Chapter 7
REVEALED
PREFERENCE
7.1 The Idea of Revealed Preference


(x1, x2) is chosen when
(y1, y2) could have been
chosen:
p1x1+p2x2≥p1y1+p2y2
(x1, x2) is directly
revealed preferred to
(y1, y2).
7.2 From Revealed Preference to
Preference

The Principle of Revealed Preference
 Suppose
there is a unique optimal bundle for each
budget set;
 Suppose the consumer always chooses the most
preferred bundle she can afford;
 Suppose the consumer has chosen (x1, x2) when
faced by (p1, p2);
 Suppose p1x1+p2x2≥p1y1+p2y2;
 we must have ( x1 , x2 ) ( y1 , y2 )
7.2 From Revealed Preference to
Preference



Suppose the consumer has chosen (y1, y2) when
faced by (q1, q2);
Suppose q1y1+q2y2≥q1z1+q2z2;
Then we know that
( x1 , x2 )

( y1 , y2 ) and ( y1 , y2 )
From transitivity we can conclude that:
( x1 , x2 )

( z1, z2 )
( z1 , z2 )
(x1, x2) is indirectly revealed preferred to (z1, z2).
7.2 From Revealed Preference to
Preference

(x1, x2) is indirectly revealed preferred to (z1, z2).
7.3 Recovering Preferences


Suppose the preference
is convex and
monotonic.
The weakly preferred
set w/r to X contains the
“smallest” monotonic
convex set that includes
X, Y, and Z.
Y
X, Z
X
7.4 The Weak Axiom of Revealed
Preference

WARP: If
 (x1, x2)
is directly revealed
preferred to (y1, y2);
 and (x1, x2)≠(y1, y2);

then (y1, y2) cannot be
directly revealed
preferred to (x1, x2).
7.4 The Weak Axiom of Revealed
Preference

Satisfying WARP
7.5 Checking WARP
bundles
prices
1
2
3
1
5
4*
6
2
4*
5
6
3
3*
3*
4
7.6 The Strong Axiom of Revealed
Preference

SARP: If
 (x1, x2)
is (directly or indirectly) revealed preferred
to (y1, y2);
 and (y1, y2)≠(x1, x2);
then (y1, y2) cannot be (directly or indirectly)
revealed preferred to (x1, x2).
 SARP is both necessary and sufficient for
rational consumer behavior.

7.7 How to Check SARP

Transform the table:
Prices

1
2
3
1
20
21
12
Bundles
2
3
10* 22(*)
20 15*
15
10
Need to look for chains of arbitrary length to see if
one observation is indirectly revealed preferred to
another.
7.8 Index Numbers

Quantity index:
wx w x
Iq 
wx w x
t
1 1
b
1 1

t
2 2
b
2 2
Paasche quantity index:
p1t x1t  p2t x2t
Pq  t b
p1 x1  p2t x2b

Laspeyres quantity index:
p1b x1t  p2b x2t
Lq  b b
b b
p1 x1  p2 x2
7.8 Index Numbers
p1t x1t  p2t x2t
Pq  t b
1
t b
p1 x1  p2 x2
p1t x1t  p2t x2t  p1t x1b  p2t x2b

The consumer is better off at t than at b.
px px
Pq 
1
px px
t
1
t
1
t
1
b
1
t
2
t
2
t
2
b
2
p1t x1t  p2t x2t  p1t x1b  p2t x2b

No inference on consumer well-being.
7.8 Index Numbers
p1b x1t  p2b x2t
Lq  b b
1
b b
p1 x1  p2 x2
p1b x1t  p2b x2t  p1b x1b  p2b x2b

No inference on consumer well-being.
p x p x
Lq 
1
p x p x
b
1
b
1
t
1
b
1
b
2
b
2
t
2
b
2
p1b x1t  p2b x2t  p1b x1b  p2b x2b

The consumer is better off at b than at t.
7.9 Price Indices

Price index:
w p  w2 p
Ip 
w p  w2 p
t
1 1
b
1 1

t
2
b
2
Paasche price index:
p1t x1t  p2t x2t
Pp  b t
p1 x1  p2b x2t

Laspeyres price index:
p1t x1b  p2t x2b
Lp  b b
b b
p1 x1  p2 x2
7.9 Price Indices
p1t x1t  p2t x2t
p1t x1t  p2t x2t
Pp  b t
 b b
M
b t
b b
p1 x1  p2 x2 p1 x1  p2 x2
p1b x1b  p2b x2b  p1b x1t  p2b x2t

The consumer is better off at b than at t.
px px
px px
Lp 

M
p x p x
p x p x
t
1
b
1
b
1
b
1
t
2
b
2
b
2
b
2
t
1
b
1
t
1
b
1
t
2
b
2
t
2
b
2
p1t x1t  p2t x2t  p1t x1b  p2t x2b

The consumer is better off at t than at b.
Indexing Social Security Payments
Indexing: Social security payments get
adjusted with price indices so that the
consumption bundle in year b is still affordable
in year t.
 Consumers are strictly better off with indexing.

Indexing Social Security Payments