ARTICLE IN PRESS
Journal of Economic Psychology xxx (2006) xxx–xxx
www.elsevier.com/locate/joep
Inequality aversion and diminishing sensitivity
Sarah A. Hill a, William Neilson
b,*
a
Division of the Humanities and Social Sciences, 228-77, California Institute of Technology,
Pasadena, CA 91125, USA
b
Department of Economics, Texas A&M University, College Station, TX 77843-4228, USA
Received 27 July 2004; received in revised form 23 December 2005; accepted 6 March 2006
Abstract
We define inequality aversion as a decision-maker disliking it when his opponents’ payoffs differ
from his own, diminishing sensitivity as this effect increasing less-than-proportionately as the opponents’ payoffs move further from the decision-maker’s, and a preference for Robin Hood redistributions as a preference for taking money from a high-payoff opponent and giving it to a low-payoff
opponent. Existing models of inequality averse preferences are unable to accommodate all three
properties. The three are not inherently inconsistent, though, and we construct a new model which
exhibits all three properties.
Ó 2006 Elsevier B.V. All rights reserved.
JEL classification: C72; D81
PsycINFO classification: 3040
Keywords: Inequity aversion; Other-regarding preferences; Fairness; Diminishing sensitivity; Redistribution
1. Introduction
Diminishing sensitivity, or the property that changes in a variable have less impact the
farther the variable is from a reference point, is pervasive in both economics and psychology. For instance, it implies the properties of diminishing marginal rates of substitution in
*
Corresponding author. Tel.: +1 979 845 7352; fax: +1 979 847 8757.
E-mail addresses: [email protected] (S.A. Hill), [email protected] (W. Neilson).
0167-4870/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.joep.2005.12.003
ARTICLE IN PRESS
2
S.A. Hill, W. Neilson / Journal of Economic Psychology xxx (2006) xxx–xxx
consumer theory, diminishing returns in producer theory, discounting in intertemporal
choice, and the pattern of risk aversion over gains and risk seeking over losses in behavior
toward risk. Even though it has been applied fruitfully to other areas of decision theory, it
has yet to be applied to other-regarding preferences, in which decision-makers care about
the payoffs of other individuals in their group.
One common hypothesis for other-regarding preferences is inequality aversion, the
property that decision-makers dislike differences between their opponents’ payoffs and
their own, as in Fehr and Schmidt (1999) and Bolton and Ockenfels (2000). Inequality
aversion implies that the decision-maker would like to take actions that move his own payoff and an opponent’s payoff closer together. Another hypothesis, examined empirically by
Engelmann and Strobel (2004) and Karni, Salmon, and Sopher (2001), and which seems
like a perfectly compatible extension, is that the decision-maker would like to take actions
that move two of his opponents’ payoffs closer to each other. After all, by taking from the
rich and giving to the poor, this sort of Robin Hood redistribution reduces inequality
within the group.
The purpose of this paper is to explore the compatibility of three properties: inequality
aversion, diminishing sensitivity, and a preference for Robin Hood redistributions. We
show that for three existing specifications of other-regarding preferences, one additive in
levels, one additive in differences, as in the Fehr–Schmidt (1999) model, and one additive
in payoff shares, as in the Bolton–Ockenfels model, the three patterns cannot coexist. Two
questions arise from this result. First, is the result special to the three functional forms
examined, or is it a general property of other-regarding preferences? Second, why should
we care about these three patterns in the first place?
To answer the second question, not only do all three patterns possess intuitive appeal,
but they all have experimental support, as discussed in Section 2. As for the first question,
we show that the incompatibility is a property of the functional specifications, and not a
property of other-regarding preferences in general, by constructing a new specification
which exhibits all three properties, and this new specification coincides with the Fehr–
Schmidt model when the decision-maker has only one opponent. The primary contribution of the paper, then, is to demonstrate that the three patterns of inequality aversion,
diminishing sensitivity, and preference for Robin Hood redistributions are not inherently
incompatible and that it is possible to allow for all three patterns in a single utility
specification.
Furthermore, an analogy with the analysis of behavior toward risk suggests that diminishing sensitivity might be the appropriate place to start when characterizing preferences.
When he first introduced what we now know as expected utility theory, Bernoulli (1738)
restricted attention to risk aversion. Risk aversion went on to become the dominant
paradigm within expected utility theory until Kahneman and Tversky (1979) provided
compelling evidence that while people tend to be risk averse over gains, they also tend
to be risk seeking over losses. This more complicated pattern is implied by diminishing
sensitivity to changes in wealth. Perhaps diminishing sensitivity can lead to the ‘‘right’’
behavioral assumptions for other-regarding preferences just as it did for preferences
toward risk.
The paper proceeds as follows. Section 2 describes the choice setting and formally
defines the notions of inequality aversion, diminishing sensitivity, and a preference for
Robin Hood redistributions. It also reviews the experimental evidence for all three patterns. Section 3 shows that the three properties are incompatible in three existing utility
ARTICLE IN PRESS
S.A. Hill, W. Neilson / Journal of Economic Psychology xxx (2006) xxx–xxx
3
specifications, and Section 4 constructs a new specification in which all three properties are
exhibited. Section 5 offers some concluding remarks.
2. Defining inequality aversion and diminishing sensitivity
A decision-maker has n partners/opponents. An allocation is a vector ðx0 ; . . . ; xn Þ 2
Rnþ1
þ , with x0 the payoff to the decision-maker and xi the payoff to opponent i, for
i = 1, . . . , n. The decision-maker has preferences over allocations that can be represented
by a utility function U : Rn+1 ! R.
To make the ideas of inequality aversion and diminishing sensitivity concrete, we use
the concept of an equal-division equivalent proposed by Neilson and Stowe (2002).
The equal-division equivalent, or EDE, of the allocation (x0, . . . , xn) is the value h that
satisfies
U ðx0 ; . . . ; xn Þ ¼ U ðh; . . . ; hÞ:
ð1Þ
In other words, the equal-division equivalent is set so that the decision-maker is indifferent
between the original allocation and everyone receiving the same payoff h. Under the unrestrictive assumption that U(h, . . . , h) is strictly increasing in h, Eq. (1) defines a function
h(x0, . . . , xn) that represents the same preference ordering as U does. Consequently, the
function h(x0, . . . , xn) can be used to gauge changes in the decision-maker’s utility.
Inequality aversion captures the idea that as xi moves farther from x0 for some
i = 1, . . . , n, inequality increases, and the decision-maker should dislike the change.1 Consequently, the EDE should fall. We say that the utility function U exhibits inequality aversion if (xi x0)oh/oxi 6 0 for all i = 1, . . . , n. If xi < x0, the decision-maker is ahead of
opponent i in the sense that his payoff is higher than i’s. An increase in xi reduces inequality by making opponent i’s payoff closer to the decision-maker’s, which should make the
EDE rise. Since xi x0 < 0 in this case, the definition of inequality aversion requires oh/
oxi P 0, and the EDE rises. When xi > x0, an increase in xi moves opponent i farther
ahead of the decision-maker, which should make the EDE fall. We say that U exhibits
strict inequality aversion if it exhibits inequality aversion and the partial oh/oxi 5 0.
There are, however, other ways to think about inequality aversion. A natural one is to
think about how the decision-maker feels about redistributing wealth between two opponents with unequal payoffs. Intuition tells us that the decision-maker would be made
strictly better off by taking money away from the richer of the two opponents and giving
it to the poorer. Consider the following scenario. The decision-maker has two opponents,
and his preferences are represented by the function U(x0, x1, x2). Assume, without loss of
generality, that x2 > x1. We are interested in the following question: Would the decisionmaker prefer to take (a marginal amount of) money away from opponent 2 and give it to
opponent 1?
Experimental evidence suggests that he would.2 Engelmann and Strobel (2004) gave
subjects a choice between three allocations, A = (12, 2, 11), B = (12, 3, 8), and
C = (12, 4, 5). Sixteen of their 30 subjects chose allocation C, reducing the total payoff
1
For an overview of the evidence supporting inequality aversion, see Camerer (2003).
The 1998 Gallup Poll Social Audit also finds evidence supporting a preference for Robin Hood redistributions
(Fong, 2001). Of 2738 respondents, 46% favored heavily taxing the rich to redistribute wealth to the poor,
including 24% of those making over $150,000 per year.
2
ARTICLE IN PRESS
4
S.A. Hill, W. Neilson / Journal of Economic Psychology xxx (2006) xxx–xxx
to opponents 1 and 2 in order to make the payoffs more equal among the two players.
They also gave subjects the choice between the allocations A 0 = (4, 5, 14), B 0 = (4, 6, 11),
and C 0 = (4, 7, 8). Ten of their 30 subjects chose the total-payoff reducing Robin Hood
redistribution C 0 . Fehr and Schmidt (forthcoming) argue that Engelmann and Strobel’s
results were biased because they used economics majors as subjects, and they found that
92 out of 164 non-economists chose C 0 .
Charness and Rabin (2002) also provide evidence pertaining to Robin Hood redistributions. In their experiment subjects were given a choice between the allocations
A = (x, 400, 400) and B = (x, 1200, 0), where x was kept hidden from the decision-maker
to keep him from comparing his own payoff to those of his opponents. Of the 22 subjects,
18 chose A, which is consistent with a preference for Robin Hood redistributions. In
another treatment subjects were given a choice between A 0 = (575, 575, 575) and
B 0 = (600, 900, 300), and 13 of the 24 subjects chose A 0 , giving up 25 in order to equalize
payoffs between the two opponents.
Finally, Karni et al. (2001) conducted an experiment in which one subject chose from a
set of vectors assigning, to himself and two opponents, probabilities of winning an indivisible prize. The resulting allocation was, of course, intrinsically unfair, but they did uncover
a willingness on the part of subjects to reduce their own probabilities of winning slightly in
order to equalize the probabilities of the two opponents winning.
The question of whether a player would like to take from a higher-payoff opponent and
give to a lower-payoff opponent can be made both more concrete and more general. Given
e > 0, define Dij(e) = oU(x0, . . . , xi + e, . . . , xj e, . . . , xn)/oe, so that Dij(e) is the marginal
impact on the decision-maker’s utility of subtracting the amount e from opponent j’s payoff and adding it to opponent i’s payoff. It follows that Dij(0) = oU/oxi oU/oxj, and this
is the impact of a marginal redistribution of wealth from opponent j to opponent i. If
xj > xi, Dij(0) measures the impact of a marginal redistribution from a higher-payoff player
to a lower-payoff player. In keeping with the logic of inequality aversion, such a Robin
Hood redistribution should make the decision-maker better off, so that Dij(0) P 0.
Accordingly, we say that the decision-maker exhibits a preference for Robin Hood redistributions if (xj xi)(oU/oxi oU/oxj) P 0 for all i, j 5 0. The preference is strict if
(xj xi)(oU/oxi oU/oxj) > 0 for all i, j 5 0 and xi 5 xj.
Diminishing sensitivity suggests that the farther something moves from a reference
point, the less additional changes should matter. Here we look at the decision-maker’s sensitivity to changes in his opponents’ payoffs, and we use as a reference point the decisionmaker’s own payoff. This reference point is appropriate because if an opponent’s payoff is
at the reference point, his payoff and the decision-maker’s are equal, and, at least within
the two-person subgroup, there is no inequality. Any movement in the opponent’s payoff
away from the reference point increases inequality, which an inequality averse decisionmaker dislikes. Diminishing sensitivity suggests that the farther the opponent’s payoff
moves from the inequality averse decision-maker’s payoff, the less additional movements
reduce the decision-maker’s utility.
Experimental support for diminishing sensitivity can be found in a study by Loewenstein, Thompson, and Bazerman (1989). They used subjects’ choices to estimate several
different social utility specifications, and their preferred estimate exhibits diminishing sensitivity with respect to both the decision-maker’s own payoff and the differences between
his opponents’ payoffs and his own. Providing further support, Andreoni and Miller
(2002) and Fisman, Kariv, and Markovits (2005) conducted a series of dictator game
ARTICLE IN PRESS
S.A. Hill, W. Neilson / Journal of Economic Psychology xxx (2006) xxx–xxx
5
experiments and used the resulting choices to estimate CES utility functions. Their estimated parameters are consistent with diminishing sensitivity.
When xi < x0, an increase in xi moves opponent i’s payoff closer to the reference point.
According to diminishing sensitivity, movements away from the reference point should be
accompanied by smaller changes in utility, which we measure as smaller changes in the
EDE h. So, as xi moves closer to x0 from below, additional increases in xi should lead
to larger changes in the EDE, which yields o2 h=ox2i P 0 when xi < x0. Similarly, when
the decision-maker is behind, increases in xi represent movements away from the reference
point. They also represent increases in inequality, and therefore a decline in h. As xi moves
farther from the reference point, the impact of additional movements should be reduced,
meaning that the corresponding changes in the EDE should be smaller, that is, less negative. Consequently, the derivative o2 h=ox2i should again be positive. Combining these ideas,
we say that an inequality averse utility function exhibits diminishing sensitivity if
o2 h=ox2i P 0, except possibly when xi = x0. The function exhibits strictly diminishing sensitivity if o2 h=ox2i > 0, except possibly when xi = x0.
3. Existing functional forms
In this section we explore the ability of three existing utility specifications, one that is
additive in payoff levels, one that is additive in payoff differences, and one that is additive
in payoff shares, to exhibit all three of the patterns discussed in the preceding section:
inequality aversion, a preference for Robin Hood redistributions, and diminishing
sensitivity.
Consider first a utility specification that is linear in payoffs:
n
X
U ðx0 ; . . . ; xn Þ ¼ uðx0 Þ þ
vðxi ; x0 Þ:
ð2Þ
i¼1
The function u(x0) captures the decision-maker’s utility from his own payoff, while the
function v(xi; x0) captures the utility or disutility generated by opponent i’s payoff, given
that his own payoff is x0. This simple functional form is used, for example, by Charness
and Rabin (2002) to analyze experimental data, and it is a variant of the form axiomatized
by Neilson and Stowe (2002).
The assumption that U(h, . . . , h) is strictly increasing in h places restrictions on the functions u and v in (2). Since U(h, . . . , h) = u(h) + nv(h; h), differentiation yields the restriction
u 0 (h) + nv1(h; h) > 0, where v1 is the partial derivative of v with respect to its first argument.
Proposition 1 establishes that when an additive-in-levels specification is used, diminishing sensitivity is incompatible with a preference for Robin Hood redistributions.
Proposition 1. Suppose that the preference function in (2) exhibits inequality aversion and
strict diminishing sensitivity. Then it cannot exhibit a preference for Robin Hood
redistributions.
Proof. By definition,
uðhÞ þ nvðh; hÞ ¼ uðx0 Þ þ
n
X
j¼1
vðxj ; x0 Þ:
ð3Þ
ARTICLE IN PRESS
6
S.A. Hill, W. Neilson / Journal of Economic Psychology xxx (2006) xxx–xxx
Differentiation with respect to xi yields
oh
v1 ðxi ; x0 Þ
:
¼ 0
oxi u ðhÞ þ nv1 ðh; hÞ
ð4Þ
Now suppose that i and j are chosen so that xi < xj. A Robin Hood redistribution takes
money away from j and gives it to i. Then Dij(0) = oU/oxi oU/oxj = v1(xi; x0) v1(xj; x0),
where the last equality follows from differentiating (2). From (4),
oh oh
0
Dij ð0Þ ¼ ½u ðhÞ þ nv1 ðh; hÞ :
ð5Þ
oxi oxj
By the restriction that U(h, . . . , h) is strictly increasing in h, the first term in brackets is positive. By strict diminishing sensitivity, oh/ox is strictly increasing in x, except possibly at
zero, and so Dij(0) < 0 when x0 < xi < xj or xi < xj < x0. The decision-maker does not prefer Robin Hood redistributions. h
Since the additive-in-levels specification is unable to accommodate all three patterns,
turn attention to an additive-in-differences specification:
n
X
U ðx0 ; . . . ; xn Þ ¼ uðx0 Þ þ
vðxi x0 Þ;
ð6Þ
i¼1
where v is zero at zero. The assumption that U(h, . . . , h) is strictly increasing in h implies
that u 0 (x) > 0. The formulation in (6) makes utility separable in the decision-maker’s
own payoff and the differences between his opponents’ payoffs and his own payoff. Furthermore, it is the specification used by Fehr and Schmidt (1999) to analyze experimental
evidence, and is axiomatized by Neilson (2006). Proposition 2 shows that once again the
three properties of inequality aversion, diminishing sensitivity, and a preference for Robin
Hood redistributions are incompatible.
Proposition 2. Suppose that the preference function in (6) exhibits inequality aversion and
strict diminishing sensitivity. Then it cannot exhibit a preference for Robin Hood
redistributions.
Proof. By definition,
uðhðx0 ; . . . ; xn ÞÞ ¼ uðx0 Þ þ
n
X
vðxi x0 Þ:
ð7Þ
i¼1
This implies that oh/oxi = v 0 (xi x0)/u 0 (h). If x0 < xi < xj or xi < xj < x0,
oh oh
0
0
0
Dij ð0Þ ¼ v ðxi x0 Þ v ðxj x0 Þ ¼ u ðhÞ :
oxi oxj
ð8Þ
By assumption u 0 (h) is strictly positive, and by strict diminishing sensitivity oh/ox is strictly
increasing. Therefore Dij(0) < 0, and the utility function cannot exhibit a preference for
Robin Hood redistributions. h
Finally, consider an additive-in-payoff-shares specification:
0
1
B x0 C
C:
U ðx0 ; . . . ; xn Þ ¼ uðx0 Þ þ vB
n
@P
A
xi
i¼0
ð9Þ
ARTICLE IN PRESS
S.A. Hill, W. Neilson / Journal of Economic Psychology xxx (2006) xxx–xxx
7
Once again the function u(x0) captures the decision-maker’s utility from his own payoff,
while here the function v(x0, . . . , xn) captures the utility from his share of the total payoff
to all players. This functional form was proposed by Bolton and Ockenfels (2000). Note
that this specification assumes that all payoffs are nonnegative, and that at least one is positive, which is why we restrict attention to allocations in Rnþ1
þ .
This model does not allow a preference for Robin Hood redistributions for the decisionmaker. With this specification Dij(0) = 0, so that the decision-maker is indifferent about any
Robin Hood redistribution. The intuition is simple in that any redistribution between opponents does not change the decision-maker’s payoff share. Thus he is actually indifferent
about any sort of reallocation among opponents as long as the total payoff does not change.
It is worthwhile to discuss further why inequality aversion, diminishing sensitivity, and
reallocating from rich to poor are incompatible in the first two specifications. Consider, for
example, the case where x0 < x1 < x2. When player 1’s payoff increases by e and player 2’s
payoff decreases by e, the inequality between the decision-maker and player 1 increases
while the inequality between the decision-maker and player 2 decreases. Because of diminishing sensitivity, however, the overall utility decreases because the decision-maker is more
sensitive to the change in player 1’s payoff. Since the decision-maker’s utility is not directly
a function of the difference in the payoffs of players 1 and 2, then any change in utility
from the decrease in inequality between the payoffs of players 1 and 2 is not accounted
for by these two models. The results are similar when x1 < x2 < x0.
Finally there is the case where x1 < x0 < x2. In this instance when player 1’s payoff
increases by e and player 2’s payoff decreases by e, the inequality between the decisionmaker and player 1 decreases and the inequality between the decision-maker and player
2 also decreases. Because there is less inequality between the decision-maker’s payoff and
the payoffs to player 1 as well as player 2, the decision-maker’s utility increases. Again,
however, this change in utility is not the result of a decrease in inequality of the payoffs
of players 1 and 2.
While these three specifications seem to have the property that diminishing sensitivity and
a preference for Robin Hood redistributions are incompatible, this may actually be due to
the construction of the models themselves. The specifications base the decision-maker’s utility function upon inequality aversion only with opponents, and none of the models consider
a direct type of inequality aversion based upon differences in the opponents’ payoffs.
4. Alternative specifications
Because the joint hypothesis of diminishing sensitivity and a desire for Robin Hood
redistributions is incompatible in the existing models, one wonders whether it is possible
to construct models in which the two are compatible.
To address this issue, we propose a new model of other-regarding preferences. Letting
N = {1, . . . , n} be the set of opponents, define IL = {i 2 Njxi < x0} and
IH = {i 2 Njxi > x0}, so that IL is the set of players whose payoffs are below the decision-maker’s and IH is the set of players whose payoffs are above the decision-maker’s.
Let nL and nH be the number of elements in IL and IH, respectively. Let fL and fH be
two increasing, real-valued functions with fL(0) = fH(0) = 0. Define
1 X
yL ¼
fL ðxi x0 Þ
ð10Þ
nL i2I L
ARTICLE IN PRESS
8
S.A. Hill, W. Neilson / Journal of Economic Psychology xxx (2006) xxx–xxx
and
yH ¼
1 X
fH ðxi x0 Þ:
nH i2I H
ð11Þ
Finally, let the decision-maker’s utility function be given by
U ðx0 ; . . . ; xn Þ ¼ uðx0 Þ þ vL ðy L Þ þ vH ðy H Þ;
ð12Þ
where vL(0) = vH(0) = 0.
This model is similar to the Fehr–Schmidt additive-in-differences model in that opponents’ payoffs enter only through differences, and it allows the decision-maker to feel differently about being ahead and behind, but it departs from the Fehr–Schmidt model in how
those payoff differences are treated. They are first transformed by the appropriate function
fL or fH, and then averaged, with yL the average transformed payoff difference for opponents with payoffs below the decision-maker’s, and yH the average transformed payoff difference for opponents with payoffs higher than the decision-maker’s. The decision-maker’s
final utility then depends on his own payoff, the average transformed payoff difference for
payoffs below his, and the average transformed payoff difference for payoffs above his.
The ability of this model to allow inequality aversion, diminishing sensitivity, and a
preference for Robin Hood redistributions simultaneously is established by the next
proposition.
Proposition 3. Let U(x0, . . . , xn) = u(x0) + vL(yL) + vH(yH), with u 0 > 0, u00 < 0, yL and yH
given by Eqs. (10) and (11), respectively, fL0 > 0, fH0 > 0, and vL(0) = vH(0) = fL(0) =
fH(0) = 0. If fL00 6 0, fH00 P 0, v0L > 0, v0H < 0, v00L =v0L P nL fL00 =ðfL0 Þ2 , and
v00H =v0H P nH fH00 =ðfH0 Þ2 , then U exhibits inequality aversion, diminishing sensitivity, and a
preference for Robin Hood redistributions.
Proof. By definition,
U ðhðx0 ; . . . ; xn ÞÞ ¼ uðx0 Þ þ vL ðy L Þ þ vH ðy H Þ:
ð13Þ
If i 2 IL, so that xi < x0, then
oh v0L ðy L ÞfL0 ðxi x0 Þ
¼
oxi
nL u0 ðhÞ
ð14Þ
and
2
oh
¼
ox2i
h
i
ðf 0 Þ2
oh
u0 ðhÞ v00L nLL þ v0L fL00 v0L fL0 u00 ðhÞ ox
i
nL ðu0 ðhÞÞ
2
:
ð15Þ
When xi < x0 inequality aversion holds if oh/oxi P 0, which is satisfied because u 0 , v0L , and
fL0 are all positive. Diminishing sensitivity holds if o2 h=ox2i P 0. The term in brackets in the
2
numerator is nonnegative by the assumption that v00L =v0L P nL fL00 =ðfL0 Þ . The second term
00
0
0
in the numerator is negative because u is negative and vL , fL , and oh/oxi are all positive. It
follows that o2 h=ox2i P 0 when xi < x0.
When xi > x0, so that i 2 IH,
oh v0H ðy H ÞfH0 ðxi x0 Þ
¼
oxi
nH u0 ðhÞ
ð16Þ
ARTICLE IN PRESS
S.A. Hill, W. Neilson / Journal of Economic Psychology xxx (2006) xxx–xxx
and
2
oh
¼
ox2i
h
i
ðf 0 Þ2
oh
u0 ðhÞ v00H nHH þ v0H fH00 v0H fH0 u00 ðhÞ ox
i
nH ðu0 ðhÞÞ
2
:
9
ð17Þ
When xi > x0 inequality aversion holds if oh/o xi 6 0, which is satisfied because u 0 and fH0
are positive while v0H is negative. Diminishing sensitivity holds if o2 h=ox2i P 0. The term in
2
brackets in the numerator is nonnegative by the assumption that v00H =v0H P nH fH00 =ðfH0 Þ .
00
0
The second term in the numerator is negative because u , vH , and oh/oxi are all negative
while fH0 is positive. It follows that o2 h=ox2i P 0 when xi > x0.
It remains to show that U exhibits a preference for Robin Hood redistributions.
Suppose that xj > xi for some i, j 5 0. There are three cases. First, if xi < xj 6 x0, since fL is
concave, raising xi and lowering xj > xi by z > 0 raises yL by Rothschild and Stiglitz (1970).
Since vL is increasing, Dij P 0. Second, if x0 6 xi < xj, since fH is convex, raising xi and
lowering xj > xi by z > 0 raises yH. Because vH is decreasing, we again have Dij P 0.
Finally, if xi < x0 < xj, adding z to xi increases yL, but subtracting z from xj reduces yH.
Since vL is increasing and vH is decreasing, Dij P 0. h
It is easiest to understand the restrictions in Proposition 3 by looking at the role each
plays. Inequality aversion is implied by the restrictions on the function v. When vL is
increasing, raising the average of the (transformed) negative payoff differences, which
reduces inequality, raises the decision-maker’s utility. By the same token, when vH is
decreasing, increasing the average of the positive payoff differences, which increases
inequality, lowers the decision-maker’s utility.
The preference for Robin Hood redistributions is governed by the averaging of the
transformed payoff differences and the use of a concave transformation function for the
negative payoff differences and a convex transformation function for positive ones. Begin
with a Robin Hood redistribution between two players whose payoffs are below the decision-maker’s. Since the transformation function for negative payoff differences is concave,
the Robin Hood redistribution increases the average transformed payoff difference. The
fact that vL is increasing then implies that the decision-maker is made better off by the
redistribution. Turning attention to redistributions among opponents with higher payoffs
than the decision-maker, the convexity of the transformation function means that the
redistribution reduces the average of the transformed probability distributions. Since vH
is decreasing, the reduction in the average increases the decision-maker’s utility.
Finally, diminishing sensitivity is implied by the restrictions on the second derivatives of
the functions vL and vH. As with Proposition 1, both vL and vH must be convex to guarantee diminishing sensitivity. However, convexity itself is not enough because of the averaging processes and the transformation functions. The restrictions show that vL must be
sufficiently convex to overcome the concavity of the transformation function fL, and vH
cannot be too convex.
It is also possible to construct a simpler specification which exhibits inequality aversion,
diminishing sensitivity, and a weak preference for Robin Hood redistributions. Define
xL = min{xijxi < x0} and xH = max{xijxi > x0}. Then let
U ðx0 ; . . . ; xn Þ ¼ uðx0 Þ þ vL ðxL x0 Þ þ vH ðxH x0 Þ
ð18Þ
with vL(0) = vH(0) = 0. It is straightforward to show that the desired properties hold if u is
increasing and concave, vL is increasing and convex, and vH is decreasing and convex. If
ARTICLE IN PRESS
10
S.A. Hill, W. Neilson / Journal of Economic Psychology xxx (2006) xxx–xxx
the player has only one opponent, then this model is the same as the Fehr–Schmidt additive-in-differences model of Eq. (6). Also, here the player has a preference for some Robin
Hood schemes because the player’s utility increases as the wealth of the lowest-payoff
opponent increases, and similarly the player’s utility increases as the wealth of the highest-payoff opponent decreases. It is interesting to note that this specification is a slight generalization of the minimax specification favored by Engelmann and Strobel (2004).
5. Conclusion
This paper concerns three behavioral patterns that can be described as follows. Suppose
that a decision-maker has two partners/opponents. One opponent has a payoff that is
lower than the decision-maker’s by 10, and the other has a payoff that is lower than that
by 10. Would the decision-maker be made better off if money was taken away from him
and given to an opponent? If so, he is inequality averse. Does taking a dollar away from
the closer opponent have more effect on his utility than taking away a dollar from the farther opponent? If so, he exhibits diminishing sensitivity. Would he be made better off by
taking a small amount away from the higher opponent and giving it to the lower opponent? If so, he prefers Robin Hood redistributions. We have shown in the paper that these
three patterns are incompatible in three prominent existing models of inequality preferences, and additionally we constructed a new model in which all three coexist.
Acknowledgements
We are grateful to Jill Stowe, Chris Starmer, and two referees for many helpful comments. Neilson thanks the Private Enterprise Research Center for financial support.
References
Andreoni, J., & Miller, J. (2002). Giving according to GARP: An experimental test of the consistency of
preferences for altruism. Econometrica, 70(2), 737–753.
Bernoulli, D. (1738). Exposition of a new theory on the measurement of risk (Translated by L. Sommer (1954)).
Econometrica, 22(1), 23–26.
Bolton, G. E., & Ockenfels, A. (2000). ERC: A theory of equity, reciprocity, and competition. American Economic
Review, 90(1), 166–193.
Camerer, C. F. (2003). Behavioral game theory: Experiments in strategic interaction. Princeton, NJ: Princeton
University Press.
Charness, R., & Rabin, M. (2002). Understanding social preferences with simple tests. Quarterly Journal of
Economics, 117(3), 817–869.
Engelmann, D., & Strobel, M. (2004). Inequality aversion, efficiency, and maximin preferences in simple
distributional experiments. American Economic Review, 94(4), 857–869.
Fehr, E., & Schmidt, K. M. (1999). A theory of fairness, competition, and cooperation. Quarterly Journal of
Economics, 114(3), 817–868.
Fehr, E., & Schmidt, K. M. (forthcoming). The role of equality, efficiency, and Rawlsian motives in social
preferences: A reply to Engelmann and Strobel. American Economic Review.
Fisman, R., Kariv, S., & Markovits, D. (2005). Individual preferences for giving. Manuscript, Columbia
University.
Fong, C. (2001). Social preferences, self-interest, and the demand for redistribution. Journal of Public Economics,
82(2), 225–246.
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47(2),
263–291.
ARTICLE IN PRESS
S.A. Hill, W. Neilson / Journal of Economic Psychology xxx (2006) xxx–xxx
11
Karni, E., Salmon, T., & Sopher, B. (2001). Individual sense of fairness: An experimental study. Manuscript,
Florida State University.
Loewenstein, G. F., Thompson, L., & Bazerman, M. H. (1989). Social utility and decision making in
interpersonal contexts. Journal of Personality and Social Psychology, 57(3), 426–441.
Neilson, W. (2006). Axiomatic reference dependence in behavior toward others and toward risk. Economic
Theory, 28(3), 681–692.
Neilson, W., & Stowe, J. (2002). Other-regarding behavior with unfamiliar opponents: A theoretical construction.
Manuscript, Duke University.
Rothschild, M., & Stiglitz, J. (1970). Increasing risk: I. A definition. Journal of Economic Theory, 2(3), 225–243.