MSc Economic Evaluation in Health Care
WELFARE ECONOMICS
Topic 5. Resolving Pareto non-comparability:
(i) The compensation principle
(ii) Social Welfare Functions
Pareto non-comparability
The strength of the Pareto Principle lies in the fact that by making the weak assumption
of ordinality, the desirability of certain social states can be determined. The weakness lies
in the fact that the Pareto ranking of states is incomplete and that there exists Pareto noncomparability (remember that Pareto non-comparable states are ones in which some
households are made better off but others are made worse off in moving from one state to
another).
This leads to two main deficiencies with the Pareto Principle:
1. The Pareto Principle is unable to rank every Pareto optimal allocation superior to a
non-optimal allocation.
2. The Pareto Principle is unable to rank different Pareto optimal allocations relative to
one another.
(i) The compensation principle and the Pareto Principle
An attempt to overcome the first deficiency and increase the power of the Pareto
Principle is known as the compensation principle. Two different versions of the
compensation principle have been proposed:
1. The compensation principle proposed by Kaldor says that state a is preferred to state
b if, in state a, it is hypothetically possible to undertake a costless lump-sum
redistribution and achieve an allocation that is superior to state b according to the
Pareto Principle. In other words, if the gainers from the move to state a can
hypothetically compensate the losers so that everyone is better off then state a is
preferred to state b.
2. The compensation principle proposed by Hicks says that state a is preferred to state b
if, in state b, it is not hypothetically possible to undertake a costless lump-sum
redistribution and achieve an allocation that makes everyone as well off as in state a.
In other words, if the losers from the proposed move cannot hypothetically bribe the
gainers not to make the move then state a is preferred to state b.
Two examples of the two compensation principles:
1. Gainers gain £2 million, losers lose £1 million
2. Gainers gain £1 million, losers lose £2 million
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We will concentrate on the Kaldor compensation principle.
The compensation principle will always rank a Pareto optimal allocation superior to a
non-optimal allocation, thus negating the first deficiency of the Pareto Principle, outlined
above.
Problems with the compensation principle
1. The redistribution is hypothetical only (if it were a real distribution then the entire
exercise would be a direct application of the Pareto Principle).
2. The hypothetical redistribution is assumed to be costless.
3. The Pareto Principle is still unable to rank different Pareto optimal allocations relative
to one another (deficiency 2, outlined above). That is, there are still Pareto noncomparable states.
4. The compensation principle may lead to contradictions (as shown by the Scitovsky
Paradox), where a move from state a to state b may satisfy the compensation
principle, but so may a move from state b to state a.
(ii) Social welfare orderings and social welfare functions
The aim of welfare economics is to provide a framework that permits meaningful
statements to be made about the desirability of certain social states (allocations of
resources) to others. We would ideally like this social ordering to be:
1. complete, so that every social state can be compared to another; and
2. consistent, so that the ranking is reflexive and transitive.
A complete and consistent ranking of social states is called a social welfare ordering
(SWO). Just as with household preferences, if a continuity assumption is also made the
SWO can be represented by a social welfare function (SWF) that assigns a number to
each social state so that they might be ranked.
We cannot achieve an SWO without someone making value judgements about the
desirability of different social states (i.e. we have to come to some decision as to how
household preferences are to be aggregated). Value judgements are statements of ethics
that cannot be found to be true or false on the basis of factual evidence. The value
judgements found in an SWO may be weak (i.e. broadly accepted) or strong (i.e.
controversial). A weak value judgement that we have used up until now to rank social
states is the Pareto Principle. However, as we have seen this does not provide a complete
ranking of social states because some states are Pareto non-comparable. Therefore the
Pareto Principle alone does not achieve the desirable properties of an SWO.
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Another weak value judgement that is commonly made in welfare economics is
individualism, which requires that the preferences of the individual households should
matter when determining the SWO. Individualism imposes certain informational
requirements on the choice of an SWO. These requirements concern each household’s
preferences over social states (the measurability requirement), and how a given level of
utility for any household compares with that of another household (the comparability
requirement). The value judgements and informational requirements limit the set of SWO
possibilities from which the decision-maker can choose.
In summary, we are concerned with the ethical problem of resolving the conflict inherent
in finding the normative solution to the distribution problem. Specifically, some
households in the economy prefer state x to state y and others prefer to y to state x. Given
that household preferences should be taken into account, how should the policy-maker
aggregate such conflicting preferences into a single SWO?
The framework
A social state is identified by an allocation of N goods to the H households in the society.
That is, an allocation x is an N*H vector where x ih is the amount of good i consumed by
household h.
The objective is to derive an SWO of social states from the households’ orderings of
social states. The means of aggregating the household orderings into the SWO is called
the social choice rule (SCR).
The most general form of the SWF is called the Bergsonian SWF (sometimes called the
Bergson-Samuelson SWF). This is simply a function f(Uh) of the utility levels of all
households such that a higher value of the function is preferred to a lower one. For a twohousehold case the Bergsonian SWF is the unspecified function f(U1, U2). The function
itself may take any form, and this will depend on the nature of the value judgements used.
The Bergsonian SWF satisfies the following three properties:
1. Welfarism, which means that social welfare depends only on the utility levels of the
households.
2. The strong Pareto Principle, which means that the social welfare function is
increasing in each household’s utility level, ceteris paribus.
3. Convexity, which means that if one household is made better off then another
household must be made worse off to maintain the same level of social welfare. The
intensity of this trade-off depends on the desirability of inequality in utility across
households. The effect of this is that SWFs are convex to the origin (just like
indifference curves).
Note that SWFs enable us to rank projects even when there are gainers and losers.
Three common SWFs are:
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1. The utilitarian SWF, where social welfare is the equal-weighted sum of household
utilities.
2. The generalised utilitarian SWF, where social welfare is the weighted sum of
household utilities.
3. The maximin SWF, where social welfare is identified with the utility of the worst-off
household.
Each of these types of SWF has different informational requirements in terms of the
measurability and comparability of households’ utility.
We may also combine the social welfare function with the utility possibilities frontier
(UPF) that bounds the set of all feasible utility distributions that can be attained for a
fixed endowment of goods. The equilibrium point occurs where the highest attainable
SWF is tangential to the UPF. This equilibrium is also called the constrained bliss point
because it represents the combination of exchange, production and distribution that leads
to the maximum attainable social welfare. The bliss point is constrained by the available
resources.
Social Welfare Functions: comparability and measurability
We wish to examine the social welfare orderings (SWOs) that are possible with different
informational requirements.
SWO possibilities with non-comparability (NC)
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.
Ordinal scale (OS) measurability
OS measurability allows only the ordering or ranking of levels of utility by households
(e.g. only statements like “Alice prefers state x to state y” are possible). Dictatorship
(strong or lexicographic) is the only possible SWO (proved by the Arrow Possibility
Theorem - to be covered in Topic 6 to follow).
Cardinal scale (CS) measurability
CS measurability 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 (e.g. “Alice has an increase in 5 units of
utility in going from state x to state y”). However, for CS measurability combined with
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NC, strong or lexicographic dictatorship is still the only possible SWO (the logic of the
Arrow Possibility Theorem is unchanged with the introduction of CS).
Ratio scale (RS) measurability
RS gives information on the proportionate change in utility in going from one
indifference curve to another (e.g. “Alice’s utility doubles in going from x to y). This
cannot be combined with NC. No units of utility are now referred to (so there is no
problem of non-commensurate utility across households), so proportionate changes must
be fully comparable across households.
Absolute scale (AS) measurability
With AS measurability a unique real number is attached to each indifference curve of the
household. This cannot be combined with NC. AS measurability implies full
comparability of utility across households.
SWO possibilities with full comparability (FC)
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. In all instances, the strong and lexicographic
dictatorship admitted with OS measurability and NC are possible.
OS measurability
OS measurability means that utility levels only can be compared and ranked by the
individual household. The fact that utility levels are comparable across households (FC)
means that all households can be ranked by utility position for any social state. This now
permits SWO possibilities based on the utility positions of the households.
The ability to compare utility levels across households allows positional dictatorships,
where the SWO is dictated not by a particular household but by the preferences of a
household occupying a particular utility position. A common example is the Rawlsian
maximin SWO, where the social welfare function (SWF) is dictated by the preferences of
the household in the lowest utility position. According to maximin SWO, if the worst-off
household in state x is better off than the worst-off household in state y then state x is
preferred. Note that a maximax SWO is also possible (where if the best-off household in
state x is better off than the best-off household in state y then state x is preferred), or any
dictatorship by the nth well-off person. However, only by adding an additional equity
axiom (i.e. determining who the positional dictator should be) does one narrow the
positional dictatorship to one specific form.
Also possible is the positional lexicographic SWO, where there is a hierarchy of
households ranked according to their utility level in some way (though not necessarily
going from the worst-off household to the best-off household or vice versa), such that the
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SWO is dictated by the first household in the hierarchy providing it has a strict
preference. If the first household does not have a strict preference (i.e. if it is indifferent),
the strict preferences of the second household in the ranking dictate the SWO, etc..
Specific forms of the positional lexicographic SWO are the leximin SWO and the leximax
SWO. The leximin SWO is a positional lexicographic SWO where the positional
hierarchy runs from the worst-off to the best-off position. For the leximax case the
hierarchy runs in the opposite direction.
CS measurability
CS measurability means that utility levels and increments in utility can both be compared
by the individual household. By FC, comparisons can also be made across households. In
addition to statements like “Alice has a 5 unit increase in utility in going from state x to
state y” (arising from CS measurability) we can also make statements such as “the
increment in Alice’s utility is greater than the increment in Bob’s utility”. The planner
now has more information, and this increases the possible SWOs.
All SWOs arising from OS measurability with FC are available, but now additional
SWOs are admitted, specifically those relying on cross-household comparisons of
changes in utility. These include the utilitarian SWO, where social welfare is the
unweighted sum of household utilities, and the generalised utilitarian SWO, where social
welfare is the weighted sum of household utilities. These are admitted because we can
now say whether the (weighted) sum of the changes in utility in going from one social
state to another is positive or negative (e.g. “is the [weighted] sum of changes greater
than, less than or equal to zero?”). This is possible since (a+x)+(b+y)=(a+b)+(x+y).
The utilitarian SWOs may be combined with the positional dictatorship (e.g., maximin,
maximax, etc.). Generally we have social welfare W given by:
W = Wu + (Wd – Wu)
Where Wu is the (generalised) utilitarian form, Wd is a positional dictatorship form
(maximin, maximax, etc.) and  is a scalar between zero and one, representing the
importance of the positional dictator’s utility to society’s welfare.
The (generalised) utilitarian SWO may also be combined with the positional
lexicographic SWO.
RS measurability
With RS measurability as with OS measurability, utility levels or rankings can be
compared by the individual household. Additionally, proportional changes in utility can
be compared by the individual household, and (by FC) utility levels and proportional
changes in utility can be compared across households. This means that statements such as
“the proportional change in Alice’s utility is greater than the proportional change in
Bob’s utility” are possible. When utility is measurable on a ratio scale, still further SWOs
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to those available with OS measurability and FC are possible. These include the
Bernouilli-Nash SWO, where social welfare is the unweighted product of household
utilities, and the generalised Bernouilli-Nash SWO, where social welfare is the weighted
product of household utilities. These are admitted because we can now say whether the
(weighted) product of the proportional changes in utility in going from one social state to
another is positive or negative (e.g. “is the [weighted] product of changes greater than,
less than or equal to one?”). This is possible since ax*by=ab*xy.
AS measurability
When utility is measured on an absolute scale and FC is assumed, the SWO possibilities
are the widest possible. AS measurability and FC therefore permit the Bergsonian SWF,
where the social welfare function may take any form.
SWO possibilities with partial comparability (PC)
PC means that only some of the households’ information is available for comparisons
across households.
OS measurability
OS measurability allows only the ordering or ranking of levels of utility by households.
Therefore, there is only one type of information available, so PC is not possible. There
may be only NC or FC.
CS measurability
With CS measurability individual households can make comparisons of both their levels
of utility and of increments in their utility. With PC, either levels or rankings of utility
only can be compared across households (called level comparability, LC), or only the
increments in utility can be compared across households (called increment comparability,
IC).
With CS measurability and LC we are in exactly the same position as with OS and FC,
and therefore positional dictatorships and positional lexicographic SWOs are possible, as
well as strong and lexicographic dictatorships admitted with OS measurability and NC.
With CS measurability and IC the utilitarian and generalised utilitarian SWOs are
possible (as well as strong and lexicographic dictatorships) since they rely only on being
able to rank changes (increments) in utility across households. Positional dictatorships
and the positional lexicographic SWOs are not possible since they require LC.
RS measurability
With RS measurability individual households can make comparisons of both their levels
of utility and of proportional changes in their own utility. With PC, either levels of utility
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only can be compared across households (this is LC), or only the proportional changes in
utility can be compared across households (called proportion comparability, PrC).
With RS measurability no units of utility are now referred to, so proportionate changes
must be fully comparable across households (i.e. PrC must be achievable at least).
Since PrC is possible, admissible SWOs are the Bernouilli-Nash SWO and generalised
Bernouilli-Nash SWO, as well as strong and lexicographic dictatorships.
AS measurability
With AS measurability a unique real number is attached to each indifference curve of the
household. This cannot be combined with PC. AS measurability implies FC across
households.
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