Individual Deservedness in Redistribution of Earned Incomes
Mingli Zheng*
Department of Economics
Faculty of Social Sciences and Humanities
University of Macau
Macau
(Email: [email protected])
Abstract
Savage framework on the consistency of preference in resource allocations
leads to the existence of weights in the social welfare function. For earned income
redistribution, the weights may depend on the information in the production stage. If
the weights in the social welfare function are positively related to the earned incomes,
the weights can be interpreted as individual deservedness. The value judgment of
deservedness will reward those with high earned incomes and punish those with low
earned incomes, and it can provide more incentive to work. We also discuss the
interaction of value judgments of deservedness and inequality aversion.
Keywords: Deservedness; inequality aversion; equity; Savage axioms; distributive
justice.
JEL classification: D63; C72; I31; H20.
*
I am grateful for discussions and comments from James Mirrlees, Dominique Thon and Porapakkarm
Ponpopje.
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1. Introduction
Recently there have been a lot of interests in the study of individual responsibility and
deservedness in the distributive justice, see Cappelen et al. (2007), Fleurbaey (2008). Among
different ideas of justice, one idea is that individuals should be rewarded for factors under their
control, and inequalities for which individuals can be held responsible are not unjust. The idea that
individuals ought to get what they deserve is also widely discussed in ethics theory.
In much of the economic analysis, the value judgment of deservedness is not an intrinsic
principle in its own right. For the widely used utilitarian or Rawlsian maximin social welfare
function, there is no place for individual responsibility and individual deservedness. Under such
utility functions, resources should be allocated equally among individuals in the first best choice.
Though we observed income inequality in real life, such inequality is not explained by individual
deservedness but by the incentive constraint under imperfect information.
However, public opinion and empirical evidence show that the value judgment of individual
deservedness does play an important role in real life. Lockean theory believes that an individual
deserves the property entitlement that is obtained through his expenditure of effort. Equity theory
argues that people allocates payments so as to equalize individual ratios between payments and
intrinsic inputs (Adams (1964)). Recent laboratory results on distributive justice also show the
existence of value judgment of responsibility and deservedness, see Hoffman and Spitzer (1985),
Rutstrom and Williams (2000), Konow (2001), Cappelen et al. (2007). Experimental subjects
behave with the consideration of deservedness or entitlement. Therefore, the value judgment of
individual deservedness might be important in its own sake rather than just a means of other goal,
and it will be useful to incorporate the value judgment of deservedness explicitly in the social
welfare function.
Researchers have suggested incorporating individual deservedness and responsibility in the
social welfare functions. For example, Arneson (2000) proposes that the social value to a given
individual is greater the more deserve he is. This suggests a more deserving individual should have
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a larger weight in the social welfare function (Fleurbaey (2008)). The weight in the social welfare
function is used to describe the value judgments in the legal decision making in Zheng and Anwar
(2006).
Apart from individual deservedness, other different value judgments in distributive justice
are discussed in the literature, including egalitarianism, inequality aversion, etc. These different
value judgments are not mutually exclusive, and it is possible to put them into a balanced single
social objective. For example, experiment data is best explained by the combination of utilitarian
and maximin social welfare function in Engleman and Strobel (2004), see also Konow (2001) for
combining different justice principle in a single function.
This paper uses the abstract Savage framework to provide an explicit way to incorporate the
value judgment of individual deservedness and inequality aversion into a social welfare function in
the form as W  i1 i u ( xi ) . The weight  i represents the decision maker’s subjective value
n
judgment about the social value of individual i (which can be interpreted as deservedness in earned
income distribution) and u represents inequality aversion of the decision maker. It is not interesting
to assume exogenously such a form of function, since there are many other forms of functions with
interesting interpretations. This paper uses the Savage framework about the consistency of
preference. Any idea of fairness and justice in distributive justice certainly requires some kind of
consistency. if some consistency requirements in resource allocations are satisfied, then the
preference can be represented by a social welfare function in the form of W  i1 i u ( xi ) .
n
The use of weights in the social welfare function already existed in literature. For example,
Brent (1984) surveyed the use of distributional weights in cost-benefit analysis. Some research used
weighted utilitarian to assign higher weights to individuals with low initial utility levels. Social
welfare function in the form of a weighted utilitarianism is also discussed in the non-transferable
utility cooperative game (see Aumann and Kurz (1977)). The abstract model in this paper suggests
the existence of individual weights in real world distributive justice. In real life, when information
about contributions, efforts, needs or other individual characteristics is available, the value
2
judgments can be very different with the case when such information is not available. One channel
for such information to take effect is that the weights in the social welfare function depend on the
available information, and the weights can be interpreted differently according to contextual detail
of the decision making. In this paper we discuss redistribution of earned income with a production
stage before the redistribution, and the information in the production stage can be used in the
redistribution (see Cappelene et al. (2007)). When the weights are positively related to the earned
income, we interpret the weights as individual deservedness. It is also possible that individuals with
higher social status are assigned higher weight and the weight can be interpreted as social power of
the individual.
As an application, this paper explores the implication of the role of value judgment of
individual deservedness in the redistribution of earned incomes. The weight  i of an individual in
the social welfare function is assumed to be proportional to his earned income. This is similar to
equity theory (Adams (1964)) in which the reward is proportional to intrinsic input. The value
judgment of individual deservedness will reward those with high earned income and increase the
incentive to work. When the effect of individual deservedness dominates the effect of inequality
aversion, income can transfer from the individual with lower earned income to those with high
earned income and incentive to work will increase.
The rest of this paper is organized as follows. Section 2 discusses briefly the social welfare
function using Savage framework and discusses the individual deservedness. Section 3 discusses
the implication of individual deservedness on earned income redistribution. Section 4 discusses the
incentive of labor supply when individual deservedness is considered.
2. Deriving Social Welfare Function Incorporating Deservedness from Savage Framework
The evaluation of distributive justice is shaped by cultural, the type of goods or burdens
being distributed, etc. In this paper, we will not consider the distributive justice in terms of the
distribution of individual utilities as in welfarist approach. Sen (1977) provides arguments in favor
3
of broader welfare concept which does not only include information about utilities. This paper
adopts a non-welfarist approach and the consequence space consists of resource allocations among
individuals. In collective decision making, there is a debate on whether there is an entity that has
consistent preference over resource allocations. To avoid such debate, it is assumed in this paper
that the preference over resource allocations is the ‘social preference’ of an individual when he is in
the imaginary position to make choices over resource allocations. Such an individual may be an
impartial spectator (Konow (2009)) or a stakeholder. Similar view is widely adopted in the study of
distributive justice.
The decision maker may allocate more resource to the poor individuals; or he may allocate
more resource to the rich individuals or those who bribed him, etc. Can we find out the social
welfare function and value judgments that are used in the decision making?
The question of preference representation can be studied with Savage framework. Savage
theorem states that consistent preference can be represented by a certain function. It uses abstract
state space  and acts space F , and preference is defined as binary relation over acts, while an act
is defined as a function from the state space to the consequences. The acts space F consists of all
acts (functions from state space to consequence space) x :   x( ) . Therefore, an act assigns a
consequence for each state  . The binary relation x  y for two acts x and y means that the
decision maker prefers x to y . If the state space is the set of individuals so that each  represents
an individual, and the consequence is the amount of resource, then an act x :   x( ) is a resource
allocation which specifies the resource assigned to each individual. The resource of individual  is
x( ) . When a set of axioms about the consistency of preferences over acts are satisfied (which we
will discuss below), there exists a unique weight function  with total weight equals to 1 and a
utility function u (unique up to an affine transformation) such that for two acts x and y , x  y if
and only if
u( x( ))( )  u( y( ))( ) .
i
i
i
i
(The discrete case is used for simplicity.)
Therefore, the preference can be uniquely represented by a social welfare function
W ( x)  u( x(i ))  (i ) .
4
For discrete case, we can write x(i ) as xi and write the weight  (i ) as  i , and the
social welfare function can be written as W ( x)   i u( xi ) . The quantity  i is the weight assigned
to individual i in the decision making. This implies that the same amount of resource allocated to
different individuals may have different social value. In the simple problem of optimal resource
allocation among n individuals with budget constraint

n
i 1 i
x  w , if u is strictly concave,
 i   j implies xi  x j by the first order condition u ' ( xi ) / u ' ( x j )   j /  i . The decision maker
allocates more resource to the individual with higher weight. The interpretation of this weight
depends on the contextual detail of the decision making. In the later part of this paper, we discuss
redistribution of earned income and interpret the weight as the value judgment of individual
deservedness if income increases with earned income. In a different context, this weight can be
explained as the social power of the individual if resources are allocated according to social status.
The function u can represent the decision maker’s attitude towards inequality aversion.
Maximization of the social welfare function also implies some kind of efficiency principle.
Therefore, such a social welfare function actually incorporates the widely discussed principle of
deservedness, inequality aversion and efficiency. Incorporating several distributive justice
principles in a single function is also discussed in Konow (2001), but in a different way from this
paper.
The above mentioned application of Savage framework is rarely seen in the literature.
Savage framework is known for its application in individual decision making with uncertainty. The
state space consists of possible states of the world of uncertainty, and an act is an action which leads
to a possible outcome in each state. If the Savage axioms are satisfied, the preference of an
individual can be represented by an expected utility U ( f )  u( f (i ))  (i ) , where the weight
 i represents the decision maker’s subjective probability belief of state  i , and the function u
represents his attitude towards risk. Individual preference can be represented completely by the
belief of probability and attitude towards risk. The use of the same mathematical model in the two
5
cases suggests that the role of value judgments of individual deservedness and inequality aversion
in distributive justice may play the role parallel to the role of probability belief and risk aversion in
the individual decision over uncertainty.
To see whether Savage axioms are appropriate for decision making over resource
allocations, we discuss these axioms briefly. The axioms are a set of consistency requirements on
the preference relation. Among many versions, we utilizes Shaffer’s version (Schaffer (1986)),
which is relatively concise and intuitive.
In Savage framework, if there are n individuals with   {1,2,, n} , a resource allocation
x can be considered as a vector ( x1 ,, xn ) , specifying the resource of each individuals. The act
space F is the space of all possible resource allocations among individuals. If the decision maker is
indifferent between two allocations x and y , then it is denoted as x  y . The axioms involves the
restriction of an allocation among subgroups. For any resource allocation x in F and any subgroup
of individuals A of  , x A is the restriction of x to the subset A , representing an allocation among
individuals in subgroup A . For example, if A  (1,2,3) , then x A  ( x1 , x2 , x3 ) , which is a resource
allocation among three individuals labeled 1, 2 and 3. A subset A of  is null if x  y whenever
x Ac  y Ac , where Ac denotes individuals out of A . The resource allocation among null subgroup
has no influence on the overall preferences.
Given a subset A of individuals and two resource allocations p and q among individuals in
A , we write p  q if x  y for every pair x and y in F such that x A  p , y A  q and x Ac  y Ac .
Let [c] denote the equal allocation in which every individual receives the same amount c of
resource.
Axiom 1. (There exists a complete ranking): Any pair of resource allocations can be ranked
by the decision maker. The relation  on F is irreflexive and transitive, and the relation  is
transitive.
Axiom 2. (The independence axiom): If x  y and x Ac  y Ac , then x A  y A .
6
Axiom 3. If A is not null, then [c] A  [d ] A if and only if [c]  [ d ] .
Axiom 4. Suppose [c]  [ d ] , x is equal to c on A and d on Ac , y is equal to c on B and
d on B c . Similarly, suppose that [c' ]  [d ' ] , x' is equal to c ' on A and d ' on Ac , y ' is equal to
c ' on B and d ' on B c . Then x  y if and only if x '  y ' .
Together with other 3 axioms A5 to A7 concerning the property of boundedness and
continuity of the welfare function (see Schaffer (1986), p468), we get:
Savage theorem for resource allocations: If the preference over resource allocations
satisfies Savage axioms A1 to A7, the preference can be represented by a social welfare function
W ( x)  E  ux( ) , i.e., x  y if and only if W ( x)  W ( y ) . In the function,  is a unique weight
function,  ( )  0 is the weight assigned to individual  , and total weight equals to 1; the function
u is bounded, continuous, and unique to an affine transformation, and E  represents the integration
with respect to  .
This theorem does not imply that the above axioms about the consistency of preference must
be satisfied in real life. It only implies that if the consistency conditions in the axioms are satisfied,
then the preference can be represented by such a simple social welfare function. The social welfare
function is unique up to an affine transformation, and what is important is the values of weights and
curvature of function u . This is similar to expected utility for individual choice: the important is the
probability belief and risk aversion.
The last three axioms not stated here do not impose significant constraints on the
consistency of preference. Axiom 1 assumes that the decision maker can rank all possible pairs of
resource allocations, including imaginary allocations. Axiom 2 (the independence axiom) states
that if two resource allocations agree on Ac , then the choice between these two allocations should
depend only on how they differ on A and should not depend on how they agree on Ac . The choice
of allocation among subgroups can ignore the allocation outside the group. Axiom 2 is one of the
natural candidates to be violated as it ignores the externality among individuals.
7
Axiom 3 considers the restriction of equal allocations on subgroups. Consider two resource
allocations: in allocation 1 each individual has the same amount c of resource while in allocation 2
each individual has the same amount d of resource. Now consider the restriction of such allocation
to a subgroup A . If the equal allocation with amount c of resource is preferred to the equal
allocation with amount d of resource for allocation among subgroup A , then this will be true for
among any other subgroup. This axiom guarantees that when restricting two equal resource
allocations to any sub-group of individuals, the preference ordering will be the same for different
subgroups. For the special case where A is a single individual, this implies the existence of a same
value function u for all individuals. The curvature of this value function can represent the
inequality aversion of the decision maker.
In axiom 4, suppose there are only two levels of income, high income level c and low
income level d . In allocation x , individuals in subgroup A have high level of income and the rest
of the individuals have lower level of income. In allocation y , individuals in subgroup B have high
level of income and the rest of the individuals have has low level of income. Let’s assume x is
preferred to y . If axiom 4 is true, the preference x  y cannot be changed when the two levels of
incomes c and d are replaced by any other pair of c ' and d ' with c'  d ' . This axiom actually
leads to individual weight in the social welfare function: the reason for preference x  y is because
the decision maker believes that members in A have higher social value than members in B .
Savage framework implies that if the decision maker’s preferences satisfy theses axioms,
then the decision maker’s preference can be represented by disentangled value judgments of
individual value (or deservedness) and the attitude towards inequality aversion. These consistency
requirements are actually used technically to guarantee such a simple representation. For example,
the second axiom makes sure the social welfare function can be written in an additive form. The
third axiom guarantees that the common value function u can be purged from individual weight.
Axiom 4 guarantees that individual weight can be discovered from social preference. Suppose that
in allocation x individual i has high level of income c while all others have low income level d ,
8
in allocation y individual j has high level of income c while all others have low level of income
d . If the decision maker prefers x to y , it implies that he assigns higher social value to the
income of i , and assign higher weight to i in the social welfare function.
It is not surprising that the axioms are not requirements of consistency or rationality. Some
of the axioms may be violated and the simple representation doesn’t hold anymore. This doesn’t
imply that such decision making is inconsistent. However, we may no longer get a simple form of
social welfare function fully described by the disentangled individual weight and the inequality
aversion.
For example, the first axiom about the existence of a complete ranking requires the decision
maker can compare any pair of resource allocations, including the imaginary ones. It is likely that in
real decision making, sometimes the decision maker is undecided: neither x  y nor y  x , and x
and y are not indifferent or substitutable. The independence axiom, axiom 2, is the most widely
criticized in Savage framework even for individual decision making with uncertainty. In social
decision setting, the independence axiom is not satisfied for maxmin social preference, thus the
widely used Rawlsian-like egalitarianism social welfare function is excluded.
In Savage framework for individual decision making, researchers have tried to relax the
independence axiom and proposed Savage axioms without the independence axiom, see Gilboa
(1987). The Savage framework for resource allocation without the independence axioms will lead
to a social welfare function that can incorporate individual deservedness, inequality aversion and
Rawlsian-like egalitarianism, see Zheng (2009). The extension in this direction will not be
discussed here.
If we look back at the Savage framework we use, the abstract framework can be applied to
general allocation problem. In the framework, there is no restriction on whether the total resource is
fixed or not or whether there exists original entitlements, or what kind of information or social
environment is available in the decision making. It can be used for imaginary position under a veil
of ignorance, but more importantly, it can be applied to situation when the veil is lifted.
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A natural question is how to determine the value judgments of individual weights and
inequality aversion. The value judgments are subjective, so there is no right or wrong value
judgment. The value judgments represent the preference over choices, so they depend on the
contextual detail of the choices, including the cultural and social environments. One element of the
context is the information available in the decision process. Such information may include the
original position, the status quo, and the information of individual characteristics. The ways of how
the individual weights depend on the contextual detail determine the interpretation of the weights.
For example, if the information of individual contribution or individual effort is available and if
weight in the social welfare function is positively related to the contribution or effort, then the
weight can be interpreted as the individual deservedness.
Savage framework only asserts that possibility of the representation of the social preference
on resource allocations; however, there is no practical ways to find out the exact form of social
welfare function in a specific allocation problem. In next section, we explore the consequence on
resource allocations for some special forms of social welfare function that incorporate the value
judgment of individual deservedness.
3. Optimal income distribution with given contributions
From the above discussion, the existence of weights in the social welfare function is more
the rule than exception. Equal weight arises because of the lack of contexture detail such as under a
veil of ignorance, or individuals happen to have the same relevant characteristics and thus are
assigned some weight. The existence of different weights will have significant effect in resource
allocations. However, the preference in resource allocations is subjective and can be very diverse.
Even among preferences with disentangled value judgment of individual deservedness and
inequality aversion, the value judgment can be very different. For example, consider a healthy
middle-age individual who does not work. Some public opinion may consider him as not deserving
public aid because he is lazy, while others may think he deserves the public aid because the reason
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that he does not work is out of his control. In the following, we consider some special form of
value judgments in distribution.
The distribution or allocation can be interpreted in a broader sense. It can be the earned
income distribution, or the reward system or profit sharing, or research grant allocation, etc. We
first consider redistribution of earned incomes without considering the incentive effect. Before the
redistribution, there is a production stage and the information in the product stage will have
influence on the preference in redistribution. Suppose that there are n individuals, and the initial
earned income of individual i is x i0 . The individuals are ordered such that x10  x20    xn0 . Total
income is w0  x10    xn0 .
Suppose the decision maker can redistribute the income among
individuals through lump sum transfer. The income after redistribution will be denoted as
x  ( x1 , , xn ) . No information about individual effort level or productivity is available.
There are many possible ways to redistribute incomes in this simple scenario. At one
extreme, some people might claim that there should be no redistribution because the income is
earned through hard work. The earned income becomes the individual’s entitlement and he deserves
to keep his own income. This is the Lockean theory of desert. At another extreme, some may
suggest to allocate total income equally to all individuals to remove any inequality. The difference
in the redistribution rules is a reflection of different value judgments used for income distribution.
If the preference of the decision maker satisfies the Savage axioms of consistency, the
optimal income distribution is determined by maximizing the social welfare function with the
budget constraint:
max W  i1 i u( xi )
n
(1)
subject to : x1   xn  w0 .
The social welfare function depends on the information in the production stage. Such
information may include the contribution of each individual, their effort level, their role in the
production, and to which degree an individual is accountable for his performance in the production,
etc. The weights (individual deservedness) and the inequality aversion can both depend on such
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information. For simplicity, we assume that the inequality aversion does not depend on the
information in the production stage, and assume inequality averse with constant absolute inequality
aversion. That is, u ( x)  ln( x) , or u ( x) 
x1
for   1 and   0 . Such functions are widely
1
used in economics. The value of    xu' ' ( x) / u ' ( x) is the Arrow-Pratt coefficient of relative
inequality aversion. High value of  represents that the decision maker is more inequality averse.
We assume that individual weights in the social welfare function depend on the information
in the production stage, and the only information is their earned incomes in that stage. If the
decision maker thinks that individuals are not responsible for the inequality of the earned incomes,
he will consider that individuals are equally deserving, and the social welfare function will be
W 
1 n
 u ( xi ) . The optimal income distribution is x1  x2    xn for any concave function u ,
n i 1
and each individual will share the total income equally.
If the decision maker thinks that individuals are accountable in some degree for his earned
income, equal distribution will not occur. Since we assume that the only information available is the
earned income, the deservedness of an individual will be a function of earned incomes. One special
interesting case is that individual deservedness is proportional to his earned income. Such a belief is
similar to the proportional rule in the equity theory (see Adams (1964)).
When the individual deservedness is proportional to his earned income, i.e., i  xi0 and
i  xi0 /( x10   xn0 ) , the first order condition u ' ( xi ) / u ' ( x j )   j /  i implies
( x j / xi )   j / i  x 0j / xi0
Thus individual i ’s income is xi 
or x j / xi  ( x 0j / xi0 )1/  .
( xi0 )1 / 
1 ( x 0j )1/ 
n
(2)
w0 , and income after redistribution should be
proportional to ( xi0 )1 /  . It is obvious that x1  x2    xn and the order of incomes after the
redistribution is the same as the order of earned incomes.
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It is easy to see that in the optimum xi  xi0 if and only if   1. Only in this case each
individual keeps his original contribution without further redistribution. This is contrary to the
thinking that there should be no redistribution once deservedness is considered.
For   1 , relative difference in the earned incomes will be reduced in the optimum
distribution. For example, if individual j ’s initial income is 2 times of that the individual i ,
x 0j  2 xi0 , then after redistribution, x j / xi  ( x 0j / xi0 )1/   21/   2 . The income of individual j
after redistribution is less than 2 times of that of the individual i . The reduction in the relative
difference is achieved by transferring income from those with high initial income to those with low
initial income.
The first order condition can be also rewritten as:
x j / x 0j
xi / xi0
(
x 0j
xi0
)1 /  1 .
(3)
If x 0j  xi0 , then the above equation implies that x j / x 0j  xi / xi0 . The ratio of the postdistribution income to the original income is a decreasing function of the original income. Since
only pure redistribution is considered, there exists an integer k 0 ( 0  k0  n ) such that
x1 / x10  x2 / x20   xk0 / xk00  1  xk0＋1 / xk00 1    xn / xn0 . Individuals with low initial earned
income are subsidized (or taxed negatively) and individuals with high earned income are taxed.
If the subsidy is considered as negative tax, the average tax rate for each individual can be
defined as   1  xi / xi0 . This tax rate is a strictly increasing function of the earned income. In most
of the literature, progressive tax is defined as the tax scheme in which the average tax rate increases
with the pretax income. Therefore, we get progressive tax for   1.
If   1, from the relation x j / xi  ( x 0j / xi0 )1/  , the redistribution will increase the relative
difference of the initial earned income. For example, if individual j ’s contribution is 2 times of that
the individual i , x 0j  2 xi0 , then after redistribution, x j / xi  ( x 0j / xi0 )1/   21/   2 . The income
of individual j after redistribution will be more than 2 times of that the individual i . The increase
13
of the relative difference is achieved by transferring the income from those with low initial income
to those with high initial income.
Similarly, If x 0j  xi0 , then x j / x 0j  xi / xi0 . The ratio of the post-distribution income to
original income is an increasing function of the original income. There exists an integer k 0
( 0  k0  n ) such that x1 / x10  x2 / x20   xk0 / xk00  1  xk0＋1 / xk00 1    xn / xn0 . Individuals with
low initial earned income are taxed positively. The lower is the earned income, the higher tax rate
he will be charged with. In most of the literature, regressive tax is defined as the system in which
the average tax rate decreases with the pretax income. Therefore, we get regressive tax. This is a
very special regressive tax: individuals with low earned incomes are punished and those with high
earned incomes are subsidized.
When we say that an individual deserves his contribution, we usually mean implicitly that
individual should keep his contribution. Our example shows that this is true only when   1. We
also usually take for granted that in an inequality-averse society, income should be transferred from
the rich to the poor, not the contrary. The result of the above analysis shows that this is not true:
even if the decision maker is inequality averse and the relative deservedness is proportional to
initial income, the redistribution by such a decision maker can increase the income inequality and
the resource can be transferred from the poor to the rich.
Why could this happen? From the above analysis, we find that value judgment of
deservedness requires rewarding those who are more deserving and punishing those who are less
deserving. If individuals with high original incomes are deemed more deserving, the decision maker
will reward individuals with higher initial income and punish those with low earned income. On
the contrary, judgment of inequality averse will reward individuals with low earned incomes and
punish those with high earned income. The parameter  of the Arrow-Pratt coefficient measures
the extent of the inequality aversion. If  is small such that 0    1 , the effect of inequality
aversion is not strong enough compared with the effect of value judgment of deservedness, and its
14
inequality reducing effect is dominated by the inequality increasing effect of the individual
deservedness. The combined effect is the increased inequality after redistribution.
If the decision maker is strongly inequality averse with   1, then the effect of inequality
reduction dominates the effect of inequality increasing caused by individual deservedness. The
combined effect is the reduction of inequality after redistribution, and income will transfer from
individuals with high earned income to those with low earned income. Only when   1 the two
effects are balanced so each individual just keep his original income without any redistribution.
The value judgment on deservedness can take other forms. For example, a more general
form of individual deservedness can be written as:
 i  ( xi0 ) h
.
with h  0
(4)
The number h can measure the elasticity of the individual deservedness to his contribution
(or his earned income). It can describe in which extend an individual is held accountable for his
contribution. Such a form of individual deservedness includes several interesting special cases. The
case of h  0 corresponds to the belief in which all individuals have the same deservedness, and an
individual is not held accountable for his contribution. When h  1, individual desert is proportional
to his contribution, and an individual is fully accountable for his contribution. As the value of h
increases, the weight assigned to the individual with high contribution will increase, and this may
suggest that some individuals are held more accountable for the total resource and play more
important role in the creation of resource. When h goes to infinity, the individual with the highest
contribution deserves all while all others deserve nothing. This becomes a “winner takes all”
scheme.
For
this
general
form
of
individual
deservedness,
the
first
order
condition
u ' ( xi ) / u ' ( x j )   j /  i becomes ( x j / xi )   j / i  ( x 0j / xi0 ) h , or x j / xi  ( x 0j / xi0 ) h /  . Therefore,
the solution is equivalent to the case when the individual deservedness is proportional to earned
income and the relative inequality aversion is  / h . From the previous discussion, if   h , the
effect of income inequality averse will dominate the effect of individual deservedness and income
15
will transfer from those with high initial income to those with low initial income, and taxation will
be progressive. When   h , the effect of individual deservedness dominates. Income will be
transferred from those with low initial income to those with high initial income, and income tax is
regressive. Only when   h , the two effects will be balanced and original income will be intact. In
this sense,  can be considered as the extent of inequality aversion and h can be considered as the
extent of deservedness.
For average tax rate 1  xi / xk00 , we get progressive tax for   h and regressive tax for
  h . Tax progressivity was justified using the equal sacrifice principle (see Mitra and OK (1996)).
In those analyses, taxation is progressive or regressive depends on whether the Arrow-Pratt
coefficient of the individual utility function u is greater than unity. The result corresponds to a
special result in our model when h  1 , in which individual deservedness is proportional to his
contribution. Our approach not only allows for interaction of the value judgments of deservedness
and inequality aversion, it also provides another explanation of tax progressivity by considering the
value judgment of individual deservedness and inequality aversion.
For more general form of function u and more general form of individual deservedness, the
income redistribution will also depend on the interaction of the two opposing effects. However, the
explicit form of the solution will be different.
IV. Incentive effect of value judgment of deservedness
Since the consideration of individual deservedness has significant effect on the
redistribution, it is interesting to see the effect of such redistribution on the incentive of labor supply
in the production stage. Intuitively, the value judgment of inequality aversion will discourage
individuals to get high initial income, while the value judgment of individual deservedness will
provide incentive to achieve high initial income. To explore their joint effect, we consider a simple
setting with two individuals and a single produced commodity. In the production period, two
individuals choose their labor supply simultaneously. For both individuals, the tradeoff between
16
consumption and leisure is described by utility function v( xi , li ) where xi is the consumption and li
is the hours of work. Their productivities are
w1 and w2 with w1  w2 . For simplicity, the
information of individual productivities and individual preferences is assumed to be public for these
two individuals. After the earned incomes are realized, then in the second period, earned incomes
from the first period will be redistributed. The decision maker of redistribution can only observe the
earned incomes from the first period. Information such as individual preferences, productivity, or
working hours is not available to the decision maker of redistribution.
To find the solution of this two stage game, we start from the second period. The preference
of the decision maker of redistribution satisfies the consistency axioms in this paper, and the
preference depends on the information of earned income in the first stage. We assume that the
weight in the social welfare function is proportional to earned income and inequality aversion is
described by
u ( x) 
x1
for   1 and   0 or u ( x)  ln( x) for   1 . Therefore, earned
1
incomes will be redistributed according to maximization of the social welfare function:
max W  i1 i u( xi ) with  i  xi0 /( x10  x20 ) subject to x1  x2  x10  x20 .
n
From the last section , the solution for this problem is:
( x20 )1 / 
( x10 )1/ 
0
0
x1  0 1/ 
( x1  x2 ) and x2  0 1 / 
( x10  x20 ) .
0 1/ 
0 1/ 
( x1 )  ( x2 )
( x1 )  ( x2 )
(5)
In the first stage, the two individuals choose their hours of work l1 and l 2 simultaneously to
maximize their utilities, by taking account that their earned incomes will be redistributed in the
second stage. If individual one chooses to work l1 hours and individual two chooses to work l 2
hours, the incomes after redistribution are:
x1 
( w1l1 )1/ 
( w2l2 )1/ 
and
(
w
l

w
l
)
x

( w1l1  w2l2 ) .
11
2 2
2
( w1l1 )1/   ( w2l2 )1/ 
( w1l1 )1/   ( w2l2 )1/ 
(6)
For simplicity, suppose the utility function for both individuals takes the special form
widely used in public economics:
17
v( x, l )  ln x  ln( 1  l ) .
It is well known that this utility function leads to the laissez-faire economy in which
l1  l2  1 / 2 and x1  w1 / 2 , x2  w2 / 2 .
The choices of the two individuals leads to two first order conditions v( x1 , l1 ) / l1  0 and
v( x2 , l2 ) / l2  0 , which can be written as:
1/ 
(1 /  ) 1
w1 l1
/
w1
1
1



0
1/ 
1/ 
l1 ( w1l1 )  ( w2l2 )
w1l1  w2l2 1  l1
(7)
1/ 
1 /  1
w2 l2
/
w2
1
1



0
1/ 
1/ 
l2 ( w1l1 )  ( w2l2 )
w1l1  w2l2 1  l2
(8)
From the above two condition, simple manipulation leads to:
l1
l
1
 2  1  . For
1  l1 1  l 2

  1 , this system of the two non-linear equations has no explicit solution. However, given values
of w1 , w2 and  , we can solve this system of equations numerically using the Fsolve function in
Matlab. We use w1  3 , w2  5 in the following example and explore how the equilibrium changes
with the value of  .
When   1 , this is the laissez faire economy, so we have l1  l2  1 / 2 and x1  1.5 ,
x2  2.5 . The utility levels are -0.2877 and 0.2231 for individual 1 and individual 2.
When  goes to infinity, earned income will be equally shared. The game has explicit
solution. The supply of labor for individual 1 and 2 are l1  1  (w1  w2 ) /(3w1 )  1/ 9 and
l2  1  (w1  w2 ) /(3w2 )  7 / 15 . Total income is l1  (l1w1  l2 w2 ) / 3  8 / 3 . When  is very small
and close to 0, the effect of inequality aversion is very small and almost all earned income will be
given to the individual with higher earned income. This will force the less productive individual to
use extreme effort. Simulation shows that when  is close to 0, the hours of work of individual 1
will be close to 1 and hours of work of individual 2 will be close to 0.6. In the equilibrium, utility
18
level of the less productive individual is very low and his income is also at very low level.
Theoretically, there is no pure strategy Nash equilibrium for the extreme case of   0 .
We present the diagram for individual’s hours of labor, income and utility for different value
of  .
Incomes of two individuals
Labor supply of two individuals
6
1
labor supply of individual 1
labor supply of individual 2
0.9
income of individual 1
income of individual 2
5
0.8
Income
hours of work
4
0.7
0.6
0.5
3
2
0.4
1
0.3
0.2
0
1
2
3
4
5
sigma
6
7
8
9
0
10
0
1
2
3
4
5
sigma
6
7
8
9
10
Utilities of two individuals
Total income
1
6
total income
income under laisser faire economy
5.5
uitlity of individual 1
utility of individual 2
0.5
0
-0.5
utility level
total income
5
4.5
4
-1
-1.5
3.5
-2
3
2.5
0
-2.5
1
2
3
4
5
sigma
6
7
19
8
9
10
-3
0
1
2
3
4
5
sigma
6
7
8
9
10
income of the less productive individual
1.6
1.5
1.4
income
1.3
1.2
1.1
1
0.9
0.8
0.7
0
1
2
3
4
5
sigma
6
7
8
9
10
From the diagrams, we can see that if individual deservedness is proportional to his earned
income, then

the supply of labor for both individual and the total income decrease strictly with  ;

For the more productive individual, his post-redistribution income and utility decrease
with  ; the utility of the less productive individual increases with  .
When 0    1 such that the effect of individual deservedness dominate the effect of
inequality averse, incentive to work increases and both individuals work harder than in the laisser
faire economy: l1  l2  0.5 . The less productive individual is forced to worker even harder. When
the effect of inequality aversion dominates with   1 , then in the equilibrium l1  l2  0.5 ; both
individual workers less harder than in the laisser faire economy.
It is interesting to see the income level for the less productive individual. His income first
increases for  between 0 and 0.6250, then it decreases for   0.6250 . For   0.4329 , the
income of the less productive individual is the same as in the laissez fair economy. Therefore, for
0.4329    1, the final incomes of both individuals are higher than in the laisser faire economy.
When the effect of the deservedness slightly dominate the effect of the income inequality
aversion, the income of both individual will increase in the equilibrium. However, this is
associated with the deterioration of the utility of the less productive individual. When the effect
20
of deservedness is dominated by the effect of inequality aversion, or when the effect of
deservedness is too strong, the income of the less productive individual will be less in the
equilibrium than in the laisser faire economy. This suggests that if the objective is to raise the
income of the less productive individual, moderate extent of individual deservedness will be
useful.
IV Conclusion
In this paper, we discuss how the value judgment of individual deservedness can be
represented in a social welfare function. The combination of the value judgments of individual
deservedness and inequality aversion can lead to very different income distribution scheme. The
consideration of individual deservedness can provide more incentive to increase the preredistribution incomes. An explicit model of individual deservedness may facilitate the application
of the idea of individual deservedness in economic studies.
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