A Utilitarian Approach of the Rawls s Difference Principle

Transcription

1 1 A Utltaran Approach of the Rawls s Dfference Prncple Hyeok Yong Kwon a,1, Hang Keun Ryu b,2 a Department of Poltcal Scence, Korea Unversty, Seoul, Korea, b Department of Economcs, Chung Ang Unversty, Seoul, Korea, Hghlghts: Economc models are establshed based on utltaransm and the Rawls s Dfference Prncple. An ndvdual capablty dstrbuton s derved usng the maxmum entropy method. Work hours are long and the ncome nequalty s large n a utltaran government. Total utltes are more or less the same n both Rawlsan and utltaran governments. Utltaran government s good for the rch and Rawlsan government s good for the poor. Abstract: Rawlsan government () maxmzes the utlty of the poorest group and utltaran government () maxmzes total utlty. Each government chooses tax parameters to acheve goals under balanced budget constrants. Indvduals have dfferent capabltes and maxmze utltes through sutable work and lesure choces. The concluson shows that Gn coeffcents are lowered but work hours ncrease n both and even f people are allowed to work whle they receve welfare payments. JEL classfcaton: D63, P52, D31 Keywords: Utltaransm, Rawls s dfference prncple, Indvdual capabltes, Gn coeffcent, Maxmum entropy method 1. Introducton Bentham (1789) and Mll (1861) ntroduced a utltaran tradton to acheve the greatest good for the greatest number. They were nterested n the aggregate sum of utltes, but not the dstrbuton of utltes. Rawls (1999) ntated the concept of dstrbutve justce. The dfference 1 Tel.: ; E-mal addresses: hkwon@korea.ac.kr 2 Correspondng author, Tel.: ; E-mal addresses: hangryu@cau.ac.kr

2 2 prncple requres that socal and economc nequaltes be organzed so that they are of the greatest beneft to the least-advantaged members of socety. Sen (2009) dscussed formal equalty of opportuntes as necessary and suffcent condtons for dstrbutve justce as havng no barrer to educaton, poston, or jobs. The Rawls s dfference prncple provdes ntutve gudelnes for welfare dstrbuton, but t s unclear on how to mpose the prncple n practcal polcy. Ths paper ntroduces a neoclasscal model. Cobb-Douglas functonal forms are used for producton and utlty functons. Indvdual capablty dstrbuton s approxmated usng the maxmum entropy method subject to the gven Gn coeffcent (US CPS data 1983). Ryu (1983, 2013) derves a probablty densty functon usng the maxmum entropy method and Ytzhak (2013) showed the equvalence of the frst moment of ncome dstrbuton wth the Gn coeffcent. Each government can acheve goals wth proper choce of tax parameter. Indvduals maxmze ther utltes wth sutable work and lesure hour choces. Two welfare dstrbuton systems are consdered of when people receve unemployment benefts: not allowed to work (the frst case) and allowed to work (the second case). The performances of and are compared. The paradox of redstrbuton thess suggests that a greater government targetng of benefts towards the poor results n a less lkelhood that poverty and nequalty wll be reduced (Korp and Palme 1998; Lndert 2004). Ths paper provdes an opposte result. The produced hgher utlty for the poorest group and lower Gn coeffcent compared to the. 2. Mathematcal Model There are 1000 persons n a socety. Each person has dfferent producton capablty as a functon of poston z [0,1]. The least capable and poorest person s located at z and the most capable and rchest person at z The dstrbuton of producton capablty s assumed to have a exp( bz ) because Ryu (2013) showed US CPS (1983) ncome shares can be well approxmated wth the above functon wth b 2.59, a b/ (exp( b) 1). Fg.1 compares approxmated shares wth the observed shares. Let the producton functon of an ndvdual depend on productvty and workng hours,

3 3 bz y a e H l. (1) Ths shows decreasng margnal productvty of labor. One day s H 24 hours. The after tax rate functon s assumed to be ATR e tz ( H l ). (2) More capable person wth hgher z pays more tax and ATR decreased. If someone faces t 0.12 and works 6 hours, then the after tax return rate becomes, ATR= e 0.3z. The tax rate s close to zero for ncapable persons but t ncreases to 52% for the most capable person. Fg.1 Approxmated and observed ncome shares Fg.2 After tax returnwth respect to z Approxmated ncome shares Observed ncome shares After tax return Consumpton functon for a person at z s Z Z C ATR y e a e H l (3) tz ( H l) bz The utlty functon s assumed to be U C l 1/3 2/3 wth lesure l. The model parameters (1/3,2/3) are chosen for convenence ln U tz( H l) ln abz 0.5ln( H l) ln l 3 3 (4) 3 Maxmze utlty, aln C(1 a) ln l, subject to wagel C wage 24. If optmal lesure s 16 hours, a 1/3.

4 4 A person at poston maxmzng utlty. z wll choose lesure hours l and work hours are H l 24 l by lnu tz 0 l 3 H l 3l (5) The optmal lesure hours are tz l ( tz H 2.5) l 2H 0 2 The optmal workng hours are level, tax amount, and utlty level can be determned. 2 ( tz 2.5) H tz H tz H l ( z) (6) 2tz H l ( z). Once the workng hours are determned, the output Output: bz y a e H l (7) Consumpton: C ATR y (8) Tax Payment: (1 ATR ) y = tz( H l) bz 1 e a e H l (9) 1 2 Utlty: ln U tz( H l ) ln abz 0.5ln( H l ) ln l 3 3 (10) 3. Welfare recpents not allowed to work Suppose a person chooses the welfare group and decdes not to work to receve welfare. There are N ( 1, 2,, N ) persons n the welfare group and 1000 N persons n the workng group n. Smlar notaton goes for and there are N welfare recpents. The boundary person has equal utlty and they ether belongs n the work group or n the welfare group. Ths boundary condton unquely determnes the number of welfare recpents ( N and N ). Total tax collectons for and are 1000 t z ( Hl ) bz (11) N 1 TAX ( N ) 1e a e H l

5 t z ( Hl ) bz (12) N 1 TAX ( N ) 1e a e H l Collected tax s evenly dstrbuted to the welfare recpents n both and. w TAX ( N ) TAX ( N ) w N N (13) Utlty of welfare recpent: 1/3 2/3 U w H (14) (24 ) 1/3 2/3 Utlty of a worker after tax payment: tz U l bz e a e 24 l l (15) If a person at the boundary can enjoy U lnu from workng, to make them to move to the welfare group, they need an equvalent consumpton level level. The needs welfare spendng of C to keep them at the same utlty 1 2 exp(3 U ) U lnu lnu ln C ln(24) C (16) (24) N C for N recpents snce everybody nsde the welfare group receves the same amount. Match the welfare spendng wth the collected tax. Ths budget constrant unquely determnes the welfare recpent number N the welfare recpents utlty s bgger than some worker s utlty f ncreases the welfare recpent number and the boundary condton wll determne. For the gven tax rate, N s very small. Ths N unquely. Fewer persons wll work and workng hours wll be short f tax parameter t s very hgh. The aggregate output and total utlty wll decrease; however, the opposte happens f tax parameter t s low. The total tax collecton was Total utltes of are 1000 t z ( Hl ) bz (17) N 1 TAX ( N ) 1e a e H l N 1000 U( z ) U( z) (18) 1 N 1

6 6 The utlty of the welfare recpent, socety total utlty, total output, total collected tax, the number of welfare recpents, and the Gn coeffcents are plotted as a functon of tax parameter () t. Fg.3 Utlty of welfare recpent Fg.4 Total utlty of the socety , ,150 Utlty of welfare recpent Total utlty 2,145 2,140 2,135 2,130 2, , ,115 In Fg. 3, the utlty of welfare recpent s maxmzed when t 0.12 for. In Fg. 4, socety total utlty s maxmzed at t Fg. 5 and Fg. 6 show people wll produce less and pay more tax as tax parameter ncreases. Fg. 7 shows the number of welfare recpent ncreases as the tax parameter ncreases. More people are dscouraged from workng as the tax parameter ncreases. Fg. 8 and Fg. 9 show that economc nequalty s lowered and people wll work less as tax parameters ncrease.

7 7 Fg. 5 Total output of socety Fg.6 Total collected tax of socety 2, , , Total output 1,900 1,800 1,700 1,600 Total collected tax , , ,300 0 Fg.7 Number of welfare recpents Fg.8 The Gn coeffcents Number of welfare recpents Gn

8 8 Fg.9 Total work hours 4,800 4,400 4,000 Total hours worked 3,600 3,200 2,800 2,400 2,000 1,600 1,200 Table 1: Performance comparson of and Number of welfare recpents Optmal tax parameter t t t Log utlty of welfare recpent Total tax collected Socety total log utlty Gn coeffcent Total work hours Total output requred long work hours and acheved larger output producton; however, the aggregate utlty was slghtly hgher than. Income transfer from the rch to the poor ncreased the utlty of the poor. The socety total utlty s more or less the same n both and.

9 9 4. Welfare recpents allowed to work Everybody partcpates n producton to earn ncome, pay tax, and receve welfare. The welfare recepton functon s desgned so that a more capable and rch person receves smaller welfare whle a less capable and poor person receves larger welfare. The welfare functon s assumed to be a monotonc decreasng functon of z. Welfare = exp[ r(1 z)] (19) The most capable and rchest person at z 1 receves no welfare wth welfare=1, but least capable and poorest person at z 0 receves 156% of hs ncome f r The consumpton level for a person at poston z s a multplcaton of earned ncome and the welfare factor. W C e a e H l e. (20) tz ( H l ) bz r(1 z ) The correspondng welfare rate ( r ) s determned by the budget constrant for the gven tax parameter t. Aggregate prvate consumpton ncludng welfare s equal to aggregate output W bz C y ae Hl (21) Suppose everyone knows ther tax parameter ( t ) and welfare parameter ( r ). Each person maxmzes the utlty functon wth a proper choce of work hours, 1 2 ln U tz( H l) ln abz 0.5ln( H l) r(1 z) ln l 3 (22) 3 The optmal choce of workng hours s the same as (6) because the welfare term come as an addton that does not affect the dfferentaton of utlty wth respect to labor choce. Total tax collecton s a functon of tax parameter (t ) tz( Hl ) bz (23) 1 TAX () t 1e a e H l Dfferent governments wll acheve goals wth dfferent optmal tax parameters. maxmzes aggregate utlty and maxmzed the utlty of the least advantaged group. The mddle ncome group s defned as the range of 50% and 150% of medan ncome. Followng Rawls (1999), all persons wth less than half of the medan ncome and wealth (p.84) s consdered the least advantaged group (poverty group). Table 2 summarzes the calculated results.

10 10 Table 2: Comparson of and when welfare recpents are allowed to work (max poverty group utlty) (max total utlty) Optmal tax parameter Redstrbuton parameter Total log utlty Total log utlty of poorest group Total collected tax Gn coeffcent Total work hours Total output producton The can maxmze the utlty of the poverty group at t The maxmzes total aggregate utlty at t Redstrbuton s large n as t maxmzed the poverty group utlty. Total utlty and output were small n because t emphaszed only the poor group and the tax burden s heavy. The capable and rch persons are dscouraged from workng long hours. The Gn coeffcent s very low n and economc equalty s well acheved. In Fg.10, s a good system for poor persons wth relatvely hgher utltes, but the same system s bad for rch persons. In Fg. 11, produced output s larger n. In Fg. 12, poor persons work long hours but rch persons work less. Fg.13 shows that the poor pay more tax n than n, but ther utlty wll ncrease wth welfare recepton (Fg. 10).

11 11 Fg.10 Indvdual log utlty Fg. 11 Indvdual output Z Z Fg.12 Indvdual work hours Fg. 13 Indvdual tax collected Z Z 5. Concluson maxmzes total utlty; therefore, government supports the rch at the cost of the poor as shown n Fg.10. The persons at z close to one are rcher and enjoy a hgher level of utltes n. maxmzes poverty group utlty; therefore, government supports the poor at the cost of the rch. At the natonal level, the dfference n the government system made a sgnfcant

12 12 dfference n total output but a small dfference n total utltes. wth no work allowance for the welfare recpents s recommended for an output-orented country because the tax burden s low and the produced output s the large. wth work allowance s recommended for an equalty-orented country because the tax burden s heavy and the Gn coeffcent s the smallest. At the ndvdual level, the produced output and utlty depended on ther capablty level and choce of government form ( vs ). Indvduals cannot change capablty levels; however, they can partcpate n the selecton of government because the poor are favored n and the rch are favored n. Acknowledgment Ths research was supported by the Natonal Research Foundaton of Korea (2014S1A3A ). References Bentham, J., Utltaransm. reproduced by Nabu Publc Doman Reprnts. Korp, W. and Palme J., The paradox of redstrbuton and strateges of equalty: welfare state nsttutons, nequalty, and poverty n the western countres, Amercan Socologcal Revew 63(5), Lndert, P. H., Growng Publc I, II. Cambrdge Unversty Press, New York. Mll, J., Utltaransm. Fraser s Magazne, reproduced by Kessnger publshng, Rawls, J., A Theory of Justce. revsed edton from 1971, Harvard Unversty Press, Cambrdge, MA. Ryu, H., Maxmum entropy estmaton of densty and regresson functons, Journal of Econometrcs 56, Ryu, H., A bottom poor senstve Gn coeffcent and maxmum entropy estmaton of ncome dstrbutons, Economcs Letters 118, Sen, A., The Idea of Justce. Harvard Unversty Press, Cambrdge, MA. Ytzhak, S., More than a dozen ways of spellng Gn, the Gn methodology, Sprnger, Reprnted from The Journal of Economc Inequalty, 1998.