Average Switching Costs in Dynamic Logit Models John Kennan January 2008

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Beryl Thornton
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1 Arag Switching Costs in Dynamic Logit Modls ohn Knnan anuary 28 Thr is an xtnsi litratur on discrt choic modls, in which agnts choos on of a finit st of altrnatis. Th mpirical rlationship btwn th charactristics of ths altrnatis and th associatd choic probabilitis can b usd to stimat th undrlying prfrncs of th agnt, and th stimatd prfrncs can b usd to prdict choics in situations not obsrd in th data. In dynamic modls, thr is typically on choic that rprsnts th status quo, and thr ar switching costs associatd with all of th othr choics. Whn switching is rar in th data, stimatd switching costs tnd to b implausibly larg. But switching dos occur, and in many cass thr is no obsrabl rason for a switch, so that th obsrd choic must b attributd to unobsrd payoff shocs, including random ariations in switching costs. Th qustion thn ariss as to how larg th actual switching costs ar, conditional on a switch bing mad. Whn th unobsrabls ar drawn from th typ I xtrm alu distribution, this qustion has a simpl answr. Suppos thr ar altrnatis, with payoffs ṽ + ζ, whr {ζ } is a st of iid xtrm alu random ariabls. If u is uniformly distributd on [,], thn y - log(u has th unit xponntial distribution, and ζ - log(y has th typ I xtrm alu distribution. Thus if u is dfind by stting ζ -log(y and y -log(u, thn {u } is a st of iid random ariabls that ar uniformly distributd on [,]. Th following two rsults ar wll nown (th proofs ar gin for compltnss, sinc thy ar quit simpl. Lmma Pr ( max Lt d b an indicator for th nt that ṽ is maximal. Thn ( ζ + ζ+ u ( y Δ y Pr Pr ( u Δ ( u ( u Pr Δ

2 whr Δ xp( -. Thus ( ( Pr d Pr d u du 2 Pr ( ζ + ζ + u du A ( u du whr A Σ > Δ. This yilds Pr ( d + A Lmma 2 ( E max γ + log whr γ is th Eulr constant. ( max Pr ( E d Intgrating this with rspct to gis ( E max c+ log whr c is constant with rspct to. In fact sinc th choic of is arbitrary, c is in fact

3 constant with rspct to for all. Ltting 6-4 for > yilds c Eζ γ. Th nxt rsult is not so wll nown, and it is somwhat surprising. In gnral, onc it is nown which altrnati was chosn, th xpctd alu of th agnt s problm is not th sam as th x ant xpctation of th maximal alu. But whn th shocs ar drawn from th xtrm alu distribution, ths two xpctd alus ar qual. Lmma 3 E( + ζ d γ + log Dfin th random ariabl X as X max u Δ > so that d iff X # u (almost surly. Th distribution function of X is 2 ( Δ A F( x Pr u x x So ( u ( ( ζ ( Pr d E d log log u df( x du log ( log ( ( u u du ( ( ( log y xp A y xp y dy A Dfin z - log((+a y. Thn z z log y xp Ay xp y dy z+ log( + A dz ( ( ( ( + A γ + log( + A + A

4 bcaus xp(-zxp(-xp(-z is th xtrm alu dnsity function, and γ is th man of th xtrm alu distribution. Thus γ + log( + A E( ζ d + A + A and ( ζ γ + log( + E d A ( ζ+ γ + log( + E d A γ + log Th point of this rsult can b illustratd as follows. Suppos thr ar two altrnatis, with 3 and 2, and suppos th payoff shocs ar drawn from a uniform two-point distribution, with support {-6,6}. Thn altrnati 2 is chosn if and only if (ζ,ζ 2 (-6,6, so that th ralizd payoff is ṽ 2 6. If altrnati is chosn, th ralizd payoff is ṽ 9 if (ζ,ζ 2 {(6,-6,(6,6} and ṽ -3 if (ζ,ζ 2 (-6,6. Thus th arag payoff gin that altrnati is chosn is (b(9+(a(-3 5. Arag Switching Costs Th nxt rsult shows that th xpctd gain from th optimal choic, rlati to an arbitrary altrnati that is not chosn, is a simpl function of th probability of choosing th altrnati. Dfin log ρ

5 Lmma 4 log( ρ E( d E( d ρ Th xpctd payoff gin that is chosn is ( E d γ + Th unconditional xpctd payoff for altrnati is E γ + Thus and ρ E( d γ + ρ ρ ( ( ( ( ( E( d γ + ρ E d + ρ E d ρ γ + + ρ E( d E( d ρ ( ρ log ρ which pros th rsult. This rsult implis that th xpctd gain rlati to altrnati is dcrasing in ρ. But th xpctd gain is nr lss than, raching this in th limit as th probability of choosing approachs. In a dynamic modl with a stat ariabl x, and switching costs Δ(x,, th arag switching cost nt of th diffrnc in payoff shocs can b writtn as ( ( ρ ( x ( x log Δ( x, E ζ ζ d + β p( x x, d p( x x, d ( x ( ρ x

6 whr th zro subscript dnots th status quo, and p is th transition probability function.

Definition A continuous random variable X on a probability space (Ω, F, P) is a function X : Ω R such that for all x R. 1 {X x} is an event, and

Definition A continuous random variable X on a probability space (Ω, F, P) is a function X : Ω R such that for all x R. 1 {X x} is an event, and Continuous random variabls I Mathmatics for Informatics 4a José Figuroa-O Farrill Lctur 7 Fbruary 22 Aftr discrt random variabls, it is now tim to study continuous random variabls; namly, thos taking valus