Download Topic 6. The Arrow Possibility Theorem

MSc in Economic Evaluation in Health Care
WELFARE ECONOMICS
Topic 6. The Arrow Possibility Theorem
Informational requirements and the social welfare ordering
We wish to restrict the choice of a social welfare ordering (SWO) to those that satisfy
certain informational requirements. If we were able to choose any SWO then any shape is
possible for the social welfare function (SWF). Imposing certain information
requirements is found to restrict the SWO possibilities drastically.
SWO possibilities can be examined under two sorts of restriction, both of which pertain
to the information that policy-makers are permitted to utilise when deriving an SWO.
These are:
1. Welfarism. This restricts the information that can be used to derive an SWO of social
states to utility information corresponding to those social states.
2. Informational requirements. These are requirements pertaining to the measurability
and comparability of the utilities of individual households.
Welfarism
An SWO has the property of welfarism if the ranking of social states depends only on the
utility levels of the households. Specifically, information about how utility levels are
obtained is irrelevant for determining how social states should be ordered. This means
that social states having the same welfare consequences are indistinguishable. Therefore,
social welfare depends only on the numerical value of the utility attained by each
individual household regardless of the measurement conventions by which numerical
utility levels are arrived at.
Three conditions are sufficient for welfarism:
1. Unrestricted domain. Any logically possible vector of H household utility functions is
admissible in determining the SWO. That is, the same SWO must be used to
aggregate individual utilities regardless of what the household utility function
happens to be. The only thing asked of household preferences is that they be able to
order social states consistently.
2. Pareto indifference. If all households are Pareto indifferent between two social states,
the SWO must rank the two states equivalently.
3. Independence of irrelevant alternatives. The social ranking of any two states x and y
must be the same whenever the utility levels attached to x and y by the individual
households are the same. This implies that the social ranking must be unchanged if
any or all households’ indifference curves are renumbered in a way that leaves the
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indifference curve numbers associated with states x and y unchanged. This also
means that the social ranking of x and y must be independent of the availability of
other social states and of the households’ preferences over social states other than
those being ranked.
The Pareto indifference condition means that all other information about states x and y
other than utility information (obtained from the consumption of x and y) must be
ignored. This is the heart of welfarism. The other two conditions generalise this finding to
strict rankings of x and y.
Informational requirements
There are two types of informational requirements (sometimes called invariance
requirements):
1. Measurability. This refers to the sense in which the actual numbers attached to a
given household’s utility levels are meaningful.
2. Comparability. This refers to the sense in which the actual numbers attached to
different households’ utility levels can be meaningfully compared. This is concerned
with whether or not utility information is commensurate across households.
There are four possible measurability assumptions:
1. Utility is measurable only with an ordinal scale (OS). In this case indifference curves
can be numbered in any arbitrary manner, but higher indifference curves must be
given higher numbers in order that the numerical scale preserves the ranking of the
indifference curves. OS measurability allows only the ordering or ranking of levels of
utility. This is the least demanding measurability assumption.
2. Cardinal scale measurability (CS), in addition to giving information on the ranking of
indifference curves, gives information on the magnitude of the change in utility in
going from one indifference curve to another. Therefore, CS allows the ordering of
changes in utility as well as the ordering of levels of utility.
3. Ratio scale measurability (RS), in addition to giving information on the ranking of
indifference curves, gives information on the proportionate change in utility in going
from one indifference curve to another. Therefore, RS measurability allows the
ordering of proportionate changes in utility as well as the ordering of levels of utility.
4. Utility is fully measurable, or measurable with an absolute scale (AS). In this case a
unique real number is attached to each indifference curve of the household. This is
the most demanding assumption.
There are three possible comparability assumptions
1. Non-comparability (NC) means that none of the information measured for individual
household utility can be used when making across-household comparisons so we
cannot say whether one household is better off than another in a particular social
state. This is the least demanding comparability assumption.
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2. Partial comparability (PC) means that only some of the household information is
available for comparisons across households.
3. Full comparability (FC) means that all of the information available for the individual
household is available for comparisons across households so we can say whether one
household is better off than another in a particular social state. This is the most
demanding comparability assumption.
The Arrow Possibility Theorem
If we set the following requirements for the SWO:
1. Ordinal scale measurability of household utility;
2. Non-comparability of household utility;
3. Welfarism (= unrestricted domain, Pareto indifference and independence of irrelevant
alternatives); and,
4. The Pareto Principle.
The only possible SWO (i.e. complete and consistent social ordering) is a dictatorship.
That is, social orderings must coincide with the preferences of some arbitrary household
in the economy regardless of the preferences of the others. This is the Arrow Possibility
Theorem. Sometimes this result is called the Arrow Impossibility Theorem if dictatorship
is precluded by assumption (i.e. if a fifth requirement is added of non-dictatorship then
the Arrow Impossibility Theorem says that no SWO is possible).
‘Intuitive’ proof
Utility of household g
II
I
v1
u1
u0 = v0
III
IV
Utility of household h
Diagram 1
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We now consider a brief ‘intuitive’ proof of the Arrow Possibility Theorem. We first
assume the third requirement for the SWO (welfarism). This means that the SWO can be
examined in terms of the rankings of social states by the utility levels of households.
Suppose for simplicity a society consisting of two households, g and h, where the utility
of g is measured on the vertical axis and the utility of h is measured on the horizontal axis
(Diagram 1). We may consider any utility point (say u0 = [u0g, u0h]) as reference point.
We can use u0 to divide the utility space into quadrants. We can immediately rank
quadrants I and III relative to u0 using the fourth requirement for the SWO (the Pareto
Principle): all allocations in I must be ranked higher than u0 and all allocations in III must
be ranked lower. The problem is to rank allocations in quadrants II and IV relative to u0.
Consider now the first requirement for the SWO (ordinal scale measurability of
household utility). This assumption means that indifference curves can be numbered in
any arbitrary manner, but higher indifference curves must be given higher numbers in
order that the numerical scale preserves the ranking of the indifference curves. In other
words, even if we change the numbers on the indifference curves the ranking of states
must remain the same.
Consider now the second requirement (non-comparability of household utility). This
means that none of the information measured for individual household utility can be used
when making across-household comparisons. This implies that households’ indifference
curves can be numbered or renumbered in any manner, and that this renumbering
mechanism may be different across households.
The effect of the first two requirements combined is that any household’s indifference
curves can be renumbered in any manner that preserves the ranking of its indifference
curves, and that different renumberings can be applied to the indifference maps of
different households.
Consider any point in quadrant II (say, u1, where u1g > u0g and u1h < u0h). u1 must be
ranked above u0, or u0 must be ranked above u1, or u1 and u0 must be ranked as
equivalent. We can show that all points in quadrant II must be ranked in the same way
relative to u0. Suppose that u1 is ranked above u0. This ranking must be preserved when
we renumber the indifference curves of households g and h (from the first requirement
for the SWO = ordinal scale measurability of household utility), and we can also apply
different renumberings to the indifferent maps of different households (from the second
requirement = non-comparability of household utility). Suppose we renumber the
households’ utility such that u0 is mapped back onto itself (v0) and u1 is mapped on to v1.
It is still the case that v1g > v0g and v1h < v0h, so this change is allowed, but it means that
all allocations in quadrant II must be ranked in the same way relative to u0 (e.g., all
allocations are ranked above u0).
We can also rule out the case that all points in II are ranked equivalently to u0. Suppose
this is the case, this means that u1 and v1 are ranked equivalently. But, by the Pareto
Principle v1 must be preferred to u1. This violates intransitivity. Therefore, all points in
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quadrant II must be ranked above u0, or u0 must be ranked all points in quadrant II. They
cannot be ranked equivalently.
By exactly the same reasoning as above we can see that all points in quadrant IV must be
ranked in the same way as well relative to u0.
It can be further established by exactly the same argument that if all points in quadrant II
are ranked above u0 (or vice versa), then u0 must be ranked above all points in quadrant
IV (or vice versa). This is because the relationship of u0 to quadrant II is the same as that
of points in IV to u0.
Finally, it is obvious that if two quadrants are ranked the same way (e.g. I and II), then
points on the boundary between the two quadrants are ranked in the same way.
Therefore, we have established so far that:
1. either quadrants I and II (and their common boundary) are preferred to u0, and u0 is
preferred to quadrants III and IV,
2. or quadrants I and IV (and their common boundary) are preferred to u0, and u0 is
preferred to quadrants II and III.
In case 1 we have not yet ranked the points along the horizontal line through u0, whereas
in case 2 we have not ranked the points along the vertical line through u0.
For illustration, let us concentrate on case 1 where we have not yet ranked the points
along the horizontal line through u0. There are two possibilities:
1. Strong dictatorship = all points above the line are preferred to u0, u0 is preferred to all
points below the line, and all points on the line are socially indifferent (i.e., the level
of social welfare is equal at all points on the line, including u0). This means that
household g is a strong dictator because if it is indifferent between any two states on
the horizontal line (as it will be because any two points on the line yield the same
level of utility to g), the states are ranked indifferent socially. Essentially the
horizontal line is a social welfare indifference curve and the entire preference map
would consist of a series of horizontal social welfare indifference curves that the
SWF would correspond with household g’s own ordinal utility function. This result is
generaliseable to a case of more than two households.
2. Lexicographic dictatorship = all points above the line are preferred to u0, u0 is
preferred to all points below the line, and on the line u0 is preferred to any point on its
left but not preferred to any point on its right. In other words, the ranking of points on
the horizontal line increases as one moves rightwards. In this case, household g is the
prior dictator but is indifferent between two states on the horizontal line (because any
two points on the line yield the same level of utility to g), so the mantle of
dictatorship falls on household h who prefers states rightwards along the horizontal
line. In this case there is no possibility of indifference between social states and the
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SWO cannot be represented by an SWF. This result is generaliseable to a case of
more than two households.
So, the conclusion is that if we assume the SWO has the characteristics of ordinal scale
measurability of household utility, non-comparability of household utility, welfarism and
the Pareto Principle, the only possible SWO is either a strong or lexicographic
dictatorship. This is the Arrow Possibility Theorem. Obviously it is not a very
satisfactory outcome.
Relaxing the informational requirements for an SWO
The Arrow Possibility Theorem applies when household utilities are measured on an
ordinal scale and are non-comparable across households. The measurability and
comparability requirements can be relaxed to widen the number of possible SWOs.
Utilities can be made more measurable (from OS to CS to RS to AS measurability).
Utilities can also be made more comparable across households (from NC to PC to FC).
In general the more measurable and comparable are household utilities the more
information the social planner will have for aggregating utilities into a social ordering,
and the wider will be the scope of possible SWFs. With the limited information used by
Arrow only a dictatorship is possible, whereas with full information (absolute scale
measurability and full comparability) any form of SWF is possible (this latter case
corresponds to the Bergsonion SWF). In between these two possibilities different
combinations of measurability and comparability assumption can lead to the possibility
of different SWFs (such as the utilitarian SWF [CS + FC], the generalised utilitarian
SWF [CS + FC], and the maximin SWF [OS + FC]).
Adding information to the Arrow framework will expand the set of possible SWFs.
Although this is helpful, it does not solve the problem as to which SWF is the appropriate
one to use. For that one needs further value judgements. In other words, the fact that, say,
household utilities are fully measurable and comparable does not imply that we know
how to aggregate them. The form of the SWF ultimately depends on the value
judgements we choose to make.
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