Chapter 7 Revealed Preference Economists infer from choices

• Chapter 7 Revealed Preference • Economists infer from choices a lot! free money on the floor my new office • Before, from w to choice • now from choice to w • this has policy content: household, university

Choices

• Chapter 7 Revealed Preference • Key Concept: what does choice reveal about preferences? • The weak axiom of revealed preference (WARP) • The strong axiom of revealed preference (SARP)

• Some maintained assumptions • A 1: The consumer’s preferences are stable over the time period for which we observe his/her choice behaviors. • A 2: There exists a unique demanded bundle for each budget set (easy to relax). • A 3: The consumer is always choosing the best she can afford (model of behavior).

• If (x 1, x 2) is chosen at (p 1, p 2, m), (y 1, y 2)≠ (x 1, x 2) and p 1 y 1+p 2 y 2≤m, then (x 1, x 2) is directly revealed preferred to (y 1, y 2). • Denote this by (x 1, x 2) d (y 1, y 2). • d is solely about choices though choices are related to preferences.

• (x 1, x 2) d (y 1, y 2). • d is solely about choices though choices are related to preferences.

• From revealed preference (d) to preference (w) • Suppose (x 1, x 2) d (y 1, y 2) and the consumer is choosing the best she can afford, then (x 1, x 2) s (y 1, y 2).

• The weak axiom of revealed preference (WARP) • If (x 1, x 2) d (y 1, y 2), then it cannot happen that (y 1, y 2) d (x 1, x 2).

• WARP is a weak and logical implication of consumers’ maximizing behaviors. • An example

obs p 1 p 2 x 1 x 2 1 1 2 2 2 1 3 1 1 2 2

Bundles Prices 1 2 3 1 5 4* 6 2 4* 5 6 3 3* 3* 4

• If (x 1, x 2) d (y 1, y 2) and (y 1, y 2) d (z 1, z 2), then we say that (x 1, x 2) is indirectly revealed preferred to (z 1, z 2). • Denote this by (x 1, x 2) id (z 1, z 2). • Allow indirect revealed preference for “chains” of observed choices longer than 3.

• If either (x 1, x 2) d (y 1, y 2) or (x 1, x 2) id (y 1, y 2), we say (x 1, x 2) is revealed preferred to (y 1, y 2). • Denote this by (x 1, x 2) r (y 1, y 2).

• Give an example to recover preferences. • How do we know whether the consumer is maximizing if we only observe choices? • We are questioning A 3 (the idea is A 1 and A 2 are OK).

• The strong axiom of revealed preference (SARP) • If (x 1, x 2) r (y 1, y 2), then it cannot happen that (y 1, y 2) r (x 1, x 2).

• SARP is a necessary and sufficient condition for optimizing behavior, but the proof is beyond the scope of this course. • Sufficiency: If choices satisfy SARP, then we can construct preferences for which the observed behavior is optimizing.

bundles prices 1 2 3 1 20 10* 22(**) 2 21(**) 20 15* 3 9* 15(**) 10

• Index numbers • Compare the consumption bundles of a consumer at two different times. • Let b stand for the base period. • Let t stand for some other period. • At t: prices (p 1 t, p 2 t), consumption (x 1 t, x 2 t) • At b: prices (p 1 b, p 2 b), consumption (x 1 b, x 2 b)

• Quantity index: compare the average consumption of these two periods, naturally could use the prices to be the weights • Laspeyres quantity index (use base price): Lq=(p 1 b x 1 t + p 2 b x 2 t)/(p 1 b x 1 b + p 2 b x 2 b), if Lq<1, at base price, base is chosen over t, so better off at base than at t (Lq>1? ) • Paasche quantity index (use t price): Pq=(p 1 t x 1 t + p 2 t x 2 t)/(p 1 t x 1 b + p 2 t x 2 b), if Pq>1, at t price, t is chosen over base, so better off at t than at base (Pq<1? )

• Price index: compare the average price of these two periods, naturally could use the quantities to be the weights • Laspeyres price index (use base q): Lp=(p 1 t x 1 b + p 2 t x 2 b)/(p 1 b x 1 b + p 2 b x 2 b) (wage adjustment) if Lp<1 (says nothing since prices different) • Paasche price index (use t q): Pp=(p 1 t x 1 t + p 2 t x 2 t)/(p 1 b x 1 t + p 2 b x 2 t) (GDP deflator)

• Define a new index of the change in total expenditure M=(p 1 t x 1 t + p 2 t x 2 t)/(p 1 b x 1 b + p 2 b x 2 b). • Lp<M: p 1 t x 1 b + p 2 t x 2 b <p 1 t x 1 t + p 2 t x 2 t, t period is better than base (intuitively, when income grows faster than prices, better off after this change) • Pp>M: p 1 b x 1 b + p 2 b x 2 b> p 1 b x 1 t + p 2 b x 2 t, base is better than t (intuitively, when prices grow faster than income, worse off)

• Social security: indexing so that base consumption is still affordable, then t cannot be worse than base

• Chapter 7 Revealed Preference • Key Concept: what does choice reveal about preferences? • The weak axiom of revealed preference (WARP) • The strong axiom of revealed preference (SARP)