# Subjective value judgments of distributive justice and legal decision-making

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## Abstract

Distributive justice is a fundamental issue in the legal decision-making. We use Savage’s framework to derive a representation of subjective value judgments of distributive justice from the coherence of the decision-maker’s preferences over income distributions. The representation function can incorporate, in a unified way, the widely held beliefs such as inequality aversion, desert, and egalitarianism. We illustrate the application of the justice representation functions in law and economics by studying the distribution rules for different beliefs of distributive justice.

## Keywords

Distributive justice Deservedness Savage axioms Taxation Contract## JEL Classification

D63 H20 K00## 1 Introduction

Distributive justice plays an important role in legal decision-making, including the design of private and public law (Scheffler 2015). In contract law, it is crucial to share the gain or loss of a partnership fairly. In public law such as in the tax law, distributive justice is a fundamental issue. The question of how people evaluate fairness and justice in resource allocations or income distributions has attracted interests in different research fields. In positive studies such as in social psychology, justice is in the eye of the beholder (Tornblom 1992). Individuals with different background can have different judgments of distributive justice for the same distribution, and the judgment of each individual may be a mixture of different principles. Conflicts in a society or within a group can be caused by conflicting beliefs of justice held by its members. Researchers have tried to collect data and report the justice principle observed in the real life.

Moral philosophy or normative study emphasizes the “correct” justice principles that people should hold (for example, see Rawls 1971; Nozick 1975). Researchers use the concepts like equality, equity, need, desert, responsibility, and many others to describe the judgments involved in distribution. Scholars have tried to convince each other about a specific concept. However, there is even no consensus over the exact meaning of each principle.

“It is well known that Savage’s formulation of decision-under-uncertainty problem may be interpreted in other ways as well. In Savage (1954), an act is a function from the states of the world into the set of consequences. But if one chooses to replace the states of the world by individuals in a population, the resulting problem is a social choice problem.”

Though Savage’s formulation for preference over income distributions is already known in the theoretical decision theory, Savage formulation and its implications are not elaborated in the study of distributive justice. We will illustrate in this paper that Savage formulation implies that the preferences satisfying some conditions of consistency can be represented by a weighted additive function, and the terms in the representation function can represent the decision-maker’s subjective belief of distributive justice, including the commonly held principles such as inequality aversion, desert, and egalitarianism.

The “decision-maker” in this paper can refer to an individual or a group of individuals who has preferences over income distributions. It could be a court, the juries, a stakeholder in a distribution problem, an outside observer, or the person (or the entity) who acts as the social planner. Decision-makers have different preferences over income distributions and different subjective perception of distributive justice. Savage’s framework implies that a representation for such subjective beliefs exists.

The representation function we obtained from Savage’s framework is a cardinal function based on individual incomes, and its form is similar to an additive utilitarian social welfare function, combined with Rawlsian-like social welfare function. Harsanyi (1953) justified additive social welfare functions by considering the uncertainty of individuals’ positions within the society. Different from Harsanyi (1953), there is no uncertainty involved in Savage’s representation of distributive justice, and the representation function is not an aggregation of individual/social wellbeing. This non-welfarist representation function represents only the subjective belief of the decision-maker. Another interesting aspect of the Savage representation of justice is that there is no ethical content imposed in the Savage axioms. The belief of justice is derived from the consistency and coherence in the preferences of the decision-maker, rather than derived from specific external ethical principles.

Different beliefs of justice will lead to different rules of resource allocation. To show that the representation of justice can shed new light in the study of distributive justice and in economic analysis of law, we use the representation function to study the rules of income distribution in the framework of taxation and subsidy under different beliefs of justice. We show that progressive or regressive taxation can arise from the perception of inequality aversion and deservedness, and a basic income or poverty line can arise naturally from the belief of egalitarianism and deservedness. Many other problems in law and economic analysis, such as sharing within a partnership, the allocation problem in bankruptcy law, can be discussed in the similar way.

In economic analysis of law, Posner criticized the utilitarian view of the objective function of the law (see Posner 2014), and proposed that the objective of law should be the maximization of total wealth. This paper illustrates that we can have an objective function which includes more justice principles in the objective of law, and at the same time this objective can leads to practical and verifiable implications in the legal decision-making.

Section 2 first presents the Savage formulation for preferences over income distributions. Section 3 discusses how different principles of justice can be represented by the representation function. Section 4 provides some interesting implications of distribution rules resulting from the representations. Section 5 is the conclusion.

## 2 Savage formulation for preferences over income distributions

In the Savage formulation, there is a state space \(\varOmega\) and a consequence space *C*. An act \(x\) is a function from the state space to the consequence space, \(x:\omega \to x(\omega )\). The act space \(F\) consists of all acts. Preference is defined as a binary relation over acts. The binary relation \(x \succ y\) for two acts \(x\) and \(y\) means that the decision-maker prefers \(x\) to \(y\).

Though it is already known for decision theorists about the possible use of the above formulation in income distributions (see Gilboa 1989), its use in income distributions is not commonly known in the study of justice. To see whether Savage framework is appropriate for decision-making over income distributions (or more general resource allocations, and we will use the term “income distribution” and “resource allocation” interchangeably in the following), we discuss the Savage axioms briefly in the framework of income distribution.

If the state space \(\varOmega\) is the set of individuals so that each \(\omega \in \varOmega\) represents an individual, and the consequence is the space of real number *R* which represents the amount of income or resource, then an act \(x:\omega \to x(\omega )\) can be considered as an income distribution or a resource allocation which specifies the income or resource assigned to each individual. The income or resource of individual \(\omega\) is \(x(\omega )\). A preference over acts becomes a preference over income distributions.

Savage axioms are a set of consistency conditions of the preferences. Among different versions of the axioms, we utilize Shafer’s version (Shafer 1986).^{1} While Shafer (1986) discussed the decision-making over uncertainty and considered the states as possible states of the world, we interpret the states as individuals. The decision-maker can be an individual or a group of individuals who has preference over distributions, either a court, a stakeholder, an outside observer, or the individual or entity acting as a social planner.

If the decision-maker is indifferent between two allocations \(x\) and \(y\), then the relation is denoted as \(x \approx y\). The axioms below also involve the restriction of an allocation among subgroups of individuals. For any resource allocation \(x\) in the act space \(F\) and any subgroup of individuals \(A\) of \(\varOmega\), \(x_{A}\) 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 \(x_{A} = (x_{1} ,x_{2} ,x_{3} )\), which is a resource allocation among three individuals labeled 1, 2 and 3. A subset \(A\) of \(\varOmega\) is null if \(x \approx y\) whenever \(x_{{A^{c} }} = y_{{A^{c} }}\), where \(A^{c}\) denotes the set of individuals who are not in \(A\). The resource allocations among null subgroup have no influence on the preferences.

Given a subset \(A\) of individuals and two resource allocations \(p\) and \(q\) among individuals in \(A\), we write \(p \succ q\) if \(x \succ y\) for every pair \(x\) and \(y\) in \(F\) such that \(x_{A} = p\), \(y_{A} = q\) and \(x_{{A^{c} }} = y_{{A^{c} }}\). 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 \(\succ\) on \(F\) is irreflexive and transitive, and the relation \(\approx\) is transitive.

Axiom 1 assumes that the decision-maker can rank all possible pairs of resource allocations, including imaginary allocations.

**Axiom 2**

(*The independence axiom*) If \(x \succ y\) and \(x_{{A^{c} }} = y_{{A^{c} }}\), then \(x_{A} \succ y_{A}\).

Axiom 2 states that if two resource allocations agree on \(A^{c}\), 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 \(A^{c}\). 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, and it excludes the maxmin preference.

**Axiom 3**

If \(A\) is not null, then \([c]_{A} \succ [d]_{A}\) if and only if \(\left[ c \right] \succ [d]\).

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 allocations to a subgroup \(A\). If for subgroup \(A\), the equal allocation with amount \(c\) of resource is preferred to the equal allocation with amount \(d\) of resource, then this will be true for any other subgroup. This axiom guarantees that when restricting two equal resource allocations to any subgroup of individuals, the ordering of the two allocations will be the same for different subgroups. For the special case where \(A\) is a single individual, this implies the existence of a common value function \(u\) for all individuals. The curvature of this value function can represent the inequality aversion of the decision-maker.

**Axiom 4**

Suppose \(\left[ c \right] \succ [d]\), \(x\) is equal to \(c\) on \(A\) and is equal to \(d\) on \(A^{c}\), \(y\) is equal to \(c\) on \(B\) and is equal to \(d\) on \(B^{c}\). Similarly, suppose that \(\left[ {c^{\prime}} \right] \succ [d^{\prime}]\), \(x^{\prime}\) is equal to \(c^{\prime}\) on \(A\) and is equal to \(d^{\prime}\) on \(A^{c}\), \(y^{\prime}\) is equal to \(c^{\prime}\) on \(B\) and is equal to \(d^{\prime}\) on \(B^{c}\). Then \(x \succ y\) if and only if \(x^{\prime} \succ y^{\prime}\).

In axiom 4, suppose there are two levels of incomes, high level \(c\) and low level \(d\) with \(c > d\). In allocation \(x\), individuals in subgroup A have high level of income and the rest of the individuals have low level of income. In allocation \(y\), individuals in subgroup B have high level of income and the rest of the individuals have low level of income. Let us assume \(x\) is preferred to \(y\). Axiom 4 requires that the preference \(x \succ y\) cannot be changed when the two levels \(c\) and \(d\) are replaced by any other pair of \(c^{\prime}\) and \(d^{\prime}\) with \(c^{\prime} > d^{\prime}\). This axiom actually implies the weights in the representation function: the reason for preference \(x \succ y\) is that the decision-maker believes that subgroup \(A\) have higher social value than subgroup \(B\). Specifically, if \(A\) includes only individual \(i\) and \(B\) includes only individual \(j\) and the decision-maker prefers \(x\) to \(y\), it implies that the decision-maker assigns individual \(i\) higher social value and thus higher weight in the representation function.

The last three axioms from axiom 5 to axiom 7 are more technical and they do not impose significant constraints on the consistency of preference.

**Axiom 5**

The nontriviality conditions. There exists at least one pair of acts in \(F\), say \(x\) and \(y\), such that \(x \succ y\).

**Axiom 6**

The continuity condition. If \(x \succ y\), then for every element \(c\) of \(C\) there is a finite partition of \(\varOmega\) such that \(x\) (or \(y\) or both) can be changed to equal to \(c\) on any single element of the partition without changing the preference.

**Axiom 7**

The dominance condition. If \(x_{A} \succ y_{A}\), then \(x_{A} \succ [y(s)]_{A}\) for some \(s\) in *A*, and \([x(s)]_{A} \succ y_{A}\) for some \(s\) in *A*.

The nontriviality condition assures that the representation is not trivial. The continuity condition implies that \(u\) is bounded. The dominance condition is not needed for the representation theorem for acts that take only finitely many values in *C*.

When axioms 1 to axiom 7 are satisfied, we get (see Shafer 1986, p 468):

**Savage theorem for preferences over resource allocations** If the preferences over resource allocations satisfy axiom 1 to axiom 7, the preferences can be represented by a function \(W(x) = E^{\mu } u\left[ {x(\omega )} \right]\), i.e., \(x \succ y\) if and only if \(W(x) > W(y)\). In the representation, \(\mu\) is a unique weight function with \(\mu (\omega ) \ge 0\) for any individual \(\omega\), and the total weight equals to 1; the function \(u\) is bounded, continuous, and unique up to an affine transformation; and \(E^{\mu }\) represents the integration with respect to \(\mu\).

For the discrete case, we can write \(x(\omega_{i} )\) as \(x_{i}\) and write the weight \(\mu (\omega_{i} )\) as \(\mu_{i}\), and the function can be written as \(W(x) = \sum {\mu_{i} u(x_{i} )}\).

It is not surprising that some of the axioms may be violated in real life, but this does not imply that the decision-making is inconsistent. However, we may no longer get a simple form of preference representation.

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 sometimes the decision-maker is undecided: neither \(x \succ y\) nor \(y \succ x\), and \(x\) and \(y\) are not indifferent or substitutable. The independence axiom, axiom 2, is the most widely criticized in Savage framework even in decision-making with uncertainty. In income distributions, the independence axiom excludes preferences with envy or altruism.

In decision theory, researchers have tried to relax the independence axiom. One relaxation is that the independence axiom is required only for comonotonic acts (see Gilboa 1987). For income distribution, comonotonic acts correspond to income distributions in which the incomes of individuals have the same raking order. For simplicity, we state the result for the preference representation in Gilboa (1987), which uses independence axiom for comonotonic acts (but similar to Shafer 1986 in other axioms).

**Savage Theorem without independent axiom** If the axioms P1–P7 in Gilboa (1987) are satisfied, then the preference can be represented by a Choquet integral of a bounded function \(u\) with respect to a unique nonlinear measure \(\nu\) such \(\nu (\emptyset ) = 0\) and \(\nu (\varOmega ) = 1\), and \(u\) is also unique up to an affine transformation.

The first-order condition is \(u^{\prime}(x_{i} )/u^{\prime}(x_{j} ) = \mu_{j} /\mu_{i}\). If \(\mu_{i} > \mu_{j}\) then \(x_{i} > x_{j}\) in the optimum when \(u\) is strictly concave. The decision-maker will allocate more income to individual with a higher weight. It is well known that the concavity of \(u\) represents inequality aversion. The existence of the weights justifies the inequality of income among individuals. The decision-maker assigns different values of a given resource to different individuals. The reason for different individual weights may come from the effect of social status, the political power, or perceived deservedness of the individual. In Zheng and Anwar (2004), the weight is used to describe the value judgments in legal decision-making. Justifications in the literature about income inequality, such as desert, responsibility or need can be described by the weights to individuals, as will be discussed in the next section.

The function in the form of \(\mathop {\hbox{min} }\limits_{s \in T} u(x(s))\) in (2) is widely known as a Rawlsian social welfare function in economics. When the coefficient \(\alpha_{T}^{\nu } > 0\) for a subset \(T\), the decision-maker assign a weight to the worst-off individual in the subgroup *T*, representing the egalitarian consideration among this group. We only consider the case with \(\alpha_{T}^{\nu } \ge 0\). Therefore, according to the representation function (2), a decision-maker’s perception is consisted of three parts: the inequality aversion, the weights to individuals representing the belief such as individual desert, and the weights to egalitarianism among groups of individuals.

Before we move to the next section, we can make a few comments about the function in the representation. First, the Savage formulation is well known for the representation of subjective beliefs. The representation function we obtained represents the belief of the beholder. This allows that different decision-makers have different preferences and thus have different beliefs of justice. (It may happen that some individuals claim that others’ beliefs are wrong). This approach is different from Harsanyi (1953), which pointed out that that the value judgment of justice must be impersonal. According to Harsanyi, if a poor prefers distribution more favorable to the poor for the sole reason that he is poor, then such judgment can hardly be considered as a genuine value judgment of justice. As we interpret justice and fairness as preference over distributions, we actually do not exclude self-interest involved in the belief of justice. A self-interested decision-maker may give higher weight to himself/herself. An extreme selfish decision-maker may even consider that the only fair distribution is the one that gives zero weight to all others. The effect of individual characteristics and self-interest on the justice belief is well documented in the social psychology and sociology studies (see, for example, Cook and Hegtvedt 1983).

Though individuals can have different subjective beliefs of distributive justice, a common justice belief can increase predictability and reduce transaction costs, while conflicting beliefs of justice among individuals may cause resentment and influence the stability of the group or the society. The justice principle that is used in actual social decision-making is the one that is dominant in the current social environment. How the dominant belief of the society is formed from different individual belief is certainly an interesting question, see List (2012).

The obtained representation function is based on individual incomes instead of individual utility. Since economists usually consider individual utility as fundamental, while income only as instrumental, this non-welfarist approach seems to ignore individual preferences and treat handicapped individuals exactly the same as other individuals. However, the ways how the decision-maker treats handicapped individuals or individuals with different tastes can be represented by the weights in the representation function. For example, should an individual with expensive taste (the taste for champagne or political ambition, for example) be given more income? Some decision-maker may say yes while others say no. The subjective belief can be represented by the weights given to the individuals with expensive tastes.

The representation function is cardinal in the sense that the weights are unique and the value function \(u\) is unique up to an affine transformation. Though the additive part of the representation function has the same form of a weighted utilitarianism, the representation function has nothing to do with individual utilities. Harsanyi (1953) justified additive social welfare function by considering the existence of uncertainty of the decision-maker’s position in the society. In Savage framework of income distribution, there is no uncertainty involved. The value of the representation function does not measure the happiness or the welfare of the society, and its numerical value is not important. What is important is the weights and the curvature of value function \(u\), which represent the belief of the decision-maker. This is similar to the expected utility for individual choice: what is important is the probability belief and risk aversion, not the numerical value of expected utility.

The function used in income distribution is usually called social welfare function in the literature. The non-welfarist representation function we obtained can still be called a social welfare function. Since the function is a representation of justice belief, it can also be called a justice evaluation function, as in Jasso (1980). We can define that the distribution \(x\) is more fair or more just that distribution *y* if and only if \(W(x) > W(y)\), thus all distributions can be ranked according to the decision-maker’s perception of fairness and justice.

More interestingly, there are no explicit ethics included in the Savage axioms. The ethics is the outcome rather than the input of the decision-maker’s coherent preferences. What Savage representation says is that the decision-maker’s belief of fairness can be derived from the consistency and coherence of his/her preference. It cannot answer the question about where the preferences come from (see an interesting paper of Dietrich and List 2013 about the formation of preference). As the weights in representation function (2) justify the income inequality and egalitarianism, it is not surprising that many researchers may ask the question: where do the weights come from? Does any axiom provide justification for special ways for assigning weights? Actually, Savage formulation does not provide information about why the weights are determined in any specific way.

This point can be illustrated by the Savage formulation for decision-making with uncertainty. In that case, the Savage framework implies that an individual’s preference can be represented by an expected utility function, with the weights being interpreted as the individual’s subjective belief of probability. Where does this belief of probability come from? The Savage formulation does not provide any answer. For both cases of income distributions and uncertainty, the belief is subjective to the decision-maker. Such beliefs may come from the decision-maker’s past experiences or social environments.

Individual weights in the representation function may represent not only individual deservedness as we discuss in the next section, it can also represent the social or political power of the individuals in the resource allocations. In a completely different approach, Aumann and Kurz (1977) derived weights as power index of individuals from cooperative game theory. Researchers in social choice, such as Barett et al. (2004), derive weight from aggregation of individual utilities; see also Sen (1977) for weights in the social welfare function.

According to Posner (see Posner 2014), the objective of the law should be the maximization of total wealth. This corresponds to a special belief in which there is no inequality aversion, no egalitarian consideration, and all individuals involved have equal weights. In most legal decision-making, individuals do not have equal weights if the activities of some individuals involved are deemed illegal. Inequality aversion and egalitarian consideration come into play even in cases of contracts and torts.

## 3 Representation of some commonly held beliefs of justice

- 1.
Egalitarianism. An egalitarian decision-maker prefers equal resource allocation to each individual. The corresponding representation function can be: \(W = \min_{i \in \varOmega } u(x_{i} )\). This function is referred to as Rawlsian social welfare function in the literature. Another belief which leads to equal distribution is the one that assigns equal weight to all individuals with \(W(x) = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {u(x_{i} )}\).

In real life, the decision-maker does not always prefer strict equality of income. Sometimes, income inequality is considered as being fair. There are different kinds of justifications for income inequality, each taking different forms in different contexts. The justifications can be represented by unequal weights in the representation function.

- 2.
Libertarian. A libertarian decision-maker prefers strict respect for existing holding of resources (entitlements), see Nozick (1975). For example, the decision-maker may consider the market as the right procedure to realize distributive justice, and the market outcome should be respected. Suppose that the market outcome for individual \(i\) is \(x_{i}^{0}\), and the total income is \(w = x_{1}^{0} + \cdot \cdot \cdot + x_{n}^{0} .\) The libertarian belief can be described by a representation function \(W(x) = \sum\nolimits_{i = 1}^{n} {\mu_{i} u(x_{i} )}\) with \(\mu_{i} = \frac{{x_{i}^{0} }}{{x_{1}^{0} + \cdots + x_{n}^{0} }}\) and \(u(x) = \ln (x)\). Under such a representation function, the first order condition \(\frac{{u^{\prime}\left( {x_{i} } \right)}}{{u^{\prime}\left( {x_{j} } \right)}} = \frac{{x_{j} }}{{x_{i} }}\) becomes \(\frac{{x_{i} }}{{x_{j} }} = \frac{{x_{i}^{0} }}{{x_{j}^{0} }}\), thus no redistribution is needed. Therefore, the belief of a libertarian decision-maker can be described by inequality aversion with value function \(u(x) = \ln (x)\) and individual weights that are proportional to individual entitlements.

- 3.
Equity. A decision-maker who believes in equity considers the just income for an individual should be proportional to his/her input, see Adams (1964). (How to measure the inputs of an individual can be subjective to the decision-maker). The representation function of the preference of the decision-maker can be \(W = \sum\nolimits_{i = 1}^{n} {\mu_{i} u(x_{i} )}\) with \(\mu_{i} = \frac{{l_{i}^{0} }}{{l_{1}^{0} + \cdots + l_{n}^{0} }}\), where \(l_{i}^{0}\) is the input of individual

*i*. When \(u(x) = \ln (x)\), the final income of an individual should be proportional to the input \(l_{i}^{0}\): \(\frac{{x_{i} }}{{x_{j} }} = \frac{{l_{i}^{0} }}{{l_{j}^{0} }}\). For more general case when \(u(x) = \frac{{x^{1 - \sigma } }}{1 - \sigma }\) for \(\sigma \ne 1\), the first-order condition becomes \(x_{j} /x_{i} = (l_{j}^{0} /l_{i}^{0} )^{1/\sigma }\). The fair income of an individual should be proportional to a power function of his/her input.When the market outcome or contribution \(x_{i}^{0}\) of individual \(i\) is considered as the input of the individual \(i\), and the valuation function is \(u(x) = \ln (x)\), then the principle of equity and the principle of libertarian lead to the same outcome.

- 4.
Need. Sometimes it is likely that the decision-maker prefers to allocate more income to the poor. As we discussed in the last section, individuals who are physically/mentally handicapped or individuals with special taste may also be considered as being entitled more income than the others. Such a belief can be represented by higher weights for the poor in the representation function, and this justification of assigning different weights can be considered as the principle of need. Need can be considered as a special form of deservedness: those who are poor or with special taste deserve more assistance. For some researchers, need is also related to the subsistence level of income that is needed for individual autonomy, which we will discuss in the next section.

- 5.
Responsibility-sensitive egalitarianism. According to responsibility-sensitive egalitarianism (Fleurbaey 2008), individuals are endowed with two different kinds of characteristics: those for which they are not responsible (called circumstances) and those for which they are responsible. Inequality derived from circumstance is undesirable, while inequality derived from responsibility is acceptable. For such a belief, individual weights are only functions of the characteristics for which individuals are held responsible, and the weights should be independent of circumstances. Accordingly, the optimal incomes of individuals are independent of the circumstance, and they depend only on characteristics of responsibility.

- 6.
Desert. The principle of desert in the real life and in some academic studies is used in a very general sense. Wagstaff (1994) incorporated equity, equality and needs into one equity principle, the “equity as desert”. In our framework, almost any justification to assign different weights in the representation function can be considered as a special form of desert. High weight of an individual in the representation function is equivalent to the perception that the individual deserve more, for reasons such as social status, contribution or effort. The belief of equity, libertarian, need, responsibility, can all be considered as special forms of desert.

In the following, we will use the term “deservedness” to refer to the different weights in the representation function. The term may refer to the belief of equity, libertarian, need and responsibility. This can be controversial, as some scholars give special meaning to the concept of deservedness.

Some researchers debated over what consists of the foundation of desert, see Lamont (1994). Similarly, researchers disagree over what should be considered as the input in the equity principle, or what consists of responsibility in responsibility-sensitive egalitarianism.^{2} Savage’s formulation implies that there is no need to find the universal foundation of deservedness or responsibility: what consists of the foundation of deservedness or responsibility is subjective to the decision-maker.

The maximization of the representation function also implies the efficiency principle discussed in some literature (see Engelmann and Strobel 2004). The efficiency principle is used in economic studies, while it is hardly seen in most of the social psychology or sociology studies.

## 4 The rules of income distribution with different justice beliefs

We discussed the representation of the perception of distribution justice. Next, we will see what new light can be shed in the study of income distributions by the representation of distributive justice. We will show that the combination of inequality aversion and deservedness can lead to different income distribution schemes and progressive or regressive taxation. When egalitarianism is also considered, a basic income or minimum standard of living will occur naturally in income distributions.

### 4.1 Income distributions with inequality aversion and deservedness

The representation function (2) shows the justice perception is a combination of principles of inequality aversion, deservedness and egalitarianism. We will explore the joint effect of such principles. We first explore the interaction of inequality aversion and the principle of deservedness. We use the term deservedness in a general sense to include commonly held principles such as equity and responsibility.

We consider a scenario in which a decision-maker is supposed to distribute certain incomes among n individuals. For example, we can consider earned income distribution or distribution of reward from a joint effort. Without loss of generality, we consider earned income distribution, and the information of earned incomes of individuals \((x_{1}^{0} , \ldots ,x_{n}^{0} )\) is known to the decision-maker, with total income \(w = x_{1}^{0} + \cdot \cdot \cdot + x_{n}^{0} .\) No other information such as the individual preferences, their ability or effort level is available for the decision-maker.

The distribution will certainly depend on the belief of justice of the decision-maker. An egalitarianism decision-maker will suggest dividing the total income equally among individuals. A libertarian decision-maker may consider the earned income as the outcome of market mechanism and will oppose any redistribution. For a responsibility-sensitive egalitarian decision-maker who thinks that individuals are only partly responsible for the outcome, another different distribution may be proposed.

We consider an inequality averse decision-maker, with \(u(x) = \frac{{x^{1 - \sigma } }}{1 - \sigma }\) for \(\sigma \ne 1\) and \(\sigma > 0\), or \(u(x) = \ln (x)\) for \(\sigma = 1\). The decision-maker has constant relative inequality aversion. Such a form of function is widely used in economics. At the same time, the decision-maker assigns weight \(\mu_{i} = \frac{{x_{i}^{0} }}{{x_{1}^{0} + \cdots + x_{n}^{0} }}\) to individual \(i\) so that the deservedness of an individual is proportional to his/her earned income. The reason that the decision-maker assigns the weights in this way may be because the decision-maker considers the earned incomes as the outcome of market mechanism, or the decision-maker holds the view of equity and considers the earned income as the input, or he/she considers individuals are responsible for the earned income. (Many other decision-makers may not hold such a view, of course).

Thus, individual *i*’s income is \(x_{i} = \frac{{(x_{i}^{0} )^{1/\sigma } }}{{\mathop \sum \nolimits_{j = 1}^{n} (x_{j}^{0} )^{1/\sigma } }}w\), and income after redistribution should be proportional to \((x_{i}^{0} )^{1/\sigma }\). The incomes after the redistribution preserve the ranking order in earned incomes.

When \(\sigma = 1\), the case becomes the strict libertarian without any redistribution. For \(\sigma > 1\), relative difference in the earned incomes will be reduced after redistribution. For example, if individual \(j\)’s initial earned income is two times of that the individual \(i\), \(x_{j}^{0} = 2x_{i}^{0}\), then after redistribution, \(x_{j} /x_{i} = (x_{j}^{0} /x_{i}^{0} )^{1/\sigma } = 2^{1/\sigma } < 2\). The income of individual \(j\) after redistribution is less than two times of that of the individual \(i\).

If \(x_{j}^{0} > x_{i}^{0}\), the above equation implies that \(x_{j} /x_{j}^{0} < x_{i} /x_{i}^{0}\) for \(\sigma > 1\). The ratio of the post-distribution income to the original earned income is a decreasing function of the original income. Since only pure redistribution is considered, if we assume that \(x_{1}^{0} \le x_{2}^{0} \le \cdots \le x_{n}^{0}\), there exists an integer \(k\) such that \(\frac{{x_{1} }}{{x_{1}^{0} }} \ge \frac{{x_{2} }}{{x_{2}^{0} }} \ge \cdots \frac{{x_{k} }}{{x_{k}^{0} }} \ge 1 \ge \frac{{x_{k + 1} }}{{x_{k + 1}^{0} }} \ge \cdots \ge \frac{{x_{n} }}{{x_{n}^{0} }}\). Individuals with low-earned income are subsidized and individuals with high-earned income are taxed.

If the subsidy is considered as negative tax (see Watts 1987), and the average tax rate is defined as \(\tau = 1 - x_{i} /x_{i}^{0}\), then the tax rate in our case is a strictly increasing function of the earned income. In the literature, a progressive tax is defined as a tax scheme in which the average tax rate increases with the pretax income. Therefore, the tax scheme is progressive for \(\sigma > 1\).

If \(\sigma < 1\), from the relation \(x_{j} /x_{i} = (x_{j}^{0} /x_{i}^{0} )^{1/\sigma }\), the redistribution will increase the relative difference in the earned incomes. For example, if individual \(j\)’s contribution is two times of that the individual \(i\), then after redistribution, \(x_{j} /x_{i} = (x_{j}^{0} /x_{i}^{0} )^{1/\sigma } = 2^{1/\sigma } > 2\). The income of individual \(j\) after redistribution will be more than two times of that the individual \(i\). The increase of the relative difference is achieved by transferring the income from those with low-earned income to those with high-earned income.

Similarly, if \(x_{j}^{0} > x_{i}^{0}\), then \(x_{j} /x_{j}^{0} > x_{i} /x_{i}^{0}\). The ratio of the post-distribution income to original income is an increasing function of the original income. There exists an integer \(k\) with \(\frac{{x_{1} }}{{x_{1}^{0} }} \le \frac{{x_{2} }}{{x_{2}^{0} }} \le \cdots \frac{{x_{k} }}{{x_{k}^{0} }} \le 1 \le \frac{{x_{k + 1} }}{{x_{k + 1}^{0} }} \le \cdots \le \frac{{x_{n} }}{{x_{n}^{0} }}\) if we assume that \(x_{1}^{0} \le x_{2}^{0} \le \cdots \le x_{n}^{0} .\) Individuals with low-earned income are taxed positively. The lower is the earned income, the higher rate he will be taxed. Therefore, the tax scheme is regressive for \(\sigma < 1\). This is a very special regressive tax: individuals with low-earned incomes are punished and those with high-earned incomes are subsidized.

**Proposition 1**

*For earned income distribution, if the decision-maker is constant relative inequality averse and assigns individual weights proportional to earned incomes, then for* \(\sigma = 1\), *no distribution is needed. For* \(\sigma > 1\)*, the poor will be subsidized and the rich will be taxed, the tax is progressive. For* \(\sigma < 1\)*, the poor will be taxed and the rich will be subsidized, and the tax is regressive.*

When we talk about deservedness, we usually mean implicitly that individual should keep what he has. The above example shows that this is true only when \(\sigma = 1\). We also take for granted that for an inequality-averse decision-maker, 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.

Why could this happen? From the above analysis, the belief of deservedness requires rewarding those who are more deserving. If individuals with high-earned incomes are deemed more deserving, they should get higher income. On the contrary, inequality aversion will help individuals with low-earned incomes. The final outcome depends on which of these two opposing effects dominates. If \(\sigma\) is small such that \(0 < \sigma < 1\), inequality-reducing effect of inequality aversion is dominated by the inequality-increasing effect of the individual deservedness. The combined effect is the increased inequality after redistribution. If \(\sigma\) converges to 0, the effect of inequality aversion almost disappears and only individual deservedness takes effect. In this case, the individual with the highest initial income will get all the income, leading to the case of “winner takes all”.

If the decision-maker is strongly inequality averse with \(\sigma > 1\), then the effect of inequality aversion dominates the effect of individual deservedness. The combined effect is the reduced inequality after redistribution, and income will be transferred from individuals with high-earned incomes to those with low-earned incomes. Only when \(\sigma = 1\) that the two effects are balanced and each individual just keeps his/her earned income.

Fairness and justice is an important consideration in designing the income redistribution scheme and tax structure. However, because of the lack of simple analytical tools to incorporate fairness and justice, fairness and justice is not commonly studied in the study of taxation. Progressive or regressive tax was considered using the equal sacrifice principle, Young (1990), Mitra and OK (1996). Our analysis shows that tax progressivity or regressivity can be determined by the justice principles of inequality aversion and individual deservedness.

### 4.2 Income distribution with egalitarian consideration

Without loss of generality, we assume that individuals are ranked according to their weights in the representation function so that \(\mu_{1} \le \mu_{2} \le \cdots \le \mu_{n}\). To avoid tedious mathematical notations, we consider the case with \(\mu_{1} < \mu_{2} < \cdots < \mu_{n}\). We also denote the income at the optimum as \(x_{1}\) to \(x_{n}\) when no confusion arises.

**Proposition 2**

- 1.
*In the optimal income distribution,*\(x_{1} \le x_{2} \le \cdots \le x_{n}\). - 2.
*There exists a unique integer*\(k^{*}\)\((1 \le k^{*} \le n - 1)\)*such that*\(x_{1} = \cdots = x_{{k^{*} }} < x_{{k^{*} + 1}} < \cdots < x_{n}\). - 3.
*The optimal income distribution can be calculated by maximizing an additive representation function*\(W = \sum\nolimits_{1}^{n} {\overline{\mu } }_{i} u(x_{i} )\)*, where*\(\overline{\mu }_{1} = \cdots = \overline{\mu }_{{k^{*} }}\)*and*\(\overline{\mu }_{j} = (1 - \theta )\mu_{j}\)*for*\(j > k^{*}\).

*Proof*

See "Appendix" for the sketch of proof.

From this proposition, an individual with a higher weight should not end up with strictly less income than those with lower weights. There exists a threshold level of income (including the special case when the threshold level is the lowest or highest level of income). All individuals with weights less than a certain value have the same income, and other individuals have income strictly increasing with their weights. This threshold level of income corresponds to the poverty line, the minimum standard of living, the subsistence level of income, or the basic income discussed in the literature.

According to proposition 2, the egalitarian consideration actually changes the individual weights used in the income distribution. A weight \(\theta\) on egalitarian consideration reduces the high individual weights proportionally by a factor of \(\theta\), and raises the rest of the low individual weights to an equal level. Proposition 2 can simplify the problem of finding the optimal income distribution.

We can show that a decision-maker who put more emphasis on egalitarian consideration in the distribution will set a higher threshold level of income, and more individuals will have income at the threshold level (the proof is omitted here). If the threshold level of income is considered as the poverty line in the income distribution, then a more egalitarian decision-maker will set a higher poverty line and categorize more individuals as “in poverty”. The minimum standard of living, or the basic income is not calculated in an objective scientific way: it is the result of the belief with egalitarian consideration. A more egalitarian decision-maker will set a higher level of basic income.

If we consider the scenario of earned income distribution as in Sect. 4.1, then according to proposition 2, the final income distribution can be obtained in two steps. In the first step, earned incomes are adjusted so that individuals from 1 to \(k^{*}\) have the same income, while other individuals have income \((1 - \theta )x_{i}^{0}\) (they are taxed at a flat rate \(\theta\)). Then, in the second step, incomes will be redistributed according to \(W = \sum\nolimits_{1}^{n} {\overline{\mu } }_{i} u(x_{i} )\), while the weight is proportional to the adjusted income obtained in the first step. Therefore, according to proposition 1, for \(u(x) = \ln (x)\), no redistribution is needed in the second step. Thus, we get an interesting distribution scheme such that individuals with high-earned income are taxed at a flat rate and other individuals have the same level of income.

For more general case of \(u(x) = \frac{{x^{1 - \sigma } }}{1 - \sigma }\) for \(\sigma \ne 1\) and \(\sigma > 0\), Proposition 2 and Proposition 1 in Sect. 4.1 imply more complicated tax structures, in which some individuals have basic income, while other individuals may be taxed progressively or regressively. As we are only interested at illustrating the use of the representation function, we do not provide the detailed tax structure here.

Though the idea of a threshold level of income such as basic income is popular, there was not an easy way to justify theoretically the existence of a threshold level of income in the framework of welfare economics. In philosophy, one justification of a subsistence level of income is the need to ensure autonomy of individuals, which is justified using the principle of need, see a discussion in chapter 10 of Fleurbaey (2008). In economics, argument of a non-divisible good or behavior analysis is used to justify the threshold level of income (see Sharif 2003). A recent paper by Fleurbaey and Maniquet (2007) considered fairness in optimal taxation using the “compensation” principle and “natural reward” principle (the compensation principle implies that inequalities not due to responsibility should be eliminated and the reward principle implies that inequalities due to responsibility should be left untouched), and they get tax structures that maximize the basic income. Our analysis shows the threshold level of income is a requirement of the general belief of distributive justice with egalitarian consideration.

## 5 Conclusion

In this paper, we illustrate how subjective beliefs in distributive justice can be represented by using the Savage framework. From the coherence of the preferences in income distributions, we obtain a simple form of representation for distributive justice, and we find that a general perception of distributive justice is a combination of commonly held principles such as deservedness, inequality aversion and egalitarian consideration. Using the representation function of distributive justice, we obtain some interesting rules of income distributions and tax schemes. Progressive or regressive taxation can be the outcome of justice belief of inequality aversion and deservedness, and a threshold level of income such as basic income arises naturally from the belief of egalitarianism and deservedness.

In this paper, we use the taxation as an illustrative example. It is easy to apply the representation of distributive justice in the contract law, such as the sharing of gain and loss in a partnership. The deservedness of a player in a partnership can be related to the contribution of the player in the partnership, and different belief of distributive justice can lead to different sharing rules. We believe that a simple analytic representation of distributive justice can allow us to incorporate the justice consideration into broad areas of researches in law and economics and other social sciences.

## Footnotes

- 1.
Shafer (1986) used a continuum of states. I use this more familiar version to show the correspondence of the two frameworks. Our analysis in the paper involves cases with discrete states (individuals). The exposition and the proof of Savage theorem need minor revision for the discrete case.

- 2.
In a famous example, it is assumed that individuals are not responsible for his preference over consumption and leisure, but they should be responsible for their choice of labor supply and leisure. However, individual’s choice of labor supply in economic theory is determined by his/her preferences over consumption and leisure, together with other constraints. If individuals are not responsible for their preferences and the constraints, they should not be held responsible for their choice of effort.

## Notes

### Acknowledgements

I thank Professor Moriki Hosoe and two anonymous referees for their useful comments. The financial support from University of Macau research grant (MYRG2014-00032-FSS) is gratefully acknowledged.

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