Seven
Revealed Chapter Preference
Revealed Preference Analysis
x
Suppose we observe the consumption choices that a consumer makes for different budgets. This reveals information about the consumer's preferences. We can use this information to ...
Revealed Preference Analysis
Test the behavioral hypothesis that a consumer chooses the most preferred bundle from those affordable. Discover the consumer's preference relation.
Assumptions on Preferences
Preferences do not change while the choice data are gathered. are strictly convex. are monotonic. x Together, convexity and monotonicity imply that the most preferred affordable bundle is unique.
x
Assumptions on Preferences
x2 If preferences are convex and monotonic (i.e. well-behaved) then the most preferred affordable bundle is unique.
x 2*
x 1*
x1
Direct Preference Revelation
x
Suppose that the bundle x is chosen when the bundle y is affordable. Then x is directly revealed as preferred to y (otherwise y would have been chosen).
Direct Preference Revelation
x2 The chosen bundle x is directly revealed as preferred to the bundles y and z. x y z x1
Direct Preference Revelation
x
That x is directly revealed as preferred to y will be written as x
e
D
y.
Indirect Preference Revelation
x
Suppose x is directly revealed preferred to y, and y is directly revealed preferred to z. Then, by transitivity, x is indirectly revealed preferred to z. Write this as x z so x
D
y and y
D
z
x
e
I
e
e
I
e
z.
Indirect Preference Revelation
x2 z is not affordable when x is chosen. x is not affordable when y is chosen. So x and z cannot be compared directly. x But xx* y y D z and y z x1
e e
D
Indirect Preference Revelation
x2 z is not affordable when x is chosen. x is not affordable when y is chosen. So x and z cannot be compared directly. x But xx* y y D z and y z x1 so x
e
I
e e
D
z.
The Weak Axiom of Revealed Preference (WARP)
x
If bundle x is directly revealed preferred to bundle y then it is never the case that y is directly revealed to preferred x; i.e. x
D
y
not (y
e
D
e
x).
The Weak Axiom of Revealed Preference (WARP)
Choice data which violate the WARP are inconsistent with economic rationality. x The WARP is a necessary condition for applying economic rationality to explain observed choices.
x
The Weak Axiom of Revealed Preference (WARP)
x
What choice data violate the WARP?
x2
The Weak Axiom of Revealed Preference (WARP)
y
x x1
x2
The Weak Axiom of Revealed Preference (WARP)
x is chosen when y is affordable so x y. y is chosen when x is affordable so y x. y x x1
e
D
e
D
x2
The Weak Axiom of Revealed Preference (WARP)
x is chosen when y is available so x y. y is chosen when x is available so y x. D These statements are inconsistent with x each other. x1
y
e
D
e
Checking if Data Violate the WARP
x
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).
x
Is the WARP violated by these data?
Checking if Data Violate the WARP
Choices Prices ($2, $2) ($2, $1) ($1, $2) (10, 1) $22 $21 $12 (5, 5) $20 $15 $15 (5, 4) $18 $14 $13
Red numbers are costs of chosen bundles.
Checking if Data Violate the WARP
Choices Prices ($2, $2) ($2, $1) ($1, $2) (10, 1) $22 $21 $12 (5, 5) $20 $15 $15 (5, 4) $18 $14 $13
Circles surround affordable bundles that were not chosen.
Checking if Data Violate the WARP
Ch o i c e s (10,1) (5,5) Prices ($2,$2) ($2,$1) ($1,$2) $22 $21 $12 $20 $15 $15 (5,4) $18 $14 $13
(10, 1) (5, 5) (5, 4) (10,1) (5, 5) (5 ,4) D D D D
Checking if Data Violate the WARP
Ch o i c e s (10,1) (5,5) Prices ($2,$2) ($2,$1) ($1,$2) $22 $21 $12 $20 $15 $15 (5,4) $18 $14 $13
(10, 1) (5, 5) (5, 4) (10,1) (5, 5) (5 ,4) D D D D
Checking if Data Violate the WARP
(10,1) is directly revealed preferred to (5,4), but (5,4) is directly revealed preferred to (10,1), so the WARP is violated by the data.
(10, 1) (5, 5) (5, 4) (10,1) (5, 5) (5 ,4) D D D D