Revealed Price Preference: Theory and Stochastic Testing
Rahul Deb, Yuichi Kitamura, John Quah and Jörg Stoye
G
†
G Toronto, † Yale, ‡ Johns
‡
Hopkins and
? Cornell
Bounded Rationality in Choice
London, July 2017
?
Introduction: Revealed preference the familiar theory
We observe a consumer's demand over some set of L goods.
At observation t , the prevailing prices are
p t = (p1t , p2t , . . . , pLt ) ∈ RL++
and the consumer purchases the bundle
x t = (x1t , x2t , . . . , xLt ) ∈ RL+ .
Formally, we have access to a data set D := {(p , x )}
t
t
T .
t=1
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Introduction: Revealed preference the familiar theory
We observe a consumer's demand over some set of L goods.
At observation t , the prevailing prices are
p t = (p1t , p2t , . . . , pLt ) ∈ RL++
and the consumer purchases the bundle
x t = (x1t , x2t , . . . , xLt ) ∈ RL+ .
Formally, we have access to a data set D := {(p , x )} .
We say the consumer revealed weakly prefers (strictly prefers) the
bundle x to the bundle x , with notation x ( ) x , if
t
t
t0
t
x
t
T
t=1
x
t0
0
p t x t ≥ (>) p t x t .
satises the generalized axiom of revealed preference (GARP) if
has no cycles containing .
D
x
x
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Introduction: Revealed preference the familiar theory
Denition: A utility function Ũ : R → R rationalizes
D = {(p , x )}
if, for all t = 1, 2, . . . , T ,
t
t
T
t=1
L
+
x t ∈ argmax{Ũ(x) : p t x ≤ p t x t }.
Afriat's Theorem: Given a data set D = {(p , x )} , the
following are equivalent:
1. D can be rationalized by a locally nonsatiated preference
2. D satises GARP.
3. D can be rationalized by a strictly increasing, continuous, and
concave utility function.
t
t
T
t=1
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Introduction: Revealed preference the familiar theory
Limitations of this approach:
Typically, we only observe the consumer's demand over a fraction
of all goods.
Expenditure on the L observed goods is endogenous (contrary to
standard textbook presentations of consumer demand).
Utility is dened over all goods and not just the L goods observed.
Therefore, empirical work requires a utility function of the form
V (U(x), y1 , y2 , y3 , · · · )
where there is a subutility U(x) over L goods, which is `detachable'
(and thus denable) from all other goods.
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Introduction: Revealed preference the familiar theory
Limitations of this approach:
Typically, we only observe the consumer's demand over a fraction
of all goods.
Expenditure on the L observed goods is endogenous (contrary to
standard textbook presentations of consumer demand).
Utility is dened over all goods and not just the L goods observed.
Therefore, empirical work requires a utility function of the form
V (U(x), y1 , y2 , y3 , · · · )
where there is a subutility U(x) over L goods, which is `detachable'
(and thus denable) from all other goods.
Lastly, this model does not tell us whether a subject is better under
one set of prices (on the L goods) versus another set of prices.
In fact, scalar multiples of prices are indistinguishable in this model.
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Revealed price preference
The familiar theory is not the only way of interpreting the data.
There is an alternative and equally natural approach.
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Revealed price preference
The familiar theory is not the only way of interpreting the data.
There is an alternative and equally natural approach.
Denition: Given a data set D = {(p , x )} , the consumer
revealed weakly prefers (strictly prefers) p to p if
t
t
t
0
0
T
t=1
t0
0
p t x t ≤ (<) p t x t .
We use the notation p ( ) p .
Motivation:
At price vector p , the consumer can buy the bundle bought at
observation t and it will cost him less.
So he must prefer the price p to the price p .
t
p
p
t0
t
0
t
t0
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Revealed price preference
The familiar theory is not the only way of interpreting the data.
There is an alternative and equally natural approach.
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Revealed price preference
The familiar theory is not the only way of interpreting the data.
There is an alternative and equally natural approach.
Given a data set D = {(p , x )} , the consumer revealed weakly
prefers (strictly prefers) the price p to p with notation
p ( ) p , if p x ≤ (<) p x .
D satises the generalized axiom of price preference (GAPP) if has no cycles containing , i.e., if there are observations t ,
t , . . . , t such that
p , and p p
p p ,p p ,. . . p
then we cannot replace with .
t
t
p
p
t0
t
t
t0
T
t=1
t
t0
t0
t0
p
p
1
n
2
t1
p
t2
t2
p
t3
p
tn−1
p
tn
tn
p
t1
p
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Revealed price preference
The familiar theory is not the only way of interpreting the data.
There is an alternative and equally natural approach.
Given a data set D = {(p , x )} , the consumer revealed weakly
prefers (strictly prefers) the price p to p with notation
p ( ) p , if p x ≤ (<) p x .
D satises the generalized axiom of price preference (GAPP) if has no cycles containing , i.e., if there are observations t ,
t , . . . , t such that
p , and p p
p p ,p p ,. . . p
then we cannot replace with .
What optimizing model is observationally equivalent to GAPP?
t
t
p
p
t0
t
t
t0
T
t=1
t
t0
t0
t0
p
p
1
n
2
t1
p
t2
t2
p
t3
p
tn−1
p
tn
tn
p
t1
p
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Expenditure-Augmented Utility
Suppose that a consumer's purchasing behavior over L goods is
guided by (1) benet he derives from the L goods and (2) disutility
of the expenditure incurred from spending money on those goods.
The consumer has an expenditure-augmented utility function (or
simply, an augmented utility function) U : R × R → R.
If consuming bundle x incurs expenditure e , then the utility is
U(x, −e).
In other words, the cost of a bundle is an attribute directly aecting
its utility.
L
+
−
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Expenditure-Augmented Utility
Suppose that a consumer's purchasing behavior over L goods is
guided by (1) benet he derives from the L goods and (2) disutility
of the expenditure incurred from spending money on those goods.
The consumer has an expenditure-augmented utility function (or
simply, an augmented utility function) U : R × R → R.
If consuming bundle x incurs expenditure e , then the utility is
U(x, −e).
In other words, the cost of a bundle is an attribute directly aecting
its utility.
We assume U(x, −e) is strictly increasing in the last argument (or,
equivalently, strictly decreasing in expenditure).
L
+
−
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Expenditure-Augmented Utility
Suppose that a consumer's purchasing behavior over L goods is
guided by (1) benet he derives from the L goods and (2) disutility
of the expenditure incurred from spending money on those goods.
The consumer has an expenditure-augmented utility function (or
simply, an augmented utility function) U : R × R → R.
If consuming bundle x incurs expenditure e , then the utility is
U(x, −e).
In other words, the cost of a bundle is an attribute directly aecting
its utility.
We assume U(x, −e) is strictly increasing in the last argument (or,
equivalently, strictly decreasing in expenditure).
At a price p, the consumer chooses x ∈ R to maximize
U(x, −p · x).
L
+
−
L
+
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Expenditure-Augmented Utility
Suppose that a consumer's purchasing behavior over L goods is
guided by (1) benet he derives from the L goods and (2) disutility
of the expenditure incurred from spending money on those goods.
The consumer has an expenditure-augmented utility function (or
simply, an augmented utility function) U : R × R → R.
If consuming bundle x incurs expenditure e , then the utility is
U(x, −e).
In other words, the cost of a bundle is an attribute directly aecting
its utility.
We assume U(x, −e) is strictly increasing in the last argument (or,
equivalently, strictly decreasing in expenditure).
At a price p, the consumer chooses x ∈ R to maximize
U(x, −p · x).
Note: This model generalizes the quasilinear utility model where x
is chosen to maximize V (x) − px .
L
+
−
L
+
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Rationalization
Denition: A expenditure-augmented utility function
U : RL+ × R− → R
rationalizes D = {(p t , x t )}Tt=1
x t ∈ argmax
if, for all t = 1, 2, . . . , T ,
n
o
U(x, −p t x) : x ∈ RL+ .
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
GAPP is also Sucient for Rationalization
Theorem 1: Given a data set D = {(p , x )} , the following are
equivalent:
1. D can be rationalized by an expenditure-augmented preference.
2. D satises GAPP.
t
t
T
t=1
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
GAPP is also Sucient for Rationalization
Theorem 1: Given a data set D = {(p , x )} , the following are
equivalent:
1. D can be rationalized by an expenditure-augmented preference.
2. D satises GAPP.
3. D can be rationalized by an augmented utility function U that
is strictly increasing, continuous, and concave.
Moreover, U is such that max U(x, −p · x) has a solution
for all p ∈ R .
t
t
T
t=1
x∈RL+
L
++
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Necessity of GAPP
The indirect utility at price p (corresponding to U ) is
(1)
V (p) := max U(x, −px).
x∈RL+
The consumer revealed weakly prefers (strictly prefers) p to p
[notation p ( ) p ] if p x ≤ (<) p x .
If consumer is maximizing U(x, −px), then p ( ) p =⇒
t0
t
t
p
p
t0
t t0
t0 t0
t
0
0
0
p
0
p
0
t0
0
V (p t ) ≥ U(x t , −p t x t ) ≥ (>) U(x t , −p t x t ) = V (p t ).
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Necessity of GAPP
The indirect utility at price p (corresponding to U ) is
(1)
V (p) := max U(x, −px).
x∈RL+
The consumer revealed weakly prefers (strictly prefers) p to p
[notation p ( ) p ] if p x ≤ (<) p x .
If consumer is maximizing U(x, −px), then p ( ) p =⇒
t0
t
t
p
t0
p
t t0
t0 t0
t
0
0
0
0
p
p
t0
0
0
V (p t ) ≥ U(x t , −p t x t ) ≥ (>) U(x t , −p t x t ) = V (p t ).
Suppose there are observations t , t , . . . , t such that
p p ,p p ,. . . p
p , and p p
Then we obtain V (p ) ≥ V (p ) ≥ ...V (p ) ≥ V (p ), which
means we cannot replace with anywhere in the cycle.
1
t1
p
t2
t1
p
t2
t1
t2
p
n
2
tn−1
p
tn
tn
tn
p
t1
t1
p
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Features of the revealed price preference model
It does not require the assumption of separability of observed goods,
i.e., does not require V (U(x), y , y , ...).
But it does require that outside prices are constant or at least that
their changes can be tracked by a composite price index of
non-observed goods.
It allows for consistent welfare comparisons between price vectors.
(Indeed, the whole theory is built on this consistency.)
It is readily generalized to a realistic random utility framework
(unlike the standard model).
1
2
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
This for a data set
Data Satises GARP
but
not GAPP
p = (2, 1) , x = (4, 0) and p
The data set with p = (2, 1) ,
x = (0, 1) satises GARP.
t
t
that violates
t GAPP.
xt
t′
′
4 0 and
= (1, 2) , x t = (0, 1).
0
12 ,
pt = ( , )
=( , )
t0
x2
b
xt
′
b
xt
x1
But notice that p x = 8 > p x = 4, so p p , and
p x = 2 > p x = 1, so p p . Data fails GAPP.
t0
t t
t0 t0
t t0
t
t0
t
p
p
t
t0
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Testing GAPP vs. GARP
Scaling Proposition: Let D = {(p , x )} be a data set and let
e = {(p , x̃ )}
D
, where x̃ = x /(p x ), be its
expenditure-normalized version.
Then the revealed preference relations are related as follows:
1. p p if and only if x̃ x̃ .
2. p p if and only if x̃ x̃ .
Proof: p ≥ p i p x ≥ p x .
t
t
t
t
t
p
p
t
t∈T
t0
t
t0
t
t xt
pt x t
0
t xt
pt 0 x t 0
t
T
t=1
t t
t
t0 t0
x
x
t0
t0
t t0
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Testing GAPP vs. GARP
Scaling Proposition: Let D = {(p , x )} be a data set and let
e = {(p , x̃ )}
D
, where x̃ = x /(p x ), be its
expenditure-normalized version.
Then the revealed preference relations are related as follows:
1. p p if and only if x̃ x̃ .
2. p p if and only if x̃ x̃ .
Proof: p ≥ p i p x ≥ p x .
t
t
t
t
t
p
p
t
t∈T
t0
t
t0
t
t xt
pt x t
0
t xt
pt 0 x t 0
t
T
t=1
t t
t
t0 t0
x
x
t0
t0
t t0
This implies that testing GAPP on D is equivalent
to scaling the
consumption bundles and testing GARP on De.
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Data Satises GARP but not GAPP
The data set:
p = (2, 1) , x
satises GARP.
This for a data set
t
4 0 and
′
′
12
pt = (2, 1) , xt = (4, 0) and p t = (1, 2) , x t = (0, 1).
t0
t0
t
that violates GAPP.
=( , )
01
p = ( , ), x = ( , )
x2
b
xt
′
b
xt
x1
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Data Satises GARP but not GAPP
Scaling bundles, it is clear that
p = (2, 1) , x = (4, 0) and p = (1, 2) , x = (0, 1)
violates GAPP. Scaled bundles: x̃ = (4, 0). x̃ = (0, 4).
This for a data set
′
t
′
pt = (2, 1) , xt = (4, 0) and p t = (1, 2) , x t = (0, 1).
t0
t
t0
after normalizing the x’s by dividing with the income.
t
x2
b
xt
t0
′
b
xt
x1
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Welfare Analysis
Rationalizable data D could potentially be rationalized by many U 's
leading to dierent indirect utilities V 's.
Notation:
V(D): set of rationalizing indirect utility functions.
( ): transitive closure of (with one relation strict).
I
I
∗
p
∗
p
p
Welfare Proposition: Suppose D = {(p , x )} is rationalizable
by an augmented utility function. Then for any p , p :
1. p p if and only if V (p ) ≥ V (p ) for all V ∈ V(D).
2. p p if and only if V (p ) > V (p ) for all V ∈ V(D).
Thus, allows for welfare comparisons under dierent prices.
t
∗
p
∗
p
t
t
t
t0
t
t0
t0
t
t0
T
t=1
t
t0
p
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Random utility in the standard model (McFadden-Richter)
In this case, an observation t consists of a distribution of demand
bundles at price p and income w .
With a nite set of such distributions, how do we check whether
the observed distributions can be generated by a population of
utility-maximizing consumers, with this requirement:
the distribution of utility functions is invariant across observations.
t
t
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Random utility in the standard model (McFadden-Richter)
In this case, an observation t consists of a distribution of demand
bundles at price p and income w .
With a nite set of such distributions, how do we check whether
the observed distributions can be generated by a population of
utility-maximizing consumers, with this requirement:
the distribution of utility functions is invariant across observations.
McFadden and Richter's crucial observation:
the number of observationally distinguishable utility-types is nite,
so this problem is equivalent to solving an appropriate linear
program.
t
t
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Stochastic GARP
B1,t
x2
B1,t
This for a data set
B1,t
′
′
x2
′
pt = (2, 1) , xt = (3, 0) and p t = (1, 2) , x t = (0, 3).
We highlight the sets where choices can be made without violating GARP.
B2,t
B2,t
This for a data set
B1,t
′
′
′
pt = (2, 1) , xt = (3, 0) and p t = (1, 2) , x t = (0, 3).
We highlight the sets where choices can be made without violating GARP.
′
B2,t
x1
B2,t
′
x1
(a) Prop. of this Preference=
(b) Prop. of this Preference=
ν1
B1,t
ν2
B1,t
ν1 + ν2
x2
x2
′
B1,t
9
B2,t
′
B1,t
10
ν2
B2,t
ν3
′
x1
(c) Prop. of this Preference=
ν1 + ν3
B2,t
B2,t
′
x1
ν3
(d) Choice Distribution
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Stochastic GARP
B1,t
ν1 + ν2
x2
B1,t
π 1,t
x2
′
ν2
π 1,t
ν1 + ν3
ν3
B2,t
B2,t
′
π 2,t
′
π 2,t
′
x1
x1
(a) Rationalizable Shares
(b) Observed Distribution
The observed distribution of choices can be stochastically
rationalized if there exist ν , ν , ν ≥ 0 such that
1
2
3
0
ν1 + ν2 = π 1,t , ν2 = π 1,t , ν3 = π 2,t , ν1 + ν3 = π 2,t
12
0
14
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Limitations of the McFadden-Richter model
The problem with the McFadden-Richter model is that data of the
type imagined is not actually observed.
Expenditure levels are not exogenously prescribed but endogenously
determined by the population of consumers.
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Limitations of the McFadden-Richter model
The problem with the McFadden-Richter model is that data of the
type imagined is not actually observed.
Expenditure levels are not exogenously prescribed but endogenously
determined by the population of consumers.
The data set is D := {(p , π̃ )} . At observation t ,
π̃ is the distribution of demand bundles on R these bundles need not generate the same expenditure at p .
A model of choice that rationalizes D must also explain the
observed distribution of expenditure levels.
t
t
t
T
t=1
L
+
t
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Rationalization by Random Augmented Utility
Let D := {(p , π̃ )} .
D is said to be rationalized by the random augmented utility model
if there exists a distribution µ over the set U of augmented utility
functions such that
π̃ (X ) = µ(U (X )) for all t ∈ T and X ⊂ R ,
where
t
t
t
T
t=1
t
t
t
(
U (X t ) :=
L
+
)
U ∈ U : argmax U(x, −p t x) ∈ X t
.
x∈RL+
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Rationalization by Random Augmented Utility
Let D := {(p , π̃ )} .
D is said to be rationalized by the random augmented utility model
if there exists a distribution µ over the set U of augmented utility
functions such that
π̃ (X ) = µ(U (X )) for all t ∈ T and X ⊂ R ,
where
t
t
t
T
t=1
t
t
t
(
U (X t ) :=
L
+
)
U ∈ U : argmax U(x, −p t x) ∈ X t
.
x∈RL+
Crucial observation:
By the Scaling Proposition, testing this model is the same as
testing the McFadden-Richter model after suitable scaling of
demand bundles.
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Observed Choices
b
b
Bt
b
b
b
b
b
b
b
b
x2
b
b
b
x2
b
b
b
b
b
Bt
b
b
x1
(a)
t
p = (1, 2)
′
x1
(b)
t
p = (2, 1)
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Scaled Choices
b
b
b
b
Bt
b
b
b
b
b
b
b
b
b
x2
b
b
b
b
x2
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
Bt
b
b
b
x1
(a)
t
p = (1, 2)
′
x1
(b)
t
p = (2, 1)
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Scaled Empirical Choice Probabilities
B1,t
b
b
x2
3
5
π̂ 1,t =
b
b
B1,t
′
b
b
b
b
b
′
b
b
1
2
b
π̂ 2,t =
2
5
π̂ 1,t =
′
π̂ 2,t =
b
b
b
b
1
2
b
b
b
b
B2,t
B2,t
′
x1
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Procedure for testing the random augmented utility model
1. Fix an arbitrary expenditure, say 1.
2. At each t , scale each observed choice x to .
3. This yields a distribution of choices on T budget planes.
4. Test stochastic GARP on these scaled empirical choice
probabilities.
t
xt
pt x t
Theorem 2: D can be rationalized by the random augmented utility
model if and only if stochastic GARP holds on the scaled data set.
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference
Concluding Remarks
We develop a simple and intuitive model that allows for consistent
welfare comparisons between prices.
This model is easily extendable into a random utility model
We implement the test of the random augmented utility model on
UK household spending data:
The model is not rejected.
We estimate the proportion of consumers who are
better/worse o between one observation and another.
These estimates are reasonably tight.
I
I
Deb, Kitamura, Quah and Stoye (2017): Revealed Price Preference