rawls, phelps, nash: e ciency curve and economic...

Rawls, Phelps, Nash:

e�ciency curve and economic justice

Louis de Mesnard

June 14, 2011

University of Burgundy and CNRS, Laboratoire d'Economie et de Gestion

(UMR CNRS 5118); 2 Bd Gabriel, B.P. 26611, F-21066 DIJON Cedex,

FRANCE. E-mail: [email protected]

Charles Gide Days : Justice & Economics

June, 16 & 17, 2011

Draft version

Abstract

This article o�ers some re�ections on the interpretation of Rawls' Theory

of Justice that Phelps gives by the e�ciency curve. In the �rst section

we demonstrate that the Phelps curve allows showing that egalitarianism

may be impossible, ine�cient or also possible but this last case excludes

and dominates the Rawlsian maximin. The debate egalitarianism vs. eq-

uity is clari�ed. We examine the e�ect of growth. Growth could make

equality easier in some cases and more di�cult to reach in some other

cases. Choosing the maximin does not guarantee that the growth is al-

ways favorable to the poor: it can be paradoxical because the poor can

be losing even when the maximin is selected. By considering that a Nash

bargaining is able to generate any point in the Phelps e�ciency curve, we

examine a new point, surplus-equality: it corresponds to an equal sharing

from the �disagreement point�, which should be considered as the origin

from the moment that the e�ciency curve is given. The transposition to n

agents is delicate: the overall maximin is not necessarily the leximin and

it is better to consider groups of agents. In conclusion, the area between

the maximin and surplus-equality should be the base of a left-wing policy

as it protects against a growing inequality.

1

JEL classi�cation. D63; H23; I31

Keywords. Rawls; Phelps; Nash; maximin; inequality; e�ciency

Abbreviated title. Rawls, Phelps, Nash

1 Introduction

The Theory of justice of John Rawls (Rawls 1971, 1993; Barry 1989; Gibbard

1991) includes two principles. We will quote them in the form most recently

stated by Rawls (1989). The �rst principle treats of freedoms:

Each person has an equal right to a fully adequate equal basic liberties

for all, which is consistent with a similar system of liberties for all.

The second principle treats inequalities:

The social and economic inequalities must satisfy two conditions:

1) they must �rst be attached to functions and positions open to all,

in areas of fair equal opportunities and

2) they must obtain the greatest bene�t to the most disadvantaged

members of society.

Concerning remunerations, the Rawlsian position is often summarized by a sim-

ple choice, as in McClelland's example: do we prefer a distribution of income

such as the average is 20000$ and the poor receive 15000$, or on the contrary

an average of 40000$, the poor receiving only 14000$? The position chosen by

Rawls corresponds to the �rst possibility (maximization of the position of the

most disadvantaged or maximin or principle of di�erence), even if McClelland

advances that the majority of American would choose the second face of the

alternative (McClelland 1990, p. 95). The Rawlsian position is often considered

as one of the most �liberal� in the American sense of the term, as close as pos-

sible to absolute (or strong) egalitarianism; this is why the maximin is called

practical justice by Kolm (1972, 1996b). However, Rawls is accused of support-

ing a feeble �ght against inequalities: in France, Minc's report (1994) has been

strongly criticized for this reason (in a country as France, egalitarianism is a

sensitive issue since the French revolution).

Phelps, Nobel prized in 2006, one of the most renown supporters of Rawls,

has very clearly explained Rawls' theory (1995). He draws the e�ciency curve

where the optimum must be chosen, which assumes a Paretian optimum, and he

posits the various points that can be chosen; among them, there is the maximin,

2

equality, the utilitarian point and the pro-rich point. As Phelps' tool is as

pedagogic as Hicks-Hansen IS-LM model is�even if most consider that IS-LM

betrays Keynes' idea�, and as the e�ciency curve undoubtedly exists, we return

in this paper to Phelps' contribution to examine what can be deduced from the

e�ciency curve, particularly for what concerns the question of the maximin, the

e�ect of growth, and overall egalitarianism and its impossibility in most cases.

Pure egalitarianism is determined by the original point where the revenue of

all individual is equal to zero (which is not Rawls' original position). However,

as the question can be considered as a sharing problem solved by bargaining,

we also introduce a �dose of Nash� in Rawls and Phelps. We consider that the

various points can be deduced by a generalized Nash bargaining (Nash 1950a

and 1950b; Rubinstein 1982; Binmore, Rubinstein and Wolinsky 1986). From

Nash comes the idea of �disagreement point� (the point where all individual

are placed before bargaining), which allows us to examine a new egalitarianism,

surplus-equality, where equality is determined by respect to the disagreement

point. This idea of equality is no more a chimera as pure equality is. We deduced

of all this that a left-wing policy must choose a point between the maximin and

surplus-equality. Nash allows an elegant generalization to the case of n agents,

etc.

2 The maximin

2.1 The maximin on the e�ciency curve

Phelps draws a graph, certainly simplifying and which could be likely to betray

the thought of Rawls by simplifying it in a neoclassical direction, 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 published

textbook, he [Rawls] said its exposition of his theory of justice was entirely

accurate� (Phelps 2011). In this graph, the Paretian e�ciency curves (see Figure

5) is the frontier curve of possible remunerations or curve of possible, which

indicates all the possible revenues that guarantee a given output (or even a

given growth rate). The important thing to be noticed is that the agents'

revenues are the argument, but not utilities: it is what makes Phelps' curve

di�erent to what is usual. As Phelps' curve uses revenues as arguments, the

problem of a�ne transformations of utilities (translation or scaling) does not

apply. Rawls (1971) himself criticizes the idea of utility: he prefers the idea

3

of primary goods. 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 ordinal, and if they are cardinal, they are only

de�ned at an a�ne transformation), the approaches that require interpersonal

comparisons of utilities obviously need cardinal utilities that are sensitive to

a�ne transformations.1 We do not think that intertemporal comparisons of

utilities and cardinal utilities can be accepted but if we refuse them, it becomes

impossible to consider some remarkable points on the e�ciency curve as the

utilitarian point. Passing by the revenues as done by Phelps allows interpersonal

comparisons and de�ning all points on the e�ciency curve.

Stated for two individuals, the e�ciency curve is R2 = f (R1), by supposing

that individual 1 is always paid best, that is, by supposing that the curve is

�attened along the Y-axis and remains on the right of the �rst bisector at least

for its e�ective part.2 Let us recall that the frontier of e�ciency corresponds

to what it is possible to obtain at best, without degrading the situation of an

individual in order to improve that of another. In the interior of the frontier,

one can increase at the same time the income of individuals 1 and 2 by going

to the top, towards the line, or both at the same time. On the frontier one

cannot increase the income of one individual without decreasing that of the

other. All the points of the frontier are equally possible, except those which

correspond to a return of the curve on itself, from where the form traced on

Figure 1: the e�cient part�or Pareto-optimal�of the curve, is the segment

(r,m): before m, or after r, the income of both individuals may increase or

decrease simultaneously. The curve (r,m) is continuous, derivable and is such

that dR2

dR1≤ 0 and d2R2

dR21≤ 0. Several typical points appear on the curve in

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 underprivileged. This point may be on the left of

m on the ine�cient part of the curve, as on Figure 5 or to the right. For Rawls,

m is the right or equitable position. Rawls defends the maximin by saying that

individuals ignore by advance 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 bad.

1The arguments of the problem are also certain: no need of von Neumann-Morgensternutilities, expected utilities, etc. On these approaches, see Harsanyi (1953, 1955); Hammond(1976, 1979, 1993), Bezembinder and van Acker (1987); Bosmans and Ooghe (2006); Miyag-ishima (2010).

2This shape of curve is also quoted by Kolm (1972, p. 33, 1.e example).

4

��

��

�Figure 1: Equality is impossible

Any intermediate solution, obtained by making a linear combination between

the incomes of the two individuals, and located betweenm and r could be chosen

and re�ects a relation of force between both: the weighting may re�ect any social

criterion. The point r is obtained if the best remunerated group is dominating,

whereas the point m gives the primacy to the less remunerated group. Phelps

(1985) says that this point is an �ideal e�ciency� while only the segment {r,m}is e�cient.

In the point u the �social income� is maximized, i.e., the sum of the incomes;

e�ciency curve's slope is there equal to −1; Gamel (2010) assimilates it to the

welfarism. It is also the point which corresponds to the Bentham's utilitarian

optimum, that is, to the maximization of the mean (this one could be weighted)

of the incomes of individuals 1 and 2. In the point e the two incomes are

equal. All this is consequentialist: only the consequences of policy decisions are

examined.

2.2 Typology of curves

The curves may adopt various forms. They may be very concentrated as in

Figure 2, left: in this case, the problem of choosing a point on the e�ciency

curve is practically evacuated. They may be very large, as in Figure 2, right:

the problem of choosing a point is increased. The e�ciency curve may be also

5

��

��

��

�� Figure 2: Typology of e�ciency curves: concentrated (left), large (right)�

��

��

�� Figure 3: Typology of e�ciency curves: atypical (left), very unequal (right)

atypical, �attened along the X-axis, as in Figure 3, left; or it can be very unequal,

�attened along the Y-axis, as in Figure 3, right.

Obviously, depending of the form of the e�ciency curve, the point u may be

closer to the point r or to the point m. However, as R1 ≥ R2, the curve is

probably �attened 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 excepted, 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 and the curve does not exist.

Proposition 1. Reaching equality can be, respectively, (i) impossible, (ii) inef-

�cient or (iii) possible but by excluding and dominating the Rawlsian optimum

6

��

��

�Figure 4: Equality is possible but ine�cient

in this last case.

Corollary 1. Among the e�cient points, the maximin is the closest to egali-

tarianism.

Proof. The proof of 1 and its corollary is done graphically.

(i) Equality is impossible when the e�ciency curve does 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 ine�cient if the e�ciency curve intersects the �rst

bisector to the left of the Rawlsian optimum as in Figure 4. Forcing equality

implies becoming under-e�cient. The price to pay for egalitarianism is ine�-

ciency. In this case, reaching equality is fanciful and the point m is again the

closest to egalitarianism.3

In both cases of Figures 1 or 4, Phelps sees a justi�cation of the maximin:

any point to the right of m on the e�ciency curve (u, r, etc.) corresponds

to more inequality. Therefore, he adopts a point of view similar to those of

Kolm (1972, 1996b) and its idea of practical justice. However, equality is itself

a judgment of value. If both individuals have the same right on the available

wealth, they have to share equally but when the individuals have di�erent claims

3On the �Pareto argument� and feasibility of equality, see Cohen (1995) and its critique byShaw (1999).

7

��

��

��

�Figure 5: Example of e�ciency curve

they share di�erently.4

(iii) When the e�ciency curve intersects the �rst bisector to the right of the

Rawlsian optimum as in Figure 5, equality is possible (it is located in the e�cient

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 individual 2 is higher than those

of individual 1: between e and m individual 2 is the richest while individual 1 is

the less favored. Therefore, the solution of the maximin is e: m is never reached

and e is selected. The point m does not correspond to the maximin anymore

and m is not the closest to egalitarianism: equality dominates and excludes the

Rawlsian maximin.

2.4 Discussing the idea of maximin

For Harsanyi and Rawls, the individuals ignore ex ante in which position they

will be ex post, which is the argument of the veil of ignorance and the original

position developed by (Harsanyi 1953, 1955, 1958, 1975) and Rawls (1971): it

is why they decide to give the larger possible revenue. However, this argument

4In the Aristotelian tradition, they share proportionally, while in the Talmudic tradition,they share in a di�erent way (Rabinovitch 1973; O'Neill 1982; Aumann and Maschler 1985;Young 1987, 1995; Moulin 2003). This shows that equality and its substitute, the maximinm, is not necessarily the most desirable point.

8

of the veil of ignorance should be quali�ed. Even if, following Dupuy (1995),

the uncertainty of life is larger today than before, in practice, the society is

largely frozen (a phenomenon known since Pareto) and the people that are in the

higher class do not spent their time in thinking that they could fall in the lowest

class tomorrow, and conversely. Thinking that the individual may consider the

right distribution of revenues before knowing their position is optimistic and

unrealistic. Moreover, asserting that this conducts the agents to favor the the

maximin, because each of them could fall in this position, it is not appropriate:

even in Rawls' perspective, the agents could as well the pro-rich point r because

they are optimistic and think that they will fall in the best position. Moehler

(2010) underlines also that Harsanyi, in its 1975 paper, argues that �a rational

individual would maximize the average utility of the di�erent positions of society.

In terms of normative decision theory, Harsanyi argues that a rational individual

would apply the principle of insu�cient reason (the Laplace rule) in the original

position, whereas Rawls argues for the maximin rule� (Moehler 2010); but for

both Gauthier (1986) and Moehler (2010), the individuals consider �rst their

own individual gains and not the utilitarian point.

For Rawls, it is possible to obtain a preferable state by modifying a given

distribution, provided that the situation of the most underprivileged is improved

(it is the principle of the maximin). He thus proposes a dynamic vision of the

optimum, since the unequal character of the situations can be modi�ed in a

direction or the other, provided that the most underprivileged �nd their interest

there (and that each individual has ex ante the same chances as the others to

be in a given situation, according to its merits).5

However, the maximin is still Paretian (since we choose a given point of the

curve of e�ciency) and, 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 towards the curve. We can thus choose a distribution that

one judges preferable rather than another�the point m rather than points u

or r for example�only ex ante before having reached the e�ciency curve when

one starts from a point in the interior to this curve. It has often been said

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

5Phelps has chosen to think ex post, when the roles have yet been attributed betweenthe two individuals; else, one does not see why one would agree to gain less than the other.Considering that the roles are attributed in advance is a hypothesis contradictory with Rawls'idea: the less favored should not be a particular person. One may qualify his point of view as�practical� or �operational�.

9

unfounded reproach, but the maximin is certainly a progress by respect to the

Pareto optimum since we now wonder which point of the curve must be retained

according to social criteria to determine.

Formally, the argument that the individuals ignore ex ante in which position

they will be ex post, which is Harsanyi and Rawls' argument of veil of ignorance

and original position (Harsanyi 1953, 1955, 1958, 1975; Rawls 1971)6 should be

quali�ed. Even if, following Dupuy (1995), the uncertainty of life is larger today

than before7, in practice, the society is largely frozen (a phenomenon known

since Pareto) and the people that are in the higher class do not spent their time

in thinking that they could fall in the lowest class tomorrow, and conversely.

Believing that people may think about the just distribution of revenues before

knowing their position is optimistic and unrealistic. Moreover, asserting that

this conducts the agents to favor the the maximin, because each of them could

fall in this position, it is not appropriate: the agents could as well the pro-rich

point r because they are optimistic and 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 e�ciency: the economy is as e�cient as in r or u. Nev-

ertheless, in Phelps' presentation of the maximin, the e�ciency curve is taken

as given: never the e�ciency curve is reconsidered, which would be considered

as obvious by many but conservative by some. However, Kolm underlines one

of the di�culties of the maximin (1972, p. 121). For example, let us assume

two states A and B such as million people are happier in A than in B and only

one person is happier in B than in A, but that this person is less happy than

all the others - according to the fundamental 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 and takes account only

of its situation, other than that of all the others. One can �nd that good. But

one can also deplore that the happiness of million is sacri�ced 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 limited impact in fact because of the form of the

feasible domain, and that consequently Rawlsian justice requires implicitly that

the size of the classes of individuals is decreasing because of their income: more

individuals in 2 that in 1, or if one prefers, more poor persons that rich persons.

6See a detailed discussion in Binmore (1989).7Even a banker may become homeless, as illustrated by the sad story of Jean-Paul Allou

(Allou 2011).

10

In practice, this is generally respected. But it remains that Kolm's argument

implies also an amount of �majority rule� within the Rawlsian reasoning, what is

awkward if we take into account the well-known limits (paradox of Condorcet)

which a�ect the majority rule. However, the Corollary 1 allows us to say that

Rawls reintroduces egalitarianism and that the maximin does not conduct to a

feeble strike against inequality.

2.5 Con�icts on the Paretian curve, stability and utilitar-

ian optimum

Within a Paretian framework, all the points are as stable the ones as the others,

or more exactly the question of their stability does not arise, since they are

located on the frontier of e�ciency. However, if one goes beyond this framework

to consider the possibility of moves along the frontier of e�ciency, i.e., the

possibility of con�icts between individuals, then the stability of the various

points is not the same one. These con�icts can logically only occur after the

choice of the social decision maker; alternately one would fall down on the case

evoked previously of insoluble con�ict. However, at the same time, is it logical

to think that there is con�ict after the choice of the social decision maker?

These con�icts suppose a type of protest against the social decision maker: that

resembles to these children who dispute after the division of a cake by their

parents.

Proposition 2. The utilitarian point u is an equilibrium point.

Proof. Consider the general case where equality is impossible or ine�cient. At

the point m one can very strongly increase the income of individual 1 by degrad-

ing very little that of individual 2, and conversely at the point r, whereas at the

point u, increasing the income of an unspeci�ed individual obliges to decrease

by as much that of the other individual. Therefore, the points m and r are in

a certain manner less stable than the point u, in that sense that con�icts will

be unbalanced there. If it is supposed that the resistance of individual i is in

inverse proportion of the elasticity ERi/Rj= dRi

dRjthen in r, individual 1, the

most favored, will tend to be less opposed to the requests of individual 2, less

favored, because 1 is far from losing when the curve is vertical. Similarly, in m,

individual 2 tends 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 already

the least favored: this is an obvious �paradox of the victim�. In every case, as

11

one approaches u, the resistance of the individual who sees his position being

degraded increases: starting from r, one will tend to stop out of u; similarly if we

start from m. Hence, the point u is at the same time a point of steady balance

and a point of accumulation. One can then think that, noting the subsequent

possibility of con�icts, the social decision maker will choose the point u rather

than the maximin m.

However, if e is in the Paretian zone of the e�ciency curve as in Figure 5,

the question of con�ict stability challenges the choice of e because u remains a

stable point of accumulation in the event of con�icts. 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 and the equilibrium will

slip towards u to stabilize itself there.

2.6 Typology of policies

In terms of simple typology of policies, the point r can be interpreted as the

point of the political hard right-wing, those that can be quali�ed as egoistic.

The point e is the point of the egalitarian left-wing that belongs to the French

tradition or can be quali�ed as being a matter for Utopian ideas because the

point e poses many problems of existence and, if it exists, excludes the point m.

The point m is the point of the �modern� left-wing as it gives the maximum to

the less favored but by remaining realistic as it is still located on the Paretian

curve. The point u provides the maximum total revenue whatever inequality

between individuals is. When one goes from the point r to the point u, the

political right-wing abandons progressively its egoistic character to tend to be

more welfarist; when one goes from m (or e) to u, the left-wing abandons its

Utopian ideas (e) or its generosity to become also more welfarist. Hence, the

point u is the limit between the political right-wing and the political left-wing:

the domain of the political right-wing goes from r to u while the domain of the

left-wing goes from u to m or e eventually. This a�rmation will be quali�ed

later.

2.7 Growth and maximin

Even if the Phelps curve can be considered as the frontier that indicates all the

possible revenues that guaranteeing a given growth rate in statics, in dynamics,

growth makes the e�ciency 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 and e�ciency curve

the revenue of the other does 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 instead of R∗2, and

conversely.

However, choosing the maximin to secure a left-wing economic policy is not

su�cient as soon as dynamics are considered. Particularly, growth may ruin all

e�orts made in favor of the less favored.

Growth may have a strong impact on economic justice. First, homothetic

growth can be quali�ed as neutral growth: both bene�t from growth. Second,

growth could make equality easier or, to the contrary, more di�cult (Figure 8,

left and right respectively). When equality is made easier, growth makes agent

2 to become sometimes the richest; it could eventually make agent 2 to become

always the richest. In homothetic growth (Figure 7), where the e�ciency domain

evolves between two straight lines, it is self-obvious that this case cannot occur.

Beyond that, it is awaited that when the maximin is selected, growth should

be favorable to the poor (see Figure 9, left). Similarly, if the pro-rich point

is selected, it is awaited (even considered 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

paradoxical 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 and e�ciency curve

��

��

��

� ��Figure 8: Growth: equality made easier (left) and more di�cult (right)

��

��

��

� � ��

Figure 9: Normal pro-poor growth (left); normal pro-rich growth (right)

14

��

��

��

��

� � ��

Figure 10: Paradoxical pro-rich growth (left); paradoxical pro-poor growth(right)��

� ��

��

��

Figure 11: Political hard right-wing and growth: the rich are losing (left); max-imin and growth: the poor are losing (right)

this. Growth may bene�t to the poor even if the point r has been selected as

in Figure 10 right: the pro-rich economic policy is a failure; growth may bene�t

to the rich even if the maximin has been selected as in Figure 10 left. In this

case, the left-wing economic policy can be considered as being a failure.

When growth makes that the curves are intersecting (in their e�cient part

or not), growth may even have an inverse e�ect, for example making the revenue

of the rich lower after growth even if a right-wing policy have been chosen (and

conversely for the poor). In Figure 11, left, the rich are losing if the points r

and r′ have been selected, but they have larger revenue if the points m and m′

have been selected! 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 selected.

Growth may make that choosing the utilitarian the point u induces a vari-

ation in the sharing between the revenues: making a �welfarist� policy is abso-

lutely not a guarantee of neutrality, as shown in Figure 12.

15

��

��

��

��

��

��Figure 12: Utilitarian point and change in revenue distribution: growth favor-able to the poor (left) and growth unfavorable to the poor (right)

3 Lessons from the Nash bargaining

3.1 Nash bargaining

If we consider the problem as a two-persons game and its generalized�i.e., by

dropping symmetry axiom8�Nash solution (Nash 1950a and 1950b; Roth 1979;

Rubinstein 1982; Binmore, Rubinstein and Wolinsky 1986; Wright no date), we

are able to generate all points in {m, r}, that is, all possible points when the

agents have di�erent claims. Here, it is applied on a curve of which arguments

are the revenues rather than utilities but we have the right to do this.910 Con-

sider the e�ciency curve R2 (R1). Denote by Rr1 and R

r2 the coordinates of the

point r on the X-axis and Y-axis respectively; denote by Rm1 and Rm2 the coor-

dinates of the point m on the X-axis and Y-axis respectively. Rm1 is individual

1's revenue when individual 2 obtains its maximum revenue, i.e., the maximin

m, and Rr2 is individual 2's revenue that when individual 1 obtains its maximum

revenue, i.e., r. We choose the point d of coordinates {Rm1 , Rr2} as disagreementpoint11 because any point outside the convex is impossible: assume that an-

other point d′ is chosen such as those of Figure 13; from there, individual 1 may

8When utilities are argument, the four axioms of the Nash bargaining are: invarianceto equivalent utility representations, symmetry, independence of irrelevant alternatives andPareto e�ciency. The �rst one is not necessary as we revenues are arguments and the secondone is dropped in the generalized Nash bargaining.

9Nash and followers consider utilities because they think in terms of set of commoditiesthat are aggregated by the idea of utility. If we think in terms of revenue, thinking in termsof utility is unnecessary.

10For the link between Nash's and Rawls' theories, see Lengaigne (2004). We do not considerthe Kalai-Smorodinsky's solution (Kalai and Smorodinsky 1975; Kalai 1977) because it doesnot satisfy the axiom of independence of irrelevant alternatives.

11The disagreement point is also called threat point or even �status quo� by Thomson (1981)or Binmore et al. (1986); it is also the point where both individuals are placed when they failto bargain.

16

��

��

��

��

��Figure 13: Disagreement point

increase its revenue up to point a without degrading those of individual 2 but as

a is not e�cient, individual 2 may also increase its revenue up to point b, which

is this time e�cient, without degrading those of individual 1; and conversely by

reversing the order of the actions of both individuals (which does not appears

in Figure 13): however, when point d is chosen, individual 1 may increase its

revenue by going to point r but individual 2 cannot make any other movement

to increase its own revenue as he is located on the e�ciency curve; conversely,

individual 2 may increase its revenue up to point m but that lets no leeway to

individual 1; conversely, any point inside the convex {m, r, d} is not the worstpoint that both agents can accept. Therefore, Rm1 and Rr2 are the minimum

revenues (while Rr1 and Rm2 are the maximum revenues): d is the worst point.

Therefore, for the Nash bargaining, we consider the convex {m, r, d}.TheNash solution is

R1 = argmax [R1 −Rm1 ]θ[R2 (R1)−Rr2]

1−θ(1)

subject to R1 ≥ Rm1 and R2 (R1) ≥ Rr2. Equation 1 is a set of hyperbolas, as

shown in Figure 14. The �rst order condition is

R′2 (R1) = −θ

1− θR2 (R1)−Rr2R1 −Rm1

(2)

17

��

��

�� Figure 14: Nash hyperbolas

When θ = 1 equation (2) is not de�ned but the Nash solution turns out to be

R1 = argmax [R1 −Rm1 ] subject to R1 ≥ Rm1 : R1 is maximized and the solution

is the point r. When θ = 0, it follows from (2) that R′2 (R1) = 0: R2 (R1) is

maximized and the solution is the point m. The utilitarian point is de�ned by

argmax [R1 +R2 (R1)], the �rst order condition being

R′2 (R1) = −1 (3)

Therefore, solution (2) corresponds to the the utilitarian point given by (3) if

θ

1− θR2 (R1)−Rr2R1 −Rm1

= 1 (4)

that is,

θ =R1 −Rm1

(R1 −Rm1 ) + (R2 (R1)−Rr2)(5)

If the curve is symmetric by respect to the bisector passing by d, then θ = 12

and u = s.12 Moreover, we remark that θ cannot be determined ex ante: it is

found only when the point u has been determined because in (5), neither R1

nor R2 (R1) are �xed but they are variable.

Remark. The above reasoning about the utilitarian point as steady balance and

12In the space of utilities, u = s always holds.

18

accumulation point is de�cient in the sense that the utilitarian point is not the

unique point which is a steady balance and an accumulation point as exposed

above.

Proposition 3. Depending on the parameter θ in a Nash generalized bargaining,

any point is an equilibrium point, depending on which θ has been chosen.

This proposition obviously includes the utilitarian point (Thomson 1981).

Proof. It is self-evident.

Remark. The role of the social decision maker could be to choose the parameter

θ. However, θ may also be considered as an indicator of individuals' relative

force. This shows that the maximin is a very particular case of bargaining

where the poorest receives all the bargaining power. We don not think that

Rawls argument about the veil of ignorance and the original position discussed

in sub-section 2.4 is su�cient to justify that the bargaining power is entirely

attributed to the less favored.

3.2 Surplus-equality

The Nash bargaining derivation of the various points along the e�ciency curve

suggests a di�erent de�nition of equality. We call this point surplus-equality,

denoted s in Figure 15: it is the point where the surplus is shared in two equal

parts, determined by the intersection of the �rst bisector that passes by the

disagreement point (Rm1 , Rr2) and the e�ciency curve. This sharing line has for

equation R2 = R1 − (Rm1 −Rr2) and is parallel to the main bisector of equation

R2 = R1. The point s is also a Nash equilibrium if the adequate value of θ is

chosen. It is easily derivable from the moment that the equation of the e�ciency

curve is known. Notice that in the world of utilities, the point s would be the

Nash equilibrium itself.

The point s generally di�ers from the egalitarian point e, even if e is on

the e�ciency curve as in Figure 5. From the moment that the e�ciency curve

is given and accepted by both individuals, the surplus-equality point is those

which shares equally what can be shared. Indeed, pure equality, the point e,

refers to the original point where both individuals receive zero, i.e., the pure

�original position� where both individuals 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 disagreement point is preferred by both, even if in that point

19

��

��

�� Figure 15: Surplus equality

a certain level of inequality has been yet introduced, which supposes that the

di�erences in talents have been recognized. In a word, individuals are not in

the position {0, 0} ex ante, even if they think that they could fall in the best

or the worst position ex post. This could seem unjust but the existence of the

e�ciency curve itself makes that di�erences in talents are predetermined. More

precisely, the e�ciency curve determines the position of the disagreement point

but the converse proposition is false: defending d as original point means that

the e�ciency curve is given and known. If such a presupposition is rejected,

and if we denote by O the point where R1 = R2 = 0, no e�ciency curve can

be drawn in O, no apportionment done except pure equality and no maximin

determined at all. In other words, there is a contradiction between taking O

as original point and considering an e�ciency curve, unless the e�ciency curve

is such that the disagreement point is confused with the origin, a very special

case where s is confused with e, which we call super-equality and is drawn in

Figure 17. The Proposition 4 and it Corollary treat this case. It is worth noting

that pure equality may be e�cient and at the same time not confused with

surplus-equality as in Figure 16. Moreover, s cannot be placed to the left of e:

this would require that d itself is to the left of the bisector {O, e}, which would

mean that R1 < R2 in d.

Proposition 4. Pure equality and surplus-equality are confused if and only if

the disagreement point shares equally the revenues.

20

��

��

�� Figure 16: Equality e�cient but not confused with surplus-equality.

��

��

�� Figure 17: s = e

21

Proof. The straight lines {O, e} and {d, s} are parallel by de�nition. Therefore,from the axiom of Euclidean geometry, the point d is located on {O, e} if andonly if the point s is on {O, e}, i.e., e and s are confused.

Corollary 2. If pure equality and surplus-equality are confused then equality is

e�cient.

Proof. Equality is e�cient if the point e is located on the e�ciency curve. As

{O, e} and {d, s} are parallel, if e = s then e is on the e�ciency curve because

s is on the e�ciency curve by construction.

For these reasons, surplus-equality s corresponds to the true equality because

it refers to the disagreement point where both individuals are placed before

bargaining. Moreover, the segment {m, s} in the e�ciency curve is those that

must be chosen for conducting a left-wing policy because it is largely insensitive

to a growing inequality.

Proposition 5. When inequality increases, that is, the e�ciency curve is in-

�nitely �attened along the Y-axis, or Rr1 →∞, m remaining �xed, then Ru1 →∞but Rs1 is �nite and equal to Rm1 +Rm2 .

Proof. The proof is obvious. We are in the normal case of Figure (18). When

Rr1 → ∞, r goes to the right, by construction. The point s is found by inter-

secting the bisector that passes by (Rm1 , Rr2) and the e�ciency curve: s moves

slightly to the right, to the limit up to the point (Rm1 +Rm2 , Rr2 +Rm2 ) because

the curve tends to be horizontal between m and s. As the e�ciency curve tends

to be very �at, the point where its slope equals −1 in�nitely goes to the right.

As in the point n, curve's slope is between −1 and zero, n goes to the right

between n and r. See Figure 18.

Therefore, while a right-wing policy corresponds to the segment {u, r}, aleft-wing policy should be restricted to the segment {m, s} (or to {e, s}, if e islocated on the e�ciency curve).

If the e�ciency curve is �normal�, that is, �attened along the Y-axis as in

Figures 1-5, the equal-surplus point is to the left of the utilitarian point (see

Figure 18): θu >12 ; s is more favorable to the poor than u. However, if the

�gure is �attened along the X-axis, which is not the standard case, the equality-

surplus and the Nash egalitarian points are to the right of the utilitarian point

as in Figure 19: θu <12 ; they are more favorable to the rich than u; s is to the

22

��

��

�� Figure 18: E�ciency curve �attened along the Y-axis

right of n: it is the most favorable to the rich. Obviously, both n and s are

the same if the curve is symmetrical by respect to the disagreement point; it is

not necessary that the e�ciency be symmetrical by respect to the origin. The

point s also di�ers from the utilitarian point u when the e�ciency curve is not

symmetrical.

Remark. Surplus equality may be di�erent. Instead of sharing equally the sur-

plus between both individuals, one may share it following a di�erent apportion-

ment rule, as the proportional rule: the surplus may be allocated proportionally

to the minimum revenues, that is, toRm

1

Rm1 +Rr

2to which corresponds a sharing line,

of equation R2 =Rm

1

Rr2R1, which passes by the origin. This type of surplus shar-

ing is highly unequal, leading to a point that can be to the right of u, very close

to r, even in a normal e�ciency curve. Surplus may also be shared following

the Talmud. Homothetic growth projects also d and ms homothetically. See

3.3 Case of more than two agents

We have considered only two agents: things are much more complicated when

three or more individuals are considered. For n agents, the �Phelps curve� is

now a surface of n − 1 dimensions. We impose a lexicographic order between

all agents: R1 > ... > Ri > ... > Rn−1 > Rn, where Rn is the revenue of the

poorest. Rn = f (R1, ..., Rn−1) is the e�ciency curve such that dRi

dRn≤ 0 and

d2Ri

dR2n≤ 0 for any i. It is handy to return to the Nash bargaining, for n-persons

23

��

��

�� Figure 19: Atypical curve: e�ciency curve �attened along the X-axis

��

��

��

��Figure 20: Homothetic growth for d and s

24

here. The Nash bargaining solves:

Ri = argmax

n−1∏i=1

[Ri −Rmi ]θi [f (R1, ..., Rn−1)−Rrn]

1−∑n−1

i=1 θi

for any i = 1, ..., n− 1 subject to Ri ≥ Rmi for any i = 1, ..., n− 1, and f (Rn) ≥Rrn. For generating the minimax, one poses θi = 0 for any i = 1, ..., n−1, which

gives

Ri = argmax

n−1∏i=1

[f (R1, ..., Rn−1)−Rrn]

for any i = 1, ..., n−1. Therefore, the di�erence between Rrn and f (R1, ..., Rn−1)

is maximized. See Figure 21 for the three individuals case. The point m is the

maximin, r is the pro-rich point and m12 is the maximin between individuals

1 and 2. The curves {r,m12}, {r,m} and {m12,m} are the e�ciency curves

between individuals 1 and 2, individuals 1 and 3 and individuals 2 and 3, re-

spectively.

However, things are a little more complicated 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 indicate where the lexicographic order is violated, we may search the

maximin between R3 and R2 for a given R1. When R1 is minimum, it is along

curve {B2,m2}; when R1 is maximum, it is along curve {B1,m1}. Therefore,

this maximin is curve {m1,m2}. When we choose the maximin between R1

and R2, it is point m2. However, nothing proves that the the revenue R3

that corresponds to m2 is higher than those of m1: {m1,m2} may perfectly

be decreasing (and m1 be the overall maximin) without violating convexity. In

other words, the overall maximin does not necessarily correspond to the leximin

(i.e., the composition of the maximin between two agents successively placed

on the scale of revenues). Obviously, surplus-equality does not poses such a

problem.

Moreover, the volume of computations become unrealistic with millions of

individuals: it becomes necessary to make a small number of groups. Unfortu-

nately, the solution is completely sensitive to the number n of agents in each

group.

25

��

��

��

��

� ��

��

��

Figure 21: Maximin for three agents.

26

��

��

��

��

��

��

Figure 22: Maximin for three agents.

27

4 Conclusion

We have examined how Phelps' e�ciency curve, where the revenues are argu-

ments of the e�ciency curve, may help understanding what is just in a Rawlsian

perspective. In the �rst part of the paper, we have shown that this curve is lim-

ited to its north-east section and identi�ed three remarkable points: the point r

which maximizes the revenue of the richest, the point m, the Rawlsian maximin,

which maximizes the revenue of the poorest and the point u which maximizes

the sum of the revenues. We have shown that egalitarianism can be impossible,

ine�cient, or possible but this last case excludes and dominates the Rawlsian

optimum. Therefore opposing egalitarianism and equity is not adequate (cf. the

debate in France about the Minc's report (1994) with Dupuy's critics (1995)):

equality is not always possible, but when it is possible, it replaces the maximin.

Overall, the maximin is the closest to equality: this is Kolm's idea of practical

justice. Growth may let anything unchanged if it is homogeneous but this is an

improbable situation. Sometimes growth will make equality easier and some-

times more di�cult to reach. Choosing the point m does not guarantee that

growth is always favorable to the poor: when the point m is chosen, growth

can be pro-poor but it can be also pro-rich, and conversely when the point r is

selected. In some situations growth may be completely paradoxical: the poor,

respectively the rich, can be losing when the maximin, respectively the point r,

is selected.

The utilitarian point u is an equilibrium point. However, a Nash generalized

bargaining�based on revenues rather than on utilities�is able to turn any point

in the e�ciency curve into equilibrium. This idea allows proposing a new point,

surplus-equality, which shares equally the surplus from the disagreement point.

While pure equality is determined by respect to an unrealistic situation where all

revenues are zero, surplus-equality is determined by respect to the disagreement

point, where are placed both individuals before bargaining. Therefore, it is the

true egalitarian point. Surplus-equality generally falls between the maximin

and the utilitarian point but, as the maximin and unlike the utilitarian point,

is rather stable by respect to a growing inequality. If the maximin, the pro-rich

point and the utilitarian point13 are relatively easy to �nd empirically, detecting

where the surplus-equality point is placed is not so easy.

Nevertheless, surplus-equality may be considered also for a left-wing policy.

We conclude that the area between maximin and surplus-equality should be the

13In the utilitarian point, giving one euro to one agent takes one euro to the other.

28

base of a left-wing policy as the maximin is the closest to egalitarianism without

being Utopian as equality is, and the area maximin - surplus-equality is rather

insensible to an increase of inequality. The point r corresponds to a political

hard right-wing policy and the point u to a Benthamite policy. We have also

examined the e�ect of the growth on these categories.

Extending the analysis to more than two agents is possible by following a

Nash bargaining but the overall maximin does not necessarily correspond to the

leximin; it needs too much information: considering groups of agents is obviously

much simpler and it raises the question of the sensitivity of the results to the

relative weights of the groups.

This study has obviously its own limitations. The Phelps curve is not easy

to compute in practical terms: the politicians must found it by trial and errors.

One concludes that all this story is more a parable for explaining the various

concepts of justice and for determining the optimum of a left-wing policy rather

than an operational tool.

Bibliographical references

Allou Jean-Paul (2011) Tous les banquiers ne �nissent pas en prison... moi,

c'était dans la rue, Paris: Michel Lafon.

Aumann R.J. and Maschler. M. (1985) �Game theoretic analysis of a

bankruptcy problem from the Talmud� , Journal of Economic Theory, 36, 2:

195-213.

Barry B. (1989) Theories of Justice, Berkeley and Los Angeles: University of

California Press.

Bezembinder Th. and P. van Acker (1987) �Factual vs representational utilities

and their interdimensional representations,� Social Choice and Welfare, 4:

79-104.

Binmore K. (1989) �Social Contract I: Harsanyi and Rawls,� The Economic

Journal, 99, 395, Supplement: Conference Papers: 84- 102.

Binmore K., A. Rubinstein and A. Wolinsky (1986) �The Nash Bargaining

Solution in Economic Modelling,� The RAND Journal of Economics, 17, 2:

176-188.

29

Bosmans K. and E. Ooghe. (2006) �A characterization of maximin,�

Downloadable at:

<http://www.econ.kuleuven.be/ew/academic/econover/Papers/wpchmax.pdf>.

Cohen G. (1995) �The Pareto Argument for Inequality,� Social Philosophy and

Policy, 12: 160-185.

Dupuy J.P. (1995) �Equité, égalité et confusion,� Libération, 27 février 1995.

Foley D. (1967) �Resource allocation and the public sector�, Yale Economic

Essays, 7: 45-90.

Gamel C. (2010) �Justice de résultat. De 'l'économie du bien-être' à

'l'égalitarisme libéral',� GREQAM Working Paper No 2010-22.

Gauthier D. (1986), Morals by Agreement, Oxford: Oxford University Press.

Gibbard A. (1991) �Constructing Justice,� Philosophy & Public A�airs, 20, 3:

264-279.

Hammond P.J. (1976) �Equity, Arrow's conditions, and Rawls' di�erence

principle,� Econometrica, 44, 4: 793-803.

Hammond P.J. (1979) �Equity in two person situations: some consequences,�

Econometrica, 47, 5: 1127-1135.

Hammond P.J. (1993) �Interpersonal Comparisons of Utility: Why and How

They Are and Should Be Made,� in: Elster J. and J.E. Roemer (1993)

Interpersonal Comparisons of Well-Being, Studies in Rationality and Social

Change, Cambridge University Press, pp. 200-254.

Harsanyi J. (1953). �Cardinal utility in welfare economics and in the theory of

risk-taking.� Journal of Political Economy, 61: 434-435.

Harsanyi, J. (1955) �Cardinal welfare, individualistic ethics, and interpersonal

comparisons of utility,� Journal of Political Economy, 63, 309�321.

Harsanyi, J. (1958) �Ethics in terms of hypothetical imperatives.� Mind, vol.

47, pp. 305-316.

Harsanyi, J. (1975) �Can the maximin principle serve as a basis for morality?

A critique of John Rawls' theory,� American Political Science Review, 69:

594-606.

30

Kalai E. (1977) �Proportional Solutions to Bargaining Situations:

Interpersonal Utility Comparisons,� Econometrica, 45, 7: 1623-1630.

Kalai E., and M. Smorodinsky. (1975) �Other Solutions to Nash's Bargaining

Problem,� Econometrica, 43: 513-518.

Kolm S.-C. (1972) Justice et équité, Monographies du séminaire d'économétrie,

CNRS.

Kolm S.-C. (1996a) A modern theory of justice, M.I.T. Press.

Kolm S.-C. (1996b) �The theory of justice�, Social Choice and Welfare, 13, 2:

151-182.

Lengaigne B. (2004) �Nash : changement de programme ?,� Revue d'Economie

Politique, 114, 5: 637-662.

McClelland P. D. (1990) The American Search for Economic Justice, Basil

Blackwell.

Minc A. (1994) La France de l'an 2000. Commissariat général du plan, Odile

Jacob.

Miyagishima K. (2010) �A characterization of the maximin social ordering,�

Economics Bulletin, 30, 2: 1-4.

Moehler M. (2010) �The (Stabilized) Nash Bargaining Solution as a Principle

of Distributive Justice,� Utilitas, 22: 447-473.

Moulin H.J. (2003) Fair Division and Collective Welfare, Cambridge: MIT

Press.

Nash J. F. (1950a) �The Bargaining Problem,� Econometrica, 18, 2: 155-162.

Nash J.F. (1950b) �Equilibrium points in n-persons games,� Proceedings of the

National Academy of Sciences of the United Stats of America, 36, 1: 48-49.

O'Neill B. (1982) �A problem of rights arbitration from the Talmud� ,

Mathematical Social Sciences, 2, 4: 345-71.

Phelps E.S. (1985) Political Economy, an introductory text, W.W. Norton.

31

Phelps (2011) �Edmund S. Phelps - Autobiography,� Nobelprize.org, 9 Mar

2011. Downloadable at:

<http://nobelprize.org/nobel_prizes/economics/laureates/2006/phelps.html>.

Rabinovitch N. (1973) Probability and Statistical Inference in medieval Jewish

Literature. Toronto: University of Toronto Press.

Rawls J. (1971) Theory of Justice, Harvard University Press.

Rawls J. (1988) �La théorie de la justice comme équité : une théorie politique

et non pas métaphysique�, in Individu et justice sociale, autour de John Rawls,

Seuil, Points Politique, pp. 279-317.

Rawls J. (1989) �Les libertés de base et leur priorité�, in �John Rawls, Justice

et libertés�, Critique, 505-506: 423-465.

Rawls J. (1993) Political Liberalism, New York, NY: Columbia University

Press, 1993.

Roth A. (1979) Axiomatic Models of Bargaining, Berlin: Springer-Verlag.

Rosen M. (1989) �Une autre argumentation en faveur du principe de

di�erence�, in �John Rawls, Justice et libertés�, Critique, 505-506: 498-505.

Rubinstein A. (1982) �Perfect equilibrium in a Nash bargaining problem,�

Econometrica, 50, 1: 97-109.

Runciman W.G. (1966) Relative Deprivation and Social Justice, Routledge.

Sen A. (1999) Commodities and Capabilities, Oxford University Press, USA.

Shaw P. (1999) �The Pareto argument and inequality,� The Philosophical

Quarterly, 49, 196: 353-368.

Thomson W. (1981) �Nash's Bargaining Solution and Utilitarian Choice

Rules,� Econometrica, 49, 2: 535-538.

Tinbergen J. (1953) Redelijke Inkomensverdeling, 2nd edition, Haarlem, De

Gulden Pers.

Wright Randall. (no date) �Chapter 9, Bargaining Theory,� Course, University

of Pennsylvania, Downloadable at:

http://www.ssc.upenn.edu/~rwright/courses/app-bar.pdf.

32

Young H. P. (1987) �On dividing an amount according to individual claims or

liabilities,� Mathematics of Operations Research, 12, 3: 398-414.

Young H. P. (1995) Equity: In Theory and Practice, Princeton (N.J.):

Princeton University Press.

33

rawls, phelps, nash: e ciency curve and economic...

Documents