Chapter 9: Welfare
9.1: Introduction
Welfare economics is the treatment of welfare issues in economics. It concerns the appropriate
allocation of resources within an economy and tries to answer the question: is there a ‘best’
allocation of resources? It is an important chapter, not least for the fact that it points out how far we
can go with economics and how much we have to rely on value judgements from others on
distributional issues. We, as economists, can say more than most people would realise but there are
limits to what we can say without outside advice.
9.2: Aggregation of Preferences
We are discussing the appropriate allocation of resources within an economy. In chapter 8 we were
able to conclude that any contract between the two individuals ‘ought’ to be on the contract curve –
because that was the locus of efficient points. Any point off the contract curve is inefficient and can
be improved upon by some movement away from that point. But this does not define a unique point
– because the contract curve is a set of points. The question that we address in this chapter is how
society chooses between these various allocations (along the contract curve).
When put in this stark way, particularly if we are dealing with an economy of two individuals, it
may seem that there is no obvious answer – unless we are prepared to say something about the
relative merits of the two individuals. We can however show that there are special cases when we
can say something. But there is a limit to how far we can go – the rest is the job of politicians. But
we should be clear why this is so.
To economists the nature of the problem with two individuals is obvious – there is a conflict of
interest between the two agents (when one gets more the other gets less). But this conflict hides a
more serious question: is there any way that we can derive the preferences of society from the
preferences of the people that make up that society. Can we aggregate in some sense the preferences
of the individuals to get the preferences of the society?
This is an important question, and one that has occupied economists and philosophers and political
scientists for some time in the past and will continue to occupy them in the future. It seems to me
that the only case where we can be sure that we can aggregate is when all the individuals have the
same preferences. In this case, society’s preferences must coincide with those of every individual’s.
But if different people have different preferences it can be shown that there can be no universally
agreed way of aggregating those preferences into society’s preferences. There is a well-known
theorem – referred to as Arrow’s Impossibility Theorem – which won Arrow a Nobel Prize in 1972
- which proves this very result. In essence, Arrow showed the following. If we accept the axioms:
(1) if individual preferences exist then so should Society's;
(2) if everyone prefers x to y then so should Society;
(3) society's preference between x and y should depend only on individual preferences between x
and y;
then it follows that Society's ranking is that of one individual. In other words, we have dictatorship.
This is an important and influential result.
Of course, in practice, societies adopt ways of aggregating preferences. These are embodied in the
constitution of the society and determine how decisions are made. For example, some societies
adopt a system of majority voting to determine what is decided.
But such mechanisms have flaws – which we can show with a particular example. Consider a
majority voting system applied in a society where there are 3 individuals A, B and C and in which
there are 3 proposals: x, y and z. One and only one proposal is to be implemented. Suppose that the
3 individuals have preferences as follows:
A prefers x to y to z
B prefers y to z to x
C prefers z to x to y
What does the majority voting society choose/prefer?
We see that there is a problem here: a majority of society (A and C) prefer x to y; a majority (A and
B) prefer y to z; and a majority (B and C) prefer z to x. So what does society choose?
The problem here is the aggregation rule. Arrow’s theorem guarantees that for any rule we can find
similarly odd examples.
9.3: Social Welfare Functions
The material above leads to the conclusion that there is in general no way that we can aggregate
preferences to get an undisputed set of preferences for society. Perhaps this is an obvious
implication of the fact that different people have different preferences and that there will, in general,
be a conflict between them. It also seems to agree with what we observe in society – if there was
such an undisputed social welfare function (a rule for aggregating individual preferences) then it
would be embodied in the constitution of that society. Moreover there would be no need for
politicians!
But there are politicians. And what they disagree about is the way preferences should be
aggregated. How they differ is precisely in this aggregation procedure. We could argue that this is
precisely the role of the politician – to define the social welfare function that they intend to use.
One way to do this is as follows: suppose that there are N people in society n = 1, 2, …, N. Let us
denote the utility of individual n by un. Then a social welfare function is simply a way of
aggregating these individual utilities into a society function. In general the welfare W of society is
given by
W = f(u1, u2, …, uN)
(9.1)
where f is some function, increasing in its various arguments.
Different political parties will have different views as to the form of this function – and indeed this
is precisely what distinguishes one party from another. For example, a party who was exclusively
concerned with the least well-off member of society would have the following function1:
1
Called a Rawlsian welfare function after its ‘originator’ John Rawls.
W = min(u1, u2, …, uN)
(9.2)
One popular function is the following ‘classic utilitarian’ form:
W = u1 + u2 + …+ uN
(9.3)
This form treats all individuals equally. Some parties think that different members of society ought
to be given different weight – so for them the social welfare function would be:
W = a1u1 + a2u2 + …+ aNuN
(9.4)
where the an are positive weights reflecting the importance of the members of society.
Once we have a social welfare function we can then choose the optimal allocation. This depends as
well on the choices available to society. These are determined by the analysis of the previous
lecture. There we agreed that, whatever allocation is chosen, it should be on the contract curve2. As
we move along the contract curve the utility of one of the two individuals increases while the utility
of the other decreases. If we know the utility functions of the two individuals we can calculate the
utility of each individual at every point on the contract curve – and hence construct the utility
possibility frontier available to society. In the example that follows the utility function of A is
assumed to be U(q1,q2) = q10.56 q20.24 and that of B U(q1,q2) = q10.54q20.36.
In this figure, individual A’s consumption is measured from the bottom left origin and individual
B’s from the top right origin. Society has a total of 100 units of good 1 and 100 units of good 2. At
the bottom left origin A’s utility is 0 and B’s utility is 1000.541000.36 = 1000.9 = 63.01; at the top
right origin A’s utility is 1000.561000.24 = 1000.8 = 39.81 whereas that of B is 0. If we draw society’s
utility possibility frontier in the next figure with A’s utility on the horizontal axis and B’s on the
vertical, this frontier goes from (39.81, 0) to (0, 63.01). In between we can calculate the utility
values at any point on the contract curve (one point is given in the figure above –with utility of 4.57
for A and 59.4 for B) and we can plot these against each other to give the frontier. It is as follows:
2
Note that all the possible welfare functions that we have listed imply the result that the point chosen will be on the
contract curve, since all the welfare functions are increasing (strictly non-decreasing) in the utility of each member of
society. Indeed it seems reasonable to argue that all reasonable social welfare functions must have this property – it
being one definition of ‘reasonable’.
Where is the best point? Well, this depends upon society’s welfare function. If we define in the
usual way indifference curves for society (defined by utility = constant) then we can draw a set of
such indifference curves and find the one that is highest – given the utility possibility frontier drawn
above.
For example, suppose we work with the classic utilitarian form (9.3) above. Then for a society with
N = 2 the indifference curves are given by u1 + u2 = constant. These are rather familiar – they are a
set of straight lines with slope3 -1. The highest attainable one of these is illustrated in the following
figure – the optimal point is around (5, 60). Individual B seems to do rather well in this society!
Obviously if we change the social welfare function we change the welfare maximising point – in
general. For example, working with the Nash welfare function appropriate for 2 people:
W = u1u2
(9.3)
we get the following figure:
3
They should remind you of perfect 1:1 substitutes – which is effectively what classic utilitarianism is saying about
different members of society – they are all equal.
Optimising with respect to this social welfare function gives us the point (21, 36) – which is
somewhat better for Individual A. Clearly changing the social welfare function changes the optimal
point as far as society is concerned.
9.4: Is Inequality Bad?
The above discussion confirms that the point chosen should be on the contract curve (for otherwise
it could be improved upon) and we note that it does not necessarily imply that the individuals
should be treated equally. Perhaps this latter point seems a bit odd and very much dependent on the
social welfare function adopted by the politicians. But to reassure you that this is not the case, let us
consider a situation in which the individuals are not treated equally but are happy nonetheless. We
take a two-person case and consider a situation in which the two individuals have different
preferences so that the contract curve is not the straight line joining the two corners of the box.
Suppose we start with the individuals being treated equally so that the initial endowment point is at
the centre of the box. Ask yourself what will happen. Obviously this depends upon the trading
mechanism but if we suppose that it is efficient (so that we end up on the contract curve) it will
clearly and necessarily be at a point where the two individuals are consuming different quantities of
the two goods. One will be consuming more of good 1 and the other more of good 2. The reason, of
course, is that we have assumed that they have different preferences. Is this unequal outcome bad?
Clearly not – as both prefer it to the original position, which you could argue was fair (because they
were both given equal endowments). So the final outcome, while unequal, can be argued to be
perfectly fair, as both individuals prefer it to the original fair situation. So inequality is not
necessarily a bad – as long as people are different.
9.5: Measuring Utility
You might have noticed a problem with section 9.3 (though not section 9.4) above: to make sense
of it, we need utility to be measurable. Unfortunately we decided in chapter 5 that that was in
general not possible: while we could represent preferences with utility functions such a
representation was not unique. Indeed we decided that we could transform a utility function with
any monotonically increasing function and it would still represent the same preferences.
To implement the social welfare material of the section above, we (or strictly speaking politicians –
for it is them who are implementing it) need to make preferences measurable and comparable. Or,
perhaps simpler, do something like making the social welfare function directly a function of
consumption. Ultimately we need to be able to compare the happiness/utility (call it what you will)
of different people. I do not really think that is something that economists can do – but at least we
can make it explicit that that is what the politicians are doing – and have to do.
9.6: Summary
What we have achieved is a little, but important. Particularly we have concluded that something is
not possible,
Arrow’s Impossibility Theorem shows that in general it is not possible to aggregate individual
preferences into a society preference.
However:
The use of social welfare functions helps us choose between allocations, but…
… this requires that utility is measurable and comparable.
The allocation implied may be unequal but we have shown that this is not necessarily unacceptable
to the members of society:
An unequal allocation is not necessarily unfair.
9.7: How can the committee chairman manipulate the outcome?
This is not really an application of economics but more a demonstration of the main result of the
chapter: that we cannot derive preferences for society solely from the preferences of the individuals
in that society. We show it in an oblique way: by showing that, given some rules for deciding
between possibilities, the person who decides the order in which decisions are taken can manipulate
the outcome. Therefore that person becomes a dictator. We therefore have a rather indirect
demonstration of the importance and relevance of Arrow’s Impossibility Theorem.
We consider a society in which there are five individuals, A, B, C, D and E and five possibilities, a,
b, c, d and e of which one must be chosen. Obviously if the five individuals have the same
preferences over the five outcomes, there is nothing to discuss, so let us assume that the preferences
differ. Specifically, let us assume that the individuals have preferences as follows:
A prefers a to b to c to d to e
B prefers b to c to d to e to a
C prefers c to d to e to a to b
D prefers d to e to a to b to c
E prefers e to a to b to c to d
You should note that a majority of the society prefers a to b, a majority prefers b to c, a majority
prefers c to d, a majority prefers d to e and a majority prefers e to a. You should note also that if we
rank each option (from 1 for the most preferred to 5 for the least preferred) in each individual’s
preferences and count the average rank of each option, we find that all five options have an average
rank of 3 and thus are ranked equally by society on this criterion.
You could do this exercise with a friend. The friend chooses the rule by which decisions are taken
and you (as the committee chairman) choose the order in which decisions are taken. You as
committee chairman have the power to make a casting vote if the outcome of the decision is a tie.
Suppose you want option a to be chosen, can you choose an order to achieve your desired outcome?
Or, alternatively, can your friend devise a set of rules that stop you achieving your desired
outcome?
Obviously it depends upon the set of rules that your friend can choose. These rules must be generic
and cannot relate to the specific problem under discussion. For example, one such rule could be the
following: decisions are taken sequentially between pairs of options; the option not chosen is
rejected and not discussed further; decisions are by majority voting. This is a very common rule
used in practice.
What you should do as chairman in this case is clear. You organise the voting in the following
order:
(1)
(2)
(3)
(4)
between d and e;
between c and d;
between b and c;
between a and b.
Note what happens: d wins the first vote (because a majority prefer it), c wins the second (because a
majority prefer it), b wins the third and a wins the fourth and last. In the meantime, all the other
options are excluded and your preferred option wins.
If your friend says that all options should be considered all at once and the decision taken by the
average rank, then all options tie and you, as chairman, have the casting vote. Again you get what
you want.
Your friend should come up with other generic rules. The question for each of these is: can you
choose the order of voting to get your way? Can, therefore, a set of rules be drawn up to stop you
manipulating the outcome? If it can in this instance, is it also robust to changes in the preferences?