Revealed Preference Analysis u Suppose we observe the

Revealed Preference Analysis u Suppose we observe the demands (consumption choices) a consumer makes for different prices. u This reveals information about his/her preferences. We can use this information to. . .

Revealed Preference Analysis – Test the behavioral hypothesis that a consumer is rational (chooses the most preferred bundle from those available). – Infer about the consumer’s preference relation.

Assumptions on Preferences u Consumer preferences: – do not change while the choice data are gathered. – are strictly convex. – are monotonic. u Together, convexity and monotonicity imply that the most preferred affordable bundle is unique.

Assumptions on Preferences x 2 If preferences are convex and monotonic (i. e. well-behaved), then the most preferred affordable bundle is unique. x 2* x 1

Direct Preference Revelation u Suppose that the bundle x* is chosen when the bundle y is affordable. Then x* is revealed directly as preferred to y (given uniqueness) u Otherwise, chosen. y would have been

Direct Preference Revelation x 2 The chosen bundle x* is revealed directly as preferred to the bundles y and z. x* y z x 1

Direct Preference Revelation u That x is revealed directly as preferred to y will be written as x p D y

Indirect Preference Revelation u 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, written as x z p I p p D y and y D z p i. e. , x x I z

Indirect Preference Revelation z is not affordable when x* is chosen. x* is not affordable when y* is chosen. So x* and z cannot be compared directly. z But x* D y* and y* z, so x* I z D x 1 p x* y* p p x 2

Two Axioms of Revealed Preference To apply revealed preference analysis, choices must satisfy two criteria: u Weak Axioms of Revealed Preference u Strong Axioms of Revealed Preference

The Weak Axiom of Revealed Preference (WARP) u 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. p D y ~ (y p x D x)

The Weak Axiom of Revealed Preference (WARP) u Choice data which violate the WARP are inconsistent with economic rationality. u The WARP is a necessary condition for applying economic rationality to explain observed choices.

The Weak Axiom of Revealed Preference (WARP) u What observed choices violate the WARP?

The Weak Axiom of Revealed Preference (WARP) p y. p x 2 x. x is chosen when y is available, so x y is chosen when x is available, so y y D D These choices are inconsistent with each other! x x 1

Checking if Data Violate the WARP u. A consumer makes the following choices at prices (p 1, p 2): – At ($2, $2): demand (x 1, x 2) = (10, 1) – At ($2, $1): demand (x 1, x 2) = (5, 5) – At ($1, $2): demand (x 1, x 2) = (5, 4) u Is WARP violated by these choices?

Checking if Data Violate the WARP * Red numbers are costs of chosen bundles.

Checking if Data Violate the WARP * Circles indicates affordable bundles that were not chosen.

Checking if Data Violate the WARP

Checking if Data Violate the WARP

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). => WARP is violated.

Checking if Data Violate the WARP x 2 p D (10, 1) p (5, 4) D (5, 4) x 1

The Strong Axiom of Revealed Preference (SARP) u 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 p p ~(y x or y D I p p D I x ).

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

SARP u Consider the following data: A: (p 1, p 2, p 3)=(1, 3, 10) & (x 1, x 2, x 3)=(3, 1, 4) B: (p 1, p 2, p 3)=(4, 3, 6) & (x 1, x 2, x 3)=(2, 5, 3) C: (p 1, p 2, p 3)=(1, 1, 5) & (x 1, x 2, x 3)=(4, 4, 3)

SARP A: ($1, $3, $10) (3, 1, 4) B: ($4, $3, $6) (2, 5, 3) C: ($1, $5) (4, 4, 3)

SARP In situation A, bundle A is directly revealed preferred to bundle C: p A D C

SARP In situation B, bundle B is directly revealed preferred to bundle A: p B D A

SARP In situation C, bundle C is directly revealed preferred to bundle B: p C D B

SARP The data do not violate the WARP.

SARP Given that p p D C, B D p A A and C D B I next, by transitivity, p p I B, B I p A I C and C I A. I

SARP D A is inconsistent p p B with A I I B. I I

SARP D C is inconsistent p p A with C I A. I I I

SARP D B is inconsistent p p C with B I C. I I I

SARP The data do not violate the WARP, but there are 3 violations of the SARP. I I I

SARP u That the observed choice data satisfy the SARP is a condition necessary and sufficient for there being a well-behaved preference relation that can rationalize the data. u So our data cannot be rationalized by a well-behaved preference relation.

Recovering Indifference Curves u Suppose we have a set of choice data that satisfy the SARP (and hence the WARP). u Then we can decide approximately where consumer’s ICs are.

Recovering Indifference Curves u Suppose we observe: A: (p 1, p 2)=($1, $1) & (x 1, x 2)=(15, 15) B: (p 1, p 2)=($2, $1) & (x 1, x 2)=(10, 20) C: (p 1, p 2)=($1, $2) & (x 1, x 2)=(20, 10) D: (p 1, p 2)=($2, $5) & (x 1, x 2)=(30, 12) E: (p 1, p 2)=($5, $2) & (x 1, x 2)=(12, 30) u Where lies the IC containing bundle A = (15, 15)?

Recovering Indifference Curves u The following table shows the direct preference revelations:

Recovering Indifference Curves Direct revelations only; the WARP is not violated by the data.

Recovering Indifference Curves u Indirect preference revelations add no extra information. u So the table showing both direct and indirect preference revelations is the same as the table showing only the direct preference revelations.

Recovering Indifference Curves * Both direct and indirect revelations; neither WARP nor SARP is violated.

Recovering Indifference Curves u Since the choices satisfy the SARP, there is a well-behaved preference relation that rationalizes the choices.

Recovering Indifference Curves x 2 A: (p 1, p 2)=(1, 1); (x 1, x 2)=(15, 15) B: (p 1, p 2)=(2, 1); (x 1, x 2)=(10, 20) C: (p 1, p 2)=(1, 2); (x 1, x 2)=(20, 10) D: (p 1, p 2)=(2, 5); (x 1, x 2)=(30, 12) E: (p 1, p 2)=(5, 2); (x 1, x 2)=(12, 30) E B A C D x 1

Recovering Indifference Curves x 2 A: (p 1, p 2)=(1, 1); (x 1, x 2)=(15, 15) B: (p 1, p 2)=(2, 1); (x 1, x 2)=(10, 20) C: (p 1, p 2)=(1, 2); (x 1, x 2)=(20, 10) D: (p 1, p 2)=(2, 5); (x 1, x 2)=(30, 12) E: (p 1, p 2)=(5, 2); (x 1, x 2)=(12, 30) E B A C D x 1 * Begin with bundles revealed to be less preferred than A.

Recovering Indifference Curves x 2 A: (p 1, p 2)=(1, 1); (x 1, x 2)=(15, 15) A is directly revealed preferred to any bundle in A x 1

Recovering Indifference Curves x 2 A: (p 1, p 2)=(1, 1); (x 1, x 2)=(15, 15) B: (p 1, p 2)=(2, 1); (x 1, x 2)=(10, 20) E B A C D x 1

Recovering Indifference Curves x 2 A: (p 1, p 2)=(1, 1); (x 1, x 2)=(15, 15) B: (p 1, p 2)=(2, 1); (x 1, x 2)=(10, 20) B A x 1

Recovering Indifference Curves x 2 A is directly revealed preferred to B B A x 1

Recovering Indifference Curves x 2 B is directly revealed preferred to all bundles in B x 1

Recovering Indifference Curves x 2 B By transitivity, A is indirectly revealed preferred to all bundles in x 1

Recovering Indifference Curves x 2 A is now revealed preferred to all bundles in the union. B A x 1

Recovering Indifference Curves x 2 A: (p 1, p 2)=(1, 1); (x 1, x 2)=(15, 15) C: (p 1, p 2)=(1, 2); (x 1, x 2)=(20, 10) E B A C D x 1

Recovering Indifference Curves x 2 A: (p 1, p 2)=(1, 1); (x 1, x 2)=(15, 15) C: (p 1, p 2)=(1, 2); (x 1, x 2)=(20, 10) A C x 1

Recovering Indifference Curves x 2 A is directly revealed preferred to C A C x 1

Recovering Indifference Curves x 2 C is directly revealed preferred to all bundles in C x 1

Recovering Indifference Curves x 2 By transitivity, A is indirectly revealed preferred to all bundles in C x 1

Recovering Indifference Curves x 2 A is now revealed preferred to all bundles in the union. B A C x 1

Recovering Indifference Curves x 2 So A is now revealed preferred to all bundles in the union. Therefore the indifference curve containing A must lie everywhere else above this shaded set. B A C x 1

Recovering Indifference Curves u Next, what about the bundles revealed as more preferred than A?

Recovering Indifference Curves x 2 A: (p 1, p 2)=(1, 1); (x 1, x 2)=(15, 15) B: (p 1, p 2)=(2, 1); (x 1, x 2)=(10, 20) C: (p 1, p 2)=(1, 2); (x 1, x 2)=(20, 10) D: (p 1, p 2)=(2, 5); (x 1, x 2)=(30, 12) E: (p 1, p 2)=(5, 2); (x 1, x 2)=(12, 30) E B A A C D x 1

Recovering Indifference Curves x 2 A: (p 1, p 2)=(1, 1); (x 1, x 2)=(15, 15) D: (p 1, p 2)=(2, 5); (x 1, x 2)=(30, 12) A D x 1

Recovering Indifference Curves x 2 D is directly revealed preferred to A. A D x 1

Recovering Indifference Curves x 2 D is directly revealed preferred to A. Well-behaved preferences are convex A D x 1

Recovering Indifference Curves x 2 D is directly revealed preferred to A. Well-behaved preferences are convex, so all bundles on the line between A and D are also preferred to A. A D x 1

Recovering Indifference Curves x 2 All bundles containing the same amount of commodity 2 and more of commodity 1 than D are preferred to D, and therefore Are also preferred to A. A D x 1

Recovering Indifference Curves x 2 Bundles revealed to be strictly preferred to A A D x 1

Recovering Indifference Curves x 2 A: (p 1, p 2)=(1, 1); (x 1, x 2)=(15, 15) B: (p 1, p 2)=(2, 1); (x 1, x 2)=(10, 20) C: (p 1, p 2)=(1, 2); (x 1, x 2)=(20, 10) D: (p 1, p 2)=(2, 5); (x 1, x 2)=(30, 12) E: (p 1, p 2)=(5, 2); (x 1, x 2)=(12, 30) E B A A C D x 1

Recovering Indifference Curves x 2 E A: (p 1, p 2)=(1, 1); (x 1, x 2)=(15, 15) E: (p 1, p 2)=(5, 2); (x 1, x 2)=(12, 30) A x 1

Recovering Indifference Curves x 2 E is directly revealed preferred to A. E A x 1

Recovering Indifference Curves x 2 E is directly revealed preferred to A. Well-behaved preferences are E convex A x 1

Recovering Indifference Curves x 2 E A E is directly revealed preferred to A. Well-behaved preferences are convex so all bundles on the line between A and E are preferred to A also. x 1

Recovering Indifference Curves x 2 E A E is directly revealed preferred to A. Well-behaved preferences are convex so all bundles on the line between A and E are preferred to A also. As well, . . . x 1

Recovering Indifference Curves x 2 all bundles containing the same amount of commodity 1 and more of commodity 2 than E are preferred to E and therefore are preferred to A also. E A x 1

Recovering Indifference Curves x 2 More bundles revealed to be strictly preferred to A E A x 1

Recovering Indifference Curves x 2 E B A Bundles revealed earlier as preferred to A C D x 1

Recovering Indifference Curves x 2 E All bundles revealed to be preferred to A B A C D x 1

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

Recovering Indifference Curves x 2 All bundles revealed to be preferred to A A x 1 All bundles revealed to be less preferred to A

Recovering Indifference Curves x 2 All bundles revealed to be preferred to A A x 1 All bundles revealed to be less preferred to A

Recovering Indifference Curves x 2 The region in which the indifference curve containing bundle A must lie. A x 1

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

Index Numbers u Two basic types of indices – price indices, and – quantity indices u Each index compares expenditures in a base period and in a current period by taking the ratio of expenditures.

Quantity Index Numbers u. A quantity index is a price-weighted average of quantities demanded; i. e. u (p 1, p 2) can be base period prices (p 1 b, p 2 b) or current period prices (p 1 t, p 2 t).

Quantity Index Numbers u If (p 1, p 2) = (p 1 b, p 2 b) then we have the Laspeyres quantity index;

Quantity Index Numbers u If (p 1, p 2) = (p 1 t, p 2 t) then we have the Paasche quantity index;

Quantity Index Numbers u How can quantity indices be used to make statements about changes in welfare?

Quantity Index Numbers If , then so consumers overall were better off in the base period than they are now in the current period.

Quantity Index Numbers u If then so consumers overall are better off in the current period than in the base period.

Price Index Numbers u. A price index is a quantity-weighted average of prices; i. e. u (x 1, x 2) can be the base period bundle (x 1 b, x 2 b) or else the current period bundle (x 1 t, x 2 t).

Price Index Numbers u If (x 1, x 2) = (x 1 b, x 2 b) then we have the Laspeyres price index;

Price Index Numbers u If (x 1, x 2) = (x 1 t, x 2 t) then we have the Paasche price index;

Price Index Numbers u How can price indices be used to make statements about changes in welfare? u Define the expenditure ratio

Price Index Numbers u If then so consumers overall are better off in the current period.

Price Index Numbers u But, if then so consumers overall were better off in the base period.

Full Indexation? u Changes in price indices are sometimes used to adjust wage rates or transfer payments. This is called indexation. u 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.

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

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

Full Indexation? Notice that this index uses current period prices to weight both base and current period consumptions.

Full Indexation? x 2 Base period budget constraint Base period choice x 2 b x 1

Full Indexation? x 2 Base period budget constraint Base period choice x 2 b Current period budget constraint before indexation x 1 b x 1

Full Indexation? x 2 Base period budget constraint Base period choice Current period budget constraint after full indexation x 2 b x 1

Full Indexation? x 2 Base period budget constraint Base period choice Current period budget constraint after indexation x 2 b Current period choice after indexation x 1 b x 1

Full Indexation? x 2 Base period budget constraint Base period choice Current period budget constraint after indexation x 2 b x 2 Current period choice after indexation t x 1 b x 1 t x 1

Full Indexation? x 2 (x 1 t, x 2 t) is revealed preferred to (x 1 b, x 2 b) so full indexation makes the recipient strictly better off if relative prices change between the base and current periods. x 2 b x 2 t x 1 b x 1 t x 1

Full Indexation? u How large is this “bias” in the US CPI? u. A table of recent estimates of the bias is given in JPE (Journal of Economic Perspectives, Volume 10, No. 4, p. 160, 1996). Some of this list of point/interval estimates are:

Full Indexation?

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