14.127 Lecture 4 Xavier Gabaix February 26, 2004
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.
Three directions: Analytical models Don t 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.
Psychological models Those models are descriptively rich, but they are unsystematic, and often hard to use.
Human computer comparison Human mind 10 15 operations per second Computer 10 12 operations per second Moore s 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, we ll need to understand how human think
1.1 Analytical models 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 don t 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
I change my decision from a 0 to a = arg maxu (a, θ t ) iff u (a, θ t ) u (a 0, θ t ) > κ
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 What s the demand function? max q i + σ i ε i i
Definition. The Gumbel distribution G is F (x) = P (ε < x) = e e x and have density f (x) = F (x) = e e x x.
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 P with q defined as e q = 1 and η is a Gumbel. ( ) max ε i + q i ln n + q + x = e e x i=1,..,n M n = n e q i.this means that max ε i + q i = d ln n + q + η i=1,..,n
Proof of Proposition 1. ( ) Call I = P max i=1,..,n ε i + q i y. Then I = P (( i) ε i + q i y) = Π n i=1 P (ε i + q i y) Thus, and ln I = P (ε i + q i y) ln P (ε i + q i y) = ln P (ε i y q i ) = e (y q i).
Thus ln I = e (y q i ) = e y e q i Using e q 1 e q i = n we have ln I = e y ne q = e [y ln n q ] which proves that I is a Gumbel. QED
Demand with noise Demand for good n + 1 equals D n+1 (q 1,..., q n+1 ) = P ( ) max ε n+1 + q n+1 > i=1,..,n ε i + q i where q i is total quality, including the disutility of price. Proposition 2. In general, e q n+1 D n+1 (q 1,..., q n+1 ) = n+1. i=1 eq i ( ) e q j D j = P ε j + q j > max ε i + q i = n+1 i=j i=1 eq i
Proof of Proposition 2. Observe that n+1 j=1 D j = 1. Note D n+1 (q 1,..., q n+1 ) = P ( ) ε n+1 > max ε i + q i=1,..,n i where q = q i q n+1. i Thus, D n+1 (q 1,..., q n+1 ) = Ee e (ε n+1 ln n q )
Call a = ln n q. Then D n+1 (q 1,..., q n+1 ) = Ee e (ε n+1 +a) = e e (x+a) f (x) dx = e e (x+a) e e x x dx = e e (x+a) e x x dx = e e x (e a +1) x dx Call H = 1 + e a and re write the above equation as D n+1 (q 1,..., q n+1 ) = = e e x ln H x dx e e x ln H (x ln H) ln H dx e
Note that b [ ] b a e e y y dy = e e y a Thus QED D n+1 (q 1,..., q n+1 ) = e ln H [ e e x ln H] + dx 1 1 = 1 = = = H 1 + e a 1 + e ln n+q 1 + ne q 1 e q n+1 e q n+1 = = = i e q n+1 + e q n+1 i n =1 eq n+1 1 + i i n =1 eq i=1 eq i 1
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.
Dividing total quality into quality and price components ( ) D 1 = P q 1 p 1 + σε 1 > max i i=2,...,n q p i + σε i where ε i are iid Gumbel, σ > 0. Then D 1 = P ( q 1 p 1 σ q ) 1 p 1 q i e + ε1 > max i p + εi σ i=2,...,n σ σ i=1 e = n q i p i This is very often used in IO.
Optimal pricing. An application example Suppose we have n firms, n 1. Firm i has cost c i and does max (p i c i ) D i (p 1,..., p n ) = π i i
Denote the profit by π i and note that and i ln π i = ln (p i c i ) + q ( i p + ln 1 1 ln π i = p i c σ + e σ ( ) q i p i p i i e σ ) ( 1 1 1 = pi c i σ + O n σ e q j p j q j p j σ )
So and unit profits p i 1 1 c i σ 0 p i c i = σ Thus decision noise is good for firms profits. See Gabaix Laibson Competition 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.
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 ( ) lim P max ε i A n + B n x = e e x. n i=1,...,n 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.
Fact 3 Power law distributions (P (ε > x) x ζ for some ζ > 0) are not in this domain.
Lemma 1. For distributions in the domain of attraction of the Gumbel F (x) = P (ε < x) take F = 1 F (x) = P (ε x), and f = F. Then A n, B n are given by F (A n ) = 1 n 1 B n = nf (An ) Lemma 2 with ( ) lim P max ε i + q i A n + B n y + q = e e y n i=1,...n n e q n /B n 1 = e q i /B n n
Proposition. ( ) D 1 = P q 1 p 1 + σε 1 > max i i=2,...,n q p i + σε i For n, lim D 1 /D 1 = 1where D 1 = e q 1 p 1 /B n σ σ n i=1 e q i p i /B n σ D 1. σ
Example. Exponential distribution f (x) = e (x+1) for x > 1 and equals 0 for x 1. then, for x > 1 = Thus F (x) = P (ε > x) = [ ] x e (x+1) dy e (x+1) = e (x+1) = f (x). x and F (A n ) = 1, n A n = 1 + ln n and B n = nf ( 1 A n ) = 1
Example 2. Gaussian. f (x) =, F (x) = e s2 2 x ds. For large x, 2π 1 the cumulative F (x) e x2 2. 2 πx Resu lt A n q ln n 1 B n 2 ln n
Optimal prices satisfy q i p i max (p i c i ) e B nσ i q j p j e Bnσ = max (p i c i ) D 1 = π i Same as for Gumbel with σ = B n σ. Thus p i c i = B n σ
Examples Gumbel p i c i = σ Exponential noise p i c i = σ Gaussian p i c = i 1 σ 2 ln n and competition almost does not decrease markup (beyond markup when there are already some 20 firms).
Example. Mutual funds market. Around 10,000 funds. Fidelity alone has 600 funds. Lots of fairly high fees. Entry fee 1 2%, every year management fee of 1 2% and if you quit exit fee of 1 2%. 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.
Is it true that if competition increases then price goes always down? l Not always. For lognormal noise B n e n n and so p i c i = σe ln n.
1.1.2 Implications for welfare measurement (sketch) Assume no noise and rational consumers. Introduce a new good which gets an amount of sales Q = pd where D is demand and p is price.. The welfare increase is ψ (η) Q where η is the elasticity of demand, the utility of consuming D is D 1 1 η, and ψ (η) denotes η 1. 1
If there is confusion, the measured elasticity ηˆ is less than the true elasticity as ln D i 1 = ηˆ p i σb n Thus, the imputed welfare gain ψ (ηˆ) Q will be bigger than the true welfare gain ψ (η) Q.