Dynamic Price Competition in Auto-Insurance Brokerage

Transcription

1 Dynamic Price Competition in Auto-Insurance Brokerage BrunoLedo LuisH.B.Braido February 15, 2012 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, derive a recursive mixed-strategy equilibrium which reproduces that empirical fact, and use it to estimate the model parameters through maximum likelihood. Using data on brokerage fees alone, we compute: (i) the number of brokers competing for each consumer;(ii) the cost of switching brokers; and(iii) the financial values of keeping an old consumer and of attracting a new one. We also perform structural policy analysis in this context. Keywords: Brokerage, fees, price competition, structural estimation, autoinsurance. JEL Classification: D4; L1. We are thankful for comments from Marcelo Medeiros, Humberto Moreira, Heleno Pioner, Marcelo Sant Anna, and seminar participants at FUCAPE, IPEA, PUC-Rio, UFRJ, USP (FEA- SP and FEA-RP), and at the following meetings: Econometric Society(Shanghai, 2010), European Economic Association(Glasgow, 2010), SBE(Salvador, 2010). UniversityofSaoPaulo;SaoPaulo,Brazil;bruno@fearp.usp.br. Getulio Vargas Foundation, Graduate School of Economics (FGV-EPGE); Praia de Botafogo 190, 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. In this sector, thousands of independent brokers offer policies from different insurance companies and add a brokerage fee to the insurance wholesale price. We analyze Brazilian data on auto-insurance contracts and document an interesting fact: the coexistence of policies being sold with zero and positive brokerage fees. Around 20% of the contracts are sold with no brokerage fee. The positive feesareverydispersewithinanintervalthatrangesfromzerotoabout40%ofthe 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 static versions of Bertrand and searching models such as Varian(1980), Stahl(1989), Sharkey and Sibley(1993), and Braido(2009) cannot be used to directly account for this feature of the data. We introduce dynamics into a model of price competition in which insurees face a switching cost when purchasing from 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 stationary 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 brokers; 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 which are difficult to obtain from data on prices. We also use this model economy to evaluate different public policies. Regulatory agencies typically interpret loss leading and negative prices as noncompetitive practices. Two interesting results of our counterfactual exercises are that increasing the number of competing brokers and forbidding negative fees both reduce consumer s welfare. Related Literature 2

3 Our theoretical 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 oligopolisticfirmssetpricesovertwoperiodsoftime,andconsumersfacedifferentcostsfor switchingfromonefirmtoanotherinthesecondperiod. Pricesarerestrictedtobe the same 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 andcapturethesameshareofthemarketineachperiod. Inanotherclassicwork, Padilla(1995) extends this analysis to an environment with overlapping generations andinfinitelymanyperiods. 1 LikeinKlemperer(1987),firmscannotpricediscriminate between old 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 stationary mixed-strategy equilibrium. In a second branch, there are papers analyzing pricing discrimination in environments with switching costs. Chen(1997) and Taylor(2003) present a model in whichtwofirmssetdifferentpricesforoldandnewconsumersoverafinitenumber of periods. 2 Consumers facedifferentswitchingcosts ineach period, accordingto 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 determinesthefractionofconsumerswhoswitchsuppliers. 3 There are several differences between our work and those in the switching cost literature. The most important is the type of price differentiation allowed in the model. Brokers usually deal to one consumeratatime. It is then natural inour setting to allow for price differentiation between old and new consumers as well as within insurees in each of these groups. Our equilibrium mixed strategies depend on whether the insuree is matched or not to the broker and are unrelated to the 1 SeeFarrelandShapiro(1988)andBeggsandKlemperer(1992)foralternativeinfinite-horizon frameworks. 2 Fudenberg and Tirole (2000) and Villas-Boas (2006) also study dynamic models of price discrimination based on consumer s purchasing history. 3 Inasomewhatdifferentcontext,ChenandRosenthal(1996)deriveamixed-strategyequilibrium for a dynamic duopoly in which consumers are loyal to firms, but loyalty varies over time according to firms realized prices and an exogenous stickiness rule. 3

4 broker s market share. Since brokers set prices to each consumer at a time, ex-ante identical consumers may randomly get very different price offers. Quantitative Analysis Theremainingofthispaperisorganizedasfollows. WeuseSection2toanalyze 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 zero brokerage fee. Moreover, our data present a strong dispersion in brokerage 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 stationary 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 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. Section 6 uses the estimated parameter values to analyze the effect of policies designed to change each of these strucutural parameters of the economy. A brief conclusion appears in Section 7. 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 auto-insurance contract that was active in Brazil during any subperiod of the 2003 first semester. Table 1 defines all variables used in our analysis. We 4

5 restrictattentiontocomprehensivepoliciescontractedbyindividuals. 4 [Table 1] Akeyaspectofthedataisthatpoliciesaresoldwithdifferentbrokeragefees. This feature is very robust and appears in many different subsamples of the data. Figure 1 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] 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 trainsinthemetropolitanareaofsaopaulo. 5 SubsampleDrefinesSubsampleCto include only contracts with positive bonus discount. Insurees with positive bonus discount are particularly interesting because they surely not new in the market(and then conform our model assumption of being initially matched to a broker). SubsamplesEandFfurtherrefineSubsampleDtoincludeonlynewcompactcarsand new Volkswagen Gol 1.0L, respectively. This vehicle model is the most popular within its class. 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 economic 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. 4 We do not consider: (i) policy endorsements, (ii) noncomprehensive contracts, (iii) policies contracted by firms, and(iv) collective policies covering multiple vehicles. 5 Thevehiclemodelsare: Peugeot2061.0L,Clio1.0L,Gol1.0L,Fiesta1.0L,Ka1.0L,Uno1.0L, Palio 1.0L, Corsa 1.0L, and Celta 1.0L. 5

6 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 dispersion of fees, as follows from the R 2 values. This suggests thatasignificantpartofthedispersioninfeesisnotrelatedtoobservedconsumer heterogeneity. [Table 2] 3 A Recursive Model of Price Competition Consider a dynamic recursive economy populated by a continuum of insurance brokers and by a finite number of identical insurees. The time horizon is represented byperiodst T {0,1,..., }. Allpoliciesinthemarketlastforoneperiodand are sold through insurance brokers. Insurees demand one insurance policy every period. They minimize expected lifetime discounted costs. Brokers access the same pool of auto-insurance companies and face the same wholesale prices. They maximize expected lifetime discounted profits, and their intertemporal discount factor is β (0, 1). In each period, brokers independently set brokerage fees q Q to be charged to each potential customer. They cannot commit to future fees. We allow for negative fees and assume there exists a finite value ḡ 0 such that Q [ ḡ,+ ). In practice, brokers are not allowed to charge negative fees. However, they can offer nonpecuniary gifts and additional services to customers. The size of those gifts are determined by social norms and the gifting technology. Thisiscapturedbytheparameterḡ. Sincegiftsarenotrecordedinthedata, the observedfeesarecensoredatq=0. In the initial node, each insurees is randomly matched to a broker. In each subsequent node, one of the following two possibilities occurs: (a) with probability λ (0,1),theinsureeremainsmatchedtothelastbrokeronehascontractedwith; and (b) with probability 1 λ, the insuree is matched to a new broker. (This 6

7 accounts for the fact that consumers might change their regular broker for external motives.) In every period, each insuree observes the offer from a matched broker and I 1offersfromunmatchedbrokers. Theyfaceafixedcostc>0toswitchtoan unmatched broker. This is the only advantage of the matched broker. Brokers need not set the same fee for all potential consumers. This is an important aspect of this market, where fees can differ over time and across identical consumers. This feature is formalized by a model in which brokers compete for one giveninsureeatatime. Recursive Game: Consider a given insuree who initially matched to a broker, say broker i=0. Nature draws I 1unmatched brokers from the continuum of brokers. These I+1 brokers set fee offers for this period. The insuree purchases onepolicy. The gameends forthebrokers whodonotmakeasale. Thewinning broker becomes matched to the consumer, and an identical game starts next period withprobabilityλ (0,1). Strategies: Thespaceof actionsineachperiodis givenby Q [ ḡ,+ ). The brokers payoffs depend on a state variable z t Z = {m,u} that describes the matched/unmatchedstatus. Conditionalonz t, thepayoffsdonotdirectlydepend on the history of the game. Although strategies could still depend on it, we will focus on equilibria in which all players use time-independent stationary strategies. Nobrokercanprofit from conditioning behavioronthehistory of thegamewhen nootherplayerdoesso. Astationarystrategyforeachplayeri I={0,1,...,I}isamapσ i :Z P(Q), where P(Q) is the space of probability distributions on Q. We use the following notationthroughoutthepaper: σ i =(σî)î i and(σ 0,...,σ I )=(σ i,σ i ). Insurees Decision Problem: Insurees minimize expected lifetime discounted costs. This becomes equivalent to minimize total costs in each period, because we focus on equilibria in which fees are determined through time-independent stationary strategies. Brokers Payoff Functions: The probability that the insuree accepts a fee q from amatchedbrokeriisgivenby: φ i,m (q,σ i )= î I\{i} (1 σ î,u(q c)). (1) 7

8 ThisistheprobabilitythatallfeessetbytheIunmatchedbrokersaregreaterthan q c. Theexpectedlife-timediscountedpayoffofbrokeriwhenfacingamatched insuree is: V i,m (σ i,σ i )= t T (λβ)t φ i,m (q,σ i ) t+1 qdσ i,m (q). (2) Q Analogously, the probability that the insuree accepts a fee q from an unmatched brokeriis: φ i,u (q,σ i )=(1 σ 0,m (q+c)) î I\{0,i} (1 σ î,u(q)), (3) wherei=0representsthematchedbroker. Inotherwords,thisistheprobability thatthematchedfeeplustheswitchingcostaswellasthei 1unmatchedoffers are all larger than q. The expected life-time discounted payoff of broker i when facing an unmatched insuree is: V i,u (σ i,σ i )= φ i,u (q,σ i )[q+λβv i,m (σ i,σ i )]dσ i,u (q), (4) Q where λβv i,m (σ i,σ i ) is the continuation value of making a sale and becoming matched next period with probability λ. Notice that V i,zi (σ i,σ i ) takes values in R {+ }. Unlike most papers in dynamic games, 6 the action space Q [ ḡ,+ ) need not be compact, and the expectations in(2) and(4) can be infinite. Stationary Equilibrium: A stationary Nash equilibrium consists of a strategy profile (σ 0,...,σ I ) such that there is no strategy σ i : Z P(Q) such that V i,zi (σ i,σ i)>v i,zi (σ i,σ i )forsome(i,z i ) I Z. Symmetric Stationary Equilibrium: A stationary Nash equilibrium is said to besymmetriciftheequilibriumstrategiesareidenticalforallplayers,i.e.,σ i =σ: Z P(Q), i I. 3.1 Deriving a Symmetric Stationary Mixed-Strategy Equilibrium FixQ=[ ḡ,+ ),whereḡrepresentsthelargestpossiblegiftinthiseconomy. Takeanarbitrarysymmetricstrategyσ i,z =σ z suchthat: 6 SeeforinstanceNowak(1985)andDuffieatal. (1994). 8

9 (a) lim q (1 σ u (q)) I q< ;and (b) lim q (1 σ m (q+c))(1 σ u (q)) I 1 q<. Condition(a)impliesV i,m (σ i,σ i )<, forallσ m P(Q); while(b)implies V i,u (σ i,σ i )<,forallσ u P(Q). Inadditionto(a)-(b),assumethat: (c) σ u isanatomlessprobabilitydistribution;and (d) thereexists g ḡsuchthatσ z ( g)=0, z {m,u}. Definethediscountedpayoffsassociatedtoeachactionq [ g,+ )ofmatched and unmatched brokers as given by: v m (q)=(1 σ u (q c)) I (q+λβv m (σ i,σ i )), (5) and v u (q)=(1 σ m (q+c))(1 σ u (q)) I 1 (q+λβv m (σ i,σ i )). (6) Similarlytobefore,theterm (1 σ u (q c)) I istheprobabilitythattheinsuree accepts a fee q from the matched broker; (1 σ m (q+c))(1 σ u (q)) I 1 is the probability that the insuree accepts a fee q from a given unmatched broker; and λβv m (σ i,σ i )isthebroker sexpecteddiscountedcontinuationvalueofmakinga sale. Unmatched-Broker Strategy: Letusnowexplicitlyderivethedistributionσ u. Setanarbitraryvalue v m 0anddefineσ u ( )suchthat: v m (q)=v m (σ i,σ i )= v m, q [c g,+ ). It follows from(5) that: v m =(1 σ u (q c)) I (q+λβ v m ), q [c g,+ ). (7) Solvingforσ u (q): [ ] v 1/I m σ u (q)=1, q g. (8) q+c+λβ v m Notice that lim q σ u (q)=1. Moreover,σ u ( g)=0ifandonlyif: v m = c g (1 λβ). (9) 9

10 One can then write: σ u (q)= [ 1 0, q< g; c g (1 λβ)q+c λβg ] 1/I, q g. (10) Matched-BrokerStrategy: Usingananalogousreasoning,define v u 0andlet σ m besuchthat: v u =(1 σ m (q+c))(1 σ u (q)) I 1 (q+λβ v m ), q [ g, ). (11) Solvingforσ m (q): [ ] v (I 1)/I u q+λβ vm σ m (q)=1, q c g. (12) q c+λβ v m v m Notice that lim σ m (q) = 1, for every I 1. Moreover, from (9), one has q =1. Therefore,σ m (c g)=0ifandonlyif: c g+λβ v m v m v u = g+λβ v m = λβc g (1 λβ). (13) Therefore,σ m iswrittenasfollows: 0, q<c g; σ m (q)= [ ] (I 1)/I 1 (λβc g) λβ+ 1 λβ c g q (1 λβ)(q c)+λβ(c g), q c g. (14) Symmetric Equilibrium: Wemusthave v m v u,sincec>0. Moreover, v u 0 ifandonlyifg λβc. Onemustthenhave: g min(ḡ,λβc). (15) It follows by construction that: v m (q) v m, q<c g; (16) v m (q)= v m, q c g; (17) 10

11 and v u (q) v u, q< g; (18) v u (q)= v u, q g. (19) Giventheunmatchedbrokers strategies,thefunctionv m isuniquelydefinedand leaves the matched broker indifferent to any action in the support of σ m, namely [c g, ). Moreover, given that all other players draw fees from σ m and σ u, the functionv u isalsouniquelydefinedandleavesanyunmatchedbrokerindifferentto anyactionin suppσ u =[ g, ). Therefore, the strategies defined by(10) and(14) describe a stationary equilibrium for economy (I,λβ,c,ḡ), for any g satisfying (15). To see this, suppose by contradiction that this is not a stationary equilibrium. In that case, there should existastrategyoutsidesuppσ z thatyieldsanexpectedpayoffhigherthan v i,z for somez Z={m,u},whichisnotpossiblegivenconditions(16)through(19). Comments on the Symmetric Equilibrium: For any possible combination of the model parameters(i, λβ, c, ḡ), there exists a continuum of symmetric stationary equilibria indexed by g min(ḡ, λβc). Equations (9) and (13) define the brokers equilibrium payoffs. They can be arbitrarily large across the multiple equilibria as g can take arbitrarily negative values. This feature of the model is related to the fact that we have imposed no reservation price for consumers. Like in static Bertrand models(seebayeandmorgan,1999),thisleadstoatypeoffolktheoreminwhich any positive payoff can be achieved in equilibrium. FromaPoperianpointofview,thisisveryinconvenientsincethemodelistoo flexible to accommodate a vast set of empirical observations. We stress however that g < 0 (i.e., large profit levels) is not be consistent with the empirically observed mass of zero brokerage fees. 7 Therefore, even though any positive payoff can be rationalized by our model, collusive(large profits) equilibria will not be consistent with our data. Remark 1: Thestrategyσ m degeneratesintoq=(1 λβ)cwheng=λβc ḡ. Inthiscase,σ u putsmassonfeeslargerthanorequalto λβc. Thisimpliesthat the matched broker will always serve the insuree and no price dispersion will be observed in equilibrium. This equilibrium is Pareto optimal (since insurees never 7 It follows from (9) that c g = v m (1 λβ) 0, and then σ m (0) = 0. Therefore, only unmatchedbrokerssetnegativefees,andthisonlyoccurswheng>0. 11

12 switch brokers), but this does not assure maximum consumer s surplus(as brokerage profits are positive). 4 Data-Generating Distribution Thedatadisplayfeesthathavebeenpaidbytheinsureesratherthanalloffers generated by σ m and σ u. That is, the data generation distribution combines the equilibriumstrategiesσ m andσ u withtheinsurees selectionrules. Lety j R + representedthebrokeragefeesrecordedinthedata,wherej index eachobservation. Theinsureesaccessoneofferq 0 drawnfromσ m andiindependent offersdrawnfromσ u. Theychoosethematchedofferq 0 overq min =min(q 1,...,q I ) wheneverq 0 q min +c. Analogously,theyprefertheofferq min =min(q 1,...,q I )over q 0 wheneverq 0 >q min +c. Therefore,eachobservationjinthesampleisgenerated as follows: q 0 q min +c y j =q 0 ; (20) q 0 >q min +c y j =max(0,q min ). (21) Notice that q 0 0since the support of σ m is [c g, ) R +. Negative fees represent gifts and services that are offered to consumers and not observed by the econometrician. Thus,y j iscensoredatthepointq=0inequation(21). Thedistributionfunctionforq min =min(q 1,...,q I )isgivenby: H(q min )=1 (1 σ u (q min )) I. (22) Define Pr( ) as the probability distribution on some Borel measurable space (Ω, Ϝ) that represents the data generating distribution. For y 0, it follows from (20)to(21)that: Pr(y j y)=pr(a B), (23) wherea={ω Ω:q 0 q min +candq 0 y}andb={ω Ω:q 0 >q min +cand q min y}. SinceAandBaredisjointsets,oneobtains: and Pr(y j y)= Pr(y j y)= y g y g (1 σ m (ỹ+c))dh(ỹ), for0 y<c g; (24) (1 σ m (ỹ+c))dh(ỹ) y + (1 H(ỹ c))dσ m (ỹ), fory c g. (25) g 12

13 Recall from equation(21) that the mass of zero-brokerage fees is generated here asacensoringoftheequilibriumstrategyσ u atthepointq=0. Theprobabilityof observingazerofeeisgivenby: Pr(y j =0)= 0 g (1 σ m (ỹ+c))dh(ỹ). (26) Now fix g < λβc such that the distribution σ m is atomless (see Remark 1 for details). WecanusetheLeibnizruletoobtain: Pr(y j y)=pr(y j =0)+ wherethedensityfunctionp(y)isgivenby: and y 0 p(ỹ)dỹ, (27) p(y)=(1 σ m (y+c)) H(y), for0<y<c g; (28) y p(y)=(1 σ m (y+c)) H(y) y 5 Maximum Likelihood Estimation +(1 H(y c)) σ m(y), fory c g. (29) y We use the maximum likelihood procedure for censored data to estimate the parameter of our model (see Meeker and Escobar, 1994). Let J = {1,...,J} the sample-indexset,anddefinej 0 ={j J:y j =0}. Thusthelikelihoodfunctionis given by: L(I,λβ,c,g)= j J 0 Pr(y j =0) j J\J 0 p(y j ). (30) 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}. Inordertoguaranteepricedispersionandzerobrokeragefeesinequilibrium,weassume0<g<λβc. Numerically, wesetg=αλβcandtakeα {0,0.01,...,0.99}. Finally,wefixg=ḡ. Thisassumptionisirrelevantforourestimationgiventhat werestrictedg tobepositive,justlikeḡ. However,sincegisanequilibriumindex and not a structural parameter, allowing for g < ḡ would preclude policy analysis inwhichoneparameterischangedtakingallothersasgiven(seesection6). 13

14 We use the Wolfram Mathematica 7 software to compute and select the highest value of (30), given our data on brokerage fees, for all possible parameter combinations. 8 Mathematica isapowerful tool whichisnecessaryherebecausethereis noclosedformulafortheintegralin(26)thatdefinespr[y j =0]. Thatintegralis 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 equivalentto142brl(i.e.,aboutus$71inpurchasingpowerparity);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. Theestimatedvalueforλβ is0.61,butthemodeldoesnotallowustodetachthe discount factor from the match-continuation probability λ. Columns(b)and(c)presentrestrictedestimationsforλβ=0.90andforI=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 obtainedfromσ m andh. Unmatchedfeesstartat g= 37.26andappearuncontestedbymatchedoffersuptothepointc g= Afterthatpoint,offersfrom matched and unmatched brokers appear together in the data, generating a second peakinthedensityfunction. Theestimatedvaluefor g isdirectlyrelatedtothe massofpolicieswithzerofees,whilecisrelatedtothesecondpeakofthedensity function. [Figure 3] 8 StatisticalresultsinotherpartsofthispaperwerecomputedinSTATA

15 5.1 Econometric Comments Thelikelihoodfunctionisnotcontinuouswhentheλβ,c,andg takearbitrary realvalues. However,when(I,λβ,c,g)isevaluatedinafiniteset,thenthelikelihood 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 emphasize that consistency and asymptotic log-likelihood tests in this paper depend on the existence of an economic justification for the assumption thatthemodelparametersarevaluedinafiniteset. A second important econometric issue is related to the identification of a unique 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(global Maximum Analysis). Figures 4-7 present the likelihood values for different (I, λβ, c, g). 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 (Local Maximum Analysis). [Table 4&Figures4-7] 5.2 Subsample with a Single Vehicle Model Wenowperformtheanalysisfromthelastsectiononasubsamplethatrefines our baseline sample to include only the most popular vehicle model in Brazil(VolkswagenGol1.0L).Theresultsaresimilartothepreviousone. Table5andFigure8 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 asobservedinthedata. [Table 5&Figure8] IdentificationofasingleoptimalpointcanbeseeninTable6andFigures9-12. TheyaretheanalogofTable4andFigures4-7presentedinthelastsubsection. [Table 6&Figures9-12] 15

16 6 Policy Analysis Switching is costly. Therefore, any allocation in which insurees switch brokers is not Pareto optimal. However, taking the oligopolistic nature of this market as given, there is a class of equilibrium strategies in which switching occurs. In these cases, policies that reduce prices might be desired by consumers even if they increase switching. We perform counterfactual exercises to analyze the impact of different economic policies. It is important to stress that, since the game has multiple equilibria, the validity our structural policy analysis depends on the assumption that the same class of symmetric stationary equilibrium will remain being played after a change in each relevant parameter of the model. WefirstarguethatanincreaseinthenumberofunmatchedbrokersI doesnot reduce but actually increases the average consumer cost(i.e., the expected brokerage feethatiscontractedbytheinsuree). Thisisbecauseahighernumberofcompeting brokers decreases the impact of a price undercut on the probability of making a sale see equations(1) and(3). In order to illustrate this aspect of the problem, we take the estimated parameters from Table 3 and compute the expected contracted fee for different levels of I (see Figure 13). Insurees are better off when there is only one unmatched broker competing with the matched broker (i.e., I = 1). Counterintuitively, a policy seeking to promote competition through exogenously increasing the number of competitors in the market would actually reduce consumer welfare in this setting. [Figures 13] Second, we analyze policies intended to change the match-continuation probability, λ. The effect of λ over the distribution of contracted brokerage fees is quite difficult to analyze once it depends on other parameters of the model and on the particularregionofthesupportoffees. Wecanseefrom(10)thatthemassofzero brokerage fees offered to each consumer decreases with λ, that is: σ u (0) λ = 1/I [ ] c g 1/I 1 (c g)βg c λβg (c λβg) 2 <0, where g < c. However, the mass of policies effectively sold with zero brokerage fee need not decrease with λ, since the effect of λ over σ m may also increase the 16

17 probability of a positive matched fee that ends up being chosen over negative unmatched fees. For the parameter values presented in Table 3, the average consumer costincreaseswithλ. ThisispresentedinFigure14. [Figure 14] Next,westudytheimpactofchangesintheswitchingcosts. Ontheonehand, higher switching costs hurts consumers directly and increase average price offered by matched brokers see equation(14). On the other hand, it reduces the average price offered by unmatched brokers see equation (10). Just like before, we take the estimated parameters from Table 3 and compute the expected contracted fee for different levels of c (see Figure 13). Intuitively, the first effect dominates and the average contracted brokerage fee increases with the switching cost. [Figure 15] Finally, we ask about the welfare effects of allowing for gifts and negative fees. These practices are typically viewed as being anticompetitive. In this setting, however, allowing for gifts reduces the average prices of unmatched brokers and reduces both matched and unmatched life-time profits. These conclusions follow from equations (9)and (13). They can also be seen in Figure 16 which display theaverage consumercostfordifferentlevelsofg=ḡ,holdingallothermodelparametersequal to their estimated values from Table 3. Notice that a marginal increase in ḡ strictly improvesinsurees welfareuptothepointatwhichḡ=λβc. Afterthispoint,additional increases in ḡ should have no welfare impact, as follows from equation(15). It is particularly important to stress that our equilibrium is Pareto optimal for any economy with ḡ λβc. As stressed in Remark 1, the observed prices collapse to q = (1 λβ)c (no price dispersion). Moreover, the insurees never switch brokers (which is socially desired since switching is costly). The optimality of allowing for arbitrary negative fees is a strong policy lesson learned from this model. [Figure 16] Toconcludethissection,itisalsoinsightfultocomparetheresultsofourcounterfactual exercises to the effect of each policy over the Pareto optimal equilibrium discussed in Remark 1. This equilibrium occurs when g = λβc ḡ and, in that case, all brokerage fees are contracted at q = (1 λβ)c (i.e., no price dispersion 17

18 and no zero brokerage fee). An increase in c increases the equilibrium fees, while changesini andḡdonotaffectthepricesinthissetting. Contrarytoourprevious numerical exercise, and increase in λ should decrease price in this Pareto optimal equilibrium. 7 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 key economic forces behind the brokerage problem and 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. In addition to account for empirical regularities, our model was also used to evaluatefourpossiblepolicies. Inordertodoso,wetaketheparametervaluesobtained in our main econometric estimation, assume that the same type of symmetric stationary equilibrium will remain being played after a change in each relevant parameter of the model, and then compute the effect of changing each parameter over the average consumer cost. We first show that increasing the number of unmatched brokers I can actually hurt consumers. Second, for our estimated parameter values, increasing the match-continuation probability λ also increase the average consumer cost. Next, we show that reducing the switching cost and allowing for negative prices reduce brokerage profits and increase consumers welfare. These last results are insightful as they challenge the conventional regulatory wisdom of increasing the number of competitors and prohibiting loss leading and negative fees. References [1] Baye, Michael R. and John Morgan (1999). A folk theorem for one-shot Bertrand games. Economics Letters, 65(1),

19 [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), [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] Nowak, Andrzej S.(1985). Existence of equilibrium stationary strategies in discounted noncooperative stochastic games with uncountable state space. Journal of Optimization Theory and Applications, 45(4),

20 [13] Padilla, A. Jorge(1995). Revisiting dynamic duopoly with consumer switching costs. Journal of Economic Theory, 67(2), [14] Sharkey, Willian W. and David S. Sibley(1993). A Bertrand model of pricing and entry. Economics Letters, 41(2), [15] Stahl, Dale O. (1989). Oligopolistic pricing with sequential consumer search. American Economic Review, 79(4), [16] Taylor, Curtis R.(2003). Supplier surfing: Competition and consumer behavior in subscription markets. RAND Journal of Economics, 34(2), [17] Varian, Hal R. (1980). A model of sales. American Economic Review, 70 (4), [18] Villas-Boas, J. Miguel (2006). Dynamic competition with experience goods. Journal of Economics and Management Strategy, 15(1),

21 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.

22 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.

23 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.

24 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.

25 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 (g), 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.

26 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%.

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

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

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

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

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

32 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 (g), 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.

33 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%.

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

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

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

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

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

39 Figure 13. Average Consumer Cost as Function of the Number of Unmatched Brokers 800 Average Consumer Cost I Expected brokerage fee contracted by insures, for different values of I, when c = , λb = 0.61, and g =

40 Figure 14. Average Consumer Cost as Function of the Match-Continuation Probability Average Consumer Cost lβ Expected brokerage fee contracted by insures, for different values of lβ, when I = 1, c = , and g =

41 Figure 15. Average Consumer Cost as Function of the Switching Cost 1000 Average Consumer Cost c Expected brokerage fee contracted by insures, for different values of c, when I = 1, λb = 0.61, and g =

42 Figure 16. Average Consumer Cost as Function of the Highest Nonpecuniary Gift Average Consumer Cost g Expected brokerage fee contracted by insures, for different values of g, when I = 1, c = , and λb = 0.61.