Revealed Preference 2011

Transcription

1 Revealed Preference 2011

2 Motivation: 1. up until now we have started with preference and then described behaviour 2. revealed preference works backwards - start with behaviour and describe preferences 3. recovering preferences - how to use observed choices to estimate the indifference curves 1

3 Revealed Preference - a tool for testing theories of optimizing behavior Given a vector of prices p t and choices x t at time t, we know that the bundle x t is preferred to another bundle x if x was affordable when x t was chosen: p t x t p t x 2

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5 Transitivity of preferences one can string together chains of these inequalities to rank bundles and bound possible indifference curves that could have generated this data. Of course, if these chains of inequalities cannot all be mutually satisfied, then the choice data fail to conform with a model of utility maximization. Hence, revealed preference is both a descriptive and a diagnostic tool. 4

6 Definition - Directly Revealed Preferred: x t is directly revealed preferred to x if p t x t p t x, and is strictly directly revealed preferred if p t x t > p t x Definition - Revealed Preferred: x t is revealed preferred to x if there is a chain of directly revealed preferred bundles linking x t to x 5

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8 Definition - Weak Axiom of Revealed Preference (WARP): If x t is directly revealed preferred to x, then x is not directly revealed preferred to x t WARP: If p 1 x 1 + p 2 x 2 p 1 y 1 + p 2 y 2, then it must happen that q 1 y 1 + q 2 y 2 q 1 x 1 + q 2 x 2 WARP is a necessary condition for behaviour to be consistent with utility maximization 7

9 Violation of WARP 8

10 Definition - Strong Axiom of Revealed Preference (SARP): If x t is revealed preferred to x, then x is not revealed preferred to x t SARP is a necessary and sufficient condition for utility maximization If the consumer is maximizing utility, then his behaviour must be consistent with SARP If his observed behaviour is consistent with SARP, then we can always find a utility function that explains the behaviour of the consumer as maximizing behaviour. 9

11 Index Numbers Suppose (p t 1, pt 2 ) are time t prices of goods 1 and 2, and the consumer chooses the bundle (x t 1, xt 2 ). Suppose in some base year b, prices were (p b 1, pb 2 ), and the consumer chose (x b 1, xb 2 ). How has the average consumption of the consumer changed? Quantity index I q = w 1x t 1 +w 2x t 2 w 1 x b 1 +w 2x b, 2 where w 1 and w 2 are some weights that go into making an average. 10

12 If I q > 1, average consumption has gone up from b to t. If I q < 1, the reverse is true. If the weights are base period weights we get Laspeyres index, and with period t prices, we get the Paasche index. Paasche quantity index P q = pt 1 xt 1 +pt 2 xt 2 p t 1 xb 1 +pt 2 xb 2 Laspeyres quantity index L q = pb 1 xt 1 +pb 2 xt 2 p b 1 xb 1 +pb 2 xb 2 11

13 Suppose P q > 1. This p t 1 xt 1 + pt 2 xt 2 > pt 1 xb 1 + pt 2 xb 2 Implication: Consumer must be better-off in period t, relative to period b, since he could have consumed the b-period bundle in period-t prices but chose not to. If P q < 1, all we know is that when he chose the (x t 1, xt 2 ), bundle (xb 1, xb 2 ) was not affordable - but this does not say anything about ranking of bundles. 12

14 If L q < 1, (x b 1, xb 2 ) is revealed preferred to (x t 1, xt 2 ) - consumer better off in period b than t. 13

15 Price Indices Price indices are weighted average of prices. Natural to choose quantities as weights for computing the average. If we choose periodt quantities, we get the Paasche price index: P p = pt 1 xt 1 +pt 2 xt 2 p b 1 xt 1 +pb 2 xt 2 Base period quantities gives us Laspeyres price index. L p = pt 1 xb 1 +pt 2 xb 2 p b 1 xb 1 +pb 2 xb 2 14

16 Define an index of change in total expenditure M = pt 1 xt 1 +pt 2 xt 2 p b 1 xb 1 +pb 2 xb 2 Ratio of total expenditure in period t to total expenditure in period b. Suppose P p > M. This pt 1 xt 1 +pt 2 xt 2 p b 1 xt 1 +pb 2 xt 2 > pt 1 xt 1 +pt 2 xt 2 p b 1 xb 1 +pb 2 xb 2 Simplifying, p b 1 xb 1 + pb 2 xb 2 > pb 1 xt 1 + pb 2 xt 2 If Paasche index > expenditure index, consumer better off in year b than in year t 15

17 If Laspeyres index < expenditure index, consumer better off in year t than in year b For price indices, what matters is whether they are greater or less than expenditure index. Intuitively, if prices rise more than income, consumer is worse off and vice versa. 16