Large losses and equilibrium in insurance markets

Abstract

We show that if losses are larger than wealth, then individuals with the option of declaring bankruptcy will not insure if the loss probability is above a threshold. In an insurance market with adverse selection, if the high risks’ loss probability is above the threshold, then no trade occurs at the Rothschild–Stiglitz equilibrium. Active trade in insurance requires cross-subsidization. When a subset of individuals with significant costs of bankruptcy and default is included in the market, then the equilibrium outcome always involves positive levels of insurance coverage for some individuals, but the parameters of the model determine whether all types receive coverage, or whether null contracts are received by both high and low risks with no bankruptcy costs or just the low risks from that group.

Appendix: Proof of Proposition 3

The Lagrangian for the maximization problem is

$$\begin{aligned} {\mathcal{L}} =& U_{L} \left( {c_{L} } \right) \, + \mu_{H} \left[ {U_{H} \left( {c_{H} } \right) \, {-}U_{H} \left( {c_{L} } \right)} \right] \, + \mu_{L} \left[ {U_{L} \left( {c_{L} } \right) \, {-}U_{L} \left( {c_{H} } \right)} \right] \\ & + \gamma \{ \lambda \left[ {w{-}p_{H} l{-} \, \left( {1 \, {-}p_{H} } \right)w_{HG} {-}p_{H} w_{HB} } \right]I_{H} \\ & + \left( {1 \, {-}\lambda } \right)\left[ {w{-}p_{L} l{-} \, \left( {1 \, {-}p_{L} } \right)w_{LG} } \right) \, - p_{L} w_{LB} )]I_{L} \} \\ & + \delta \left[ {U_{H} \left( {c_{H} } \right) \, {-}U_{H} \left( {c_{H}^{*} } \right)} \right], \\ \end{aligned}$$

where μ_H, μ_L, γ, and δ are the Lagrange multipliers. The first-order conditions are

$$\partial {\mathcal{L}}/\partial w_{LG} = \, \left( {1 \, + \mu_{L} } \right)\partial U_{L} \left( {c_{L} } \right)/\partial w_{LG} - \mu_{H} \partial U_{H} \left( {c_{L} } \right)/\partial w_{LG} {-}\gamma \left( {1 \, {-}\lambda } \right)\left( {1 \, {-}p_{L} } \right) \, = \, 0$$

(6)

$$\partial {\mathcal{L}}/\partial w_{LB} = \, \left( {1 \, + \mu_{L} } \right)\partial U_{L} \left( {c_{L} } \right)/\partial w_{LB} - \mu_{H} \partial U_{H} \left( {c_{L} } \right)/\partial w_{LB} {-}\gamma \left( {1 \, {-}\lambda } \right)p_{L} = \, 0$$

(7)

$$\partial {\mathcal{L}}/\partial w_{HG} = \, \left( {\delta + \mu_{H} } \right)\partial U_{H} \left( {c_{H} } \right)/\partial w_{HG} - \mu_{L} \partial U_{L} \left( {c_{H} } \right)/\partial {\text{w}}_{HG} {-}\gamma \lambda \left( { 1 { }{-}p_{H} } \right) \, = \, 0$$

(8)

$$\partial {\mathcal{L}}/\partial w_{HB} = \, \left( {\delta + \mu_{H} } \right)\partial U_{H} \left( {c_{H} } \right)/\partial w_{HB} - \mu_{L} \partial U_{L} \left( {c_{H} } \right)/\partial w_{HB} {-}\gamma \lambda p_{L} ) \, = \, 0,$$

(9)

along with the complementary slackness conditions.

• A. (i) The resource constraint is binding: If at least one type buys insurance, then I_i =1 for some i, and non-satiation implies the resource constraint is binding. If neither type buys insurance, then I_H = I_L = 0 and the resource constraint becomes 0 = 0. (ii) Either both IR constraints are binding or both IR constraints are slack: We need to show (a) IR_H binding implies IR_L binding, (b) IR_L binding implies IR_H binding, (c) IR_H slack implies IR_L slack, and (d) IR_L slack implies IR_H slack. Observe that (a) and (d) are equivalent and (b) and (c) are equivalent. We first prove (b). Assume IR_L is binding. Suppose, by way of contradiction, that IR_H is slack, U_H(c_H) > U_H($E_{0}$). Since the Hs purchase a policy with positive coverage, either p_H < p* or the Hs are subsidized. The first possibility is ruled out by hypothesis. Since the Ls do not purchase insurance, the Hs cannot be subsidized without violating the resource constraint. We now prove (d), that if IR_L is slack then IR_H is slack. If IR_L is slack, then U_L(c_L) > U_L($E_{0} )$. Assume, by way of contradiction, the IR_H is binding, U_H(c_H) = U_H($E_{0}$). Since c_L must break even, it lies on the type L fair odds line above (w, b). But this implies U_H(c_L) > U_H($E_{0}$), which is the desired contradiction.

• B. If both IR constraints are slack, then (i) The SS_H constraint must be binding: Let (c_H, c_L) be a proposed solution to the maximization problem such that the resource constraint is binding. Suppose that SS_H is slack, U_H(c_H) > U_H(c_L). Let c_L = (w_LG, w_LB) and c_L′ = (w_LG − ε₁, w_LB + ε₂), where ε₂ = (1 − p_L)/p_L]ε₁, ε₁ > 0. If ε₁ is small enough then SS_H is still slack. But c_L′ is a mean preserving contraction relative to c_L, so U_L(c_L′) > U_L(c_L). So (c_H, c_L) cannot be a solution to the maximization problem. (ii) The high risks must be fully insured: If both self-selection constraints are binding, the equilibrium must be at a pooled policy c_H = c_L. Suppose the pooled policy offers less than full insurance. Then both types are better off at c_L′, so (c_H, c_L) cannot be a solution to the maximization problem. Any pooled equilibrium must be at the population pooled policy c_P. Now suppose SS_L is slack and SS_H is binding, U_H(c_H) = U_H(c_L), and c_H offers less than full insurance. Let c_H′ be a mean preserving contraction from c_H. Then there is a c_L′ such that SS_H is binding, U_H(c_H′) = U_H(c_L′). Both types are better off at (c_H′, c_L′), so (c_H, c_L) cannot be a solution to the maximization problem.

  (iii) Equation (5) in the text gives the slope of the CSL: If both self-selection constraints are slack, then the expected utility of both types can be increased. So at least one self-selection constraint must be binding. From (8) and (9), we have

  $$\frac{{\left( {1 - p_{H} } \right)u^{\prime}(w_{HG)} }}{{p_{H} u^{\prime}\left( {w_{HB} } \right)}} = \frac{{\mu_{L} \left( {1 - p_{L} } \right)u^{\prime}\left( {w_{HG} } \right) + \gamma \lambda \left( {1 - p_{H} } \right)}}{{\mu_{L} p_{L} u^{\prime}\left( {w_{HB} } \right) + \gamma \lambda p_{H} }}.$$

  (10)

  Evaluated at any full insurance contract (w_H, w_H), this becomes

  $$\frac{{1 - p_{H} }}{{p_{H} }} = \frac{{\mu_{L} \left( {1 - p_{L} } \right)u^{\prime}\left( {w_{H} } \right) + \gamma \lambda \left( {1 - p_{H} } \right)}}{{\mu_{L} p_{L} u^{\prime}\left( {w_{H} } \right) + \gamma \lambda p_{H} }}.$$

  (11)

  This can hold as an equality if and only if μ_L = 0. If both self-selection constraints are binding, then both types must be at the pooled contract c_P. Then high-risk utility is U_H(c_P). From (6) and (7), we have

  $$\frac{{\left( {1 - p_{L} } \right)u^{\prime}(w_{LG)} }}{{p_{L} u^{\prime}\left( {w_{LB} } \right)}} = \frac{{\mu_{H} \left( {1 - p_{H} } \right)u^{\prime}\left( {w_{LG} } \right) + \gamma \left( {1 - \lambda } \right)\left( {1 - p_{L} } \right)}}{{\mu_{H} p_{H} u^{\prime}\left( {w_{LB} } \right) + \gamma \left( {1 - \lambda } \right)p_{L} }}.$$

  (12)

  Evaluated at c_P, this becomes

  $$\frac{{1 - p_{L} }}{{p_{L} }} = \frac{{\mu_{H} \left( {1 - p_{H} } \right)u^{\prime}\left( {w_{H} } \right) + \gamma \left( {1 - \lambda } \right)\left( {1 - p_{L} } \right)}}{{\mu_{H} p_{H} u^{\prime}\left( {w_{H} } \right) + \gamma \left( {1 - \lambda } \right)p_{L} }}.$$

  (13)

  This holds as an equality if and only if μ_H = 0. Then (11) and (13) imply that μ_H = μ_L = 0 at c_P. If U_H(c_H) < U_H(c_P), then we have μ_L = 0, μ_H > 0, so that the low-risk self-selection constraint is slack and the high-risk self-selection constraint is binding.

  Using (8) to eliminate the Lagrangian multiplier γ in (12) yields

  $$\frac{{\left( {1 - p_{L} } \right)u'\left( {w_{LG} } \right)}}{{p_{L} u'\left( {w_{LB} } \right)}} = \frac{{\mu_{H} \left( {1 - p_{H} } \right)u^{\prime}\left( {w_{LG} } \right) + [\lambda \left( {1 - p_{H} } \right]^{ - 1} \left( {\mu_{H} + \delta } \right)\left( {1 - \lambda } \right)\left( {1 - p_{L} } \right)u'\left( {w_{H} } \right)}}{{\mu_{H} p_{H} u^{\prime}\left( {w_{LB} } \right) + [\lambda \left( {1 - p_{H} } \right]^{ - 1} \left( {\mu_{H} + \delta } \right)\left( {1 - \lambda } \right)p_{L} u'\left( {w_{H} } \right)}}.$$

  (14)

Rearranging (14) yields (5) in the text.

• C. If both IR constraints are binding, then (i) Both SS constraints are binding and (ii) both types obtain the null contract: This is Proposition 2. Observe that the IR and SS constraints coincide, so the SS constraints are binding and the null contract is unique.

• D. The solution is unique: Suppose that there are two distinct solutions to the constrained maximization problem, $(\tilde{c}_{H} ,\tilde{c}_{L}$) and $\left( {\hat{c}_{H} , \hat{c}_{L} } \right).$ Then we have U_L($\tilde{c}_{L}$) = U_L($\tilde{c}_{L}$). Since the CSL is downward sloping, one of the low-risk contracts, say,$\tilde{c}_{L} ,$ must have more coverage than the other low-risk contract. But then the tax on the low risks must be higher, and, since net transfers must balance, the subsidy to the high risks must also be higher. This implies $\tilde{w}_{H}$ > $\hat{w}_{H}$ making the high risks better off. Then the solution ($\tilde{c}_{H} ,\tilde{c}_{L}$) Pareto dominates $\left( {\hat{c}_{H} , \hat{c}_{L} } \right)$. Therefore, $(\tilde{c}_{H} ,\tilde{c}_{L}$) is the unique solution.