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 oers 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. equity is clari ed.
We examine the eect 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 always 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 simple 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 dierence ), 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 possible 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 iseven 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 eect 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 dierent 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 eective 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 partor Pareto-optimalof the curve, is the segment
(r, m):
before
m,
or after
r,
the income of both individuals may increase or
decrease simultaneously. The curve
dR2 that dR1
≤ 0
d2 R2 and dR12
Figure 5. In point
r
≤ 0.
(r, m)
is continuous, derivable and is such
Several typical points appear on the curve in
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 m
on the ine cient part of the curve, as on Figure 5 or to the right. For Rawls, 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.
1 The
arguments of the problem are also certain: no need of von Neumann-Morgenstern
utilities, expected utilities, etc. On these approaches, see Harsanyi (1953, 1955); Hammond (1976, 1979, 1993), Bezembinder and van Acker (1987); Bosmans and Ooghe (2006); Miyagishima (2010).
2 This
shape of curve is also quoted by Kolm (1972, p. 33, 1.e example).
4
Z
Ϯ
ŵ Ƶ ƌ Z
Ϭ
ϭ
Figure 1: Equality is impossible
Any intermediate solution, obtained by making a linear combination between the incomes of the two individuals, and located between
m and r could be chosen
and re ects a relation of force between both: the weighting may re ect any social criterion. The point whereas the point
m
r
is obtained if the best remunerated group is dominating, 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
ZϮ
ZϮ
ŵ ŵ ƌ ƌ Zϭ
Ϭ
Zϭ
Ϭ
Figure 2: Typology of e ciency curves: concentrated (left), large (right)
Z
Z
Ϯ
Ϯ
ŵ
ŵ ƌ
ƌ Z
Ϭ
ϭ
Z
Ϭ
ϭ
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 closer to the point
r
or to the point
m.
However, as
R1 ≥ R2 ,
u
may be
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.
Reaching equality can be, respectively, (i) impossible, (ii) inef cient or (iii) possible but by excluding and dominating the Rawlsian optimum Proposition 1.
6
Z
Ϯ
Ğ
ŵ
Ƶ ƌ Z
ϭ
Ϭ
Figure 4: Equality is possible but ine cient
in this last case. Corollary 1.
tarianism. Proof.
Among the e cient points, the maximin is the closest to egali-
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 closest to egalitarianism.
m
is again the
3
In both cases of Figures 1 or 4, Phelps sees a justi cation of the maximin: any point to the right of to more inequality.
m
on the e ciency curve (u,
r,
etc.)
corresponds
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 dierent claims
3 On
the Pareto argument and feasibility of equality, see Cohen (1995) and its critique by
Shaw (1999).
7
Z
Ϯ
ŵ
Ƶ
Ğ
ƌ Z
Ϭ
ϭ
Figure 5: Example of e ciency curve
they share dierently.
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
m the revenue of individual 2 is higher than those 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 the left of the rst bisector, in
of individual 1: between
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
4 In
the Aristotelian tradition, they share proportionally, while in the Talmudic tradition,
they share in a dierent way (Rabinovitch 1973; O'Neill 1982; Aumann and Maschler 1985; Young 1987, 1995; Moulin 2003). This shows that equality and its substitute, the maximin
m,
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 dierent 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
5
be in a given situation, according to its merits).
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 anotherthe point or
r
for exampleonly
ex ante
m
rather than points
u
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.
5 Phelps
has chosen to think
ex post,
It is perhaps an
when the roles have yet been attributed between
the 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
7
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. 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
u.
or
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
one person is happier in
B
than in
A,
A than in B
and only
but that this person is less happy than
all the others - according to the fundamental preferences - in each of the two states. state
Practical justice (the maximin) results in preferring the state
A,
B
with
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.
6 See a 7 Even
detailed discussion in Binmore (1989). 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 aect 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 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 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.
Proof.
The utilitarian point u is an equilibrium point.
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 degradr, whereas at the
ing very little that of individual 2, and conversely at the point point
u,
increasing the income of an unspeci ed individual obliges to decrease
by as much that of the other individual. Therefore, the points a certain manner less stable than the point
u,
m
and
r
in that sense that con icts will
be unbalanced there. If it is supposed that the resistance of individual inverse proportion of the elasticity
ERi /Rj =
are in
dRi dRj then in
r,
i
is in
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 start from
m.
Hence, the point
u
r, one will tend to stop out of u ; similarly if we 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 than the maximin However, if
e
m.
u
rather
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
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
point
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
e
m
(or
e)
to
u,
the left-wing abandons its
Utopian ideas ( ) 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 left-wing goes from
u
to
m
or
e
r
to
u
while the domain of the
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
ZϮ Ğ͛ ŵ͛
ZϮΎΎ ZϮΎ
Ƶ͛
Ğ ŵ
Ϭ
Ƶ
ƌ
ƌ͛ Zϭ
ZϭΎ Figure 6: Growth and e ciency curve
the revenue of the other does not change. For example, for the same level of agent's 1 revenue, it is possible to pay more agent 2:
R2∗∗
instead of
R2∗ ,
R1∗
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 eorts 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
Z
Ϯ
ŵ͛ Ƶ͛
ŵ
Ƶ
ƌ͛
ƌ
Z
Ϭ
ϭ
Figure 7: Homothetic growth and e ciency curve
ZϮ
ZϮ
Zϭ
Ϭ
Zϭ
Ϭ
Figure 8: Growth: equality made easier (left) and more di cult (right)
ZϮ ZϮ
ŵ͛
ŵ
ƌ͛ ƌ
Ϭ
Zϭ
Ϭ
Zϭ
Figure 9: Normal pro-poor growth (left); normal pro-rich growth (right)
14
ZϮ
ZϮ
ŵ͛
ƌ͛
ŵ
ƌ Ϭ
Zϭ
Figure 10:
Zϭ
Ϭ
Paradoxical pro-rich growth (left); paradoxical pro-poor growth
(right) ZϮ
Z Ϯ ŵ͛ ŵ
ƌ͛
ŵ
ŵ͛
ƌ Zϭ
Ϭ
ƌ͛
ƌ
Zϭ
Ϭ
Figure 11: Political hard right-wing and growth: the rich are losing (left); maximin 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 eect, 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 and
r
0
have been selected, but they have larger revenue if the points
m
and
r
m0
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 losing if the point
m
r
is chosen but they are
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 absolutely not a guarantee of neutrality, as shown in Figure 12.
15
ZϮ
ZϮ
Ƶ͛ Ƶ
Ƶ
Ϭ
Zϭ
Ƶ͛
Ϭ
Zϭ
Figure 12: Utilitarian point and change in revenue distribution: growth favorable 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 generalizedi.e., by
8
dropping symmetry axiom 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 dierent claims. Here, it is applied on a curve of which arguments
910 Con-
are the revenues rather than utilities but we have the right to do this.
r sider the e ciency curve R2 (R1 ). Denote by R1 and
R2r the coordinates of the m m point r on the X-axis and Y-axis respectively; denote by R1 and R2 the coorm dinates of the point m on the X-axis and Y-axis respectively. R1 is individual 1's revenue when individual 2 obtains its maximum revenue, i.e., the maximin
m, and R2r
is individual 2's revenue that when individual 1 obtains its maximum
revenue, i.e., point
r.
We choose the point
d of coordinates {R1m , R2r } as disagreement
11 because any point outside the convex is impossible: assume that an-
other point
8 When
d0
is chosen such as those of Figure 13; from there, individual 1 may
utilities are argument, the four axioms of the Nash bargaining are:
invariance
to equivalent utility representations, symmetry, independence of irrelevant alternatives and Pareto e ciency. The rst one is not necessary as we revenues are arguments and the second one is dropped in the generalized Nash bargaining.
9 Nash
and followers consider utilities because they think in terms of set of commodities
that are aggregated by the idea of utility. If we think in terms of revenue, thinking in terms of utility is unnecessary.
10 For the link between Nash's and Rawls' theories,
see Lengaigne (2004). We do not consider
the Kalai-Smorodinsky's solution (Kalai and Smorodinsky 1975; Kalai 1977) because it does not satisfy the axiom of independence of irrelevant alternatives.
11 The
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 fail to bargain.
16
Z
Ϯ
ŵ
Zŵ Ϯ
Ě
Zƌ Ϯ
ď
Ě͛
ƌ Ă Z
ϭ
Ϭ
Zƌ
Zŵ ϭ
ϭ
Figure 13: Disagreement point
increase its revenue up to point
a
a without degrading those of individual 2 but as
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 revenue by going to point
r
d
is chosen, individual 1 may increase its
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
individual 1; conversely, any point inside the convex point that both agents can accept. revenues (while
R1r
and
R2m
but that lets no leeway to
{m, r, d}
m Therefore, R1 and
are the maximum revenues):
is not the worst
R2r are the minimum d
is the worst point.
Therefore, for the Nash bargaining, we consider the convex
{m, r, d}.The
Nash solution is
θ
1−θ
R1 = arg max [R1 − R1m ] [R2 (R1 ) − R2r ] subject to
R1 ≥ R1m
and
R2 (R1 ) ≥ R2r .
(1)
Equation 1 is a set of hyperbolas, as
shown in Figure 14. The rst order condition is
R20 (R1 ) = −
θ R2 (R1 ) − R2r 1 − θ R1 − R1m 17
(2)
ZϮ
ŵ ƌ
Ě
Zϭ
Ϭ
Figure 14: Nash hyperbolas
When
θ=1
equation (2) is not de ned but the Nash solution turns out to be
R1 = arg max [R1 − R1m ] subject to R1 ≥ R1m : R1 is the point
r.
When
θ = 0,
it follows from (2) that
maximized and the solution is the point
arg max [R1 + R2 (R1 )],
is maximized and the solution
m.
R20 (R1 ) = 0: R2 (R1 )
is
The utilitarian point is de ned by
the rst order condition being
R20 (R1 ) = −1
(3)
Therefore, solution (2) corresponds to the the utilitarian point given by (3) if
θ R2 (R1 ) − R2r =1 1 − θ R1 − R1m that is,
θ=
R1 − R1m (R1 − R1m ) + (R2 (R1 ) − R2r )
If the curve is symmetric by respect to the bisector passing by and
u=
s.12 Moreover, we remark that
found only when the point nor
R2 (R1 )
Remark. 12 In
(4)
u
θ
(5)
d,
cannot be determined
then
θ=
ex ante :
1 2
it is
has been determined because in (5), neither
R1
are xed but they are variable.
The above reasoning about the utilitarian point as steady balance and
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.
Depending on the parameter θ in a Nash generalized bargaining, any point is an equilibrium point, depending on which θ has been chosen.
Proposition 3.
This proposition obviously includes the utilitarian point (Thomson 1981).
Proof.
It is self-evident.
Remark. θ.
The role of the social decision maker could be to choose the parameter
However,
force.
θ
may also be considered as an indicator of individuals' relative
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 dierent de nition of equality. denoted
s
We call this point
surplus-equality,
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 equation
R2 =
R2 = R1 .
(R1m , R2r )
R1 − (R1m
The point
s
−
and the e ciency curve. This sharing line has for
R2r ) and is parallel to the main bisector of equation
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 diers 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
Z
Ϯ
ŵ
Ɛ
Ƶ
Ě
ƌ Z
Ϭ
ϭ
Figure 15: Surplus equality
a certain level of inequality has been yet introduced, which supposes that the dierences in talents have been recognized. the position
{0, 0}
In a word, individuals are not in
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 dierences in talents are predetermined. More precisely, the e ciency curve determines the position of the disagreement point but the converse proposition is false: defending the e ciency curve is given and known. and if we denote by be drawn in
O,
O
the point where
d
as original point means that
If such a presupposition is rejected,
R1 = R2 = 0,
no e ciency curve can
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, this would require that mean that
R1 < R2
in
d
s
cannot be placed to the left of
itself is to the left of the bisector
{O, e},
e:
which would
d.
Pure equality and surplus-equality are confused if and only if the disagreement point shares equally the revenues.
Proposition 4.
20
Z Ϯ
Ğ Ɛ Ě
Ƶ ƌ
Ϭ
Zϭ
Figure 16: Equality e cient but not confused with surplus-equality.
Z Ϯ
ƐсĞ Ƶ Ě
ƌ
Ϭ
Zϭ Figure 17:
21
s=e
Proof.
The straight lines
{O, e} and {d, s} are parallel by de nition.
from the axiom of Euclidean geometry, the point
s
only if the point Corollary 2.
e cient.
is on
{O, e},
i.e.,
e
and
is located on
{O, e}
if and
are confused.
If pure equality and surplus-equality are confused then equality is
Proof.
Equality is e cient if the point
{O, e}
and
s
s
d
Therefore,
{d, s}
are parallel, if
e=s
e
is located on the e ciency curve. As
then
e
is on the e ciency curve because
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.
When inequality increases, that is, the e ciency curve is in nitely attened along the Y-axis, or R1r → ∞, m remaining xed, then R1u → ∞ but R1s is nite and equal to R1m + R2m . Proposition 5.
Proof. R1r
The proof is obvious. We are in the normal case of Figure (18). When
→ ∞, r
goes to the right, by construction. The point
s
is found by inter-
m r secting the bisector that passes by (R1 , R2 ) and the e ciency curve: s moves m m r m slightly to the right, to the limit up to the point (R1 + R2 , R2 + R2 ) because the curve tends to be horizontal between
m and s.
to be very at, the point where its slope equals As in the point between
n
and
n,
r.
curve's slope is between
−1
As the e ciency curve tends
−1
in nitely goes to the right.
and zero,
n
goes to the right
See Figure 18.
{u, r}, a {e, s}, if e is
Therefore, while a right-wing policy corresponds to the segment left-wing policy should be restricted to the segment
{m, s}
(or to
located 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 >
1 2;
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 equalitysurplus and the Nash egalitarian points are to the right of the utilitarian point as in Figure 19:
θu
... > Ri > ... > Rn−1 > Rn ,
where
Rn
is the revenue of the
Rn = f (R1 , ..., Rn−1 ) is the e ciency curve such that ≤ 0 for any i. It is handy to return to the Nash bargaining,
poorest. 2
n
23
dRi dRn for
≤ 0 and n-persons
ZϮ
ŵ
Ƶ Ɛ
Ě
ƌ Zϭ
Ϭ
Figure 19: Atypical curve: e ciency curve attened along the X-axis
Z Ϯ
ŵ͛ Ɛ͛ ŵ Ɛ ƌ͛
Ě͛ Ě
ƌ
Ϭ
Zϭ Figure 20: Homothetic growth for
24
d
and
s
here. The Nash bargaining solves:
n−1 Y
Ri = arg max
θ
1−
[Ri − Rim ] i [f (R1 , ..., Rn−1 ) − Rnr ]
Pn−1 i=1
θi
i=1 for any
i = 1, ..., n − 1
subject to
Ri ≥ Rim
for any
Rnr . For generating the minimax, one poses
i = 1, ..., n − 1,
and
f (Rn ) ≥
θi = 0 for any i = 1, ..., n − 1,
which
gives
Ri = arg max
n−1 Y
[f (R1 , ..., Rn−1 ) − Rnr ]
i=1 for any
i = 1, ..., n−1.
Therefore, the dierence between
Rnr and f (R1 , ..., Rn−1 )
is maximized. See Figure 21 for the three individuals case. The point maximin,
r
1 and 2.
The curves
is the pro-rich point and
m12
{r, m12 }, {r, m}
m
is the
is the maximin between individuals
and
{m12 , m}
are the e ciency curves
between individuals 1 and 2, individuals 1 and 3 and individuals 2 and 3, respectively. 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 curve
{B2 , m2 };
R3
and
when
this maximin is curve and
R2 ,
it is point
that corresponds to be decreasing (and
R1
for a given
m2 .
R1 .
When
R1
is minimum, it is along
is maximum, it is along curve
{m1 , m2 }.
m2
m1
R2
{B1 , m1 }.
Therefore,
When we choose the maximin between
R1
However, nothing proves that the the revenue
R3
is higher than those of
m1 : {m1 , m2 }
may perfectly
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. Unfortunately, the solution is completely sensitive to the number group.
25
n
of agents in each
Zϯ
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
Ĩ;Zϭ͕͙ZŶͲϭͿͲZϯƌ
ƌ
ZϮ Figure 21: Maximin for three agents.
26
Zϭ
Zϯ
00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 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00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 Ϯ00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00
Ĩ;Zϭ͕͙ZŶͲϭͿͲZϯƌ
ϭ
Zϭ
ZϮ Figure 22: Maximin for three agents.
27
4
Conclusion
We have examined how Phelps' e ciency curve, where the revenues are arguments 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-
r m, the Rawlsian maximin, which maximizes the revenue of the poorest and the point u which maximizes ited to its north-east section and identi ed three remarkable points: the point
which maximizes the revenue of the richest, the point
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 sometimes 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
bargainingbased on revenues rather than on utilitiesis 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 point
13 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
13 In
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 hard right-wing policy and the point
u
r
corresponds to a political
to a Benthamite policy. We have also
examined the eect 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,
Journal, 99, 395, Supplement:
The Economic
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: . Cohen G. (1995) The Pareto Argument for Inequality,
Policy, 12:
Social Philosophy and
160-185.
Dupuy J.P. (1995) Equité, égalité et confusion, Libération, 27 février 1995. Foley D. (1967) Resource allocation and the public sector,
Essays, 7:
Yale Economic
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:
Gibbard A. (1991) Constructing Justice,
Oxford University Press.
Philosophy & Public Aairs, 20, 3:
264-279. Hammond P.J. (1976) Equity, Arrow's conditions, and Rawls' dierence 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, 309321.
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:
Econometrica, 45, 7:
Interpersonal Utility Comparisons,
1623-1630.
Kalai E., and M. Smorodinsky. (1975) Other Solutions to Nash's Bargaining Problem,
Econometrica, 43:
Kolm S.-C. (1972)
513-518.
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 ?,
Politique, 114, 5:
Revue d'Economie
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, Moulin H.J. (2003)
Utilitas, 22:
447-473.
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: Phelps E.S. (1985)
345-71.
Political Economy, an introductory text, W.W. Norton. 31
Phelps (2011) Edmund S. Phelps - Autobiography, Nobelprize.org, 9 Mar 2011. Downloadable at: . Rabinovitch N. (1973)
Literature.
Probability and Statistical Inference in medieval Jewish
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:
Rawls J. (1993)
423-465.
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 dierence, 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) Sen A. (1999)
Relative Deprivation and Social Justice, Routledge.
Commodities and Capabilities, Oxford University Press, USA.
Shaw P. (1999) The Pareto argument and inequality,
Quarterly, 49, 196:
The Philosophical
353-368.
Thomson W. (1981) Nash's Bargaining Solution and Utilitarian Choice Rules,
Econometrica, 49, 2:
Tinbergen J. (1953)
535-538.
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:
Young H. P. (1995)
398-414.
Equity: In Theory and Practice, Princeton (N.J.):
Princeton University Press.
33
