Public versus private insurance system with (and without) transaction costs: optimal segmentation policy of an informed monopolist

ABSTRACT

Computer-mediated transactions allow insurance companies to customize their contracts, while transaction costs limit this tendency toward customization. To capture this phenomenon, we develop a complete-information framework in which it is costly to design a new market segment when the segmentation policy (number and design of segments) is endogenously chosen. Both the case of a private and a public insurer are considered. Without transaction costs, these two insurance systems are equivalent in terms of social welfare and participation. With transaction costs, this equivalence is no longer present, and the analysis of this difference is the subject of this article.

KEYWORDS:

• Insurance
• direct segmentation
• transaction costs
• social welfare

JEL CLASSIFICATION:

• D04
• D42
• D60

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ (Rejda 2014) offer an overview of the various (practical) methods for selling and marketing insurance products for different types of insurance, including health, life, property and casualty.

² It is frequently noted that with big data, adverse selection could indeed be reversed (Siegelman 2014); i.e. the insurer would be able to know more about the risk than the agent herself. See, e.g. (Villeneuve 2000) for such a model, although his argument is not related to big data.

³ Such a situation is generally called perfect (or first-degree) price discrimination in Economics.

⁴ For instance, with genetic testing, an insurer may be completely informed about the type of a potential policyholder. However, such genetic testing is not allowed in many jurisdictions; see the Oviedo conventions in Europe http://www.coe.int/en/web/bioethics/oviedo-convention. An interesting discussion of genetic testing and insurance is provided in (Durnin, Hoy, and Ruse et al. 2012).

⁵ Since 2012, in the European Union, insurance companies have to charge the same price to men and women for the same insurance products, without distinction on the grounds of sex.

⁶ Interestingly, note that contrary to most goods, an insurance contract is nominative and thus cannot be subject to parallel trade; i.e. those who pay a low premium cannot resell their contract to those who pay a high premium. As a result, the costs associated with the prevention and mitigation of parallel trade, which may be important for goods such as pharmaceuticals, books, computers, and cars (see, e.g. (Braouezec 2012), and see, also, (Braouezec 2016)) are non-existent for insurance contracts.

⁷ It would be strange to design a group $G_i$, i.e., to choose ${\theta }_i$ and ${\theta }_{i+1}$, and a contract for that group $C_i=(P_i,L)$ such that a fraction of the agents of that group do not accept the contract. Since information is complete, the monopolist knows in advance that this fraction of agents of $G_i$ will not accept the contract. As a result, the insurer should either decrease the premium $P_i$ or change the design of the group. We thus assume throughout this paper that the design of the group $G_i$ is consistent with the contract offered $C_i$, i.e., each agent assigned to the group $G_i$ accept the contract $C_i$.

⁸ Up to an elementary permutation, for each $n\ge 1$, one can move from the ordered vector $({\theta }_{min}^{(n)},...,{\theta }_{min}^{(1)})$ to the ordered vector $({\theta }_1^{\ast \ast },...,{\theta }_n^{\ast \ast })$.

⁹ This approach cannot, by assumption, maximize the total surplus. However, as it maximizes the total expected profit, it maximizes the amount used to subsidize another group of agents.

¹⁰ See, e.g. the proposition 2 in M Mrešević, (2008), ‘Convexity of the inverse function’, The teaching of Mathematics, Vol $XI$, 21–24.

¹¹ Some of the results are interesting, but we have not found them suitable for our model. If $A=(a_{i,j})\in {\mathrm{R}}^{n,n}$ is a tridiagonal matrix, few theorems presented show that the positive definiteness of $A$ turns out to be related to $\mathrm{c}o{s}^2(\pi n+1)$.