Information Cost, Broker Compensation,

Transcription

1 Bidding Insurance Strategies Maret Annette Hofmann/Martin Nell* Information Cost, Broer Compensation, and Collusion in Insurance Marets ** Abstract We examine the impact of intermediation on insurance maret transparency and performance. In a differentiated insurance maret under imperfect information, consumers can gain information about product suitability by consulting an intermediary. We analyze current broer compensation methods: commissions and fees. Although insurers equilibrium profits are equivalent under both systems, social welfare is always higher under a fee-for-advice system than under a commission system. Both systems offer the opportunity to increase profits via collusion. Under a commission system, collusion enables insurers to separate consumers into groups purchasing different contracts. Insurers may then extract additional rents from some consumers. This advantage can explain why broers tend to be compensated by insurers. JEL-Classification: D43, G22, L13. Keywords: Collusion; Insurance Maret; Intermediation. 1 Introduction Common economic models of insurance marets usually consider either a monopolistic insurer or a perfectly competitive insurance maret. These contrarian analyses suppose two implicit assumptions. The first assumption is that insurance is a homogeneous good. The second is that any insurer might be able to serve the whole maret demand. Given these assumptions, if an insurer offers a marginally smaller price than its competitors, then it gains (and serves) the whole maret demand. Therefore, if at least two * Annette Hofmann, post-doc, University of Hamburg, Institute for Ris and Insurance, Von-Melle-Par 5, 2146 Hamburg, Germany, hofmann@econ.uni-hamburg.de. Martin Nell, Chair, University of Hamburg, Institute for Ris and Insurance, Von-Melle-Par 5, 2146 Hamburg, Germany, martin.nell@wiso. uni-hamburg.de. ** We would lie to than two anonymous referees for helpful comments and suggestions. Moreover we wish to than participants in the Asian-Pacific Ris and Insurance Association Annual Meeting in Sydney, August 28, as well as in the 35 th Seminar of the Group of Ris and Insurance Economists in Toulouse, September 28, for valuable discussions. sbr 63 July

2 A. Hofmann/M. Nell insurers compete in premiums, Bertrand competition yields zero profits and insurance premiums are actuarially fair. This so-called Bertrand Paradox is named after Bertrand (1883) who argued that if duopolistic firms compete via price, the non-cooperative equilibrium price would fall to marginal cost since the firms would eep undercutting each other. Today, the Bertrand Paradox seems unrealistic and might be inappropriate as a basic theoretical concept for economic reasoning with regard to (rather oligopolistic) competition in insurance marets. Most insurance models interpret an insurance contract as a pair of only two parameters, the insurance premium paid by the insured regardless of state and the indemnification payment paid by the insurer in case of loss. As pointed out by Schlesinger and Schulenburg (1991), in reality such an insurance contract is a long description of contingencies for which the contract does or does not pay out. Thus, the individual should mae a reasonable distinction between the insurance contract and the insurance product. The insurance product represents a service that may differ from other services. Therefore, the actual fit of the product to the needs of a consumer also differs. The service may include claims settlement and ris management services, the availability of local agents, and method-of-payment options. Warranties and embedded options in life insurance are also a means by which we can differentiate insurance products. For instance, consider life insurance policies with the same premium but different maturities or embedded options. In health insurance, insurance companies often wor together with care providers who can offer different inds of services (managed care). Policyholders can become member of a preferred provider organization, one in which an insurer reimburses the member for the cost of treatment. In summary, as pointed out by Schlesinger and Schulenburg (1991): Product characteristics differ between insurers even though insurance contracts seem to be identical. Schlesinger and Schulenburg (1993) find empirical evidence of perceived product heterogeneity in insurance marets in Germany. Despite many potential sources of information about insurance products available today, some consumers who are looing for insurance coverage are often not well-informed about the product or insurance company that best suits their individual preferences. Although consumers tend to be well-informed about the existence and prices of insurance products (and maybe even some attributes that differentiate them), due to a lac of information they might not be easily able to determine their best matching product. A reason for this phenomenon could be that information is not easy to get and is sometimes too costly to obtain. In fact, insurance is actually a complex and multidimensional product, and consumers are usually poorly informed about its true characteristics (usually contained in the contract s so-called fine print ). Given imperfectly informed consumers in insurance marets, it is plausible to introduce intermediaries who may improve maret transparency and efficiency by servicing consumers. Intermediaries can act as experts with superior information about contract 288 sbr 63 July

3 Insurance Maret conditions and product features. Yet, increasing maret transparency is often thought to promote price competition (see, e.g., Stigler (1964) or Tirole (1988)), and it might not be in the interest of insurers to support intermediaries. In differentiated insurance marets, however, the broers function in improving maret transparency implies that uninformed consumers become aware of product varieties. Therefore, product differentiation is increased which maes the maret less competitive and might imply higher expected profits for insurance companies in the long run. As a result, intermediation is generally profitable for insurers. However, there is an ongoing discussion about who should actually pay the intermediary. Current intermediary compensation systems are the fee-for-advice system, under which broers are paid by consumers, and the commission system, when broers are paid by insurance companies. In actual practice, intermediaries usually earn commissions paid by insurance companies. Commission payments are mainly conditioned on premiums. But this compensation structure may offer insurance companies and broers incentives for collusion. Indeed, recent events in the United States and Germany illustrate these considerable incentive problems. In 24, New Yor Attorney General Eliot Spitzer sued Marsh & McLennan, the largest insurance broer in the world, over alleged collusion with insurance companies (see Spitzer (24)). In 25, a cartel of German insurers considerably suppressed price competition and successfully increased premiums (see EC (27)). Therefore, commissions paid by insurance companies have recently been under attac and potential regulatory interventions have become an important ingredient in political debate (see, for instance, Ladbury (27)). We examine these maret imperfections. In our study s framewor, consumers can observe insurance prices but are uninformed about product features. Consumers may engage in costly search for product features or consult an intermediary. In particular, some consumers information cost might be very high. Intermediaries act as information broers; they might improve the matching of those consumers to insurance companies, and do so at a lower cost. We evaluate current broer compensation systems and their economic implications by taing into account the incentive structures of all maret participants. The paper is organized as follows. In Section 2, we discuss related literature. Sections 3 and 4 introduce our model. As a point of reference, we first study a differentiated insurance maret without intermediation in Section 5. Section 6 introduces intermediation. Social welfare is studied in Section 7. Section 8 introduces collusion. The last section concludes. 2 Related Literature Previous research on intermediation addresses several functions that intermediaries might generally fulfill in marets with imperfect information. For instance, Biglaiser (1993) presents a basic bargaining framewor in which intermediaries can reduce inefficiencies that arise from adverse selection problems. Rubinstein and Wolinsy (1987), Posey and Yavas (1995), Cosimano (1996), Posey and Tennyson (1998) as well as Seog sbr 63 July

4 A. Hofmann/M. Nell (1999) study broers as search agents whose function is to match trading partners. Rubinstein and Wolinsy (1987) present a general framewor in which time-consuming negotiations between buyers, sellers, and intermediaries tae place. Cosimano (1996) introduces a monopolist intermediary who lowers the probability of an unsuccessful trade among buyers and sellers. Posey and Yavas (1995) and Posey and Tennyson (1998) refer to insurance marets. These authors offer search models in which insurers and consumers can engage in costly search for matching partners. Both insurers and consumers can be of either high or low search cost type. Insurers may transfer the search tas to broers. Broers then fulfill a search function by looing for potential customers at a commission fee to be paid by the insurer. Seog (1999) examines an insurance maret in which consumers are poorly informed about the price distribution. To find a long-run equilibrium where dependent and independent broers might coexist, Seog focuses on dynamic aspects of price search by intermediaries. Cummins and Doherty (26) study an insurance maret in which insurers cannot observe loss propensities but do now the overall average loss probability. In this setting, broers might prevent maret failure, which is liely due to ris selection over time, since their superior information allows them to inform insurers about loss probabilities of their customers. Gravelle (1994), building on his earlier research written in 1991 and 1993, offers a model for a competitive insurance maret with intermediation. Although Gravelle (1994) also deals with transaction costs and broer compensation in insurance marets, he is not concerned with matching. In his framewor, insurance products are homogeneous and consumers differ only in their willingness to pay for insurance coverage that is determined by some unnown random variable. This variable is observed only by broers. Therefore, the broers main service consists of advising consumers on whether they should purchase insurance coverage. In contrast, we allow for product differentiation and product-specific mismatching. Gravelle assumes that insurance contracts brea-even implying that neither broer compensation system affects insurers expected profits. Focht, Richter, and Schiller (29) build on Schultz (24) s product differentiation framewor. As we do in this study, they loo at compensation customs of intermediaries and analyze these customs from a profitability and welfare viewpoint. In a model with uninformed consumers who can neither observe prices nor product varieties, it is shown that a pure-strategy equilibrium exists if all consumers have rational and identical expectations about prices and product varieties. Without intermediation, equilibrium profits of insurance providers are higher than under intermediation. When they introduce intermediation, the authors find that both fee-based and commission-based broer remuneration systems are payoff-equivalent for insurers. Our assumptions and results differ from Focht et al. in two important respects: First, because we assume that prices are observable but product varieties are not. The rationale underlying this assumption is that price information seems quite easily available but information about individual product match is less obvious and often requires expertise in finance, actuarial science, 1 See Gravelle (1991; 1993). 29 sbr 63 July

5 Insurance Maret law or engineering. Second, because consumers are able to search for information about their best-fitting product and do not necessarily need a broer s advice. Schiller (29) studies the incentive effects of different broer compensation systems for independent agents. He shows that in a situation in which broers report private information truthfully, a commission system is wealy superior to a fee-for-advice system given that both matching and ris classification are of major importance in the maret. The superiority of the commission system results from the lac of the insurer s ability to identify ris types. Mismatching will not occur in maret equilibrium, since a good match is more valuable than a bad one. Given truthful reporting, a commission system implies perfect matching and ris classification. 3 Model Structure Our model follows Hotelling (1929), who formulated the following model of location and price choice in duopoly: The model is a two-stage game between the two firms. In the first stage, each firm (simultaneously) selects a location on the unit line, and in the second stage, having observed the locations selected, each firm (simultaneously) offers a price. Consumers are uniformly distributed on the unit line. A single good is produced at zero cost by two firms, each of which selects a location in the unit line and a price. Consumers have travel costs proportional to the distance to the firms, and buy one unit of the good from the firm for which price plus travel cost is lowest. Hotelling s model has been discussed in different contexts by many authors. Our setting follows d Aspremont, Gabszewicz, and Thisse (1979). We consider an insurance maret with a continuum of consumers. The number of consumers in the maret is normalized to one. Consumers are uniformly distributed on a line of unit length. There are two insurers in the maret, j {1, 2} each of which offers some variant of an insurance product. Insurers are located at the extreme points of the unit line: Provider 1 is located at zero and provider 2 at one. This assumption is without loss of generality. The position of a consumer on the Hotelling line represents his individual preferences for the insurance products offered. A consumer located at x (, 1) cannot have a perfectly matching product, so there is some disutility involved in purchasing this product. Consumers cannot observe product varieties in order to determine which insurance product actually offers the best match for their type. Technically, they do not now their individual position on the Hotelling line. Following the principle of insufficient reason, the consumers have rational expectations of being in 2 See Cummins and Doherty (26, 392). 3 Our results generalize to the case of N firms in an insurance maret. Maintaining the symmetric variety pattern of the Hotelling duopoly maret, the N firm case can be analyzed in a circular street maret as in Salop (1979) where product varieties are symmetric. 4 The locations and 1 are actually chosen by insurers in equilibrium as we show in the Appendix. In order to simplify exposition, we refer to locations and 1 here. See also the conclusions by d Aspremont, Gabszewicz, and Thisse (1979). sbr 63 July

6 A. Hofmann/M. Nell the middle of the maret. Hence, observing prices, they tend to purchase insurance at the cheapest provider. Consumers are uniformly distributed on locations. This is common nowledge. All consumers are ris-averse with respect to the insurable ris and ris-neutral with respect to mismatch ris. Consumers differ in their preferences for insurance products (i.e., their location on [, 1]). Thus, the consumers individual suitability of insurance products also differs. Ris-neutral insurers offer premiums P i = E + p i, where E is expected loss of the policy and p i represents the premium loading. Since premiums are marginally fair, i.e., they include a fixed loading fee, full insurance is optimal for consumers as long as the cost of insurance net of expected loss (the sum of premium loading and individual disutility of mismatch) does not exceed their individual ris premium r. We interpret the expected loss plus the ris premium as the individual willingness to pay for full ris reduction of a consumer. The amount paid by policyholders above actuarial cost is the price of insurance. Hence, we refer to p i as the price for provider i s insurance product. Given some positive transportation cost t >, an informed consumer located at x [, 1] faces disutility tx 2 from purchasing an insurance product from insurer one. He faces disutility t(1 x) 2 when he purchases from insurer 2. The heterogeneity or transportation cost parameter t represents the marginal disutility of mismatch and thus measures the intensity of product differentiation. In our setting, we refer to t as disutility. In doing so, we emphasize the utility loss from purchasing an insurance product that does not perfectly match a consumer s type. Figure 1 depicts the sequence of play. It shows that we might introduce a stage zero into our game, in which we allow insurers first to simultaneously compete in locations (pro- 5 The principle of insufficient reason states that if an agent is ignorant of the ways an event might occur (and therefore has no reason to believe that one way will occur rather than another), the event will occur equally liely. See Keynes (1921, 52-53). 6 Note that all consumers are expected utility maximizers in our model. We abstract from potential violations of rational behavior. However, experimental evidence suggests that consumers may not always act rationally in practice. For instance, ris perceptions are often biased. These biases may very well be relevant in this maret. However, as a benchmar, we find it useful to understand the implications within a rational choice framewor here. It seems interesting to study how our results would vary when these biases are taen into account. 7 Introducing ris aversion with respect to mismatch ris complicates the analysis without providing further insight. 8 See Pratt (1964), Arrow (1963), Mossin (1968), Smith (1968), or Doherty (1975). 9 It might seem arbitrary that we use quadratic dependence. The assumption that unit disutility rises more than proportionately with distance to an insurance provider is for technical reasons only. Given linear transportation cost, the so-called principle of maximum differentiation (i.e., firms locate as far from each other as possible in equilibrium) is replaced by the principle of minimum differentiation. See, for instance, Martin (22, 99). However, one should not expect minimum differentiation as advocated by Hotelling. This is because the price subgame in Hotelling s model fails to have a pure strategy equilibrium if firms are located too close to each other (but not at the same point). Correcting the non-existence problem in Hotelling s original model, d Aspremont, Gabszewicz, and Thisse (1979) verify that with quadratic transportation cost a price equilibrium exists for all possible locations. Economides (1986) shows that maximal differentiation also results for more general convex functions. 292 sbr 63 July

7 Insurance Maret duct varieties) on the Hotelling-line (before competing in premiums). For expositional simplicity, do not explicitly consider this stage. The locations of insurers are not very interesting from our point of view, since we are mainly interested in gaining insight into the performance of broer remuneration systems. However, we note that insurance providers will actually choose these extreme locations zero and one, respectively (see Appendix). Our solution concept is subgame-perfect equilibrium and the solution method is bacward induction. Figure 1: Sequence of play () Insurers select their location on [, 1] Consumers decide whether to become informed (about varieties) Insurers announce premiums Consumers choose an insurance provider 4 Costly Consumer Search Consumers can engage in costly searches for suitable insurance products. We suppose that consumers differ in individual information or search cost θ [, θ], θ >, according to an arbitrary continuous density f(θ). If a consumer decides to search for a suitable insurance product, then search reveals his position on the Hotelling line perfectly. Since consumers are initially uninformed, they have to mae a choice: each consumer must decide whether he wants to engage in costly search or not. This decision depends, of course, upon individual search cost and expected value of information. Consumers have rational expectations and base their decisions to become informed on expected mismatch and information cost 1. Expected (mismatch) cost of an uninformed consumer is 1 EMC U = tx 2 dx = _ t, (1) 3 1 It might seem implausible that prices are not taen into account in the consumers decision process. This is because at this stage of the game, prices are not (yet) set. It should be noted, however, that consumers would expect prices to be symmetric due to the symmetry of the problem. Therefore, prices do not play a major role in this decision. sbr 63 July

8 A. Hofmann/M. Nell and expected mismatch cost in case of information is given by 11 1/2 EMC I = 1 tx 2 dx + 1/2 t(1 x) 2 dx = t. (2) Hence, consumers decide to engage in costly searches if expected reduction in transport cost is higher than the individual search costs: θ + t 12 _ t 3 θ θ := _ t. (3) 4 Depending on the distribution of search cost in the consumer population, we can identify two cases: the first is θ t/4 and the second is θ > t/ Maret Equilibria without Intermediation We assume that the maret is completely covered, i.e., that consumers are sufficiently ris-averse and no consumer prefers not to purchase insurance. An informed consumer purchases insurance from provider 1 if his net utility that is, his utility resulting from individual transport cost plus the net price of insurance coverage is higher than net utility when purchasing from provider 2: r (p 1 + tx 2 ) r (p 2 + t(1 x) 2 ) x p 2 p 1 + t, (4) where p 1 + tx 2 represents the total cost of purchase for a x-type consumer when purchasing at provider 1. Similarly, p 2 + t(1 x) 2 when the consumer purchases at provider 2. A consumer is indifferent if he is located at ˆx := p 2 p 1 + t. (5) 2t ˆx is provider 1 s demand of informed consumers while there is Bertrand price competition for uninformed consumers. First, we assume that θ < θ. In this case, some consumers search for information about product varieties and some consumers prefer to stay uninformed. Proposition 1: If some consumers search cost is higher than the expected reduction in transport costs ( θ < θ), then some consumers stay uninformed and a symmetric equilibrium in mixed strategies results. Increasing consumers information (i.e. increasing the number of informed consumers) increases insurers equilibrium expected profits. 11 A rational consumer expects to be located at x = 1/2. Due to the symmetry of the problem, he expects prices to be symmetric or drawn from a symmetric distribution. After information, he nows which insurer best matches his individual preferences. 2t 294 sbr 63 July

9 Insurance Maret To see this result, note that given θ < θ, consumers with θ θ decide to engage in costly search while consumers with θ > θ prefer to stay uninformed. Thus, depending on the distribution of search costs in the consumer population, a fraction F( θ) of consumers decides to search for a suitable insurance provider while (1 F( θ)) prefer to remain uninformed. Here F(.) is the cumulative distribution function of search cost. To simplify notation, we write δ for the share of informed consumers, i.e. we define δ := F( θ). Uninformed consumers, represented by (1 δ), minimize expected cost of purchase by choosing the cheapest provider. Hence, in the group of uninformed consumers, we have classic Bertrand competition. The fraction δ of informed consumers could be interpreted as a measure of maret transparency: the higher is δ the more transparent is the insurance maret 12. Therefore, the maret demand of insurer 1 is given by 1 if p 1 < p 2 (6) D 1 (p 1, p 2, δ) = δ p 2 p 1 { + t + (1 δ) 2t 1/2 if p 1 = p 2 if p 1 > p 2 and demand of insurer 2 is similar. Given the maret share of provider 1, its profit is Π 1 (p 1, p 2, δ) = p 1 D 1 (p 1, p 2, δ) (7) and similar for insurer 2. Observation suggests that the price game has no (subgame perfect) equilibrium in pure strategies. This nonexistence of equilibrium seems natural, since the presence of uninformed consumers provides insurers with an incentive to slightly undercut the rival, thus capturing all uninformed consumers without losing informed ones. Undercutting seems worthwhile as long as the price is not too low. Then, it may be profitable for an insurer to become a niche player and set a high price supplying only informed consumers, which again maes undercutting worthwhile for the rival insurer, and so on. Polo (1991) first showed that Hotelling s game under imperfect information had no pure-strategy equilibrium in prices. However, there is a symmetric equilibrium in mixed strategies (also shown by Polo (1991)) 13. Equilibrium prices are defined over a finite interval which lies below the full information equilibrium prices and shifts down as the mass of uninformed consumers increases, i.e., δ decreases (see Polo (1991), proposition 4, p. 78 and proposition 5, p. 711). Increasing consumers information (i.e. increasing δ) increases insurers equilibrium expected profits. Second, we assume now that θ θ. In this case, all consumers become informed about product varieties and we obtain 12 See Schultz (24, 175). 13 For our framewor and results, it is sufficient to assume that the maret is completely covered (see Polo (1991, 73, ). This assumption is standard and avoids the possibility of an issue discussed by Wang and Yang (1999), which may be interpreted as follows. When ris aversion of consumers reaches a sufficiently low level, less than maximum differentiation might result in the pure-strategy equilibrium since insurers might have an incentive to move towards the middle of the maret in order to capture consumers in the central area. Since our focus is on the second stage of the original game, this problem is not interesting from our view and we thus avoid it. sbr 63 July

10 A. Hofmann/M. Nell Proposition 2: If search cost is below the expected reduction in transport cost for all consumers, θ θ, then all consumers engage in costly searches and become informed about product varieties and δ = F(θ) = 1. Then, a symmetric full-information maret equilibrium results. Equilibrium prices are p * 1 = p * 2 = p * = t and profits are Π * 1 = Π * 2 = Π * = t _ 2. Proof: The proof follows Focht, Richter, and Schiller (29). See Focht, Richter, and Schiller (29) and set c ϕ. As a result, if search costs are below expected reduction in transport costs for all consumers, θ θ then the result is a symmetric full-information equilibrium associated with highest expected profits for insurers (compared to all possible mixed strategy equilibria, i.e., δ (, 1)). There is no room for intermediation here since all consumers are informed and purchase their best-fitting product variety. However, because consumers are often rather uninformed, we loo at the more realistic case of a somewhat intransparent insurance maret. In such a maret, information broers usually have some (transaction) cost advantage over consumers and might thus improve the matching of consumers to insurers in the maret. Given that a ris analysis is usually quite expensive requiring expertise not only in finance, but also in actuarial science, law and engineering information intermediaries may increase consumer welfare. 6 Insurance Maret with Intermediation Generally, there are two remuneration systems under which the broer can be paid: the broer might be paid for his service by the insurer (we refer to this system as the commission remuneration system) or the broer might be paid directly by consumers (we refer to this system as the fee or fee-for-advice system). We analyze each remuneration system separately by taing it as exogenously given. Under a commission system, insurance products are sold via intermediaries who earn a commission paid by insurers. Under a fee-for-advice system, products are sold by insurers and intermediaries are paid by consumers for their information service. The case where an insurer may use alternative distribution channels for different groups of consumers is discussed in the concluding section. Under intermediation, consumers can become informed about product varieties either by consulting a costly intermediary or by searching for suitable insurance products themselves. The new sequence of play is depicted in Figure sbr 63 July

11 Insurance Maret Figure 2: Sequence of events with intermediation () Insurers select their location on [, 1] Broers announce price m for ris analysis (fee system) Consumers decide whether to become informed (individual search or broer) Insurers announce premiums Consumers choose an insurance provider 6.1 Commission System We assume that a broer incurs some variable cost > for performing a ris analysis and reveals the position of a consumer on the Hotelling-line perfectly. The broer maret is competitisve, and therefore marginal cost pricing leads to the broers fee m =. For example, in the United States, there is intense competition in the intermediary maret, especially for small and medium-sized riss. Overall competitiveness tends to vary by maret segment 14. Under the commission remuneration system, requesting information from a broer is free for consumers since the insurer bears the commission. Consequently, all consumers prefer to become informed by a broer, even those for whom private information cost θ is lower than the broers cost. Therefore, we obtain Proposition 3: Under a commission system, there is a full-information maret equilibrium in pure strategies. Equilibrium prices and expected profits of insurance providers are t + and t/2. Proof: The proof follows Focht, Richter, and Schiller (29). See Focht, Richter, and Schiller (29) and set c. By informing all consumers about product suitability via insurance intermediaries, the commission system leads to a symmetric pure-strategy full-information equilibrium. Insurers mae positive profits and share maret demand equally. We conclude that intermediation is desirable for both consumers and insurers because consumers are optimally matched to product varieties and insurers earn higher expected profit than without intermediation. Equilibrium profits would be the same in a Hotelling-maret 14 See Cummins and Doherty (26, ). sbr 63 July

12 A. Hofmann/M. Nell with full information and no intermediation. This result suggests that costs of commissions are entirely allocated to policyholders via insurance premiums, a phenomenon that can be empirically observed Fee-for-Advice System Under the fee-for-advice system, consumers must pay the intermediary for the service. Again, uninformed consumers prefer to search for suitable insurance products themselves if individual search costs plus expected mismatch costs are lower than expected mismatch costs without information. Of course, if > θ, then there is no demand for the broers service. Therefore, the broer maret can only exist if < θ. First, we assume > θ. Since consumers base their decisions on the expected value of information, asing a broer is only worthwhile if the cost of information is not higher than the benefit (in terms of expected mismatch cost advantage): + t 12 t _ 3 t _ 4 = θ. (8) Therefore, if > θ there is no demand for the broers services. In this case, our results above when there is no intermediation apply (see proposition 1). An intermediary maret is sustainable if θ. In this case, consumers with θ < search for suitable insurance products themselves and consumers with θ as for information from a broer. Hence, the proportion δ := F() invests in individual search while the proportion (1 δ) ass for a broer s advice. Proposition 4: Under a fee-for-advice system, θ leads to a full-information maret equilibrium in pure strategies. Equilibrium prices and expected profits of insurance providers are t and t/2. 7 Social Welfare To compare social welfare (or social costs) under both broer compensation systems, we represent social costs by the sum of disutilities of mismatch over all consumers and the broers and consumers information costs. From a social welfare viewpoint, a feefor-advice system leads to the social welfare optimum when the intermediaries costs are low. Then, social costs are minimized. Under a commission system, in contrast, becoming informed is included in the price so that all consumers as for the broers information, even those for whom private information costs are smaller than the broers costs. Hence, the commission system misses the social optimum. We summarize this intuition in 15 See Cummins and Doherty (26, ). 298 sbr 63 July

13 Insurance Maret Proposition 5: Social welfare is always higher under a fee-for-advice system than under a commission system. If the intermediaries information cost is low, a feefor-advice system maximizes social welfare. A commission system leads to a crowding-out of private search activities. Proof: Under a commission system, all (previously uninformed) consumers as for a ris analysis by a broer. Social costs are then 16 S c = t + df(θ) = t +. (9) 12 θ Under a fee-for-advice system, social costs are 12 S f = t + θdf(θ)+ df(θ). (1) 12 Comparing (9) to (1), we obtain θ S c S f = t + df(θ) [ t + θdf(θ)+ df(θ) ] (11) 12 = θ 12 ( θ)df(θ), i.e., social costs given commissions are higher than given fees (note that only in the special case = social cost would be the same). As a result social welfare is indeed higher under a fee-for-advice system than under a commission system. We can also compare social costs under a fee system to social costs in case without intermediation. Taing into account that θ = t/4, we obtain ˆθ S w/o S f = δ t 12 + (1 δ) t _ 3 + θdf(θ) [ + θ t θdf(θ) + 12 df(θ)] (12) ˆθ = θ (1 δ) + θdf(θ)+ θdf(θ) (13) ˆθ θ θdf(θ)+ df(θ)+ F(θ) ] [ 16 Note that F(θ) = and F(θ) = 1. θ = (1 δ) = (1 δ) ( θ ) + (θ )df(θ), θ θ 16 Note that F(θ) = and F(θ) = 1. sbr 63 July

14 A. Hofmann/M. Nell i.e., social costs without intermediation are higher than social costs under a fee system (or equal in the special case that θ). When we loo at the difference S w/o S c, we find ˆθ S w/o S c = δ t 12 + (1 δ) t _ 3 + θ θdf(θ) [ t + 12 df(θ)] (14) = (1 δ) ( θ ) + (θ )df(θ), (15) ˆθ where the last expression is positive for and negative for θ. Indeed, the difference S w/o S c depends on the distribution of θ and. As a result, we obtain S w/o S c S f if is low. This completes the proof. We conclude that both remuneration systems are associated with equal equilibrium expected profits but that the fee system is superior from a social welfare viewpoint. Remunerating intermediaries via a commission system implies that more consumers are informed by an intermediary than would be socially optimal. There is a socially undesirable crowding-out of private search activities. If the intermediaries information cost is high, then introducing insurance intermediation may even worsen overall welfare. 8 Collusion In practice, intermediaries usually earn commissions paid by insurance companies. This compensation structure is interesting, given that it might offer insurance companies and broers incentives for collusion. In this context, collusion refers to a setting where insurance companies and broers cooperate and act as one player in the maret. Therefore, insurance companies and broers maximize joint profits when cooperation is possible. If such cooperation between insurers and broers becomes possible, then collusion is worthwhile for insurers since it offers them the opportunity to sustain monopoly prices and ensure higher expected profits 17. Cooperation with big broer firms seems a particularly important issue. For insurance companies, it maes sense to cooperate mainly with the global players who already have considerable maret power which may be useful to gain higher maret shares. Remember the political debate triggered by the well-nown attac of attorney general of New Yor, Eliot Spitzer, who forced abandon of some well-established broer compensation customs, so called contingent commissions 18. These commissions are ex post payments of insurance companies to broers based on performance criteria (i.e. profitability or volume of business). Spitzer found that a coalition of commercial insurance companies was paying such commissions. In 17 In Germany, for instance, a cartel of German commercial insurance firms successfully increased premiums in this way. The cooperative agreements were found illegal by antitrust authorities in 25. See Bundesartellamt (25). 18 See Ruquet (24) and Spitzer (24). 3 sbr 63 July

15 Insurance Maret return for these payments, broers offered expensive pseudo-offers, which they actually received from other coalition members, to unsuspecting clients. 8.1 Collusion Under a Fee-for-Advice System We first consider collusion under a fee-for-advice system. We note that the consumers ris premium r represents their maximum willingness to pay for (full) insurance coverage. Rationing consumers is not profitable, so the maximum premium that insurers can set without losing demand is 19 p * I = r ( 1 _ 2 ) 2 t. (16) This maximum price ensures that all consumers want to purchase insurance. This price cannot be increased without losing customers. Since the broer maret is competitive, broers are paid at marginal cost and mae zero profits. As a result, profits of the coalition of insurers are Π * f = p * I (see eq. (16)). Collusion is worthwhile compared to the equilibrium situation without collusion if Π * f = r _ ( 1 2 ) 2 t > 2 _ t 2 r > 5 _ t+. (17) As a result, collusion is a profitable strategy if (17) holds, that is, if consumers are sufficiently ris-averse. It should be noted that deviating might be a profitable strategy for an insurer in a oneshot perspective involving a critical discount rate. However, in the long run, the (Nash) equilibrium will result which is inferior for both insurers. As long as insurers can mae binding collusive contracts, which may include some ind of sanction in case of deviation offsetting the impact of the one-shot gain, the collusive coalition is a stable situation. In such a case it is not worthwhile for insurers to deviate from this agreement and long-run collusive profits are possible Collusion under a Commission System Under a commission system, profits under collusion would not remain the same as above. Instead, profits might be even higher, since insurers could price-discriminate between consumers. By charging higher prices, insurers might extract additional rents from some consumers. Broers might wor together with insurers and help to separate consumers into two groups: informed consumers who (search for product variety themselves and) buy insurance directly from insurers, and uninformed consumers who 19 An informed consumer located at x = 1/2 has highest transport cost. He or she just about purchases insurance at this price. is the maximum cost of information and all consumers with lower information cost search for information themselves. sbr 63 July

16 A. Hofmann/M. Nell buy insurance using the broers advice. If marets for informed and uninformed consumers are separated, then the group of informed consumers is optimally charged the collusive monopoly price p * I (see eq. (16)). All consumers have to mae a choice: they might stay uninformed and purchase insurance directly at the price p * I for informed consumers (and choose an insurer at random) or they might consult a broer and mae a well-informed purchase decision. Without loss of generality, we assume that an uninformed consumer purchases the policy for his type when he is indifferent. Thus, the uninformed consumer s incentive constraint is: r p * U t 12 r p* I _ t, (18) 3 where p * U indicates the price of the intermediated policy. Substituting for p * I and rearranging terms gives the optimal price of the intermediated policy for separate marets 2 p * U = r > p * I. (19) By setting collusive prices p * I and p * U, the coalition of insurers and broers maes higher profits than under the fee system, due to Π * c [ = δ r _ ( 1 2 ) 2 t ] = r δ t _ 4 > r t _ 4 = Π* f. + (1 δ) (r ) (2) However, the broers need an incentive to participate in this collusive agreement because they could still offer their information services directly to consumers under a fee-for-advice system. Consumers would then pay the broers fee and purchase insurance at the price for informed consumers, which leaves insurers with lower profits under a fee system. To motivate broers to participate in this collusion, insurers might pay them, say + ε, where is some tiny positive number. As a result, the commission system offers the advantage of separating consumers into informed and uninformed groups, each of which purchases different contracts. In this way, collusion of insurers and broers can extract additional rents from uninformed consumers, resulting in higher potential profits. 2 Note that a consumer pays p * U p * I = t/4 in order to mae an informed purchase decision. Thus we still have δ informed consumers since getting the information from the broer is worthwhile as long as θ θ. 32 sbr 63 July

17 Insurance Maret This result is of practical interest in view of the recent debate on collusive agreements between insurance companies and broers. However, when we loo at social welfare, we find that welfare is still lower compared to a maret with a fee system and no collusion. A public regulator might consider increasing welfare by fighting against collusion or by simply prohibiting commissions paid by insurance companies. We note that such regulatory interventions are currently discussed in many countries, and some countries have even recently legally established such interventions. For instance, Finland and Denmar have implemented a ban on commissions paid by insurance companies Discussion and Conclusion Introducing product differentiation into insurance demand broadens the range of possible maret equilibria compared to the standard Bertrand models. In our framewor, intermediation results in a symmetric maret equilibrium that exhibits higher expected profits for insurance companies and higher information for consumers compared to a situation without intermediation. Hence, intermediation is profitable for both insurers and consumers. The rationale for the latter is that increasing maret transparency via intermediation increases product differentiation, since previously uninformed consumers become informed ones. Therefore, increased transparency leads to less tight competition. Although higher transparency on the consumer side is usually thought to promote competition, our framewor, in which consumers become informed about product match via intermediation, suggests that increasing consumers information maes the maret less competitive, which implies higher profits. We evaluate current broer compensation customs. We find that a fee-for-advice system leads to higher welfare than does a commission-based compensation system. A commission system implies that more consumers are informed by an intermediary than would be socially optimal. Private information activities are crowded out. This effect raises the question whether a commission system is indeed stable. An insurer may have an incentive to opt out of the system and directly target the group of low-cost consumers without using an intermediary. This strategy might be profitable, and it seems to be common practice in some insurance marets to use intermediated and direct distribution channels separately. However, the coexistence of direct and intermediated distribution channels may lead to separate marets. In this case, low-cost consumers would use the direct distribution channel while high-cost consumers would use the intermediated maret. We obtain borderline cases for our analysis and intermediation (in the maret with high-cost consumers) leads to a symmetric full-information maret equilibrium again. Our results for the commission system depend on the implicit assumption that customers pay the same price regardless of whether they purchase through a broer. This implies that in every case somebody performs a ris analysis. We assume that in this set-up a rational consumer ass for a share in the overall commission () and informs 21 See Ladbury (27). sbr 63 July

18 A. Hofmann/M. Nell himself. Since his information cost is θ, he will demand not less than θ. If he demands θ, he is not better off. Thus, he would as for κ > θ. It is rational for the consumer to demand the maximum share possible. If he demands, which constitutes the maximum a broer could afford spending without maing negative profits, he would gain θ. As a consequence, welfare under commissions would be equivalent to welfare under fees. However, we note that such commission sharing may not be legal, but instead constitutes a criminal act (as in Germany). The consumer would then need to tae the fraudulent action into account when deliberating over asing for a share in the broer s commission. Both broer compensation customs appear to be equivalent from a viewpoint of insurers profits in a non-cooperative setting. Yet, when cooperation between insurers becomes possible, incentives for collusion are high. Under a commission system, collusion with broers seems very attractive, since it allows broers to separate consumers into informed and uninformed groups, each of which purchases different contracts. Using this strategy, a coalition of insurers and broers could extract additional rents from the uninformed group (those consumers asing for the intermediated policy) resulting in the highest possible profits. This result may explain why intermediaries are usually compensated by insurance companies. Indeed, a commission system, which is inferior from a social welfare viewpoint, is liely to be established in the industry due to its property of being associated with very high profits under collusion with broers. Any regulatory intervention, such as a ban of commissions, would then increase social and individual welfare. Appendix Product characteristics can be represented by the location q i of provider i, i {1, 2}, on the Hotelling line. Throughout our analysis, we suppose that providers are located at and 1, respectively. Our concern here is to verify these equilibrium locations, which result from the game when there is a stage zero. Without loss of generality, we assume that provider 1 is to the left of provider 2: q 1 q 2 1. Of course, at q 1 = q 2, we have no product differentiation: Products are identical and Bertrand competition yields the Bertrand result. Under the commission system, all consumers as for a ris analysis from a broer and the structure of the game can thus be reduced as follows: In a first stage, firms choose their product characteristics, and then, in a second stage, firms compete in prices. D Aspremont, Gabszewicz, and Thisse (1979) show that in case of such a two-stage game where firms choose a location in an initial stage, and then compete in prices in the second stage, there will be maximum differentiation, i.e., the firms select locations as far apart from each other as possible. We show that maximum differentiation also holds in our framewor under the commission system. Given some positive transportation cost t >, a consumer located at x [, 1] faces disutility t(q i x) 2 from purchasing a product from provider i. However, by choosing different locations, providers can ensure themselves positive profits. Given providers locations q 1 and q 2 and prices p 1 and p 2, the marginal consumer (who is indifferent between purchasing from either provider) is located at 34 sbr 63 July

19 Insurance Maret ˆx = p 2 p 1 + t(q 2 q 2 1). (A1) 2t(q 2 q 1 ) Assuming again that ris aversion r is sufficiently high, demand for providers 1 and 2 is then ˆx and 1 ˆx, respectively. Insurers profits are sales minus the cost of uninformed consumers ris analyses by broers: and Π 1 (p 1, p 2, q 1, q 2 ) = (p 1 ) p 2 p 1 + t(q 2 2 q2 1 ) 2t(q 2 q 1 ) Π 2 (p 1, p 2, q 1, q 2 ) = (p 2 ) [1 p 2 p 1 + t(q 2 2 q2 1 ) ]. 2t(q 2 q 1 ) (A2) (A3) Given q 1 and q 2, maximizing Π 1 and Π 2 with respect to each provider s price yields their second stage price reaction functions: and p R 1 (p 2 ) = p t(q 2 2 q2 1 ) 2 (A4) p R 2 (p 1 ) = p t(q 2 1 q2 2 ) + 2t(q 2 q 1 ). (A5) 2 Thus each provider s best-reply price is a linearly increasing function of the other provider s price. The (Nash) equilibrium prices in the second stage of the game are then and p * 1 (q 1, q 2 ) = 3 + 2t(q 2 q 1 ) + t(q 2 2 q2 1 ) 3 (A6) p * 2 (q 1, q 2 ) = 3(1 δ) + 4t(q 2 q 1 ) + t(q 2 1 q2 2 ). (A7) 3 We can now derive providers profits in the second stage, which are their reduced profit functions in the first stage, by substituting equilibrium prices in (A2) and (A3). Then, we obtain and Π red. 1 (q 1, q 2 ) = t(q 2 q 1 )(2 + q 1 + q 2 ) 2 18 (A8) Π red. 2 (q 1, q 2 ) = t(q 2 q 1 )(4 q 1 q 2 ) 2. (A9) 18 sbr 63 July

20 A. Hofmann/M. Nell We see that insurance providers choose so called maximum differentiation. Consider and Π red. 1 = t(2 + 3q 1 q 2 )(2 + q 1 + q 2 ) < q 1 18 (A1) Π red. 2 = t(3(q 1 q 2 ) 4)(q 1 q 2 4) >. (A11) q 2 18 Π red. 1 is decreasing in q 1 for any q 1 [, 1] while Π red. 2 is increasing in q 2 for any q 2 [, 1]. Hence, insurance providers will optimally select locations as far apart from each other as possible. Given that they are restricted to locate in [, 1], they will choose q * 1 = and q * 2 = 1, respectively. The marginal consumer is located at the middle point ˆx * = 1/2 and insurers thus share maret demand equally in symmetric equilibrium. References Arrow, Kenneth J. (1963), Uncertainty and the Welfare Economics of Medical Care, American Economic Review 53, Bertrand, Joseph (1883), Théorie Mathématique de la Richesse Sociale, Journal des Savants (Paris), Biglaiser, Gary (1993), Middlemen as Experts, RAND Journal of Economics 24, Bundesartellamt (25), Bundesartellamt verhängt 3 Mio. Euro Bussgelt gegen Industrieversicherer, Press Release, March 23 rd. Cosimano, Thomas F. (1996), Intermediation, Economica 63, Cummins, David J. and Neil A. Doherty (26), The Economics of Insurance Intermediaries, Journal of Ris and Insurance 73, D Aspremont, Claude, Jean J. Gabszewicz, and Jacques-Francois Thisse (1979), On Hotelling s Stability in Competition, Econometrica 47, Doherty, Neil A. (1975), Some Fundamental Theorems of Ris Management, Journal of Ris and Insurance 42, EC (27), Business Insurance Sector Inquiry Interim Report, European Commission. Economides, Nicholas (1986), Minimal and maximal product differentiation in Hotelling s duopoly, Economics Letters 21, Focht, Uwe, Andreas Richter, and Jörg Schiller (29), Intermediation and Matching in Insurance Marets, Woring Paper (Ludwig-Maximilians-Universität München). Gravelle, Hugh (1993), Product Price and Advice Quality: Implications of the Commission System in Life Insurance, Geneva Papers on Ris and Insurance Theory 18, Gravelle, Hugh (1994), Remunerating Information Providers: Commissions versus Fees in Life Insurance, Journal of Ris and Insurance 61, Gravelle, Hugh (1991), The Welfare Economics of Controls on Broers Commissions, Geneva Papers on Ris and Insurance Issues and Practice 16, Hotelling, Harold (1929), Stability in Competition, Economic Journal 39, Keynes, John M. (1921), Fundamental Ideas, A Treatise on Probability, London: MacMillan. 36 sbr 63 July