Rawls, Phelps, Nash: eciency curve an economic justice Louis e Mesnar June 14, 2011 University of Burguny an CNRS, Laboratoire 'Economie et e Gestion (UMR CNRS 5118); 2 B Gabriel, B.P. 26611, F-21066 DIJON Ceex, FRANCE. E-mail: louis.e-mesnar@u-bourgogne.fr Charles Gie Days : Justice & Economics June, 16 & 17, 2011 Draft version Abstract This article oers some reections on the interpretation of Rawls' Theory of Justice that Phelps gives by the eciency curve. In the rst section we emonstrate that the Phelps curve allows showing that egalitarianism may be impossible, inecient or also possible but this last case exclues an ominates the Rawlsian maximin. The ebate egalitarianism vs. equity is clarie. We examine the eect of growth. Growth coul make equality easier in some cases an more icult to reach in some other cases. Choosing the maximin oes not guarantee that the growth is always favorable to the poor: it can be paraoxical because the poor can be losing even when the maximin is selecte. By consiering that a Nash bargaining is able to generate any point in the Phelps eciency curve, we examine a new point, surplus-equality: it correspons to an equal sharing from the isagreement point, which shoul be consiere as the origin from the moment that the eciency curve is given. The transposition to n agents is elicate: the overall maximin is not necessarily the leximin an it is better to consier groups of agents. In conclusion, the area between the maximin an surplus-equality shoul be the base of a left-wing policy as it protects against a growing inequality. 1
JEL classication. D63; H23; I31 Keywors. Rawls; Phelps; Nash; maximin; inequality; eciency Abbreviate title. Rawls, Phelps, Nash 1 Introuction The Theory of justice of John Rawls (Rawls 1971, 1993; Barry 1989; Gibbar 1991) inclues two principles. We will quote them in the form most recently state by Rawls (1989). The rst principle treats of freeoms: Each person has an equal right to a fully aequate equal basic liberties for all, which is consistent with a similar system of liberties for all. The secon principle treats inequalities: The social an economic inequalities must satisfy two conitions: 1) they must rst be attache to functions an positions open to all, in areas of fair equal opportunities an 2) they must obtain the greatest benet to the most isavantage members of society. Concerning remunerations, the Rawlsian position is often summarize by a simple choice, as in McClellan's example: o we prefer a istribution of income such as the average is 20000$ an the poor receive 15000$, or on the contrary an average of 40000$, the poor receiving only 14000$? The position chosen by Rawls correspons to the rst possibility (maximization of the position of the most isavantage or maximin or principle of ierence), even if McClellan avances that the majority of American woul choose the secon face of the alternative (McClellan 1990, p. 95). The Rawlsian position is often consiere as one of the most liberal in the American sense of the term, as close as possible to absolute (or strong) egalitarianism; this is why the maximin is calle practical justice by Kolm (1972, 1996b). However, Rawls is accuse of supporting a feeble ght against inequalities: in France, Minc's report (1994) has been strongly criticize for this reason (in a country as France, egalitarianism is a sensitive issue since the French revolution). Phelps, Nobel prize in 2006, one of the most renown supporters of Rawls, has very clearly explaine Rawls' theory (1995). He raws the eciency curve where the optimum must be chosen, which assumes a Paretian optimum, an he posits the various points that can be chosen; among them, there is the maximin, 2
equality, the utilitarian point an the pro-rich point. As Phelps' tool is as peagogic as Hicks-Hansen IS-LM moel iseven if most consier that IS-LM betrays Keynes' iea, an as the eciency curve unoubtely exists, we return in this paper to Phelps' contribution to examine what can be euce from the eciency curve, particularly for what concerns the question of the maximin, the eect of growth, an overall egalitarianism an its impossibility in most cases. Pure egalitarianism is etermine by the original point where the revenue of all iniviual is equal to zero (which is not Rawls' original position). However, as the question can be consiere as a sharing problem solve by bargaining, we also introuce a ose of Nash in Rawls an Phelps. We consier that the various points can be euce by a generalize Nash bargaining (Nash 1950a an 1950b; Rubinstein 1982; Binmore, Rubinstein an Wolinsky 1986). From Nash comes the iea of isagreement point (the point where all iniviual are place before bargaining), which allows us to examine a new egalitarianism, surplus-equality, where equality is etermine by respect to the isagreement point. This iea of equality is no more a chimera as pure equality is. We euce of all this that a left-wing policy must choose a point between the maximin an surplus-equality. Nash allows an elegant generalization to the case of n agents, etc. 2 The maximin 2.1 The maximin on the eciency curve Phelps raws a graph, certainly simplifying an which coul be likely to betray the thought of Rawls by simplifying it in a neoclassical irection, but which is very eloquent for the comparison about the various optima (Phelps 1985, p. 159). Nevertheless, Phelps says... writing to me about my just publishe textbook, he [Rawls] sai its exposition of his theory of justice was entirely accurate (Phelps 2011). In this graph, the Paretian eciency curves (see Figure 5) is the frontier curve of possible remunerations or curve of possible, which inicates all the possible revenues that guarantee a given output (or even a given growth rate). The important thing to be notice is that the agents' revenues are the argument, but not utilities: it is what makes Phelps' curve ierent to what is usual. As Phelps' curve uses revenues as arguments, the problem of ane transformations of utilities (translation or scaling) oes not apply. Rawls (1971) himself criticizes the iea of utility: he prefers the iea 3
of primary goos. Sen (1999) also criticizes utility. If the Nash bargaining problem is insensitive to the scaling of utilities (this is fortunate because it is known that utilities are essentially orinal, an if they are carinal, they are only ene at an ane transformation), the approaches that require interpersonal comparisons of utilities obviously nee carinal utilities that are sensitive to ane transformations. 1 We o not think that intertemporal comparisons of utilities an carinal utilities can be accepte but if we refuse them, it becomes impossible to consier some remarkable points on the eciency curve as the utilitarian point. Passing by the revenues as one by Phelps allows interpersonal comparisons an ening all points on the eciency curve. State for two iniviuals, the eciency curve is R 2 = f (R 1 ), by supposing that iniviual 1 is always pai best, that is, by supposing that the curve is attene along the Y-axis an remains on the right of the rst bisector at least for its eective part. 2 Let us recall that the frontier of eciency correspons to what it is possible to obtain at best, without egraing the situation of an iniviual in orer to improve that of another. In the interior of the frontier, one can increase at the same time the income of iniviuals 1 an 2 by going to the top, towars the line, or both at the same time. On the frontier one cannot increase the income of one iniviual without ecreasing that of the other. All the points of the frontier are equally possible, except those which correspon to a return of the curve on itself, from where the form trace on Figure 1: the ecient partor Pareto-optimalof the curve, is the segment (r, m): before m, or after r, the income of both iniviuals may increase or ecrease simultaneously. The curve (r, m) is continuous, erivable an is such that R2 R 1 0 an 2 R 2 0. Several typical points appear on the curve in R1 2 Figure 5. In point r the richest receives more: we call it the pro-rich point. The Rawlsian optimum, or maximin, is the point m, which consists in giving as much as possible to the most unerprivilege. This point may be on the left of m on the inecient part of the curve, as on Figure 5 or to the right. For Rawls, m is the right or equitable position. Rawls efens the maximin by saying that iniviuals ignore by avance in which position they will fall, the bets or the worst. Therefore, it is better for them to make that the worst position is not too ba. 1 The arguments of the problem are also certain: no nee of von Neumann-Morgenstern utilities, expecte utilities, etc. On these approaches, see Harsanyi (1953, 1955); Hammon (1976, 1979, 1993), Bezembiner an van Acker (1987); Bosmans an Ooghe (2006); Miyagishima (2010). 2 This shape of curve is also quote by Kolm (1972, p. 33, 1.e example). 4
Figure 1: Equality is impossible Any intermeiate solution, obtaine by making a linear combination between the incomes of the two iniviuals, an locate between m an r coul be chosen an reects a relation of force between both: the weighting may reect any social criterion. The point r is obtaine if the best remunerate group is ominating, whereas the point m gives the primacy to the less remunerate group. Phelps (1985) says that this point is an ieal eciency while only the segment {r, m} is ecient. In the point u the social income is maximize, i.e., the sum of the incomes; eciency curve's slope is there equal to 1; Gamel (2010) assimilates it to the welfarism. It is also the point which correspons to the Bentham's utilitarian optimum, that is, to the maximization of the mean (this one coul be weighte) of the incomes of iniviuals 1 an 2. In the point e the two incomes are equal. All this is consequentialist: only the consequences of policy ecisions are examine. 2.2 Typology of curves The curves may aopt various forms. They may be very concentrate as in Figure 2, left: in this case, the problem of choosing a point on the eciency curve is practically evacuate. They may be very large, as in Figure 2, right: the problem of choosing a point is increase. The eciency curve may be also 5
Figure 2: Typology of eciency curves: concentrate (left), large (right) Z Z Figure 3: Typology of eciency curves: atypical (left), very unequal (right) atypical, attene along the X-axis, as in Figure 3, left; or it can be very unequal, attene along the Y-axis, as in Figure 3, right. Obviously, epening of the form of the eciency curve, the point u may be closer to the point r or to the point m. However, as R 1 R 2, the curve is probably attene along the Y-axis. Hence, the point where the slope of the curve is equal to 1, that is, the point u, is probably closer to r than to m along the X-axis. This shows that utilitarianism, which is the main point in the Anglo-Saxon culture, Rawls excepte, is probably very favorable to the rich. 2.3 On egalitarianism The point e of equality of incomes may not exist if the intersection between the bisector an the curve oes not exist. Proposition 1. Reaching equality can be, respectively, (i) impossible, (ii) inef- cient or (iii) possible but by excluing an ominating the Rawlsian optimum 6
Figure 4: Equality is possible but inecient in this last case. Corollary 1. Among the ecient points, the maximin is the closest to egalitarianism. Proof. The proof of 1 an its corollary is one graphically. (i) Equality is impossible when the eciency curve oes not intersect the rst bisector as in Figure 1. In this case, reaching equality is impossible. The point m is the closest to egalitarianism (i.e., the rst bisector). (ii) Equality is possible but inecient if the eciency curve intersects the rst bisector to the left of the Rawlsian optimum as in Figure 4. Forcing equality implies becoming uner-ecient. The price to pay for egalitarianism is ineciency. In this case, reaching equality is fanciful an the point m is again the closest to egalitarianism. 3 In both cases of Figures 1 or 4, Phelps sees a justication of the maximin: any point to the right of m on the eciency curve (u, r, etc.) correspons to more inequality. Therefore, he aopts a point of view similar to those of Kolm (1972, 1996b) an its iea of practical justice. However, equality is itself a jugment of value. If both iniviuals have the same right on the available wealth, they have to share equally but when the iniviuals have ierent claims 3 On the Pareto argument an feasibility of equality, see Cohen (1995) an its critique by Shaw (1999). 7
Figure 5: Example of eciency curve they share ierently. 4 (iii) When the eciency curve intersects the rst bisector to the right of the Rawlsian optimum as in Figure 5, equality is possible (it is locate in the ecient zone) but prevents the Rawlsian optimum from existing. As the point m is to the left of the rst bisector, in m the revenue of iniviual 2 is higher than those of iniviual 1: between e an m iniviual 2 is the richest while iniviual 1 is the less favore. Therefore, the solution of the maximin is e: m is never reache an e is selecte. The point m oes not correspon to the maximin anymore an m is not the closest to egalitarianism: equality ominates an exclues the Rawlsian maximin. 2.4 Discussing the iea of maximin For Harsanyi an Rawls, the iniviuals ignore ex ante in which position they will be ex post, which is the argument of the veil of ignorance an the original position evelope by (Harsanyi 1953, 1955, 1958, 1975) an Rawls (1971): it is why they ecie to give the larger possible revenue. However, this argument 4 In the Aristotelian traition, they share proportionally, while in the Talmuic traition, they share in a ierent way (Rabinovitch 1973; O'Neill 1982; Aumann an Maschler 1985; Young 1987, 1995; Moulin 2003). This shows that equality an its substitute, the maximin m, is not necessarily the most esirable point. 8
of the veil of ignorance shoul be qualie. Even if, following Dupuy (1995), the uncertainty of life is larger toay than before, in practice, the society is largely frozen (a phenomenon known since Pareto) an the people that are in the higher class o not spent their time in thinking that they coul fall in the lowest class tomorrow, an conversely. Thinking that the iniviual may consier the right istribution of revenues before knowing their position is optimistic an unrealistic. Moreover, asserting that this conucts the agents to favor the the maximin, because each of them coul fall in this position, it is not appropriate: even in Rawls' perspective, the agents coul as well the pro-rich point r because they are optimistic an think that they will fall in the best position. Moehler (2010) unerlines also that Harsanyi, in its 1975 paper, argues that a rational iniviual woul maximize the average utility of the ierent positions of society. In terms of normative ecision theory, Harsanyi argues that a rational iniviual woul apply the principle of insucient reason (the Laplace rule) in the original position, whereas Rawls argues for the maximin rule (Moehler 2010); but for both Gauthier (1986) an Moehler (2010), the iniviuals consier rst their own iniviual gains an not the utilitarian point. For Rawls, it is possible to obtain a preferable state by moifying a given istribution, provie that the situation of the most unerprivilege is improve (it is the principle of the maximin). He thus proposes a ynamic vision of the optimum, since the unequal character of the situations can be moie in a irection or the other, provie that the most unerprivilege n their interest there (an that each iniviual has ex ante the same chances as the others to be in a given situation, accoring to its merits). 5 However, the maximin is still Paretian (since we choose a given point of the curve of eciency) an, in that sense, it remains conservative (in the political sense of the term) because one cannot move on the curve but only from the interior of this curve towars the curve. We can thus choose a istribution that one juges preferable rather than anotherthe point m rather than points u or r for exampleonly ex ante before having reache the eciency curve when one starts from a point in the interior to this curve. It has often been sai that the maximin supports a feeble ght against inequalities: this is the basis of the critics against Minc's report (Minc 1994) in France. It is perhaps an 5 Phelps has chosen to think ex post, when the roles have yet been attribute between the two iniviuals; else, one oes not see why one woul agree to gain less than the other. Consiering that the roles are attribute in avance is a hypothesis contraictory with Rawls' iea: the less favore shoul not be a particular person. One may qualify his point of view as practical or operational. 9
unfoune reproach, but the maximin is certainly a progress by respect to the Pareto optimum since we now woner which point of the curve must be retaine accoring to social criteria to etermine. Formally, the argument that the iniviuals ignore ex ante in which position they will be ex post, which is Harsanyi an Rawls' argument of veil of ignorance an original position (Harsanyi 1953, 1955, 1958, 1975; Rawls 1971) 6 shoul be qualie. Even if, following Dupuy (1995), the uncertainty of life is larger toay than before 7, in practice, the society is largely frozen (a phenomenon known since Pareto) an the people that are in the higher class o not spent their time in thinking that they coul fall in the lowest class tomorrow, an conversely. Believing that people may think about the just istribution of revenues before knowing their position is optimistic an unrealistic. Moreover, asserting that this conucts the agents to favor the the maximin, because each of them coul fall in this position, it is not appropriate: the agents coul as well the pro-rich point r because they are optimistic an think that they will fall in the best position. The beauty of the Rawlsian maximin is that it favors the poor without implying any loss in eciency: the economy is as ecient as in r or u. Nevertheless, in Phelps' presentation of the maximin, the eciency curve is taken as given: never the eciency curve is reconsiere, which woul be consiere as obvious by many but conservative by some. However, Kolm unerlines one of the iculties of the maximin (1972, p. 121). For example, let us assume two states A an B such as million people are happier in A than in B an only one person is happier in B than in A, but that this person is less happy than all the others - accoring to the funamental preferences - in each of the two states. Practical justice (the maximin) results in preferring the state B with state A, which puts all the weight on the least happy an takes account only of its situation, other than that of all the others. One can n that goo. But one can also eplore that the happiness of million is sacrice to that of only one, even unhappy that is this one. The argument scores a bull's-eye even if Kolm thinks that it has a limite impact in fact because of the form of the feasible omain, an that consequently Rawlsian justice requires implicitly that the size of the classes of iniviuals is ecreasing because of their income: more iniviuals in 2 that in 1, or if one prefers, more poor persons that rich persons. 6 See a etaile iscussion in Binmore (1989). 7 Even a banker may become homeless, as illustrate by the sa story of Jean-Paul Allou (Allou 2011). 10
In practice, this is generally respecte. But it remains that Kolm's argument implies also an amount of majority rule within the Rawlsian reasoning, what is awkwar if we take into account the well-known limits (paraox of Conorcet) which aect the majority rule. However, the Corollary 1 allows us to say that Rawls reintrouces egalitarianism an that the maximin oes not conuct to a feeble strike against inequality. 2.5 Conicts on the Paretian curve, stability an utilitarian optimum Within a Paretian framework, all the points are as stable the ones as the others, or more exactly the question of their stability oes not arise, since they are locate on the frontier of eciency. However, if one goes beyon this framework to consier the possibility of moves along the frontier of eciency, i.e., the possibility of conicts between iniviuals, then the stability of the various points is not the same one. These conicts can logically only occur after the choice of the social ecision maker; alternately one woul fall own on the case evoke previously of insoluble conict. However, at the same time, is it logical to think that there is conict after the choice of the social ecision maker? These conicts suppose a type of protest against the social ecision maker: that resembles to these chilren who ispute after the ivision of a cake by their parents. Proposition 2. The utilitarian point u is an equilibrium point. Proof. Consier the general case where equality is impossible or inecient. At the point m one can very strongly increase the income of iniviual 1 by egraing very little that of iniviual 2, an conversely at the point r, whereas at the point u, increasing the income of an unspecie iniviual obliges to ecrease by as much that of the other iniviual. Therefore, the points m an r are in a certain manner less stable than the point u, in that sense that conicts will be unbalance there. If it is suppose that the resistance of iniviual i is in inverse proportion of the elasticity E Ri/R j = Ri R j then in r, iniviual 1, the most favore, will ten to be less oppose to the requests of iniviual 2, less favore, because 1 is far from losing when the curve is vertical. Similarly, in m, iniviual 2 tens to satisfy more 1's requests at the beginning because when the curve is horizontal, he is far from losing, while at the same time he is alreay the least favore: this is an obvious paraox of the victim. In every case, as 11
one approaches u, the resistance of the iniviual who sees his position being egrae increases: starting from r, one will ten to stop out of u; similarly if we start from m. Hence, the point u is at the same time a point of steay balance an a point of accumulation. One can then think that, noting the subsequent possibility of conicts, the social ecision maker will choose the point u rather than the maximin m. However, if e is in the Paretian zone of the eciency curve as in Figure 5, the question of conict stability challenges the choice of e because u remains a stable point of accumulation in the event of conicts. In Figure 5, even if e is the point of equality, agent 2 will be less able to resist at the requests of agent 1 than agent 1 is able to resist the requests of agent 2 an the equilibrium will slip towars u to stabilize itself there. 2.6 Typology of policies In terms of simple typology of policies, the point r can be interprete as the point of the political har right-wing, those that can be qualie as egoistic. The point e is the point of the egalitarian left-wing that belongs to the French traition or can be qualie as being a matter for Utopian ieas because the point e poses many problems of existence an, if it exists, exclues the point m. The point m is the point of the moern left-wing as it gives the maximum to the less favore but by remaining realistic as it is still locate on the Paretian curve. The point u provies the maximum total revenue whatever inequality between iniviuals is. When one goes from the point r to the point u, the political right-wing abanons progressively its egoistic character to ten to be more welfarist; when one goes from m (or e) to u, the left-wing abanons its Utopian ieas (e) or its generosity to become also more welfarist. Hence, the point u is the limit between the political right-wing an the political left-wing: the omain of the political right-wing goes from r to u while the omain of the left-wing goes from u to m or e eventually. This armation will be qualie later. 2.7 Growth an maximin Even if the Phelps curve can be consiere as the frontier that inicates all the possible revenues that guaranteeing a given growth rate in statics, in ynamics, growth makes the eciency frontier to go to the North-East of the gure (see Figure 6) because in a growing economy, it is possible to pay more one agent if 12
Figure 6: Growth an eciency curve the revenue of the other oes not change. For example, for the same level R 1 of agent's 1 revenue, it is possible to pay more agent 2: R 2 instea of R 2, an conversely. However, choosing the maximin to secure a left-wing economic policy is not sucient as soon as ynamics are consiere. Particularly, growth may ruin all eorts mae in favor of the less favore. Growth may have a strong impact on economic justice. First, homothetic growth can be qualie as neutral growth: both benet from growth. Secon, growth coul make equality easier or, to the contrary, more icult (Figure 8, left an right respectively). When equality is mae easier, growth makes agent 2 to become sometimes the richest; it coul eventually make agent 2 to become always the richest. In homothetic growth (Figure 7), where the eciency omain evolves between two straight lines, it is self-obvious that this case cannot occur. Beyon that, it is awaite that when the maximin is selecte, growth shoul be favorable to the poor (see Figure 9, left). Similarly, if the pro-rich point is selecte, it is awaite (even consiere as immoral by many) that growth is favorable to the rich (see Figure 9, right). However, the main question with growth is that it is possible to have a paraoxical evolution, namely a pro-poor growth when the point r has been chosen or a pro-rich growth when the maximin has been chosen. Let's illustrate 13
Figure 7: Homothetic growth an eciency curve Figure 8: Growth: equality mae easier (left) an more icult (right) Figure 9: Normal pro-poor growth (left); normal pro-rich growth (right) 14
Figure 10: (right) Paraoxical pro-rich growth (left); paraoxical pro-poor growth Figure 11: Political har right-wing an growth: the rich are losing (left); maximin an growth: the poor are losing (right) this. Growth may benet to the poor even if the point r has been selecte as in Figure 10 right: the pro-rich economic policy is a failure; growth may benet to the rich even if the maximin has been selecte as in Figure 10 left. In this case, the left-wing economic policy can be consiere as being a failure. When growth makes that the curves are intersecting (in their ecient part or not), growth may even have an inverse eect, for example making the revenue of the rich lower after growth even if a right-wing policy have been chosen (an conversely for the poor). In Figure 11, left, the rich are losing if the points r an r have been selecte, but they have larger revenue if the points m an m have been selecte! In Figure 11, right, the poor are losing if the maximin is chosen in the new curve: they are winning if the point r is chosen but they are losing if the point m is selecte. Growth may make that choosing the utilitarian the point u inuces a variation in the sharing between the revenues: making a welfarist policy is absolutely not a guarantee of neutrality, as shown in Figure 12. 15
Figure 12: Utilitarian point an change in revenue istribution: growth favorable to the poor (left) an growth unfavorable to the poor (right) 3 Lessons from the Nash bargaining 3.1 Nash bargaining If we consier the problem as a two-persons game an its generalizei.e., by ropping symmetry axiom 8 Nash solution (Nash 1950a an 1950b; Roth 1979; Rubinstein 1982; Binmore, Rubinstein an Wolinsky 1986; Wright no ate), we are able to generate all points in {m, r}, that is, all possible points when the agents have ierent claims. Here, it is applie on a curve of which arguments are the revenues rather than utilities but we have the right to o this. 910 Consier the eciency curve R 2 (R 1 ). Denote by R r 1 an R r 2 the coorinates of the point r on the X-axis an Y-axis respectively; enote by R m 1 an R m 2 the coorinates of the point m on the X-axis an Y-axis respectively. R1 m is iniviual 1's revenue when iniviual 2 obtains its maximum revenue, i.e., the maximin m, an R r 2 is iniviual 2's revenue that when iniviual 1 obtains its maximum revenue, i.e., r. We choose the point of coorinates {R m 1, R r 2} as isagreement point 11 because any point outsie the convex is impossible: assume that another point is chosen such as those of Figure 13; from there, iniviual 1 may 8 When utilities are argument, the four axioms of the Nash bargaining are: invariance to equivalent utility representations, symmetry, inepenence of irrelevant alternatives an Pareto eciency. The rst one is not necessary as we revenues are arguments an the secon one is roppe in the generalize Nash bargaining. 9 Nash an followers consier utilities because they think in terms of set of commoities that are aggregate by the iea of utility. If we think in terms of revenue, thinking in terms of utility is unnecessary. 10 For the link between Nash's an Rawls' theories, see Lengaigne (2004). We o not consier the Kalai-Smoroinsky's solution (Kalai an Smoroinsky 1975; Kalai 1977) because it oes not satisfy the axiom of inepenence of irrelevant alternatives. 11 The isagreement point is also calle threat point or even status quo by Thomson (1981) or Binmore et al. (1986); it is also the point where both iniviuals are place when they fail to bargain. 16
Figure 13: Disagreement point increase its revenue up to point a without egraing those of iniviual 2 but as a is not ecient, iniviual 2 may also increase its revenue up to point b, which is this time ecient, without egraing those of iniviual 1; an conversely by reversing the orer of the actions of both iniviuals (which oes not appears in Figure 13): however, when point is chosen, iniviual 1 may increase its revenue by going to point r but iniviual 2 cannot make any other movement to increase its own revenue as he is locate on the eciency curve; conversely, iniviual 2 may increase its revenue up to point m but that lets no leeway to iniviual 1; conversely, any point insie the convex {m, r, } is not the worst point that both agents can accept. Therefore, R1 m an R2 r are the minimum revenues (while R1 r an R2 m are the maximum revenues): is the worst point. Therefore, for the Nash bargaining, we consier the convex {m, r, }.The Nash solution is R 1 = arg max [R 1 R1 m ] θ [R 2 (R 1 ) R2] r 1 θ (1) subject to R 1 R1 m an R 2 (R 1 ) R2. r Equation 1 is a set of hyperbolas, as shown in Figure 14. The rst orer conition is R 2 (R 1 ) = θ 1 θ R 2 (R 1 ) R r 2 R 1 R m 1 (2) 17
Figure 14: Nash hyperbolas When θ = 1 equation (2) is not ene but the Nash solution turns out to be R 1 = arg max [R 1 R m 1 ] subject to R 1 R m 1 : R 1 is maximize an the solution is the point r. When θ = 0, it follows from (2) that R 2 (R 1 ) = 0: R 2 (R 1 ) is maximize an the solution is the point m. The utilitarian point is ene by arg max [R 1 + R 2 (R 1 )], the rst orer conition being R 2 (R 1 ) = 1 (3) Therefore, solution (2) correspons to the the utilitarian point given by (3) if θ R 2 (R 1 ) R2 r 1 θ R 1 R1 m = 1 (4) that is, θ = R 1 R m 1 (R 1 R m 1 ) + (R 2 (R 1 ) R r 2 ) (5) If the curve is symmetric by respect to the bisector passing by, then θ = 1 2 an u = s. 12 Moreover, we remark that θ cannot be etermine ex ante: it is foun only when the point u has been etermine because in (5), neither R 1 nor R 2 (R 1 ) are xe but they are variable. Remark. The above reasoning about the utilitarian point as steay balance an 12 In the space of utilities, u = s always hols. 18
accumulation point is ecient in the sense that the utilitarian point is not the unique point which is a steay balance an an accumulation point as expose above. Proposition 3. Depening on the parameter θ in a Nash generalize bargaining, any point is an equilibrium point, epening on which θ has been chosen. This proposition obviously inclues the utilitarian point (Thomson 1981). Proof. It is self-evient. Remark. The role of the social ecision maker coul be to choose the parameter θ. However, θ may also be consiere as an inicator of iniviuals' relative force. This shows that the maximin is a very particular case of bargaining where the poorest receives all the bargaining power. We on not think that Rawls argument about the veil of ignorance an the original position iscusse in sub-section 2.4 is sucient to justify that the bargaining power is entirely attribute to the less favore. 3.2 Surplus-equality The Nash bargaining erivation of the various points along the eciency curve suggests a ierent enition of equality. We call this point surplus-equality, enote s in Figure 15: it is the point where the surplus is share in two equal parts, etermine by the intersection of the rst bisector that passes by the isagreement point (R1 m, R2) r an the eciency curve. This sharing line has for equation R 2 = R 1 (R1 m R2) r an is parallel to the main bisector of equation R 2 = R 1. The point s is also a Nash equilibrium if the aequate value of θ is chosen. It is easily erivable from the moment that the equation of the eciency curve is known. Notice that in the worl of utilities, the point s woul be the Nash equilibrium itself. The point s generally iers from the egalitarian point e, even if e is on the eciency curve as in Figure 5. From the moment that the eciency curve is given an accepte by both iniviuals, the surplus-equality point is those which shares equally what can be share. Inee, pure equality, the point e, refers to the original point where both iniviuals receive zero, i.e., the pure original position where both iniviuals have the same chance of being rich or poor. However, no one wants to be in that place because he receives no revenue there: the isagreement point is preferre by both, even if in that point 19
Figure 15: Surplus equality a certain level of inequality has been yet introuce, which supposes that the ierences in talents have been recognize. In a wor, iniviuals are not in the position {0, 0} ex ante, even if they think that they coul fall in the best or the worst position ex post. This coul seem unjust but the existence of the eciency curve itself makes that ierences in talents are preetermine. More precisely, the eciency curve etermines the position of the isagreement point but the converse proposition is false: efening as original point means that the eciency curve is given an known. If such a presupposition is rejecte, an if we enote by O the point where R 1 = R 2 = 0, no eciency curve can be rawn in O, no apportionment one except pure equality an no maximin etermine at all. In other wors, there is a contraiction between taking O as original point an consiering an eciency curve, unless the eciency curve is such that the isagreement point is confuse with the origin, a very special case where s is confuse with e, which we call super-equality an is rawn in Figure 17. The Proposition 4 an it Corollary treat this case. It is worth noting that pure equality may be ecient an at the same time not confuse with surplus-equality as in Figure 16. Moreover, s cannot be place to the left of e: this woul require that itself is to the left of the bisector {O, e}, which woul mean that R 1 < R 2 in. Proposition 4. Pure equality an surplus-equality are confuse if an only if the isagreement point shares equally the revenues. 20
Figure 16: Equality ecient but not confuse with surplus-equality. l Figure 17: s = e 21
Proof. The straight lines {O, e} an {, s} are parallel by enition. Therefore, from the axiom of Eucliean geometry, the point is locate on {O, e} if an only if the point s is on {O, e}, i.e., e an s are confuse. Corollary 2. If pure equality an surplus-equality are confuse then equality is ecient. Proof. Equality is ecient if the point e is locate on the eciency curve. As {O, e} an {, s} are parallel, if e = s then e is on the eciency curve because s is on the eciency curve by construction. For these reasons, surplus-equality s correspons to the true equality because it refers to the isagreement point where both iniviuals are place before bargaining. Moreover, the segment {m, s} in the eciency curve is those that must be chosen for conucting a left-wing policy because it is largely insensitive to a growing inequality. Proposition 5. When inequality increases, that is, the eciency curve is in- nitely attene along the Y-axis, or R1 r, m remaining xe, then R1 u but R1 s is nite an equal to R1 m + R2 m. Proof. The proof is obvious. We are in the normal case of Figure (18). When R1 r, r goes to the right, by construction. The point s is foun by intersecting the bisector that passes by (R1 m, R2) r an the eciency curve: s moves slightly to the right, to the limit up to the point (R1 m + R2 m, R2 r + R2 m ) because the curve tens to be horizontal between m an s. As the eciency curve tens to be very at, the point where its slope equals 1 innitely goes to the right. As in the point n, curve's slope is between 1 an zero, n goes to the right between n an r. See Figure 18. Therefore, while a right-wing policy correspons to the segment {u, r}, a left-wing policy shoul be restricte to the segment {m, s} (or to {e, s}, if e is locate on the eciency curve). If the eciency curve is normal, that is, attene along the Y-axis as in Figures 1-5, the equal-surplus point is to the left of the utilitarian point (see Figure 18): θ u > 1 2 ; s is more favorable to the poor than u. However, if the gure is attene along the X-axis, which is not the stanar case, the equalitysurplus an the Nash egalitarian points are to the right of the utilitarian point as in Figure 19: θ u < 1 2 ; they are more favorable to the rich than u; s is to the 22
Figure 18: Eciency curve attene along the Y-axis right of n: it is the most favorable to the rich. Obviously, both n an s are the same if the curve is symmetrical by respect to the isagreement point; it is not necessary that the eciency be symmetrical by respect to the origin. The point s also iers from the utilitarian point u when the eciency curve is not symmetrical. Remark. Surplus equality may be ierent. Instea of sharing equally the surplus between both iniviuals, one may share it following a ierent apportionment rule, as the proportional rule: the surplus may be allocate proportionally R to the minimum revenues, that is, to m 1 R to which correspons a sharing line, 1 m+rr 2 of equation R 2 = Rm 1 R R 2 r 1, which passes by the origin. This type of surplus sharing is highly unequal, leaing to a point that can be to the right of u, very close to r, even in a normal eciency curve. Surplus may also be share following the Talmu. Homothetic growth projects also an ms homothetically. See 3.3 Case of more than two agents We have consiere only two agents: things are much more complicate when three or more iniviuals are consiere. For n agents, the Phelps curve is now a surface of n 1 imensions. We impose a lexicographic orer between all agents: R 1 >... > R i >... > R n 1 > R n, where R n is the revenue of the poorest. R n = f (R 1,..., R n 1 ) is the eciency curve such that Ri 2 R i R 2 n R n 0 an 0 for any i. It is hany to return to the Nash bargaining, for n-persons 23
Figure 19: Atypical curve: eciency curve attene along the X-axis Figure 20: Homothetic growth for an s 24
here. The Nash bargaining solves: n 1 R i = arg max [R i Ri m ] θi [f (R 1,..., R n 1 ) Rn] r 1 n 1 i=1 θi i=1 for any i = 1,..., n 1 subject to R i Ri m for any i = 1,..., n 1, an f (R n ) Rn. r For generating the minimax, one poses θ i = 0 for any i = 1,..., n 1, which gives n 1 R i = arg max [f (R 1,..., R n 1 ) Rn] r i=1 for any i = 1,..., n 1. Therefore, the ierence between R r n an f (R 1,..., R n 1 ) is maximize. See Figure 21 for the three iniviuals case. The point m is the maximin, r is the pro-rich point an m 12 is the maximin between iniviuals 1 an 2. The curves {r, m 12 }, {r, m} an {m 12, m} are the eciency curves between iniviuals 1 an 2, iniviuals 1 an 3 an iniviuals 2 an 3, respectively. However, things are a little more complicate when the highest point along the Z-axis is not unique as it is in Figure 21. For instance, in Figure 22, where the gray areas inicate where the lexicographic orer is violate, we may search the maximin between R 3 an R 2 for a given R 1. When R 1 is minimum, it is along curve {B 2, m 2 }; when R 1 is maximum, it is along curve {B 1, m 1 }. Therefore, this maximin is curve {m 1, m 2 }. When we choose the maximin between R 1 an R 2, it is point m 2. However, nothing proves that the the revenue R 3 that correspons to m 2 is higher than those of m 1 : {m 1, m 2 } may perfectly be ecreasing (an m 1 be the overall maximin) without violating convexity. In other wors, the overall maximin oes not necessarily correspon to the leximin (i.e., the composition of the maximin between two agents successively place on the scale of revenues). problem. Obviously, surplus-equality oes not poses such a Moreover, the volume of computations become unrealistic with millions of iniviuals: it becomes necessary to make a small number of groups. Unfortunately, the solution is completely sensitive to the number n of agents in each group. 25
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000 000000000000000000000000000000000 000000000000000000000000 000000000000000 000 000 0000000000 0000000000000000000 0000000000000000000000000000 00000000000000000000000000000000000000 00000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 Figure 21: Maximin for three agents. 26
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000 000000000000000000000000000000000 00000000000000000000000 0000000000000 00 00 0000000000 0000000000000000000 00000000000000000000000000000 00000000000000000000000000000000000000 000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 Figure 22: Maximin for three agents. 27