ethicsmath.DVI


Optimality Conditions for Distributive Justice

J. N. Hooker
Carnegie Mellon University

john@hooker.tepper.cmu.edu

Abstract

We analyze utilitarian and Rawlsian criteria for distribution
of limited resources by deriving optimality conditions for ap-
propriate optimization problems. We assume that some indi-
viduals are more productive than others, so that an inequitable
distribution of resources creates greater overall utility. We de-
rive conditions under which a distribution of wealth (a) max-
imizes utility, (b) maximizes a utility function that accounts
for the social cost of inequality, and (c) satisfies a lexmax
criterion that reflects the Rawlsian difference principle. We
show that a utilitarian solution (a) can distribute resources
equally only when all individuals have the same marginal pro-
ductivity. Equality is possible under (b) in a diverse popula-
tion when the cost of inequality is sufficiently large. Equality
is possible under the Rawlsian option (c) when no segment
of society has a much greater average productivity than the
rest. Equality is more likely to be consistent with Rawlsian
justice when there are rapidly decreasing returns to greater
investment in productivity, when the most productive indi-
viduals are not much more productive than the average, and,
ironically, when people are more interested in getting rich.

1 Introduction
Utilitarianism and the Rawlsian difference principle imply
different criteria for distributive justice, but both can be
viewed as mathematical optimization problems. Utilitari-
anism maximizes a social utility function whose arguments
represent wealth distributed to individuals. The Rawlsian
difference principle calls for a lexicographic maximum of
the utilities allotted to individuals. This suggests that the
theory of optimization might provide some insight into the
conditions under which a distribution of wealth satisfies a
utilitarian or a Rawlsian criterion.

In particular, we use classical optimality conditions to an-
alyze distributions over nonidentical individuals. This is
a departure from most axiomatic treatments of distributive
justice, which assume that individuals are indistinguishable
(Blackorby, Bossert, & Donaldson 2002). This capability
allows us to study one of the perennial issues of distribu-
tive justice—the extent to which an efficient distribution of
wealth requires inequality. It is sometimes argued that more
utility is created when greater shares of wealth are alloted to

The author retains the copyright for this article.

individuals who are more talented, more productive, or work
harder.

We use the modeling device of assigning to each individ-
ual i a productivity function ui(α) that measures the total
utility eventually created when individual i is initially al-
loted wealth α. We then find the distribution of initially
available wealth that ultimately results in the greatest to-
tal utility. We investigate the degree of inequality that is
required to maximize utility, and well as conditions under
which a completely egalitarian distribution maximizes util-
ity. We perform a similar analysis when the calculation of
utility accounts for the fact that excessive inequality may
disrupt social harmony and ultimately reduce total utility. In
particular, we determine when the cost of inequality is high
enough so that an egalitarian distribution maximizes utility.

The Rawlsian difference principle states roughly that in-
equality should be tolerated only when it is necessary and
sufficient to result in greater utility for everyone. We follow
the common practice of interpreting this as an imperative to
find a lexmax distribution. To do so we suppose that indi-
viduals have a common utility function v(α) that measures
the personal utility that results when an individual is alloted
wealth α. We further suppose that the fraction of the total
utility that is eventually enjoyed by an individual is propor-
tional to the utility of that individual’s initial wealth allo-
cation. Thus we view the initial allocation of resources to
individuals as assigning social status and privilege. We de-
rive conditions under which a distribution of wealth satisfies
the lexmax criterion, as well as conditions under which the
lexmax distribution is completely egalitarian.

2 Utilitarian Distribution
We first formulate the utilitarian problem. Let the utility
generated by person i from wealth xi be ui(xi). If the total
resource budget is 1, the problem of distributing wealth to
maximize utility is

max
n∑

i=1

ui(xi) (a)

n∑

i=1

xi = 1 (b)

xi ≥ 0, all i (c)

(1)



If we associate Lagrange multiplier λ with the constraint
(1b), any optimal solution of (1) in which each xi > 0 must
satisfy

u′i(xi) − λ = 0, i = 1, . . . , n
Eliminating λ yields

u′1(x1) = · · · = u
′
n(xn) (2)

Thus a wealth distribution is optimal only when the marginal
productivity of wealth is the same for everyone.

Assume that individuals 1, . . ., n are indexed by increas-
ing marginal productivity:

u′i+1(α) ≥ u
′
i(α) for all α ≥ 0 and i = 1, . . ., n − 1 (3)

In this case, (2) is satisfied only if x1 ≤ · · · ≤ xn. Thus the
less productive individuals receive less wealth, as one might
expect. Furthermore, a utilitarian distribution is completely
egalitarian (x1 = · · · = xn = 1/n) only when the marginal
productivities are equal:

u′1(1/n) = · · · = u
′
n(1/n)

To obtain some idea of how skewed the wealth distribu-
tion might be, it is helpful to assume a specific form

ui(xi) = cix
p
i (4)

for the utility functions, where p ≥ 0 and each ci ≥ 0. Here
ci indicates the productivity of person i. When p = 1, per-
son i produces utility in proportion to the wealth received.
When 0 < p < 1, greater wealth has decreasing marginal
utility, and p = 0 indicates inability to use wealth to cre-
ate utility. If individuals are indexed in order of marginal
productivity, we have that c1 ≤ · · · ≤ cn.

Since an optimal solution of (1) in which each xi > 0
must satisfy (1b) and (2), it is

xi = c
1

1−p
i




n∑

j=1

c
1

1−p
j




−1

(5)

when 0 ≤ p < 1. When p ≥ 1, it is clear on inspection
that an optimal solution sets xn = 1 and xi = 0 for i =
1, . . . , n − 1.

Then the optimal distribution is completely unequal when
utility generated is proportional to wealth ( p = 1). The
most productive member of society receives all the wealth.
The distribution becomes increasingly egalitarian as p ap-
proaches zero, reaching in the limit a distribution in which
each person i receives wealth in proportion to ci. Thus the
most egalitarian distribution that is possible in this utilitarian
model is one in which people receive wealth in proportion to
their productivity. Moreover, this occurs only in the limiting
case when the utility generated is independent of the wealth
received (p = 0). When 0 < p ≤ 1, a utilitarian distribution
can be completely egalitarian (x1 = · · · = xn) only when
c1 = · · · = cn. When p > 1, one individual must receive all
the wealth even when c1 = · · · = cn.

Using this model, more egalitarian distributions are less
efficient. In an optimal distribution with 0 ≤ p < 1, the
total utility is (

n∑

i=1

c
1

1−p
i

)1−p
(6)

In a completely egalitarian distribution, each xj = 1/n, and
the total utility is (

1
n

)p n∑

i=1

ci (7)

The ratio (7)/(6) indicates the utility cost of an egalitarian
distribution.

3 Cost of Inequality
The rudimentary utilitarian model above implies that a utili-
tarian solution can result in considerable inequity when indi-
viduals have different abilities. A classical defense of utili-
tarianism, however, is that excessive inequity generates disu-
tility by contributing to social disharmony. The model (1)
does not account for any such cost of inequality. A more
adequate model may result in utilitarian wealth distributions
that are more equitable.

A simple way to try to capture the cost of inequity is to
model it as a proportional to the total range of incomes. The
model (1) becomes

max
n∑

i=1

ui(xi) − β
(
max

i
{xi} − min

i
{xi}

)

n∑

i=1

xi = 1

xi ≥ 0, all i

(8)

Presumably, a positive cost factor β could result in utilitarian
solutions that distribute wealth more equally. It is also inter-
esting to derive how large β must be to result in a completely
egalitarian distribution.

The analysis is easier if we linearize problem (8) using
the following lemma. We again assume that individuals are
indexed by increasing marginal productivity, as in (3).

Lemma 1 If the utility functions ui satisfy (3), and (8) has
an optimal solution, then the following problem has the same
optimal value as (8):

max
n∑

i=1

ui(xi) − β(xn − x1) (a)

n∑

i=1

xi = 1 (b)

xi ≤ xi+1, i = 1, . . . , n − 1 (c)
xi ≥ 0, all i (d)

(9)

Proof. Let x∗ be an optimal solution of (8) with optimal
value U ∗. If x∗j > x

∗
k for some j, k with j < k, then create a

new solution x1 defined by x1j = x
∗
k, x

1
k = x

∗
j , and x

1
i = x

∗
i

for i 6= j, k. If U1 is the objective function value of solution
x1 in (8), then

U1 = U
∗ + uj(x

∗
k) − uj (x

∗
j ) + uk(x

∗
j ) − uk(x

∗
k) (10)

But due to (3),

uk(x
∗
j ) − uk(x

∗
k) ≥ uj(x

∗
j ) − uj(x

∗
k)



because j < k. This and (10) imply that U1 ≥ U ∗. Now
if x1j > x

1
k for some j, k with j < k, create a new solution

x2 in the same manner, and observe again that the objec-
tive function of (8) does not decrease. Continue with the
sequence x1, . . . , xt until xt1 ≤ · · · ≤ xtn. Then xt is feasi-
ble in the problem

max
n∑

i=1

ui(xi) − β
(
max

i
{xi} − min

i
{xi}

)

n∑

i=1

xi = 1

xi ≤ xi+1, i = 1, . . . , n − 1
xi ≥ 0, all i

(11)

and has an objective function optimal value no less than U ∗.
But (11) has an optimal value no greater than U ∗ because it
is more highly constrained than (8). Thus (8) and (11) have
the same optimal value. But (11) is obviously equivalent to
(9), which implies that (8) and (9) have the same optimal
value, as claimed.

To characterize optimal solutions of (9), we associate La-
grange multiplier λ with (9b) and multipliers µ1, . . . , µn−1
with the constraints in (9c). The Karush-Kuhn-Tucker
(KKT) optimality conditions imply that x is optimal in (9)
only if there are a value of λ and nonnegative values of
µ1, . . ., µn−1 such that

u′1(x1) + β − λ − µ1 = 0
u′i(xi) − λ + µi−1 − µi = 0, i = 2, . . . , n − 1
u′n(xn) − β − λ + µn−1 = 0

(12)

where µi = 0 if xi < xi+1 in the solution.
We first examine the case in which each individual re-

ceives a different wealth allotment xi. In this case each
µi = 0, and we can eliminate λ from (12) to obtain

u′2(x2) = · · · = u′n−1(xn−1)
u′1(x1) = u

′
2(x2) − β

u′n(xn) = u
′
2(x2) + β

Thus all individuals who are not at the extremes of the distri-
bution have equal marginal productivity in a utilitarian dis-
tribution, just as they do in the solution of the original model
(1). The individual at the bottom of the distribution, how-
ever, has marginal productivity that is β smaller than that
of those in the middle, while the individual at the top has
marginal productivity that is β larger than that of those in the
middle. This tends to result in somewhat larger allotment for
the individual at the bottom, and a smaller allotment for the
one at the top. Since the remaining individuals are forced
to lie between these extremes, the net result is a distribution
that is less skewed than in the original model.

4 Equality in a Utilitarian Distribution
We can also determine what value of β results in a com-
pletely egalitarian model. In this case the multipliers µi can

be nonzero. Again eliminating λ from the KKT conditions
(12), we get

2µ1 − µ2 = d1
µ1 + µi − µi+1 = di, i = 2, . . . n − 2
µ1 + µn−1 = dn−1

(13)

where

di = u′1(x1) − u
′
i+1(xi+1) + β, i = 1, . . ., n − 1

dn−1 = u′1(x1) − u
′
n(xn) + 2β

(14)

It can be checked that the following solves (13)

µk =
k

n

n−1∑

i=k

di −
(

1 −
k

n

)k−1∑

i=1

di (15)

for k = 1, . . . , n − 1. Substituting (14) into (15), we get

µk = β +
(

1 −
k

n

) k∑

i=1

u′i(xi) −
k

n

m∑

i=k+1

u′i(xi) (16)

for k = 1, . . . , n − 1.
We now consider an egalitarian solution, in which each

xi = 1/n. Since each µi ≥ 0 in an optimal solution, we
obtain the following from (16).

Theorem 2 Suppose that individuals are indexed in order
of increasing marginal productivity. Then an utilitarian dis-
tribution in the model (8) is egalitarian ( x1 = · · · = xn)
only if

β ≥
k

n

m∑

i=k+1

u′i(1/n) −
(

1 −
k

n

) k∑

i=1

u′i(1/n) (17)

for k = 1, . . . , n − 1.

This may be easier to interpret for the specific productiv-
ity functions defined earlier.

Corollary 3 If the productivity function ui are given by (4),
a utilitarian distribution in the model (8) is egalitarian only
if

β ≥
p

np
k(n − k)

(
1

n − k

n∑

i=k+1

ci −
1
k

k∑

i=1

ci

)

for k = 1, . . . , n − 1.

Thus to determine the minimum β required to ensure
equality, we examine each group of k smallest coefficients
c1, . . . , ck. The value of β depends on the difference be-
tween the average of these coefficients and the average of the
remaining coefficients. Thus if there is a group of individ-
uals who are much less productive on the average than the
remaining individuals, relative to the overall range of pro-
ductivities, a larger β is required to ensure inequality. This
could occur in a two-class society with a relatively homoge-
neous underclass and relatively homogenous elites, for ex-
ample.



5 Rawlsian Distribution
A lexmax (lexicographic maximum) model can be used to
represent a wealth distribution that satisfies the Rawlsian
difference principle. As before we let ui(xi) be the social
utility generated by a person i who initially receives wealth
xi. We also suppose that the fraction of total utility received
by person i is proportion to the personal utility v(xi) of per-
son i’s initial wealth allocation. Thus everyone has the same
personal utility function, even though different people may
have different productivity functions.

If yi is the utility enjoyed by person i, any solution of the
following problem is a Rawlsian distribution:

lexmax y (a)
yi
y1

=
v(xi)
v(x1)

, i = 2, . . . , n (b)

n∑

i=1

yi =
n∑

i=1

ui(xi) (c)

n∑

i=1

xi = 1 (d)

xi ≥ 0, i = 1, . . . , n (e)

(18)

where y = (y1, . . . , yn). By definition, y∗ solves (18) if
and only if y∗i solves problems L1, . . . , Ln, where Lk is the
problem

max min {yk, . . . , yn}
(y1, . . ., yk−1) = (y∗1 , . . ., y

∗
k−1)

(18b)–(18e)
(19)

The lexmax solution is frequently defined with respect to
a particular ordering y1, . . . , yn of the variables, in which
case L1 maximizes yk rather than min{yk, . . ., yn}. This is
inappropriate for the Rawlsian problem because we do not
know in advance how the solution values y∗k will rank in size.

Suppose, however, that persons 1, . . . , n are indexed by
increasing marginal productivity as in (3). Then we can
assume without loss of generality that persons with less
marginal productivity are nearer the bottom of the distribu-
tion.

Lemma 4 Suppose that (3) holds and that v(α) is monotone
nondecreasing for α ≥ 0. Then if (18) has a solution, it has
a solution in which y1 ≤ · · · ≤ yn.

Proof. Since v is monotone, it suffices to show that (18)
has a solution (x̄, ȳ) in which x̄1 ≤ · · · ≤ x̄n. For this it
suffices to exhibit a solution (x̄, ȳ) that solves Lk for k =
1, . . . , n and for which x̄1 ≤ · · · ≤ x̄n.

Let (x∗, y∗) be a solution of (18), and let (x0, y0) =
(x∗, y∗). If x01 ≤ x

0
i for i = 2, . . . , n, then x

0 solves L1
and we let x1 = x0. Otherwise we suppose x0k = mini{x

0
i }

and define x1 by x11 = x
0
k, x

1
k = x

0
1, and x

1
i = x

0
i for

i 6= 1, k. We define y1 to satisfy (18b)-(18c). We can see
as follows that (x1, y1) solves L1. If U0 =

∑
i ui(xi) is

the total utility for solution (x0, y0), then the total utility for
solution (x1, y1) is

U1 = U0 + uk(x
0
1) − uk(x

0
k) + u1(x

0
k) − u1(x

0
1)

But we have from (3) that

uk(x
0
1) − u1(x

0
k) ≥ u1(x

0
1) − u1(x

0
k)

Thus U1 ≥ U0, and x1 generates no less total utility than x0.
Since utility is allotted to the y1i s in proportion to v(x

1
i ), and

v is monotone nonincreasing, we get y11 ≤ y01. Thus (x1, y1)
solves L1.

Now if x11 ≤ x
1
i for i = 2, . . . , n, then (x

1, y1) solves
L1, L2 and we let (x2, y2) = (x1, y1). Otherwise we sup-
pose x1k = mini≥2{x

1
i } and define x

2 by x21 = x
1
k , x

2
k = x

1
1,

and x2i = x
1
i for i > 2 and i 6= k. We can show as above

that (x2, y2) solves L1, L2. In this fashion we construct the
sequence (x1, y1), . . . , (xn, yn) and let (x̄, ȳ) = (xn, yn).
By construction, x̄1 ≤ · · · ≤ x̄n. Since (x̄, ȳ) solves
L1, . . . , Ln, it solves (18).

To analyze solutions of (18), it is convenient to eliminate
the variables yi from each Lk . Using constraints (18b)–
(18c), we get

yi = v(xi)

n∑

i=1

ui(xi)

n∑

i=1

v(xi)

, i = 1, . . ., n

Using Lemma 4, Lk can be written

max v(xk)

n∑

i=1

ui(xi)

n∑

i=1

v(xi)
(a)

(x1, . . . , xk−1) = (x∗1, . . ., x
∗
k−1) (b)

n∑

i=1

xi = 1 (c)

xk ≤ · · · ≤ xn (d)
xk ≥ 0 (e)

(20)

where x∗1, . . . , x
∗
k−1 are previously computed solutions of

L1, . . . , Lk−1, respectively.
We focus first on L1. Associating Lagrange multipliers

µ1, . . . , µn−1 with the constraints in (20d), the KKT opti-
mality conditions imply that a solution x with each xi > 0
is optimal in (20) only if there are nonnegative values of
µ1, . . . , µn−1 such that

v′(x1)
Σu
Σv

+ v(x1)
u′1(x1)Σv − v′(x1)Σu

(Σv)2
− λ − µ1 = 0

v(x1)
u′i(xi)Σv − v′(xi)Σu

(Σv)2
− λ + µi−1 − µi = 0,

i = 2, . . . , n − 1

v(x1)
u′n(xn)Σv − v′(xn)Σu

(Σv)2
− λ + µn−1 = 0

(21)
where

Σu =
n∑

i=1

ciui(xi), Σv =
n∑

i=1

v(xi)



and where µi = 0 if xi < xi+1 in the solution.
We begin by examining the case in which each individual

receives a different allotment xi. Here each µi = 0, and (21)
implies

v′(x1)
v(x1)

+
u′1(x1)

Σu
−

v′1(x1)
Σv

=
u′i(xi)
Σu

−
v′i(xi)
Σv

for i = 1, . . . , n−1, assuming v(x1) > 0. This says that the
marginal difference between productivity and personal util-
ity is the same for everyone except the lowest ranked indi-
vidual, for whom the difference is somewhat less. This tends
to increase the allotment to the lowest individual, reducing
the gap between this person and the others. The optimal-
ity conditions for L2 are similar and likewise move the sec-
ond closest individual closer to those who are more highly
ranked. Thus in general, the lexmax solution results in a
distribution that is more egalitarian than one in which the
marginal difference between productivity and personal util-
ity is the same for everyone.

6 Equality in a Rawlsian Distribution
We now examine conditions under which a Rawlsian dis-
tribution can be egalitarian. We found earlier that a util-
itarian distribution with utility functions ui(xi) = cix

p
i ,

v(xi) = x
q
i cannot be egalitarian unless individuals are iden-

tical in their productivity. We will show that a Rawlsian dis-
tribution can, under certain conditions, be egalitarian in a
more diverse population.

In an egalitarian distribution any µi can be nonzero. We
eliminate λ from the optimality conditions (21) for L1 to
obtain

v′(x1)
v(x1)

+
u′1(x1)

Σu
−

v′(x1)
Σv

−
1

v(x1)
Σv
Σu

µ1

=
u′i(xi)
Σu

−
v′(xi)
Σv

+
1

v(x1)
Σv
Σu

(µi−1 − µi)
(22)

‘ for i = 2, . . ., n − 1, and
v′(x1)
v(x1)

+
u′1(x1)

Σu
−

v′(x1)
Σv

−
1

v(x1)
Σv
Σu

µ1

=
u′n(xn)

Σu
−

v′(xn)
Σv

+
1

v(x1)
Σv
Σu

µn−1

(23)

This yields the following.

Theorem 5 Suppose the productivity functions are given by
ui(α) = ciαp and the utility function by v(α) = αq. Then
L1 has an egalitarian solution ( x1 = · · · = xn) only if

1
k

k∑

i=1

ci ≥
1

n − k

n∑

i=k+1

ci −
q

p
·
n − k

k

n∑

i=1

ci (24)

or equivalently,

1
k

k∑

i=1

(
1 +

q

p
·
n − k

k

)
ci ≥

1
n − k

n∑

i=k+1

(
1 −

q

p
·
n − k

k

)
ci

(25)
for k = 1, . . . , n − 1.

Proof. The equations (22)–(23) can be written as (13)
where

di = v(x1)
Σu
Σv

(
v′(x1)
v(x1)

−
u′i+1(xi+1) − u′1(x1)

Σu

+
v′(xi+1) − v′(x1)

Σv

)

for i = 1, . . . , n − 1. Substituting x1 = · · · = xn = 1/n
and the functions ui, v as given above, we obtain

di = qn
−p

n∑

j=1

cj − pn−p (ci+1 − c1) (26)

Since (15) solves (13), we can substitute (26) into (15) and
get

µk = p
k(n − k)

n1+p

(
q

pk

n∑

i=1

ci +
1
k

k∑

i=1

ci −
1

n − k

n∑

i=k+1

ci

)

for k = 1, . . ., n − 1. The KKT conditions imply that xk =
· · · = xn = 1/n can be an optimal solution only if µk ≥ 0
for k = 1, . . . , n − 1, which implies (24).

An egalitarian solution (x1 = · · · = xn) solves L1 if and
only if it solves the lexmax problem (18). If it solves L1,
then a lexmax solution must have x1 = 1/n, which implies
by (18d) that x2 = · · · = xn = 1/n. If an egalitarian
solution does not solve L1, then some distribution with x1 <
1/n solves L1, which implies that x1 < 1/n in any lexmax
solution. Thus we have

Corollary 6 If the productivity functions are given by
ui(α) = ciαp and the utility function by v(α) = αq, then
a lexmax distribution is egalitarian ( x1 = · · · = xn) only if
(24) and (25) hold.

Thus a Rawlsian distribution is completely egalitarian
when the gap between the average productivity of the k least
productive individuals and that of the remaining population
is not too great for any k. The maximum gap is proportional
to q/p and (n − k)/k. This means that a smaller gap is re-
quired when the marginal utility of wealth decreases rapidly
with the level of wealth (q is small), and when the opposite
is true of marginal productivity ( p is large). Thus an inegal-
itarian distribution is more likely when individuals do not
care very much about getting rich and are satisfied with a
moderate level of prosperity. Inequality is also more likely
when allocating greater advantages to talented or industrious
people reaps consistently greater rewards.

An egalitarian distribution also requires a smaller produc-
tivity gap between the highest class and the remaining pop-
ulation (i.e., when k = n−1 ) than between the lowest class
and the remaining population ( k = 1). Thus if the distribu-
tion of talents and industry has a long tail at the upper end,
as is commonly supposed, the condition for equality could
be hard to meet.

7 Conclusion
We find that a utilitarian distribution of wealth can result in
substantial inequality when some individuals are more pro-
ductive than others. The distribution is completely egali-
tarian only when every individual has the same marginal



productivity. When marginal productivities are unequal,
the most egalitarian distribution that is possible is one in
which individuals are allocated wealth in proportion to their
marginal productivity, and this occurs only when there are
rapidly decreasing marginal returns for greater allocations
of wealth.

A more egalitarian distribution results when the utility
function includes a penalty to account for social dysfunc-
tion that inequality may cause. In particular, if the penalty
is proportional to the gap between the richest and poorest
individuals, we can calculate a constant of proportionality
that results in a completely egalitarian distribution. This
constant tends to be larger when there is large gap in av-
erage productivity between two segments of society. That
is, there a group of individuals that have a much smaller
average marginal productivity than the remaining individu-
als, relative to the overal range of productivities. This may
occur, for example, when elites and common people form
fairly homogenous groups separated by a large gap in aver-
age productivity.

Finally, the Rawlsian difference principle can result in a
completely egalitarian distribution when no two segments
of society have a large gap in average productivity. Equality
is more likely to occur when there are decreasing returns
for placing greater investment in talented and industrious
people. Somewhat surprisingly, equality is also more likely
when people are nearly as concerned about getting rich as
about living a minimally comfortable lifestyle. When people
want riches more, a privileged class is less likely to be con-
sistent with Rawlsian justice. Finally, equality is more likely
when the most talented and industrious individuals are not
much more productive than the average person, even though
the least productive individuals may fall far below the mean.

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