Optimality Conditions for Distributive Justice

Transcription

1 Optmalty Codtos for Dstrbutve Justce Joh Hooker Carege Mello Uversty Aprl

2 Just Dstrbuto The problem: How to dstrbute resources Tax breaks Medcal care Salares Educato Govermet beefts 2

3 Justce ad Optmzato The problem s ot to satsfy prefereces, but to acheve justce. 3

4 Justce ad Optmzato The problem s ot to satsfy prefereces, but to acheve justce. Two classcal crtera for dstrbutve justce: Utltarasm Dfferece prcple of Joh Rawls 4

5 Justce ad Optmzato The problem s ot to satsfy prefereces, but to acheve justce. Two classcal crtera for dstrbutve justce: Utltarasm Dfferece prcple of Joh Rawls Both ca be vewed as mathematcal optmzato problems. 5

6 Justce ad Optmzato Utltarasm seeks dstrbuto of wealth to dvduals that maxmzes total utlty. 6

7 Justce ad Optmzato Utltarasm seeks dstrbuto of wealth to dvduals that maxmzes total utlty. The Rawlsa dfferece prcple calls for a lexcographc maxmum of utltes allotted to dvduals. 7

8 Justce ad Optmzato We aalyze dstrbutos over odetcal dvduals. Ulke most mathematcal/axomatc treatmets of socal welfare. 8

9 Justce ad Optmzato We aalyze dstrbutos over odetcal dvduals. Ulke most mathematcal/axomatc treatmets of socal welfare. Dstrbuto of greater resources to more productve dvduals may crease overall utlty..e., to dvduals who are more taleted or work harder. 9

10 Justce ad Optmzato We aalyze dstrbutos over odetcal dvduals. Ulke most mathematcal/axomatc treatmets of socal welfare. Dstrbuto of greater resources to more productve dvduals may crease overall utlty..e., to dvduals who are more taleted or work harder. To what extet does effcecy requre equalty the utltara ad Rawlsa models? 10

11 Justce ad Optmzato Ethcs caot be reduced to mathematcs, but Optmzato theory ca provde some sght to whe a dstrbuto of wealth s just. Allow us to calculate just allocatos of resources. 11

12 Outle Utltara prcple Utltara rule Basc utltara model Mathematcal aalyss Utltara model wth cost of socal dsharmoy Mathematcal aalyss Dfferece prcple Aalyss of Joh Rawls Lexmax model of the dfferece prcple Mathematcal aalyss 12

13 Utltara Prcple 13

14 Utltara Prcple Socal polcy should be chose to maxmze total utlty across all persos. Utlty = happess, pleasure, or well-beg some sese. Hstorc example: pushmet of crmals The crmal justce system should maxmze total utlty rather tha exact retrbuto (Jeremy Betham). Push crme whe postve utlty of deterrg future crme outweghs egatve utlty of the pushmet. 14

15 Utltara Prcple Whe dstrbuto of wealth s at ssue, we assume that every dvdual has a utlty fucto v(x), where x s the wealth allocato to the dvdual. Idvdual Utlty Fucto Utlty Wealth 15

16 Utltara Prcple A just dstrbuto of wealth s oe that maxmzes total expected utlty. Example: redudat workers. A compay must decde whether to lay off workers durg bad tmes. A layoff creates egatve expected utlty for the redudat workers but postve expected utlty for the stockholders, etc. 16

17 Utltara Prcple Let x = wealth of perso wth layoff. Let x = wealth wthout layoff. Lay off the workers f v( x ) v( x ) 17

18 Basc Utltara Model Let x = wealth tally allocated to perso u (x ) = utlty evetually produced by perso 18

19 Basc Utltara Model The utlty maxmzato problem: max u ( x ) = 1 x x = 1 = 1 0, all Total budget 19

20 Basc Utltara Model To solve t: max u ( x ) = 1 x x = 1 = 1 0, all Assocate Lagrage multpler λ wth ths costrat 20

21 Basc Utltara Model To solve t: max u ( x ) = 1 x x = 1 = 1 0, all Assocate Lagrage multpler λ wth ths costrat Ay soluto ( whch each x 0) satsfes L( x, λ) = u ( x ) λ x = u ( x ) λ = 0, all x x 21

22 Basc Utltara Model So u ( x ) = = u ( x ) 1 1 Margal productvty Dstrbute wealth so as to equalze margal productvty. 22

23 Basc Utltara Model So u ( x ) = = u ( x ) 1 1 Margal productvty Dstrbute wealth so as to equalze margal productvty. If we assume persos are dexed order of margal productvty,.e., u ( ) u + 1 ( ), all The x x 1 Less productve dvduals receve less wealth. 23

24 4 p = 0.5 Utlty maxmzg dstrbuto Producto fuctos u (x ) for 5 dvduals Producto u5 u4 u3 u2 u Wealth allocato 24

25 p = 0.5 Utlty maxmzg dstrbuto Producto u5 u4 u3 u2 u Wealth allocato 25

26 Basc Utltara Model A egaltara dstrbuto x 1 = = x s optmal oly whe u (1/ ) = = u (1/ ) 1 So, equalty s optmal oly whe everyoe has the same margal productvty a egaltara dstrbuto. 26

27 Basc Utltara Model u ( x ) = c x p Let where p 0 The the optmal wealth dstrbuto s p 1 p x = c c j j =

28 Basc Utltara Model The the optmal wealth dstrbuto s p 1 p x = c c j j = 1 Whe p 1: x = 1 ad all other x = 0. The most productve dvdual gets everythg. 1 28

29 Basc Utltara Model The the optmal wealth dstrbuto s p 1 p x = c c j j = 1 Whe p 1: x = 1 ad all other x = 0. The most productve dvdual gets everythg. Whe p < 1: Dstrbuto s completely egaltara oly f c 1 = = c Otherwse the most egaltara c dstrbuto occurs whe p 0: x 1 = c j j 29

30 Basc Utltara Model So f productvty s at least proportoal to put (p 1), the most productve class gets everythg. Otherwse, the most early egaltara dstrbuto that ca be optmal s oe whch people receve wealth proporto to c. Ad ths occurs oly whe productvty very sestve to vestmet (p 0). 30

31 Utlty maxmzg wealth alllocato Wealth p = 0.9 p = 0.8 p = 0.7 p = 0.5 p = Productvty coeffcet c 31

32 Basc Utltara Model Let s see how wealth s dstrbuted a multclass socety 32

33 Sze of socal classes Frequecy Populato Productvty coeffcet c 33

34 Wealth allocato, p = Frequecy, relatve wealth Populato Wealth Productvty coeffcet c 34

35 Wealth allocato, p = Frequecy, relatve wealth Populato Wealth Productvty coeffcet c 35

36 Wealth allocato, p = Frequecy, relatve wealth Populato Wealth Productvty coeffcet c 36

37 Wealth allocato, p = Frequecy, relatve wealth Populato Wealth Productvty coeffcet c 37

38 Wealth allocato, p = Frequecy, relatve wealth Populato Wealth Productvty coeffcet c 38

39 Utlty Loss Due to Equalty Utlty wth equalty/max utlty p 39

40 Utlty Loss Due to Equalty Whe output s proportoal to vestmet, equalty has hgh cost (cuts utlty half) Utlty wth equalty/max utlty p 40

41 As p 0, optmal utlty requres hghly uequal allocato, but equal allocato s oly slghtly suboptmal Utlty Loss Due to Equalty Utlty wth equalty/max utlty p 41

42 Socal Dsharmoy Model Utltaras argue that a hghly uequal dstrbuto caot be optmal, due to socal dsharmoy. Utlty s ot a addtvely separable fucto. 42

43 Socal Dsharmoy Model Utltaras argue that a hghly uequal dstrbuto caot be optmal, due to socal dsharmoy. Utlty s ot a addtvely separable fucto. Let s model cost of equalty as proportoal to total rage of comes. 43

44 Socal Dsharmoy Model Utltaras argue that a hghly uequal dstrbuto caot be optmal, due to socal dsharmoy. Utlty s ot a addtvely separable fucto. Let s model cost of equalty as proportoal to total rage of comes. Now maxmze utlty: = 1 = 1 ( { } { }) max u ( x ) β max x m x x x = 1 0, all Coeffcet of socal dsharmoy 44

45 Socal Dsharmoy Model Does a postve β result a more egaltara dstrbuto of wealth? How large must β be to force equalty a utlty maxmzg dstrbuto? 45

46 Socal Dsharmoy Model Theorem. If we ca rewrte the model u ( ) u + 1 ( ), all = 1 = 1 ( { } { }) max u ( x ) β max x m x x x = 1 0, all as max u ( x ) β ( x x ) = 1 = 1 = 1 x x, = 1,, 1 x x + 1 0, all 1 46

47 Socal Dsharmoy Model Theorem. If we ca rewrte the model u ( ) u + 1 ( ), all = 1 = 1 ( { } { }) max u ( x ) β max x m x x x = 1 0, all as max u ( x ) β ( x x ) = 1 = 1 = 1 x x, = 1,, 1 x x + 1 0, all 1 Assocate Lagrage multplers λ µ 47

48 Socal Dsharmoy Model The Karush-Kuh-Tucker (KKT) optmalty codtos mply that x s optmal oly f there are λ ad µ 1,, µ 1 0 such that u ( x ) + β λ µ = u ( x ) λ + µ µ = 0, = 2,, 1 u 1 ( x ) β λ + µ = 0 1 where µ = 0 f x < x +1 the soluto. 48

49 Socal Dsharmoy Model Frst suppose that everyoe gets a dfferet wealth allotmet x. The each µ = 0 ad u ( x ) u ( x ) u ( x ) u ( x ) β smaller tha those the mddle All equal β larger tha those the mddle 49

50 p = 0.5 Utlty maxmzg dstrbuto Producto fuctos u (x ) for 5 dvduals Producto u5 u4 u3 u2 u Wealth allocato 50

51 4 p = 0.5 Utlty maxmzg dstrbuto β = Producto u5 u4 u3 u2 u Wealth allocato 51

52 4 p = 0.5 Utlty maxmzg dstrbuto β = Producto u5 u4 u3 u2 u Wealth allocato 52

53 4 p = 0.5 Utlty maxmzg dstrbuto β = Slope = 3.52 Producto u5 u4 u3 u2 u Wealth allocato 53

54 4 p = 0.5 Utlty maxmzg dstrbuto β = Slope = Slope = 3.52 Slope 2.5 = Producto u5 u4 u3 u2 u Wealth allocato 54

55 4 p = 0.5 Utlty maxmzg dstrbuto β = Producto u5 u4 u3 u2 u Wealth allocato 55

56 4 p = 0.5 Utlty maxmzg dstrbuto β = Producto u5 u4 u3 u2 u Wealth allocato 56

57 4 p = 0.5 Utlty maxmzg dstrbuto β = Producto u5 u4 u3 u2 u Wealth allocato 57

58 Socal Dsharmoy Model How large must β be to force equalty? Here each µ > 0. Elmate λ from KKT codtos & get equatos of the form 2µ µ = d µ + µ µ = d, = 2,, µ + µ = d

59 Socal Dsharmoy Model How large must β be to force equalty? Here each µ > 0. Elmate λ from KKT codtos & get equatos of the form 2µ µ = d µ + µ µ = d, = 2,, µ + µ = d These equatos have a partcularly smple soluto: k k µ = d 1 1 k 1 d k = k = 1 59

60 Socal Dsharmoy Model I ths case d = u ( x ) u ( x ) + β, = 1,, 1 d = u ( x ) u ( x ) + 2β So, k k( k) 1 1 µ k = β + u (1/ ) u (1/ ) k = k + 1 k = 1 Average over k most productve dvduals Average over k least productve dvduals 60

61 Socal Dsharmoy Model Theorem. The utltara dstrbuto s egaltara oly f each µ k 0, thus oly f for all k, β k k( k) 1 1 u (1/ ) u (1/ ) k = k + 1 k = 1 Average over k most productve dvduals Average over k least productve dvduals 61

62 Socal Dsharmoy Model Theorem. The utltara dstrbuto s egaltara oly f each µ k 0, thus oly f for all k, β k k( k) 1 1 u (1/ ) u (1/ ) k = k + 1 k = 1 If u (x ) = c x p, we have equalty oly f for all k, p k( k) 1 1 β c p 1 k = k + 1 k = 1 k c Average over k most productve dvduals Average over k least productve dvduals 62

63 Socal Dsharmoy Model If u (x ) = c x p, we have equalty oly f for all k, p k( k) 1 1 β c p 1 k = k + 1 k = 1 k c So β must be larger to eforce equalty whe there s a large gap betwee k least productve people ad the rest. β s more sestve to the gap whe k /2, because k( k) s larger. 63

64 Umodal productvty dstrbuto 40 Sze of socal classes Total utlty of egaltara dstrbuto, gorg cost of socal dsharmoy Frequecy Populato U U egal max 27.6 = = Productvty costat c c avg = 0 c avg = 2.88 Maxmum total utlty, gorg cost of socal dsharmoy β U max

65 Umodal productvty dstrbuto 40 Sze of socal classes Total utlty of egaltara dstrbuto, gorg cost of socal dsharmoy Frequecy Populato U U egal max 27.6 = = Productvty costat c c avg = 0.8 c avg = 3.25 Maxmum total utlty, gorg cost of socal dsharmoy β U max

66 Umodal productvty dstrbuto 40 Sze of socal classes Total utlty of egaltara dstrbuto, gorg cost of socal dsharmoy Frequecy Populato U U egal max 27.6 = = Productvty costat c c avg = 1.5 c avg = 3.92 Maxmum total utlty, gorg cost of socal dsharmoy β U max

67 Umodal productvty dstrbuto 40 Sze of socal classes Total utlty of egaltara dstrbuto, gorg cost of socal dsharmoy Frequecy Populato U U egal max 27.6 = = Productvty costat c c avg = 1.97 c avg = 4.6 Maxmum total utlty, gorg cost of socal dsharmoy β U max

68 Umodal productvty dstrbuto 40 Sze of socal classes Total utlty of egaltara dstrbuto, gorg cost of socal dsharmoy Frequecy Populato U U egal max 27.6 = = Productvty costat c c avg = 2.35 c avg = 5.29 Maxmum total utlty, gorg cost of socal dsharmoy β U max

69 Umodal productvty dstrbuto 40 Sze of socal classes Total utlty of egaltara dstrbuto, gorg cost of socal dsharmoy Frequecy Populato U U egal max 27.6 = = Productvty costat c c avg = 2.63 c avg = 6 Maxmum total utlty, gorg cost of socal dsharmoy β U max

70 Umodal productvty dstrbuto 40 Sze of socal classes Total utlty of egaltara dstrbuto, gorg cost of socal dsharmoy Frequecy Populato U U egal max 27.6 = = Productvty costat c To eforce equalty, let Maxmum total utlty, gorg cost of socal dsharmoy β U max

71 Bmodal productvty dstrbuto Sze of socal classes Frequecy Populato U U egal max 15.9 = = Productvty costat c c avg = 0 c avg = 1.77 β U max

72 Bmodal productvty dstrbuto Sze of socal classes Frequecy Populato U U egal max 15.9 = = Productvty costat c c avg = 0.87 c avg = 4 β U max

73 Bmodal productvty dstrbuto Sze of socal classes Frequecy Populato U U egal max 15.9 = = Productvty costat c c avg = 1 c avg = 5.54 β U max

74 Bmodal productvty dstrbuto Sze of socal classes Frequecy Populato U U egal max 15.9 = = Productvty costat c c avg = 1.02 c avg = 5.75 β U max

75 Bmodal productvty dstrbuto Sze of socal classes Frequecy Populato U U egal max 15.9 = = Productvty costat c c avg = 1.06 c avg = 5.91 β U max

76 Bmodal productvty dstrbuto Sze of socal classes Frequecy Populato U U egal max 15.9 = = Productvty costat c c avg = 1.1 c avg = 6 β U max

77 Bmodal productvty dstrbuto Sze of socal classes Frequecy Populato U U egal max 15.9 = = Productvty costat c To eforce equalty, let β U max

78 Bmodal productvty dstrbuto Sze of socal classes Frequecy Populato U U egal max 15.9 = = Productvty costat c To eforce equalty, let Uβ max Larger tha 9.68, the rato for a umodal productvty dstrbuto. 78

79 Dfferece Prcple 79

80 Problems wth Utltarasm A utlty maxmzg dstrbuto may be ujust. Dsabled or oproductve people may be eglected. Less taleted people who work hard may receve meager wage. 80

81 Rawlsa Dfferece Prcple The root dea s that whe I make a decso for myself, I make a decso for ayoe smlar crcumstaces. It does t matter who I am. So whe I choose polces for dstrbutg wealth, I should preted that I do t kow who I am. I make decsos (formulate a socal cotract) a orgal posto, behd a vel of gorace as to who I am. 81

82 Rawlsa Dfferece Prcple Rawls argues that ths mples a prcple for dstrbutve justce: Dfferece prcple: A just dstrbuto of wealth creates oly as much equalty as s ecessary to mprove everyoe s welfare. Ths refers to equalty of opportuty, ot outcome. As dstrbuto of salares, tax burde, medcal beefts, etc. 82

83 Rawlsa Dfferece Prcple Example: salary dffereces I ca ratoally agree oly to salary dffereces that are ecessary to make everyoe better off. Perhaps because f salares were more equal, people would be lazy, the ecoomy would shrk, ad everyoe would be worse off. So f I fd myself wth a low salary, I ca reaso that I would be eve worse off f salares were more equal. 83

84 Lexmax Prcple The dfferece rule mples a lexmax prcple. If we also assume that a dstrbuto s just oly f there s o Pareto mprovemet. Pareto mprovemet = some people are better off, o oe s worse off. Lexmax (lexcographc maxmum) prcple: Maxmze welfare of least advataged class the ext-to-least advataged class ad so forth. 84

85 Lexmax Prcple Why the dfferece prcple mples lexmax prcple: Ca mprove lowest class wthout hurtg ayoe else 85

86 Lexmax Prcple Why the dfferece prcple mples lexmax prcple: The there s a Pareto mprovemet Ca mprove lowest class wthout hurtg ayoe else 86

87 Lexmax Prcple Why the dfferece prcple mples lexmax prcple: Ca mprove lowest class, but t hurts someoe else 87

88 Lexmax Prcple Why the dfferece prcple mples lexmax prcple: The we created equalty that does ot make everyoe better off. Ca mprove lowest class, but t hurts someoe else 88

89 Lexmax Prcple Why the dfferece prcple mples lexmax prcple: Argue ductvely for the other classes. 89

90 Lexmax Model We wll model the lexmax prcple as a utlty maxmzato problem wth a lexmax objectve fucto. 90

91 Lexmax Model We wll model the lexmax prcple as a utlty maxmzato problem wth a lexmax objectve fucto. Let v(x ) = persoal utlty of wealth x. We assume everyoe has the same utlty fucto, but ot the same productvty fucto. 91

92 Lexmax Model We wll model the lexmax prcple as a utlty maxmzato problem wth a lexmax objectve fucto. Let v(x ) = persoal utlty of wealth x. We assume everyoe has the same utlty fucto, but ot the same productvty fucto. We assume each perso s share of total utlty s proportoal to the utlty of hs/her tal wealth allocato. Thus dvduals wth more educato, salary have greater access to socal utlty. 92

93 Lexmax Model The utlty maxmzato problem: Wealth allocato to perso lexmax ( y,, y ) y y = 1 = 1 = 1 v( x ) =, = 2,, v( x ) y = u ( x ) x x = 1 0, all 93

94 Lexmax Model The utlty maxmzato problem: Utlty allocato to perso Wealth allocato to perso lexmax ( y,, y ) y y = 1 = 1 = 1 v( x ) =, = 2,, v( x ) y = u ( x ) x x = 1 0, all 94

95 Lexmax Model The utlty maxmzato problem: Utlty allocato to perso Wealth allocato to perso lexmax ( y,, y ) y y = 1 = 1 = 1 v( x ) =, = 2,, v( x ) y = u ( x ) x x = 1 0, all Budget 95

96 Lexmax Model The utlty maxmzato problem: Utlty allocato to perso Wealth allocato to perso lexmax ( y,, y ) y y = 1 = 1 = 1 v( x ) =, = 2,, v( x ) y = u ( x ) x x = 1 0, all y s sum to total utlty produced Budget 96

97 Lexmax Model The utlty maxmzato problem: Utlty allocato to perso Wealth allocato to perso lexmax ( y,, y ) y y = 1 = 1 = 1 v( x ) =, = 2,, v( x ) y = u ( x ) x x = 1 0, all Proportoal allocato of total utlty y s sum to total utlty produced Budget 97

98 Lexmax Model The utlty maxmzato problem: lexmax ( y,, y ) y y = 1 = 1 = 1 v( x ) =, = 2,, v( x ) y = u ( x ) x x = 1 0, all To defe lexmax: Let L k be the problem of maxmzg { y y } m,, k subject to ths ad ( y,, y ) = ( y *,..., y *) 1 k 1 1 k 1 Optmal soluto of L k 1 98

99 Lexmax Model The utlty maxmzato problem: lexmax ( y,, y ) y y = 1 = 1 = 1 v( x ) =, = 2,, v( x ) y = u ( x ) x x = 1 0, all To defe lexmax: Let L k be the problem of maxmzg { y y } m,, k subject to ths ad ( y,, y ) = ( y *,..., y *) 1 k 1 1 k 1 The y* solves lexmax problem f (y 1 *,,y k *) solves L k for k = 1,,. 99

100 Lexmax Model The utlty maxmzato problem: lexmax ( y,, y ) y y = 1 = 1 = 1 v( x ) =, = 2,, v( x ) y = u ( x ) x x = 1 0, all To defe lexmax: Let L k be the problem of maxmzg { y y } m,, k subject to ths ad ( y,, y ) = ( y *,..., y *) 1 k 1 1 k 1 Note: The lterature ofte defes L k to maxmze y k --ot ths 100

101 Lexmax Model The utlty maxmzato problem: lexmax ( y,, y ) y y = 1 = 1 = 1 v( x ) =, = 2,, v( x ) y = u ( x ) x x = 1 0, all Theorem. If u ad v(. ( ) u + 1 ( ) ) s odecreasg, ths has a optmal soluto whch y y 1 101

102 Lexmax Model So L k s { y y } max m,, k ( x,, x ) = ( x *,, x *) y y 1 k 1 1 k y = u ( x ) = 1 = 1 = 1 v( x ) = v( x ) x = 1 x 0, = 1,, k 1 102

103 Lexmax Model So L k s max ( y,, y ) = ( y *,, y *) y y 1 k 1 1 k y = u ( x ) = 1 = 1 = 1 y k = 1 x 0, = 1,, k 1 v( x ) = v( x ) x y k y Apply the theorem 103

104 Lexmax Model So L k s max v( x ) k = 1 = 1 u ( x ) ( x,, x ) = ( x *,, x *) 1 k 1 1 k 1 v( x ) Elmate y s = 1 x x k k x = 1 0 x 104

105 4 p = 0.5 Utlty maxmzg dstrbuto β = Producto u5 u4 u3 u2 u Wealth allocato 105

106 p = q = 0.5 Lexmax dstrbuto Producto u5 u4 u3 u2 u Wealth allocato 106

107 p = 0.5 Utlty Maxmzg Dstrbuto 7 6 Producto u5 u4 u3 u2 u Wealth allocato 107

108 p = q = 0.5 Lexmax Dstrbuto 7 6 Producto u5 u4 u3 u2 u Wealth allocato 108

109 Lexmax Model Whe does the Rawlsa model result equalty? That s, whe do we have x 1 = = x the soluto of the lexmax problem? The lexmax problem forces equalty f ad oly f L 1 forces equalty. 109

110 Lexmax Model = 1 L 1 s max v( x ) 1 = 1 u ( x ) v( x ) = 1 x x 1 k x = 1 0 x 110

111 Lexmax Model u ( x ) L 1 s = 1 max v( x1) v( x ) = 1 = 1 x x 1 k x = 1 0 x Assocate Lagrage multplers µ 1,, µ 1 111

112 Lexmax Model Remarkably, the KKT codtos have the same form as for the socal dsharmoy model: d 2µ µ = d µ + µ µ = d, = 2,, µ + µ = d where ths case cu ( x ) v ( x1) u + 1 ( x + 1) u 1 ( x1) v ( x + 1) v ( x1) = v( x ) v( x ) + v( x1) cu ( x ) v( x ) 112

113 Lexmax Model Theorem. If u (x ) = c x p ad v(x ) = x q, the the lexmax dstrbuto s egaltara (x 1 = = x ) oly f for k = 1,, k q k c c c k k p k = k + 1 = 1 = 1 113

114 Lexmax Model Theorem. If u (x ) = c x p ad v(x ) = x q, the the lexmax dstrbuto s egaltara (x 1 = = x ) oly f for k = 1,, k q k c c c k k p k = k + 1 = 1 = 1 Average of k largest c s Average of k smallest c s 114

115 Lexmax Model Theorem. If u (x ) = c x p ad v(x ) = x q, the the lexmax dstrbuto s egaltara (x 1 = = x ) oly f for k = 1,, k q k c c c k k p k = k + 1 = 1 = 1 Equalty s more lkely whe p s small. That s, whe greater vestmet a dvdual yelds rapdly decreasg margal returs. 115

116 Lexmax Model Theorem. If u (x ) = c x p ad v(x ) = x q, the the lexmax dstrbuto s egaltara (x 1 = = x ) oly f for k = 1,, k q k c c c k k p k = k + 1 = 1 = 1 Equalty test s more sestve at upper ed (large k). Equalty s ulkely whe dvduals at the top are much more productve tha average. Equalty s stll possble eve whe dvduals at the bottom are much less productve tha the average. 116

117 Lexmax Model Theorem. If u (x ) = c x p ad v(x ) = x q, the the lexmax dstrbuto s egaltara (x 1 = = x ) oly f for k = 1,, k q k c c c k k p k = k + 1 = 1 = 1 Equalty s less lkely whe q s small. That s, whe greater wealth yelds rapdly decreasg margal utlty. That s, whe people do t care much about gettg rch. 117

118 Umodal productvty dstrbuto Sze of socal classes Frequecy Populato Rawlsa justce requres equalty whe p q Productvty costat c 118

119 Bmodal productvty dstrbuto Sze of socal classes Frequecy Populato Rawlsa justce requres equalty whe p q Productvty costat c 119

120 Lexmax Model I a lexmax model, there s a more equal dstrbuto of resources whe: Productvty s sestve to vestmet. The productvty dstrbuto has a short upper tal. The lower tal does t matter. The productvty dstrbuto s umodal rather tha bmodal. People wat to get rch. 120

121 Lexmax Model Lexmax dstrbuto ca be egaltara a lassezfare socety. Govermet cotrols oly a few resources, such as hgher educato subsdes or tax breaks. Hstorcally uderprvleged dvduals rema much less productve tha eltes. Large dffereces betwee dvdual productvty fuctos. 121

122 Lexmax Model Lexmax dstrbuto ca be egaltara a lassezfare socety. Govermet cotrols oly a few resources, such as hgher educato subsdes or tax breaks. Hstorcally uderprvleged dvduals rema much less productve tha eltes. Large dffereces betwee dvdual productvty fuctos. Lexmax dstrbuto should be more egaltara a socalst system Govermet cotrols a wder rage of resources. Small dffereces betwee dvdual productvty fuctos. 122

123 Lexmax Model A socety that cotrols a wder rage of resources s oblged to dstrbute those resources more equally tha f t cotrolled fewer resources. 123