Sibling Rivalry and Strategic Parental Transfers
Author(s): Yang-Ming Chang and Dennis L. Weisman
Source: Southern Economic Journal, Vol. 71, No. 4 (Apr., 2005), pp. 821-836
Published by: Southern Economic Association
Stable URL: http://www.jstor.org/stable/20062082
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Southern Economic
Journal
821-836
2005, 71(4),
Sibling Rivalry and Strategic
Parental Transfers
and Dennis
Chang*
Yang-Ming
L. Weisman|
a noncooperative
two siblings compete
Nash model
in which
for their parents'
paper develops
contest"
and using a Tullock-Skaperdas
financial
transfers. Treating
sibling rivalry as a "rent-seeking
more
are
success
we derive
resources
contest
the conditions
under which
financial
function,
This
transferred
that
the
to the sibling with lower earnings. We
as an institution
serves
as an
family
find
that parental
"income
are compensatory
a sequential
transfers
equalizer."
transfers and
Within
and
game
we characterize
of parental
link it to parents'
the endogeneity
framework,
income,
or child services).
and children's
altruism,
supply of merit goods
(e.g., parent-child
companionship
We
show that merit goods are subject to a "moral hazard" problem
from the parents' perspective.
Classification:
JEL
Dl,
C7
H3,
1. Introduction
with
Beginning
to
paid
Several
analyzing
empirical
studies
documented
altruistic
purely
Another
inter
of
resources
are more
amounts
transfer
transfers
(i.e.,
to be
likely
living
to children
lends
child's
recipient
with
and Ohlsson,
earnings
recipient
the
between
transferred
1995; Hochguertel
and
and
transfers
transfers
attention has been
1981), considerable
(1974,
parental
vivos
(McGarry and Schoeni,
between
relationship
between
relationship
that financial
income
high
the seminal works of Becker
the
earnings.
low
rather
This
2000).
to Becker's
support
have
persons)
than
inverse
model
of
transfers.
prominent
on parents
study
and
children
is by Becker
and Tomes
(1976).
examine
They
investment in the human capital of children and its implications for earnings capabilities and for the
distribution
intrafamily
with
children
transfers
larger
inter
(e.g.,
of
income
vivos
to achieve
gifts)
to achieve
The seminal works of Becker
not
allow
for
"merit
care). The pioneering
that
argue
*
goods"
Department
f Department
that
transfers
of Economics, Kansas
corresponding author.
of Economics, Kansas
they
Specifically,
in human
"efficiency"
in income
"equity"
show
capital
that parents
invest
more
investment
and
then
are
render
to
their
parents
(e.g.,
(1976), however, do
services,
to
the
exchange
between
or parental
visits,
Shleifer, and Summers (1985) and Cox
related
in
use
distribution.
(1974, 1976, 1981) and Becker and Tomes
children
studies of Bernheim,
intergenerational
ymchang@ksu.edu;
children.
among
endowments
parent
(1987) further
and
a child
for
State University,
319 Waters
Hall, Manhattan,
KS
66506-4001,
USA,
E-mail
State University,
246 Waters
Hall, Manhattan,
KS,
66506-4001,
USA,
E-mail
weisman@ksu.edu.
The authors are grateful to the editor, Laura Razzolini, and an anonymous referee for numerous
and critical insights that led to substantial improvements in the paper. The usual caveat applies.
Received December 2003; accepted October 2004.
constructive
suggestions
821
822
Yang-Ming
merit
family-specific
Pollak's
(1988)
and Dennis
Chang
as
such
goods
child
that parental
finding
of
by
are
studies
Empirical
a "bribe"
to induce
inter
of
vivos
Most
theoretical
and
develop
models
children
an
without
alternative
and
parents
offspring
behavior.
income,
strategic
services.
success
children's
the underlying
Cox
child
evidence
that
income
among
the
to parents.
altruistic
transfer
to
attempt
and
toward
We
rent-seeking
We
family.
siblings
goods
between
siblings.
in
a financial
of
equally
of merit
are
the
we
In this paper,
between
is common
within
between
relationship
interactions
interactions
which
choice
siblings
the
the children.
intergenerational
children
by
parents'
supply
that
premise
only
redistributing
the
the
Moreover,
provide
emphasize
function,"
activities
in
exchange
between
intragenerational
characterize
and
altruism,
on
Based
emphasizes
not
transfers
also
transfers
interactions
a "contest
parental
We
strategic
considering
"transfer-seeking"
of
role
the
also
analysis
to examine
examine
that
or
goods
related (e.g., Cox 1987; Cox and Rank 1992; Cox
intergenerational
explicitly
but
the
into
of
approach
children,
incorporate
literature,
parents'
parallels
of particular
to provide
them
and
transfers
transfers and recipient earnings may be positively
and Jakubson 1995; Lillard andWillis 1997).
parents
consideration
consumption
(1989) contend that parents attempt tomanipulate
andMorris
offering
to a child's
tied
This
services.
(1992) contend that parental transfers can be interpreted as a way of "buying"
and Rank
services.
children
their
or
companionship
transfers
services that parents value. Kotlikoff
behavior
L. Weisman
in affecting
link
it to
parents,
the
and
their
sibling-rivalry model developed in this study implies that children whose earning capabilities or
earnings are higher supply less services (or exert less effort) in acquiring financial resources from their
parents.
In other
parental
transfers
words,
more
transfer
parents
are compensatory
resources
in the Beckerian
to children
sense,
with
lower
the fact
despite
In this
earnings.
that
the models
case,
of Becker
(1974, 1981) and Becker and Tomes (1976) do not allow for merit goods. We compare differences in
income between siblings before and after parental transfers and find that in equilibrium the income
differential is reduced. This finding supports the proposition that the family as an institution serves as
an "income
(e.g.,
good
lower
ability
under
conditions
altruistic
hazard
by
transferring
sibling
transfers
cooperative
with
problems
merit
a recipient
and
for parental
rivalry
parents
more
the parents.
We
further
in transfers
and
children's
by
solution
equal,
being
child's
may
analyze
also
services.
also
utility
discuss
related.
maximizing
Nash
attention
Special
"insurance"
be positively
earnings
the noncooperative
compare
of
to that child. We
we
other
ability,
a type
provide
resources
transfers,
a child's
for
proxy
good
then
proportionately
amount
the transfer
to examining
financial
the
child
which
In addition
is a reasonably
capability
in labor markets)
fortune
for the
with
If earnings
equalizer."
factors
equilibrium
is paid
to moral
goods.
The remainder of the paper is organized as follows. Section 2 develops a simple model of sibling
competition
for
implications
of
characterize
the
alternative
parental
resources
parental
transfers
endogeneity
games
a family
Consider
make
a total
have
their
proportion
of Sibling
transfers
of
time
children
two
of M
each
sibling
play.
compete
to distribute
Rather,
expends
the
In
game.
between
and
transfers
4
Section
for Parental
siblings
dollars
unconditionally.
that
may
Nash
differentials
financial
Rivalry
in which
amount
income
parents'
and
that parents
2. A Nash Model
parents
of
in a noncooperative
for
this
In
siblings.
the
compare
summarizes
we
section,
3,
of
the
outcomes
and
discuss
section
we
two
concludes.
Transfers
for financial
to the
parents
in rendering
The
siblings.
divide
services
from
transfers
the
parents,
"prize"
to
their
M
their
The
parents.
however,
do
not
according
to
the
parents.
Specifically,
Sibling Rivalry and Parental Transfers
the amount
let et denote
of
the prize
time
of
and
is pi(ei,e2),
=
/, /
that sibling
for
2,
sibling
to his
For
1, 2, devotes
parents.1
=
1
where
P\(e\9e2),
success
1 is a "contest
literature
Tullock
(e.g.,
1980;
*
e\ +
In other
1996).
(!)
e2
to
similar
function"
Skaperdas
1, the share
sibling
it is p2(eue2)
Pi(eue2)=
Equation
823
those
words,
in
used
commonly
rent-seeking
a "transfer-seeking
orchestrate
parents
the
contest" between siblings to induce their supply of services and to determine the distribution of the
financial
transfer.
to Equation
According
and
ex,
services,
on
positively
on his time of
1, sibling l's share of the prize, M, depends positively
on
negatively
time
2's
sibling
ex. It can
on
e2 and negatively
of
services,
e2. Similarly,
be verified
easily
that
2's
sibling
the marginal
effect
share depends
of et on ph p\ =
?
= 1, 2, /
dpi(e\9e2)ldei
^ j.
ejl(e\ + e2)2, is positive but is subject to diminishing returns, where /, j
For analytical simplicity, each sibling is assumed to be risk neutral and has T units of time
for working
available
of
capabilities
market
wage
The
the
rate
siblings
for
the
service
for
to
services
providing
rates
the wage
by
/ be w? >
their
and
family
are reflected
sibling
choose
siblings
of
outside
they
command
to maximize
their
the
in the
Earning
parents.2
labor markets.
Let
the
0.
allocations
individual
incomes,
expected
which are given by
Fi
=
(T -ex)wx
+ aLlel, (2)
+pl(eue2)M
and
Y2 =
where
of
We
the altruism
time
spent with
coefficient,
ai9 represents
the parents.
Note
that 0 <
assume
(T -e2)w2-\-
a,- <
wi9 which
that
[1-pl(eue2)]M-\-ct2e2,
the monetary
if a?- >
ensures
0,
an
valuation
sibling
interior
(3)
that
/ "enjoys"
spending
(et >
solution
sibling
0). The
on
/ places
each
unit
time with
the parents.
first-order
conditions
(FOCs) for sibling l's and sibling 2's optimization problems are given respectively by
6l
JT=,
dei
(4)
,2M-Wl+<*!=(),
{ex+ e2y
and
eX
ffi=
de2
(ei + e2f
.M-w2 + ?2=0. (5)
The FOCs indicate that each sibling's service time is optimally chosen so that the expected marginal
benefit of exerting one more unit of service time equals itsmarginal cost (in terms of wage income
forgone)
for
net
a maximum
of
the altruistic
coefficient
(i.e.,
are
as a result
of
satisfied
p\M
the
=
strict
wt
?
ol?). The
concavity
of
sufficient,
the contest
second-order
success
conditions
functions.
1
The variable e?may be variously interpreted as time, frequency of parental visits, or the level of care supplied
The results of the analysis will not be affected if T is normalized to unity.
2
to parents.
824
Yang-Ming Chang and Dennis L. Weisman
It follows from Equations 4 and 5 that
6 implies
Equation
5 that
-
if wx
wx
e\
w2
c*i
?
(6)
cc2
amount
the relative
coefficients,
ax >
w2
?
of
services
supplied
the
by
oc2 then
>
H^M'
(ex+ e2)
r^r^M
(ex+ e2)
which
?
related to their relative market wage. Furthermore, it follows from Equations 4
siblings is negatively
and
the altruism
that, given
e2
(7)
implies that
ei <
amount
the
Hence,
of
child
services
is
e2. (8)
to
related
inversely
the market
of
(net
wage
the
altruism
coefficient).
we
Next,
FOCs
examine
in Equations
ex
respectively,
the equilibrium
of
behavior
4 and 5 implicitly define
=
and
ex(e2,M,wx,0Lx)
e2
=
the
and
siblings
its economic
two
The
e2(ex'9M,w29a2).
The
implications.
the reaction functions for sibling
reaction
1 and sibling 2,
functions
the noncooperative Nash equilibrium solution, denoted by the two-tuple
4
and 5, we solve for the Nash equilibrium levels of child services:
Equations
determine
jointly
(e\9e\). Using
(9)
-H^-_Mi
[(wx ai) + (w2 -a2)]
and
e\ =-^=^--2M.
[(wi -oii)
(10)
+
(h>2-a2)]
It follows from Equations 9 and 10 that
(H^-wO + ^-Ob)
e*_4=
Equations 9 and 10 further imply the following
Be*
k<0>
where
/, j =
from
following
(a) The
Be*
Be*
^>0<
^>(=)<0<
(12)
^>o,
a?>H<o,
their
parents
Nash
game
to a Tullock-Skaperdas
according
two
in which
contest
success
a financial
for
compete
siblings
we
function,
have
the
results:
of
supply
services
a
by
sibling
decreases
with
his wage
rate and
increases
with
his
altruism
ce ter is paribus.
in the
increase
supply
of
total
services,
The findings
related
comparative static derivatives:
Be*
1. In a noncooperative
coefficient,
(b) An
Be*
_ (u)
(w2-a2)]
1, 2, / ^ j. This analysis suggests our first proposition.
Proposition
transfer
+
[(wx-ai)
to a child's
amount
in Proposition
earnings
of parental
transfer
encourages
both
siblings
to increase
their
ce ter is paribus.
1 imply that the supply of child services to parents is inversely
capability
but
is positively
related
to
the
child's
altruism
toward
the
Sibling Rivalry and Parental Transfers
Also,
parents.
strategically
who
parents
induce
a greater
have
("bribe")
more
to supply
resources
more
to allocate
ability
children
their
to
their
825
can
children
services.
that the expected parental transfer to sibling / isM* = p*M, where p* = e?l(e\ + e*2), it
follows from Equations 9 and 10 that
Given
m;
=
7?^F^?-m,
(wx -ai) +
(w2-a2)
(is)
and
AC=7-^T1-&.
K
-ai)
"?>?.
Boii
fUo,
awj
aw i
^<o,
/, j =
1,2, / ^ y. Inspection of Equations
***
ha*
M\-M2=)-i-?j-'-M.
are
There
(14)
(w2-a2)
13 and 14 yield the following comparative static derivatives:
Equations
where
+
cases
three
of
"?<*
Boij
a?
BM
^>o,
13 and 14 reveals that
(^2-Wi) + (0li -0l2)
(wx oil) + (w2 a2)
(16)
interest:
=
ot2,then e\ < e2andM\ < M2. (17)
(i) If w\ > w2 and oti
(ii) If w\ > w2 and oti < ot2,then e\ < e2andM\ < M2. (18)
?
(iii) If wx > w2 and ax > a2 + (wi w2),then e\ > e*2andM\ > M2.
One
is that the expected
observation
equilibrium
for each
compensation
unit
of time
(19)
spent with
par
ents is equalized across the siblings. To see this, divide M* (=p*M) by e* to obtain the following result:
M*
M*
?i =
?^=(wl-OL1)
e2
e\
We
+
(w2
-
oe2)> 0.
(20)
summarize these findings in the following proposition.
Proposition
2.
(i) If both the high-wage and the low-wage siblings are equally altruistic toward their parents, the
more
transfer
parents
resources
to the
child
low-wage
than
the high-wage
child,
(ii) If the high-wage child is less altruistic toward his parents than the low-wage child, the result in
(i) is compounded,
(iii) If the high-wage
child,
(iv)
child is "sufficiently" more altruistic toward the parents than the low-wage
the parents
Parents
transfer
a greater
that have
more
ability
resources
to offer
to the high-wage
a
larger
prize,
sibling,
transfer
more
resources
to their
children, or dM*/dM > 0.
(v) In the noncooperative Nash equilibrium with a transfer-seeking contest, the financial
compensation per unit of time of child services is equalized across the children.
The findings in (i) and (ii) of Proposition 2 imply that there is a negative relationship between
parental
resulting
transfers
from
and
ability
a
recipient
and
human
child's
capital
earnings.
differentials
Parents
by
compensate
transferring
for
more
labor
earnings
financial
differentials
resources
to the
Yang-Ming Chang and Dennis L. Weisman
826
less-endowed
sense.
children.
The
Under
economic
these
intuition
circumstances,
behind
is straightforward.
services to parents is higher for the high-wage
less
proportionately
in caring
time
for
are compensatory
transfers
parental
this finding
The
opportunity
the high-wage
child. Consequently,
the parents
and
a lower
receives
concomitantly
in the Beckerian
cost
of rendering
child supplies
transfer.
parental
The finding in Proposition 2(iii) implies that there can be a positive relationship between parental
transfers and the earnings of the recipient child when the high-ability child is sufficiently altruistic
his
toward
In this
parents.
to the
payments
simple
The finding
resources
for
for
children
transfers,
parental
the
are
and more
compensatory
of
reflective
parents tend to transfer more
indicates that high-income
low-income
parents
less
to the parents.
rendered
2(iv)
than do
are
transfer
parental
services
in Proposition
to their
rivalry
case,
special
child
Moreover,
parents.
toward
"unbiased"
within
their
the framework
children.
This
of
sibling
nondiscrimination
result derives from the fact that the "equilibrium price" of child services?measured
in terms of
to
of
the
child
for
the
for
each
unit
time
with
the
identical
spent
parents?is
compensation
siblings.
we
Finally,
each
compare
income
sibling's
the financial
and after
before
to determine
transfer
whe
ther the implied participation constraints are satisfied. Let the posttransfer income for child /be denoted by
Y*. Substituting e\ and e\ in Equations 9 and 10 into the objective functions in Equations 2 and 3 yields
,
v* = t
Yx
v* =
Y2
(w2-a2)2M
Twx
H-2
[(wx-oii) + (w2-oi2)]
>L
-\-2
t
Tw2
(wx-ctx)2M
+
[(wi -ai)
21
Equations
and
22
that w? >
provided
or F* >
that each
indicate
these
a,-. Under
> Twx, (21)
a financial
has
sibling
each
conditions,
>
Tw2. (22)
(H>2-a2)]
to participate
incentive
sibling's
posttransfer
in the "contest"
income
expected
increases,
Tw?.
It is instructive to examine how the income differentials between the siblings are affected by the
to the transfer-seeking
transfers.
Prior
?
After
the transfer-seeking
Twx
Tw2.
parental
simply
contest,
the
income
contest,
the
income
differential
between
is
differential
the
?
Yx
Y2.
is
siblings
It follows
from Equations 20 and 21 that
+ (?i
V ;= [^-^)
-*2)]m.v J(23)
-rw2)
[(wi-oii) + (w2-a2)]
V
'
2)
(y;-yd-(ty*
For
the case
implies
wx >
in which
that
Conversely,
w2
income
differential
for
the case
in which
ot2, the sign
between
wx >
the siblings actually widens. We
w2
the
and
of
the expression
siblings
o^ >
a2
in Equation
is reduced
-f (wx
the
by
?
the
w2)9
23
success
contest
function,
parental
transfers
may
or may
not
reduce
the
income
the siblings, depending upon their wage differential and the difference
toward
when
children
parents.
the
Using
differential
in the degree of
the parents.
the symmetric
M\ ?M\
between
differential
thus have the following.
altruism
and Y\
This
contest.
transfer-seeking
income
between
For
is negative.
3. If sibling rivalry for a financial transfer from their parents is based on a Tullock
Proposition
Skaperdas
otx <
and
the
?
in which
oti
=
=
w2, we
ot2 and Wi
have
=
e\
e\
and
In this
p\ =p*2.
case,
are equally divided among children
Y\. These results imply that parental transfers
are
the
case
equally
symmetric
interesting behavioral
productive
case
as
in the
a benchmark,
labor markets
our
findings
and
are
equally
in Propositions
implications for the role of the family. Assuming
altruistic
2
and
toward
3
the
suggest
that children are equally
827
Sibling Rivalry and Parental Transfers
toward
altruistic
to
more
transfer
resources
financial
transfers
compensating
of a contest
the design
their parents,
to
the
supports
the
idea
with
child
the
that
sense
in the Tullock-Skaperdas
lower
earnings
as an
family
leads
institution
the parents
This
capabilities.
may
of
finding
as
serve
an
income
equalizer.
Another
is
implication
interesting
potentially
that
parents
the high-wage
nature,
is also
child
ability child proportionately more
financial protection for lower wage
if high-wage
Finally,
instead
may
parents
a positive
In
the
have
reality,
parents
Consequently,
We
sibling
contest.3
parents
commit
and
we
of
timing
a specified
as an
the
compensate
low
the former is exogenously
or nonmarket
intrahousehold
the
of money
or
insurance
levels
In
of
are
the
child's
exists
earnings.
in the model.
transfer
the
stage,
that maximize
their
that maximizes
of
the financial
In the first
to their
distributed
In
in a utility-maximizing
is as follows.
second
services
there
case,
exogenous
amount
transfer
game
be
children,
low-wage
this
Game
the endogeneity
two-stage
that will
function.
the
total
In
the recipient
of a financial
to characterize
or
and
transfers
parental
then
parents
children.
transfer
the amount
sequential
success
their
high-wage
the parental
that
game
the
toward
the
to endogenize
choose
simultaneously
child because
in a Sequential
assumed
amount
contest
of
to determine
a sequential
employ
to
Transfers
it is important
The
If the parents
child.
altruistic
the amount
the discretion
a Tullock-Skaperdas
transfer
resources
analysis,
previous
utility.
framework.
more
of Parental
3. Endogeneity
or
"insurance"
(ability) children.
are more
between
serve
transfers
children
transfer
relationship
the high-ability
than the high-ability
then parental
by nature,
disadvantaged
of
type
that the ability of a child is a random draw
protection for children with differential abilities. Given
from
a
provide
children
children
their
stage,
the
according
to
for
compete
functions.
objective
in
prize
the
The
parents do not distribute the prize until after they have received services from their children. This, in
is the basic
essence,
As
the parents
equilibrium
commit
in the first
in the previous
parents'
For
preference
theory,
subgame
the analysis
altruistic
that Hirshleifer
stressed:
(1977)
we
use
backward
decision
analytical
of
stage
simplicity,
the
size
assume
and
services,
the game.
We
section.
on
of
choice
then
of
As
the prize,
that
the
yp
is the parents'
and y represents
the
the "last
for
word."
the subgame
perfect
Nash
induction, we first solve for the
solve
for
second
to the first
the
stage
stage
total
of
of
amount
transfer
the game
the game
that
to
is identical
to examine
the
M.
parents
have
collectively
initial
income,
the utility
valuation
e2) + ?yFi + ?y72,
?
u(yp
M,ex,e2)
the
following
altruistic
is their own
utility
(24)
as a function
of
income
after
1) is the altruism coefficient attached to each child's utility,
that the parents
This section is the result of a suggestion by an anonymous
of a financial transfer in a sequential game.
4
have
to solve
function4:
transfer and children's services, ? (0 < ? <
3
then
such,
proceed
U = u(yp -M,ex,
where
Parents
induction
in the sequential game. Consistent with backward
equilibrium
children's
idea
in game
is standard
place
on each
child's
referee and the editor that we
income.5
Define
income
after
include an analysis of parents' choice
5
An additively separable utility function has been widely adopted to analyze various issues such as the "rise and fall of families"
(Becker and Tomes 1979, 1986), the economic analysis of fertility (Becker and Barro 1988), the biological origin of altruism
(Mulligan 1997), and residential choice of family members (Konrad, et al. 2002).
Parents may be "biased" in that they place different values of ? and y across children. We rule out this possibility
analysis on differences in children's earnings capabilities and their implications for parental transfers.
and focus the
828
to 1. For
normalized
strictly
concave
u?
positive
/=
but
M.
yp
=
own
?
B^u/BI2
0,
composite
?
u(yp
uei
?
M,ex,e2)
the following
adopt
<
is a Hicksian
there
function,
further
=
>
B2u/BlBei
that
utility
e2. We
un
Bu/BI>0,
assume
We
the parents'
in /, eX9 and
uiei
where
?
I=
as /, where
transfer
L. Weisman
Dennis
and
Chang
Yang-Ming
=
Bu/Be?
u(I,ex,e2),
and
notation
>
=
ue?e?
0,
good
whose
we
assume
is
price
that
it is
assumptions:
<
c^u/Be2
0,
0,
1,2. The marginal utility of consumption of either the private good or the merit good is
The
diminishing.
that
assumption
is nonnegative
uIei
implies
that
the parents'
marginal
utility of consuming the private good is nondecreasing in the merit good supplied by child /.
The objective of the parents is to choose M to maximize the altruistic preference function in
Equation 24, where ex and e2 are given by Equations 9 and 10, and Yx and Y2 are given by Equations
21 and 22. The parents' FOC is
=
m
dU
del
-u^^m
+
de2
^m
+
?
o dY\
+ n &Y2= 0^
^BM ^BM
(25)
where
w2
Bex
[(h>i
BM
cti) +
wx
Be2
?
ol2
(w2
?
-
'
a2)]2
oti
BYX_
BM~
'
oil) + (h>2 a2)]2
?
(w2 ot2)
>0,
[(Wl_ai) + (H;2-oi2)]2
BY2_
dM
[(Wl_ai)
BM
[(ivi
-
and
Assuming
(wx-ax)2
+ (w2-a2)]2
an interior and unique solution forM,
>o
the SOC (denoted as J) requires that
^
Bex Be2
Be2
dex fdex\ +
+ U f^e2\ K ?J = u?- 2u">
M + """[dM) 2"-2dMdM
dM 2"'\dMJ
Let
amount
the equilibrium
of
the prize
that
the parents
commit
be
denoted
/^^n
(26)
as M*.
It follows from the FOC in Equation 25 that the effect of changes in the parents' income yp on
the prize is
?') > p.
^ +^^
gg = 1L - "?'(?>
(27)
}(
+
dyP
J\"
[(Wl ?,)
(w2 a2)f
It is straightforward to show that
?
(w2 ot2)
)+ (W2
[(wx OLX
y JBM*
??
J\
?
oil)
(wi
+ 77-^-^-^
+
Oli
)
(W2
[(WX
0l2)]2
The findings of the analysis are summarized by the following proposition.6
6
See the Appendix
for detailed derivations
of the results in Proposition
4.
>0.
CL2)Y
(28)
Sibling Rivalry and Parental Transfers
4. Higher
Proposition
children
than
lower
children
than
less
Note
consider.
that
altruistic
How
does
Benthamite
parents
do. Also,
the
solution?8
choose
the actions
The
BU
=
this
+
?y
BM
more
services
their
by
resources
uex+ ?y
of
to their
Nash
we
game
because
self-enforcement
with
compare
first
M,
ex, and
are given
FOCs
=
in the noncooperative
the property
we
choose
each
characterize
to a cooperative
solution
e2 to maximize
respectively
or
the cooperative-utilitarian,
the
the altruistic
preference
by
(29)
0,
ei
BU
Be]
parents
the child
equilibrium
question,
parents'
-u}
to induce
transfer
self-interest.7
Nash
sequential
of
has
game
simultaneously
24.
resources
financial
altruistic
do.
answer
the parents
in Equation
more
that maximizes
To
more
offer
noncooperative
the two-stage
in which
function
parents
do not
behavior
pursues
income
parents
the parents
Moreover,
individual
game
income
829
?
M
wx +
?ye2M
a?
{e\ + e2y
=
(30)
0,
{e\ + e2)'
and
BU
=
uei + ?y
Be2
Denote
the
compares
to the cooperative
solution
to
the Nash
equilibrium
BU_
BM
-u?(yp-M
ei
?
M
w2 +
?yexM
a2
0i + e2)'
as
game
(M,
ex, e2). To
we
M*9
transfer,
(31)
0i +e2y
rewrite
see how
the
parents'
the cooperative
FOC
transfer,
in Equation
M,
25
as
follows:
account
the derivative
Equation
BU_
BM
25',
BM*
in Equation
+
?
u^
B
+ M*
BM*
+
\e\ + e%
e\
e*2
OL2]
-(w2
BU/BM
we
BM
u^j^
Bel
?
+ ?y
Evaluating
+
^ej
29
at the Nash
0.
equilibrium,
(M*,
ei*9e2*)9
BM*
BM*
\e* +
(25')
taking
into
have:
Bel
(
Bel
(M*,e*e*2)
+?y
Bel
-(w2 -ot2)
BM*
+ M*
e"
B
<
BM* \e* + e*2J
0.
(32)
It follows from Equation 32 and the strict concavity of the altruistic utility function that
M <M\
7
(33)
(1974) was the first to introduce parental altruistic preferences into the analysis of family economics. Becker (1991,
their own utility subject to the family constraints, they may be labeled
p. 279) further observes that because parents maximize
Pollak (1988) proposes the use of "paternalistic" preferences to
"selfish," not altruistic, in terms of utility maximization.
replace altruistic preferences in analyzing parent-child relationships and tied transfers.
Becker
8
We
are grateful to an anonymous
referee for suggesting
that we examine
this important question.
830
As
Yang-Ming Chang and Dennis L. Weisman
for
children's
we
services,
compare
4 and
Equations
ex >
1
and
find
?ye2M
if ue* >-?~
e\
30
(34)
& + G)2
that
.
Similarly, a comparison between Equations 5 and 31 reveals that9
e2> e\
We
?ye*M
if ue;>
/^ /
2- (35)
(e\ + e2
thus have the following:
5.
Proposition
then
renders,
If parental
amount
the total
are
transfers
transferred
on
based
a
under
the proportion
of
services
that
each
sibling
a noncooperative
than under
is less
game
cooperative
Nash game. If the parents' marginal utility of enjoying children's services is critically "high," other
things
being
than
the
equal,
levels
of
services
in the noncooperative
those
Nash
The finding in Proposition
children's
the
in which
consuming
than
those
and
35 are
satisfied,
not
despite
merit
completely
to determine
"large,"
the amounts
Nash
game.
are
goods
are greater
game
and
sufficient
undersupplied
hazard"
of
from
problem
interest
between
when
services.
the
For
from
gains
utility
are greater
the parents
in Equations
conditions
Nash
case
of
supply
their
set by
the goods
if the
the
and
goods
in a noncooperative
that a conflict
surprising
of
to the
compared
transfers
merit
Alternatively,
to a "moral
is subject
goods
in the cooperative
manner,
on
decisions
power
the
critically
in a noncooperative
of merit
perhaps
are
the goods
in a noncooperative
their parents'
have
parents
choose
game.
behave
they
the parents
5 implies that parents allocate more financial resources to induce
whatever
accept
simply
case
supply
when
services
children
that
34
game.
Consequently,
the
the parents'
perspective.
It is
parents
and
children
arise
may
altruism.
parental
Becker (1974, 1976, 1981, and 1991) contends that a child, no matter how selfish, is induced by
his
altruistic
allocation
parent
decisions.
or
family
head
In reply
to
to
the
the
internalize
comments
The
solution
equilibrium
each would
Nevertheless,
a
family
Pollak
"ultimatum
ten Kid
sions
of
children
cessively,
9
the parent's
other
resource
on
economists
the
(1977)
states
that
(1985) contends
game"
(p. 506)
equilibrium,
with
aspect
unsatisfactory
?
Theorem
however
a
that Becker's Rotten Kid Theorem
take-it-or-leave-it
commitment.
of my discussion,
however,
?
lamentable
be
they may
altruism.
"merit
By
goods,"
"merit goods"
and
that parents care about: whether
...
marry well
(p. 10)
See the Appendix
and
(1977)
on
(1977) argues that the theorem is the solution to
edition of his Treatise responds to the critiques by concluding
The most
actions
as a static Cournot-Nash
in my
is most
paper
simply
interpreted
noncooperative
If an altruist and each beneficiary
maximized
the social income of the altruist,
position.
be maximizing
his own utility,
the behavior
of the others; hence
the solution
given
be a noncooperative
would
Becker
game.
noncooperative
of his
of Hirshleifer
validity of the "Rotten Kid Theorem," Becker
a Cournot-Nash
effects
for the detailed derivations
they are
lazy,
Becker
(1991)
in the
enlarged
that
is not
but
incorrect
the failure
I mean
study
should be interpreted as
hard
of the results that lead to Proposition
particular
at school,
5.
application
to combine
of
the Rot
the discus
traits or behavior
visit
often,
drink
of
ex
Sibling Rivalry and Parental Transfers
In reviewing
Becker's
The
Becker's
for
of
(1977)
and
response
to the critiques
interactions
to as the "last
transfers
But
to the standard
Compared
two
these
is no
there
games
Alternatively,
incentive
of a noncooperative
advantage
can use
what
do not choose
or
for "contract"
in a sequential
game
follows
is relevant
Hirshleifer
We
game.
a cooperative
to move
mechanism
analyzing
further
independently on the supply of themerit
mechanism
Nash
have
in the sequential
the parents
one
for
essentially
equilibrium
the parents
transfer
methodology,
a well-defined
require
determined
endogenously
The
solution.
analysis.
generally
the analysis
First,
Second,
a financial
in choosing
principal-agent
move
in the sequential
children
never
that Becker
move
noncooperative
members.
family
in a sequential
is twofold.
consider the case that the children make their own decisions
good.
contends
(2003)
game
that the Cournot-Nash
between
word"
Nash
noncooperative
interactions,
characterizing
refers
a
employing
parent-child
(1977)
Pollak
economics,
as a game.
model
advantage
rivalry,
sibling
to family
contributions
the altruist
formulates
831
actions
of
their
game.
bargaining
enforcement
because
to the cooperative
or
move
for examining
framework
bargaining
family behavior with distinct preferences lies in the realization that theNash game of sibling rivalry is
and
self-enforcing
Becker's
the parents
have
a financial
to make
the discretion
transfer.
(1974) assertion that parental altruism necessarily ensures efficient family choices by
all members,
is not
however,
a matter
As
valid.
generally
of
on
fact,
the
first
his
of
page
book
A Treatise on the Family, Becker (1981) observes that "Conflict between the generations has become
more
less
that parental
suggest
findings
are now
and parents
open,
confident
that they
is not
altruism
can
the behavior
guide
to conflict
immune
of
or moral
their
hazard
children."
Our
an
problems,
implication consistent with Stark (1995), Chami (1998), Bergstrom and Bergstrom (1999), and Cox
(2002). By explicitly taking into account a merit good in a sequential game, our analysis of sibling
and moral
rivalry
hazard
an alternative
offer
may
problems
and potentially
promising
to the
approach
analysis of parent-child interactions. Our findings complement the studies by Lindbeck andWeibull
(1988), Bergstrom (1989, 1996), Bruce and Waldman (1990), and Pollak (2003), that show that
altruism does not automatically imply efficiency for child behavior and intergenerational interactions
from
the parent's
perspective.
4. Concluding
This
Remarks
affecting
the
nature
In addition
is emphasized
siblings
parental
transfers
altruism
toward
of
literature
and
their
to children
this
sibling
on
to focusing
in the
between
of parental
In
behavior.
offspring
characterizing
game.10
the role
examines
paper
on
vary
analysis,
we
apply
for
competition
economics,
we
to compete
for
family
inversely
with
a
also
between
investigate
parental
the children's
income
transfers.
earnings
among
or
rent-seeking
transfers
parental
interactions
intergenerational
incentives
in redistributing
transfers
in
a
parents
and
ability
in
to
approach
Nash
noncooperative
which
children,
interactions
intragenerational
Our
and
siblings
contest
findings
suggest
that
with
their
and directly
the parents.
Though based on a simple model of a contest within the family, the sibling-rivalry analysis offers
some
unique
perspectives
linking
parental
transfers
to children's
earnings
and
altruism.
The
analysis
also has implications for the design of effort-inducing schemes (or contests) for the role of parents
in redistributing financial resources between children, and for intergenerational family relations
10
An extension
of the analysis for the case of N > 2 siblings is straightforward.
... ,eN)= e?/(ei + ... + eN), where /? 1, 2, ..., N.
the form of Pi(ex,
In this case, the contest success function takes
Yang-Ming Chang and Dennis L. Weisman
832
(i.e.,
serve
to reduce
altruistic
Also,
We
also
in which
game
between
more
altruistic
determine
equilibrium
merit
more
parental
goods
resources
amount
transfer
and
to their
the
we
transfer,
is an
increasing
children
in a noncooperative
equilibrium
the
both
of a financial
choice
the children's
transfer
parents
the Nash
compare
parents
children,
the parent's
to induce
used
altruistic
children.
that determines
game
resources
income.
parents.
a cooperative
that for equally
differentials
of financial
of parental
function
less
the amount
that
income
of a sequential
In the analysis
find
It is noteworthy
interaction).
parent-child
transfers
than
with
game
levels
of
children's
merit goods. We find that the amount of resources required to induce children's supply of merit goods
under
a
under
is greater
moral
which
hazard
from
undersupplied
equalizer
for
problems
in the
Nash
noncooperative
with
a
within
the
family
Thus,
despite
perspective.
different
of merit
supply
arise
problems
the parent's
children
under
than
game
abilities,
goods
the fact
to their
children
in
There
are
conditions
that
children's
merit
goods
that
the family
serves
an
is not
altruism
parental
from
game.
cooperative
immune
to moral
are
income
hazard
parents.
Appendix
of Proposition
Proof
parents' FOC
in Equation
4. To prove Proposition
4 for the comparative
static results of the sequential game, we rewrite the
25 as
= 0.
+
-ul{yp-M,eue2)^uei(yp-M,e[le2)^
Taking
the partial derivative
ue2{yp-M1eue2)^+^^+^y^
of the FOC with
respect to yp yields
1
dM* __
J
dyD
where M*
is the Nash
equilibrium
-Un+
(Al)
U?'?t)+U?\dM
transfer. Substituting
de,
dM
W2
?
0L2
[(W]_ai)
+
(W2_a2);
[(wj _ai)
+
(H,2-d2)f
and
de2
dM
into (Al) yields
dM'? ? ?1
a2) + m*2/(wi
oti)l
\j Uj]-^?ueiI(w2
> > J )""
+
dyP
oti)
(w2
[(wi
0L2)f
Next,
taking the partial derivative
respect to ? yields
of the parents' FOC with
dM* _
dyp
1
J
(A2)
({dMJ y\dM
Substituting
dYx
dM
(w2
?
0L2)
+
[(w, -ai)
>
0
(w2 -oc2)f
and
dY2 _
dM
(wi-oti)2
+ (>v2-a2)]2
> 0
[(W]-0tl)
into (A2) yields
dM*
d?
/
1 [(w!
-
(w2 -a2f
ai) + (w2
Q(2)]2
[(w,
(h>!-oci)2
ai) + (w2
> 0.
ot2)f
QED.
Sibling Rivalry and Parental Transfers
5. To prove Proposition 5, we first compare parental transfers in the two alternative games.
game, the parents' choice of a financial transfer M satisfies the following FOC (see Eqn. 29):
Proof of Proposition
cooperative
dO_
dM'
In the sequential game,
? dU
,.*
el
\ de*
=
-U?(yp-M*,e\,e*2)
d
wt
+ ?y + uei
^
+ uei
^
in Equation A3
dU/dM
we have
dU
dM
de\
1
dM*
(M*,e\,e*2)
dM*
(see Eqn. 25):
. d
dM *\e\+e*2)
e\ + e*2
\
dM*
dM* \e\ + e*2
(A4)
_
(Af*, e\, e*2) and subtracting Equation A4
equilibrium
de*2
'dM
<9M*
dM* \e* +
<
-(w2-a2)--2- <9M*+ M* ,<9M*
V^?+ e2
?y
from the
el
d f
de*
The first two terms on the right-hand
FOC
e*
x del
+ ?y
at the Nash
In the
(A3)
satisfies the following
e*
(
+ dM*
dM* M*?f^T)l=0.
-(w2-o(2)^
\e\ +e*2J
the derivative
resulting expression,
transfer M*
de*
* *x
? de*7
+ ?y
+ ue2
,eve2) + uex?
ldM
?y
+ ?y
+ ?Y = 0.
-ui(yp -Ai,ei,<?2)
the parents' choice of a financial
,
=-u,(yp-M
/
Evaluating
833
(A5)
side of Equation A5 are negative. To show that the derivative
dU
dM
in Equation A5
term, which is
it suffices
is negative,
to show that both the third and the fourth terms are negative.
x de*
/
?y
dM* \e* + e*2)
?
w2
ot2
-ai)
+
ai )
(w2 a2)]2
[ l(wi
-?y(w2
?y(w2
+
(wi -ai)
given
M*[(wi
a2)
-
(wi
-
a2)
M*
?
oti) +
a2
(w2
-
M*
?
+
0C2)]2 (h>1 Oti))
+
?0
(w2 a2)]
(wl
otQ
[(w!
oti) + (w2 a2)]
(w1-a1)+H
[
-
w2
-M*
W! -a
at
\\
-?y(w2-oc2W---??
look at the third
(V4
dM*
-?y<-(wi
=
Iirt d
First, we
(W2
?M2^
a2)
1
(w2-a2)J
?
) < o,
(A6)
(w2 -a2)
that
-MMy
dM1
and that the Nash
equilibrium
(h>!-ai)
(see Eqn.
14). Because
(wi-ai)
?
+
a2
(w2-a2)
transfer to sibling 2 is
M* =
negative.
w2
(M*)2 > M*
>
M2
implies
(wi -ai)
+ (w2 -a2)
that M* -
(M*2/M*)
M*
>
0, the sign in Equation A6
is unambiguously
834
Yang-Ming Chang and Dennis L. Weisman
Similarly, we
look at the last term in Equation A5, which
-
?r
?-fa -^2)S:dM*
L
{
+
dM* (-?-,
M'^\e\ +
+
[(wl -ai)
w2
-?y(^i
-ai)
-
=
-?y(wi
(w2
-
(wi -ai)
oti) +
l
(wi
ai) +
(w2
(w2
-
a2)
a2)J
-a2)]-(w2-a2)]
a2)]
[(Wl-a,) +
\
(wi
a2)]2
a2 M1
r+
(w2 a2)]2
+
?M*[(wi -a1)
cxi)
?
"*
'-AT
-
(w:
(w2
o?j)+
[(wi
?
e*2
w\ - ai
-(w2-a2)--??
=
is
(w2
?y(wi -ai)
+ (w2 -a2)
(A7)
(wi -ai)
given
that
g
\
( 4
wi -aj
dM* \?\ + e*2)
and that the Nash
equilibrium
transfer to sibling
+
(wi -ai)
(w2-a2)
1 is
(w2
M\ 1
-
a2)
+
(wi-ai)
(see Eqn. 13). Because (M*) > M* > M? implies thatM*
It follows from Equations A5, A6, and A7 that
M*
(w2-a2)
(M\lM*)
> 0, the sign in Equation A7
is unambiguously
dU\
Um<,^)<0
dM
Strict concavity
of the parents' altruistic preference
function
(A8)
implies that
M <M*.
(A9)
Next, we compare children's services in the two alternative games. In the cooperative
children satisfy the following FOCs (see Eqns. 30 and 31):
du
negative.
+ ?Y ?-?2-M-wi
U~ex
(ei +e2)
game, the amounts of services by the
?ye2M
+ai
(A10)
{?x+ e2)2
and
dU_
de2
In the sequential game,
e\
: +
uh
?y ?-?2M-w2
?yeiM
+ a2
=0.
(All)
(h +ei)2
the amounts of children's
satisfy the following
services
-
FOCs
(see Eqns. 4 and 5):
wi + a! = 0,
(A12)
M* ? w2 + a2 = 0.
(A13)
-?e-L?2M*
and
(e\+e*2y
Evaluating the derivatives dU/dei and dU/de2
account Equations A12 and A13, we have
dU
dex
in Equations
>
rv^2) (=)(<)0
A10
and All
if"< >(=)(<)
at the Nash
?ye2M*
and
dU
<?v;,<;>>(=)(<)0
if ue. >
(=)(<)-M^L
(e'+
e'2)
equilibrium
(M*, e\, e*2\ taking into
Sibling Rivalry and Parental Transfers
of the parents' altruistic preference
Strict concavity
implies that
function
>
e\
835
if ue. > - $ye*2M
and
e2>e*2
iiue2
>
l
;r
QED.
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