Pool size and the sustainability of optimal risk-sharing agreements

Abstract

We study a risk-sharing agreement where members exert a loss-mitigating action which decreases the amount of reimbursements to be paid in the pool. The action is costly and members tend to free-ride on it. An optimal risk-sharing agreement maximizes the expected utility of a representative member with respect to both the coverage and the (collective) action such that efficiency is restored. We study the sustainability of the optimal agreement as equilibrium in a repeated game with indefinite number of repetitions. When the optimal agreement is not enforceable, the equilibrium with free-riding emerges. We identify an interesting trade-off: welfare generated by the optimal risk-sharing agreement increases with the size of the pool, but at the same time the pool size must not be too large for collective choices to be self-enforcing. This generates a discontinuous effect of pool size on welfare.

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Notes

  1. 1.

    We refer the reader to footnote 4 and to the end of Sect. 8 for a discussion about the consequences of relaxing the assumption of uniform wealth in the pool.

  2. 2.

    For a brief overview see http://www.igpandi.org/Group+Agreements/The+Pooling+Agreement, consulted in November 2015.

  3. 3.

    Contracts with fixed premium are typically offered by traditional for profit companies and correspond to policies where the risk is transferred, in full or in part, to the insurer.

  4. 4.

    The specification of identical wealth for all individuals simplifies the analysis. It allows us to focus on agents with identical risk perceptions. Heterogeneity would call for further assumptions on risk aversion (whether it is increasing or decreasing in wealth), on the collective objective of the pool (how the agreement aggregates utility of different members) and on the negotiation process. In the conclusion we provide a discussion on how relaxing such assumption would affect the analysis.

  5. 5.

    As an example, suppose that individuals are facing the risk of illness. Prevention is the loss-reducing action implying an ex ante utility cost and having a benefit in terms of a lower health shock in the case of illness. In Ehrlich and Becker (1972), a consumer’s action decreasing the amount of a possible loss is called a self-insurance measure.

  6. 6.

    Clearly, with a non-linear contract paying \(-\infty \) if the loss is higher than the one associated with the desired action, the efficient action would be implementable.

  7. 7.

    This represents a major difference with respect to the standard insurance policy with fixed premium that we analyze in Appendix 9.5.

  8. 8.

    In this sense this is also the symmetric Pareto efficient agreement.

  9. 9.

    From an analytical point of view, full coverage is not optimal anymore in the risk-sharing agreement with free-riding because the Envelope Theorem cannot be applied in this case. See, in particular, Eqs. (14) and (27) in Appendix 9.1 and 9.4, respectively.

  10. 10.

    Lee and Ligon (2001) analyze a risk-sharing agreement where a non-contractible self-protection action is available to policyholders (i.e., in the case of an action that decreases the probability of the loss). Using “Cournot conjectures” they show that full coverage is optimal. Our result differs from theirs since we solve the problem using the concept of Nash equilibrium. This implies that, when choosing the reimbursement level q in the first stage, individuals anticipate the effect of such a coverage on the choice of other members of the pool.

  11. 11.

    In fact, under full coverage \(\left( q=1\right) ,\) we would observe a positive action in the risk-sharing agreement with free-riding (as mentioned below Eq. 6), whereas we would observe no-action in the case of a linear policy with fixed contribution.

  12. 12.

    Simulations’ files are available upon request.

  13. 13.

    Simulations with CRRA are available upon request to the authors.

  14. 14.

    We will study the sustainability of the optimal risk-sharing agreement with respect to individual deviations by members of the pool. Hence we do not study a cooperative game which would instead require the specification of a value function for each possible coalition of agents and the analysis of collective deviations.

  15. 15.

    Note that, by focusing on full commitment on the policy, our model is mostly suited to describe formal risk-sharing agreements. See Sect. 7 for a real-world example.

  16. 16.

    In this sense the number of periods over which the game is repeated is indefinite (see Roth and Murnighan 1978).

  17. 17.

    See Neyman (1999) for a model in which uncertainty on the duration of the game is interpreted as a small departure from the common knowledge assumption on the number of repetitions.

  18. 18.

    Such a Grim trigger strategy perfectly fits a non-competitive environment, where there is no choice for individuals except that of staying in the pool. We refer the reader to the end of this section for a discussion about exclusion as a punishment strategy, and about punishment when an outside option exists, for example, because profit and nonprofit organizations compete in the market.

  19. 19.

    With respect to the setting analyzed in Friedman (1971), our model has two specific features: (a) players do not directly observe the action of other players but they infer deviation through the contribution required for membership and from their knowledge of the number of claims; (b) detection only occurs if the loss of the deviator realizes, that is with probability p.

  20. 20.

    In Fig. 2, the graphs showing the left and the right-hand side of (10) are not represented for the pool size \(n=1\) because of the scale of the picture.

  21. 21.

    Green and Porter (1984) examine the nature of cartel self-enforcement in the presence of demand uncertainty. In particular, in their setting, demand fluctuations (that are not directly observed by firms) make the detection of deviation difficult to infer. The collusive equilibria are then less likely and unstable industry performances can occur. Reversionary episodes, where price cut is performed by all firms in the cartel as a punishment strategy, can sometimes happen with no firm really defecting, simply because of low demand.

  22. 22.

    This is possible, for example, if the pool reveals to its members the list of individuals entitled for reimbursement, together with the specific amount paid to each of them.

  23. 23.

    Notice that, after the exclusion of the deviator, the pool will be composed by \(n-1\) members, so that sustainability of the optimal agreement will be more likely.

  24. 24.

    One of the objectives of the Affordable Care Act, recently approved in the USA, is to protect policyholders from the insurers’ practice of refusing policy renewal in case of serious health conditions.

  25. 25.

    Interestingly, long-term contracts are instead offered by standard companies in other insurance markets. For example, front-loaded contracts in life insurance generate a partial lock-in of consumers: contracts that are more front-loaded have a lower present value of premiums over the period of coverage (see Hendel and Lizzeri 2003).

  26. 26.

    These simulations are available upon request to the authors.

  27. 27.

    See the Global Mutual Market Share report 2010, available at http://www.icmif.org/mms2010.

  28. 28.

    Similarly, Kerleau (2009) shows that the market for mutual contracts is characterized by low concentration in France, as the 5 biggest mutuals in 2005 represented only 20 % of the market share and the 30 biggest ones only 44 %.

  29. 29.

    The beneficial matching between agents characterized by a similar “mission” (or social attitude) has been analyzed by Besley and Ghatak (2005). They show that fewer incentives are required if employer and employee share the same mission.

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Correspondence to Giuseppe Pignataro.

Appendix

Appendix

Proof of Proposition 1

Denote \(\widetilde{W}_{i}\left( \vec {a},q,n\right) \) the (stochastic) wealth of individual i under a risk-sharing agreement characterized by n members choosing actions \(\vec {a}\), with coverage level q:

$$\begin{aligned} \widetilde{W}_{i}\left( \vec {a},q,n\right) \equiv \left( \left( 1-q\right) \left( w- \tilde{\delta }_{i}L\left( a_{i}\right) \right) +q\left( w-\frac{\sum _{j=1}^{n} \tilde{\delta }_{j}L\left( a_{j}\right) }{n}\right) \right) \end{aligned}$$
(11)

Importantly, the second term \(q\left( w-\frac{\sum _{j=1}^{n}\tilde{\delta } _{j}L\left( a_{j}\right) }{n}\right) \) is the same for all members.

The expected utility achieved by member i in the agreement (see equation 1) is then simply written as:

$$\begin{aligned} u_{i}\left( \overrightarrow{a},q,n\right) =\mathbb {E}\left[ U\left( \widetilde{W} _{i}\left( \overrightarrow{a},q,n\right) \right) \right] -C\left( a_{i}\right) \end{aligned}$$

We now write \(v\left( a,q,n\right) \equiv u_{i}\left( [a,\ldots ,a],q,n\right) \) as the utility obtained by each individual when all participants play the same action a (this utility does not depend on i, since all the members face the same individual risk):

$$\begin{aligned} v\left( a,q,n\right)\equiv & {} u_{i}\left( [a,\ldots ,a],q,n\right) =\mathbb {E} \left[ U\left( \widetilde{W}_{i}\left( \overrightarrow{a},q,n\right) \right) \right] -C\left( a\right) \nonumber \\= & {} \mathbb {E}\left[ U\left( w-\left( \left( 1-q\right) \tilde{\delta }_{i}+ \frac{q}{n}\sum _{j=1}^{n}\tilde{\delta }_{j}\right) \right) L\left( a\right) \right] -C\left( a\right) \end{aligned}$$
(12)

Thanks to the assumptions on \(U\left( \cdot \right) \), \(C\left( \cdot \right) \) and \(L\left( \cdot \right) \), function v is strictly concave in a and q.

In the optimal agreement, the action and the coverage are chosen simultaneously by the pool. In particular, the optimal collective action \( a^{C}\) and the optimal coverage \(q^{*C}\) solve the two first-order conditions. The one for a is:

$$\begin{aligned} \frac{\partial v}{\partial {a}}= & {} \left[ -\mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{i}\right] \left( 1-q+\frac{q}{n }\right) -\sum _{j\ne i}\mathbb {E}\left[ U^{\prime }\left( \widetilde{W} _{i}\right) \widetilde{\delta }_{j}\right] \frac{q}{n}\right] L^{\prime }\left( a\right) -C^{\prime }\left( a\right) =0, \nonumber \\ \end{aligned}$$
(13)

where the term \(-\mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{i}\right] \left( 1-q+\frac{q}{n}\right) L^{\prime }\left( a\right) \) indicates the total effect of the action chosen by individual i on his/her utility from consumption, whereas the term \( -\sum _{j\ne i}\mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{j}\right] \frac{q}{n}L^{\prime }\left( a\right) \) measures the effect of the action chosen by all the other members of the pool on the contribution paid by the same individual i. The first-order condition for q is:

$$\begin{aligned} \frac{\partial v}{\partial {q}}= & {} \mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \left( \widetilde{\delta }_{i}-\frac{\sum _{j=1}^{n} \widetilde{\delta }_{j}}{n}\right) \right] L\left( a\right) \nonumber \\= & {} \frac{n-1}{n} \left[ \mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{i}\right] -\mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{j}\right] \right] L\left( a\right) =0, \end{aligned}$$
(14)

where the second equality comes from the fact that, because of i.i.d of the deltas: \(\forall i,\forall k\ne i,\ell \ne i,\mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{k}\right] =\mathbb {E} \left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{\ell } \right] \).

Using definition (11) we observe that, for \(q=1,\) \( \widetilde{W}_{i}=w-\frac{\sum _{j=1}^{n}\tilde{\delta }_{j}}{n} L\left( a\right) =\widetilde{W}\) does not depend on i Hence, for \(q=1,\) \( \mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta } _{i}\right] =\mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{j}\right] \) .

Hence, \(q^{*C}=1\) is the unique solution of (14).

The first-best linear contract with fixed contribution

We consider here the optimal linear contract with fixed contribution (premium) when the action can be directly controlled by the insurer. The individual receives qL(a) when the loss occurs and, under fair pricing, the contribution equals pqL(a). The expected utility becomes:

$$\begin{aligned} \mathbb {E}U\left( w-\left[ \left( 1-q\right) \widetilde{\delta }+pq\right] L\left( a\right) \right) -C\left( a\right) \end{aligned}$$
(15)

By maximizing the previous expected utility with respect to q, we find full coverage. Maximizing with respect to the action and considering that \( q^\mathrm{FB}=1,\) the action \(a^\mathrm{FB}\) is the implicit solution of:

$$\begin{aligned} -pU^{\prime }\left( w-pL\left( a\right) \right) L^{\prime }\left( a\right) =C^{ \prime }\left( a\right) \end{aligned}$$
(16)

The left-hand side of (16) indicates the marginal benefit while the right-hand side represents the marginal cost of the action. Importantly, the contribution \(pqL(a)=pL(a)\) implies that individuals perfectly internalize the beneficial effect of their action on the contribution (see the term \(U^{\prime }(w-pL(a))\)). In particular, they take into account that a higher action, by decreasing the contribution, has a positive impact on marginal utility in both the possible states of nature. Marginal benefit is increasing in p and in \(-L^{\prime }(a)\), i.e., the efficiency of the action technology. Individuals’ welfare is maximized and corresponds to:

$$\begin{aligned} \mathbb {E}U^{FB}=U\left( w-pL(a^\mathrm{FB})\right) -C(a^\mathrm{FB}) \end{aligned}$$
(17)

Proof of Lemma 1

We develop the proof in three steps: first we show existence, second we prove symmetry and uniqueness.

  • Existence: since \(u_{i}(.)\), as function of \(\overrightarrow{a}\), is continuous and concave and the set of actions is convex, a Nash equilibrium exists.

  • Best response: for all \(i\in \{1,\ldots ,n\},\) consider for an individual i the equation defining his/her best response:

    $$\begin{aligned} a_{i}= & {} BR_{i}(\overrightarrow{a}_{-i}) \nonumber \\\Leftrightarrow & {} \mathbb {E}\left[ U^{\prime }\left( \left( 1-q+\frac{1}{n} \right) \left( w-\widetilde{\delta }_{i}L(a_{i})\right) +q\left( w-\frac{ \sum _{k\ne i}\widetilde{\delta }_{k}L(a_{j})}{n}\right) \right) \widetilde{ \delta }_{i}\right] \nonumber \\&\times \left( 1-q+\frac{q}{n}\right) +\frac{C^{\prime }(a_{i})}{ L^{\prime }(a_{i})}=0 \end{aligned}$$
    (18)

    or

    $$\begin{aligned} a_{i}=BR_{i}(\overrightarrow{a}_{-i})\Leftrightarrow \mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\left( \overrightarrow{a}\right) \right) \widetilde{\delta }_{i}\right] \left( 1-q+\frac{q}{n}\right) +\frac{ C^{\prime }(a_{i})}{L^{\prime }(a_{i})}=0,\;i=1,\ldots n \end{aligned}$$

    Fix i and j, and consider the partial function \(f_{i}:a_{j}\rightarrow BR_{i}\left( a_{j},\overrightarrow{a}_{-\left\{ i,j\right\} }\right) \), where \(\overrightarrow{a}_{-\left\{ i,j\right\} }\) is given:

    $$\begin{aligned} f_{i}^{\prime }\left( a_{j}\right)= & {} \frac{\mathbb {E}\left[ U^{^{\prime \prime }}\left( \widetilde{W}_{i}\left( \overrightarrow{a}\right) \right) \widetilde{\delta }_{i}\widetilde{\delta }_{j}\right] \left( 1-q+\frac{q}{n} \right) \frac{q}{n}}{\mathbb {E}\left[ -U^{^{\prime \prime }}\left( \widetilde{W}_{i}\left( \overrightarrow{a}\right) \right) \widetilde{\delta } _{i}^{2}\right] \left( 1-q+\frac{q}{n}\right) ^{2}+\left( -\frac{C^{\prime } }{L^{\prime }}\right) ^{\prime }\left( a_{i}\right) } \\= & {} \frac{p^{2}\mathbb {E}\left[ U^{^{\prime \prime }}\left( \widetilde{W} _{i}\left( \overrightarrow{a}\right) \right) /\widetilde{\delta }_{i} \widetilde{\delta }_{j}=1\right] \left( 1-q+\frac{q}{n}\right) \frac{q}{n}}{p \mathbb {E}\left[ -U^{^{\prime \prime }}\left( \widetilde{W}_{i}\left( \overrightarrow{a}\right) /\widetilde{\delta }_{i}=1\right) \right] \left( 1-q+\frac{q}{n}\right) ^{2}+\left( -\frac{C^{\prime }}{L^{\prime }}\right) ^{\prime }\left( a_{i}\right) } \end{aligned}$$

    which is, given our assumptions on \(C(\cdot )\), \(L(\cdot )\) and \(U\left( .\right) \), negative. Actions are then strategic substitutes. Moreover we have:

    $$\begin{aligned} \left| f_{i}^{\prime }\left( a_{j}\right) \right|= & {} \frac{p^{2} \mathbb {E}\left[ -U^{^{\prime \prime }}\left( \widetilde{W}_{i}\left( \overrightarrow{a}\right) \right) /\widetilde{\delta }_{i}\widetilde{\delta } _{j}=1\right] \left( 1-q+\frac{q}{n}\right) \frac{q}{n}}{p\mathbb {E}\left[ -U^{^{\prime \prime }}\left( \widetilde{W}_{i}\left( \overrightarrow{a} \right) /\widetilde{\delta }_{i}=1\right) \right] \left( 1-q+\frac{q}{n} \right) ^{2}+\left( -\frac{C^{\prime }}{L^{\prime }}\right) ^{\prime }\left( a_{i}\right) } \\<&\frac{p^{2}\mathbb {E}\left[ -U^{^{\prime \prime }}\left( \widetilde{W} _{i}\left( \overrightarrow{a}\right) \right) /\widetilde{\delta }_{i} \widetilde{\delta }_{j}=1\right] \left( 1-q+\frac{q}{n}\right) \frac{q}{n}}{p \mathbb {E}\left[ -U^{^{\prime \prime }}\left( \widetilde{W}_{i}\left( \overrightarrow{a}\right) /\widetilde{\delta }_{i}=1\right) \right] \left( 1-q+\frac{q}{n}\right) ^{2}} \\ \left| f_{i}^{\prime }\left( a_{j}\right) \right|<&\frac{p\mathbb { E}\left[ -U^{^{\prime \prime }}\left( \widetilde{W}_{i}\left( \overrightarrow{a}\right) \right) /\widetilde{\delta }_{i}\widetilde{\delta } _{j}=1\right] \left( 1-q+\frac{q}{n}\right) \frac{q}{n}}{\mathbb {E}\left[ -pU^{^{\prime \prime }}\left( \widetilde{W}_{i}\left( \overrightarrow{a} \right) /\widetilde{\delta }_{i}\widetilde{\delta }_{j}=1\right) -\left( 1-p\right) U^{^{\prime \prime }}\left( \widetilde{W}_{i}\left( \overrightarrow{a}\right) /\widetilde{\delta }_{i}=1,\widetilde{\delta } _{j}=0\right) \right] \left( 1-q+\frac{q}{n}\right) ^{2}} \\ \left| f_{i}^{\prime }\left( a_{j}\right) \right|< & {} \frac{p\mathbb { E}\left[ -U^{^{\prime \prime }}\left( \widetilde{W}_{i}\left( \overrightarrow{a}\right) \right) /\widetilde{\delta }_{i}\widetilde{\delta } _{j}=1\right] \left( 1-q+\frac{q}{n}\right) \frac{q}{n}}{p\mathbb {E}\left[ -U^{^{\prime \prime }}\left( \widetilde{W}_{i}\left( \overrightarrow{a} \right) /\widetilde{\delta }_{i}\widetilde{\delta }_{j}=1\right) \right] \left( 1-q+\frac{q}{n}\right) ^{2}}=\frac{\frac{q}{n}}{\left( 1-q+\frac{q}{n} \right) }\le 1 \end{aligned}$$

    So that \(0\ge f_{i}^{\prime }\left( a_{j}\right) >-1\).

  • Symmetry: suppose that there exists a Nash equilibrium for which there are two individuals i and j such that \(a^{\prime }=a_{i}^{N}\ne a_{j}^{N}=a^{\prime \prime }.\) Then, by symmetry, \(a_{i}=a^{\prime \prime }\), \(a_{j}=a^{\prime }\) and \(a_{k}=a_{k}^{N}\) is also a Nash equilibrium. That means that \(a^{\prime }=f_{i}\left( a^{\prime \prime }\right) \) and \(a^{\prime }=f_{i}\left( a^{\prime \prime }\right) \), so that \(\frac{f_{i}\left( a^{\prime }\right) -f_{i}\left( a^{\prime \prime }\right) }{a^{\prime }-a^{\prime \prime } }=-1\). But by Rolles theorem, this implies that there exists a such that \( f_{i}^{\prime }(a)=-1\), which is impossible.

  • Now, the condition defining the symmetric Nash equilibrium \(a^{N}(q,n)\) reduces to :

    $$\begin{aligned} E\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{i} \right] \left( 1-q+\frac{q}{n}\right) +\frac{C^{\prime }\left( a\right) }{ L^{\prime }\left( a\right) }=0 \end{aligned}$$
    (19)

    Once again, given our assumptions on \(C(\cdot )\), \(L(\cdot )\) and \(U\left( \cdot \right) \), this equation has a unique solution.

Proof of Proposition 2

Using the definition of \(\widetilde{W}_{i}\left( \vec {a},q,n\right) \) in (11) above (see Appendix 9.1), the expected utility achieved by member i in the agreement, described by Eq. (1) in the main text, can be rewritten in a compact form as:

$$\begin{aligned} u_{i}\left( \overrightarrow{a},q,n\right) =\mathbb {E}\left[ U\left( \widetilde{W}_{i}\left( \overrightarrow{a},q,n\right) \right) \right] -C\left( a_{i}\right) \end{aligned}$$

Recall that, in the first stage, individuals collectively choose the level of coverage q and, in the second stage, each individual plays his/her best reply \(a^{N}(q,n)\) to the actions chosen by the other members of the pool. From Lemma 9.3, the optimal action \(a^{N}(q,n)\) is unique and symmetric and solves \(\frac{\partial u_{i}}{\partial a_{i}}\left( \left[ a,a\ldots ,a\right] ,q,n\right) =0,\) or it satisfies Eq. (19).

Before focusing on the choice of q, we prove here that \(\frac{\partial a^{N}}{\partial q}(q,n)<0,\) or that the optimal action in the Nash equilibrium decreases in the amount of coverage provided by the pool. Denote \(g(a,q,n)=\frac{\partial u_{i}}{\partial a_{i}}\left( \left[ a,a\ldots ,a\right] ,q,n\right) \) expressed in (19). By totally differentiating g(aqn) with respect to a and q one finds:

$$\begin{aligned} \frac{\partial g}{\partial q}(a,q,n)= & {} \mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{i}\right] (1-\frac{1}{n} )L^{\prime }\left( a\right) \\&+\,\mathbb {E}\left[ U^{\prime \prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{i}\left( \widetilde{\delta }_{i}L\left( a\right) -\frac{ \sum _{j=1}^{n}\tilde{\delta }_{j}L\left( a\right) }{n}\right) \right] \nonumber \\&\times \left( 1-q+\frac{q}{n}\right) L^{\prime }\left( a\right) <0 \end{aligned}$$

The first term is negative because \(L^{\prime }\left( a_{i}\right) <0\), while the second term is negative because \(U^{\prime \prime }\left( \widetilde{W}_{i}\right) <0\). Moreover,

$$\begin{aligned} \frac{\partial g}{\partial a}(a,q,n)= & {} -\mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \tilde{\delta }_{i}\right] (1-q+\frac{q}{n} )L^{\prime \prime }\left( a\right) \\&+\,\mathbb {E}\left[ U^{\prime \prime }\left( \widetilde{W}_{i}\right) \tilde{ \delta }_{i}\right] \left( 1-q+\frac{q}{n}\right) \left( L^{\prime }\left( a\right) \right) ^{2}-C^{\prime \prime }\left( a\right) <0, \end{aligned}$$

where all terms are negative. Therefore, \(\forall q\in [0,1]\),

$$\begin{aligned} \frac{\partial a^{N}}{\partial q}(q,n)=-\frac{\partial g(a,q,n)}{\partial q}/ \frac{\partial g(a,q,n)}{\partial a}<0 \end{aligned}$$
(20)

Let us consider now the choice of q in the first stage of the game and recall that \(v\left( a,q,n\right) \) is the utility achieved by each individual when all participants play the same action a (see also 12):

$$\begin{aligned} v\left( a,q,n\right)\equiv & {} u_{i}\left( [a,\ldots ,a],q,n\right) \nonumber \\= & {} \mathbb {E} \left[ U\left( w-\left( \left( 1-q\right) \tilde{\delta }_{i}+\frac{q}{n} \sum _{j=1}^{n}\tilde{\delta }_{j}\right) \right) L\left( a\right) \right] -C\left( a\right) \end{aligned}$$
(21)

The action is now \(a=a^{N}(q,n).\) Importantly, when collectively choosing q in the first stage, the pool anticipates that all members will choose the action \(a^{N}(q,n)\) without taking into account the positive impact that the action of other members exerts on the contribution to be paid. In different words and contrary to the case analyzed before (see Appendix 9.1), the action \(a^{N}(q,n)\) is not the optimal one from the point of view of the pool in the first stage and, hence, the Envelope Theorem does not apply.

To see that, let us consider members’ indirect utility \(v\left( a^{N} \left( q,n\right) ,q,n\right) \). The optimal coverage here solves:

$$\begin{aligned} \frac{\partial v}{\partial q}\left( a^{N}(q,n),q,n\right)= & {} \frac{\partial v}{ \partial q}\left( a^{N}(q,n),q,n\right) \nonumber \\&+\frac{\partial v}{\partial a^{N}} \left( a^{N}(q,n),q,n\right) \frac{\partial a^{N}}{\partial q}(q,n)=0, \end{aligned}$$
(22)

where \(\frac{\partial v\left( a^{N}(q,n),q,n\right) }{\partial a^{N}}\ne 0.\) In fact, as mentioned before, \(v\left( a^{N}\left( q,n\right) ,q,n\right) \) \( \ne y\left( q,n\right) \) \(=\) \(v\left( a^{C}\left( q,n\right) ,q,n\right) \) \(=\) \( \max _{a}v\left( a,q,n\right) \).

From (14),

$$\begin{aligned} \frac{\partial v}{\partial q}(a^{N}\left( q,n\right) ,q,n)=\frac{n-1}{n} \left( \mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{i}\right] -\mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{j}\right] \right) L\left( a^{N}\right) . \end{aligned}$$

Moreover, when \(q=1,\) we showed in Appendix 9.1 that \( \forall i,j,\widetilde{W}_{i}=\widetilde{W}_{j}\) , so that :

$$\begin{aligned} \frac{\partial v}{\partial q}(a^{N}\left( 1,n\right) ,1,n)=0 \end{aligned}$$
(23)

In addition, using (13):

$$\begin{aligned}&\frac{\partial v}{\partial {a}}\left( a,q,n\right) \nonumber \\&\quad =\left( -\mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{i}\right] \left( 1-q+\frac{q}{n}\right) -\sum _{j\ne i}\mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{j}\right] \frac{q}{n} \right) L^{\prime }\left( a\right) -C^{\prime }\left( a\right) \ne 0\nonumber \\ \end{aligned}$$
(24)

In particular, for \(q=1,\) (24) becomes:

$$\begin{aligned} \frac{\partial v}{\partial {a}}\left( a^{N}(1,n),1,n\right)= & {} \left( -\frac{ 1}{n}\mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{ \delta }_{i}\right] -\frac{n-1}{n}\mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{j}\right] \right) L^{\prime }\left( a^{N}(1,n)\right) \nonumber \\&-\, C^{\prime }\left( a^{N}(1,n)\right) \nonumber \\= & {} -\mathbb {E}\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{ \delta }_{i}\right] L^{\prime }\left( a^{N}(1,n)\right) -C^{\prime }\left( a^{N}(1,n)\right) \end{aligned}$$
(25)

Using (13), action \(a^{N}(1,n)\) is such that:

$$\begin{aligned} -\frac{1}{n}E\left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{ \delta }_{i}\right] L^{\prime }\left( a^{N}(1,n)\right) -C^{\prime }\left( a^{N}(1,n)\right) =0 \end{aligned}$$

By substituting the previous expression in (25) we find:

$$\begin{aligned} \frac{\partial v}{\partial {a}}\left( a^{N}(1,n),1,n\right) =-\frac{n-1}{n}E \left[ U^{\prime }\left( \widetilde{W}_{i}\right) \widetilde{\delta }_{i} \right] L^{\prime }\left( a^{N}(1,n)\right) >0 \end{aligned}$$
(26)

Using (23), (26) and (20), we are now in the position to evaluate (22) when \(q=1\):

$$\begin{aligned} \frac{\partial v}{\partial q}\left( a^{N}(1,n),1,n\right) =\frac{\partial v}{ \partial a}\left( a^{N}(1,n),1,n\right) \frac{\partial a^{N}}{\partial q} (1,n)<0 \end{aligned}$$
(27)

which proves that \(q=1\) does not satisfy (22). Moreover, (27) shows that the function \(v\left( a^{N}(q,n),q,n\right) \) is decreasing in q when \(q=1,\) which implies that the optimal coverage is \( q^{*N}\left( n\right) <1.\)

The second-best linear contract with fixed contribution

Consider now the optimal linear contact with fixed contribution (premium) when the action cannot be controlled. Since the action is not contractible and moral hazard has bite, we call this policy the second-best contract with fixed contribution.

A policyholder pays the premium P and receives the reimbursement qL(a) if the loss realizes. The individual’s expected utility is:

$$\begin{aligned} \mathbb {E}U\left( w-\left[ \left( 1-q\right) \widetilde{\delta }_{i}\right] L\left( a\right) -P\right) -C\left( a\right) \end{aligned}$$
(28)

The timing of actions is the following: first, the for profit insurer offers the contract (Pq); second, the individual accepts the contract and chooses the action level; finally, the risk is realized.

Solving backward, the optimal choice of the action, given the contract (Pq) , is the solution of:

$$\begin{aligned} -p(1-q)U^{\prime }\left( w-\left( 1-q\right) L\left( a\right) -P\right) L^{\prime }\left( a\right) =C^{\prime }\left( a\right) \end{aligned}$$
(29)

Obviously, with \(q=1,\) the optimal action is zero because the action does not bring any benefit in this case. Hence, the insurer will never offer full insurance. By comparing (16) and (29) we observe that, under the second-best policy, the policyholder does not internalize the positive impact that the action has on his premium. In particular, in the left-hand side of (29) only the beneficial effect of the action on the potential loss is taken into account.

In the first stage, the insurer maximizes the policyholder’s utility (28) subject to the resources constraint (\(P=pqL(a)\)) and the individual’s incentive constraint (29). As is well known, the optimal level of coverage q is lower than 1 (partial coverage), which means that the usual trade-off between risk-sharing and incentives arises.

Note that the fair contribution \(P=pqL(a)\) is coherent both with the case of a benevolent monopolistic insurer (i.e., a public/social insurance) and with the case of a large number of for profit insurers in a competitive market.

Proof of Proposition 3

Since \(q^{*C}=1\), the level of utility achieved in the optimal risk-sharing agreement is written

$$\begin{aligned} u^{C}\left( n\right) =\max _{a}\mathbb {E}\left[ U\left( w-\frac{\sum _{i=1}^{n} \widetilde{\delta }_{i}}{n}L\left( a\right) \right) \right] \end{aligned}$$

Now, as the \(\tilde{\delta }_{i}\) are i.i.d. with finite expectation, we know that \(w-\frac{\sum _{i=1}^{n}\widetilde{\delta }_{i}}{n}L\left( a\right) \) is a mean preserving spread of \(w-\frac{\sum _{i=1}^{n+1}\widetilde{\delta }_{i} }{n+1}L\left( a\right) \) for all a so that:

$$\begin{aligned} \mathbb {E}\left[ U\left( w-\frac{\sum _{i=1}^{n}\widetilde{\delta }_{i}}{n} L\left( a\right) \right) \right] \le \mathbb {E}\left[ U\left( w-\frac{ \sum _{i=1}^{n+1}\widetilde{\delta }_{i}}{n+1}L\left( a\right) \right) \right] \ \ \ \forall a \end{aligned}$$

Therefore:

$$\begin{aligned} u^{C}\left( n\right)= & {} \max _{a}\mathbb {E}\left[ U\left( w-\frac{\sum _{i=1}^{n} \widetilde{\delta }_{i}}{n}L\left( a\right) \right) \right] \le u^{C}\left( n+1\right) \\= & {} \max _{a}\mathbb {E}\left[ U\left( w-\frac{\sum _{i=1}^{n+1} \widetilde{\delta }_{i}}{n+1}L\left( a\right) \right) \right] . \end{aligned}$$

Proof of Proposition 4

We show below that the left-hand side (lhs) of (10) is equal to its right-hand side for \(n=1;\) whereas, the lhs is lower than the right-hand side (rhs) for \(n=+\infty .\)

  • For \(n=1,\) the sole individual will always choose the optimal action level. Thus, \(u^{C}\left( n\right) =u^{D}\left( n\right) )=u^{P}\left( n\right) \) and condition (10) holds with equality.

  • When \(n\rightarrow +\infty \) the impact of one individual’s action on the contribution is negligible so that deviation is always profitable. This can be seen by rewriting inequality (10) with \(n\rightarrow +\infty \)

    $$\begin{aligned}&\frac{1}{1-\sigma }\left[ U(w-pL\left( a^{C}\right) )-C\left( a^{C}\right) \right] \\&\quad \ge \frac{1}{1-\sigma \left( 1-p\right) }\left[ U(w-pL\left( a^{C}\right) )+ \frac{p\sigma }{1-\sigma }u^{N}(0,1,\infty )\right] , \end{aligned}$$

    where

    $$\begin{aligned} u^{N}(0,1,\infty )=u^{P}\left( \infty \right) =U\left( w-pL(0)\right) \end{aligned}$$

    because incentives to free-ride are the highest as possible when the size of the pool is infinite and the optimal action is \(a^{N}(1,\infty )=0\), implying \(C\left( a^{N}\right) =0.\) Rearranging we can write:

    $$\begin{aligned} C\left( a^{C}\right) \le p\sigma \left[ U\left[ w-pL(0)\right] -\frac{ U(w-pL\left( a^{C}\right) )}{1-\sigma \left( 1-p\right) }\right] . \end{aligned}$$

    Since \(\frac{1}{1-\sigma \left( 1-p\right) }>1\) and \(U\left( w-pL(0)\right) <U(w-pL\left( a^{C}\right) ),\) the rhs of the previous inequality is negative so that the latter is never satisfied. Thus, deviation is always profitable for \(n\rightarrow +\infty \) and (10) does not hold.

From Proposition 3 we also know that \(u^{C}\left( n\right) \) is monotonically increasing in n.

We conclude that either the rhs of (10) is always above its lhs and the optimal risk sharing agreement is never enforceable, or it exists a pool size \(\hat{n}>1\) such that the rhs crosses from below the lhs in \(\hat{n}\) and the rhs lies above the lhs for \( n>\hat{n}.\) Thus, a necessary condition for the optimal risk-sharing to be enforceable is that \(n<\hat{n}.\) The condition \(n<\hat{n}\) is necessary but not sufficient because it is possible that the rhs crosses the lhs more than once so that subsets of n belonging to the interval \(\left( 1,\hat{n} \right) \) and such that the lhs lies above the rhs may in principle exist. However, in our simulations with CARA and CRRA utility functions, the lhs and rhs cross just once.

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Barigozzi, F., Bourlès, R., Henriet, D. et al. Pool size and the sustainability of optimal risk-sharing agreements. Theory Decis 82, 273–303 (2017). https://doi.org/10.1007/s11238-016-9573-9

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Keywords

  • Optimal risk-sharing agreement
  • Loss-mitigating actions
  • Repeated interactions
  • Collectively optimal vs Nash behaviors