Molly W. Dahl Georgetown University Econ 101 – Spring 2009

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Revealed Preference

Molly W. DahlGeorgetown UniversityEcon 101 – Spring 2009

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Revealed Preference Analysis

Suppose we observe the demands (consumption choices) that a consumer makes for different budgets. This reveals information about the consumer’s preferences. We can use this information to Test the behavioral hypothesis that a consumer

chooses the most preferred bundle from those available.

Discover the consumer’s preference relation.

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Assumptions on Preferences Preferences

do not change while the choice data are gathered.

are strictly convex.are monotonic.

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Assumptions on Preferences

x2

x1x1*

x2*

If preferences are strictly convex and monotonic then the most preferredaffordable bundle is unique.

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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).

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x2

x1

x*

y

The chosen bundle x* isrevealed directly as preferredto the bundles y and z.

z

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That x is revealed directly as preferred to y will be written as x DRP y.

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Indirect Preference Revelation Suppose x is revealed directly preferred

to y, and y is revealed directly preferred to z. Then, by transitivity, x is revealed indirectly as preferred to z. Write this as x IRP z

so x DRP y and y DRP z x IRP z.

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Indirect Preference Revelation

x2

x1

x*

z

z is not affordable when x* is chosen.

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x2

x1

x*y*

z

x* is not affordable when y* is chosen.

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z is not affordable when x* is chosen.x* is not affordable when y* is chosen. So x* and z cannot be compared directly.


x2

x1

x*y*

z

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z is not affordable when x* is chosen.x* is not affordable when y* is chosen. So x* and z cannot be compared directly.


x2

x1

x*y*

zBut x* DRP y*and y* DRP z so x* IRP z.

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Two Axioms of Revealed Preference

To apply revealed preference analysis, choices must satisfy two criteria -- the Weak and the Strong Axioms of Revealed Preference.

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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 DRP y not (y DRP x).

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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.

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The Weak Axiom of Revealed Preference (WARP)

What choice data violate the WARP?

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The Weak Axiom of Revealed Preference (WARP)x2

x1

xy

x is chosen when y is availableso x DRP y.

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The Weak Axiom of Revealed Preference (WARP)x2

x1

xy These statements are

inconsistent with each other.

x is chosen when y is availableso x DRP y.

y is chosen when x is availableso y DRP x.

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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?

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Checking if Data Violate the WARP

ChoicesPrices (10, 1) (5, 5) (5, 4)

($2, $2) $22 $20 $18

($2, $1) $21 $15 $14

($1, $2) $12 $15 $13

Red numbers are costs of chosen bundles.

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ChoicesPrices (10, 1) (5, 5) (5, 4)

($2, $2) $22 $20 $18

($2, $1) $21 $15 $14

($1, $2) $12 $15 $13

Circles surround affordable bundles thatwere not chosen.

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( 1 0 , 1 ) ( 5 , 5 ) ( 5 , 4 )

( 1 0 , 1 ) D D

( 5 , 5 ) D

( 5 , 4 ) D

Ch o i c e s P r i c e s ( 1 0 , 1 ) ( 5 , 5 ) ( 5 , 4 )

( $ 2 , $ 2 ) $ 2 2 $ 2 0 $ 1 8

( $ 2 , $ 1 ) $ 2 1 $ 1 5 $ 1 4

( $ 1 , $ 2 ) $ 1 2 $ 1 5 $ 1 3

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( 1 0 , 1 ) ( 5 , 5 ) ( 5 , 4 )

( 1 0 , 1 ) D D

( 5 , 5 ) D

( 5 , 4 ) D

(10,1) is directlyrevealed preferredto (5,4), but (5,4) isdirectly revealedpreferred to (10,1),so the WARP isviolated by the data.

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(5,4) DRP (10,1)

(10,1) DRP (5,4)

x1

x2

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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 DRP y or x IRP y

not ( y DRP x or y IRP x ).

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The Strong Axiom of Revealed Preference

What choice data would satisfy the WARP but violate the SARP?

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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)

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ChoicePrices 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).

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ChoicesPrices 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 data do not violate the WARP.

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A B C

A D

B D

C D

The data do not violate the WARP but ...

We have thatA DRP C, B DRP A and C DRP B

so, by transitivity,A IRP B, B IRP C and C IRP A.

I

I

I

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A B C

A D

B D

C D

I

I

I

B DRP A is inconsistent

with A IRP B.

The data do not violate the WARP but ...

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A B C

A D

B D

C D

I

I

I

The data do not violatethe WARP but there are3 violations of the SARP.

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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 data cannot be rationalized by a well-behaved preference relation.

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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?

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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)?

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The table showing direct preference revelations is:

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Direct revelations only; the WARPis not violated by the data.

A B C D EA D DBCD D D DE D D D

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In this example, indirect preference revelations add no extra information. The table showing both direct and indirect preference revelations is the same as the table showing only the direct preference revelations:

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Recovering Indifference CurvesA B C D E

A D DBCD D D DE D D D

Both direct and indirect revelations; neitherWARP nor SARP are violated by the data.

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Since the choices satisfy the SARP, there is a well-behaved preference relation that “rationalizes” the choices.

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Recovering Indifference Curvesx2

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 revealedto be less preferred than bundle A.

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x1

A

A: (p1,p2)=(1,1); (x1,x2)=(15,15).

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x1

A

A: (p1,p2)=(1,1); (x1,x2)=(15,15).

A is directly revealed preferredto any bundle in

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x1

A

B

E

C D

A: (p1,p2)=(1,1); (x1,x2)=(15,15)B: (p1,p2)=(2,1); (x1,x2)=(10,20).

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x1

A

B

A: (p1,p2)=(1,1); (x1,x2)=(15,15)B: (p1,p2)=(2,1); (x1,x2)=(10,20).

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x1

A

B

A is directly revealed preferredto B and …

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x1

B

B is directly revealed preferredto all bundles in

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x1

B

so, by transitivity, A is indirectlyrevealed preferred to all bundles in

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x1

B

so A is now revealed preferredto all bundles in the union.

A

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x1

A

B

E

C D

A: (p1,p2)=(1,1); (x1,x2)=(15,15)

C: (p1,p2)=(1,2); (x1,x2)=(20,10).

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x1

A C

A: (p1,p2)=(1,1); (x1,x2)=(15,15)

C: (p1,p2)=(1,2); (x1,x2)=(20,10).

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x1

A C

A is directly revealedpreferred to C and ...

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x1

C

C is directly revealed preferredto all bundles in

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x1

C

so, by transitivity, A isindirectly revealed preferredto all bundles in

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x1

C

so A is now revealed preferredto all bundles in the union.

BA

Therefore the indifferencecurve containing A must lie everywhere else above this shaded set.

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Now, what about the bundles revealed as more preferred than A?

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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

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x1

D

A: (p1,p2)=(1,1); (x1,x2)=(15,15)

D: (p1,p2)=(2,5); (x1,x2)=(30,12).

A

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x1

D

D is directly revealed preferredto A.

A

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x1

D

D is directly revealed preferredto A.Well-behaved preferences areconvex

A

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x1

D

D is directly revealed preferredto A.Well-behaved preferences areconvex so all bundles on the line between A and D are preferred to A also.A

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x1

D

D is directly revealed preferredto A.Well-behaved preferences areconvex so all bundles on the line between A and D are preferred to A also.A

As well, ...

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x1

D

all bundles containing thesame amount of commodity 2and more of commodity 1 thanD are preferred to D and therefore are preferred to A also.A

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x1

DA

bundles revealed to be strictly preferred to A

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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

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x1

A

E

A: (p1,p2)=(1,1); (x1,x2)=(15,15)

E: (p1,p2)=(5,2); (x1,x2)=(12,30).

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x1

A

E

E is directly revealed preferredto A.Well-behaved preferences areconvex so all bundles on the line between A and E are preferred to A also.

As well, ...

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x1

A

E

all bundles containing thesame amount of commodity 1and more of commodity 2 thanE are preferred to E and therefore are preferred to A also.

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x1

A

EMore bundles revealed to be strictly preferred to A

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x1

A

B

C

E

D

Bundles revealedearlier as preferredto A

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x1

B

C

E

D

All bundles revealedto be preferred to A

A

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Now we have upper and lower bounds on where the indifference curve containing bundle A may lie.

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x1

All bundles revealedto be preferred to A

A

All bundles revealed to be less preferred to A

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x1

The region in which the indifference curve containing bundle A must lie.

A

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Molly W. Dahl Georgetown University Econ 101 – Spring 2009