The Social Satisfaction: a Fairness Theory about Income Distribution with Applications in China
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Transcript of The Social Satisfaction: a Fairness Theory about Income Distribution with Applications in China
The Social Satisfaction: a Fairness Theory about Income Distribution
with Applications in China
Ouyang KuiInstitute 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:
duuWdu
uWdu
uWdW
N
2
21
1
duuWdu
uWdu
uW
N
2
21
1
),,(),,( 11 uuWttutuW kN
• Theorem 2.3 The only homogeneous, symmetrically differentiable SWF is the linear power transformation of Nash SWF:
0,),,,(0
21
CuCuuuW N
kN
iiN
3 The Social Satisfaction• 3.1 The SF: a fuzzy measure of utility Definition 3.1 The individual satisfaction function (SF):
S: RN→[0,1].• 3.2 The SSF: a normative on SWF Definition 3.2.1 The SWF W(S1, …, SN) is a social satisfaction
function (SSF) if we have
Theorem 3.2.2 The unique homogeneous and symmetrically differentiable SSF is the geometric average of individual SFs.
NjiSSWSSji jiji ,,2,1,,,,
• 3.3 The invariance properties of SWF
),,(
)(
),,(
),,(),,(
''11
1
'
1111
111
''1
1
'
11
NNNN
N
iii
N
ii
N
iii
N
ii
N
iiiii
NNN
N
N
iii
N
iiiN
uuW
dufdufudf
uuW
uuWdufdufuuW
4 Fairness and equality in income distribution
• 4.1 The Nash bargaining problem:
The Nash solution to the bargaining problem (impartiality):
j
j
ji
i
i xS
SxS
S
11
0)(..
)(max
1
1
N
ii
NN
iii
xGts
xSW
• 4.2 The general form of satisfaction function:
• 4.3 The social satisfaction index (SSI) of income distribution
The Nash solution:),exp()(i
iii x
rxS )1exp(1
N
i i
i
xr
NW
jj
ii x
rrx
))(exp()(0x
dttfCxS
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:
ji SS
jxjixi SS11
jxjS
jixiS
i SS
),min( 21 SSW
21SSW
21 21
21 SSW
Inequality in different solutions(r1=1, r2=1.2) (r1=1, r2=1.6) (r1=1, r2=2) (r1=1, r2=4)
x S W x S W x S W x S W
E4.54555.4545
.8025
.8025 .80253.84626.1538
.7711
.7711 .77113.33336.6667
.7408
.7408 .74102.00008.0000
.6065
.6065 .6065
N4.77235.2277
.8110
.7949 .80294.41525.5848
.7973
.7509 .77384.14215.8579
.7855
.7108 .74723.33336.6667
.7408
.5488 .6376
U4.80035.1996
.8119
.7939 .80294.50005.5000
.8007
.7476 .77424.28415.7159
.7918
.7048 .74833.76286.2372
.7666
.5266 .6466
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
)(')(''
)(')('')(')(')()(
xSxS
xSxSxSxSxSxS
j
j
i
ijiji
0,)(
')( rrxInSxInSx
i
i
• 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)Si(xi)<Sj(xj); (b) xi-yi=yj-xj>0; (c) xk=yk for all k≠i,j, we have: (1)If xi<xj, then W(X) > W(Y); (2)More generally, if yi<yj, then W(X) > W(Y).
N
ii
i
iN xNxxW
11 .0,0),1exp(),,(
NixxS iiii ,,2,1,0,0),exp()(
• 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, Sn×m) is a replication of another society M(X, Sm), Y=Xn, X=xm, then W(X) = W(Y).
• Property 5.7 (Geometric Decomposability) In a society of N agents, for N = n + m, then
Generally, for N =n1+ …+nm, then we have
mnmn
nn
NN WWW 1
1
mm
nn
NN WWW
• 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)xi<xj; (b) xi-yi=yj-xj>0; (c) xk=yk for all k≠i,j, then W(Y) > W(X).
N
iiN x
NxxW
11 .0,0),exp(),,(
• 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(xi)=Si(xi). Then the unique index W of income distribution satisfies the geometric decomposability (Property 5.7) for all N ≥1 is
N
i
iN
xN
xxW1
1 .0,0),exp(),,(
.)(),,( 11N N
i iiNN xSxxW
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?
9872.,4739. 2
)3392.65( RBeijingHunan
,)(2126.3392. 1994)1775.14()9977.31( tDUrbanUrbanUrbanRural
1994;01994;1
,9869.2
tt
DR t
• Figure 1 Impartial regional income distribution
1980 1990 2000 20100
5
10
15
20
25
30
35
40
45
50
year
aver
age
wag
e(103 y
uan)
Beijing
Hunan
0 10 20 300
5
10
15
20
25
30
35
40
45
50
wage in Hunan(103 yuan)
wag
e in
Bei
jing(
103 yua
n)
• Figure 2 Unfair income distributions between urban and rural
1980 1990 2000 20100
5
10
15
20
25
year
inco
me(
103 yua
n)
urban
rural
0 2 4 60
5
10
15
20
25
rural income(103 yuan)
urba
n in
com
e(10
3 yua
n)
• 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:
NixNx
xSN
nntt
it
titit ,,2,1,1),exp()(
12
)1exp(1
2
N
i it
t
xNW
N
i it
Nttt
xNxxW
12
1 1),min(exp(
• Table SSI in ChinaⅡyear China Beijing Hunan Max Min year China Beijing Hunan Max Min
1981 .4291 .4768 .3727 .6593 .3170 1995 .4215 .7098 .3723 .7679 .2644
1982 .4253 .4708 .3689 .7193 .3214 1996 .4229 .7343 .3363 .7941 .2565
1983 .4292 .5021 .3747 .7360 .3190 1997 .4298 .7707 .3279 .7848 .2662
1984 .4293 .4936 .3746 .7532 .3201 1998 .4353 .7713 .3921 .8038 .2493
1985 .4404 .5766 .4055 .7708 .3051 1999 .4284 .7739 .3837 .8329 .2526
1986 .4327 .5517 .3961 .7823 .3139 2000 .4183 .7847 .3748 .8280 .2581
1987 .4417 .5740 .4344 .7712 .3010 2001 .4243 .7911 .3952 .8343 .2530
1988 .4386 .5728 .4304 .7333 .2912 2002 .4175 .7877 .3877 .8304 .2582
1989 .4378 .5681 .4182 .7027 .2897 2003 .4104 .7963 .3763 .8222 .2591
1990 .4396 .5911 .4103 .6933 .2901 2004 .4113 .8059 .3756 .8193 .2588
1991 .4435 .5941 .4027 .6850 .2807 2005 .4083 .8065 .3587 .8081 .2614
1992 .4343 .6038 .4004 .7263 .2841 2006 .4121 .8159 .3579 .8245 .2600
1993 .4437 .6611 .4263 .7679 .2592 2007 .4215 .8089 .3718 .8281 .2579
1994 .4446 .6977 .4028 .7563 .2606
• Figure 3 Inequality of SSI
1980 1985 1990 1995 2000 2005 20100.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
year
SS
I
China
Beijing
Hunan
Max
Min
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
• References • Arrow, K.J.: Social choice and individual values, 2d ed. New York: Wiley (1963).• d’Aspremont, C., Gevers, L.: Equity and the informational basis of collective choice. Review of Economic
Studies 44, 199-209 (1977)• Atkinson, A.B.: On the measure of inequality, Journal of Economic Theory 2, 244-263 (1970)• Bergson, A.: A reformulation of certain aspects of welfare economies. Quarterly Journal of Economics 52,
310-334 (1938)• Dalton, H.: The measurement of the inequality of incomes, Quarterly Journal of Economics 30, 348-361
(1920)• Dasgupta, P., Sen, A., Starrett, D.: Notes on the measurement of inequality. Journal of Economic Theory 6,
180-187 (1973)• Dubins, L.E., Spanier, E.H.: How to cut a cake fairly. The American Mathematical Monthly 68, 1-17 (1961)• Easterlin, R.A.: Income and happiness: Towards a unified theory. The Economic Journal 111, 465-484 (2001)• Foster, J.E.: An axiomatic characterization of the Theil measure of income inequality. Journal of Economic
Theory 31, 105-121(1983)• Kahneman, D., Krueger, A.B.: Developments on the measurement of subjective well-being. The Journal of
Economic Perspectives 20, 3-23(2006)• Mandler, M.: Cardinality versus ordinality: a suggested compromise. The American Economic Review 96,
1114-1136 (2006)• Nash, J.F.: The bargaining problem. Econometrica 18, 155-162 (1950)• Pigou, A.C.: Wealth and welfare. Macmillan Co., London (1912)• Samuelson, P.: Foundations of economic analysis. Harvard University Press, Cambridge Mass. (1947)• Samuelson, P.: The problem of integrability in utility theory. Economica 17, 355-385 (1950)