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Chapter Seven

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Chapter Seven

Revealed Preference

- 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.

- Preferences
- do not change while the choice data are gathered.
- are strictly convex.
- are monotonic.

- Together, convexity and monotonicity imply that the most preferred affordable bundle is unique.

x2

If preferences are convex andmonotonic (i.e. well-behaved)then the most preferredaffordable bundle is unique.

x2*

x1*

x1

- 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).

x2

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

x*

z

y

x1

p

D

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

p

p

D

D

- 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 zso x y and y z x z.

p

I

p

I

x2

z is not affordable when x* is chosen.

x*

z

x1

x2

x* is not affordable when y* is chosen.

x*

y*

z

x1

x2

z is not affordable when x* is chosen.x* is not affordable when y* is chosen.

x*

y*

z

x1

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*

y*

z

x1

p

D

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 x*x*y*

y*

z

x1

p

p

D

D

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 x*x*y*and y* z

y*

z

x1

p

p

D

D

x2

x*

But x*x*y*and y* z so x* z.

y*

z

p

x1

I

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

p

p

D

D

- 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).

- 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.

- What choice data violate the WARP?

x2

y

x

x1

p

D

x2

x is chosen when y is availableso x y.

y

x

x1

p

p

D

D

x2

x is chosen when y is availableso x y.

y is chosen when x is availableso y x.

y

x

x1

p

p

D

D

x2

x is chosen when y is availableso x y.

y is chosen when x is availableso y x.

These statements are inconsistent with each other.

y

x

x1

- 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?

Red numbers are costs of chosen bundles.

Circles surround affordable bundles thatwere not chosen.

Circles surround affordable bundles thatwere not chosen.

Circles surround affordable bundles thatwere not chosen.

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

p

p

D

D

x2

(5,4)(10,1)

(10,1)(5,4)

x1

p

p

D

D

p

I

- If the bundle x is revealed (directly or indirectly) as preferred to the bundle y and x ¹ 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 ).

p

I

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

- 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)

A: ($1,$3,$10) (3,1,4).

B: ($4,$3,$6) (2,5,3).

C: ($1,$1,$5) (4,4,3).

p

D

In situation A,bundle A is

directly revealedpreferred tobundle C;

A C.

p

D

In situation B,bundle B is

directly revealedpreferred tobundle A;

B A.

p

D

In situation C,bundle C is

directly revealedpreferred tobundle B;

C B.

The data do not violate the WARP.

p

p

p

D

D

D

p

p

p

I

I

I

We have thatA C, B A and C Bso, by transitivity,A B, B C and C A.

The data do not violate the WARP but ...

p

p

p

D

D

D

p

p

p

I

I

I

We have thatA C, B A and C Bso, by transitivity,A B, B C and C A.

I

I

I

The data do not violate the WARP but ...

p

D

p

I

B A is inconsistentwith A B.

I

I

I

The data do not violate the WARP but ...

p

D

p

I

A C is inconsistentwith C A.

I

I

I

The data do not violate the WARP but ...

p

D

p

I

C B is inconsistentwith B C.

I

I

I

The data do not violate the WARP but ...

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

I

I

I

- 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.

- Suppose we have the choice data satisfy the SARP.
- Then we can discover approximately where are the consumer’s indifference curves.
- How?

- 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)?

- The table showing direct preference revelations is:

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

- Indirect preference revelations add no extra information, so the table showing both direct and indirect preference revelations is the same as the table showing only the direct preference revelations:

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

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

x2

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).

E

B

D

A

C

x1

x2

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).

E

B

D

A

C

x1

Begin with bundles revealedto be less preferred than bundle A.

x2

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

A

x1

x2

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

A

x1

x2

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

A is directly revealed preferredto any bundle in

A

x1

x2

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

E

B

D

A

C

x1

x2

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

B

A

x1

x2

A is directly revealed preferred

to B and …

B

A

x1

x2

B is directly revealed preferredto all bundles in

B

x1

x2

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

B

x1

x2

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

B

A

x1

x2

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

E

B

D

A

C

x1

x2

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

A

C

x1

x2

A is directly revealedpreferred to C and ...

A

C

x1

x2

C is directly revealed preferredto all bundles in

C

x1

x2

so, by transitivity,Aisindirectly revealed preferredto all bundles in

C

x1

x2

soAis now revealed preferredto all bundles in the union.

B

A

C

x1

x2

soAis now revealed preferredto all bundles in the union.

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

B

A

C

x1

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

x2

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).

E

B

A

D

C

A

x1

x2

A: (p1,p2)=(1,1); (x1,x2)=(15,15)D: (p1,p2)=(2,5); (x1,x2)=(30,12).

A

D

x1

x2

D is directly revealed preferredto A.

A

D

x1

x2

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

A

D

x1

x2

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

D

x1

x2

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

D

As well, ...

x1

x2

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

D

x1

x2

bundles revealed to be strictly preferred to A

A

D

x1

x2

E

B

A

D

C

A

x1

x2

A: (p1,p2)=(1,1); (x1,x2)=(15,15)E: (p1,p2)=(5,2); (x1,x2)=(12,30).

E

A

x1

x2

E is directly revealed preferredto A.

E

A

x1

x2

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

E

A

x1

x2

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.

E

A

x1

x2

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.

E

A

As well, ...

x1

x2

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

E

A

x1

x2

More bundles revealed to be strictly preferred to A

E

A

x1

x2

Bundles revealedearlier as preferredto A

E

B

A

C

D

x1

x2

All bundles revealedto be preferred to A

E

B

A

C

D

x1

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

x2

All bundles revealedto be preferred to A

A

x1

All bundles revealed to be less preferred to A

x2

All bundles revealedto be preferred to A

A

x1

All bundles revealed to be less preferred to A

x2

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

A

x1

- Over time, many prices change. Are consumers better or worse off “overall” as a consequence?
- Index numbers give approximate answers to such questions.

- 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.

- 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).

- If (p1,p2) = (p1b,p2b) then we have the Laspeyres quantity index;

- If (p1,p2) = (p1t,p2t) then we have the Paasche quantity index;

- How can quantity indices be used to make statements about changes in welfare?

- If thenso consumers overall were better off in the base period than they are now in the current period.

- If thenso consumers overall are better off in the current period than in the base period.

- 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).

- If (x1,x2) = (x1b,x2b) then we have the Laspeyres price index;

- If (x1,x2) = (x1t,x2t) then we have the Paasche price index;

- How can price indices be used to make statements about changes in welfare?
- Define the expenditure ratio

- Ifthenso consumers overall are better off in the current period.

- But, ifthenso consumers overall were better off in the base period.

- 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.

- Since prices do not all increase at the same rate, relative prices change along with the “general price level”.
- A common proposal is to index fully Social Security payments, with the intention of preserving for the elderly the “purchasing power” of these payments.

- The usual price index proposed for indexation is the Paasche quantity index (the Consumers’ Price Index).
- What will be the consequence?

Notice that this index uses currentperiod prices to weight both base andcurrent period consumptions.

x2

Base period budget constraint

Base period choice

x2b

x1

x1b

x2

Base period budget constraint

Base period choice

x2b

Current period budgetconstraint before indexation

x1

x1b

x2

Base period budget constraint

Base period choice

Current period budgetconstraint after full indexation

x2b

x1

x1b

x2

Base period budget constraint

Base period choice

Current period budgetconstraint after indexation

x2b

Current period choiceafter indexation

x1

x1b

x2

Base period budget constraint

Base period choice

Current period budgetconstraint after indexation

x2b

Current period choiceafter indexation

x2t

x1

x1b

x1t

x2

(x1t,x2t) is revealed preferred to(x1b,x2b) so full indexation makesthe recipient strictly better off if

relative prices change betweenthe base and current periods.

x2b

x2t

x1

x1b

x1t

- So how large is this “bias” in the US CPI?
- A table of recent estimates of the bias is given in the Journal of Economic Perspectives, Volume 10, No. 4, p. 160 (1996). Some of this list of point and interval estimates are as follows:

- So suppose a social security recipient gained by 1% per year for 20 years.
- Q: How large would the bias have become at the end of the period?

- So suppose a social security recipient gained by 1% per year for 20 years.
- Q: How large would the bias have become at the end of the period?
- A: so after 20 years social security payments would be about 22% “too large”.