This paper strengthens the framework of stochastic revealed preferences introduced by Bandyopadhyay et al. (1999, 2004) for stochastic homothetically revealed preferences for tight stochastic demand functions.Homotheticity, Stochastic Revealed Preference

Jan Heufer

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 3 
  
Stochastic homothetically revealed preference for tight stochastic demand 
functions 
  
 
 
Jan Heufer  
TU Dortmund University and Ruhr Graduate School in Economics 
Abstract 
This paper strengthens the framework of stochastic revealed preferences introduced by Bandyopadhyay et al. (1999, 
2004) for stochastic homothetically revealed preferences for tight stochastic demand functions.
I am grateful to my advisor Wolfgang Leininger for his support and comments and to an anonymous referee for helpful suggestions. 
Citation: Jan Heufer, (2009) ''Stochastic homothetically revealed preference for tight stochastic demand functions'', Economics Bulletin, Vol. 
29 no.3 pp. 2472-2477. 
Submitted: Aug 18 2009.   Published: September 30, 2009. 
 
     1 Introduction
Bandyopadhyay, Dasgupta, and Pattanaik (1999) initiated a line of investigation in which
they explored choice behavior of a consumer who chooses in a stochastic fashion from
dierent budget sets. In Bandyopadhyay, Dasgupta, and Pattanaik (2002) this approach
was extended by an interpretation of tuples of deterministic demand functions of dierent
consumers as a stochastic demand function. They dene a weak axiom of stochastic
revealed preference which is implied by but does not imply rationalizability in terms of
stochastic orderings. In Bandyopadhyay, Dasgupta, and Pattanaik (2004), the authors
show that if a consumer spends his entire wealth their weak axiom is equivalent to a
restriction they call stochastic substitutability. Heufer (2008) transformed the problem of
nding a probability measure on orderings over the available commodity bundles into a
problem of nding a probability measure on orderings over the budgets on which choices
are observed.
In this paper we strengthen the framework for rationalizability in terms of stochastic
homothetic orderings. We focus on the case in which a consumer spends his entire wealth.
It is shown that rationalizability in terms of stochastic homothetic orderings implies but
is not implied by a homothetic version of stochastic substitutability.
2 Notation and Denitions
Let ` denote the number of commodities, `  2, and let R+ and R++ denote the set of
non-negative real numbers and positive real numbers, respectively. The consumption set
is R`
+. A price-wealth situation is a pair (p;w) 2 Z, where Z = R`
++  R++ is the set of
all possible price-wealth situations. Given a price-wealth situation (p;w), the budget set
of a consumer is dened as

x 2 R`
+ : w  p  x
	
, where  denotes the dot product. The
boundary of a budget is dened as

x 2 R`
+ : w = p  x
	
. The budget sets corresponding
to price-wealth situations (p;w), (p0;w0), etc. will be denoted B(p;w), B(p0;w0), etc.
Denition 1 (Bandyopadhyay et al. 2004) A deterministic demand function
(DDF) is a rule d, which, for every (p;w) 2 Z, species exactly one bundle x in B(p;w).
A stochastic demand function (SDF) is a rule D which, for every (p;w) 2 Z species
exactly one nitely additive probability measure q over the class of all subsets of B(p;w).
Let D be a SDF, and let q = D(p;w). Then for every subset A of B(p;w), q(A) is
interpreted as the probability that the consumer choses a bundle contained in the set A
for every price-wealth situation (p;w). We denote q0 = D(p0;w0) etc.
Denition 2 (Bandyopadhyay et al. 2004) A DDF d is tight if, for every (p;w) 2
Z, pd(p;w) = w. An SDF D is tight if, for every (p;w) 2 Z, q (fx 2 B(p;w) : p  x = wg) =
1.
We follow the same notation as in Bandyopadhyay et al. (2004):
1x
y
o
a0
a
a00
b00 b b0
g
h
k
Figure 1: oab is the budget set B(p;w), oa0b0 is the budget set B(p0;w0), oa00b00 is the budget set
B(p0;w0). I is the singleton set containing g. G is ag excluding the point g. H is gb excluding the point
g. G0 is gb0 excluding the point g. H0 is a0g excluding the point g. I is the singleton set containing h.
G is ah excluding the point h. H is hb excluding the point h. G0
 is hb00 excluding the point h. H0
 is
a00h excluding the point h. I0

  1


is the singleton set containing k. G0

  1


is kb0 excluding the point k.
Notation 1 Given two price-wealth situations (p;w) and (p0;w0), let
I =

x 2 R
`
+ : p  x = w and p
0  x = w
0	
G =

x 2 R
`
+ : p  x = w and p
0  x > w
0	
H =

x 2 R
`
+ : p  x = w and p
0  x < w
0	
G
0 =

x 2 R
`
+ : p  x > w and p
0  x = w
0	
H
0 =

x 2 R
`
+ : p  x < w and p
0  x = w
0	
:
Furthermore, we introduce the following notation:
Notation 2 Given two price-wealth situations (p;w) and (p0;w0), let
I =

x 2 R
`
+ : p  x = w and p
0  x = w
0	
G =

x 2 R
`
+ : p  x = w and p
0  x > w
0	
H =

x 2 R
`
+ : p  x = w and p
0  x < w
0	
G
0
 =

x 2 R
`
+ : p  x > w and p
0  x = w
0	
H
0
 =

x 2 R
`
+ : p  x < w and p
0  x = w
0	
:
Denition 3 Given a set A  R`
+ and a scalar  > 0, A() is dened as the
projection of A which is obtained by multiplying the coordinates of all elements of A by .
See Figure 1 for an illustration of Notations 1 and 2.
3 Stochastic Homothetic Substitution
We rst recall the relevant part from the work of Bandyopadhyay et al. (1999, 2004).
2Notation 3 (Bandyopadhyay et al. 1999) Let T be the set of all orderings over
R`
+, the elements of T being denoted by J;J0, etc. For all subsets G and H of R`
+ such
that G is non-empty, let T(G;H) be the set of all J 2 T such that J has a unique greatest
element in G and this unique greatest element in G belongs to H.
Denition 4 (Bandyopadhyay et al. 1999) An SDF D satises rationalizability
in terms of stochastic orderings (RSO) if there exists a probability measure r dened
on the set of all subsets of T such that, for every price-wealth situation (p;w) and every
subset A of B(p;w), q(A) = r[T(B(p;w);A)], where q = D(p;w).
Denition 5 (Bandyopadhyay et al. 2004) A tight SDF D satises stochastic
substitutability (SS) if for every ordered pair of price-wealth situations (p;w) and (p0;w0)
and for all A  I we have
q
0(G
0) + q
0(A)  q(H) + q(A): (1)
Proposition 1 (Bandyopadhyay et al. 1999,Bandyopadhyay et al. 2004) For
tight stochastic demand functions, rationalizability in terms of stochastic orderings implies
but is not implied by stochastic substitutability.
Remark 1 Bandyopadhyay et al. (1999) show that RSO implies but is not implied
by their weak axiom of stochastic revealed preference (WASRP). Bandyopadhyay et al.
(2004) show that WASRP is equivalent to SS for tight SDF.
Next we introduce the relevant notations and denitions to strengthen the framework
for homothetic orderings.
Denition 6 An ordering J is homothetic if xJy implies xJy for any scalar  > 0.
Notation 4 Let Th be the set of all homothetic orderings over R`
+, the elements of
Th being denoted by Jh;J0
h, etc. Otherwise the notation is analogous to Notation 3.
Denition 7 An SDF D satises rationalizability in terms of stochastic homothetic
orderings (RSHO) if there exists a probability measure r dened on the set of all subsets
of Th such that, for every price-wealth situation (p;w) and every subset A of B(p;w),
q(A) = r[Th(B(p;w);A)], where q = D(p;w).
Lemma 1 Rationalizability in terms of stochastic homothetic orderings implies that
for every price-wealth situation (p;w) and all A  B(p;w) we have q (A) = q (A()),
where q = D(p;w).
We omit the proof for Lemma 1 as it is obvious.
Denition 8 A tight SDF D satises stochastic homothetic substitutability (SHS) if,
for every ordered pair of price-wealth situations (p;w) and (p0;w0) and for all  > 0 and
all A  I we have
q
0

G
0


1


+ q
0

A

1


 q(H) + q(A): (2)
Proposition 2 Rationalizability in terms of stochastic homothetic orderings implies
but is not implied by stochastic homothetic substitutability.
3Proof. Suppose a tight SDF D satises RSHO. RSHO implies RSO, and RSO implies SS,
hence SS must be satised. Let (p00;w00) = (p0;w0), with  > 0. Then by SS, for the pair
h(p;w);(p00;w00)i we have for every A  I
q
00(G
00) + q
00(A)  q(H) + q(A); (3)
where
G
00 =

x 2 R
`
+ : px > w and p
00x = w
00	
I =

x 2 R
`
+ : px = w and p
00x = w
00	
H =

x 2 R
`
+ : px = w and p
00x < w
00	
:
We have G00 = G0
, H = H, I = I, and q00 = q0
. Therefore (3) is equivalent to
q
0
(G
0
) + q
0
(A)  q(H) + q(A) (4)
for every A  I. By Lemma 1 we have with RSHO that q0
(G0
) = q0  
G0

 
1


and
q0
(A) = q0  
A
 
1


. This holds for every  = , and we obtain SHS.
Liu and Wong (2000) showed that acyclicity of the homothetic closure of a revealed
preference relation is equivalent to rationalizability in terms of deterministic homothetic
preferences. Because asymmetry is a weaker condition than acyclicity, asymmetry does
not imply rationalizability. To see that SHS does not imply RSHO, it can be easily
checked that for a deterministic demand function, which is induced by a degenerate SDF
(see Bandyopadhyay et al. 2004, Denition 2.2), SHS is equivalent to asymmetry of the
homothetic closure.
4 Discussion
Dasgupta (2005) introduced a consistency postulate for input-output choices by rms
and generalized the approach to stochastic supply functions in Dasgupta (2009). An
analogous expansion of the work in this paper to homothetic production functions would
be a natural subject for future work.
Nandeibam (2009) examined rationalizability of a choice system in a general choice
context. Where his universal set of alternatives is an ane space, we can consider
homothetic utility functions. In his notation, this would imply the condition that P B(S) =
P B()(S()), i.e. that the probability that the choice set from the feasible set B belongs
to S (S being an element of an algebra on the set of all nonempty subsets of B) equals the
probability that the choice set from B() belongs to S(), where B() and S() are dened
as in Denition 3. This condition, however, might already be entailed in Nandeibam's
rationalizability condition (Theorem 1) if the set of allowable utility functions of an agent is
restricted to homothetic utility functions. In particular, let u denote elements of the set of
homothetic utility functions, then fu : argmaxBfu(x)g 2 Sg = fu : argmaxB()fu(x)g 2
S()g. While Nandeibam's denition of rationalizability would not need to be strengthened
for homothetic utility functions beyond the aforementioned restriction on the set of utility
functions, an analysis of what additional (if any) conditions homothetic rationalizability
implies remains a subject for future investigation.
4References
Bandyopadhyay, T., I. Dasgupta, and P. K. Pattanaik (1999): \Stochastic Revealed
Preference and the Theory of Demand," Journal of Economic Theory 84(1):95{110.
Bandyopadhyay, T., I. Dasgupta, and P. K. Pattanaik (2002): \Demand Aggregation and
the Weak Axiom of Stochastic Revealed Preference," Journal of Economic Theory
107(2):483{489.
Bandyopadhyay, T., I. Dasgupta, and P. K. Pattanaik (2004): \A General Revealed
Preference Theorem for Stochastic Demand Behavior," Economic Theory 23(3):589{
599.
Dasgupta, I. (2005): \Consistent Firm Choice and the Theory of Supply," Economic
Theory 26(1):167{175.
Dasgupta, I. (2009): \Supply Theory sans Prot Maximization," The B.E. Journal of
Theoretical Economics: Contributions 9(1):Article 26.
Heufer, J. (2008): \Stochastic Revealed Preference and Rationalizability," Ruhr Economic
Papers #70, TU Dortmund University, Discussion Paper.
Liu, P.-W. and K.-C. Wong (2000): \Revealed Homothetic Preference and Technology,"
Journal of Mathematical Economics 34(3):287{314.
Nandeibam, S. (2009): \On Probabilistic Rationalizability," Social Choice and Welfare
32(3):425{437.
5

Stochastic homothetically revealed preference for tight stochastic demand functions

http://www.accessecon.com/Pubs/EB/2009/Volume29/EB-09-V29-I3-P92.pdf

Stochastic homothetically revealed preference for tight stochastic demand functions

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics