14.13 Lecture 8. Xavier Gabaix. March 2, 2004

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1 14.13 Lecture 8 Xavier Gabaix March 2, 2004

2 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: there are many attempts but none is developed in cumulative fashion.

3 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 (Modelling bounded rationality, MIT Press) 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.

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

5 Human - computer comparison (see Kurzweil, The Age of Spritual Machine) Human mind operations per second Computer 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

6 1.1 Analytical models Bounded Rationality as noise. Consumer sees a noisy signal q = q + σε of quantity/quality q, noise σε has standard deviation σ and mean 0. 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.

7 Bounded Rationality as adjustment cost. Let θ denote the state of the world. Now I am doing a 0 and κ = cost of decision/change I change my decision from a 0 to a =argmaxu (a, θ t ) iff u (a,θ t ) u (a 0,θ t ) >κ

8 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 i q i + σ i ε i

9 Definition. The Gumbel distribution G is F (x) =P (ε <x)=e e x and have density f (x) =F 0 (x) =e e x x.

10 If ε has the Gumbel distribution then Eε = γ>0, whereγ ' is the Euler constant. Proposition 1. Suppose ε i are iid Gumbel. Then P Ã max ε i + q i ln n + qn + x i=1,..,n! = e e x with q n defined as e q n = 1 n P e q i.this means that and η is a Gumbel. M n = max ε i + q i = d ln n + qn + η i=1,..,n

11 ProofofProposition1. Call I = P ³ 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, ln I = X P (ε i + q i y) and ln P (ε i + q i y) =lnp (ε i y q i )= e (y qi).

12 Thus ln I = X e (y q i) = e y X e q i Using e q n = 1 n X e q i we have ln I = e y ne q n = e [y ln n q n] which proves that I is a Gumbel. QED

13 Demand with noise Demand for good n +1equals D n+1 (q 1,...,q n+1 ) = P Ã ε n+1 + q n+1 > where q i is total quality, including the disutility of price. max i=1,..,n ε i + q i! Proposition 2. In general, D j = P D n+1 (q 1,...,q n+1 )= eq n+1 P n+1. i=1 eq i Ã ε j + q j > max i6=j ε i + q i! = e q j P n+1 i=1 eq i

14 ProofofProposition2. Observe that P n+1 j=1 D j =1. Note D n+1 (q 1,..., q n+1 )=P Ã ε n+1 > max i=1,..,n ε i + q 0 i! where q 0 i = q i q n+1. Thus, D n+1 (q 1,..., q n+1 )=Ee e (ε n+1 ln n q n)

15 Call a = ln n q n.then D n+1 (q 1,..., q n+1 )=Ee e (ε n+1 +a) = = Z Z e e (x+a) f (x) dx = Z e e (x+a) e x x dx = e e (x+a) e e x x dx Z 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 ) = = Z Z e e (x ln H) x dx e e (x ln H) (x ln H) e ln H dx

16 Note that Z b a e e y y dy = e e y b a Thus D n+1 (q 1,..., q n+1 )=e ln H e x ln H + e = = QED dx = 1 H 1 1+e a = 1 1+e ln = 1 n+q n 1+ne = 1 q n 1 1+ P n i=1 e q i q n+1 = 1+ P n i=1 e q0 i e q n+1 e q n+1 + e q n+1 P n i=1 e q = eq n+1 i q n+1 P n+1 i=1 eq i

17 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.

18 Dividing total quality into quality and price components D 1 = P Ã q 1 p 1 + σε 1 > where ε i are iid Gumbel, σ>0. max i=2,...,n q i p i + σε i! Then D 1 = P Ã q1 p 1 σ + ε 1 > max i=2,...,n q i p i σ + ε i! = eq 1 p 1 σ P ni=1 e q i p i σ This is very often used in IO.

19 Optimal pricing. An application example Supposewehaven firms, n À 1. Firm i has cost c i and does max p i (p i c i ) D i (p 1,..., p n )=π i

20 Denote the profit byπ i and note that and ln π i =ln (p i c i ) e q 1 p 1 σ P ni=1 e q i p i σ =ln(p i c i )+ q i p i σ p i ln π i = 1 ln p i c i 1 σ e nx j=1 = 1 p i c i 1 σ + O µ 1 n qi p ³ i σ P e q j p j σ q j p j e σ

21 So and unit profits 1 p i c i 1 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 chargedmuchmorethansophisticatedpeople.

Lecture 4. Xavier Gabaix. February 26, 2004

Lecture 4. Xavier Gabaix. February 26, 2004 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: