They then regressed the rank correlation between the orders of acquisition of two countries on the absolute value of the difference between the equalized median incomes of the two countries compared. The coefficient of the absolute value of this difference was negative and significant, and the R-squared of this regression was equal to 0.50.

4.4 Comparing the Different Approaches to the Derivation of the Order of Acquisition of Durable Goods: A New Study

This empirical investigation is based on data from the 2019 Eurobarometer survey. We apply the Rasch model (see Section 3.4 for more details on Item Response Theory and the Rasch model). We used data covering thirty-five countries: France (1), Belgium (2), The Netherlands (3), Former West Germany (4), Italy (5), Luxembourg (6), Denmark (7), Ireland (8), United Kingdom (9) Greece (10), Spain (11), Portugal (12), Former East Germany (13), Finland (14), Sweden (15), Austria (16), Cyprus Republic (17), Czech Republic (18), Estonia (19), Hungary (20), Latvia (21), Lithuania (22), Malta (23), Poland (24), Slovakia (25), Slovenia (26), Bulgaria (27), Romania (28). Turkey (29), Croatia (30), Cyprus TCC(31), North Macedonia (32), Montenegro (33), Serbia (34), and Albania (35).

The following information was available concerning the durables and services available in the households: Phone (Good 1), Mobile Phone (Good 2), Television (Good 3), DVD (Good 4), CD player (Good 5), Computer (Good 6), Laptop (Good 7), Tablet (Good 8), Smartphone (Good 9), Internet (Good 10), Car (Good 11), and House which is already paid for (Good 12).

Table 4 gives the most common order of acquisition for each of the thirty-five countries under study, when using the Paroush approach. The columns indicate which good (or service) has the first, second, third rank, and so on. Table 5 is similar to Table 4, but it is derived from Item Response Theory.

We can then use each of these two tables to compute, for each binary comparison of countries, the rank correlation between two countries in the order of acquisition of these goods or services. These correlations are presented in Appendix B in Table B.1 for the Paroush approach and in Table B.2 for the approach based on Item Response Theory. We observe a great variety of values for these correlation coefficients as some correlations are quite low (less than 0.2) and some others very high (close to 1). A quick look at these correlations seems to indicate that the correlations are much higher when the two countries compared are of a somewhat similar development level. This was confirmed by a regression analysis, which will be mentioned below.

We then computed for each of the thirty-five countries on which we had information the correlation coefficient between the order of acquisition derived from the Paroush approach and that obtained when implementing Item Response Theory. These correlations are presented in Table 6. It turns out that this coefficient of correlation was never below 0.85, most of the time above 0.9, if not 0.95. Therefore, it does not seem to matter very much whether we use the Paroush approach or Item Response Theory to derive the most common order of acquisition of these goods and services.

Finally, we regressed the rank correlation between the orders of acquisition of two countries on the absolute value of the difference between the per capita GDP of these two countries. One regression was based on the rank correlation derived from the Paroush approach and the other from that obtained with Item Response Theory. These regressions are given in Table 7. It turns out that the coefficient of the absolute value of this per capita GDP difference is negative and significant, and the R-squared of this regression is equal to 0.38 when using the Paroush approach and to 0.46 when implementing Item Response Theory.

Note that in estimating these regressions we took account of the fact that we work with dyadic regressions. As stressed by Reference Cameron and MillerCameron and Miller (2014), regression models with paired or dyadic data have a complicated pattern of error correlations. Reference Cameron and MillerCameron and Miller (2014) observed that even after including country fixed effects, standard errors that properly control for dyadic error correlation can be several times those being reported using regular methods.Footnote ¹³

4.5 Concluding Comments

Two main conclusions may be drawn from the new empirical investigation that was conducted in this section. First, it probably does not matter whether the order of acquisition of durable goods and assets is derived via the use of the Paroush approach or using Item Response Theory. Both techniques seem to lead to very similar results. Second, the order of acquisition of durable goods and assets depends, at least partly, on the standard of living of a country since we found that the rank correlation between the orders of acquisition of two countries is a negative and significant function of the absolute gap between the per capita GDP of these two countries. This leads one to conclude that the order of acquisition of durable goods and assets could be considered as a proxy for the standard of living of a country although clearly other factors, such as geographical and cultural considerations, most likely have also an impact on such an order of acquisition.

5.1 Introduction

In this section we show that deriving the order of acquisition of durable goods and assets may allow one to draw conclusions concerning the extent of inequality and poverty in a country and even to compare the levels of welfare in different countries. This is possible because such an order of acquisition tells us which percentage of the individuals (households) surveyed have no durable good, one, two, three …, all the durable goods. This information then allows one to estimate the extent of inequality and poverty in a country, given that important work has shown in recent years how to obtain measures of inequality and poverty when only ordinal variables are available. It is even possible to estimate the extent of welfare, using recent work deriving achievement indices that also take account of the inequality of achievements. This recent literature focusing on ordinal variables is reviewed in Section 5.2 for the measurement of inequality, in Section 5.3 for the derivation of poverty indices, and in Section 5.4 for the estimation of achievement indices, and more generally of welfare. Finally, in Section 5.5 an empirical illustration, based again on data from the 2019 Eurobarometer Survey, is presented where the different inequality, poverty, and achievement indices are computed for the various countries that participated to this survey.

5.2 Ordinal Variables and the Measurement of Inequality

In a pathbreaking article, Reference Allison and FosterAllison and Foster (2004) stressed the fact that traditional measures of inequality, such as the Gini coefficient, the index of Reference AtkinsonAtkinson (1970), the entropy-related measures introduced by Reference TheilTheil (1967), or the variance, are based on the mean since in one way or the other they turn out to be deviations from the mean or they are indices which are normalized by the mean. However, when the variable under study is ordinal, the notion of mean is not well defined. Reference Allison and FosterAllison and Foster (2004), taking self-assessed health, an ordinal variable, as illustration, showed that in measuring health inequality in different populations, via traditional inequality indices, completely different results could be obtained, depending on the scale chosen for the ordinal categories.

This is the reason why Reference Allison and FosterAllison and Foster (2004) advocated a median-based approach. They then defined what they called a S-curve,Footnote ¹⁴ which allows ranking distributions of ordinal variables in the same way as the Lorenz curve represents the Lorenz ranking of income distributions. The S-curve, like the Lorenz curve, gives only a partial ordering since when two S-curves cross, no conclusion may be drawn. The authors then indicated that several useful indexes of “spread” can be derived from the S-curve in the same way as the Gini index may be derived from the Lorenz curve.

Three years later, in a quite original article, Reference ApoueyApouey (2007) noted that the literature on polarization (see Reference WolfsonWolfson, 1994; Reference Esteban and RayEsteban and Ray, 1994; Reference Duclos, Esteban and RayDuclos et al., 2004) also puts the emphasis on the median. This literature deals, however, with cardinal variables, but the two principles that it stresses are also relevant in the case of ordinal variables, as will be shown in the next paragraphs.

The first principle (increasing spread) states that moving from the middle position (the median) to the tails of the distribution will make the distribution more polarized. In the case of a cardinal variable such as income, this principle implies that a rank-preserving increment in incomes above the median or a rank-preserving reduction in income below the median will widen the distribution, that is, increase the distance between the group of individuals whose income is smaller than the median income and the group of people whose income is higher than the median income. Consequently, polarization in society will increase because the rich become richer and the poor poorer.

The second principle is called increased bipolarity. It refers to the case where the incomes below the median or those above the median become closer to each other. The literature on polarization states then that there has been some “bunching” of the two groups; that is, the gaps between the incomes below the median (or those above the median) have been reduced. If this is so, bipolarization is said to increase.

It should therefore be clear that there is an important difference between the notions of “inequality” and “bipolarization.” While any regressive transfer will increase inequality, it will increase bipolarization only in the case where such a transfer takes place across the median. When it takes place on the same side of the median, bipolarization will decrease.

Reference ApoueyApouey (2007) used these two principles of increased spread and bipolarity to derive polarization indices that could be used in the case of ordinal variables. Moreover, the transfer principle was interpreted, in the case of ordinal variables, as implying a movement of individuals from one category to another. Reference ApoueyApouey (2007) also indicated that such a polarization index should vary between 0 and 1. Taking as illustration the order of acquisition of durable goods, it is assumed that there is no polarization when everyone has the same number of durable goods and polarization will be maximal (the index being equal to 1) when half the population owns no durable good and the other half has all the durable goods.

Assume $\left(K+1\right)$ categories so that $k=0$ corresponds to the case where an individual does not own any durable good while $k=K$ refers to the case where the individual owns all the $K$ durable goods. The ordinal inequality index Apouey proposed is then defined as

(13)$I_{APOUEY}=1-\frac{2^{\alpha }}{K}{{\sum }_{k=0}^K\left|P_k-0.5\right|}^{\alpha }$

where $P_k$ refers to the cumulative frequency corresponding to the case when k durable goods are owned. Note that Apouey suggested to calibrate α in such a way that it will be equal to 0.5 in the case of a uniform distribution (the case where there is the same number of individuals in each of the possibilities considered, that is from $k=0$ to $k=K$ ). The idea is that a uniform distribution is a kind of intermediate state between the extreme cases where polarization is minimal or maximal. Apouey then listed some other useful properties of her index.Footnote ¹⁵

In another important paper Reference Abul Naga and YalcinAbul Naga and Yalcin (2008) characterized the entire class of inequality indices founded on the ordering defined by Allison and Foster. The ordinal inequality measures they derived have also the property that inequality is maximal when half of the population is in the lowest category and the other half is in the highest one. These ordinal inequality indices were shown by the authors to satisfy the properties of continuity, scale invariance, normalization, and aversion to median preserving spreads.Footnote ¹⁶

Taking again as illustration the order of acquisition of durable goods, let $p_k$ be the proportion of individuals in category k. Assume that the various categories are ordered by increasing number of durable goods owned. Define then $P_k$ as the cumulative values of the probabilities $p_k$ , that is,

$P_k=p_0+p_1+\dots +p_k$ . The first ordinal inequality index suggested by Abul Naga and Yalcin is then defined as

$I_{ABUNAGAYALCIN}=1-\left[\frac{\left(2{\sum }_{k=0}^K\left|P_k-0.5\right|-1\right)}{K}\right]$(14)

Note that this index $I_{ABUNAGAYALCIN}$ has the four properties mentioned previously.

Abul Naga and Yalcin introduced then a generalization of the index defined in (14), which depends on two parameters. These two parameters allow one to assume asymmetry, that is, to obtain different results when a given deviation from 0.5 take place below or above the median.Footnote ¹⁷

Reference Kobus and MilosKobus and Milos (2012) also proposed a generalization of the index introduced by Reference Abul Naga and YalcinAbul Naga and Yalcin (2008).Footnote ¹⁸

Another approach was taken by Reference ReardonReardon (2009). The focus of his analysis was on the measurement of ordinal segregation. However, it turns out that these measures may also be used to measure ordinal inequality (see Reference Lazar and SilberLazar and Silber, 2013). As before, let $\left(K+1\right)$ refer to the number of categories (remember that $k=0$ corresponds to the case where an individual does not own any durable good(. Define then a function $v$ as

$v=\left(\frac{1}{K}\right){\sum }_{k=0}^{K-1}f\left(P_k\right)$(15)

Reference ReardonReardon (2009) introduces then the following four functions $f\left(P_k\right):$

$f_1\left(P_k\right)=-\left[P_k{\log }_2{P}_k+\left(1-P_k\right){\log }_2\left(1-P_k\right)\right]$(16)

$f_2\left(P_k\right)=4P_k\left(1-P_k\right)$(17)

$f_3\left(P_k\right)=2\sqrt{P_k\left(1-P_k\right)}$(18)

$f_4\left(P_k\right)=1-\left|2P_k-1\right|$(19)

If we combine (15) with one of the four functions defined in equations (16) to (19), we obtain, in fact, measures of ordinal inequality which satisfy the four desirable properties stressed by Reference Abul Naga and YalcinAbul Naga and Yalcin (2008): continuity, scale invariance, normalization, and aversion to median preserving spreads. Note that it can be shown that the index proposed by Reference Abul Naga and YalcinAbul Naga and Yalcin (2008) in (13) is, in fact, a combination of (15) and (19).

Reference Lazar and SilberLazar and Silber (2013) combined the approach of Reference ReardonReardon (2009) with the generalization of the index introduced by Abul Naga and Yalcin to derive another measure of ordinal inequality.Footnote ¹⁹

Reference Lv, Wang and XuLv et al. (2015) took a somewhat different approach to the measurement of ordinal inequality. Taking again the case of the order of acquisition of durables as illustration,Footnote ²⁰ these authors first proposed a measure of the inequality between any two durables ownership categories. Then, in a second stage, these inequalities are aggregated via a weighted sum, in which the further apart the two durables ownership categories are, the higher the weight attached to the inequality between these two ownership categories. In fact, Reference Lv, Wang and XuLv et al. (2015) derived axiomatically two ordinal inequality indices. The first one is expressed as

(20)$I_{LWX1}={\sum }_{k=0}^K{\sum }_{j\ne k}^{}\left(\frac{2}{K}\right)\left|k-j\right|f_k{f}_j$

where, as before, $\left(K+1\right)$ is the number of possible durables ownership categories and $f_k$ and $f_i$ are the proportions of individuals belonging to ownership categories k and j.

The second index that Reference Lv, Wang and XuLv et al. (2015) proposed is defined as

$I_{LWX2}={\sum }_{k=0}^K{\sum }_{j\ne k}^{}{}^{K-\left|k-j\right|}{f}_k{f}_j$(21)

with $0<<1$

Reference Lv, Wang and XuLv et al. (2015) showed that the two previous indices obey a certain number of desirable axioms.Footnote ²¹

Reference Cowell and FlachaireCowell and Flachaire (2017) took a completely different approach to the measurement of ordinal inequality, one that is based on the notion of “status.” The idea is to link a categorical data structure to the notion of “status.” More precisely, the approach of Cowell and Flachaire includes three main elements: the notion of status within a distribution, a reference point, and a set of axioms. Note that in their view status can be downward- or upward-looking, depending on the context of the analysis. These authors characterize then a family of indices that depends on a sensitivity parameter and a reference point. This reference point can be either the maximal or the minimal possible value of the status.Footnote ²²

5.3 Measuring Poverty with Ordinal Variables: The Counting Approach

Following Reference SenSen (1976), the traditional approach to unidimensional poverty measurement makes a distinction between an identification and aggregation stage. First a decision needs to be made concerning the way an individual will be classified as poor or non-poor. Then this information is aggregated to derive an overall measure of poverty.

However, in a multidimensional approach to poverty measurement there will be three and not two stages. First, for each variable, it is necessary to decide whether the individual or household is deprived with respect to this variable. If the focus of the analysis is on the number of durable goods or assets owned, the answer to such a question will be easy to derive since the individual either owns the durable good or asset under study, or he/she does not. In a second stage, it is necessary to determine the number of durable goods or assets which an individual or a household needs to own to be considered as “not deprived” or “not poor.” Finally, in a third stage, the conclusions reached for each individual (whether he/she is “poor” or not) will be aggregated to derive an overall measure of the extent of multidimensional poverty in the society.

The Alkire and Foster (2011) Approach with Dichotomous Variables

Given the available data on the ownership of various assets, one may wonder how to aggregate such an information (individuals either own or do not own a given asset) to obtain an overall measure of poverty. There are in fact several ways of looking at this issue. First there is the so-called “union” approach. It assumes that the various assets are perfect complements, so that, as soon as one asset is missing, the individual or household will be considered as poor. The intersection approach, on the contrary, assumes that the assets are perfect substitutes, so that an individual or a household will be considered as poor only if he/she does not own any asset. Reference Alkire and FosterAlkire and Foster (2011) proposed an intermediate approach, where, given that K refers to the total number of assets, an individual (or household) will be considered as poor only if the number of assets he/she owns is smaller than or equal to $k_{critical}$ , with $1\le k_{\mathrm{critical}}\le K$ .

Now let H refer to the proportion of individuals or households defined as “poor.” Let N refer to the total number of individuals (households) and $N_P$ to the number of poor. H is then computed as

$H=\left(\frac{N_P}{N}\right)$(22)

Among those $N_P$ individuals (households) considered as poor, let A refer to the proportion of assets that these poor individuals (households) do not have. Let $I\left(x_{ik}\right)$ be equal to 1 if individual (household) i does not have asset k, to 0 otherwise. We may then write that

$A=\frac{\left\{{\sum }_{i=1}^{N_P}\left[{\sum }_{k=1}^K\frac{I\left(x_{ik}\right)}{K}\right]\right\}}{N_P}$(23)

Reference Alkire and FosterAlkire and Foster (2011) combined the two indicators H and A to define a “dimension adjusted headcount ratio” $M_0$ where

(24)$M_0=H\times A=\left(\frac{N_P}{N}\right)\left\{\left[{\sum }_{i=1}^{N_P}\left[{\sum }_{k=1}^K\frac{I\left(x_{ik}\right)}{K}\right]\right]/N_P\right\}=\left(\frac{1}{N}\right)\frac{{\sum }_{i=1}^{N_P}\left[{\sum }_{k=1}^{K}I\left(x_{ik}\right)\right]}{K}=\left(\frac{1}{N}\right){\sum }_{i=1}^{N_p}{c}_i$

where $c_i$ refers to the proportion of assets that individual i does not have.

In other words, $M_0$ is equal to the ratio of the total number of assets that the individuals (households) classified as poor do not have, over the maximal number $\left(NK\right)$ of assets that the total population could be deprived of.

Reference Alkire and FosterAlkire and Foster (2019) then generalized their approach and defined an index

$M_0^{}=\frac{1}{N}{\sum }_{i=1}^N{c}_i^{\gamma }\left(k\right)for\gamma \ge 0$(25)

where $c_i^{\gamma }\left(k\right)=c_i^{\gamma }$ if individual $i$ is multidimensionally poor $(c_i\ge k)$ and $c_i^{\gamma }\left(k\right)=0$ otherwise.

Alternative Counting Approaches

The approach of Reference Alkire and FosterAlkire and Foster (2011) is not the only way of deriving a measure of poverty when only binary or ordinal variables are available. Reference AtkinsonAtkinson (2003) gave a nice introduction to the counting approach and stressed that it is an important topic because in many cases the data available on the various dimensions of poverty are binary variables.

Let $c_i$ be defined as before. Reference Dhongde, Li, Pattanaik and XuDhongde et al. (2016) called $c_i$ the “nominal deprivation” of individual i while they defined the “real deprivation” of individual i as $r_i=g\left(c_i\right)$ .

The extent r of “real deprivation” in the population is then expressed as

$r=\left(\frac{1}{N}\right){\sum }_{i=1}^{N}g\left(c_i\right)$(26)

Reference YalonetzkyYalonetzky (2012), and Reference Silber, Yalonetzky, Betti and LemmiSilber and Yalonetzky (2013) extended expression (26) to obtain, as special case, the approach of Reference Alkire and FosterAlkire and Foster (2011). Calling $R{D}_i$ the “real deprivation” of individual (household) $i,$ as defined by Reference Silber, Yalonetzky, Betti and LemmiSilber and Yalonetzky (2013), they defined $R{D}_i$ as

$R{D}_i={\psi }_i{r}_i={\psi }_{i}g\left(c_i\right)$(27)

In (29) ${\psi }_i$ refers to some poverty identification function for individual $i$ . One possibility, as implicitly suggested by Reference Alkire and FosterAlkire and Foster (2011), is to assume that ${\psi }_i=1$ if $c_i,$ the extent of “nominal deprivation” (that is, the weighted number of assets that individual or household i does not have), is higher than or equal to $k_{critical}$ , the threshold defined previously, and that ${\psi }_i$ will be equal to 0 otherwise.

Several different functions ${\psi }_i$ and $r_i$ have been proposed in the literature (see Reference Silber, Yalonetzky, Betti and LemmiSilber and Yalonetzky, 2013, for illustrations). One interesting suggestion is that of Reference RippinRippin (2012), who assumed that ${\psi }_i=1$ and $r_i=c_i^{1+\gamma }$ so that

$R{D}_i^{Rippin}=\frac{1}{N}{\sum }_{i=1}^N{c}_i^{\gamma +1}.$(28)

Note that the measure proposed by Reference RippinRippin (2012) takes into account the degree of inequality, between the individuals (households) classified as poor, in the number of assets they do not have, while the measure introduced by Reference Alkire and FosterAlkire and Foster (2011) ignores such an inequality.

5.4 Inequality-Sensitive and Additive Achievement Measures Based on Ordinal data

In an important paper Reference AtkinsonAtkinson (1970) defined the concept of “equally distributed equivalent income,”Footnote ²³ which is the level of income that, if received by every individual, would put society at a level of welfare identical to the actual level of welfare. Reference Apouey, Silber and XuApouey, Silber, and Xu (2020) attempted to obtain a somewhat similar result for the case where only ordinal variables are available. They derived axiomatically new classes of measures of the level of achievement in a population when the achievement variable is ordinal. With K durables, there are (K+1) possible durables ownership categories so that in such a case the social achievement index they proposed would be written as

$h\left(s\right)=\frac{1}{N}{\sum }_{k=1}^K{p}_k\left(s\right)\frac{1-{\alpha }^{\left(K+1\right)-k}}{1-{\alpha }^{\left(K+1\right)-1}}=\frac{1}{N}{\sum }_{k=1}^K{p}_k\left(s\right)\frac{1-{\alpha }^{\left(K+1\right)-k}}{1-{\alpha }^K}$(29)

with $0<\alpha <1$ and where s refers to the achievements, ranked by decreasing levels, (K+1) to the number of achievement categories (since there are K durable goods), $p_k\left(s\right)$ to the number of individuals with achievement level k, and N to the total number of individuals.Footnote ²⁴