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14.127 Lecture 4 Xavier Gabaix February26, 2004

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1 Bounded Rationality Three reasons to study: Hope that it will generate a unified framework for behavioral economics Some phenomena should be captured: difficult-easy difference. It would be good to have a metric for that Artificial intelligence Warning a lot of effort spend on bounded rationality since Simon and few results.

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Three directions: Analytical models Dont get all the fine nuances of the psychology, but those models are tractable. Process models, e.g. artificial intelligence Rubinstein direction. Suppose we play Nash, given your reaction function, my strategy optimizes on both outcome and computing cost. Rubinstein proves some existence theorems. But it is very difficult to apply his approach.

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Psychological models Those models are descriptively rich, but they are unsystematic, and often hard to use.

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Human - computer comparison Human mind 10 15 operations per second Computer 10 12 operations per second Moores law: every 1.5 years computer power doubles Thus, every 15 years computer power goes up 10 3 If we believe this, then in 45 years computers can be 10 6 more powerful than humans Of course, well need to understand how human think

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1.1 Analyticalmodels Bounded Rationality as noise. Consumer sees a noisy signal q˜= q +σε of quantity/quality q. Bounded Rationality as imperfect monitoring of the state of the world. People dont think about the variables all the time. They look up variable k at times t 1,..., t n Bounded Rationality as adjustment cost. Call by θ the parameters of the world. Now I am doing a 0 and κ = cost of decision/change

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I change my decision from a 0 to a = arg max u(a,θ t ) iff u(a,θ t )u(a 0,θ t )> κ

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1.1.1 Model of Bounded Rationality as noise Random utility model Luce (psychologist) and McFadden (econometrician )who provided econometric tools for the models) n goods, i = 1,..., n. Imagine the consumer chooses Whats the demand function?

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Definition. The Gumbel distribution G is and have density

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If ε has the Gumbel distribution then Eε = γ > 0, where γ 0.59 is the Euler constant. Proposition 1. Suppose ε i are iid Gumbel. Then with q defined as This means that and η is a Gumbel.

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Proof of Proposition 1. Call Then Thus, and

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Thus Using we have which proves that I is a Gumbel. QED

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Demand with noise Demand for good n +1 equals where q i is total quality, including the disutility of price. Proposition 2. In general,

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Proof of Proposition 2. Observe that Note where Thus,

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and rewrite the above equation as Call Then

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Thus QED

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Demand with noise cont. This is called discrete choice theory. It is exact for Gumbel. It is asymptotically true for almost all unbounded distributions you can think off like Gaussian, lognormal, etc.

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Dividing total quality into quality and price components where ε i are iid Gumbel, Then This is very often used in IO.

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Optimal pricing. An application example Suppose we have n firms, Firm i has cost c i and does

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Denote the profit by π i and note that and

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So and unit profits Thus decision noise is good for firms profits. See Gabaix-Laibson Com-petition and Consumer Confusion Evidence: car dealers sell cars for higher prices to women and minorities than to white men. Reason: difference in expertise. There is lots of other evidence of how firms take advantage of consumers. See paper by Susan Woodward on mortgage refinancing markets: unsophisticated people are charged much more than sophisticated people.

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What about non-Gumbel noise? Definition. A distribution is in the domain of attraction of the Gumbel if and only if there exists constants A n, B n such that for any x when ε i are iid draws from the given distribution. Fact 1. The following distributions are in the domain of attraction of a Gamble: Gaussian, exponential, Gumbel, lognormal, Weibull. Fact 2. Bounded distributions are not in this domain.

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Fact 3 Power law distributions (P(ε>x) xζ for some ζ > 0) are not in this domain.

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Lemma 1. For distributions in the domain of attraction of the Gumbel takeand Then A n, B n are given by Lemma 2 with

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Proposition For where

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Example. Exponential distribution equals and for Thus and then,

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Example 2. Gaussian.For large the cumulative Result

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Optimal prices satisfy Same as for Gumbel with Thus

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Examples Gumbel Exponential noise Gaussian and competition almost does not decrease markup (beyond markup when there are already some 20 firms).

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Example. Mutual funds market. Around 10,000 funds. Fidelity alone has 600 funds. Lots of fairly high fees. Entry fee 12%, every year management fee of 12% and if you quit exit fee of 12%. On the top of that the manager pays various fees to various brokers, that is passed on to consumers. The puzzle how all those markups are possible with so many funds? Part of the reason for that many funds is that Fidelity and others have incubator funds. With large probability some of them will beat the market ten years in a row, and then they can propose them to unsophisticated consumers.

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Is it true that if competition increases then price goes always down? Not always. For lognormal noiseand so

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1.1.2 Implications for welfare measurement (sketch) Assume no noise and rational consumers. Introduce a new good which gets an amount of sales where D is demand and p is price.. The welfare increase is where η is the elasticity of demand, the utility of consuming D is and denotes

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will be bigger than the true welfare Thus, the imputed welfare gain gain If there is confusion, the measured elasticityis less than the true elasticity as

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