revealed meta-preferences: axiomatic foundations of ... · revealed meta-preferences: axiomatic...

Draft: 27/05/2011 -1- © Sebastian Silva-Leander

Revealed Meta-Preferences: Axiomatic Foundations of Normative Assessments in the Capability Approach

“It is a mistake to set up physics as a model and pattern for economic research. But those committed to this fallacy should have learned one thing at least: that no physicist ever believed that the clarification of some of the assumptions and conditions of physical theorems is outside the scope of physical research." (Ludwig von Mises, 1949, p. 4)

George Stigler described Adam Smith’s The Wealth of Nations as a “stupendous palace erected on

the granite of self-interest” (Stigler, 1982, p. 136). Indeed, no assumption has been more central to

the development of modern economic theory than this one, and in the eyes of many of its

detractors, as well as its supporters, this assumption has come to define the discipline1. As Prof. A.K.

Sen has pointed out, however, there is nothing intrinsically “econonomic” about the assumption of

selfishness and there is no reason why the study of the allocation of scarce resrouces could not start

from a richer representation of the complexity of human nature (Sen A. K., Rational Fools: A Critique

of the Behavioural Foundations of Economic Theory, 1977; Sen A. K., Equality of What?, 1979; Sen A.

, Rationality and Freedom, 2002, p. 21; Sen A. K., Choice, Welfare and Measurement, 1997). Sen’s

point poses a fundamental challenge, not only for mainstream neoclassical economists, but also for

proponents of his own so-called “capability approach”, as it calls for the reconstruction, from

sounder foundations, of an edifice that could rival the internal coherence and solidity that has been

achieved by classical and neoclassical theory through more than two centuries of theoretical

refinement.

This paper focuses on a small piece of this vast problem, one that was pointed out by Martin

Ravallion in a recent piece in which he criticised what he perceived as the arbitrariness of various

“mashup” indices of dubious conceptual credentials that have sprung up in recent years (Ravallion,

2010) – particularly as part of the effort to operationalise the “capability approach”. Unlike

conventional economic measures, such as GDP, which are constructed on the basis of a body of

economic literature, he argues, these “mashup” indices contain “key decision variables that the

1 The most famous example of such a confusion of the discipline for its axioms is probably found in Becker’s exploration of the “economics” of issues pertaining to the fields of biology, criminology and even religion (Becker, 1976).


analyst is free to choose, largely unconstrained by economic or other theories intended to inform

measurement practice” (Ravallion, 2010, p. 3). In particular, he argues that the weights chosen, for

instance, in UNDP’s Human Development Index and more recent Multidimensional Poverty Index, do

not rest on the sort of theoretical justification that allows neoclassical economists to claim that

“market prices are defensible weights on quantities in measuring national income” (Ravallion, 2010,

p. 3).

Indeed, much of the efforts of welfare economics over the past couple of centuries have been

concerned precisely with the problem of linking the common variables that furnish economic theory

(e.g. prices, markets, goods) with the normative concepts (e.g. utility, efficiency, maximisation) that

can infuse moral value into these otherwise inert terms. The problem, however, is that all these

technical innovations have aimed to link normative welfare economics back to one particular ethical

framework, namely the utilitarian school, of which Adam Smith was one of the main proponents (Sen

A. K., On ethics and economics, 1987). Hence, once all the technical artifices of the theory are

removed, one is left with an argument that in fine relies on the acceptance of the notion of utility as

an ultimate standard of value for assessing the “goodness” of individual or social outcomes. And this

is one of Sen’s main contentions against neoclassical welfare economics (Sen & Williams,

Utilitarianism and Beyond, 1982). Indeed, as Sen has shown, there is no particular reason why

normative assessments of economic phenomena could not be made in terms of, say, Kantian or

Aristotelian ethics. Why, for instance, should we consider that a market equilibrium is efficient when

it maximises aggregate utility or balances marginal utilities, rather than, say, when it ensures that

fundamental social and economic rights are fulfilled?

Section 1 of this paper offers a very succinct review of the trail of theoretical innovations that have

been brought into economic theory over the past two centuries to justify the use of prices as

normative weights. In section 2, we look at Sen’s critique of a key step in this chain, namely the

revealed preference theory, and its implications for the formalisation of the same. In section 3,

finally, we propose a response to Ravallion, in the form of two normative ranking rules that are

consistent with Sen’s axiomatic representation of human nature.

The Normative Value of Market Prices

Utility, Prices and Market Equilibria

The fundamental arguments that constitute the pillars of neoclassical welfare economics can be

traced back to the founder of classical economics in The Wealth of Nations. Chief among these is the

idea that the interaction of self-interested individuals in a free market place will, in time, generate a


natural equilibrium, which is not only stable, but – more importantly from a normative point of view

– optimal in the sense that it is the most conducive to the common good:

“[Every individual] generally, indeed, neither intends to promote the public interest, nor knows how much he is promoting it. By preferring the support of domestic to that of foreign industry, in such a manner as its produce may be of the greatest value, he intends only his own gain, and is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention. Nor is it always the worse for the society that it was no part of it. By pursuing his own interest he frequently promotes that of the society more effectually than when he really intends to promote it.”(Smith, Book IV: Of Systems of Political Economy, 1776, p. 400). 2

The notion of “common good” is here defined in utilitarian terms, that is, in Bentham’s definition

“the greatest happiness of the greatest number” (Bentham, 1789). In the original formulations

proposed by Bentham, the assessment of the amount of “common good” generated by various social

arrangements was to be made through a very specific so-called “felicific calculus”, involving a

cardinal comparison of the net amounts of pleasure and pain experienced by each individual.

(Bentham, 1789, p. chap. 4). Bentham’s proposal faltered, however, due to the unobservable nature

of the notion of utility and the consequent impossibility to measure and compare a unit of pleasure

and pain across individuals.

The marginalist revolution in economic theory, in particular the work of William Jevons (Jevons,

1871) and Alfred Marshall (Marshall, 1890) helped to circumvent this difficulty by shifting the

attention away from aggregate utility towards marginal utility. Although the total intensity of utility

generated by a given good could not be measured, the marginalist revolution made it possible to

make binary comparisons between goods of the sort: an additional unit of increases utility more

than an additional unit of . This opened the door for replacing Bentham’s demanding cardinal

felicific calculus with the significantly more modest ordinal principle of Pareto-optimality, whereby a

social state would be considered optimal not if it maximised the “greatest happiness of the greatest

number” but if it was such that “no individual could be made better off without making someone

else worse off” (Pareto, 1906).

Another marginalist, Leon Walras, used the analysis of marginal utilities and production costs to

study, from a positive rather than normative perspective, the way in which markets for goods and

services achieve general equilibrium across all markets in an economy by adjusting relative prices so

2 It is worth noting, in passing, that the “Invisible hand” is none other than the hand of God, who has the power to – somehow – transform the uncoordinated actions of self-interested individuals into an outcome that is both stable and beneficial: “The happiness of mankind, as well as of all other rational creatures, seems to have been the original purpose intended by the author of Nature, when he brought them into existence. No other end seems worthy of that supreme wisdom and divine benignity which we necessarily ascribe to him; and this opinion, which we are led to by the abstract consideration of his infinite perfection is still more confirmed by the examination of the works of nature, which seem all intended to promote happiness, and to guard against misery.” (Smith, The Theory of Moral Sentiments, 1761, p. 237).


as to eliminate surplus demand or supply (Walras, 1874). In the so-called Walrasian equilibrium,

relative prices will reflect consumers’ marginal rates of (utility) substitution between goods, as well

as producers’ marginal rate of transformation (or opportunity cost).

The first fundamental theorem of welfare economics, then, simply states that the Walrasian

equilibrium is – by construction – Pareto-efficient in the sense that individuals cannot increase their

marginal utility further by exchanging goods that confer them low marginal utility for goods that

generate higher marginal utility (Arrow & Debreu, Existence of a Competitive Equilibrium for a

Competitive Economy, 1954). The first fundamental theorem of welfare economics has thus been

widely interpreted as a formalisation Adam Smith’s original insight – albeit, reformulated in terms of

ordinal rather than cardinal utility.

Revealed Preference Theory

There are numerous criticisms that have been formulated against this model for the simplistic and

idealised assumptions it makes about the functioning of complete and competitive markets (e.g.

(Greenwald & Stiglitz, 1986)). In this paper, however, we will choose to disregard those and focus,

instead, on problems associated with the second part of this normative defence of market prices as a

standard of relative welfare. First, of course, the notion of Pareto-efficiency falls far short, as a

normative standard, of the original utilitarian social objective of aggregate welfare maximisation.

However, in the absence of observable or cardinally measureable utility, Pareto efficiency has

prevailed as the closest possible approximation to Bentham’s initial proposition that may be

compatible with the minimal empiricist requirements of verifiability (Hicks, A Revision of Demnand

Theory, 1956, p. 6).

A more serious problem with this formulation, as Samuelson pointed out, was that, in the absence of

observable preferences, even the minimal normative claims of Pareto efficiency seemed tenuous and

liable to the same criticism that marginalists had formulated against classical economists for their

normative reliance on the notion of utility:

Consistently applied, however, the modern criticism [of classical utility theory] turns back on itself and cuts deeply. For just as we do not claim to know by introspection the behaviour of utility, many will argue we cannot know the behaviour of ratios of marginal utilities or of indifference directions. Why should one believe in the increasing rate of marginal substitution, except in so far as it leads to the type of demand functions in the market which seem plausible? (Samuelson, A Note on the Pure Theory of Consumer's Behaviour, 1938, p. 61)

In order to address this flaw, Samuelson proposed a key piece of theoretical engineering which has

come to be known under the term of revealed preference theory. Samuelson initially claimed that


the purpose of his contribution was to rid economic theory “of the last vestiges of utility analysis”

(Samuelson, A Note on the Pure Theory of Consumer's Behaviour, 1938, p. 62) by creating a theory

that would be based on nothing else than observable behaviour. However, as Sen argued, if

successful, this theory would quickly empty consumer choices of the normative power they derived

from utilitarian postulates about the value of utility: if behaviour didn’t reveal anything about

underlying preferences, price ratios would not tell us anything about marginal rates of substitution

and market equilibria would thus tell us nothing about Pareto-optimality. From a normative

perspective, therefore “the rationale of the revealed preference approach” must lie in the

“assumption of revelation and not in doing away with the notion of underlying preference” (Sen A.

K., Behaviour and the Concept of Preference, 1973, p. 244).

As long as individuals behave consistently, Samuelson argued, we could infer from their choice of

over , when is available, that they prefer over (i.e. that gives them more utility than , if we

assume utility maximisation). Hence, while utility itself may be unobservable, sufficient fragments of

it can be indirectly observed to allow us to make the type of ordinal claims needed for normative

welfare assessments (Little, 1949; Houthakker, 1950). Later Samuelson, building on Little (Little,

1949), went on to show how indifference curves could be approximated through the iterative

comparison of binary preference relations over choice bundles (Samuelson, Consumption Theory in

Terms of Revealed Preference , 1948).

Here, we will build on the notation used by Sen (Sen A. K., Choice Functions and Revealed

Preference, 1971) and Arrow (Arrow, Rational Choice Functions and Orderings, 1959) to formalise

the argument. Let be the set of all possible options and the set of all non-empty and finite

subsets, , of . The non-empty “choice set” ( ) represents elements of any given subset of

that have been chosen by individual . Henceforth, we will write ’s choice set over simply as for

short. The decision maker ’s preferences are represented by transitive and binary relations, ,

defined over the elements of contained in . Let = {1, … , , … , } be the set of all individuals

holding preferences . Then Γ = { , … , } represents the set of all possible preference

relations over .

The preference for over is designated by , meaning that “ is at least as good as

according to the preference relation ”. The superscript indicates that preference relations are

based on utility maximising calculus. Hence, is equivalent to saying that confers with at

least as much utility as , and can be interpreted in marginalist terms, if and represent units of

and for given quantities. The symmetric and asymmetric elements of will be designated,

respectively, by and .


The axiom of revealed preference (ARP) then states in its original, weak, form:

∀( , ) ∈ ∀ ∈ : ∈ ∈ \ ⇒ ¬ (1)

Plainly put, if both and are part of ’s opportunity set, but only is chosen, we can conclude, at

the least, that does not prefer to .

Sen’s Critique

Plural Motivations

Implicit in the revealed preference axiom is the assumption that choice is motivated by one thing and

one thing only, namely utility maximisation. Consequently, if an option is observed to have been

chosen, it must be because it was deemed by the chooser to provide more utility than any of the

available alternatives. It is this assumption that allows the revealed preference axiom to reconnect

price ratios with marginal rates of substitution, and it is this assumption that Sen questions. In

practice, he argues, though self-interest maximisation is undoubtedly one of the most powerful

driving forces of human conduct, it is but one of the factors that motivate action, alongside things

like commitment, compliance with moral rules, religious principles or social norms. It is this capacity

to disregard one’s own narrow self-interest and act in the interest of the common good that allows

individuals to avoid suboptimal outcomes in problems such as the prisoners’ dilemma, as well as

numerous other social problems, such as taxation, provision of public goods, etc. (Sen A. K.,

Behaviour and the Concept of Preference, 1973, p. 250).

Importantly, Sen is adamant to point out that he is not interested in the sympathy or altruism that

the prisoners may feel for each other, which may lead them to incorporate the other’s welfare into

their own maximisation function. Instead, he insists, it is their capacity to “suspen[d] calculations

geared towards individual rationality” (i.e. self-interest maximisation) and to “behave as if they are

maximizing a different welfare function from the one that they actually have” (i.e. to see a problem

from someone else’s perspective) that makes the solution interesting (Sen A. K., Behaviour and the

Concept of Preference, 1973, p. 252)3.

For the purpose of the present argument, make a distinction between two fundamental types of

motivations described by Sen, namely (1) self-centred motivations, which can include altruism,

3 This ability of individuals to dissociate themselves from their own preferences and interests, in order to pursue a higher common objective, is what makes the difference between what Rousseau calls “the General Will” and “the will of all” (Sen A. K., Behaviour and the Concept of Preference, 1973, p. 250).


sympathy, etc., and (2) reasoned motivations, which we denote by ∈ Γ = { , … , }. The

former type of motivations are idiosyncratic in the sense that they are defined by a person’s

particular interests, inclinations, desires, fears, etc. – more broadly captured by the term utility. The

latter type of motivations is here defined as those that allow us to disregard our own particular

circumstances and act, as Sen calls it, “as if” we are maximising someone else’s utility. The latter is

derived from our capacity for universalisation, that is, the faculty of reason to abstract from our own

circumstances in order to act in accordance with a law or a moral principle4. It is also reason that

allows us to generalise from the observation of our own behaviour that if we are able to distance

ourselves from our own self-interest, others should be able to do the same. It is thus reason that

allows the prisoners to resolve their dilemma by abstracting from their particular circumstances in

order to reach an agreement that transcend the individual interests of each prisoner. As Sen points

out, it is also this capacity that allows us to reach mutually beneficial agreements through politics and

even in many market transactions (Sen A. K., Markets and freedoms, 1991).

While we do not exclude the possibility that different individuals’ idiosyncratic preferences may

coincide, due to sympathy or convergence of interests, for instance, we must recognise the

possibility that they may diverge due to divergence of interests and tastes. Therefore, we define

as :

Definition 1: Γ = | ∃( , ) ∈ , ≠ , . . ≠ .

In addition to utility-based preferences, Sen argues, individuals have reasoned preferences, also

called “preferences over preferences” or “ ‘metarankings’, reflecting what [an individual] would like

his preference to be” (Sen A. K., The Journal of Philosophy, 1983, p. 25). Without going as far as

claiming that the latter motivations must concord with a universal law of reason, we will define them

as the ability of individuals, who are part of a group, to take a bird’s eye view on a situation, and

recognise that there is a common interest that transcends their individual interests. We thus define

by their convergence across individuals in a given group. Let be the set of all non-empty

subsets of and let ∈ designate a group of individuals = (1, … , ):

Definition 2: Γ = | ∀ ∈ : ( , ) ∈ ⇒ = .

4 Formally, Kant defines pure reason as the “faculty of the unity of the rules of understanding under principles” (Kant, Kritik der reinen Vernuft, 1781, p. A302/B359), that is the capacity for abstraction or universalisation. Judgement, then is the capacity “of thinking the particular as contained under the universal” (Kant, Critique of the Power of Judgement, 1790, p. 5:179), that is to link a particular situation to a universal rule.


Once we admit the possibility of plural motivations, the technical trick of revealed preferences, which

allowed us to draw inference about underlying utility from observable behaviour, ceases to be

effective, since observed behaviour could equally well have been cause by moral commitment, norm

following or even self-sacrifice, neither of which tell us anything about a person’s level of utility5. The

table below indicates various possible outcomes that can result from a combination of preferences

over the outcomes and :

Table 1: Possible choices resulting from combinations of utility and meta-preferences

Utility

Reason

∈ ,∈ \

∈ ,∈ \

( , ) ∈ ,( , ) ∈

∈ ,∈ \

∈ ,∈ \

( , ) ∉ ,( , ) ∈ \

∈ ,∈ \

( , ) ∉ ,( , ) ∈ \

( , ) ∈ ,( , ) ∈

∈ ,∈ \

∈ ,∈ \

∈ ,∈ \

∈ ,∈ \

( , ) ∉ ,( , ) ∈ \

∈ ,∈ \

( , ) ∉ ,( , ) ∈ \

In this example, we are assuming that strict preferences will prevail over weak or indifferent

preferences in cases where there is a conflict between utility and reason. In cases where the

individual has strict preferences for both and simultaneously, he will choose both. In cases where

the individual is indifferent both in terms of utility and reasoned preferences, he chooses neither.

While it is possible to conceive of other plausible combinations of preference relations to outcomes,

the precise form of the relation will not matter here. Instead, we are interested in showing that, in

the presence of dual motivations it will no longer be possible to infer underlying preference relations

with certainty from observed choices, as was done in ARP, since each choice pattern will be

associated with, not one, but several plausible combinations of underlying preference relations.

Depending on the assumptions we choose about the form of underlying decision making mechanism,

5 There have been attempts to circumvent the problem of inference by expanding the definition of preferences to include anything sought by the individual, be it for religious reasons, moral reasons, pathological desires or some other form of “want” (Hicks, Value and Capita, 1939). Such an expansion of the definition of preference, however, would quickly empty the term of its normative content by equating it with the descriptive notion of “behaviour”(Harsanyi, 1997). If preferences are defined as “behaviour” the problem of inference simply gets pushed up by one level from preferences to underlying motivations, without enabling us to reconnect observed behaviour with the underlying notion of utility from which any utilitarian justification ultimately derives its normative force.


we will still be able to make certain weaker claims based on observed choices. Table 1, for instance,

allows us to make the following claim, which is a close but weaker relative to the ARP:

∀( , ) ∈ ∀ ∈ : ∈ ∈ \ ⇒ ¬ Ψ ∪ Ψ(2)

where Ψ and Ψ describe the events and , respectively. The axiom now states that if

is observed to choose over , he cannot possibly have a strong preference for over . Given the

uncertainty we now face about underlying preference relations, it may be more appropriate to

represent these claims as probabilities. Axiom (2), for instance, would thus become:

∀( , ) ∈ ∀ ∈ : Ψ ∪ Ψ | ∈ ∈ \ = 0(3)

Henceforth, we will write the probability of an event having occurred, given that we observe that

has chosen over (i.e. given ∈ ∈ \ ), simply as ( ∙ | ).

Let χ and χ designate the events and , respectively. We will assume that ∀( , ) ∈and ∀( , ) ∈ , ( | ) and ( | ) satisfy the following intuitive properties, which are

consistent with definitions 1 and 2:

1. Existence: | > 02. Completeness: ⋃ | = 13. Uncertainty: | = |4. Anonymity: | = |5. Independence : ∙⋂ ∙ | ⋂ = ( ∙| ) × ∙ |

Loosely put, property 1 simply states that the probability that an observed choice, , is motivated by

selfish motives is strictly greater than 0 (i.e. there is a possibility that even a seemingly virtuous act is

motivated by selfish motives). Property 2 states that a given act, , must have been motivated either

by utility based motives or by reasoned meta-preferences. Property 3 captures the fact that, with no

additional information about underlying preferences, we cannot know whether a given behaviour is

motivated by self-interested utility-based considerations or by moral concerns. Property 4 states that

the probability that a given act is motivated by selfish motives is equal across all individuals (i.e. there

are no intrinsically more virtuous individuals). Property 5, finally, simply makes the point that the


probability of any event occurring is independent across individuals (i.e. what motivates me is not

dependent on what motivates my neighbour).

Meta-Preferences and Consensus

In the “rationalist”6 perspective advocated by Prof. Sen, it is not underlying levels of utility that

constitute a legitimate standard of value, but rather the reasoned assessment that individuals make

of their own preferences, i.e. their metarankings (Sen A. , Rationality and Freedom, 2002). Indeed,

utility-based preferences may be determined by culture, upbringing, genetics or other factors over

which the individual has no control (Nussbaum, 2003) and they may be adaptive, in the sense of

being moulded by prejudice, fear or simply resignation to an unjust fate (Sen A. K., Resources, Values

and Development, 1984). The question that is of interest for the purpose of operationalisation of the

capability approach is thus whether meta-preferences, , can be, if not observed, at least inferred

from looking at choices . in the same way as Samuelson’s revealed preference theory had allowed

us to infer unobservable utility-preferences from observed choices.

At first sight, the answer to this question would appear to be negative: in the same way as it was

impossible to infer from the observation of choice, whether was caused by rather than , we

cannot infer whether was caused by rather than . Importantly, we do not exclude the

possibility that = , such that idiosyncratic preferences , may coincide with moral

preferences, , either due to chance or sympathy (e.g. deriving utility from someone else’s

achievement). A same behaviour , e.g. charitable giving, could therefore be caused by either self-

interested motives (e.g. attempting to secure loyalties or relieving one’s own displeasure at the sight

of poverty) or moral motivations (e.g. a social obligation towards other less fortunate members of

the group). There is, however, a peculiarity in Sen’s concept of meta-preferences, which is captured

in definition 2, that may allow us to, if not resolve, at least manage this problem. To understand how,

we shall return to the predicament of our prisoners.

In the case presented by Sen, it would be easy to imagine that the two prisoners, having worked

closely together in crime and endured hardship jointly as a result, may have developed feelings of

sympathy for each other, and that these feelings may lead them to naturally include the other

prisoner’s utility in their own maximisation problem and thus choose the optimal solution, even

though it may not be the most rational from a narrow perspective of self-interest. But imagine now

6 Sen’s alternative conception of human nature links up to a non-utilitarian tradition, which he has referred to elsewhere as the “rationalist” tradition, including non-utilitarian thinkers, such as Kant, Rousseau among others (Sen 1999, p.255).


that instead of two prisoners, the District Attorney is trying an organised gang of ten criminals, each

of whom have participated in the crime. As before, each prisoner has an overwhelming incentive to

denounce the other prisoners, but now the likelihood of there being a close bond of friendship

between each of the 10 prisoners is significantly diminished. Furthermore, the risk faced by each of

the prisoners that one of his co-inmates will defect has now increased ninefold.

The likelihood that sympathy alone will be sufficient to generate the desired outcome will now, as

neoclassical choice theory rightly predicts, be small. But we can also look at the problem from the

other angle and say that in the event that the ten prisoners are able to reach agreement, then it is

likely that something other than pure self-interest was at work. Of course, we can never know with

certainty by observing a given outcome whether the motivation behind it was mutual sympathy, cold

calculus of self-interest or reason’s recognition of duty under the moral law. But, the same logic that

tells us that ten individuals are less likely than two to reach agreement through self-regarding

motives alone, also implies that, as the consensus broadens, the likelihood that sympathy alone is at

work decreases and, consequently, the likelihood that the observed consensus reflects some form of

reasoned meta-preferences or common good increases.

This idea is captured in the following lemma, which follows from definitions 1-2 and properties 1-5.

We will represent the consensus between a group ∈ of individuals, = (1, … , ), as , which

we define as follows: = ⋂ ≠ ∅. We designate by the set of all possible non-empty and

finite consensus choice sets, , held by groups ∈ . The cardinality of a group of individuals

= (1, … , ),. is designated by | | = .

Lemma 1: ∀( , ) ∈ , 1 < | |, | | < ∞, with individual preference defined as 1-2, and satisfying

properties 1-5: ⊃ ⇒ ( ) ≥ ( ).

See Annex 1 for a proof of Lemma 1.

In the limit, as tends to ∞, we can think of a hypothetical consensus involving an infinite number of

individuals with all possible combinations of preference relations. Such a consensus would, by

construction, need to coincide with a universal law of reason since that is the only law that any

reasonable individual could voluntarily be expected to subject themselves to, regardless of their

personal circumstances, inclinations and interests. In other words, the probability of utility-based


interests converging over an infinite number of individuals is 0. This idea is represented by the

following corollary, which follows from Lemma 1:

Corollary 1: ∀ ∈ , = { | = (1, … . , )}, 1 ≤ < ∞, with individual preference defined as 1-2,

and satisfying properties 1-5: lim→ ( ∪ | ) = ( ) = 1.

See Annex 2 for a proof of Corollary 1.

Normative Meta-Rankings

Probability-Based Ranking

In an ideal world, we would have liked to be able to reach categorical (yes/no) conclusions regarding

reasonableness and, why not, make cardinal comparisons of value (e.g. decision is three time more

reasonable than decision ), in the same way as utilitarians would have wished to make cardinal

comparisons of utility. While this may not be possible, the transient and contingent truths generated

by the process of inter-rational validation should at least allow us to make ordinal binary

comparisons between outcomes based on the nature and extent of the consensus on collective

choices. Based on Lemma 1, we propose the following general re-formulation of Axiom (3), which we

will call the Axiom of Revealed Meta-preferences (ARM):

∀( , ) ∈ ∀ ∈ , . . = { | = (1, … , )}: ( | ) = (∙, )(4)

Where (∙, ) is a monotonically non-decreasing function of . The axiom of revealed meta-

preferences simply states that the probability that an observed consensus, , has been generated by

a meta-preferences will be a function of the number of individuals reaching consensus. An outcome

over which there is a consensus among members of a group will thus be said to be revealed

meta-preferred to outcome with probability ( | ). In this perspective, the original revealed

axiom becomes a special case of this general axiom, in which = 1. In this case, we know from

property 3 that | = | , meaning that the observed choice has equal probabilities of

having been caused by either motive. Other special cases that have been discussed above, include

the case in which → ∞, in which case corollary 1 tells us that ( | ) = 1. A final special case is

that in which there is no consensus, that is = 0. By definition 2, an outcome over which there is no


consensus, such as the public provision of amusement parks in society , for instance, is one in which

we know that individuals have been unable to overcome their divergence of interests in order to

agree on a common objective. Hence, in this case, we have ( ) = 0. All other cases, such that

1 < < ∞ are covered by Lemma 1.

At the moment, the axiom does not make any normative claim. Hence, except in the hypothetical

case of a universal consensus, the fact that a group, , reveals a meta-preference for, say, healthcare

( ) over amusement parks ( ) does not necessarily mean that healthcare is absolutely and

universally a more valid social objective than amusement parks. Indeed, another society, ,may

reach the opposite consensus to support amusement parks instead of than healthcare. In order to

compare the choices of societies and , we need to turn the probabilistic logic of Lemma 1 into a

normative rule. If we consider that value stems from reason rather than from utility, it would be

logical to make the normative value of a given choice directly proportional on the probability that it is

caused by rather than , using the following axiom:

Probability-Based Ranking (PBR): ∀( , ) ∈ ∀( , ) ∈ ∶1. ≻ ⇔ ( | ) > |2. ∼ ⇔ ( | ) = |

(5)

Where ≽ , meaning “is at least as good as”, describes the normative ranking of social states– not to

be confused with the positive description of individuals’ actual preferences over those states,

denoted and . Hence, whereas indicated that a group meta-preferred to , ≽indicates that it is also, and for that reason, preferable from a normative point of view. We will

denote the asymmetric and symmetric elements of ≽ by ≻ and ∽, respectively.

When combined with definitions 1-2 and propositions 1-5, this yields the following proposition:

Proposition 1: If ≽ is a binary transitive ranking over satisfying PBR and ARM, and if properties 1-5

hold, then ∀ , ∈ ∀( , ) ∈ , . . 1 < | |, | | < ∞ ∶ | | > | | ⇒ ≽ .

See appendix 3 for a proof of proposition 1.


It is important to note that we are ranking choice sets, rather than the options that may be contained

in the choices that have been made. Hence, if groups and both agree on the public provision of

healthcare, we could have ≻ if | | > | |, which means that the moral force of ’s preference

for healthcare is greater than that of ’s preferences for healthcare. The ranking will therefore be

transitive over but may not be transitive over .

As formulated here, the normative rule now allows us to resolve cases in which there is a

disagreement between different normative standards, such as, say a tribal or customary law and a

national law or international convention. Although we can never reach certainty that one is

universally and unconditionally superior to the other, the logic developed in this paper implies that

the international consensus reached about fundamental human rights, for instance, will be more

likely to approximate some version of a universal law of reason than, say, the norms that govern a

small isolated community or tribe. This is because the former involves agreement between a large

number of different individuals with widely divergent views, cultures and interests, whereas in the

latter case it is more likely that agreement may have been facilitated by a strong convergence of

interests and experiences. Consequently, international human rights norms will carry more

normative weight than customary or tribal law. By the same logic, a national constitution should also

carry more normative weight than national legislation, insofar as it is protected by additional

procedural guarantees (e.g. two thirds majority or double majority) that mean that a broader

consensus would be needed to reach agreement on the former than on the latter.

This need not mean that lower order norms are unfit for normative assessments. When judging each

others’ actions, members of the tribe or the village may be content with, and indeed prefer, doing so

in accordance with their own set of rules and norms that they have all consented to. However, when

comparing members of different tribes or communities, or when asked to judge the validity of a

tribal norm or custom such as female circumcision, we will need to appeal to higher order consensus,

such as a national legislation or international human rights.

Consensus-Based Ranking

Finally, we propose an alternative normative ranking rule that captures the rationale of the above

discussion, but presents the advantage of being based entirely on the external and thus observable

properties of the consensus reached between different agents. Needless to say that the normative

power of this ranking ultimately is derived from the assumed link of consensuses to underlying meta-

preferences, much in the same way as the normative power of choice in the neoclassical framework

was derived from its assumed link to underlying utility. By avoiding references to unobservable


properties of the individual decision-making mechanism, however, we make it possible to centre the

axiomatic discussion on the desirable political properties of a consensus-based ranking. We propose

the four following simple axioms to characterise our ranking:

Equality (EQU):

∀( , ) ∈ : ∽(6)

Monotonicity (MON):

∀( , ) ∈ , . . ( , ) ≠ ∅: ⊃ ⇒ ≻(7)

Independence (IND):

∀( , ) ∈ , ∀ℎ ∈ \( ⋃ ), . . ⋂ ≠ ∅, ⋂ ≠ ∅: ≽ ⇒ ⋂ ≽ ⋂(8)

Irrelevance of non-consensual preferences (INC):

∀( , ) ∈ , . . ⊃ ∀ℎ ∈ \ : ⋂ = ∅ ⇒ ⋂ ∼ ⋂(9)

The EQU axiom states that the choice of any given individual has the same normative force as that of

any other individual. The MON axiom states that the consensus reached by any given group of

individuals has more normative force than the consensus reached by any of its subgroups. IND states

that a ranking between two consensuses will not be affected by the inclusion of an additional

individual who agrees with both. These axioms echo fairly closely the axioms proposed by Pattanaik

and Xu (Pattanaik & Xu, 1990) to characterise a cardinal ranking rule. However, they exclude cases in

which individuals are unable to reach consensus. To complete the ranking we therefore need to add

a fourth axiom, to deal with those cases. We do this by using the INC axiom, which states that the

expansion of a group that has been unable to reach consensus will not increase the group’s

normative power.

Based on these four axioms, we are able to construct a ranking rule that is dependent solely on the

number of individuals reaching consensus.


Theorem 1: If ≽ is a binary and transitive ranking over , satisfying EQU, MON, IND and INC, then

∀ ( , ) ∈ ∀( , ) ∈ ∶ ≽ ⇔ | | ≥ | |.

See Annex 4 for a proof of theorem 1.

It is easiest to think of and as sets of individuals (e.g. electorates). However, since the consensus-

based ranking is not directly dependent on the properties of individual motivational structures, it is

also possible to think of a different unit, such that and representing sets of countries, for instance.

This would, for instance, be relevant for international human rights legislation, where currently

countries have equal weights in the decision-making process regardless of their population. Based on

this logic, we could for instance state that the Universal Declaration of Human Rights’ endorsement

of the right to healthcare carries more normative force than the recognition of the same right in the

European Convention on Human Rights, since the group having agreed on the latter (all European

countries) is a strict subset of the group having agreed on the former (all countries in the World).

From a strict perspective of revealed meta-preferences, however, as well as from the perspective of

other normative frameworks, such decision-making mechanisms that does not take the individual as

its starting point may violate the principle of equality of individuals.

Finally, we should note that by taking the existing consensus as the starting point of the assessment,

and thus disregarding the decision-making process, we are implicitly imposing a number of

assumptions that will need to be clarified. In particular, we have assumed that the decisions reached

reflect a genuine consensus in the sense that no individual has been subjected to any form of

coercion, intimidation or deceit. We may also need to add assumptions regarding the nature of the

decision makers. In particular, we need to assume that no individual has more power than other and

can influence the decisions of his followers. Finally, we are, of course, assuming that everyone has

access to the same level of information and has sufficient information to make free and informed

decisions. All of these assumptions may be optimistic and none may in fact ever be met in reality.

However, they are not fundamentally less plausible than the basic set of assumptions needed to

prove the fundamental theorems of welfare economics. In fact these assumptions echo quite well

the ones that characterise the Weberian general equilibrium, such as perfect and competitive

markets with perfect information and no monopolies. As such, they may provide an adequate basis

for exploring the conditions under which these ideals may be realised and the reasons for which they

fail to do so.


Conclusion

Let us conclude this discussion by attempting an answer to the valid query that had been raised

about measures, such as the Human Development Index or Multidimensional Poverty Index, which

attempt to capture key insights of Sen’s so-called “capability approach”. What, Ravallion asked,

would be acceptable normative standards for selecting and weighting indicators in such an index?

The argument developed in this paper allows us, first of all, to discard the use of market prices as

acceptable normative weights in Sen’s framework. Market prices are the product of an iterative

process involving a very large number of individual decisions taken by consumers, individually, based

on tastes, needs and perhaps even moral concerns (and/or producers individually, based on

production costs, etc.). At no point does this process require individuals to reach out and seek

compromise with other individuals who have different interests and inclinations. Consequently,

market prices do not tell us anything about people’s reasoned assessments of the various options

they face, and may, at best, provide some fragments of information about individuals’ marginal rates

of utility substitutions between different options. From the perspective of Sen’s theory, this thus

undermines the normative justification for using a measure, such as GDP, which effectively rely on

the market price to weigh different goods or dimensions of wellbeing.

Let us now consider another measure that assigns relative monetary values to different goods,

namely a public budget that has been voted by a democratically elected parliament. Unlike market

prices, a budget will required lengthy discussions and a process of inter-rational validation in order to

agree on which priorities the community should invest in an in what proportions. The weights

provided y public spending ratios in a national budget could therefore, unlike market prices,

constitute a valid normative standard in the construction of an aggregate index.

It is possible to think of other standards that may carry information about weights. The Millennium

Declaration, for instance, describes an implicit weighting system, as it defines eight equally important

goals, each of which is composed of a number of targets, which in turn are measured by various

indicators. The Universal Declaration of Human Rights also makes reference to weights and even to

elasticities of substitution when it states that all rights are indivisible and inalienable. More

importantly, it also implicitly assigns a weight of zero to a large number of rights, namely those that

are not included in the declaration.


Which of these standards should be used in constructing a normative index of welfare will depend on

the purpose and the scope of the assessment. A national budget may, for instance, provide an

adequate standard for a punctual assessment of national policies, whereas a more demanding

standard, such as the Universal Declaration, may be required when making a comparison between

various countries and over several years. Finally, an assessment geared specifically towards

development issues, may be content with a more focused normative standard, such as the

Millennium Declaration.


Annex 1: Proof of Lemma 1

Let ( , ) ∈ be two groups of individuals, such that ⊂ and 1 < | |, | | < ∞. Further, let

= \ .

Definition 1 combined with property 1 imply 0 < ( | ) < 1 for any > 1 since we

know that exists (property 1) and that there is at least one pair of individuals ( , ) ∈ for which ≠ .

Similarly, we know from property 1 that 0 < ( | ) ≤ 1 , since ⊂ , which implies

| | > | |, and hence, | | − | | > 0.

Consequently, and assuming independence across individuals (property 5), we have :

( | ) = | × ( | ) ≤ |(10)

Let us now look at the implication for ( | ). We know that:

( ⋃ | ) = ⋃ ⋂( ⋃ )| ⋂(11)

Applying the second distributive law,(11) is equivalent to:

( ⋃ | ) = ⋂ ⋃( ⋂ )| ⋂(12)

By the general property of additivity, (12) is equivalent to:

( ⋃ | ) = ⋂ | ⋂ + ( ⋂ )| ⋂ − ⋂ ⋂( ⋂ )| ⋂(13)

By the property of commutativity, (13) is equivalent to:

( ⋃ | ) = ⋂ | ⋂ + ( ⋂ )| ⋂ − ⋂ ⋂( ⋂ )| ⋂(14)

By the property of independence (property 5), (14) becomes:


( ⋃ | ) = | ( | ) + ( | ) ( | ) − ⋂ | ( ⋂ | )(15)

By the property of completeness (property 2), we then have:

( ⋃ | ) = | ( | ) + ( | ) ( | ) − ⋂ | ( ⋂ | ) = 1(16)

We can now solve for ( | ) ( | ):

( | ) ( | ) = 1 − | ( | ) + ⋂ | ( ⋂ | )We know from (10) that | × ( | ) ≤ | . Furthermore, by definition of an

intersection, we know that ⋂ | ≤ | < 1 and ( ⋂ | ) ≤ ( | ) ≤ 1.

Consequently, we have:

( | ) ( | ) = 1 − | ( | ) + ⋂ | ( ⋂ | )≥ 1 − | + ⋂ | = ( | )

By property 5, finally, we have:

( | ) × ( | ) = ( ⋂ | ⋂ ) = ( | )And consequently,

( | ) ≥ ( | )


Annex 2: Proof of corollary 1

Let ⋂ denote the consensus between two individuals, and , on the preferable choice in ,

such that ( , ) ∈ , ≠ and ⋂ ≠ ∅. For ⋂ to be non-empty, it must be the case that the

preferences of and overlap such that both and would choose over , either because they

derive utility from the same outcomes, hence and have both occurred, or because they are

using their meta-rankings, hence has occurred for both and , or a combination of these two.

By the general property of additivity of probabilities, we can write the probability of any of these

events having occurred as:

( ∪ | ) = | + ( | ) − ( ∩ | )(17)

Using the second distributive law, we can write the probability of this event having occurred jointly

for and as:

( ⋃ )⋂( ⋃ )| ⋂ = ( ⋂ )⋃ ⋂ | ⋂(18)

By the general property of additivity of probabilities (18) is equivalent to:

( ⋂ )⋃ ⋂ | ⋂= ⋂ | ⋂ + ⋂ | ⋂ − ⋂ ⋂ ⋂ | ⋂

(19)

By the property of commutativity, (19) is equivalent to:

( ⋂ )⋃ ⋂ | ⋂= ⋂ | ⋂ + ⋂ | ⋂ − ⋂ ⋂ ⋂ | ⋂

(20)

By applying property 5, we can re-write (20) as:

( ⋂ )⋃ ⋂ | ⋂= | | + | | − ⋂ | ⋂ |

(21)


This can be generalised for the case of a group of individuals as:

( ∪ | ) = ( | ) + ( | ) − ( ⋂ | )

(22)

Where describes the event that simultaneously for all ∈ . In other words it describes the

case in which there is a concordance of interests, sympathy etc. describes the event that

simultaneously for all ∈ . The limit of equation (22) as tends towards ∞, is:

lim→ ( ∪ | ) = lim→ ( | ) + lim→ ( | ) − lim→ ( ⋂ | )

(23)

From definition 2, we know that is equal for all ∈ , which means that ⋂ = = .

Definition 1 tells us that ⋂ = ∅ , since ≠ for some ( , ) ∈ . Hence, as tends to

infinity, lim→ ∏ ( | ) = (∅) = 0 and lim→ ∏ ( ⋂ | ) = (∅⋂ | ) =(∅) = 0. Consequently, in the limit, equation (23) will boil down to:

lim→ ( ∪ | ) = ( | )(24)

Since must have been motivated either by utility-based preferences or by reasoned meta-

preferences (property 2), we have lim→ ( ∪ | ) = ( ) = 1.


Appendix 3: Proof of Proposition 1

Let ( , ) ∈ be two groups of individuals, such that | | > | | and let ⊂ such that | | = | |.By Lemma 1, we have:

( | ) ≥ ( | )By definition of ≽ and ≥, we have ≻⇒≽ and >⇒≥.

We can thus apply PBR1 to get ( | ) ≥ ( | ) ⇒ ≽ .

Further, we know that | | = | | > 1 . We can therefore choose two pairs of individuals

((ℎ, ), ( , )) ∈ , such that (ℎ, ) ∈ and ( , ) ∈ .

By property 4, we have ∀( , ) ∈ : | = | and ∀(ℎ, ) ∈ : | =| .

By property 5, we can combine these two pairs to obtain: ⋂ | ⋂ = | ×| = | × | = ⋂ | ⋂ .

Similarly, by using properties 4 and 5, we can add all remaining elements in and without

affecting the equality, to obtain: | = | .

By PBR2, we then have | = | ⇒ ∼ .

Since ≽ is transitive over : ≽ ∼ ⇒ ≽ .


Appendix 4: Proof of Theorem 1.

It is easy to see that a ranking of consensus sets in terms of the number of individuals reaching

consensus would be transitive and would satisfy (EQU), (POC), (IND) and (INC).

In order to prove the theorem, we need to prove that for any binary transitive ranking ≽ satisfying

the four axioms:

∀( , ) ∈ , , ≠ ∅: | | = | | ⇒ ∼(25)

And

∀( , ) ∈ , , ≠ ∅: | | > | | ⇒ ≻(26)

Let us start with a proof of (25) by induction:

Axiom EQU gives us the basis of the induction for the case where = 1, as it tells us that for any

individuals ({ }, { }) ∈ : ∽ . Our inductive hypothesis is that this holds for all . In order to

prove the inductive step, we need to show that this holds true also for + 1. Consider the sets

( , ) ∈ , such that ≠ ∅ and ≠ ∅.

Let | | = | | = + 1 , with 1 ≤ ≤ | |. Further, let { } ∈ and let ⊂ , such that = \{ }.

Since | | = + 1 and since |{ }| = 1 by definition of a singleton, it follows that | | = .

Logically, { } must either be a member of or not be a member of . If { } ∈ , we know, by

construction of that ⋂ ≠ ∅. However, if { } ∉ it can either be the case that ⋂ ≠ ∅ or

that ⋂ = ∅. This gives rise to 3 possible cases:

Case 1 ({ } ∈ ):

Let ⊂ such that = \{ }.

By construction of and we know that ≠ ∅ and ≠ ∅.

Since | | = + 1 and |{ }| = 1, it follows that | | = .


Since we know that | | = = | |, we can apply the inductive hypothesis to get ∽ .

By applying the IND axiom, we then get: ⋂ ∽ ⋂ , which,

By construction of and is equivalent to ∽ .

Case 2 ({ } ∉ ⋂ ≠ ∅):

If { } ∉ and | | = | | = + 1, then there must be a { } ∈ , such that { } ∉ .

Let ⊂ such that = \{ }.

By construction we know that ≠ ∅.

Since| | = + 1 and |{ }| = 1 by definition of a singleton, it follows that | | = .

We can thus apply the inductive hypothesis to get: ∽ , and then

Apply IND to obtain: ⋂ ∽ ⋂ .

By construction of , this is equivalent to ∽ ⋂ .

Let { } ∈ .

Since ⊂ it follows from { } ∉ that { } ∉ , and from { } ∈ that { } ∈ .

Consequently, we have: |( \{ })⋃{ }| = and | \{ }| = .

By applying the induction hypothesis, we get: \ ⋂ ∽ \ .

We can then apply IND to obtain: ⋂ ∽ .

We have now shown that ∽ ⋂ and that ⋂ ∽ .

Since ≽ is transitive, this implies that: ∽ .

Case 3 ({ } ∉ ⋂ = ∅):

This is a special case in which, by definition of a non-consensus, we have = 0 for ⋂ . So we need

to show that the indifference holds for all = 0 and + 1 = 1.

If it is the case that ⋂ = ∅, then there must be at least one element, { } ∈ , such that

⋂ = ∅.


By INC, we have: ⋂ ∼ ⋂Since we know that ∈ , and that ⋂ = ∅ we can also use INC to expand around to

obtain: ⋂ ∼ ⋂ .

Since ≽ is transitive on , we have: ⋂ ∼ ⋂ ⋂ ∼ ⋂ ⇒ ⋂ ∼ ⋂The choice sets of individuals and can be compared using the EQU axiom, which yields: ∼ . By

definition of a singleton consensus set, we have: ⋂ ≠ ∅ and ⋂ ≠ ∅ , which covers the case

when = 1.

Proof for (26):

Let | | > | |. Now, let ⊂ , such that | | = | |.

By construction of , we know that ≠ ∅.

By (25), we have: ∼ .

Further, by MON, we have: ≻Finally, we can apply transitivity of ≽ to obtain: ≻ .

QED.

Bibliography

Arrow, K. J. (1959). Rational Choice Functions and Orderings. Economica , 26 (102), 121-127.

Arrow, K. J., & Debreu, G. (1954). Existence of a Competitive Equilibrium for a Competitive Economy.

Econometrica , 22 (3), 265-290.

Atkinson, A. B. (1999). The Contributions of Amartya Sen to Welfare Economics. The Scandinavian

Journal of Economics , 101 (2), 173-190.


Becker, G. S. (1976). The Economic Approach to Human Behavior. Chicago: Chicago University Press.

Bentham, J. (1789). The Principles of Morals and Legislation. Oxford: Clarendon Press.

Greenwald, B. C., & Stiglitz, J. E. (1986). Externalities in Economies with Imperfect Information and

Incomplete Markets. Quarterly Journal of Economics , 90, 229-264.

Harsanyi, J. C. (1997). Utilities, preferences, and substantive goods. Social Choice and Welfare , 14,

129-145.

Hicks, J. R. (1956). A Revision of Demnand Theory. Oxford: Oxford University Press.

Hicks, J. R. (1939). Value and Capita. Oxford, England: Clarendon Press.

Houthakker, H. S. (1950). Revealed Preference and the Utility Function. Economica , 17 (66), 159-174.

Jevons, S. W. (1871). The Theory of Political Economy (Fourth Edition: 1911 ed.). London: Macmillan

and Company, Ltd.

Kant, I. (1790). Critique of the Power of Judgement. In P. Guyer (Ed.). Cambdridge, New York:

Cambridge University Press.

Kant, I. (1781). Kritik der reinen Vernuft. In P. Guyer (Ed.). Cambridge: Cambridge University Press.

Little, I. M. (1949). A Reformulation of the Theory of Consumer's Behaviour. Oxford Economic Papers

, 1.

Ludwig von Mises. (1949). Human Action. New Haven, CT: Yale University Press.

Marshall, A. (1890). Principles of Economics: An Introductory Volume (8th edition ed.). New York:

Macmillan.

Nussbaum, M. (2003). Capabilities as Fundamental Entitlements: Sen and Social Justice. Feminist

Economics , 9 (2), 33-59.

Pareto, V. (1906). Manual of Political Economy (1927 ed.). New York: Kelley.

Pattanaik, P., & Xu, Y. (1990). On Ranking Opportunity Sets in Terms of Freedom of Choice.

Recherches Economiques de Louvain , 56, 383–390.

Ravallion, M. (2010). Mashup Indices of Development. Policy Research Working Paper, World Bank,

Washington D.C.


Samuelson, P. A. (1938). A Note on the Pure Theory of Consumer's Behaviour. Economica , 5 (17), 61-

71.

Samuelson, P. A. (1948). Consumption Theory in Terms of Revealed Preference . Economica , 15 (60),

243-253.

Sen, A. K. (1973). Behaviour and the Concept of Preference. Economica , 40 (159), 241-259.

Sen, A. K. (1971). Choice Functions and Revealed Preference. The Review of Economic Studies , 38 (3),

307-317.

Sen, A. K. (1997). Choice, Welfare and Measurement. Cambridge Massachusetts: Harvard University

Press.

Sen, A. K. (1985). Commodities and Capabilities. Amsterdam: Elsevier.

Sen, A. K. (1979). Equality of What? Stanford University.

Sen, A. K. (1992). Inequality Reexamined. Cambridge: Harvard University Press.

Sen, A. K. (1991). Markets and freedoms. London School of Economics and Political Science,

Development Economics Research Programme. London: Suntory-Toyota International Centre for

Economics and Related Disciplines.

Sen, A. K. (1987). On ethics and economics. Oxford: Blackwell.

Sen, A. K. (1977). Rational Fools: A Critique of the Behavioural Foundations of Economic Theory.

Philosophy and Public Affairs , 6, 317-344.

Sen, A. K. (1984). Resources, Values and Development. Oxford: Basil Blackwell.

Sen, A. K. (1983). The Journal of Philosophy. Liberty and Social Choice , 80 (1), 5-28.

Sen, A. K. (1987). The Standard of Living. In Hawthorn, The Standard of Living. Cambridge: Cambridge

University Press.

Sen, A. (2002). Rationality and Freedom. Cambridge, London: The Belknap Press of Harvard

University Press.

Sen, A. (2002). Rationality and Freedom. Cambridge, London: The Belknap Press of Harvard

University Press.

Sen, A., & Williams, B. (1982). Utilitarianism and Beyond. New York: Cambridge University Press.


Smith, A. (1776). Book IV: Of Systems of Political Economy. In E. R. Seligman (Ed.), An Inquiry into the

Nature and Cause of the Wealth of Nations (1957 ed., Vol. 1). London: J.M. Dent & Sons. Ltd.

Smith, A. (1761). The Theory of Moral Sentiments. London: A. Millar in the Strand.

Stigler, G. J. (1982). The Economist as a Preacher, and other essays. The University of Chicago: Basil

Blackwell Publishers.

Walras, L. (1874). Elements of Pure Economics – Or the Theory of Social Wealth. In O. Editions (Ed.).

Philadelphia, PA: Homewood, Ill.: Richard D. Irwin Inc.

revealed meta-preferences: axiomatic foundations of ... · revealed meta-preferences: axiomatic...

Documents