The Social Satisfaction: a Fairness Theory about Income Distribution with Applications in China Ouyang Kui Institute of Quantitative & Technical Economics, Chinese Academy of Social Sciences

1 Introduction Economic development, income growth and social welfare The Dalton-Atkinson’s approach (Dalton, 1920; Atkinson, 1970) The choice of SWFs and the choice of utility functions The dictatorship conclusion(d’Aspremont & Gevers, 1977) and Arrow’s impossibility theorem (Arrow, 1963) The Nash SWF(Nash, 1950) Revealed preferences and subjective satisfaction: Ordinalism vs Cardinalism (Mandler,2006) The axiomatic characterization of the measure of income distribution

2 The Nash SWF: A differential equation approach Definition 2.1 A SWF is homogeneous of degree k: Definition 2.2 A SWF is symmetrically differentiable:

Theorem 2.3 The only homogeneous, symmetrically differentiable SWF is the linear power transformation of Nash SWF:

3 The Social Satisfaction 3.1 The SF: a fuzzy measure of utility Definition 3.1 The individual satisfaction function (SF): S: R N →[0,1]. 3.2 The SSF: a normative on SWF Definition The SWF W(S 1, …, S N ) is a social satisfaction function (SSF) if we have Theorem The unique homogeneous and symmetrically differentiable SSF is the geometric average of individual SFs.

3.3 The invariance properties of SWF

4 Fairness and equality in income distribution 4.1 The Nash bargaining problem: The Nash solution to the bargaining problem (impartiality):

4.2 The general form of satisfaction function: 4.3 The social satisfaction index (SSI) of income distribution The Nash solution:

4.4 Be equal of welfare or income? Inequality in different solutions The Egalitarian SSF: The Egalitarian solution: The Nash SSF: The Nash solution: The Utilitarian SSF: The Utilitarian solution:

Inequality in different solutions (r 1 =1, r 2 =1.2)( r1=1, r 2 =1.6)(r 1 =1, r 2 =2)(r 1 =1, r 2 =4) xSWxSWxSWxSW E N U

5 The axiomatic characterization of the SSI of income distribution Theorem 5.1 If for all x ∈ R, the SF is second- order differentiable, S(0) = 0, S(+∞) = 1, then we have Definition 5.2 A SF has logarithmic constant elasticity if for all we have

Theorem 5.3 A SF has logarithmic constant elasticity if and only if it can be generated from If all SFs have logarithmic constant elasticity, then the SSI can be expressed as: Property 5.4 (Transfers principle) Y is obtained from X and for some i and j, (a)S i (x i ) 0; (c) x k =y k for all k≠i,j, we have: (1)If x i W(Y); (2)More generally, if y i W(Y).

Property 5.5 (Independent of income units) If W(Y) = W(X), then for t > 0, W(tY) = W(tX). Property 5.6 (Replication principle) If a society N(Y, S n×m ) is a replication of another society M(X, S m ), Y=X n, X=x m, then W(X) = W(Y). Property 5.7 (Geometric Decomposability) In a society of N agents, for N = n + m, then Generally, for N =n 1 + …+n m, then we have

If we set for all i, i.e. the SSI is symmetric, then the SSI can be defined as: Property 5.8 (Symmetry) If Y is obtained from X by a permutation of incomes, then W(Y) = W(X). Property 5.9 (Pigou-Dalton transfers principle) If is obtained from such that for some i and j, (a)x i 0; (c) x k =y k for all k≠i,j, then W(Y) > W(X).

Property 5.10 (Population principle) If Y is a replication of X, then W(X) = W(Y). Property 5.11 (Homogeneity) If Y = tX, t>0, then W(Y) = W(X) if we set Theorem 5.12 Let W(x i )=S i (x i ). Then the unique index W of income distribution satisfies the geometric decomposability (Property 5.7) for all N ≥1 is

6 A simple application in China Does the Chinese Reform and Open Policy practice generate a fair income distribution? How much had Chinese people been satisfied by the great increase in national income in the past several decades?

Figure 1 Impartial regional income distribution

Figure 2 Unfair income distributions between urban and rural

Inspired by supported evidences for that more income brings greater satisfaction among income groups at a point in life and a cohort’s satisfaction remains constant throughout the life span (Easterlin, 2001), we thus construct the following SF and SSI:

Table Ⅱ SSI in China yearChinaBeijingHunanMaxMinyearChinaBeijingHunanMaxMin

Figure 3 Inequality of SSI

7 Conclusions First, the uniqueness of homogenous and symmetrically differentiable SSF shows that economists might be more unified about the analytic form of SWF. Second, the concept of social satisfaction is similar to individual satisfaction as well as the social welfare and the individual utility. Another fact is that the equality on welfare does not necessarily mean the equality on income distribution. In addition, the evidence from China shows that the social welfare may not increase even if the social income level increases a lot. Finally, an important but unresolved question is that whether the impartial income distribution can lead to an equal distribution of income or satisfaction. After all, the fairness concept should be about both impartiality and equality.

Appendices A Proof of theorem 2.3 B Proof of theorem 5.1 C Proof of theorem 5.3 D Original datasets

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