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Chapter Three
Measurements and Theories of Welfare
Economics
Welfare Economics for MSc Students of PPM

2
I Welfare of the Household
Objective
 In welfare economics we are interested in being able
to rank different social states (or allocations of
resources).
 Society’s welfare ultimately depends on the welfare
of its constituent households.
 Therefore to make value judgements of the
desirability of different social states to society we
need to have a theory of household behaviour.
 Put another way, we are interested in how
households rank different social states.

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Two key assumptions
 Welfarism = social welfare depends only on the
welfare of households, which depends on the bundle
of commodities consumed.
 Non-paternalism = individualism = the welfare of
the household must correspond with the household’s
own view of its welfare, or at least be consistent with
the household’s preferences = social welfare must
respect household preferences

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Part I: Welfare of the Household
Preference orderings over alternative bundles of
commodities
 Simple economic model = a household must choose
how to spend its income on different goods. This is
the household’s utility maximisation problem.
 The household chooses among available bundles of
commodities on the basis of its preferences.
Commodity bundles
 x = (x1, x2, …., xn ) [1]

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Utility functions
 The utility function depicts the relationship between
the level of satisfaction reached by a household and
the amounts of different commodities it consumes.
 U = u(x1, x2, …., xn ) [2]
 The utility function can be used to compare any
number of commodity bundles.
Indifference curves
 Indifference curves provide a locus of points
representing combinations of two commodities (x1
and x2) between which the consumer is indifferent
(i.e. has equal utility).
 There are an infinite number of indifference curves,
each corresponding to a given level of utility, and
they cannot intersect.

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Marginal rate of substitution (MRS)
 MRS is the amount of good y that must be given up
per unit of x gained if the consumer is to remain at
the same level of utility
 MRS is equal to the slope of the indifference curve
at any one point
 Algebraically, MRS = [3]
 Convexity of indifference curves represents a
diminishing MRS
dx
dy


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The budget set and budget constraint
 A household receives an income y and faces a set of
prices p for commodities given by
 p = p(p1, p2, …., pn ) [4]
for each good x = (x1, x2, …., xn )
 The budget constraint is given by
[5]
 The budget set is the set of different bundles that it
is feasible to consume so that
[6]
y
x
p
n
i
i
i 

1
y
x
p
n
1
i
i
i 



Budget Set and Constraint for Two Commodities
x2
x1
Budget constraint is
p1x1 + p2x2 = y.
y/p1
Budget
Set
the collection
of all affordable bundles.
y/p2

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 A formal statement of the utility maximisation
problem
 Maximise U = u(x1, x2, …., xn ) [7]
Subject to
 The solution to the utility maximisation problem
requires that the MRS between goods must equal
their price ratio. E.g.
MRS = [8]
y
x
p
n
1
i
i
i 


1
2
dx
dx

2
1
p
p


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The demand function
 Each time the budget constraint changes (due to
changes in prices or incomes) there will be a new
equilibrium. This can be used to derive a relationship
between the optimum amount of a good purchased
(i.e. quantity demanded) and prices and income.
 The demand function is given by
xi = x(p1, p2, …, pn, y) [9]
 The demand function can be thought of as the
solution to the household utility maximisation
problem.
 Given the prices and income facing the household
the demand functions determine the bundles of
goods that yield the highest value of the utility
function.

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The indirect utility function
 If those demand functions are substituted into the
utility function, the results is the indirect utility
function, which shows the maximum utility that can
be achieved for any set of prices and income.
v(p, y) = v[x1(p, y), x2(p, y), …., xn(p, y)] [10]

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II. INTRODUCTION TO WELFARE MEASUREMENTS
 Welfare economics focuses on using resources optimally
to achieve the maximum well-being for the individuals
in society.
 But, unfortunately, agreement cannot always be
reached on what is optimal.
 Ethical Assumptions
 They are ethical assumptions or value judgments with
which economists may legitimately disagree.
 Two of these ethical assumptions are, however,
sufficiently widely accepted that they provide the
foundations for a large part of applied welfare
economics and policy evaluation. They are that
 the welfare status of society must be judged solely by
the members of society (also called fundamental ethical
postulate or the principle of individualism) (Quirk and
Saposnik 1968, p. 104); and

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 the notion that society is better off if any member of
society is made better off without making anyone else
worse off (Pareto principle after the founder of the
principle, Vilfredo Pareto (1896)).
 Measurement issues
 A difficulty with welfare economics is that economic
‘welfare’ is not an observable variable like the number
of machines, houses, market prices or profits.
 The economic welfare status of an individual is formally
represented by his or her utility level, a term generally
used synonymously with happiness or satisfaction.
 Utility is measured by ‘utils’, which is an imaginary and
not a metric unit.
 But one cannot measure the increase in utility by
additional utils obtained from consumption.

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 Consequently, positive economics assumes that utility is
only ordinally measurable, however, normative
economics seeks to measure welfare cardinally. Hence,
 Ordinality = the ability to (only) rank alternatives
according to the utility they provide
 Cardinality = indicate the magnitude of the change in
utility in moving from one alternative to another (like a
temperature scale)
 A cardinal system specifies exactly how much utility
each affected individual would gain or lose from a
proposed policy decision.
 Such information would surely be helpful to those
concerned with determining the maximum well-being
for society and would simplify the subject of welfare
economics substantially.

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 Measurability of utility, however, is not sufficient to
determine optimal social choices.
 The point is that, even if utility were measurable, there
would still be the problem of how to weight individuals:
welfare weightings
 No objective way exists for solving this problem of
interpersonal comparisons.
 Since utility is not measurable, an alternative measure
must be chosen.

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III The Pareto Principle and Pareto Optimality
Background
 So far we have been concerned with measuring the
welfare of individual households who are concerned
with maximizing their utility subject to a budget
constraint.
 In practice most economies are populated by
millions of households, most of whom have different
tastes and different budget constraints.
 Somehow we need to distil welfare statements about
an event from its effects on millions of different
households.
 One possible solution is to aggregate household
welfare in a very direct manner by simply adding the
utility of each household together to get a total
utility score for a particular state.

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 In this way the welfare effects of a particular public
policy could be measured by whether or not the sum
of utilities was raised or lowered. This type of
aggregation requires two very strong and restrictive
assumptions:
1) Cardinality (so that utilities can be measured on a
scale which says by how much a household prefers
one state to another); and,
2) Comparability among household utilities (so that
adding household utilities is possible, like adding
apples and apples rather than apples and oranges
which is meaningless)
 Vilfredo Pareto, a great Italian economist was the
first to define concepts of society’s welfare without
having to invoke cardinality and comparability.

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The Pareto Principle
 Any allocation of goods and services across the
many households in the economy is referred to as a
state of the economy.
 Associated with each state is an H element utility
vector (u1, u2, u3, …uH) that gives the level of utility
for every household where H is the number of
households in the economy.
 The Pareto Principle allows us to compare the social
welfare of two states by determining whether the
utility vector of one state dominates that of another
state: weak vs strong pareto criterion

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1) According to the weak Pareto criterion, if the utility of
every household is higher in state x than state y then
state x yields a higher level of societal welfare than
state y and is preferred.
2) According to the strong Pareto criterion, if the utilities
of some households are higher in state x than in state
y and the utility of no household is lower in state x
than state y then state x yields a higher level of
societal welfare than state y and is preferred.
 NB: these Pareto criteria require only very weak
assumptions about the nature of household utilities:
I. only ordinality of utility is needed (not cardinality); &,
II. we do not need to make comparisons about the
utilities of different households, so that units and
levels of utilities need not be comparable across
households.

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Pareto optimality
 If state x allows a welfare improvement over state y
according to the Pareto criterion, then state x is said
to be Pareto superior to state y, and state y is said
to be Pareto inferior to state x.
 If all households enjoy the same level of utility in
state x and state y then states x and y are Pareto
indifferent.
 If state x is neither Pareto superior, nor Pareto
inferior, not Pareto indifferent to state y then states
x and y are Pareto non-comparable. 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.

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 A feasible state is one that can be achieved given
the economy’s resource constraints.
 Any feasible state for which no feasible Pareto
superior state exists (i.e. there is no scope for Pareto
improvement) is said to be Pareto optimal.
 If a state is Pareto optimal then there is no change
that can be made in the economy, given current
resource constraints, that can make any household
better off without making another household worse
off.
 There are many Pareto optimal states for a set of
feasible states and all Pareto optimal states are
Pareto non-comparable.

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Problems with the Pareto Principle
1. The Pareto ranking of states is incomplete (there
exists Pareto non-comparability);
2. The Pareto Principle is neutral to the distribution of
utility – a state of extreme utility disparity can be
superior to one of utility equality providing
somebody is better off and nobody is worse off in
utility terms; and,
3. The Pareto Principle can conflict with liberalism.

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IV COMPENSATING AND EQUIVALENT VARIATIONS
 The two most widely used willingess-to-pay welfare
measures proposed by John R. Hicks (1943, 1956) are
the compensating variation and the equivalent
variation.
 The motivation for the Hicksian measures is that an
observable alternative for measuring the intensities of
preferences of an individual for one situation versus
another is the amount of money the individual is willing
to pay or willing to accept to move from one situation to
another.
 This principle has become a foundation for modern
applied welfare economics.
 The two most important WTP measures are
compensating and equivalent variations.

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 Compensating variation is the amount of money which,
when taken away from an individual after an economic
change, leaves the person just as well off as before.
 For a welfare gain, it is the maximum amount that the
person would be willing to pay for the change.
 For a welfare loss, it is the negative of the minimum
amount that the person would require as compensation
for the change.

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 Equivalent variation is the amount of money paid to an
individual which, if an economic change does not
happen, leaves the individual just as well off as if the
change had occurred.
 For a welfare gain, it is the minimum compensation that
the person would need to forgo the change.
 For a welfare loss, it is the negative of the maximum
amount that the individual would be willing to pay to
avoid the change.

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