Chapter Seven Revealed Preference

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

Direct Preference Revelation x2x2 x1x1 x*x* y The chosen bundle x * is revealed directly as preferred to the bundles y and z. z

Direct Preference Revelation u That a bundle x is revealed directly as preferred to another bundle y will be written as x y. D 

Indirect Preference Revelation u Suppose that x is revealed directly preferred to y, and that y is revealed directly preferred to z. Then, by transitivity, x is revealed indirectly as preferred to z. That x is revealed indirectly as preferred to z will be written as x z so x y and y z x z. D  D  I  I 

Indirect Preference Revelation x2x2 x1x1 x* z z is not affordable when x* is chosen.

The Weak Axiom of Revealed Preference (WARP) u The WARP is: If the bundle x is revealed directly as preferred to the bundle y then it is never the case that the bundle y is revealed directly as preferred to the bundle x; i.e. x y not (y x). D  D 

The Weak Axiom of Revealed Preference (WARP) u What sort of choice data would violate the WARP?

The Weak Axiom of Revealed Preference (WARP) x2x2 x1x1 x y These statements are inconsistent with each other. x is chosen when y is available so x y. y is chosen when x is available so y x. D  D 

Recovering Indifference Curves u Suppose we have a set of choice data which satisfy the SARP. u Then we can discover approximately where are the consumer’s indifference curves. u How?

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 There are two basic types of indices –price indices, and –quantity indices u Each type of index compares expenditures in a base period and in a current period by taking the ratio of these 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 u 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 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? x2x2 x1x1 x2bx2b x1bx1b Base period budget constraint Base period choice Current period budget constraint before indexation

Full Indexation? x2x2 x1x1 x2bx2b x1bx1b Base period budget constraint Base period choice Current period budget constraint after full indexation

Full Indexation? x2x2 x1x1 x2bx2b x1bx1b Base period budget constraint Base period choice Current period choice after indexation Current period budget constraint after indexation x2tx2t x1tx1t

Full Indexation? x2x2 x1x1 x2bx2b x1bx1b x2tx2t x1tx1t (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.