Econ 439
Non-cooperative games in the family?
A number of models:
1. "separate spheres" model of Lundberg and
Pollak (1993)
2. Chen and Woolley (2001) "A Cournot Nash
model of family decision making"
3. Engineer and Welling (1999) "Human
capital, true love, and gender roles"
1. Separate spheres
reference: Lundberg and Pollak
(Dec/93) "Separate spheres bargaining
and the marriage market" Journal of
Political Economy pp 988-1011
- basic idea: partners specialize in
household - traditional gender
roles
- if cooperation breaks down, noncoop outcome within marriage
results
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Econ 439
Model:
- each individual consumes three
goods: two public, one private
- 4 goods in total
- individuals are selfish
- each partner totally responsible for
one of the public goods
- in cooperative game, jointly decide
on quantities (of all four goods),
then produce
- in non-coop (nc) game, each
partner chooses own private
good, and amount of one public
good to provide
- this provides threat point in
(coop've) bargaining game
- each chooses allocation of own
income/resources between
private and public good - ignoring
benefit of public good to other
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Econ 439
Simple model:
- Alice (A), Ben (B)
- Private goods ( x , x ); public ( y1, y2 )
- income ( I A , I B )
A B
- U A = ln xA + α1 ln y1 + α2 ln y2
- U B = ln xB + β1 ln y1 + β2 ln y2
- nc game: decisions are:
- Alice: choose ( xA , y1) to max U A ,
taking as given (I A , y2 )
- Ben: choose ( xB , y2 ) to max U B ,
taking as given (I B , y1)
- Nash equilibrium to the non-coop game:
xA =
IA
αI
, y1 = 1 A
(1 + α1 )
(1 + α1 )
xB =
IB
β2 IB
, y2 =
(1 + β 2 )
(1 + β 2 )
- NE not efficient: solving
max
xA, xB , y1, y2
ρU A + (1− ρ )U B
subject to xA + xB + y1 + y2 = I A + I B
yields quantities of all four goods which
depend on the sum of Alice and Ben's incomes,
as well as all the preference parameters.
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Econ 439
Main result:
- threat point for coop game depends
on individual incomes in marriage
- therefore changes in individual control
over resources within marriage will
change allocation in coop've game
2. Chen and Woolley, Economic Journal
Oct 2001: "A Cournot-Nash Model of
Family Decision Making"
- 2 adults: m, f
- two private goods, one household
public good
- two sided altruism, variable s 0 ≤ s ≤ 1
Wm = [u( xm ) + v( y)] + s[u( x f ) + v( y)]
W f = [u( x f ) + v( y)] + s[u( xm ) + v( y)]
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Econ 439
- sequential game:
- 1. income transfers t : Im > I f
- 2: independent decisions on
consumption - private good,
contribution to public good
Questions:
- who purchases what?
- how much will be transferred?
- will income (re)distribution affect
household decisions?
Model uses backward induction:
Second stage solved first:
- given incomes, what is chosen?
- choices functions of own incomes,
transfer, and given amount of
other's contribution to public good
First stage: given purchase plans,
determine transfer
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Econ 439
C-W consider 3 cases:
1. No transfer: t=0 "pre-transfer equilibrium"
Nash equilibrium: each adult chooses own
private good and contributions to
household public good to maximize own
welfare, taking as given choices of
partner
Two types of solutions:
Corner: only male (higher income spouse)
purchases household goods; female
purchases only own private goods
Interior: both purchase both goods
Equilibrium moves from interior solution to
corner solution as I ↓
f
At corner:
Interior:
I f ↑⇒ x f ↑ ; I m ↑⇒ xm , ym , y ↑;
Ii ↑⇒ xi , x j , yi ↑, y j ↓ ;
total y constant
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Econ 439
2. Voluntary income transfers:
At interior solution, small transfer has no
effect at all.
At corner, for any s such that 0<s<1, a small
transfer raises m's welfare if f's income
is sufficiently low. If f's income is below
this threshold, m will make a transfer amount decreases as f's income rises,
increases as m's income rises.
3. Nash-bargained income transfers:
Choose transfer (t) to maximize
[W f (t ) − W f (0)]a [Wm (t ) − Wm (0)]1−a
"Threat point" is
[W f (0),Wm (0)]: t = 0
When will transfer occur? How will it affect
household expenditures?
First: only when f's income is sufficiently low
Second; when s = 1: cooperative outcome
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Econ 439
Third: There is some critical level of s above this, may move from only
male contributing to household good
to both, or only female
Fourth: greater caring leaves male worse off
-disagreement point less attractive.
Comparative statics wrt income?:
For low s,
I f ↑⇒ t ↓, xm , ym ↑; x f ≤ 0? ≥ 0?
Comparative statics for income
redistribution?
No change at interior solution, or if no
transfer;
If transfer positive, change in division of
family income will in general change
expenditure on household good
(Ex: if household public good is children….)
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Econ 439
Related literature - game theoretic:
Engineer and Welling (1999)
- "game" here is pre-marital investment
in human capital
- this paper: training chosen by parents
- once married ("true love" - random),
efficient task specialization within
marriage (separate spheres)
- when are "gender roles" an
equilibrium outcome?
- When is that outcome efficient?
- What is individuals differ in aptitude
for various types of work?
- what if multiple equilibria - role for
policy?
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