Social Preferences and the
Efficiency of Bilateral Exchange
Daniel J. Benjamin
Cornell University
Summer Econometric Society Meetings
June 6, 2009
Motivation
• When will a transaction be Pareto efficient?
– Enforceable contract (Coase theorem)
– Repetition/reputation (folk theorems)
– Social preferences
• I study one-shot exchange, no contracts
–
–
–
–
E.g., gift-exchange game, trust game
Lays bare role of social preferences
Very simple cases: tipping, honor system
Similar forces apply in contexts with contracts and
repetition (e.g., Hart & Moore 2008, Bewley 1999, Akerlof 1982)
Relation to Existing Literature
• I study properties of social preferences, rather
than particular functional forms.
(e.g., Fehr, Klein, & Schmidt 2007)
– Assess sensitivity of results to controversial
properties.
– Understand role of assumptions implicit in models.
– Clarify relationship between literatures from family
economics and behavioral economics.
• A key result: Rotten Kid theorem logic applies
in “market” transactions, not just within family.
(Becker 1974; Bergstrom 1989)
• I do not address signaling or intentions-based
models. (e.g., Levine 1998; Rabin 1993)
The Bilateral Exchange Game
Two players: First-mover (FM) and second-mover (SM).
Timing:
1. FM can take action a 1 that transfers a commodity from
herself to SM.
2. SM can choose action a 2 that transfers a commodity
from himself to FM.
3. Either player can choose not to trade.
No contracts: Players have no extrinsic incentive to make
transfers. Starkest case.
Material Payoff Functions
a 1 , a 2 a 2 a 1 .
FM’s material payoff: 
1
SM’s material payoff: 
a 1 , a 2 v
a 1 c
a2 
.
2
v
strictly concave with lima 1  v 
a 1 .
c
strictly convex with lima 2  c 
a 2 .
FM maximizes 1 a 1 , a 2 .
SM maximizes Ua 1 , a 2 U1 a 1 , a 2 , 2 a 1 , a 2 .
Normalizations: v0c0U0, 00, v 0c0.
Pareto and Material Efficiency
A transaction a 1 , a 2 is utility Pareto efficient if
 
there is no other transaction a 1 , a 2 such that
 



a 1 , a 2 
a 1 , a 2 and U
a 1 , a 2 U
a1 , a2 
,
1
1
at least one inequality strict.
A transaction a 1 , a 2 is materially Pareto
 
efficient if there is no other transaction a 1 , a 2 
 
such that 1 a 1 , a 2 1 a 1 , a 2 and
 


a1 , a2 
, at least one strict.
2 a 1 , a 2 
2
SM’s Social Preferences
The SM’s utility U

1,
2 may have some or
all of the following properties:
1.
2.
3.
4.
Joint-monotonicity
Quasi-concavity
Normality
Fairness-kinked

1

2
Monotonic social preferences weakened to…

1
Joint-Monotonicity

2
…joint-monotonic social preferences.
SM’s Social Preferences
The SM’s utility U

1,
2 may have some or
all of the following properties:
1.
2.
3.
4.
Joint-monotonicity
Quasi-concavity
Normality
Fairness-kinked

1



I;p 
1
N
I;p 
I
Normality

2
measures (local) income effect.
SM’s Social Preferences
The SM’s utility U

1,
2 may have some or
all of the following properties:
1.
2.
3.
4.
Joint-monotonicity
Quasi-concavity
Normality
Fairness-kinked

1
region of
disadvantageous
unfairness
equal-split
fairness rule
region of
advantageous
unfairness

2
Fairness-Kinked

1
region of
disadvantageous
unfairness
(U = UA)
fairness rule
region of
advantageous
unfairness
(U = UB)

2
Let

a 1 , a 2 arg maxa 1 ,a 2 |1 a 1 ,a 2 0U

a1 , a2 
,
a 1 , a 2 
1
2
be called SM’s favorite transaction.
Theorem 1: Suppose U is continuous, jointmonotonic, and quasi-concave. SM’s favorite
transaction exists and is unique. Moreover, a
transaction is utility Pareto efficient if and only
if it is materially Pareto efficient and satisfies
 




a
,
a

a
.
2 1 2
2 1 , a2 

1
utility Pareto
efficiency
frontier
indifference curve
second-mover’s
favorite transaction
set of individuallyrational material
payoffs
material Pareto-efficiency
frontier

2
U=0
Necessary Conditions for Efficiency
Theorem 2: Suppose U is joint-monotonic, quasi-concave,
continuously twice-differentiable, and satisfies (TA),
and suppose c 0. Then no equilibrium is materially
Pareto efficient. Furthermore, at any interior
equilibrium, the marginal inefficiency is:
c 
a2
1


v
a 1 c 
a 2  NIa ,a ;pa  2
0.
d 
1
2
2
2
d 1 2
UU a 1 ,a 2
Action a 1 would induce 
a 1 , a 2 
, but FM can do better:
• “IE” from a small deviation in a1 is second-order.
• Hence “SE” allows FM to gain material payoff.
One of the following is necessary for efficiency:
d 2


2
c 0 (cf., Bergstrom 1989), or
.
2
d

1
UU
a 1 ,a 2 

1
equilibrium
a1 = a1’
a1 = a1*
material
Paretoefficiency
frontier

2
Efficient Case I:
Transferable Material Payoffs
Theorem 3: Suppose U is continuous, jointmonotonic, quasi-concave, and normal. If
c
a 2 1 2 a 2 for some 1 and 2 0, and if
 


a
1 1 , a 2 0, then the unique equilibrium
transaction is a 1 , a 2 (and is efficient).
SM’s action ensures both players’ material payoffs
increasing in size of pie → FM maximizes pie.
Applies to commercial transactions, not just family.

1
a1 = a1*
equilibrium
a1 = a1’
material
Paretoefficiency
frontier

2
Efficient Case II:
Fairness-Kinked Social Preferences
Theorem 4: Suppose U is fairness-kinked, with
U A and U B being joint-monotonic, quasi-concave,
normal, continuously twice-differentiable, and
 
satisfying (TA). Let a 1 , a 2 denote the
transaction with 
a 1 a 1 such that
 
 
 

a 1 , a 2 
a 1 , a 2 and U
a 1 , a 2 0.
1
1
If 1 a 1 , a 2 0, and if:

UA


1
U A U B
UA
 
c a 2 
0

2

UB


1
UB
  
c 
a 2 

2
0
at a 1 , a 2 , then the unique equilibrium transaction
is a 1 , a 2 .
SM follows fairness rule → FM maximizes pie.
Any fairness rule works, even if self-serving.

1
fairness rule
Material
Paretoefficiency
frontier
equilibrium

2
a1 = a1’’
a1 = a1’’’
Discussion
Social preferences generate efficient exchange if:
1. Material payoffs are quasi-linear, i.e., SM’s
payment for the good has a discretionary
monetary component, or
2. SM follows a fairness rule, such as equal-split
or a customary rate of exchange.
A key property of social preferences is normality.
Monotonicity is not crucial; hence results about
altruism within the family also apply to
fairness concerns in commercial transactions.