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La Crema: A Case Study of utual Fre Insurance Antono Cabrales y Anton Calvó-Armengol z atthew O. Jackson x Ths Draft: December 12, 2000 Abstract e analyze a mutual re nsurance mechansm used n Andorra,

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La Crema: A Case Study of utual Fre Insurance Antono Cabrales y Anton Calvó-Armengol z atthew O. Jackson x Ths Draft: December 12, 2000 Abstract e analyze a mutual re nsurance mechansm used n Andorra, whch s called La Crema n the local language. Ths mechansm reles on households announced property values to determne how much a household s rembursed n the case of a re and how payments are apportoned among other households. The only Pareto e cent allocaton reachable through the mechansm requres that all households honestly report the true value of ther property. However, such honest reportng s not an equlbrum except n the extreme case where the property values are dentcal for all households. Nevertheless, as the sze of the socety becomes large, the bene ts from devatng from truthful reportng vansh, and all of the non-degenerate equlbra of the mechansm are nearly truthful and approxmately Pareto e cent. Keywords: nsurance, contract theory, mechansm desgn, truthful revelaton. JEL Class caton: A13, C72, D64, D80 e are grateful to Lus Corchón and Joel Sobel for ther comments. e also gratefully acknowledge the nancal support of Span s nstry of Educaton under grant PB and the Generaltat de Catalunya under grant 1999SGR and the Natonal Scence Foundaton under grant SES y Department of Economcs, Unverstat Pompeu Fabra, Ramon Tras Fargas 25-27, Barcelona, Span. Emal: z Department of Economcs, Unversdad Carlos III, C/ adrd 126, Getafe (adrd), Span and CERAS-ENPC, 28 rue des Sants-Pères, Pars, France. Emal: x Dvson of Humantes and Socal Scences , Calforna Insttute of Technology, Pasadena, Calforna 91125, USA. Emal: 1 1 Introducton utual nsurance companes wrte large proportonsof nsurance polces n many sectors. 1 They have been very successful for several reasons. Frst, as alnvaud (1973) ponts out, future markets provde only a remote dealzaton to the actual mechansmforrsk allocaton snce the deal market system s too costly to mplement. On the contrary, poolng ndvdual rsk by means of mutual nsurance polces permts substantal economzng on market transactons (Cass, Chchlnsky and u, 1996). Another mportant reasonforthe success of mutual nsurance s that they can solve through peer montorng some moral hazardproblemsthat plague ncorporated nsurance companes. 2 3 hle these problems are well understood, mutual nsurance arrangements also solve other nformatonal problems relatng to the dscovery of the value of nsured property, as we show here. In ths paper we present and analyze a real-lfe mutual re nsurance mechansmthat has been functonng n a rural mountanous area of estern Europe for well over a century and a half. In ths mechansm, called La Crema n the local language, each partcpatng household must report a value. In case there sa re, the ownerof the burnedhouseholdreceves her reported value, whch s pad by all partcpatng households (ncludng herself) n proporton to ther reported values. e focus on the rules of La Crema because they are partcularly clear from a game-theoretc pont of vew, and they are by no means exceptonal, and the mechansm has some remarkable propertes. 4 1 Advance premum mutuals wrte almost 40 percent of the lfe nsurance n force and almost 23 percent of the property and lablty nsurance premums. (llams, Smth and Young, 1998). 2 utuals seem to have been more e ectve than stock companes n constructng such ncentve systems, partcularly n the early phases of ther hstory. Indvdual ndustralsts were sometmeslargeenough to makenvestment n research on repreventon worthwhle, but stock companes dscouraged theprovson of publcgoods by appropratng too much of the savng from decreased re losses (Hener, 1985). 3 Obvously, mutuals have problems of ther own, or they would be the only organzatonal form. From a nancal perspectve, the key mpedment to mutual lfe company stablty, growth and development, s that equty captal can be rased only through retaned earnngs from the company s operatons, (Garber, 1993). Also, mutuals are very d cult to take over, whch makes the corporate governance problem harder to solve, especally n large mutuals. 4 A smlar proporton rule s adopted, for nstance, n marne nsurance clubs: At the begnnng of the year the shpowners are gven an estmate of the amount (call) they wll 2 Inpartcular, the propertes of the La Crema mechansm that we explore concern ts e cency characterstcs and the ncentves t provdes for truthful reportng of property values. th regards to e cency, the mechansm places strong constrants on the possble rsk sharng that can take place snce rembursements and payments are both scaled drectly n terms of the announced property values. For nstance, f households have constant (and dentcal) relatve rsk averson, the only Pareto e cent allocaton that s reachable through the game requres that all households truthfully report the value of ther property. Thngs are even worse wth constant (and dentcal) absolute rsk averson as then no Pareto e cent allocaton s obtanable as an outcome n the game regardless of how the announcements are vared. 5 th regards to the ncentves that the mechansm provdes for truthful reportng of property values, we show that there s anequlbrumwhere all households report the true value of ther nsured property f and only f these valuatons are exactly the same acrosshouseholds. Apart fromths extreme case of dentcal property values, we show that households wth relatvely hgh property values have an ncentve to overreport ther value (to ncrease rembursement fromothers when needed) and households wth low property valueshave anncentve to underreport ther value (to decrease payment to others when asked for). The analyss descrbed above appears to be n con ct wth the conventonal wsdom among the actual partcpants n the game, who are happy wth the functonng of the mechansm and consder that the only natural thng one can do s to report the true value of the property. Snce the mechansmhas exsted fora long tme one would thnk that tradton or therown experence could furnsh enough nformaton for agents to know ther best response. In fact, the ncentve and e cency propertes that the mechansm exhbts are qute appealng and closely n lne wth local wsdom once we examne large enough socetes and consder approxmate rather than exact e cency. From the perspectve of larger socetes, we rst show that households n large enough socetes have arbtrarly small ncentves to devate from be requred to pay nto the [Protecton and Indemnty] Club. However, the eventual call s dependent upon the clam made by all members: each member knows only the proporton [emphass n theorgnal] ofthetotal cost they wll be requred to bear. (Bennett, 2000). 5 As one would expect, by the nature of the mechansm, where only property values are reported, d erences n rsk averson do not seem to be the answer ether. 3 honest reportng, ornotherwords, truthsan -nashequlbrum. Second, we show that n large enough socetes, the (exact) Nash equlbra of the La Crema mechansm nvolve reports that are arbtrarly close to the truth. 6 Thrd, the Nash equlbra (and -Nash equlbra) are arbtrarly close to beng Pareto e cent n large enough socetes. Fnally, we show that for reasonable parameterzatons of utlty functons what s needed n the above statements n terms of large enough socetes, can actually be reasonably small. The nterest of ths nsttutons manfold, and qute d erent fromother studes of rsk sharng nsttutons. 7 Frst of all, the La Crema nsttuton refers to a specalzed type of rsk, whch lmts the potental explanatons for observed behavor. Secondly, the transfer rules are qute explct and regulated. Fnally, the rural socety under consderatons relatvely rch durng the whole perod of the mechansm s operaton (for example, there are no nstances of famnes durng ts exstence). The remander of the paper proceeds as follows. Secton 2 descrbes the mechansm (nformally and formally) and gves some background on the socety where the nsttuton operates. Secton 3 dscusses the equlbrum and e cency propertes of the mechansm. Secton 4 provdes results characterzng the equlbra and approxmate e cency of the mechansmn large socetes. Secton5 concludes. 6 One should note, of course, that whle the larger scale of socety may solve the reportng problem of La Crema, t may create other problems, as provdng adequate re preventon can become now a worse publc good problem. Today s P&I [Protecton and Indemnty] Clubs are global n scale, wth the largest contanng over 20% of the world s oceangong eet. Communal responsblty may be unrealstc n such large-scale nsttutons because free rder problems become more d cult to montor and control as group sze and dsperson ncrease. (Bennett, 2000). 7 Such as the ones mentoned n ccloskey (1989), Townsend (1993) or Fafchamps (1999). Besley, Coate and Loury (1993, 1994) examne the allocatve performance of a smple, easly organzed and wdely observed nsttuton for nancal ntermedaton called rosca (rotatng savngs and credt assocatons). 4 2 La Crema 2.1 The nsttuton of La Crema In 1882, and under the ntatve of the local prest, the 102 farms of Canllo n Prncpalty of Andorra 8 organzed themselves nto a re nsurance cooperatve named La Crema. By that tme, Andorra was mostly a rural area lvng n quas-autarchy, and La Crema was conceved as a rsk-sharng nsttuton to cope wth re damages that were a source of major worres to farmers n mountanous Canllo where snuous and steep roads dd not allow for quck nor e ectve re brgades. Snce ts early begnnngs, the role of La Crema was twofold: as a logstc structure, to organze the local reman forces; as a nancal structure, to guarantee pecunary compensatons to farms su erng re destructons. 9 The organzaton of La Crema s as follows. Once a year, the cooperatve members meet n a general assembly, the consell de La Crema (La Crema councl). The meetng s xed on the Sunday that falls two weeks before the carnval and attendance s compulsory for all members. 10 The meetng s supervsed by two permanent secretars (secretares) who are elected for lfe. Durng ths general assembly, each farmer announces a value for each of the buldng that he or she owns (farm, barn, cow-shed, stable, etc.). Conventonal wsdom suggests that farmers report the true and total value of ther property, and La Crema cooperatve members typcally do so. Ths amount s noted n three d erent books: each secretar keeps a copy at home and a thrd book s stored at the parsh town-hall. 11 In the case of a re, the owner 8 The Prncpalty of Andorra, located n the heart of the Pyrenees between France and Span, s both one of the smallest and the oldest states n estern Europe: the natonal terrtory s 468 km 2 and today s fronters were de ntely settled n The country s dvded admnstratvely nto seven parshes: Canllo, Ordno, La assana, Encamp, Andorra la Vella, Sant Julà de Lòra and Escaldes-Engordany. Agrculture has been the major economc actvty of Andorra untl the end of the 19 th century; toursm, commerce and nancal servces are now the basc natonal economc actvtes. In 1999, the GDP per capta was 20,252 $. See for more detals. 9 La Crema s stll actve and ntervened recently to nancally compensate Cal Soldevla whose barn partally burned n August of 1998 and Cal Batsta for smlar damages n July of An absent member wthout a good excuse s ned. The last ne dates back to e are ndebted to one secretar, Josep Torres Babot (Cal Jep), and to the Canllo publc lbraran a Dolors Calvó Casal (Cal Soldevla) for ther nvaluable help n provdng thorough nformaton about La Crema durng long conversatons. 5 of the damaged buldng receves a compensaton equal at most to the value noted n the book for the current year, dependng on the extent of the damages. Ths nancal compensaton s made by the other cooperatve members, who pay n proporton to the share ther own announced property value represents wth respect to the total of all values announced by the La Crema members. An early reference and bref descrpton of the La Crema transfer rule can also be found n Brutals (1904): Comme dans toute les populatons aux prses avec une nature ngrate, la soldarté est développée parm les Andorrans; elle a donné nassance à des socétés d assurances mutuelles contre l ncende. Les socétés d assurances sont généralement ouvertes aux habtants d un vllage; les assocés peuvent refuser d admettre au béné ce de l assurance les mmeubles dont les rsques dépassent la moyenne. En cas de snstre, chacun pae, pour ndemnser le proprétare, au prorata de la somme pour laquelle lu-même est assuré (p. 42). 12 Durng the yearly meetng, four comssonats (commssoners) andthree recaudadors (money-collectors) are elected for one year. The comssonats are responsble for the logstc and techncal actvtes. Frst, they guarantee that all cooperatve members take the approprate precautonary measures to prevent possble res by reportng to the consell de La Crema carelessness n farm and buldng mantenance. Second, they are n charge of the re ghtng materal owned by the cooperatve ( re-hoses, etc.). Fnally, n case of re, the comssonats x, n accordance wth the concerned farmer, the total value of the damages to be rembursed (dependng on the extent of the damages and not exceedng the value noted n the book) and submt t to the consell for approval. The three elected recaudadors represent each a d erent geographcal area: Canllo, la Rbera and Prats. 13 In case of re, and once the amount to be transferred to the damaged farm s xed by the consell under proposton of the comssonats, the recaudadors are responsble for collectng the contrbutons of the La Crema members wthn ther area of 12 As t s often the case wth socetes lvng n nhosptable areas, soldarty s hghly developed among theandorrans, and has gven rse n partcular to mutual re-nsurance assocatons. Inhabtants of a same vllagecan usually all becomensurancesocety fellows. Nonetheless, buldngso erng re-rsksabove average may bedened nsurance coverage. In caseof damage, all fellows pay to compensate the owner for her loss, and they do so n proporton to the value for whch they are themselves nsured. 13 The rst regon, Canllo, corresponds to the man town wth the same name. The second regon, la Rbera, ncludes the followng vllages: Els Plans, Els Vlars, El Tarter, L Aldosa, L Armana, Ransol and Soldeu. Fnally, the last regon, Prats, ncludes: El Forn, ertxell, olleres and Prats. 6 nterventon. 2.2 The La Crema game There s a setn of households, wth jnj=n. Each household has a utlty functonu and a wealthw 2[c;C] wherec c 0. Let = P 2Nw. 14 e take eachu to be twce contnuously d erentable and strctly concave. LetS=2 N be the set of possble states. In partcular,s 2S s a lst of farms that burned. For nstance,s=f2;7;12g denotes that farms 2, 7 and 12 (and only those farms) burned. Let S (k) = fs j #s = kg be the set of states where exactlyk farms burn. Note thats= [ n k=0s (k). For any 2N, let S denote the set of states for whch farm burns (perhaps along wth some other farms), ands (k) be the set of states for whchk farms n addton to farmburn. Letp s be the probablty of states. e assume that all states where an dentcal numberk of farms burn are equally lkely. That s, for all s;s 0 2S (k),p s =p s 0 and we denote ths probablty byp k. A specal case of ths swhere eachfarmburnswthanndependent anddentcalprobablty. Note, however, that t s not requred that the burnngs be ndependent. As an extreme example, t could be thatp 0 0andp n 0 andp k =0for all otherk. Ths mght be an example where all the farms le close to each other n a forest, so that ether all farms burn or none burns. All we assume s that p k 0 for somek 0, so that there s some chance of a re. e now descrbe formally the rules of the La Crema game. Each household sends a messagem 2[0;2C]; to the coordnator, whch s nterpreted to be an announcement of ther property value. 15 Letm=(m 1 ;:::;m n ) 2 [0;2C] n be a vector of messages. Let = P 2Nm and for all s 2 S, let s = P 2Nnsm. The allocaton rule used by the coordnator s the followng: n state s 2 S, household 2 s receves m s ; whereas each ( household j 2 Nns receves w j m s) j : One can easly check that ( Pj2Nnsm s) j = P s 2sm, namely that the sum of the contrbutons by householdsj2nns whose farms dd not burn s equal to the sum that households 2 s receve as a compensaton for ther losses. Note that f announcements are truthful (m =w ), then n each statesthe undamaged 14 e treat wealth as the property that may potentally burn. Utlty functons may, of course, be normalzed so that ths s wthout loss of generalty. 15 The upper bound on announcements s arbtrarly set at twce the hghest magnable property value. Any upper bound would do. 7 property s e ectvely dstrbutedamong all households n proporton to ther wealths (so the nal allocatons are s w ). 3 Dscusson of the game 3.1 Equlbra The rst proposton says that truthful announcements are a Nash equlbrum only n the case where all wealths are dentcal. Proposton 1 The La Crema game has a Nash equlbrum n pure strateges wherem =w for all 2N f and only fw =w j, 8;j 2N. The proof of Proposton 1 appears n the appendx. The ntuton behnd the Proposton s roughly as follows. Increasng m has two e ects. Frst, t ncreases the rembursement that household receves n the case of a re that consumes s property. Second, t ncreases the lablty that faces n the event that some other household s property burns. Some heurstc calculatons help llustrate the relatve sze of these two e ects and the ncentves that households have as a result. For smplcty, consder a stuaton where at most one household wll have a re, and so we need only consder states of the form fg, where s property s destroyed. 16 Consder what happens f rases m by some small amount 0. Ths ncreases s rembursement by (approxmately) f the state s fg (where recall that = P j6=m j and = P jm j ). It also ncreases the payments that has to make to householdj 6=n state fjg bym j. Note that summng across states, these cancel each other out. That s, = P j6=m j. So, by lowerng the announcement m, householdtransfers wealth from state fg to the other states fjg,j 6=; and vce versa from rasng the announcement. So what are the households ncentves n the game? Gven therrsk averson, they wsh to come as close as possble to smoothng ther wealth across the states. If all households have exactly the same wealth, then at a truthful announcement n the La Crema game household gets nal wealth w s n state s, and gven the equal startng wealths s equal across each 16 The state where no farm burns has no mpact snce no payments are made. States where several farms burn have analogous calculatons as those dscussed here, as the consderaton s what happensf s farmburnsversus some other farmburns (on themargn). 8 state s = fkg. Thus, the households wealths are evenly spread acrossthese states and they have no ncentves to change ther announcements. Next, consder the case where households do not have the same wealth. Order them so that w n w n 1 w 1, and w n w 1. Then notce that farmer 1 consumes the hghest amount n the state where her property burnsw 1 1, versus w j 1 n some state j 6= 1, snce 1 2 n. By lowerng m 1 a lttle, household 1 decreases consumpton n the state f1g where farm 1 burns, and dstrbutes a commensurate ncrease among other states fjg, where farmj 6=1 burns. As households are rsk averse, ths strctly bene ts household 1. Conversely, farmern consumes less n the state where farm n burns compared to states where some other farm burns. By rasng m n, farmer n shfts wealth fromstates fjg,j 6= n, to state fng. Roughly, households wth below average property value wll bene t from underreportng, and those wth above average property value wll bene t fromoverreportng. The proposton tells us that the game does not have an equlbrum where households report the true value of ther property f there s any heterogenety n household value. The case of heterogenety s arguably the nterestng case, as t would be hard to see the reason for an elaborate mechansm (whch s not costless to admnster) unless there were some knd of heterogen

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