DynamicPriceCompetitioninAuto-InsuranceBrokerage

Transcription

1 DynamicPriceCompetitioninAuto-InsuranceBrokerage BrunoAurichio LuisH.B.Braido June 23, 2010 Abstract Auto-insurance data present an intriguing fact: the coexistence of policies sold with zero and positive brokerage fees. We propose a dynamic model of price competition with switching costs that reproduces this empirical fact. We derive a recursive mixed-strategy equilibrium in which brokers discriminate prices across old and new consumers. Using data on brokerage fees alone, we perform a structural maximum-likelihood estimation that identifies: (i) the number of brokers effectively bidding to a given consumer;(ii) the insuree s cost of switching brokers; and(iii) the broker s expected present values of keeping an old consumer and of attracting anewone. Keywords: Brokerage, fees, price competition, structural estimation, auto-insurance. JEL Classification: D4; L1. WearethankfulforcommentsfromMarceloMedeiros,HumbertoMoreira,HelenoPioner,Marcelo Sant Anna, and seminar participants at IPEA, PUC-Rio, and USP. FUCAPEBusinessSchool;Vitoria,Brazil;aurichio@fucape.br. GetulioVargasFoundation,GraduateSchoolofEconomics(FGV-EPGE);PraiadeBotafogo190, s.1100, Rio de Janeiro, RJ , Brazil; lbraido@fgv.br. 1

2 1 Introduction Brokers intermediate sales in many different markets such as real state and insurance. In the automobile insurance market, for instance, they act as retailers for the insurance companies and help consumers to fill out reimbursement papers in the case of a claim. This sector is very competitive. Thousands of independent brokers offer policies from different insurance companies and add a brokerage fee to the insurance wholesale price. In this paper, we analyze Brazilian data on auto-insurance contracts and document a interesting fact: the coexistence of policies being sold with zero and positive brokerage fees. Around20%ofthecontractsaresoldwithnobrokeragefee. Thepositivefeesare very disperse within an interval that ranges from zero to about 40% of the contract s final price. This evidence is not consistent with static models of price competition since Nash equilibrium requires that strategies played with positive mass display the same expected payoff. Therefore, recent versions of Bertrand and searching models such as Varian(1980), Stahl(1989), Sharkey and Sibley(1993), and Braido(2009) cannot be usedtodirectlyaccountforthisfeatureofthedata. We introduce dynamics into a model of price competition in which insurees face a switching cost when signing a contract with a new (unmatched) broker. This incumbent s advantage drives unmatched brokers to set fees aggressively and generates a mass of policies with zero brokerage fees. We explicitly derive a symmetric recursive Nash equilibrium in which brokers discriminate prices across old and new consumers. We then use the mixed-strategy equilibrium distribution to perform a maximum likelihood estimation of the model parameters. Using data on brokerage fees alone, this structural econometric procedure allows us to identify: (i) the number of insurance quotes effectively accessed by insurees before purchasing a policy;(ii) the insuree s cost of switching; and(iii) the broker s expected present values of keeping an old customer and of attracting a new one. These are important concepts in industrial organization whicharedifficulttoobtainfromdataonprices. Our approach differs from those used in the literature of oligopolistic price competition with switching costs. We refer the reader to Farrell and Klemperer (2007) for an extensive survey on this literature. We use the next paragraphs to describe two branches of this literature that are closely related to our work. In a pioneering paper, Klemperer(1987) introduces a model in which oligopolistic firms set prices over two periods of time, and consumers face different costs for switching from one firm to another in the second period. Prices are restricted to be the same 2

3 for old and new consumers. Thus, the second-period pricing strategies depend on the firm s stock of old customers. In equilibrium, firms charge equal prices and capture thesameshareof themarketineachperiod. Inanotherclassicwork, Padilla(1995) extends this analysis to an environment with overlapping generations and infinitely manyperiods. 1 LikeinKlemperer(1987),firmscannotpricediscriminatebetweenold and new consumers, and the equilibrium pricing depend on each firm s market share. Similarly to our paper, consumers face an identical switching cost, and the authors focus on computing a recursive mixed-strategy equilibrium. The possibility of price discrimination is the key difference between our work and this first branch of the switching cost literature. Brokers usually deal independently with each consumer. It is then natural in our setting to allow for price differentiation between old and new consumers and within consumers in each of these groups. The pricing mixed strategies are different for new and old insurees, and they do not depend on the broker s market share. In equilibrium, ex-ante identical consumers may get very different price offers as a consequence of the mixed-strategy realizations. There are also a few papers analyzing pricing discrimination in environments with switchingcosts. Chen(1997)andTaylor(2003)presentamodelinwhichtwofirmsset differentpricesforoldandnewconsumersoverafinitenumberofperiods. 2 Consumers face different switching costs in each period, according to i.i.d. draws from an exogenous distribution function. In equilibrium, the two firms offer the same price for their own old customers and the same discount price for their rival s consumers. The exogenous distribution of consumers switching costs determines the fraction of consumers who switch suppliers. Different from this second branch of the switching cost literature, we model an arbitrary number of brokers competing for identical consumers over an infinite time horizon. There is no pure-strategy equilibrium for our game. When consumers are homogeneous, suppliers are led to use mixed strategies in order to be unpredictable to each other. As a result, equilibrium prices charged to ex-ante identical consumers endupbeingverydisperse. 3 Theremainingofthis paper is organizedasfollows. WeuseSection2to analyze 1 See Farrel and Shapiro (1988) and Beggs and Klemperer (1992) for alternative infinite-horizon frameworks. 2 FudenbergandTirole(2000)andVillas-Boas(2006)alsostudydynamicmodelsofpricediscrimination based on consumer s purchasing history. 3 In a somewhatdifferentcontext, Chen and Rosenthal(1996)deriveamixed-strategy equilibrium foradynamicduopolyinwhichconsumersareloyaltofirms,butloyaltyvariesovertimeaccordingto firms realized prices and an exogenous stickiness rule. 3

4 a thorough data set recorded by the Brazilian Superintendence of Private Insurance (SUSEP) which conveys population information on auto-insurance contracts. As mentioned before, we notice that about 20 percent of the analyzed contracts have no brokerage fee. Moreover, our data present a strong dispersion in fee values. Our attempt to rationalize this evidence relies on a game theory model in which brokers use mixed strategies to set different offers to identical insurees. Section 3 formally describes our dynamic game of price competition and derives the distribution of price offers that is consistent with a symmetric recursive Nash equilibrium. In Section 4, we stress that the database displays solely the contracted fees, not all bids accessed by the insuree. We then use the mixed-strategy equilibrium distribution and the individual decision rules to derive the data-generating distribution. This distribution is used in Section 5 to identify the parameters of the model through a maximum likelihood estimation. Our results identify that: (i) insurees typically access two offers before purchasing an insurance policy; (ii) they face a switching cost equivalent to 142 BRL (approximately US$ 71 in purchasing power parity); and (iii) the broker s expected lifetime discounted profit is about 269 BRL when dealing with a matched consumer, and about 127 BRL when dealing with an unmatched consumer. These values are economically meaningful. 2 Data We use data recorded by the Brazilian Superintendence of Private Insurance(SUSEP) which contain information on brokerage fees and insuree characteristics for every autoinsurance contract signed in Brazil during the first semester of Table 1 defines all variables used in our analysis. As is stressed there, we restrict attention to comprehensivepoliciescontractedbyindividuals. 4 [Table 1] Akeyaspectofthedataisthatpoliciesaresoldwithdifferentbrokeragefees. This featureisveryrobustandappearsinmanydifferentsubsamplesofthedata. Figure1 presents histograms for nominal brokerage fees expressed in BRL, and Figure 2 displays the broker s percentage fees as a fraction of the policy total premium. [Figures 1-2] 4 We do not consider: (i) policy endorsements, (ii) noncomprehensive contracts, (iii) policies contracted by firms, and(iv) collective policies covering multiple vehicles. 4

5 Six different subsamples are presented. Subsample A shows the fees paid to brokers across all insurance contracts in the country. Subsample B focuses on contracts in the metropolitan area of Sao Paulo(the largest in the country). Subsample C presents fees on policies covering nine comparable compact cars with identical power trains in the metropolitanareaofsaopaulo. 5 Next,SubsampleDrefinesSubsampleCtoinclude only contracts with positive bonus discount. Insurees with positive bonus discount are particularly interesting for our purposes because they must have signed with a broker intheperiodbefore. SubsamplesEandFfurtherrefineSubsampleDtoincludeonly new compact cars and new Volkswagen Gol 1.0L, respectively. Note that the data refer to2003yearmodelsandthatthevolkswagengol1.0laccountsfor30percentofits class within our data set. On average, about 20 percent of the insurance policies are sold with zero fees. (This is omitted in Figure 1 for scale reasons.) Furthermore, there is large dispersion in both nominal and percentage brokerage fees. The shapes of the histograms are surprisingly similar across the different subsamples. This suggests that there might be some relevant feature behind the dispersion in brokerage fees paid by consumers. We will use subsample E in our quantitative analyses throughout the paper and subsample F to check the robustness of our results in Section 5.2. The choice of analyzing more homogeneous subsamples is justified by the fact that we attempt to understand the economics behind the dispersion in brokerage fees which are unrelated to consumer heterogeneity. In line with this argument, Table 2 presents reduced-form OLS regressions for three different variables. First, we analyze how observable characteristics are correlated to the choice of zero brokerage fees. The following two regressions consider the subsample of contracts with positive fees to analyze the correlation between characteristics and fees in both nominal and percentage values. In all three regressions, we use an enormous amount of dummy variables to capture different characteristics of contracts and insurees. This large number of control variables account for 10 to 23 percent of the dispersionoffees,asfollowsfromther 2 values. Thissuggeststhatasignificantpart of the dispersion in fees is not related to observed consumer heterogeneity. [Table 2] 5 The vehicle models are: Peugeot L, Clio 1.0L, Gol 1.0L, Fiesta 1.0L, Ka 1.0L, Uno 1.0L, Palio 1.0L, Corsa 1.0L, and Celta 1.0L. 5

6 3 A Dynamic Model of Price Competition Consider a dynamic recursive economy populated by a continuum of insurance brokers and by a finite number of homogeneous insurees. These two sets brokers and insurees are fixed over an infinite time horizon. Insurees demand one insurance policy perperiod,andtheymustbuyitthroughabroker. Theyminimizetotalcostsineach period. Since we will focus on recursive Nash equilibria, this is equivalent to minimizing life-time discounted costs. Allpoliciesinthemarketlastforoneperiod. Brokersaccessthesamepoolofautoinsurance companies and face the same wholesale prices. They maximize expected life-time discounted profits, and their intertemporal discount factor is β (0, 1). Ineachperiod,brokerssetindependentbrokeragefeesq Q Rtobechargedto each potential customer. In principle, brokers are not allowed to charge negative prices. In practice, however, they could do so by offering nonpecuniary gifts to consumers. We thenallowfornegativefeesandassumethatthedatasetiscensoredatq=0. That is, the values of such gifts are not observed by the econometrician. Intheinitialnode, eachinsureeisrandomlymatchedtoabroker. Ineachsubsequent node, one of the following two possibilities occurs: (a) with probability λ (0, 1), theinsureeremainsmatchedtothelastbrokeronehascontractedwith;and(b)with probability(1 λ),theinsureeisrandomlymatchedtoanewbroker. Theparameter λ accounts for the fact that some consumers exogenously enter and leave the market ineveryperiod. 6 Ineveryperiod,eachinsureeobservestheofferfromamatchedbrokerandI 1 offers from unmatched brokers. Insurees seek the smallest total cost and, in addition to the brokerage fee, they face a fixed cost c > 0 to switch from a matched to an unmatched broker. This is an important advantage of the matched broker. This environment can be seen as if I+1 brokers were setting prices to attract a single consumer. This switching-cost version of the Bertrand game is repeatedly and independently played for each consumer. We formally describe this as a recursive game, inthesenseofduffieetal. (1994,p. 771). RecursiveGame. ConsideragamewithI+1brokersandasingleconsumer. Nature playsintheinitialnodeanddefinesonebrokertobematchedtothatconsumer. Next, all brokers set their fees. Thegame ends for those whodonot make a sale. For the 6 Since the probability of the scenarios (a) and (b) add up to one, the number of consumers (just like the set of brokers) remains constant over time. 6

7 winning broker, an identical game starts in the next period with probability λ. In this case,thatbrokerissurelyselectedbynaturetobematchedtotheconsumer. Ineachperiod,thespaceofactionsisQ R. Thebrokers payoffsdependonastate variable z Z ={m, u} that describes the matched/unmatched status. Conditional onz,thepayoffsdonotdependontheprevioushistoryofthegame. A recursive strategy for each player i I = {0,1,...,I} is a map σ i : Z, where is the space of probability distributions on Q. We use the following notation throughoutthepaper: σ i =(σî)î i and(σ 0,...,σ I )=(σ i,σ i ). Payoff Functions. For each broker i, define v i,m and v i,u as the expected life-time discounted payoff function when i faces a matched and unmatched insuree, respectively. Thefunctionv i :Z Q Risdefinedby: v i,m (q)= î I\{i} (1 σî,u (q c)) ( q+λβ ) v i,m ( q)dσ i,m ( q), (1) and v i,u (q)=(1 σ i,m(q+c)) î I\{i,i} ( (1 σî,u (q)) q+λβ ) v i,m ( q)dσ i,m ( q), (2) whereσ i,m andσ i,u aretherecursivestrategiesofbrokeriwhendealingwithamatched andanunmatchedinsuree. Notethati representsthematchedbrokerinequation(2). To understand equation(1), notice that (1 σî,u (q c))isthe probability î I\{i} that the insuree accepts a fee q from the matched broker. In other words, it is the probability that all fees set by the I unmatched brokers are greater than q c. Moreover, thetermλβ v i,m (q)dσ i,m ( q)representsbrokeri sexpecteddiscountedcontinuation value of making a sale today. (Recall that, after a purchase, the consumer becomes matched to the broker with probability λ.) Analogously, to understand equation(2), notice that the probability that the insuree acceptsafeeq fromtheunmatchedbrokeris (1 σ i,m(q+c)) (1 σî,u (q)). î I\{i,i} Inotherwords,thisistheprobabilitythatqissmallerthanthematchedbroker sfee plustheswitchingcostaswellassmallerthanallotheri 1unmatchedoffers. Like in equation (1), the expected discounted continuation value of making a sale is also λβ v i,m (q)dσ i,m ( q). Recursive Nash Equilibrium. A recursive Nash equilibrium consists of a strategy profile(σ 0,...,σ I )suchthat,foreachi I: 7

8 (i) Thereexistsauniquevaluefunctionv i associatedwith(σ i,σ i),definedby(1) and(2),foranypossiblerecursivestrategyσ i :Z. [ ] (ii) Thereisnodeviationσ i :Z suchthate σ v i i,z i (q) >E σi [v i,zi (q)],where v i isthecorrespondingvaluefunctionassociatedwith(σ i,σ i)andz i describes playeri scurrentstatusinz={m,u}. Itisimportanttostressthat,unlikeDuffieatal. (1994),thespaceofactionQneed notbecompact. Asaconsequence,thepayofffunctionmaynotbewell-definedforall strategies in. This is why the payoff functions are part of our equilibrium concept. Propositions A1 and A2 in the Appendix show two alternative sets of conditions for the existence and uniqueness of a value function satisfying(1) and(2). When computing an equilibrium in the next section, we will focus on candidates satisfying the conditions presented in Proposition A2. Remark 1. Ifc=0wereallowed,thisgamewouldbecomeaversionofthestandard Bertrand model. In that case, all brokers setting q = 0 would be a pure-strategy equilibrium. Notice however that no pure strategy equilibrium exists when c > 0 and q R. 3.1 Deriving a Symmetric Recursive Equilibrium LetQ=Randdefineasymmetricstrategyσ i =σ(identicalforalli)asfollows: (i) lim q (1 σ u(q)) I 1 q< ;and(ii)foreachz {m,u},σ z isanatomlessprobability distributionsuchthatσ z (a)=0,forsomea>. Given this strategy profile, the expected life-time discounted payoffs associated to eachactionq [a,+ )ofthematched(v m )andunmatched(v u )brokersare: ) v m (q)=(1 σ u (q c)) (q+λβ I v m ( q)dσ m ( q), (3) and v u (q)=(1 σ m (q+c))(1 σ u (q)) (q+λβ I 1 ) v m ( q)dσ m ( q). (4) ItfollowsfromPropositionA2intheappendixthatv m andv u areuniquelywelldefined in [a,+ ). Noticethat theterm (1 σ u (q c)) I standsforthe probability thattheinsureeacceptsafeeqfromthematchedbroker; (1 σ m (q+c))(1 σ u (q)) I 1 istheprobabilitythattheinsureeacceptsafeeqfromagivenunmatchedbroker;and 8

9 λβ v m ( q)dσ m ( q) represents the broker s expected discounted continuation value of making a sale. Unmatched-BrokerStrategy. Letusnowexplicitlyderivethedistributionσ u. Set arbitraryvaluesfora Rand v m >0,anddefineσ u ( )suchthat: (1 σ u (q c)) I (q+λβ v m )= v m, q [a+c,+ ). (5) Solvingforσ u (q): [ ] v 1/I m σ u (q)=1, q a. (6) q+c+λβ v m Notice that lim q σ u (q)=1. Moreover, for any a R, one has σ u (a) = 0 if and only if: v m = c+a (1 λβ). (7) One can then write: σ u (q)= [ 1 0, q<a; c+a (1 λβ)q+c+λβa ] 1/I, q a. (8) Matched-BrokerStrategy. Usingananalogousreasoning,define v u >0andletσ m be such that: [ ] v (I 1)/I m (1 σ m (q+c)) (q+λβ v m )= v u, q [a, ). (9) q+c+λβ v m Solvingforσ m (q): [ ] v (I 1)/I u q+λβ vm σ m (q)=1, q a+c. (10) q c+λβ v m v m Notice that lim q σ m (q) = 1, for every I 1. Moreover, from (7), one has [ a+c+λβ vm v m ]=1. Therefore,σ m (a+c)=0ifandonlyif: v u =a+λβ v m = λβc+a (1 λβ). (11) 9

10 Therefore,σ m iswrittenasfollows: 0, q<a+c; σ m (q)= 1 (λβc+a)[λβ+1 λβ c+a q](i 1)/I (1 λβ)(q c)+λβ(c+a), q a+c. (12) SymmetricEquilibrium. Wealwayshave v m > v u,sincec>0. Moreover, v u >0if andonlyif: a> λβc. (13) We also have by construction that: v m (q) v m, q<a+c; (14) v m (q)= v m, q a+c; (15) and v u (q) v u, q<a; (16) v u (q)= v u, q a. (17) Giventheunmatchedbrokers strategies, thefunctionv m isuniquely definedand leaves the matched broker indifferent to any action in the support of σ m, namely (a+c, ). Moreover, given that all other players draw fees from σ m and σ u, the functionv u isalsouniquelydefinedandleavesanyunmatchedbrokerindifferenttoany actionin suppσ u =(a, ). Therefore, the payoff functions that are uniquely defined by equations(3) and(4) joint with the strategies defined by (8) and (12) describe a recursive equilibrium for any given(i, λβ, c, a) satisfying the positive-profit condition(13). To see this, suppose by contradiction that they are not a recursive equilibrium. In that case, there should existastrategyoutsidesuppσ z thatyieldsanexpectedpayoffhigherthan v i,z forsome z Z={m,u},whichisnotpossiblegivenconditions(14)through(17). Comments on the Symmetric Equilibrium. The parameters of the model are (I,λβ,c). For each parameter value and for any a> λβc, the payoffs (v m,v u ) and strategies(σ m,σ u )definedby(3),(4),(8),and(12)characterizeasymmetricrecursive equilibrium. Therefore, for any vector of parameters(i, λβ, c), there exists a continuum ofequilibriaindexedbya> λβc. 10

11 Itfollowsfrom(7)thata+c= v m (1 λβ)>0,andthenσ m (0)=0. Therefore, only unmatched brokers set negative fees. This occurs when a < 0. Moreover, equations (7) and(11) define the equilibrium payoffs for matched and unmatched brokers. Since there is a continuum of a > λβc that are consistent with a recursive equilibrium, there is also a continuum of possibilities for the equilibrium profits. This feature of the model is related to the fact that we have imposed no reservation price for consumers. Like in static Bertrand models (see Baye and Morgan, 1999), the absence a reserve leads to this type of Folk Theorem in which any finite positive payoff can be achieved in equilibrium. From a Poperian point of view, this isveryinconvenientsincethemodelistooflexibletoaccommodateavastsetofempirical observations. However, from a statistical perspective, this flexibility is welcome andisusedtoestimatetheparametersofthemodelinconjunctionwithaparticular equilibrium(indexed by a) which most likely could generate the observed data. 4 Data-Generating Distribution Thedatadisplayfeesy j R + thathavebeenpaidbytheinsureesratherthanthe offersgeneratedbyσ m andσ u. Thatis,thedatagenerationdistributioncombinesthe mixed-strategiesequilibriumσ m andσ u withtheinsurees selectionrules. Eachinsureeinthesampleaccessesoneofferq 0 drawnfromσ m andi independent offers drawn from σ u. They accept the offer q 0 from their matched broker whenever q 0 min(q 1,...,q I )+c. Moreover,theyprefertheofferq min =min(q 1,...,q I )whenever q 0 >min(q 1,...,q I )+c. Therefore, each observation j in the sample is generated as follows: q 0 q min +c y j =q 0 ; (18) q 0 >q min +c y j =max[0,q min ]. (19) Thedistributionfunctionforq min =min(q 1,...,q I )isgivenby: H(q min )=1 (1 σ u (q min )) I. (20) Let Pr( ) be a conditional probability measure defined on some arbitrary measurable space(ω, ) that represents the data generating distribution. Notice from(18) to(19)that,foranyy 0,onehas: Pr[y j (a,y)]=pr[a B], (21) 11

12 where A = {ω Ω:q 0 q min +candq 0 y} and B = {ω Ω : q 0 > q min +c and q min y}. SinceAandBaredisjointsets,oneobtains: and Pr[y j (a,y)]= y a (1 σ m (ỹ+c))dh(ỹ), fory<a+c; (22) Pr[y j (a,y)]= y a (1 σ m (ỹ+c))dh(ỹ) + y a (1 H(ỹ c))df m (ỹ), fory a+c. (23) We have interpreted negative fees as being nonpecuniary gifts and services that are offered to consumers and not observed by the econometrician. Thus, a mass of zero-brokeragefeesisgeneratedhereasacensoringoftheequilibriumstrategyσ u at thepointq=0. Theprobabilityofobservingazerofeeisgivenby: Pr[y j =0]= 0 a (1 σ m (ỹ+c))dh(ỹ). (24) Moreover,fory>0,weusetheLeibnizruletofind: Pr[y j (0,y)]= wherethedensityfunctionp(y)isgivenby: and y 0 p(ỹ)dỹ, (25) p(y)=(1 σ m (y+c)) H(y), for0<y<a+c; (26) y p(y)=(1 σ m (y+c)) H(y) y 5 Maximum Likelihood Estimation +(1 H(y c)) σ m(y), fory a+c. (27) y We use the maximum likelihood procedure for censored data to estimate the parameterofourmodel(seemeekerandescobar,1994). LetJ={1,...,J}thesample-index set,anddefinej 0 ={j J:y j =0}. Thusthelikelihoodfunctionisgivenby: L(I,λβ,c,a)= j J 0 Pr[y j =0] j J\J 0 p(y j ). (28) 12

13 We define the following grids for the structural parameters: I {1,2,...,20}; λβ {0.01,0.02,...,0.99}; c {0.01,0.02,...,500}; and a = αλβc, where α {0.01, 0.02,..., 0.99}. We then use the Wolfram Mathematica 7 software to compute andselectthehighestvalueof(28), givenourdataonbrokeragefees,forallpossible parametercombinations. 7 Mathematica isapowerfultoolwhichisnecessaryherebecausethereisnoclosedformulafortheintegralin(24)thatdefinespr[y j =0]. This is solved by means of generalized hypergeometric functions. Table 3 presents estimation results. Column (a) displays the estimates for the unrestricted version of the model. Results suggest that: (i) insurees typically access two offers before purchasing an insurance policy;(ii) they face a switching cost equivalent to 142 BRL(i.e., about US$ 71 in purchasing power parity); and(iii) the brokers expected lifetime discounted profit is about 269 BRL when dealing with a matched consumer, and about 127 BRL when dealing with an unmatched consumer. The estimated value forλβ is0.61,butthemodeldoesnotallowustodetachthediscountfactorfromthe probability λ. Columns (b) and (c) present restricted estimations for λβ = 0.90 and for I = 2. Log-likelihood ratio tests indicate that these two restricted models are statistically different from the unrestricted model. Interestingly, the switching cost and brokerage profitability do not vary much across these different versions of the model. The obtained values are economically reasonable for the city of Sao Paulo. [Table 3] Figure 3 graphs the density function for the parameters of the unrestricted estimation against the empirical histogram. For scale reasons, we graph only positive brokerage fees. The empirical and estimated mass at q = 0 are, respectively, 15.73% and 10.73%. Notice that the density curve combines two hyperbolic functions obtained from σ m and H. Unmatched fees start at a = and appear uncontested by matched offers up tothe point a+c= After that point, offers from matched and unmatched brokers appear together in the data, generating a second peak in the density function. The parameter a is directly related to the mass of policies with zero fees,whilecisrelatedtothesecondpeakofthedensityfunction. [Figure 3] 7 StatisticalresultsinotherpartsofthispaperwerecomputedinSTATA

14 5.1 Econometric Comments The likelihood function is not continuous when the parameters λβ, c, and a take arbitraryrealvalues. However,whentheparametervector(I,λβ,c,a)isevaluatedina finite set, then the likelihood function is continuous. (Any function with a finite domain is continuous.) In this case, standard results from M-estimators guarantee consistency and asymptotic normality of our maximum-likelihood estimator. We do not explore these issues in further detail. We simply want to emphasize that consistency and asymptotic log-likelihood tests in this paper depend on the existence of an economic justification for the assumption that the model parameters are valued in a finite set. A second important econometric issue is related to the identification of a single maximum value in the estimation process. We have computed likelihood values for all possible combinations of the parameters. Table 4 displays the parameters associated with the ten highest likelihood values. Notice that the maximal point is unique and its likelihood value is substantially higher than that of the second best combination of parameters. Figures 4-7 present the likelihood values for different(i, λβ, c, a). Each figure freezes three parameters at their optimal point and varies the fourth parameter around its optimal value. These figures illustrate the marginal effect of each parameter over the likelihood function around the optimal value. [Table 4&Figures4-7] 5.2 Subsample with a Single Vehicle Model Wenowperformtheanalysisfromthelastsectiononasubsamplethatrefinesour baseline sample to include only the most popular vehicle model in Brazil (Gol 1.0L). The results are similar to the previous one. Table 5 and Figure 8 are the analog of Table 3 and Figure 3. This confirms that potential heterogeneity across insurees is not the main element behind the large variation in brokerage fees as observed in the data. [Table 5&Figure8] Identification of a single optimal point can be seen in Table 6 and Figures TheyaretheanalogofTable4andFigures4-7presentedinlastsubsection. [Table 6&Figures9-12] 14

15 6 Conclusion We noticed that data on brokerage fees exhibited wide dispersion which was not strongly correlated to observables. We propose a theory that accounts for the key economic forces behind the brokerage problem, then take that model to the data. Based on this structural procedure, we have been able to infer important structural parameters such as the number of offers accessed by each insuree, the individual costs of switching to a new (unmatched) broker, and the brokers profitability when dealing with old (matched) and new(unmatched) customers. Our dynamic model with switching costs replicates stylized facts that the existing models of price competition cannot generate. A Appendix: Value Function In this appendix, we present two alternative sets of conditions that guarantee that equations (1) and (2) define a unique value function v i. We first consider the case whereqisacompactintervaloftherealline. Next,weanalyzethecasewhereQ=R and players strategies are well behaved when q approaches and +. Proposition A1. Fix Q = [a, b] R. There exists a unique measurable bounded functionv i,m :[a,b] Rthatsatisfiescondition(1),foranystrategyprofile(σ 1,...,σ I ). Proof. Let bethecompletemetricspaceofσ i,m -measurableboundedfunctions mapping[a,b]into[a,b]. LetusendowthisspacewithsupnormanddefineT : as follows: Tv i,m (q)= î I\{i} ( (1 σî,u (q c)) q+λβ ) v i,m ( q)dσ i,m ( q). (29) Being that q [a,b], Tv i,m defines a bounded function for any bounded v i,m. Since λβ<1,onehas: ( sup Tv i,m Tv i,m ) ( <λβ sup v i,m v i,m ), ( v i,m,v i,m). (30) q [a,b] q [a,b] The result follows then from the contraction mapping theorem. Noticethat, whenq=[a,b],thefunctionv i,m :[a,b] Risuniquelydefinedfor eachstrategyprofile(σ 0,...,σ I ). WhenextendingthedomaintoQ=(,+ ),restrictionsonstrategiesarenecessaryinordertohaveawell-definedv i,m. Animportant 15

16 caseusedinthispaperiswhenabrokerifacesstrategiesforwhichthereexistsa> suchthatσî,z (a)=0and lim q + (1 σ î,z(q))q<+,foranyî iandz {m,u}. Proposition A2. Let q Q = R and assume that for each player î i and lim q + î I\{i} each state z {m,u}: (i) there exists a > such that σî,z (a) = 0; and (ii) (1 σî,u (q c))q < +. Then, any action q i < a is strictly dominated by q i = a. Moreover, there exists a unique measurable bounded function v i,m :[a,+ ) Rthatsatisfiescondition(1),foranystrategyprofileσ=(σ 1,...,σ I ). Proof. Any action q i < a is strictly dominated by q i = a, since qî > a almost surely, for all î i. Then, equation (1) defines a unique bounded function v i,m : [a,+ ) R. TheproofforthatisanalogoustotheproofofProposition1. Definethe operatort asin(29)andnoticethatt isacontractionmapofmodulusλβ<1that maps the space of σ i,m -measurable bounded functions on R, into itself. Notice that Tv i,m is σ i,m -measurable and bounded for any σ i,m -measurable bounded v i,m, since (1 σî,u (q c))q < +. Thus the result once again follows from the lim q + î I\{i} contraction mapping theorem. Toconcludethisappendix,notethatforeachgivenv i,m ( ),thefunctionv i,u ( )is also uniquely defined by(2). References [1] Baye, Michael R. and John Morgan(1999). A folk theorem for one-shot Bertrand games. Economics Letters, 65(1), [2] Beggs, Alan and Paul Klemperer(1992). Multi-product competition with switching costs. Econometrica, 60(3), [3] Braido, Luis H.B.(2009). Multiproduct price competition with heterogeneous consumers and nonconvex costs. Journal of Mathematical Economics, 45(9-10), [4] Chen, Yongmin and Robert W. Rosenthal(1996). Dynamic duopoly with slowly changing customer loyalties. International Journal of Industrial Organization, 14 (3),

17 [5] Chen, Yongmin (1997). Paying customers to switch. Journal of Economics and Management Strategy, 6(4), [6] Duffie, Darrell, John D. Geanakoplos, Andreu Mas-Colell, and Andrew M. McLennan(1994). Stationary Markov equilibria. Econometrica 62(4), [7] Farrell, Joseph and Paul Klemperer(2007). Coordination and lock-in: Competition with switching costs and network effects. In Handbook of Industrial Organization, Volume 3, Chapter 31, M. Armstrong and R. Porter(eds.), Amsterdam: North- Holland. [8] Farrell, Joseph and Carl Shapiro (1988). Dynamic competition with switching costs. RAND Journal of Economics, 19(1), [9] Fudenberg, Drew and Jean Tirole(2000). Customer poaching and brand switching. RAND Journal of Economics, 31(4), [10] Klemperer, Paul(1987). Markets with consumer switching costs. Quarterly Journal of Economics, 102(2), [11] Meeker, William Q. and Luis A. Escobar(1994). Maximum likelihood methods for fitting parametric statistical models to censored and truncated data. In Probabilistic and Statistical Methods in the Physical Sciences, J. Stanford and S. Vardeman (eds.), New York: Academic Press. [12] Padilla, A. Jorge (1995). Revisiting dynamic duopoly with consumer switching costs. Journal of Economic Theory, 67(2), [13] Sharkey,WillianW.andDavidS.Sibley(1993).ABertrandmodelofpricingand entry. Economics Letters, 41(2), [14] Stahl, Dale O.(1989). Oligopolistic pricing with sequential consumer search. American Economic Review, 79(4), [15] Taylor, Curtis R.(2003). Supplier surfing: Competition and consumer behavior in subscription markets. RAND Journal of Economics, 34(2), [16] Varian,HalR.(1980).Amodelofsales.AmericanEconomicReview,70(4), [17] Villas-Boas, J. Miguel(2006). Dynamic competition with experience goods. Journal of Economics and Management Strategy, 15(1),

18 Table 1. SUSEP/AUTOSEG Auto-Insurance Data Variable Description Total Insurance Premium (a) Broker s Percentage Fee (PERC_CORR) Brokerage Fee Total premium paid by the insuree to the insurance company (in BRL). Fraction of the total premium that is transferred from the insurance company to the broker. Product of the Broker s Percentage Fee and the Total Insurance Premium. Vehicle Type (COD_MODELO) Vehicle Model Year (ANO_MODELO) Geographic Region (REGIAO) Coverage Type (COBERTURA) Deductible Type (TIPO_FRANQ) Physical Damage Coverage (IS_CASCO and IS_RCDM) Personal Injury Coverage (IS_APP and IS_RCDP) Bonus Discount (PERC_BONUS) Gender (SEXO) Date of Birth (DATA_NASC) Zip Code (CEP) (a) (b) Alpha-numerical variable describing the make, model, and power train of the insured vehicle. For the main estimation we restrict attention to nine models of compact cars with 1.0L (1,000cc) engines (namely, Celta, Clio, Corsa, Fiesta, Gol, Ka, Palio, Peugeot 206, and Uno). Year of the vehicle model. When performing the estimations, we restrict attention to new vehicles (model year 2003). Numerical variable ranging from 1 to 41 that describes the geographic region of the insured policy. For our estimations, we restrict the sample to the metropolitan region of Sao Paulo (11). Qualitative variable describing the type of insurance coverage for first-party vehicular damages. Code 1 is used for comprehensive coverage; 2 for coverage against fire and theft; 3 for fire only; 4 for write-off, collision, and theft; 9 for others. We restrict attention to contracts with comprehensive coverage (1). Qualitative variable describing the level of deductible for physical damages to the insured vehicle. Codes 1, 2 and 3 indicate low, regular, and high deductible levels, respectively. Code 9 indicates no deductible. Maximum insurance coverage for first-party (IS_CASCO) and third-party (IS_RCDM) vehicular damages. Values are expressed in BRL. Maximum insurance covered (in BRL) for personal injuries caused to passengers of the insured vehicle (IS_APP) and to passengers of third-party vehicles and pedestrians (IS_RCDP). (b) Percentage discount over the premium for insurance against physical damages to the insured vehicle. (c) For the estimations, we restrict attention to contracts with positive bonus discount. Qualitative variable describing the gender of the vehicle's primary driver, as declared in the contract. The code "M" is used for male; "F" for female; and "J" for corporate entities. We exclude policies code "J". Primary driver s date of birth as stated in the contract. The value " " is used for fleet vehicles (code "J"). Zip code pertaining to the primary driver s address as stated in the contract. We use the first five digits of this eight-digit code. The sum of the following variables: PRE_CASCO, PRE_RCDM, PRE_APP, PRE_RCDP, and PRE_OUTROS. The coverage limits include medical expenses and death/disability benefits that extend the minimum coverage. (c) This bonus depends on the insuree s previous claims record which is publicly accessible by all companies. Each agent starts with no bonus and becomes eligible for a 10 percent discount after one year without a claim. The discount bonus is increased by 5 percentage points after each additional year without a claim, peaking at around 40 percent, depending on the insurance company. In the case of a claim, the insuree's rating is reduced by one level. In case of multiple claims in a same year, the classification might be reduced by more than one level, depending on the insurance company.

19 Figure 1: Positive Brokerage Fees (in BRL) Density Subsample A: Brazil (8 mi obs.; 24.7% zero fees) Density Subsample B: Sao Paulo (2 mi obs.; 22.9% zero fees) Density Subsample C: Sao Paulo, Compact Cars (869,545 obs.; 22.4% zero fees) Density Subsample D: Sao Paulo, Bonus, Compact Cars (591,833 obs.; 15.1% zero fees) Density Subsample E: Sao Paulo, Bonus, New Compact Cars (16,044 obs.; 15.7% zero fees) Density Subsample F: Sao Paulo, Bonus, New Gol 1.0L (2,932 obs.; 18.4% zero fees) For scale reasons, we only graph positive fees. Histogram Width: 5 BRL.

20 Figure 2: Broker s Percentage Fees Density 0.25 Subsample A: Brazil (8 mi obs.; 24.7% zero fees) Density Subsample B: Sao Paulo (2 mi obs.; 22.9% zero fees) Density Subsample C: Sao Paulo, Compact Cars (869,545 obs.; 22.4% zero fees) Density Subsample D: Sao Paulo, Bonus, Compact Cars (591,833 obs.; 15.1% zero fees) Density Subsample E: Sao Paulo, Bonus, New Compact Cars (16,044 obs.; 15.7% zero fees) Density Subsample F: Sao Paulo, Bonus, New Gol 1.0L (2,932 obs.; 18.4% zero fees) Zero fees included. Histogram Width: 1 Percentage Point.

21 Table 2. Reduced-Form OLS Regressions Zero-Brokerage Dummy Positive Fees (in BRL) Positive Percentage Fees Primary Driver Gender (male = 1) * * Zero-Deductible Dummy * * Low-Deductible Dummy * * High-Deductible Dummy Maximum Coverage for Damages to the Insured Vehicle (BRL) Maximum Coverage for Damages to Third-Part Vehicles (BRL) Maximum Coverage for Personal Injury to Passengers of Third-Part Vehicles and Pedestrians (BRL) Maximum Coverage for Personal Injury to Passengers of Insured Vehicle (BRL) -2.16e * 9.93e-05 * -2.45e-06 *.0010 * 8.96e-05 * 8.17e-07 * * -2.75e-05 * 1.31e * -1.57e-05 * Zip Code Fixed Effects 828 dummies 802 dummies 802 dummies Vehicle Model Year Fixed Effects 8 dummies 8 dummies 8 dummies Primary Driver Age Fixed Effects 72 dummies 72 dummies 72 dummies Bonus-Category Fixed Effects 16 dummies 16 dummies 16 dummies Sample Size 16,044 13,521 13,521 R % 23.38% 16.96% * Significant at 1%; Robust standard errors; All regressions include a constant term.

22 Table 3. Structural Maximum-Likelihood Estimation Unrestricted Estimation Restricted Model: λβ = 0.90 Restricted Model: I = 2 Structural Parameters (a) (b) (c) Adjusted Discount Factor (λβ) Number of Unmatched Offers (I) Switching Cost (c), in BRL Highest Nonpecuniary Gift (-a), in BRL Sample Size 16,044 16,044 16,044 Likelihood , , ,181 Log-Likelihood Ratio 4, * * Profitability Expected Life-Time Discounted Profit per Matched Customer ( v m ), in BRL Expected Life-Time Discounted Profit per Unmatched Customer ( v u ), in BRL * The 1% and 10% critical values for the log-likelihood ratio test are 6.64 and 2.71, respectively. Thus, the two restricted models are statistically different from the unrestricted model.

23 Figure 3. Empirical and Estimated Density Functions Density Brokerage Fees Vertical bars for the empirical histogram and the line for the estimated density function. For scale reasons, we only graph positive brokerage fees. Histogram Width: 5 BRL. Sample Size: 16,044; Empirical Fraction of Zero Fees: 15.73%; Estimated Fraction of Zero Fees: 10.73%.

24 Table 4. Global Maximum: Top-10 Likelihood Values Likelihood ( 10-39,026 ) I λb c -a Sample Size: 16,044

25 Figure 4. Local Maximum: Switching Cost (c) L(c I=1, λb=0.61, -a=37.26) Likelihood ( 10-39,026 ) for different switching costs when I = 1, λb = 0.61, and -a = Global Maximum: ,026 ; Sample Size: 16,044. c

26 Figure 5. Local Maximum: Adjusted-Discount Factors (λb) L(λb I=1, c=142.06, -a=37.26) Likelihood ( 10-39,026 ) for different adjusted-discount factors when I = 1, c = and a = Global Maximum: ,026 ; Sample Size: 16,044. λb

27 Figure 6. Local Maximum: Highest Gift (-a) L(-a I=1, λb=0.61, c=142.06) Likelihood ( 10-39,026 ) for different values of -a when I=1, λb=0.61, and c= Global Maximum: ,026 ; Sample Size: 16,044. -a

28 Figure 7. Local Maximum: Unmatched Offers (I) L(I λb=0.61, c=142.06, -a=37.26) Likelihood ( 10-39,026 ) for different values of I when λb=0.61, c=142.06, and a = Global Maximum: ,026 ; Sample Size: 16,044. I

29 Table 5. Structural Estimation: Single-Model Subsample Unrestricted Estimation Restricted Model: λβ = 0.90 Restricted Model: I = 2 Structural Parameters (a) (b) (c) Adjusted Discount Factor (λβ) Number of Unmatched Offers (I) Switching Cost (c), in BRL Highest Nonpecuniary Gift (-a), in BRL Sample Size 2,932 2,932 2,932 Likelihood , , ,211 Log-Likelihood Ratio * * Profitability Expected Life-Time Discounted Profit per Matched Customer ( v m ), in BRL Expected Life-Time Discounted Profit per Unmatched Customer ( v u ), in BRL * The 1% and 10% critical values for the log-likelihood ratio test are, respectively, 6.64 and Thus, the two restricted models are statistically different from the unrestricted model.

30 Figure 8. Empirical and Estimated Density Functions: Single-Model Subsample Density Brokerage Fees Vertical bars for the empirical histogram and the line for the estimated density function. For scale reasons, we only graph positive brokerage fees. Histogram Width: 5 BRL. Sample Size: 2,932; Empirical Fraction of Zero Fees: 18.42%; Estimated Fraction of Zero Fees: 13.44%.

31 Table 6. Global Maximum: Top-10 Likelihood Values Single Model Subsample Likelihood (â10-7,179 ) I λb c -a Sample Size: 2,932.

32 Figure 9. Local Maximum: Switching Cost (c) Single-Model Subsample L(c I=1, λb=0.67, -a=61.65) Likelihood ( 10-7,179 ) for different switching costs when I = 1, λb = 0.67, and a = Global Maximum: ,179 ; Sample Size: 2,932. c

33 Figure 10. Local Maximum: Adjusted Discount Factor (λb) Single-Model Subsample L(λb I=1, c=170.4, -a=61.65) Likelihood ( 10-7,179 ) for different adjusted-discount factors when I = 1, c = 170.4, and a = Global Maximum: ,179 ; Sample Size: 2,932. λb

34 Figure 11. Local Maximum: Highest Gift (-a) Single-Model Subsample L(a I=1, λb=0.67, c=170.4) Likelihood ( 10-7,179 ) for different values of -a when I = 1, λb = 0.67, and c = Global Maximum: ,179 ; Sample Size: 2,932. -a

35 Figure 12. Local Maximum: Unmatched Offers (I) Single-Model Subsample L(I λb=0.67, c=170.4, -a=61.65) Likelihood ( 10-7,179 ) for different values of I when λb = 0.67, c = 170.4, and a = Global Maximum: ,179 ; Sample Size: 2,932. I