Social Justice
Topics in Political Economy I
Ana Fernandes
University of Bern
Fall 2011
• Income redistribution involves taking resources from some people and giv-
ing those to other people
— By nature, this cannot be a Pareto-improving policy:
— The givers are worse off, those who receive income are better off
— Involuntary taxes are also contrary to the natural right of possession
• How, then, are involuntary income transfers justified?
— And, if these transfers are justified, how much income should be trans-
ferred among people?
• We will provide answers to these questions from the point of view of social
justice, while specifying the definition of social justice used
— As before, efficiency and social justice will often be conflicting goals
1 Social Justice and Insurance
• We will consider social justice from the point of view of ex-post equality
• We can relate this definition of social justice to the concept of insurance:
Social justice is achieved when people are provided with complete insurance
whereby people who fare well in life fully compensate those who do not
fare well — because of the full compensation, after the income transfers
predicated on insurance, all people have equal ex-post outcomes
• Insurance is thus the ex-ante sharing of risk and also the ex-post sharing
of incomes
— Complete insurance (leading to complete ex-post equality) means that
people are completely compensated by for having low incomes by people
with high incomes
— As a result, everyone has the same income ex-post
• We begin without a government, in circumstances where people do not
know their future income
— With future income uncertain, insurance protects against being desti-
tute or having low income
• To describe uncertain personal income, we divide future income in two
parts, one known and one unknown
— The known part is y A
— Actual future income, denoted y, is the sum of y A and a random
component of income, µ:
y = yA + µ
— The distribution of µ determines the pre-tax income distribution in a
society
• For simplicity, we will consider the case in which the random part of income
µ can take only two values
µ ∈ {µL, µH } , µH > 0 > µL
— Thus, people for whom µ = µH have high income, and those for whom
µ = µL have low income
— We assume that, in the case of receiving the negative income draw µL,
people still have positive income:
y A > y A + µL > 0
— The probabilities of the two µ-values are identical and equal to 0.5
— Further, the two income shocks are symmetric:
µH = −µL
• Because of the symmetry, the expected value of income Ey equals y A,
the nonrandom component: ³ ´ ³ ´
Ey = 0.5 y A + µH + 0.5 y A + µL
= y A + 0.5 (µH + µL) = y A
• Thus, people know with certainty the income Ey that they will have on
average
— But they do not know whether they will receive the high income draw
or the low one
• For the moment, we will take the random component of income µ to be
exogenous to personal actions
— This means that the realizations of µ are the exclusive consequence of
fate or luck
— In particular, they do not depend on nor are affected by actions of the
individuals earning the income
• Later on, the random component of income µ will be endogenous to per-
sonal actions
— This means that individuals influence the likelihood that µH (or µL)
takes place
— This influence is exerted through effort in seeking to secure a good
outcome or in avoiding a bad one
• Personal income is then subject to moral hazard
— Moral hazard characterizes a relationship between two parties under
information asymmetries that develop after the stage of contracting
— Further, under moral hazard, one of the parties can undertake “hidden”
actions that influence contractual outcomes
• For example, when an insurance company and an individual sign an insur-
ance contract, they both know that the likelihood that the car is not stolen
depends on how careful the driver is (not leaving the door unlocked, for
example)
— At the time of signing the contract, they have identical information
— But after that, only the insured knows whether or not s/he took rea-
sonable caution in looking after the car
• Under endogenous effort, moral hazard occurs because the presence of
insurance influences personal effort decisions
— There is no moral hazard if income is the outcome of fate or luck, as
we now consider
1.1 Uncertain Incomes and the Demand for Insurance
• Good or bad luck — expressed in the good and bad income realizations
µH > µL — can originate in a multitude of situations
— People are born with different abilities that later materialize into an
ability to earn higher or lower incomes
— There is luck also in whether or not your family encourages you to
study — or even gives you financial support for studying
— People may have to care for small children in the absence of a partner
— Individuals may face unemployment
— Genetic factors affect how good a health one has and bad luck with
health also affects the ability to earn income
— ...
A Veil of Ignorance
• To express the uncertainties over which people have no control, we use the
metaphor of a veil of ignorance:
The veil of ignorance is a metaphor for conditions under which people know
nothing about their future selves
• This metaphor is of course counterfactual because people know who they
are
— By resorting to this metaphor we nonetheless consider the decisions
that people would make under conditions of anonymity behind a veil
of ignorance in which they know nothing about their future selves — or
as if they had not yet been born
• People behind a veil of ignorance do know the income distribution of the
society into which they will be born:
— They know that half of the people will have high income whereas the
other half will have low income
— But they ignore who they will be within society
Risk-aversion and Insurance
• People are in general risk-averse: they are prepared to pay to avoid risk
— For example, individuals normally would be willing to pay to avoid
facing a fair gamble (where they stand to win or lose a sum of money
with equal probability)
— Someone willing to pay nothing to avoid a fair gamble is risk-neutral
• Suppose we offer someone a lottery where s/he can win or lose CHF1000
with equal probability
— This lottery has an expected value of zero
— A risk-averse person would be willing to pay to avoid it
— The more s/he were prepared to pay, the higher the degree of risk-
aversion
• A person who is prepared to pay to avoid a fair gamble values additions to
income less than income losses
— Therefore, for risk-averse people, the marginal utility of income is
declining
• In figure 7.2, individual utility U (y) increases with personal income y
— The slope of U (·) is the marginal utility of income and is portrayed as
decreasing, indicating risk-aversion
— The fair gamble offered is that of getting yL ≡ y A + µL or the high
income realization yH ≡ y A + µH with equal probability
— The utility of getting yL is U (yL) and that of getting yH is U (yH )
— Expected utility from the gamble EU is the average of these two util-
ities
• The expected value of income under this lottery is y A
— Because we are considering risk-averse individuals, it follows that
³ ´
EU 0 in order to avoid the gamble
— The fact that ρ is positive confirms risk-aversion
Ex-ante Equality and Ex-post Inequality
• People behind a veil of ignorance are equal ex-ante:
— Everyone has the same utility function and faces the same income
uncertainty about future income
— People will not be equal ex-post after emerging from behind the veil
of ignorance because some people will experience µH while others will
experience µL
— There is ex-ante equality but ex-post inequality
Insurance Markets
• An insurance company is prepared to bear the risk associated with income
uncertainty
— The insurance company provides everyone with certain income y ≥
¯
yc so that individual incomes no longer depend on whether a person
experiences µH or µL
• Behind the veil of ignorance and therefore before knowing whether they
will be lucky or unlucky, people agree to transfer their income, yH or yL,
in exchange for y with certainty
¯
— In bearing all risk and providing income y to everybody with certainty,
¯
the insurance company provides complete insurance
— Complete insurance spreads risk and results in ex-post income equality:
Complete insurance makes personal income independent of whether a
person experiences a bad or good outcome
• The insurance company receives an average payment of y A = Ey from
each person
— Because the insurance company deals with a large number of people,
we invoke a law of large numbers and assume that the income received
equals its expectation
— That is, under the law of large numbers, the insurance company will
effectively receive income Ey per person
— Its profits per person will then be the difference between Ey and the
certain income paid out, y¯
— Individuals will accept this insurance contract as long as y ≥ yc
¯
— Competition in the insurance market will drive y close to Ey and above
¯
yc
— The payment y might be lower than Ey (the value of y that would drive
¯ ¯
profits of the insurers to zero) due to the costs of providing insurance
(administrative, etc.)
• Risk-averse individuals are strictly better off under the insurance arrange-
ment as long as y > yc
¯
— If competition in the insurance market drives y all the way up to Ey,
¯
the gain for each individual is the income ρ
Mutual Risk-Sharing Contracts
• Rather than use the insurance market to buy insurance, risk-averse individ-
uals could agree to a contract whereby they cooperate to pool (or share)
risk associated with personal incomes
— They would turn in all their income to one person who would then pay
out some constant amount to all participants in this arrangement
• This exactly duplicates what the insurance company does:
— If the income paid out is Ey, the gain per person is ρ
Efficiency and Social Justice Through Insurance
• The voluntary pooling of risk through either competitive insurance markets
or mutual risk-sharing arrangements is efficient:
— Insurance is complete and there is no additional uncertainty to eliminate
or pool
— Further, Pareto improvement takes place because all people gain (at
most) ρ from the risk-sharing
• People who are equal ex-ante behind the veil of ignorance are also equal
ex-post
— Complete insurance gives them y independently of whether or not they
¯
experienced µH or µL
• With social justice defined as ex-post equality after the realization of µ:
Insurance markets efficiently ensure social justice
Revealed Information
• People seek insurance because of uncertainty or risk
— That is, at the time of signing an insurance contract, they do not know
whether they will have high or low income
• The more information they have about the actual realization of their future
income, the narrower the possibilities for insurance or pooling risk
— For example, if you knew that you were going to have colon cancer for
sure, and this information were public, in most countries no insurance
company would give you health insurance
— We conclude that
Revealed personal information is an impediment to insurance
Social Insurance
• Behind a veil of ignorance, personal information is not revealed and insur-
ance companies would be willing to provide insurance contracts
— Insurance companies would then agree to provide certain income y after
¯
people see their income realizations
• However, in real life there are no insurance companies behind the veil of
ignorance that could write contracts with people also behind the veil
— Stated differently, the people who would sign insurance contracts are
already born and knowledgeable about their income
— All the relevant information has already been revealed
• It is generally not possible for someone to make binding legal promises on
our behalf
— That is, our parents could not write contracts stipulating what we
would do with our income once born:
— Enforceable contracts are only those written by people in possession of
their faculties (e.g. already born)
• Thus, even if contracts behind the veil of ignorance could be signed (for
example by our parents), they would not be enforceable
— As such, there is no additional scope for insurance from the point of
view of an insurance company:
— The only people who would show up to sign the contract would be
those with low income
• The only way to provide insurance against income luck is to introduce
government
— After µ is determined and people know whether they have high or low
incomes (e.g. when they are born), a government takes the respon-
sibility of enforcing compulsory social insurance as if the insurance
contract had been determined when people were anonymous behind a
veil of ignorance
Insurance through government is social insurance because everyone in
society is included in the social-insurance contract and participation is
compulsory
• Through the legal monopoly on coercion and the corresponding ability
to compel payment of taxes, governments ensure that the people who
experienced income µH compensate those with µL
— This is the contract to which risk-averse individuals behind the veil of
ignorance would have agreed when confronting risk about their future
incomes
2 Social Welfare Functions and Social Insurance
Contracts
The choice of a social-insurance contract behind the veil of ignorance — to be
implemented through government when people will know who they are — is
correspondingly the choice of a social welfare function
A Social Welfare Function
• When we examined efficiency, we looked at outcomes that maximized net
social surplus W = B − C
— Disregarding the cost side for the moment, we were concerned with the
maximization of the sum across consumers of their individual surpluses
n
X
B= Bi
i=1
— Because the coefficient of each Bi terms is unity, this means that we
weighted the welfare of individual consumers equally
• There is no reason why this would have to be so: society might value the
utility of particular individuals more than that of others
• In the pursuit of social justice and the optimal social-insurance contract,
we consider a more general social welfare function W
— This function includes general judgements about the distribution of
benefits and costs across people in the population
— Thus, it need no longer be the case that individuals are weighed sym-
metrically
• With n people in society, a social welfare function measures total welfare
W of the population as:
W = f (U1, U2, · · · , Un) (1)
— Total social welfare W thus depends on the values of individual utilities
Ui in the population
— Because social welfare as expressed here depends only on personal
utilities, the social welfare function is described as utilitarian
• Welfare function W is general enough to include concern with producers
— Companies — the producers of output — are owned by shareholders
whose utility is contemplated in W (·)
— Thus, the “cost-side” of the economy is also included in W by valuing
the utilities/profits of shareholders
— Beyond the impact of higher profits on the utility of a company’s share-
holders, efficient production would be a constraint in the maximization
of W (·)
— Because we are considering exogenous income processes, we do not
have to worry about the previous item
Social Welfare and Pareto Improvements
• Social welfare function W (·) is such that Pareto improvements increase
social welfare
— That is, if the utility of a person Ui increases and everyone else’s stays
constant:
∂W
= ωi > 0, i = 1, . . . , n (2)
∂Ui
— The number ωi is an individual’s social weight in the measurement of
social welfare
— The result in (2) says that everyone has positive weight in the mea-
surement of society’s welfare
— If any one person is better off and no one is worse off, society is better
off
• Figure 7.3a shows a social welfare function for a society composed of two
people
— Social welfare is constant along the indifference curves W1 and W2,
where W2 indicates higher social welfare
— Consistent with (2), Pareto-improving change from point 1 to 2 in-
creases social welfare
— However, Pareto improvement is not necessary for social welfare to
increase
— A move from 1 to 3 still increases social welfare even though person 1
is now worse off than before
— Society is also indifferent between points 1 and 4 even though person
1 is better off at 1 and person 2 is better off at 4
Interpersonal Comparisons of Utility
• In figure 7.3a, interpersonal comparisons of utility are being made
— An interpersonal comparison of utilities is a judgement about compar-
ative values of different people’s utilities
— Interpersonal comparisons allow judgements about how changes in dif-
ferent people’s utilities affect social welfare
— This is what allows us to draw indifference curves
• We are using a common utility function across people to compare the
implications of changes in welfare in the population
— We are also assuming that the measurement of utility is cardinal
— That is, that the difference in utilities across two different situations
has meaning (like differences in height or temperature)
• When the measurement of utility is ordinal, different utility numbers (over
high and low income, for example) simply provide rankings of utility
— With ordinal utility, we could not compare the consequences of distrib-
uting income in different ways among people
— When Pareto-improvement takes place, however, we could nonetheless
conclude that welfare had improved with ordinal utility
• Because the measurement of utility is cardinal, we can compare the impli-
cations of taking income from person 1 and giving it to person 2
— This income redistribution causes a reduction in the utility of person 1
and an increase in that of person 2
— The cardinality of utility makes these changes comparable
• The common utility function used rules out the possibility of a judgement
that two people with the same income benefit differently or have different
utilities
— A common utility function preempts privilege or prejudice that would
be present if some people were regarded as being capable of benefiting
more from income than others
— Of course it also prevents differencing across people in situations when
that were adequate (think of a very ill person’s marginal utility of
income...)
Anonymity
• Privilege or prejudice is also preempted by anonymity in the social welfare
function
— Figure 7.3b shows a social welfare contour that is symmetric around
the 45◦ line
— The symmetry ensures that the shape of the social contour is indepen-
dent of the identities of the people whose utilities are measured on the
axis
— It does not matter, from the viewpoint of treatment by the social
welfare function whether someone is (or will be) person 1 or person 2
Distribution of Predetermined Income
• The social welfare function can be used to describe a decision behind the
veil of ignorance about how a predetermined amount of income Y will be
divided once people have emerged from behind the veil
— Person 1 will receive income y1 while person 2 will receive y2 with
y1 + y2 = Y
— The utilities of the two people from the division of Y into y1 and y2
evaluated at the common utility function are
U1 = U (y1) , U2 = U (y2)
• Figure 7.4a shows the utility possibilities frontier
— It maps all the utility possibilities for the two people in the economy
from alternative distributions of Y
• All points on this line (all distributions of Y among the two people) are
Pareto efficient:
— At any point along the frontier, no person can be made better off
without making the other worse off
— Movements along the frontier are not Pareto improving change: one
person must be made necessarily worse off for the other to gain
• If all income is given to person 1, society is at point S; if Y is in turn given
to person 2, point V is attained
— Because utility functions are identical across individuals, these utilities
are also identical: OS = OV
— The remainder points on the frontier result from dividing income be-
tween the two people
• The concavity of the frontier is due to declining marginal utility from in-
come
— Reductions in person 1’s utility accomplish smaller and smaller incre-
ments in the utility of person 2
• Because of identical utility functions and diminishing marginal utility of
income, the frontier is symmetrically concave around the 45◦ line
— Maximal social welfare W2 is achieved at point E on the 45◦ line by
equal division of income
— It achieves ex-post equality of income and utility