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New Developments in the Theory of Adverse Selection in Competitive Insurance

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Abstract

We provide an overview of the paths taken to understand existence and efficiency of equilibrium in competitive insurance markets with adverse selection since the seminal work by Rothschild and Stiglitz (1976). A stream of recent work reconsiders the strategic foundations of competitive equilibrium by carefully modelling the market game.

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Notes

  1. Rothschild and Stiglitz (1997).

  2. Note that this is different from market unravelling in the sense of Akerlof where an equilibrium exists; however, it is characterised by no trade. See Hendren (2014).

  3. Rothschild and Stiglitz (1976).

  4. The non-existence problem does not only appear in competitive insurance markets, but more generally in competitive markets with asymmetric information and common values such as credit markets or the screening versions of job market models with adverse selection.

  5. Netzer and Scheuer (2014), Picard (2014), Mimra and Wambach (2010, 2011), Guerrieri et al. (2010), Dosis (2014).

  6. Throughout, we focus on the essential Rothschild and Stiglitz (1976) set-up with a binary loss and no uncertainty/ambiguity where insurance buyers are expected utility maximisers.

  7. Under asymmetric information, regulation out of efficiency considerations seems warranted if the market does not achieve second-best efficient outcomes. Note that among second-best efficient allocations, the allocation that pools both risk types at the contract with full insurance has the nice property that risk-sharing in the market is efficient. However, selection among second-best efficient allocations is a distributional but not an efficiency consideration.

  8. In the screening game underlying the RS analysis presented here and in most of subsequent work, n ⩾ 2 is sufficient to obtain the results.

  9. Smart (2000), Villeneuve (2003) and Wambach (2000) introduce an additional dimension of asymmetric information by assuming that consumers furthermore differ in wealth/risk aversion. With two dimensions of asymmetric information, the single crossing property may be violated. The authors all consider single contract offers and show that due to violation of single crossing there might be contracts with positive profits offered in equilibrium. For a critical discussion, see Snow (2009).

  10. Insurance purchase is exclusive, that is a consumer is assumed to buy at most one insurance contract at one firm. For non-exclusive contracting see footnote 12.

  11. A second-best efficient allocation is Pareto efficient among those that satisfy self-selection conditions and resource constraints, see Crocker and Snow (1985). The resource constraint here translates to non-negative profits on the whole population.

  12. Riley (1979).

  13. An easy solution there is to assume a mass point at the higher endpoint.

  14. Hendren (2013).

  15. Note that with full support near p=1 in any allocation different from the endowment some types are cross-subsidised, which is incompatible with a competitive equilibrium with positive insurance in the simple screening game.

  16. Hendren (2014).

  17. Dasgupta and Maskin (1986).

  18. For the screening version of Spence’s (1973) signalling model of education that exhibits a similar structure as the RS model, Rosenthal and Weiss (1984) derive an equilibrium in mixed strategies. A somewhat unsatisfying characteristic of that equilibrium is that although there is no profitable deviation for existing firms in the market, an entrant can earn positive expected profits.

  19. Wilson (1977).

  20. Jaynes (1978), Hellwig (1987), Engers and Fernandez (1987), Asheim and Nilssen (1996), Ania et al. (2002), Mimra and Wambach (2011), Netzer and Scheuer (2014)).

  21. Inderst and Wambach (2001); Faynzilberg (2006); Mimra and Wambach (2010); Picard (2014).

  22. Most of these contributions only discuss existence and uniqueness in the two-type case. We will thus continue to concentrate on the two-type case.

  23. Lacker and Weinberg (1999).

  24. Dubey and Geanakoplos (2002).

  25. Bisin and Gottardi (2006).

  26. Spence (1973).

  27. Maskin and Tirole (1992).

  28. For the modelling of competition between the uninformed market side when the informed buyer proposes contract menus in the first stage see Dosis (2014).

  29. Wilson (1977) is the seminal contribution on competitive insurance markets with adverse selection besides Rothschild and Stiglitz (1976) and was reportedly not completed later than Rothschild and Stiglitz’s work.

  30. With single contract offers, anticipatory equilibrium does not always yield a unique equilibrium allocation: If the low risk type is indifferent between his RS contract and the Wilson pooling contract, both the RS allocation and the Wilson pooling allocation can be sustained in an anticipatory equilibrium.

  31. Miyazaki (1977).

  32. Spence (1978).

  33. Crocker and Snow (1985).

  34. Engers and Fernandez (1987).

  35. To be precise, any allocation that yields non-negative profits when all firms offer it can be sustained as outcome of a perfect Nash equilibrium. This for example. includes the MWS allocation, but as well the monopoly outcome. Profit making allocations can be sustained as final outcomes, as any deviating offer by an insurer will be followed by (sequences of) retaliating offers by other insurers, which makes deviation unattractive.

  36. Hellwig (1987).

  37. Jaynes (1978).

  38. Hellwig (1988).

  39. In Jaynes (1978) and Hellwig (1988) firms can offer sets of linear contracts; however, menus of nonlinear contracts are not discussed. Ales and Maziero (2013) and Attar et al. (2014) consider non-exclusive contracting in RS environments without information sharing. In both models, a pure strategy equilibrium may fail to exist. The key insight in these models is that in equilibrium, if it exists, contracts are linear and offer positive insurance only for the high risk type.

  40. Asheim and Nilssen (1996).

  41. There exist other symmetric equilibria involving renegotiation; however, they all yield the MWS allocation.

  42. Note that the overall result depends on the assumption that insurers cannot renegotiate contracts individually, which seems unrealistic once consumers have chosen an insurance contract. To our knowledge, there is to date no work on competitive markets with adverse selection where renegotiation is unrestricted.

  43. Netzer and Scheuer (2014).

  44. As an example, some firms offer the RS contracts, while some firms offer MWS contracts. With zero withdrawal costs, if there is no deviation, the latter withdraws the MWS contracts in the second stage and becomes inactive, and customers receive RS contracts.

  45. Mimra and Wambach (2011).

  46. In a methodologically different approach, Ania et al. (2002) model dynamics in insurance markets using evolutionary game theory. Insurers do not have perfect knowledge about the market and imitate successful behaviour, that is they copy the most profitable contract on the market and in addition, they experiment with their own contracts. RS contracts are the long-run outcome of the evolutionary game if insurers experiment only locally. This is because even if a pooling contract is preferred, the RS contracts cannot be destabilised by small modifications in contract terms, whereas pooling contracts can.

  47. Inderst and Wambach (2001).

  48. Guerrieri et al. (2010).

  49. Faynzilberg (2006).

  50. Mimra and Wambach (2010).

  51. Picard (2014).

  52. Picard (2010) also analyses the n risk type case and thereby shows coexistence of participating and non-participating contracts, which could be interpreted as coexistence of mutuals and stock insurers.

  53. In the screening version of the standard job market models with adverse selection, von Siemens and Kosfeld (2014) assume that high productivity workers prefer to be pooled with their own type, for example because of bonus payments when a performance target is met in team production. This creates an analogous externality that guarantees equilibrium existence.

  54. Faynzilberg (2006), Mimra and Wambach (2010).

  55. Note that the MWS allocation has the nice property of being robust to perturbations of the type space precisely because it allows for cross-subsidisation, whereas the RS allocation is not.

  56. For an overview and discussion of empirical work, see for example Cohen and Siegelman (2010) and Einav et al. (2010).

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Authors and Affiliations

  1. Center of Economic Research, ETH Zurich, Zuerichbergstrasse 18, Zuerich, 8092, Switzerland

    Wanda Mimra

  2. Department of Economics, Staatswissenschaftliches Seminar, University of Cologne, Gyrhofstr. 8c, Cologne, 50931, Germany

    Achim Wambach

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  1. Wanda Mimra
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  2. Achim Wambach
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Mimra, W., Wambach, A. New Developments in the Theory of Adverse Selection in Competitive Insurance. Geneva Risk Insur Rev 39, 136–152 (2014). https://doi.org/10.1057/grir.2014.11

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