Faruk Gul

Wolfgang Pesendorfer

English

Research Papers in Economics

The Revealed Preference Implications of
Reference Dependent Preferences†
Faruk Gul
and
Wolfgang Pesendorfer
Princeton University
December 2006
Abstract
K¨ oszegi and Rabin (2005) propose a theory of reference dependent utility in which the
ultimate choice also serves as the reference point. They analyze dynamic choice problems
with uncertainty and interpret their model as a description of the individuals psychological
process. We focus on static and deterministic choice problems and identify the revealed
preference implications of their model.
† This research was supported by grants from the National Science Foundation. The content of this
note was previously part of the paper “The Case for Mindless Economics.”1. Reference Dependent Utility
A reference dependent utility function U, associates a utility with each reference point
z ∈ X and each choice object x ∈ X.K ¨ oszegi and Rabin (2005) (henceforth K¨ oszegi-Rabin)
propose a novel theory to determine the reference point. Let X be a ﬁnite set1 and let
U : X × X → I R, where U(x,z) is the utility of x given the reference z. A personal
equilibrium (K¨ oszegi (2004)) for a decision-maker facing the choice set A is any x ∈ A
such that
U(x,x) ≥ U(y,x) (2)
for all y ∈ A. Hence, K¨ oszegi-Rabin deﬁne the reference point as the x that ultimately
gets chosen. It follows that an alternative x ∈ A is optimal (i.e., a possible choice) for
aK ¨ oszegi-Rabin decision-maker if (and only if) condition (2) above is satisﬁed. K¨ oszegi-
Rabin assume that U has the form
U(x,y)=
 
k∈K
uk(x)+
 
k∈K
µ(uk(x) − uk(y)) (3)
where µ is an increasing function with µ(0) = 0 and K is some ﬁnite set indexing the
relevant hedonic dimensions of consumption. K¨ oszegi-Rabin also require that
U(x,y) ≥ U(y,y) implies U(x,x) >U(y,x) (4)
for all x,y ∈ X.
In this note, we provide a revealed preference analysis of the K¨ oszegi-Rabin model.
Let Y be the set of all nonempty subsets of X. A function c : Y → Y is a choice function
if c(A) ⊂ A for all A ∈ Y . Given any state dependent utility function U, deﬁne C(·,U)a s
follows:
C(A,U)={x ∈ A|U(x,x) ≥ U(y,x)∀y ∈ A}
A choice function c is a general K¨ oszegi-Rabin choice function if there exists a reference
dependent utility function U such that c = C(·,U). If the U also satisﬁes (3) and (4)
1 An element x ∈ X may be uncertain (i.e., may be a lottery).
1then c is a a special K¨ oszegi-Rabin choice function. For any binary relation  , deﬁne the
function C  as follows:
C (A)={x ∈ A|x   z∀z ∈ A}
It is easy to construct examples such that C (A)=∅ unless certain assumptions are
made on  . We say that the choice function c is induced by the binary relation   if
c(A)=C (A) for all A ∈ Y . It is well-known that C  is a choice function whenever   is
complete (x   y or y   x for all x,y ∈ X) and transitive (x   y and y   z implies x   z
for all x,y,z ∈ X). However, transitivity is not necessary for C  to be a choice function.
The proposition below characterizes K¨ oszegi-Rabin choice functions2:
Proposition: The following three conditions are equivalent:
(i) c is a general K¨ oszegi-Rabin choice function
(ii) c is a choice function induced by some complete binary relation
(iii) c is a special K¨ oszegi-Rabin choice function
Proof: See Appendix
Note that c = C  is a choice function implies   is complete. Hence, we may omit the
word complete in the above proposition. The equivalence of (i) and (ii) establishes that
abandoning transitivity is the only revealed preference implication of the K¨ oszegi-Rabin
theory. The equivalence of (ii) and (iii) implies that the particular functional form (3) and
condition (4) are without loss of generality.
The revealed preference analysis answers the following question: suppose the modeler
could not determine the individual ingredients that go into the representation, how can
he check whether or not the decision-maker behaves in a manner consistent with such a
representation? Or to put it diﬀerently, how is the behavior of a K¨ oszegi-Rabin decision
maker diﬀerent from a standard decision-maker? For the case of deterministic choice, the
answer is that the K¨ oszegi-Rabin decision-maker may fail transitivity.
2 Kim and Richter (1986) provide a condition on a demand function that is equivalent to maximizing
a (possibly) nontransitive binary relation over a standard neoclassical consumption set.
22. Proof
First, we will show that (i) implies (ii): Suppose c is a general K¨ oszegi-Rabin choice
function. Then, there exists a reference dependent utility function U such that c = C(·,U).
Deﬁne   as follows: x   y if U(x,x) ≥ U(y,x). Then, for all A ∈ Y ,
c(A)=C(A,U)={x ∈ A|U(x,x) ≥ U(y,x)} = {x ∈ A|x   y} = C (A)
as desired.
To prove that (ii) implies (iii), assume that c = C  and let n be the cardinality of X. Recall
that   is a complete, reﬂexive, binary relation. We write x   y for x   y and y    x. Let
K = X × X.F o r k =( w,z) ∈ K, we deﬁne the function u(w,z) : X →{ − 2,0,2,3} as
follows:
u(w,z)(x)=

 
 
3i f x = w = z
2i f x = w and w   z
−2i f x = z and w   z
0 otherwise.
Deﬁne the function µ as follows:
µ(t)=
 
16nt if t ∈{ − 4,−3,4}
t if t ∈{ − 2,0,2,3}
Clearly, µ is strictly increasing and µ(0) = 0. Let
U(x,y)=
 
k∈K
uk(x)+
 
k∈K
µ(uk(x) − uk(y))
To complete the proof, we will show that C  = C(·,U); that is x   y iﬀ U(x,x) ≥ U(y,x)
for all x,y ∈ X, and
U(x,y) ≥ U(y,y) implies U(x,x) >U(y,x) (4)
Note that
2n ≥
 
k =(x,x)
uk(x) ≥− 2n (∗)
3Let Kx,y = K\{(y,y),(x,x),(x,y),(y,x)} and note that for k ∈ Kx,y
2 ≥ uk(x) − uk(y) ≥− 2( ∗∗)
Equations (*) and (**) and the deﬁnition of µ imply that
4n ≥
 
Kx,y
(uk(x) − uk(y)) =
 
Kx,y
µ(uk(x) − uk(y)) ≥− 4n
Let x   y. Note that µ(u(x,y)(y)−u(x,y)(x)) ≤ 0 and, since x   y, we also have µ(u(y,x)(y)−
u(y,x)(x)) ≤ 0. It follows that
U(x,x) − U(y,x)=
 
k∈K
(uk(x) − uk(y)) −
 
k∈K
µ(uk(y) − uk(x))
≥−4n −
 
Kx,y
µ(uk(x) − uk(y))−
− µ(u(x,x)(y) − u(x,x)(x)) − µ(u(y,y)(y) − u(y,y)(x))
≥−8n +4 8 n − 3 > 0
Conversely, let y   x. Then, µ(u(x,y)(y)−u(x,y)(x)) = 0 and µ(u(y,x)(y)−u(y,x)(x)) = 64n.
Therefore,
U(x,x) − U(y,x)=
 
k∈K
(uk(x) − uk(y)) −
 
k∈K
µ(uk(y) − uk(x))
≤ 4n −
 
Kx,y
µ(uk(y) − uk(x)) − µ(u(x,x)(y) − u(x,x)(x))
− µ(u(y,y)(y) − u(y,y)(x)) − µ(u(y,x)(y) − u(y,x)(x))
≤ 8n − 3+4 8 n − 64n<0
Finally, suppose U(x,y) − U(y,y) ≥ 0. Then,
U(x,x) − U(y,x) ≥ U(x,x) − U(y,x) − U(x,y)+U(y,y)
= −
 
k∈K
[µ(uk(y) − uk(x)) + µ(uk(x) − uk(y))]
= −2(µ(−3) + µ(3)) = 2(48n − 3) > 0
completing the proof that (ii) implies (iii). That (iii) implies (i) is immediate.
4References
Kim, T. and M. K. Richter, (1986) “Nontransitive-nontotal Consumer Theory,” Journal of
Economic Theory, 38, 324–363.
K¨ oszegi, B. (2005). “Utility from Anticipation and Personal Equilibrium,” mimeo.
K¨ oszegi, B. and M. Rabin, (2005). “A Model of Reference-Dependent Preferences,” Quar-
terly Journal of Economics, forthcoming.
5

The Revealed Preference Implications of Reference Dependent Preferences

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The Revealed Preference Implications of Reference Dependent Preferences

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