Chapter Seven
Revealed Preference
显现性偏好
What Is the Question of the
Chapter?
Suppose we observe a collection of
a consumer’s choices,can we
discover his/her preference?
Is this possible at all? If so,under
what conditions is this possible?
(This is the reverse question of finding
the optimal choice)
Digression,Science vs,Engineering
(对比科学家思维与工程师思维）
找问题与解问题的不同
–要对问题提问题
务虚与务实的不同
解释世界与改造世界的不同
–旁观者与参与者的不同
Maintained Assumptions on
Preferences
Preferences
–do not change while the choice
data are gathered.
–are strictly convex.
–are monotonic.
Assumptions on Preferences
x2
x1x1*
x2*
If preferences are convex and
monotonic (i.e,well-behaved)
then the most preferred
affordable bundle is unique.
Direct Preference Revelation
Suppose that the bundle x* is chosen
when the bundle y is affordable,
Then x* is revealed directly as
preferred to y (otherwise y would
have been chosen).
Direct Preference Revelation
x2
x1
x*
y
The chosen bundle x* is
revealed directly as preferred
to the bundles y and z.
z
Direct Preference Revelation
That x is revealed directly as
preferred to y will be written as
x y,
D
p
Indirect Preference Revelation
Suppose x is revealed directly
preferred to y,and y is revealed
directly preferred to z,Then,we call
x is revealed indirectly as preferred
to z,Write this as
x z
so x y and y z x z.
D
p
D
p
I
p
I
p
z is not affordable when x* is chosen.
x* is not affordable when y* is chosen.
So x* and z cannot be compared
directly.
Indirect Preference Revelation
x2
x1
x*
y*
z
But x*x* y*
and y* z
so x* z.
D
p
D
p
I
p
Two Axioms of Revealed
Preference
To apply revealed preference
analysis,choices must satisfy two
criteria -- the Weak and the Strong
Axioms of Revealed Preference.
The Weak Axiom of Revealed
Preference (WARP)
If the bundle x is revealed directly as
preferred to the bundle y then it is
never the case that y is revealed
directly as preferred to x; i.e.
x y not (y x).
D
p
D
p
The Weak Axiom of Revealed
Preference (WARP)
Choice data which violate the WARP
are inconsistent with economic
rationality.
The WARP is a necessary condition for
applying economic rationality to
explain observed choices.
The Weak Axiom of Revealed
Preference (WARP)
What choice data violate the WARP?
The Weak Axiom of Revealed
Preference (WARP)
x2
x1
xy
y is chosen when x is available
so y x.
x is chosen when y is available
so x y.
D
p
D
p
Checking if Data Violate the WARP
A consumer makes the following
choices:
–At prices (p1,p2)=($2,$2) the choice
was (x1,x2) = (10,1).
–At (p1,p2)=($2,$1) the choice was
(x1,x2) = (5,5).
–At (p1,p2)=($1,$2) the choice was
(x1,x2) = (5,4).
Is the WARP violated by these data?
Checking if Data Violate the WARP
(5,4) (10,1)
(10,1) (5,4)
x1
x2
D
p
D
p
The Strong Axiom of Revealed
Preference (SARP)
If the bundle x is revealed (directly or
indirectly) as preferred to the bundle
y and x? y,then it is never the case
that the y is revealed (directly or
indirectly) as preferred to x; i.e.
x y or x y
not ( y x or y x ).
D
pD
p
I
p
I
p
The Strong Axiom of Revealed
Preference
What choice data would satisfy the
WARP but violate the SARP?
The Strong Axiom of Revealed
Preference
Consider the following data:
A,(p1,p2,p3) = (1,3,10) & (x1,x2,x3) = (3,1,4)
B,(p1,p2,p3) = (4,3,6) & (x1,x2,x3) = (2,5,3)
C,(p1,p2,p3) = (1,1,5) & (x1,x2,x3) = (4,4,3)
The Strong Axiom of Revealed
Preference
Choice
Pr ices
A B C
A $46 $47 $46
B $39 $41 $46
C $24 $22 $23
A,($1,$3,$10)
(3,1,4).
B,($4,$3,$6)
(2,5,3).
C,($1,$1,$5)
(4,4,3).
The Strong Axiom of Revealed
Preference
Ch oices
Pr ices
A B C
A $46 $47 $46
B $39 $41 $46
C $24 $22 $23
In situation A,
bundle A is
directly revealed
preferred to
bundle C;
A C.Dp
The Strong Axiom of Revealed
Preference
Ch oices
Pr ices
A B C
A $46 $47 $46
B $39 $41 $46
C $24 $22 $23
A B C
A D
B D
C D
The Strong Axiom of Revealed
Preference
A B C
A D
B D
C D
The data do not violate the WARP but,..
We have that
A C,B A and C B
so,by transitivity,
A B,B C and C A.
D
p
D
p
D
p
I
p
I
p
I
p
I
I
I
The Strong Axiom of Revealed
Preference
That the observed choice data satisfy
the SARP is a condition necessary
and sufficient for there to be a well-
behaved preference relation that
“rationalizes” the data.
So our 3 data cannot be rationalized
by a well-behaved preference
relation.
Recovering Indifference Curves
Suppose we have the choice data
satisfy the SARP.
Then we can discover approximately
where are the consumer’s
indifference curves.
How?
Recovering Indifference Curves
Suppose we observe:
A,(p1,p2) = ($1,$1) & (x1,x2) = (15,15)
B,(p1,p2) = ($2,$1) & (x1,x2) = (10,20)
C,(p1,p2) = ($1,$2) & (x1,x2) = (20,10)
D,(p1,p2) = ($2,$5) & (x1,x2) = (30,12)
E,(p1,p2) = ($5,$2) & (x1,x2) = (12,30).
Where lies the indifference curve
containing the bundle A = (15,15)?
Recovering Indifference Curves
The table showing direct preference
revelations is:
Recovering Indifference Curves
Direct revelations only; the WARP
is not violated by the data.
A B C D E
A D D
B
C
D D D D
E D D D
Recovering Indifference Curves
A B C D E
A D D
B
C
D D D D
E D D D
Both direct and indirect revelations; neither
WARP nor SARP are violated by the data.
Recovering Indifference Curves
Since the choices satisfy the SARP,
there is a well-behaved preference
relation that,rationalizes” the
choices.
Recovering Indifference Curves
x2
x1
A
B
E
C D
A,(p1,p2)=(1,1); (x1,x2)=(15,15)
B,(p1,p2)=(2,1); (x1,x2)=(10,20)
C,(p1,p2)=(1,2); (x1,x2)=(20,10)
D,(p1,p2)=(2,5); (x1,x2)=(30,12)
E,(p1,p2)=(5,2); (x1,x2)=(12,30).
Begin with bundles revealed
to be less preferred than bundle A.
Recovering Indifference Curves
x2
x1
A
A,(p1,p2)=(1,1); (x1,x2)=(15,15).
A is directly revealed preferred
to any bundle in
Recovering Indifference Curves
x2
x1
A
B
E
C D
A,(p1,p2)=(1,1); (x1,x2)=(15,15)
B,(p1,p2)=(2,1); (x1,x2)=(10,20).
Recovering Indifference Curves
x2
x1
B
B is directly revealed preferred
to all bundles in
Recovering Indifference Curves
x2
x1
B
so,by transitivity,A is indirectly
revealed preferred to all bundles
in
Recovering Indifference Curves
x2
x1
B
so A is now revealed preferred
to all bundles in the union.
A
Recovering Indifference Curves
x2
x1
C
so A is now revealed preferred
to all bundles in the union.
B
A
Therefore the indifference
curve containing A must lie
everywhere else above
this shaded set.
Recovering Indifference Curves
Now,what about the bundles
revealed as more preferred than A?
Recovering Indifference Curves
x2
x1
A
B
E
C D
A,(p1,p2)=(1,1); (x1,x2)=(15,15)
B,(p1,p2)=(2,1); (x1,x2)=(10,20)
C,(p1,p2)=(1,2); (x1,x2)=(20,10)
D,(p1,p2)=(2,5); (x1,x2)=(30,12)
E,(p1,p2)=(5,2); (x1,x2)=(12,30).A
Recovering Indifference Curves
x2
x1
D
D is directly revealed preferred
to A.
Well-behaved preferences are
convex
A
Recovering Indifference Curves
x2
x1
D
D is directly revealed preferred
to A.
Well-behaved preferences are
convex so all bundles on the
line between A and D are
preferred to A also.A
Recovering Indifference Curves
x2
x1
D
D is directly revealed preferred
to A.
Well-behaved preferences are
convex so all bundles on the
line between A and D are
preferred to A also.A
As well,...
Recovering Indifference Curves
x2
x1
D
all bundles containing the
same amount of commodity 2
and more of commodity 1 than
D are preferred to D and
therefore are preferred to A
also.A
Recovering Indifference Curves
x2
x1
D
A
bundles revealed to be
strictly preferred to A
Recovering Indifference Curves
x2
x1
A
B
E
C D
A,(p1,p2)=(1,1); (x1,x2)=(15,15)
B,(p1,p2)=(2,1); (x1,x2)=(10,20)
C,(p1,p2)=(1,2); (x1,x2)=(20,10)
D,(p1,p2)=(2,5); (x1,x2)=(30,12)
E,(p1,p2)=(5,2); (x1,x2)=(12,30).A
Recovering Indifference Curves
x2
x1
B
C
E
D
All bundles revealed
to be preferred to A
A
Recovering Indifference Curves
Now we have upper and lower
bounds on where the indifference
curve containing bundle A may lie.
Recovering Indifference Curves
x2
x1
All bundles revealed
to be preferred to A
A
All bundles revealed to be less preferred to A
Recovering Indifference Curves
x2
x1
The region in which the
indifference curve containing
bundle A must lie.
A
Index Numbers
Over time,many prices change,Are
consumers better or worse off
“overall” as a consequence?
Index numbers give approximate
answers to such questions.
Index Numbers
Two basic types of indices
–price indices,and
–quantity indices
Each index compares expenditures
in a base period and in a current
period by taking the ratio of
expenditures.
Quantity Index Numbers
A quantity index is a price-weighted
average of quantities demanded; i.e,
(p1,p2) can be base period prices (p1b,p2b)
or current period prices (p1t,p2t).
I
p x p x
p x p x
q
t t
b b
1 1 2 2
1 1 2 2
Quantity Index Numbers
If (p1,p2) = (p1b,p2b) then we have the
Laspeyres quantity index;
L
p x p x
p x p x
q
b t b t
b b b b
1 1 2 2
1 1 2 2
Quantity Index Numbers
If (p1,p2) = (p1t,p2t) then we have the
Paasche quantity index;
P
p x p x
p x p x
q
t t t t
t b t b
1 1 2 2
1 1 2 2
Quantity Index Numbers
(study the rest of the chapter on your
own)
How can quantity indices be used to
make statements about changes in
welfare?
Quantity Index Numbers
If then
so consumers overall were better off
in the base period than they are now
in the current period.
L
p x p x
p x p x
q
b t b t
b b b b
1 1 2 2
1 1 2 2
1
p x p x p x p xb t b t b b b b1 1 2 2 1 1 2 2
Price Index Numbers
A price index is a quantity-weighted
average of prices; i.e,
(x1,x2) can be the base period bundle
(x1b,x2b) or else the current period
bundle (x1t,x2t).
I
p x p x
p x p x
p
t t
b b
1 1 2 2
1 1 2 2
Price Index Numbers
If (x1,x2) = (x1b,x2b) then we have the
Laspeyres price index;
L
p x p x
p x p x
p
t b t b
b b b b
1 1 2 2
1 1 2 2
Price Index Numbers
If (x1,x2) = (x1t,x2t) then we have the
Paasche price index;
P
p x p x
p x p x
p
t t t t
b t b t
1 1 2 2
1 1 2 2
Price Index Numbers
How can price indices be used to
make statements about changes in
welfare?
Define the expenditure ratio
M
p x p x
p x p x
t t t t
b b b b
1 1 2 2
1 1 2 2
Price Index Numbers
If
then
so consumers overall are better off in
the current period.
L p x p x
p x p x
p
t b t b
b b b b?
1 1 2 2
1 1 2 2

p x p x
p x p x
M
t t t t
b b b
1 1 2 2
1 1 2 2
p x p x p x p xt b t b t t t t1 1 2 2 1 1 2 2
Full Indexation?
Changes in price indices are
sometimes used to adjust wage rates
or transfer payments,This is called
“indexation”.
“Full indexation” occurs when the
wages or payments are increased at
the same rate as the price index
being used to measure the aggregate
inflation rate,
Summary
The Key to this chapter is the concept of
revealed preference,if x is chosen while
y is choosable,then x is revealed
preferred to y,
Under the Strong Axiom of Revealed
Preference (SARP),one can recovered the
preference (and indifference curve) of the
consumer.
Full Indexation?
The usual price index proposed for
indexation is the Paasche quantity
index (the Consumers’ Price Index).
What will be the consequence?
Full Indexation?
Notice that this index uses current
period prices to weight both base and
current period consumptions.
P
p x p x
p x p x
q
t t t t
t b t b
1 1 2 2
1 1 2 2
Full Indexation?
x2
x1
x2b
x1b
Base period budget constraint
Base period choice
Current period budget
constraint before indexation
Full Indexation?
x2
x1
x2b
x1b
Base period budget constraint
Base period choice
Current period budget
constraint after full indexation
Full Indexation?
x2
x1
x2b
x1b
Base period budget constraint
Base period choice
Current period choice
after indexation
Current period budget
constraint after indexation
Full Indexation?
x2
x1
x2b
x1b
x2t
x1t
(x1t,x2t) is revealed preferred to
(x1b,x2b) so full indexation makes
the recipient strictly better off if
relative prices change between
the base and current periods.