1 Is There a ‘ Society’?
In the last chapter I introduced the concept of
the social welfare function – that is, a function describing what
variables contribute to the welfare of society as a whole. But the
British Prime Minister in the 1980s, the late Baroness Thatcher, is
famously misreported as having said that there is no such thing as
‘society’. What she actually said is ‘Who is society? There is no
such thing! There are individual men and women and there are
families…’1 It is not
clear what exactly she had in mind. One possibility is that she was
simply referring to the view expressed by Jeremy Bentham in 1789
that ‘the community is a fictitious body, composed of the individual
persons who are considered as constituting as it were its
members’. 2 Alternatively, she may have had in
mind Pareto’s view that it
makes more sense to talk about the welfare for society, rather than the welfare
of society.3 Simon Blackburn has suggested that
she was reflecting ‘the ideology of the self-interested agent in
eternal competition with others’.4 Another – and more charitable –
interpretation is that she was making a philosophical objection to
the metaphysical notion that one can attribute states of mind to
‘society’ of the same kind that one can to individuals. It is not
for me to attempt to adjudicate between these various
interpretations and the reader must judge for himself.
But whatever the correct interpretation of the
late Baroness Thatcher’s view there is no doubt that it is linked
to a serious aggregation problem. This is the problem of how to go
from individual preferences to the preferences of society. It is a
well-known problem in welfare economics and in political philosophy
which we shall discuss in more detail in the next section. The
problem arises because the whole edifice of welfare economics is
built up on a fairly clearly defined theory of individual choice,
some components of which – such as the way consumers respond to
changes in prices – has been empirically confirmed. But even if one
could assume that these choices correspond to their welfares it is
not clear how far one can go from this to a theory of what
constitutes the welfare of a group of individuals. How does one
weigh together the welfare of individuals to arrive at the welfare
of the group?
Another problem, which is less obvious and far
less important for practical purposes, follows from the fact that
our theory of whether the utility of an individual has risen or
fallen is in terms of whether the individual is on a higher or
lower ‘indifference’ curve (or ‘behaviour line’ as it is sometimes
called). Hence, strictly
speaking, one cannot compare the welfare of societies
comprising different people with different indifference curves.
This gives rise to what is known as the ‘constituency principle’,
namely that comparisons of welfare are only valid if they refer to
identical groups – that is, a ‘constituency’ of people. But this
would even mean that if the GDP of some country were much higher
today than yesterday one could not say that its economic welfare
had risen. For some people would have died during the previous 24
hours and others would have been born. So we would be comparing
societies with different populations, which, strictly speaking, is not possible
within the framework of economic theory.
But all such limitations on the possibility of
passing judgement on the economic welfare of different societies
are nonsense, of course. For example, strictly speaking, the
constituency principle would exclude our saying that welfare is
higher in a country where people are free, well-fed, healthy and
happy than in some other country where people are oppressed, hungry
and miserable. But we know that this is nonsense.
2 Social Choice Theory and the Impossibility Theorem
So what do we mean when we talk about the
aggregate welfare of ‘society’? It is intuitively plausible to
assume that society does not have a ‘mind’ in the ordinary sense of
the word so that one cannot attribute to ‘society’ the sort of
mental thought processes that correspond exactly to those that an
individual goes through when making choices. This means that, for
example, society does not follow thought processes such as those
that correspond, ideally, to the rational choice process that an
individual would follow. There is no such thing as a ‘society’ that
has a mind that could say to itself, ‘Oh dear, silly me! Since I
preferred X to Y and Y to Z how stupid of me to prefer Z to X’.
This is particularly inconvenient for economists – who are the
guardians of ‘rational choice’ and who like to equate rational
choice with the maximisation of something, such as a utility
function, or a social welfare function, that respects some criteria
of rationality.
One implication of this is that whereas one can
make empirically verifiable positive propositions about the
preference rankings of any individual, it is not possible to make
similar propositions about the collective preference rankings of
any collection of
individuals. For example, if we assert that Jones thinks he prefers
situation 1 to situation 2, we are making a positive proposition
that could be tested by asking Jones. If he says, ‘Yes, I think I
prefer 1 to 2’, our proposition has been confirmed, unless we have
some reason to suspect he is lying. But we cannot make a similar
testable proposition about
the combined preferences of a number of individuals. We can, of
course, add up how many prefer one or other of the two options,
like in a referendum. But that would simply amount to defining
society’s preference as being what the majority want.
Or, as in a market economy, we could check the
amounts of money that the agents were prepared to pay for the two
alternatives and add them up. But to conclude from these that
‘society’ prefers one situation rather than another merely means
that we define the statement that ‘society prefers one situation
rather than another’ in terms of survey results or people’s
expenditure patterns.5
Nevertheless, decisions have to be made all the
time about various policies or projects, at all levels of public
life. The classic examples are public goods such as national
defence, or law and order. These have certain well-known
characteristics, namely ‘non-excludability’ or ‘non-rivalry’.
Non-excludability means that if the service is provided for one or
more persons it is technically impossible, or very difficult, to
prevent other people from using it. If we have armed forces to
protect the British Royal Family from invasion by the French it is
impossible to exclude everybody else in the country from this
invasion, however much some of them may have welcomed it.
Non-rivalry means that once the service is provided to some people
then, within limits, it costs no more to provide it to
others.
For example, if one person admires the flowers in
a park he does not prevent others from doing so, as he would do if
he ate them instead. So his admiration for the flowers imposes no
cost on anybody else. In addition, there are services, such as
education or health, where there are good reasons for believing
that positive externalities or asymmetrical information requires
some public provision. And, even where the technical properties of
the goods in question may not require public provision there are
often cases for some sort of public intervention in the market –
for example, to cater for the well-known weak links in the
choice-to-welfare chain such as those discussed in Chapter 5.
In addition, there are also
the arguments discussed earlier concerning ‘the moral limits’ of
the market. Thus, for various well-known reasons, there is a
prima facie case for the
view that the provision of certain goods or services ought not to
be left to the market mechanism, and some other process is needed
in order to aggregate individual preferences into some concept of
society’s choices. Social choice theory is concerned with this
aggregation of people’s preference rankings between the outcomes –
or ‘different states of society’ – that will follow from different
policies. It is concerned with the aggregation of people’s
preference ranking between different collective choice, such as
where to site an incinerator to burn local garbage, or whether to
open a new school or shut down some local hospital or build a new
road and so on.
This raises the problem of collective – or social
– choice theory, to which major contributions have been made since
Ken Arrow’s seminal work in 1951 set out to answer the question of
whether ‘…it is formally possible to construct a procedure for
passing from a set of known individual tastes to a pattern of
social decision-making…it being assumed, on the way, that
individuals’ choices are rational’.6
Arrow tackled the question of whether it was
possible to find a ‘constitution’ – that is, a set of rules – by
which society could rank various social states and that satisfied
some apparently appealing ethical axioms, such as no dictatorship,
or the weak version of the Pareto principle to the effect that, if
everybody prefers situation A to B then A is better than B, or the
requirement that rational social choice should respect the same
transitivity axiom required of rational choice for an
individual.7 What
Arrow did was to prove that no possible rules could be found for
reaching such an aggregate ranking in the light of individual
rankings without violating one or other of his proposed axioms, all
of which have great intuitive appeal. To prove such an
‘impossibility’ theorem is, of course, a very disconcerting and
fundamental result. Although one may question some of the
assumptions required to reach this result (such as no interpersonal
comparisons of utilities) it was nevertheless an extraordinary
achievement to prove that, given these assumptions, ‘…attempts to
form social judgments by aggregating individual expressed
preferences always lead to the possibility of paradoxes. Thus there
cannot be a completely consistent meaning to collective
rationality’.8
3 An Example: Local Air Pollution
For example, suppose that there is some
unpleasant localised air pollution from factories, such as smoke or
sulphur dioxide. Consider three methods of dealing with it (apart
from just ignoring it). One of them is simple regulation. The other
two both recognise the scope for a market in ‘property rights’. But
these could take the form either of rights to clean unpolluted air
that could be given to the potential victims of the pollution, or
rights to pollute the air that could be given to the owners of the
factories that cause the pollution. There are thus three
possibilities, namely:
X = use the market by giving property rights in clean air to the poor people who are the ones usually most affected by the pollution. They would then negotiate to sell some of their rights to clean air to the factory owners at a mutually acceptable price and quantity.Y = use bureaucratic regulation.Z = use the market by giving a certain amount of ‘air pollution rights’ either to the rich owners of the factories who might be willing to reduce their use of clean air in return for suitable compensation, or to the victims of the pollution who might be willing to sell some of their rights to clean air to the factory owners. Both sides could then negotiate a mutually acceptable deal.
A typical bureaucratic way
of dealing with such a problem would be to set up a committee,
since, as Art Buchwald once put it, committees are places where,
for most people, conversation is a substitute for the boredom of
work and the loneliness of thought. The committee’s remit would be
to decide which of these three methods to adopt. This might consist
of a pollution expert, a health expert, the owners of the local
factories, representatives of the citizens (e.g. local
councillors), a lawyer, a moral philosopher and two economists (in
order to ensure three opinions). Such a committee would have to
begin by agreeing on their decision rules, such as whether the
ranking of all the possible alternative decisions should be taken
by a simple vote, or by some transferable vote system. Social
choice theory is addressed to the rules that the committee would
draw up for ranking different possible decisions. Arrow has shown
that no rules could satisfy apparently compelling axioms of
collective rational
choice.
The flavour of the argument can, perhaps, be
illustrated in the following simplified example, in which the local
authorities set up a small sub-committee of only three members,
namely:
Mr LWF-M has left-wing tendencies but is in favour of free markets, up to a point, so wants to promote his distributional judgement that one ought to make income distribution more equal.Mrs RWF-M, who has right-wing tendencies and also favours free markets, thinks that the pollution rights ought to be given to the factory owners.Miss BB is a bossy bureaucrat so naturally prefers regulation since it gives her more power and responsibility.
Their preference orderings are likely to be as
follows:
-
LWFM’s preferences are likely to be X > Y > Z.
-
RWFM’s preferences are likely to be Z > X > Y.
-
BB’s preferences might be Y > Z > X.
Suppose they vote on the choice between X and Y.
Two of them rank X > Y. And if they also vote on the choice
between Y and Z, two of them will prefer Y to Z. So it seems that a
majority vote would show that X > Z. But if the committee then
vote on the choice between X and Z they will find that two of them
rank Z > X. So their ranking is X > Y > Z > X!!!! In
other words, the democratic procedure of majority voting in this
instance does not satisfy the transitivity axiom of rational choice
that we met Chapter
4 in connection with individuals’ choices. This
paradox, to which Arrow referred, is known as the ‘Condorcet
paradox’ in honour of the Marquis de Condorcet who drew attention
to it over 200 years ago.
A big industry on social choice theory has
developed, of course, since Arrow’s pioneering contribution, some
of which disputes the significance of Arrow’s work. One of the most
fundamental critiques was made back in 1954 by Nobel Laureate James
Buchanan.9 He argued
that one should not expect a ‘social decision rule’ to satisfy the
same criteria of rationality – notably transitivity – as for a
single mind (such as an individual’s mind). He argued that
‘rationality or irrationality as an attribute of the social group
implies the imputation to
that group of an organic existence apart from that of its
individual components.…’
In other words, one might expect that rationality
would be the criterion of the preference ordering of a single
rational mind. But we cannot go from that to expecting transitivity
to be the characteristic of collective preference orderings. Thus
Buchanan was not worried by the implications of the result that any
decision rule can lead to apparently intransitive orderings, for he
regarded this as a necessary feature of democratic decision-making.
He argued that ‘Correctly speaking, majority decision must be
viewed primarily as a device for breaking a stalemate and for
allowing some collective action to be taken’.
That is all very well and may even set one’s mind
at rest. But most of us would have liked to have believed that, if
only one could find it, there must surely be some way in which a
group could reach decisions that would satisfy our conception of
rationality. But Arrow showed that – disconcerting though it may be
– this is simply not the case. As he put it ‘…there are well-known
fundamental dilemmas in any concept of social good…for reasons that
I think are intrinsic to the logic of the subject. The root facts
here are the incommensurability and incomplete communicability of
human wants and values’.10
Thus we have lost our innocence and we now have
to abandon our belief in some ‘rational’ collective choice.
According to Partha Dasgupta, ‘It seems to me we simply have to
live with Arrow’s theorem and do the best we can…Just as circles
can’t be squared, ideal voting rules don’t exist, ideal markets are
a pleasant myth, and ideal governments can’t be conjured up because
governments are run by people’.11 One cannot refuse to do anything at
all on account of some known ambiguities in the notion of an
aggregate, or collective, ‘social’ welfare. So what do we do?
Presumably we eschew referenda on every important decision in the
domain of economic policy. Instead we have to fall back on the best
approach we have, namely old-fashioned welfare economics, while
bearing in mind its limitations.
4 The Welfare Economics Approach
The welfare economics approach, as exemplified,
for example, in a CBA, is far less ambitious. It is limited to
trying to promote the state of affairs that would have been
achieved in a free market in the absence of the usual market
imperfections, such as monopolistic conditions, taxes,
externalities and so on. In this way it might even be able to take
some account of the intensity of people’s preferences, which is, of
course, absent in the social choice approach. As explained in
Chapter
5, we know that people’s preferences do not
always accurately reflect what is good for their welfare. But let
us assume, for present purposes, that we are dealing with
situations in which this divergence between preference satisfaction
and welfare is not very important. How then would conventional
welfare economic analysis handle the aforementioned problem of
dealing with local air pollution?
In the previous example, two out of the three
members of the committee might have preferred solution Y to
solution X, but the intensity of the preference of the third person
for solution X may have been very great, whereas the intensity of
preference of the two members of the committee for solution Y may
have been very slight. In ordinary market transactions the
intensity of people’s preferences is represented by the prices
people are prepared to pay for the options open to them.
For example if a vote were to be taken in some
office about whether smoking were to be permitted in common areas,
there may well be a majority for smoking to be permitted. However,
it is possible that the non-smokers would have been prepared to pay
a much larger sum for smoking to be prohibited than the smokers
would have been prepared to pay for it to have been permitted. In
that case, there would have been scope for a mutually satisfactory
deal between the smokers and the non-smokers in which some agreed
compensation would have been paid by the non-smokers to the smokers
that would have left both parties better off. A CBA would have
indicated the scope for a Pareto-optimising move.
The welfare economics approach focuses our
attention on the causes of market failure in any particular
instance and possible ways of rectifying them. One criticism of the
cost-benefit approach is that – as mentioned earlier – allowance
for the intensity of preferences by taking account of how much they
are prepared to pay for different things means giving greater
weight to wealthier people’s preferences. But a good CBA could take
some account of the extent to which costs and benefits are
influenced by the distribution of incomes, as well as by other
institutional features of society that had a major influence on
relative prices.12
Thus in the aforementioned example, welfare
economics would start by identifying whether there was an
externality which established a prima facie case for some sort of
public intervention. It would then try to establish what would be
the optimal amount of the pollution permitted, taking account of
the social costs of reducing it and the social damage done by the
pollution. (This could make some allowance for the extent to which
people’s preferences might be biased by incorrect information,
etc.) One obvious method for reducing pollution to the optimal
amount would be to impose a tax on the pollution equal to the
marginal cost or marginal social damage at the point where these
two magnitudes are equal.13 Another method would be to mimic a
market system by giving pollution rights to either the victims of
the pollution or the factory owners, along the lines indicated in
Section 3, and
then let them negotiate a price with each other.
The next step would be to go beyond the basic
theory of Pareto optimality and to take account of distributional
considerations. For example, the victims of the air pollution in
the previous example might be mainly poor people, since on the
whole, polluting factories tend to be located in poorer
neighbourhoods. On the other hand, even if the factories were not
located in particularly poor areas they might have employed mainly
poor people whose livelihood would be threatened if the pollution
charges made their factories uncompetitive. In either case the
distributional judgements represented in the social welfare
function of the decision makers would come into play and should
influence the outcome.
Another subjective judgement
that would need to be made would be how far one attached intrinsic
value to the consequences of any actions compared to the intrinsic
value of the processes by which any particular outcome is achieved
and hence of the ‘fairness’ or ‘justice’ of these processes. But
there are different
interpretations of ‘consequentialism’ and of the extent to which
processes ought to be included in the consequences of any action.
Standard CBA does not necessarily incorporate information about
whether the process leading up to any particular consequences
involved violation of anybody’s rights. Hence, it tends to be
frequently overlooked.
But if the domain of welfare economics is
extended to incorporate aspects of processes, such as respect for
‘rights’, into the assessment of the outcome of any policy, it
raises a whole host of other value judgements. In the
aforementioned example, judgements would have to include the
intrinsic value of ‘rights’ to the clean air that was being
polluted, since this would determine who is to pay and who is to
receive the proceeds. For example, if the local residents had lived
in the area before the factories had arrived, it might be argued
that they enjoyed the property rights in the clean air and so
should receive compensation from the factory owners. But if the
factories had arrived first their owners might argue that there had
been no pollution – in the sense that anybody was actually harmed –
before the local residents came to live there. So, it could be
argued, it was the arrival of the poor residents who are to blame
for causing the pollution. In the words of the old song:
It’s the same the ‘ole world overIt’s the poor wot gets the blameIt’s the rich wot gets the pleasureAin’t it all a bloomin shame!
Thus, one way or another, although the welfare
economics approach avoids the logical problems of arriving at a
decision that can be said to represent some ‘collective’ rational
preference ranking, it raises numerous difficult choices between
the intrinsic value one attaches to aspects – consequences or
processes – of the choices in question. But both welfare economics
and collective decisions have crucial roles to play. It is widely
accepted that the unfettered market mechanism is unlikely to
provide a perfect mechanism for the promotion of society’s economic
welfare. Hence, there may be room for some form of social
intervention. The problem then is what form this should take
starting from individuals’ preferences. There is no simple
algorithm that can tell us how to reach the decisions. Meanwhile,
decisions have to be made about policies and projects that will
affect the whole of some society.
5 An Example: The ‘Summers Memorandum’
It was asserted in the Introduction to this book
that, in most practical policy decisions, the basic value
judgements implied in welfare economics may often not play as a
significant role in decision-making as does the knowledge about the
facts. This, it was argued, was because one is never starting from
an optimal initial situation. Instead, one is always concerned with
relatively small moves from whatever initial situation we happen to
have started. But there are often cases where basic value
judgements are important.
One well-known example is the memorandum written
in 1991 by Larry Summers, who was the chief economist at the World
Bank at the time. The memorandum that he wrote to some colleagues
soon became famous.14 This is because he raised the
question of whether the export of polluting industries to
less-developed countries (LDCs) ought to be encouraged. He outlined
the conventional economic case for doing so which was based on
certain propositions, notably:
- (i)
If valued by conventional methods, any economic valuation of morbidity and mortality in such countries would be much lower than in advanced countries.
- (ii)
In some sparsely populated countries a given amount of air pollution would have a far less serious impact.
- (iii)
In countries that have much shorter life expectancy anyway, some of the very long-term possible effects of air pollution would be relatively less important.
His famous memorandum has, inevitably, been
heavily criticised and scrutinised, and it would be out of place
here to rehearse again many of the points made.15 But it should be pointed out that
in the final paragraph of his memorandum Summers did indicate that
various objections could be made to this proposal, including
objections on moral grounds. And he pointed out that similar
objections could be made to many other World Bank proposals for
trade liberalisation. He was, therefore, raising a question about
consistency in Bank policy rather than making a specific proposal
about the export of polluting activities. (It is not his fault that
he sometimes raises questions that most people would prefer to
leave unasked, presumably because they know that they would not
like the answers.)
But it does serve as a particularly good example
of five of the weaknesses in Pareto optimality that have been
identified earlier (and of which Summers was, of course, perfectly
aware, as the final paragraph of his memo indicated). They would
be:
- (i)
The trade in question would be a Pareto-optimising in the sense that both parties would be moving to a preferred position. But it was only the original unequal distribution of endowments that induced poor countries to find it beneficial in spite of its hazards. If they had been much richer the first and third of the aforementioned Summers’ points would not have applied.
- (ii)
The citizens of the LDCs are likely to be very poorly informed about the possible harmful health effects of the polluting activities in question.
- (iii)
The trades in question are likely to have been negotiated with the authorities of the LDCs, or other powerful interests therein, who, even if they were well informed of the health consequences, may not have worried much about the interests of the population as a whole. In other words, it would be classic example of the agent-principal problem – that is, that certain organisations are often run in the interests of the ‘agents’ in charge of them instead of the ‘principals’ who own them or whose interests they are supposed to serve.
- (iv)
In any case, it is extremely unlikely that the actual victims of the pollution in question would be properly compensated for their loss of welfare (or loss of life in some cases, such as the infamous Bhopal disaster), so that potential Pareto optimality would certainly be irrelevant.
- (v)
How far rich countries ought to take account of the likely loss of welfare of the poor countries raises the issue of how one ought to draw the boundaries around the society whose welfare one is trying to maximise. Thus, in the modern world the most serious problem about what is meant by ‘society’ is the question of how we draw the boundary around the society the welfare of which (or for which) we are seeking to maximise. It is what is often known in moral philosophy discourse as the problem of ‘moral distance’. For example, how far should one take account of distant people or distant generations? Hence, the ‘moral distance’ problem is examined in greater detail in Chapters 17 and 18 in connection with international and intergenerational justice. It would be nice if welfare economic theory could avoid getting mixed up in this particular value judgement, which is a matter of political philosophy. But this problem has become increasingly important in today’s world of globalisation, mass emigration and environmental change.
6 Conclusions
The way that the value judgements implicit in
welfare economics enter into the decision-making process is
illustrated in Fig.
3.1 in Chapter
3. Some specific value judgements have already
been mentioned – such as the intrinsic value of consumers’ freedom
to make their own mistakes, or of equality of incomes, or of
economic prosperity – and others will be discussed in more detail
in later chapters.
It might already appear at this point that
welfare economics cannot take us very far in making rational
choices among different policy options. But this would be a
mistake. Welfare economics and its application in cost-benefit
analysis provide a valuable framework and organising principle for
taking account of the effect of any economic policy. It clarifies
criteria that help to evaluate the relative economic merits of
alternative policies and projects. This conceptual framework
enables one to identify the value judgements involved so that one
can take explicit account of them, rather than to leave them to
dwell unacknowledged in some murky background.
This does not mean that decisions become easy to
make. For there are probably no clear-cut ethical theories that are
in harmony with all our ethical intuitions and attitudes. Insofar
as we hold plural values, some of which often conflict, one cannot
expect to find an over-riding ethical system that reconciles the
different preferences of different people.
On the other hand, as pointed out previously,
most policy choices are about marginal changes in the allocation of
resources, starting from a ‘second-best’ point. In practice,
therefore, what is usually required will be factual information,
and there will be little point in wringing one’s hands over the
normative significance of the starting point. There are many
situations where some numerical evaluation of alternative courses
of action along cost-benefit lines that take account of
distributional considerations will make a useful – if not decisive
– contribution to rational decision-making that would be preferable
to total reliance on other decision-making processes. An example of
this in the UK context is given at the end of Chapter 19, and no doubt innumerable
others could be found, not to mention the way some decisions are
made in other parts of the world. In most situations the ethical
limitations on CBA and its welfare economics underpinning it are
small beer compared to the role of uncertain facts, administrative
failures, pressure groups and empire-building officials, not to
mention downright corruption, so that to jib at the former while
accepting the latter is a clear case of swallowing a camel while
straining at a gnat.
But, because some of the value judgements
implicit in welfare economics are usually only in the background,
the extent to which they qualify the whole structure of standard
welfare economics is not always adequately appreciated. As a
result, the valuable achievements of welfare economics can be put
to the wrong use, as is sometimes the case, for example, with bad
applications of CBA. The function of normative economics is to help
improve the allocation of resources in the economy in the pursuit
of its objectives. But this should not take the form of providing
ready-made formulae that can be applied mechanically. As far as
possible it should take account of the deviations between market
prices and costs, on the one hand, and ‘social’ prices and costs,
on the other. It must also take account of historical experience,
of psychological motivations that are poorly understood, of
political and social circumstances, and of ethical values that are
usually only dimly recognised. Otherwise it becomes a sterile – and
possibly harmful – activity.
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Footnotes
3
Pareto did not actually use the term ‘welfare’
(or its French equivalent, ‘bien-etre’ since he was writing in
French) but carefully distinguished between his concept of
‘ophelimity’ and ‘utility’, the latter being a much broader concept
than the former insofar as it refers to the whole ‘property (of
something) which makes a thing favourable to the development and
well-being of an individual, a community or the whole human
species’. See Finer,1966:254 and
99–103.
5
It should not be thought that the economist’s
device of the ‘compensating variation’, which owes its origins to
the Hicks/Kaldor/Scitovsky compensation tests, overcomes the
aggregation problem. For it does not deal with the welfare
implications of different distributions of incomes.
7
See Hausman and McPherson, 2006:ch 14, or Hargreaves Heap, 1989:190ff, for
more comprehensive expositions of the essential features of social
choice theory, including a full enumeration and explanation of
Arrow axioms.
13
There is not actually a unique optimal point,
since different means of reducing the pollution will have different
income effects on the parties concerned, which will feed back on
the cost and benefit schedules.