Revealed Preference

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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

Preferencesdo 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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Direct Preference Revelation

x2

x1

x*

y

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

z

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Direct Preference Revelation

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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Indirect Preference Revelation

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.

Indirect Preference Revelation

x2

x1

x*y*

z

12

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*

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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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

Circles surround affordable bundles thatwere not chosen.

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Checking if Data Violate the WARP

( 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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Checking if Data Violate the WARP

( 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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Checking if Data Violate the WARP

(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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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)

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The Strong Axiom of Revealed Preference

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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The Strong Axiom of Revealed Preference

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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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 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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The Strong Axiom of Revealed Preference

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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The Strong Axiom of Revealed Preference

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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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 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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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)?

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Recovering Indifference Curves

The table showing direct preference revelations is:

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Recovering Indifference Curves

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

A B C D E

A D D

B

C

D D D D

E D D D

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Recovering Indifference Curves

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 D

B

C

D D D D

E D D D

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

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Recovering Indifference Curves

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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Recovering Indifference Curvesx2

x1

A

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

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Recovering Indifference Curvesx2

x1

A

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

A is directly revealed preferredto any bundle in

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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).

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Recovering Indifference Curvesx2

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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Recovering Indifference Curvesx2

x1

A

B

A is directly revealed preferredto B and …

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Recovering Indifference Curvesx2

x1

B

B is directly revealed preferredto all bundles in

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Recovering Indifference Curvesx2

x1

B

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

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Recovering Indifference Curvesx2

x1

B

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

A

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Recovering Indifference Curvesx2

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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Recovering Indifference Curvesx2

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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Recovering Indifference Curvesx2

x1

A C

A is directly revealedpreferred to C and ...

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Recovering Indifference Curvesx2

x1

C

C is directly revealed preferredto all bundles in

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Recovering Indifference Curvesx2

x1

C

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

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Recovering Indifference Curvesx2

x1

C

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

B

A

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

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Recovering Indifference Curves

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

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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).A

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Recovering Indifference Curvesx2

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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Recovering Indifference Curvesx2

x1

D

D is directly revealed preferredto A.

A

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Recovering Indifference Curvesx2

x1

D

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

A

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Recovering Indifference Curvesx2

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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Recovering Indifference Curvesx2

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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Recovering Indifference Curvesx2

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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Recovering Indifference Curvesx2

x1

DA

bundles revealed to be strictly preferred to A

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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).A

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Recovering Indifference Curvesx2

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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Recovering Indifference Curvesx2

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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Recovering Indifference Curvesx2

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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Recovering Indifference Curvesx2

x1

A

E

More bundles revealed to be strictly preferred to A

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Recovering Indifference Curvesx2

x1

A

B

C

E

D

Bundles revealedearlier as preferredto A

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Recovering Indifference Curvesx2

x1

B

C

E

D

All bundles revealedto be preferred to A

A

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Recovering Indifference Curves

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

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Recovering Indifference Curvesx2

x1

All bundles revealedto be preferred to A

A

All bundles revealed to be less preferred to A

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Recovering Indifference Curvesx2

x1

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

A