# Self-insurance and market insurance substitutability: an established tenet reconsidered

Jean-Marc Bourgeon, Pierre Picard

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Jean-Marc Bourgeon, Pierre Picard. Self-insurance and market insurance substitutability: an established tenet reconsidered. 2025. hal-04401333v2

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Distributed under a Creative Commons Attribution 4.0 International License
Self-insurance and market insurance substitutability:
an established tenet reconsidered

Jean-Marc Bourgeon* Pierre Picard†

September 17, 2025

**Abstract**

We analyze the interaction between self-insurance and market insurance when accident losses are multivalued. We show that self-insurance and market insurance may be complementary when self-insurance expenses do not affect much the probability distribution of large losses and the loading factor is high. This contrasts sharply with the conclusion of Ehrlich and Becker (1972) who establish the substitutability between self-insurance and market insurance when the cost of an accident is single-valued.

**Keywords:** insurance, self-insurance, contracts.

**JEL Classification Numbers:** D86, G22, D80.

<footnote>*INRAE and CREST- Ecole Polytechnique. E-mail: bourgeon@agroparistech.fr</footnote>

<footnote>†CREST- Ecole Polytechnique. E-mail: pierre.picard@polytechnique.edu.</footnote>
# 1 Introduction

The article of Ehrlich and Becker (1972) on risk prevention and insurance is a cornerstone of insurance economic theory. Ehrlich and Becker (1972) distinguish self-insurance from self-protection. Self-insurance (or loss reduction) consists in making preventive expenses that decrease the damages caused by accidents that may occur, while self-protection (or loss prevention) refers to costly measures that reduce the probability of such accidents.<fnref n="1" /> They show that self-insurance and market insurance are substitutable, while self-protection and market insurance may be complementary. In concrete terms, when a decrease in the price of market insurance makes it more attractive, individuals reduce self-insurance and they may increase self-protection. The objective of this paper is to reconsider this alleged substitutability between market insurance and self-insurance in a broader setting than Ehrlich and Becker (1972) and to highlight its lack of robustness.

As illustrated by debates on the insurability of climate-related risks in the global warming context, this is not a purely academic issue: when the price of market insurance is affected by taxes and subsidies, while self-insurance exerts externalities on social welfare, their interaction is a crucial element in government decision-making. We will show that the complementarity between market insurance and self-insurance tends to occur when insurance is particularly costly to policyholders. High insurance prices take the form of a high loading factor in the insurance terminology, i.e. of a larger gap between actuarial premiums and insurance prices charged to policyholders.<fnref n="2" /> This is particularly relevant for large risks, such as natural hazards, industrial catastrophes, cyber risks or high-value corporate liabilities, that involve significant uncertainty and potential for massive financial losses. Insurers must set higher loading factors to cover

<footnote n="1">¹The expression “self-insurance” has a variety of meanings, depending on the context in which it is used. It sometimes refers to the losses not covered by an insurance policy, such as the small losses in a deductible contract or the copayment in health insurance. In macroeconomics, self-insurance may refer to the intertemporal smoothing of consumption and investment through liquidity management and borrowing. In corporate risk management, it corresponds to insurance captives that allow large firms to pool the idiosyncratic risks of their subsidiaries. As far as we are concerned, following Ehrlich and Becker (1972), we analyze self-insurance as a loss-reduction device: the upfront expenses that allow individuals or businesses to incur lower losses, should some adverse operative event occur. For example, a farmer who faces the risk of long drought or pest infestation may self-insure by purchasing crop and seed varieties that are more costly, but resistant to such events.</footnote>

<footnote n="2">²The insurance loading is the extra charge that the policyholder has to pay on top of its actuarial premium. It accounts for the insurer's administrative expenses, including distribution, underwriting and claim processing costs, and it also reflects the cost of the insurance solvency capital. Another major component is the cost of reinsurance, which helps insurers manage imperfectly diversifiable risks, but adds to their overall expenses. Finally, it also depends on the intensity of competition in the insurance market, similarly to pricing mechanisms in oligopoly markets.</footnote>
the additional costs associated with the underwriting of these risks.<fnref n="3" /> However, a too high loading factor may make insurance coverage prohibitively expensive. Insurability may then require governmental intervention, either through State-funded subsidies or various forms of public-private insurance programs.

Ehrlich and Becker (1972) analyze the interaction between market insurance and self-insurance in a two-state model, in which a risk-averse individual faces the risk of an accident that may occur with a given probability. He may protect himself against the potential damages of such an accident, through market insurance or self-insurance. Both risk-management tools consist in substituting wealth from the no-accident state to the accident state. In the case of market insurance, this substitution goes through the payment of an insurance premium paid in both states in compensation for an indemnity received in the accident state. In the case of self-insurance, preventive expenses also borne in both states allow the individual to reduce the damages that would result from an accident. In such a setting, it comes as no surprise that an increase in the price of market insurance leads the individual to rely more on self-insurance through a standard effect of relative prices on customer's optimal choices.

In what follows, we show that this substitutability between market insurance and self-insurance does not extend beyond the two-state model considered by Ehrlich and Becker (1972). Indeed, consider the case where the insurance contract is a straight deductible, i.e. an insurance policy that covers the difference (if positive) between the incurred damages and the deductible. We know from Arrow (1971) that such a contract is optimal when the price of insurance is affected by a constant loading factor and the policyholder's self-insurance expenditures can be verified by the insurer. To fix ideas, assume that the losses incurred in case of an accident may be small, medium or large, and that the loading factor is so large that only large losses are covered by an optimal contract (i.e. the deductible is larger than medium-sized losses). Then, by a standard price effect, a decrease in the loading factor leads the individual to purchase more insurance, i.e., to choose a contract with a lower deductible: he receives a larger compensation for large losses, while, if the change in the loading remains small, there is still no compensation for small or medium-sized losses. Assume furthermore that self-insurance shifts probability weights from medium-sized losses to low-sized losses, without significant effect on the probability of large losses. In that

<footnote n="3">³Insuring large risks requires significant solvency capital and expertise in risk assessment, which increases insurance costs and also limits the number of insurers willing and able to participate, leading to a more concentrated market with less competition. As a result, large risks often have fewer coverage options and premiums are higher due to the limited number of insurers willing to take on such exposures.</footnote>
case, the improvement in the coverage of large losses makes the individual more willing to self-insure. This is because self-insurance expenses are made before knowing the state that will occur, and the decrease in the deductible mitigates the consequences of the worse-case scenario, i.e. a large-scaled accident. This mitigation incentivizes the risk-averse individual to spend more money on self-insurance. We will show that the complementarity between market insurance and self-insurance emerges in such a case: a decrease in the loading factor (i.e. a cheaper insurance coverage) leads the individual to purchase more market insurance, by a standard price effect, and this improvement in the insurance coverage (that boils down to increasing the worst-case bound of final wealth in the above three-state example) stimulates the individual's willingness to pay for self-insurance.

One of the main advantages of the two-state framework introduced by Ehrlich and Becker (1972) lies in its clear conceptual distinction between self-insurance and self-protection. In this setup, self-insurance modifies the policyholder's state-dependent final wealth, whereas self-protection influences the probability of each state. This clean separation, however, becomes blurred when the loss outcome involves multiple severity levels such as small, medium, or large losses, as in the example above. In such multi-scenario cases, self-insurance may influence the probabilities of different loss magnitudes (e.g., shifting weight from medium to small losses, possibly with an unchanged probability of the no-accident state). In other words, the simple conceptual distinction between self-insurance as a change in the state-dependent wealth on one side, and self-protection as a change in the probability of each state on the other, is no more valid when accident losses are multi-valued. Thus, extending the analysis of Ehrlich and Becker (1972) to a multi-state setting invites us to reinterpret any form of risk prevention as a costly intervention that induces a first-order stochastic dominance shift in the loss distribution. Pure self-insurance arises when the probability mass assigned to the no-loss state remains unchanged, while pure self-protection increases the likelihood of avoiding a loss altogether, leaving the conditional distribution of losses untouched. In reality, both mechanisms often operate simultaneously. While our analysis focuses primarily on self-insurance, our conclusions regarding its complementarity with market insurance hold if it is pure self-insurance, or whether it coexists with self-protection.<fnref n="4" /> We will also show that the first-order stochastic-dominance effect of self-insurance can be reformulated through a deterministic function (called the *residual*

<footnote n="4">⁴Lee (1998) and Jullien et al. (1999) study the relationship between risk aversion and effort when the two modes of prevention coexist. As far as we are concerned, we will explore a setting in which more self-insurance can possibly, but not necessarily, lead to more self-protection.</footnote>
*loss function*) that transforms the random losses in the absence of self-insurance into the actual losses incurred under self-insurance. A simple condition on this function will allow us to characterize the case where self-insurance and market insurance are complements.

In the main part of the paper, we make the following assumptions to facilitate a tractable analysis. On the one hand, information is symmetrically shared between the insured and the insurer, so that self-insurance effort is verifiable and thus contractible. On the other hand, the insurable risk is fully diversifiable within the portfolio of a risk-neutral insurer, with transaction costs proportional to indemnity payments. Under these assumptions, as demonstrated by Arrow (1971), the optimal insurance risk-sharing agreement between the insured and the insurer takes the form of a straight deductible policy, i.e. the indemnity is set equal to the excess of the loss over the deductible. When these assumptions do not hold, or when additional frictions affect the insured-insurer relationship, some forms of coinsurance may become optimal, with losses, regardless of their size, being shared in fixed or variable proportions between the insured and the insurer. One key justification for coinsurance arises from the (non-)verifiability of risk prevention efforts. Verifiable prevention measures are often specified in insurance contracts, and indemnification can then be made contingent on compliance with these measures.<fnref n="5" /> If prevention effort is not verifiable, the optimal contract becomes subject to (ex ante) moral hazard, and the insured must be incentivized to take appropriate preventive actions. In such settings, as shown by Holmström (1979), the optimal insurance contract typically combines a deductible with coinsurance.<fnref n="6" /> Secondly, coinsurance helps mitigate ex post moral hazard, which may involve inflated payments to service providers (e.g., health care providers in health insurance) or claim build-up when the insurer cannot detect falsification.<fnref n="7" /> Thirdly, coinsurance reduces the variability of claim costs, which is especially valuable in cases of high-value exposures or catastrophe-prone lines of business, where frictional capital costs reduce underwriting capacity and threaten insurers’ financial stability.<fnref n="8" /> In these various set-

<footnote n="5">⁵Insurers may include risk management or warranty clauses in policies requiring policyholders to implement and maintain certain safety measures to qualify for coverage or receive full indemnification. Audits can be conducted before issuing the policy (as a risk assessment), periodically during the policy term, or after a claim is filed. If the policyholder is found to have violated the agreed-upon measures, the insurer may reduce the payout, impose higher premiums in the future, or even cancel the policy.</footnote>

<footnote n="6">⁶A straight deductible policy remains optimal only when prevention efforts affect the probability of an accident but not the probability distribution of losses conditional on an accident.</footnote>

<footnote n="7">⁷See Parra and Winter (2024) for a survey on ex post moral hazard, and Martinon et al. (2018) for a focus on the design of health insurance contracts. See Crocker and Morgan (1998) on the optimality of coinsurance under costly state falsification.</footnote>

<footnote n="8">⁸This is related to the literature on how frictions in imperfect capital markets induce risk aversion</footnote>
tings, coinsurance emerges as a means to mitigate inefficiencies, or in other words, as a second-best solution in environments where frictions cannot be fully eliminated. It is therefore important to examine the robustness of our findings regarding the substitutability and complementarity between market insurance and self-insurance when a straight deductible policy is no more optimal. This is the purpose of the final section of the paper, which begins with the implications of moral hazard and then assesses the properties of coinsurance as a second-best insurance mechanism.

The economic analysis of risk prevention, despite a rich body of theoretical work, still lacks interpretable and generalizable conditions that unambiguously determine whether prevention acts as a substitute for or a complement to market insurance (see Courbage et al., 2025, for a comprehensive literature review). Very few contributions extend the self-insurance framework to settings with multiple loss levels. Hiebert (1989) and Lee (2010a) explore how changes in risk aversion affect self-insurance; Lee (1998) establishes sufficient conditions under which individuals exhibiting higher degrees of risk aversion engage in greater self-insurance efforts; and Lee (2010b) investigates the role of initial wealth. Other authors have considered settings where the single-loss framework of Ehrlich and Becker (1972) is extended by introducing additional sources of uncertainty, such as uncertain prevention effectiveness (Briys et al., 1991), background risk (Briys and Schlesinger, 1990; Mahul, 1997), technological uncertainty (Li and Peter, 2021), or endogenous information (Peter et al., 2017; Crainich, 2015).⁹ In fact, in many studies that move beyond the framework of Ehrlich and Becker (1972), conclusions about the relationship between insurance and prevention remain ambiguous and highly context-dependent. For example, self-protection may act either as a complement to or a substitute for market insurance depending on the insurance loading, individual risk preferences, or characteristics of the prevention technology (Chang and Ehrlich, 1985; Boyer and Dionne, 1989). These ambiguities are compounded by the scarcity of closed-form solutions and the lack of transparent comparative statics, limiting our ability to derive broadly applicable policy insights. By contrast, our setting will be similar to a standard principal-agent model combining market insurance and

and thus larger risk premiums charged to policyholders, starting with Masson (1972) and extended to financial institutions by Froot and Stein (1998). See Bauer et al. (2025) on insurance pricing under capital market frictions.

⁹Several recent contributions adopt the same modeling approach as Lee (1998), who developed the concept of self-insurance-cum-protection (SICP) to analyze how agents simultaneously mitigate the probability and severity of loss. Wong (2016) examines precautionary risk-reduction behavior in the presence of background risk, highlighting how uncertainty over potential losses motivates increased investment in SICP. Similarly, Peter and Hofmann (2024) explore how individuals coordinate savings and precautionary efforts to manage risk over time.
self-insurance, with a continuum of losses but without additional specific features. Our objective is to derive conclusions that are as robust as possible within such a general framework.

We find that the interaction between self-insurance and market insurance critically depends on the insurance price measured by the loading factor.<fnref n="10" /> Specifically, the two instruments tend to act as complements under high loading and as substitutes when the loading is low. This duality arises from a conflict between several consequences of an increase in the insurance price, which operate either directly or indirectly through changes in the optimal insurance policy fully characterized by the value of the deductible. First, a higher loading factor increases the premium discount that a policyholder receives for verifiable self-insurance efforts. Additionally, it induces the policyholder to choose a higher deductible, thereby enlarging the range of losses for which no indemnity is paid. This enhances the usefulness of self-insurance. These two mechanisms –referred to as the direct insurance premium effect and the insurance coverage effect, respectively –encourage greater self-insurance when market insurance becomes more expensive. In other words, they push the interaction toward substitutability. Conversely, a higher deductible also reduces the premium discount that can be achieved through self-insurance efforts. Moreover, due to risk aversion, it diminishes the monetary equivalent of the expected utility gain from increased self-insurance. These effects –termed the indirect insurance premium effect and the risk-aversion effect, respectively– make self-insurance less attractive when the deductible rises in response to higher insurance prices. Thus, they push the interaction toward complementarity. Our main conclusion is that the risk-aversion effect dominates the others when the price of market insurance is high and self-insurance does not significantly alter the probability distribution of the largest losses.

It is also worth emphasizing the economic policy dimension of this issue. The case of agriculture is of particular importance, since farmers and other agricultural businesses face significant risks from natural hazards, pests and other unforeseen events, which are all the more likely to occur with extreme events. High transaction costs, and the raise in the insurers' capital costs due to climate change, make crop insurance unaffordable, and there is intense debate about whether and to what extent crop insurance should

<footnote n="10">¹⁰As usual in the insurance terminology, loading refers to the charges added to the actuarial premium (i.e. the expected indemnity) to cover the frictional transaction costs, including administrative and capital costs, that affect the insured-insurer relationship. Insurance price is measured by the loading factor, which is the percentage increase in the actual premium charged to policyholders over their actuarial premium.</footnote>
be subsidized.<fnref n="11" /> Some of these debates, starting in the 1990's, centered on whether
crop insurance could be a substitute to pesticides, fertilizers, crop specialization or
irrigation, viewed as alternative ways to protect farmers against risks affecting their
crop. This line of research, particularly Weber et al. (2016), suggests that, in practice,
expanded insurance coverage has little effect on technologies used by farmers. Our
approach is coherent with these findings: pesticides and other risk-reducing devices
may be considered as self-insurance, and there is no reason to think that they should
be substitutable to insurance. In a more theory-oriented perspective and in line with
recent works in ecology, Quaas and Baumgärtner (2008) consider the contribution of
farmers to ecosystem quality and biodiversity as self-insurance.<fnref n="12" /> They develop a model
inspired from Ehrlich and Becker (1972) in which this form of self-insurance and market
insurance are substitutes, with the ultimate conclusion that improving access to market
insurance leads to a lower level of ecosystem quality. Our model takes a different view
on this issue, by putting into question the alleged substitutability between these two
forms of insurance.

Beyond the case of crop insurance, the insurability of large risks is increasingly
becoming a major concern due to the high capital costs required to cover risks with a
systemic dimension (see Hartwig et al., 2020, in the case of economic losses caused by
pandemics). The question of whether or not the government should provide funding
for individuals and firms to buy insurance in those cases, along with implementing
policies such as insurance mandates and post-disaster relief programs, is becoming
increasingly important (see Boomhower et al., 2023, for an analysis of these issues in
the case of wildfire insurance). In such cases, understanding how government funding
affects self-insurance choices is of major importance.

The rest of the paper is organized as follows: in Section 2, we describe our setup.
Section 3 studies the interaction between market insurance and self-insurance in the
constant absolute risk aversion (CARA) case when self-insurance decisions are con-
tractible and straight-deductible policies are optimal. In Section 4, we extend our

<footnote n="11">¹¹See for instance Gao (2023) in the case of the US crop insurance program. In the context of a recent crop insurance reform in France, Rozan and Spaeter (2024) analyze how conditioning insurance subsidies on self-insurance decisions may encourage farmers to simultaneously purchase more insurance and to increase their self-insurance effort.</footnote>

<footnote n="12">¹²In ecology and theoretical biology, biodiversity is often viewed as a “natural insurance” device that acts either by guaranteeing that some species maintains when other fails as in portfolio management, or by creating asynchronicity between fluctuations in the development of species when the ecosystem may be affected by environmental shocks such as heatwaves or rain deficit. See Yachi and Loreau (1999), Cottingham et al. (2001) and Shanafelt et al. (2015) for illustrative examples of this line of research.</footnote>
analysis to the case where the individual is prudent, without postulating constant
absolute risk aversion. Section 5 assesses the validity of our conclusions, firstly in a
setting with moral hazard and non-contractible self-insurance, and next in the case
where insurers offer coinsurance contracts as a second-best insurance mechanism. The
last section wraps up. All proofs are in the Appendix.

2 The model

Consider a risk-averse individual (household or business) who has initial wealth $w$ and may incur a loss $x \in (0, \bar{x}]$ in the case of an accident. Should an accident occur, the extent of the loss depends on the individual's behavior: he can reduce it by taking appropriate self-insurance measures. We denote by $a \ge 0$ the corresponding expenditures and by $F(x|a)$ the resulting cumulative probability distribution function of the loss $x$. There is a (possibly zero) mass of probability $F(0|a) = \bar{F}(a)$ for the no-accident state, and $F(x|a)$ is differentiable with respect to $x$ when $x > 0$. Here and in what follows subscripts refer to partial derivatives, and we denote $f(x|a) = F_x(x|a)$. Hence, $f(x|a)/[1 - \bar{F}(a)]$ is the density of the loss, conditionally on an accident occurring. We assume that increasing $a$ improves this distribution in the sense of the First-Order Stochastic Dominance (FOSD), i.e. that $F_a(x|a) \ge 0$ with strict inequalities in a subset of $(0, \bar{x})$ of positive measure. We also assume $F_{aa}(x|a) \le 0$, which corresponds to the decreasing marginal efficiency of self-insurance. The case $F(0|a) = \bar{F} \ge 0$ for all $a$, a constant, corresponds to the “pure” self-insurance case as defined by Ehrlich and Becker (1972). It means that the probability of a loss occurring, $1 - \bar{F}$, does not depend on $a$. In what follows, we assume that $F(0|a)$ may depend on $a$ (i.e. $\bar{F}(a)$ with $\bar{F}'(a) > 0$ for some $a$), a setting in which self-insurance measures also affect self-protection.<fnref n="13" />

We assume $d\mathbb{E}[x|a]/da < -1$ at $a = 0$, and furthermore we have $d^2\mathbb{E}[x|a]/da^2 = -\int_0^{\bar{x}} F_{aa}(x|a)dx \ge 0$: hence, an increase in self-insurance expenses reduces the expected loss by an amount larger than its expected cost (at least when $a$ is small), but with a decreasing marginal efficiency. We also assume that $a$ is verifiable, i.e., there is no moral hazard impeding the contractual relationship between the insurer and the insured. The case of moral hazard will be considered in Section 5.

<footnote n="13">¹³For the sake of notational simplicity, we assume that the loss upper limit $\bar{x}$ does not depend on $a$, and that $f(x|a)$ is twice-differentiable almost everywhere in $\mathbb{R}_+^2$ with $f(\bar{x}|a) = 0$. We consider the case where $\bar{x}$ may depend on $a$ in Section 3.4. We also assume that $x \mapsto F_a(x|a)$ is continuous in $\mathbb{R}_+^*$ (and right continuous at $x = 0$), for all $a \ge 0$.</footnote>
The policyholder's preferences are characterized by Von Neumann-Morgenstern utility function $u(w_f)$ that satisfies $u' > 0, u'' < 0$ at any final wealth level $w_f$. The market insurance contract specifies a premium $P$ paid at the outset, and an indemnity $I(x)$ for all $x \in [0, \bar{x}]$, and thus $w_f = w - x - P - a + I(x)$. Since the policyholder's self-insurance decision can be verified, the contractual arrangement between insurer and insured includes the insurance contract stricto sensu, i.e. the premium $P$ and the indemnity schedule $I(x)$, but also the self-insurance level $a$. We assume that the insurance market is competitive with risk-neutral insurers, and also, as usual in insurance demand models, that insurance costs are reflected by a constant loading factor $\sigma$.<fnref n="14" />

Under competitive pressure, the equilibrium contractual agreement maximizes the policyholder's expected utility subject to the insurer's break-even constraint. Thus, it is an optimal solution to

$$ 
\begin{array}{l}
\max_{I(\cdot), P, a} \mathbb{E}u = \int_{0}^{\bar{x}} u(w - x - P - a + I(x))dF(x|a) \\
\text{s.t.} \\
P \ge (1 + \sigma) \int_{0}^{\bar{x}} I(x)dF(x|a), \\
I(x) \ge 0 \text{ for all } x \in [0, \bar{x}].
\end{array}
$$

We know from Arrow (1971) that, in that case, the optimal insurance contract entails full coverage over a deductible $D > 0$, leading to $I(x) = \max\{0, x - D\}$.<fnref n="15" />

<footnote n="14">¹⁴We assume that all insurance costs (including taxes, underwriting costs and solvency costs) are proportional to the expected indemnity. The loading factor $\sigma$ reflects all these costs incurred by insurers for assuming risk $I(\cdot)$. The perfect competition assumption is made to simplify the presentation, but imperfect competition in the insurance market can result in a higher loading factor, similarly to higher margins charged by firms in oligopoly markets with differentiated products.</footnote>

<footnote n="15">¹⁵See Eeckhoudt et al. (2005) for an intuitive presentation of this property. It reflects the priority that risk-averse individuals grant to the coverage of large losses. A sketch of the proof is as follows. Assume a *contrario* that $I(x) > 0$ when $x \in [x_1, x_2]$ and $I(x) = 0$ when $x \in [x_3, x_4]$ with $0 < x_1 < x_2 < x_3 < x_4 < \bar{x}$. Then, a small positive indemnity payment when $x \in [x_3, x_4]$ compensated by a decrease in $I(x)$ when $x \in [x_1, x_2]$ so that the expected indemnity and the insurance premium $P$ are unaffected is favorable to the risk-averse policyholder because $u(\cdot)$ is a concave function. Thus, there exists a threshold $D \in [0, \bar{x}]$ such that $I(x) > 0$ if and only if $x > D$. By a similar argument, the retained loss $x - I(x)$ should be constant when $I(x) > 0$, which leads to $I(x) = \max\{0, x - D\}$. Finally, the fact that $D > 0$ if and only if $\sigma > 0$ is an extension of the conclusion reached by Mossin (1968) in the coinsurance model, according to which partial insurance is optimal if and only if the loading factor is positive. As shown by Karni (1992), Machina (1995) and Gollier and Schlesinger (1996), the optimality of straight deductible contracts goes beyond the expected utility model: it holds as soon as policyholders dislike mean-preserving spreads in the distribution of their final wealth.</footnote>
The corresponding insurance premium is given by

$$P = \hat{P}(D, a, \sigma) \equiv (1 + \sigma) \int_{D}^{\bar{x}} (x - D) dF(x|a),$$

and the individual's program can be rewritten as

$$\max_{a,D} \mathbb{E}u \equiv \int_{0}^{D} u(w - x - \hat{P}(D, a, \sigma) - a)dF(x|a) + [1 - F(D|a)]u(w - D - \hat{P}(D, a, \sigma) - a). \quad (1)$$

Let $a^*(\sigma)$ and $D^*(\sigma)$ be the optimal solution to this problem when the loading factor is $\sigma$.<fnref n="16" /> An increase in $\sigma$ makes insurance more expensive and this reduces the demand for market insurance under the form of a higher deductible. This direct price effect may be compensated by a wealth effect when absolute risk aversion changes with wealth, hence an ambiguous global effect in that case. Furthermore, the change in the optimal self-insurance expenses $a^*(\sigma)$ induced by the increase in $\sigma$ modifies the individual's risk exposure, which also affects the optimal deductible $D^*(\sigma)$. If the direct price effect dominates the indirect consequences of the wealth effect and of the change in risk exposure, then $D^*(\sigma)$ is an increasing function. In that case, market insurance and self-insurance are substitutable risk management tools when $a^*(\sigma)$ is increasing, and they are complementary when $a^*(\sigma)$ is decreasing.

# 3 Interaction between market insurance and self-insurance

We will start by focusing attention on the two polar cases that correspond to large and small values of the loading factor $\sigma$, with large and small values of the optimal deductible $D^*(\sigma)$, respectively. The case of constant absolute risk aversion will allow us to show that market insurance and self-insurance may be complementary or substitutable, according to the value of $\sigma$. We then examine the full scope of the loading factor, i.e. from 0 to the largest value beyond which the individual does not purchase market insurance, and present conditions that allow for a monotonic deductible and a

<footnote n="16">¹⁶In what follows, it is assumed that $a^*(\cdot)$ and $D^*(\cdot)$ are continuous functions from $\mathbb{R}_+$ to $\mathbb{R}_+$. In full generality, we cannot exclude that $a^*(\sigma)$ and/or $D^*(\sigma)$ are multivalued for some $\sigma$. By a standard continuity argument of consumer demand theory (see Proposition 3.AA.1 in Mas-Colell et al., 1995), $(a^*(\cdot), D^*(\cdot))$ would be an upper hemi-continuous correspondence in that case. However, since $a^*(\sigma)$ and $D^*(\sigma)$ are uniquely defined when $\sigma$ is either equal to 0 or large enough, we can conclude that $a^*(\cdot)$ and $D^*(\cdot)$ are continuous functions at least when $\sigma$ is small or large enough.</footnote>
bell-shaped (i.e. first increasing and then decreasing) self-insurance expenditure. We illustrate these theoretical results through simulations before examining a case where losses are multi-valued and nevertheless, self-insurance and market insurance are substitutable for all values of $\sigma$. We show that the complementarity or substitutability between market insurance and self-insurance reflects the trade-off between the direct insurance premium effect and the insurance coverage effect on one side, and the indirect insurance premium effect and the risk-aversion effect on the other. When losses are single-valued, the interaction between self-insurance and market insurance is fully driven by the insurance premium effect, which leads to substitutability, as shown by Ehrlich and Becker (1972). Finally, our results will be interpreted in a very intuitive way through the notion of residual loss function.

## 3.1 Two polar cases

First, let us focus attention on the case where more self-insurance (i.e. an increase in $a$) does not affect significantly the probability of incurring the largest losses.<fnref n="17" /> This is obviously the case if, for any $a$, there exists $\hat{x}(a)$ smaller than $\bar{x}$ such that an increase in $a$ does not affect $F(x|a)$ when $x$ is in the interval $[\hat{x}(a), \bar{x}]$. Fig. 1 illustrates this extreme case: an increase in self-insurance only reduces the probability of incurring medium-sized losses: an increase in $a$ shifts $F(x|a)$ upwards when $x$ is smaller than $\hat{x}(a)$, without change for loss levels larger than $\hat{x}(a)$. In what follows, more generally, the relative weakness of self-insurance in the prevention of the largest risks corresponds to the case where $F_a(x|a)$ is low (more precisely, smaller than $f(x|a)$) when $x$ is close to $\bar{x}$.

Market insurance is attractive when the loading factor is not too large, and we denote $\bar{\sigma}$ its upward limit, above which the individual does not purchase insurance, that is $\bar{\sigma} = \sup\{\sigma \in \mathbb{R}_+ | D^*(\sigma) < \bar{x}\} > 0$, with $D^*(\sigma)$ increasing when $\sigma$ is close to $\bar{\sigma}$ and $D^*(\bar{\sigma}) = \bar{x}$. We also have $D^*(0) = 0$ because full coverage is optimal when insurance is purchased at actuarial price, and $D^*(\sigma) > 0$ when $\sigma > 0$, which implies that $D^*(\sigma)$ is increasing when $\sigma$ is small. In other words, at least when $\sigma$ is close to 0 and $\bar{\sigma}$, the direct price effect on $D^*(\sigma)$ dominates the indirect consequences of changes

<footnote n="17">¹⁷This case is illustrated by ? in his study of wealth effects in self-insurance: as he writes, the construction code may significantly reduce losses resulting from a small earthquake, because design constraints enable buildings to withstand earthquake forces up to some limit, but a strong earthquake may destroy all buildings and make these constraints almost useless. Similarly, in ecosystem management, preserving biodiversity and adopting less intensive agricultural practices may protect farmers from short-term droughts, but this would not protect them from the consequences of a catastrophic long-term drought.</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 1: Shifts in the p.d.f. and in the c.d.f. when self-insurance increases from 0 to $a > 0$.</figcaption></figure>

in wealth and risk exposure when $\sigma$ changes. For intermediate values of $\sigma$, $D^*(\sigma)$ may not be monotonic, although we may find assumptions under which this will be the case (see Section 3.3). This section restricts attention to the case of constant absolute risk aversion (i.e. $u$ is a CARA function) where the wealth effect on optimal choices vanishes. The more general case of a prudent individual will be considered in Section 4. Proposition 1 shows that market insurance and self-insurance are complementary when self-insurance does not affect significantly the probability of incurring the largest-size losses and the loading factor $\sigma$ is large enough.

**Proposition 1** Assume that, for all $a > 0$, there exists $\hat{x}(a)$ smaller than $\bar{x}$ such that $F_a(x|a) < f(x|a)$ if $x \in [\hat{x}(a), \bar{x}]$. If the individual displays constant absolute risk aversion, then $D^*(\sigma)$ is increasing and $a^*(\sigma)$ is decreasing when $\sigma$ is close to $\bar{\sigma}$.

The condition on the distribution of losses in Proposition 1 stipulates that self-insurance does not strongly affect the likelihood of the largest possible loss values. More precisely, let us define the “global cost of risk” as the sum of the loss and the self-insurance expenses, $z \equiv x + a$, which is distributed according to c.d.f. $G(z|a) = F(z - a|a)$, with $G(a|a) = \bar{F}$, over the support $[a, a + \bar{x}]$. We thus have $G_a(z|a) = F_a(z - a|a) - f(z - a|a)$ and we will say that self-insurance is locally weak (resp. strong) at loss level $x = x_0$ when $G_a(x_0 + a|a)$ is negative (positive). The weak or strong power of self-insurance thus characterizes its marginal efficiency depending on the level of loss considered. In words, self-insurance is strong at loss level $x_0 \in (0, \bar{x}]$ when increasing $a$ reduces the probability that the global cost of risk will be larger than $x_0 + a$, and it
is locally weak otherwise. Clearly, self-insurance is locally weak in the neighborhood of $\bar{x}$ in the extreme case described in Fig. 1, but of course, the validity of Proposition 1 is broader.

In order to intuitively interpret Proposition 1, consider how market insurance affects the individual's willingness to pay for self-insurance.<fnref n="18" /> Denote by $\bar{V}$ his expected utility in a status quo situation where he self-insures for an amount $a$ while the deductible is $D$, and consider the supplementary self-insurance spending $\delta$ that the individual would be willing to make (for $D$ unchanged) to reduce his risk exposure from $F(\cdot|a)$ to $F(\cdot|a + \varepsilon)$, $\varepsilon > 0$. In other words, $\delta \equiv \delta(\varepsilon, a, D, \sigma)$ is defined by

$$ \bar{V} = \int_{0}^{D} u(w - x - \hat{P}(D, a + \varepsilon, \sigma) - a - \delta)dF(x|a + \varepsilon) \\ + [1 - F(D|a + \varepsilon)]u(w - D - \hat{P}(D, a + \varepsilon, \sigma) - a - \delta), $$

with $\delta(0, D, a, \sigma) = 0$. Differentiating $\bar{V}$ with respect to $\varepsilon$ and $\delta$ yields the marginal willingness to pay (MWP) for an improvement in the risk exposure when the status quo is $(D, a)$ and the loading factor is $\sigma$:

$$ \begin{aligned} MWP(D, a, \sigma) &\equiv \frac{\partial \delta(\varepsilon, D, a, \sigma)}{\partial \varepsilon} \Big|_{\varepsilon=0} \qquad (2) \\ &= \frac{\int_{0}^{D} [u(w - x - P - a) - u(w - D - P - a)]dF_{a}(x|a)}{\partial \mathbb{E}u/\partial w} \\ &\quad - \frac{\partial \hat{P}(D, a, \sigma)}{\partial a}, \end{aligned} $$

where $\partial \mathbb{E}u/\partial w > 0$ and $P = \hat{P}(D, a, \sigma)$ with

$$ \frac{\partial \hat{P}(D, a, \sigma)}{\partial a} = -(1 + \sigma) \int_{D}^{\bar{x}} F_{a}(x|a) dx \leq 0. \qquad (3) $$

Equation (2) shows that the marginal willingness to pay for self-insurance is the sum of two positive terms. The first term is the monetary equivalent of the expected utility gain that results from the downward shift in losses, for a given insurance premium. This term cancels when $D = 0$ and it is positive when $D > 0$. The second term corresponds to the decrease in the insurance premium due to the decreased probability of losses above the deductible.

At the optimum of the insured's program, the marginal willingness to pay for self-

<footnote n="18">¹⁸For more details on the following results, see appendix D.</footnote>
insurance is equal to its marginal cost, i.e. $MWP = 1$. This equality holds for all $\sigma \in [0, \bar{\sigma}]$ when $a = a^*(\sigma)$ and $D = D^*(\sigma)$, and thus we have

$$ \frac{dMWP}{d\sigma} = \frac{\partial MWP}{\partial\sigma} + \frac{\partial MWP}{\partial D} D^{*'}(\sigma) + \frac{\partial MWP}{\partial a} a^{*'}(\sigma) = 0, $$

which yields

$$ a^{*'}(\sigma) = \frac{-1}{\partial MWP / \partial a} \left( \frac{\partial MWP}{\partial \sigma} + \frac{\partial MWP}{\partial D} D^{*'}(\sigma) \right) $$

where $\partial MWP / \partial a < 0$ from the local second-order optimality condition with respect to $a$ at the optimal solution $a = a^*(\sigma)$.

This formulae reveals that an increase in the loading factor $\sigma$ affects self-insurance directly and indirectly. The direct effect $\partial MWP / \partial \sigma$ may be viewed as a direct insurance premium effect: for a given deductible $D$, the larger the loading, the larger the insurance premium discount that the policyholder may benefit from if he increases his self-insurance expenditures. Indeed, we have

$$ \frac{\partial MWP}{\partial \sigma} = \int_{D}^{\bar{x}} F_{a}(x|a) dx > 0, $$

and thus this direct insurance premium effect pushes toward the substitutability between market insurance and self-insurance. The indirect effect $\partial MWP / \partial D \times D^{*'}(\sigma)$ results from the increase in the optimal deductible when the loading factor increases (i.e. $D^{*'}(\sigma) > 0$), and it goes itself through several channels. Most obviously, the larger the deductible, the larger the interval $[0, D]$ in which the losses entirely remain at the policyholder's expenses, and thus where he benefits from a first-order negative shift in his risk exposure. This insurance coverage effect is a reason for increasing self-insurance when the deductible increases. However, two other effects work in the opposite direction. Firstly, because of the policyholder's risk aversion, the larger the deductible, the lower the money equivalent of the expected utility gain drawn from an increase in self-insurance: this is the risk-aversion effect. Intuitively, this is because self-insurance expenditures are borne in all states, including the cases where the loss is larger than the deductible (what we have called the worse case scenario), and a larger deductible makes this cost more painful to the risk-averse policyholder. Secondly, the larger the deductible, the lower the reduction in the insurance premium when self-insurance increases: this is the indirect insurance premium effect.

In a nutshell, the direct insurance premium effect and the insurance coverage effect push in the direction of substitutability between self-insurance and market insurance,
while the risk aversion effect and the indirect premium effect move in the direction of complementarity. If $F_a(x|a) = 0$ when $x$ is close to $\bar{x}$ (as in the case of Figure 1), then the direct and indirect insurance premium effects and the insurance coverage effect vanish when $\sigma$ is close to $\bar{\sigma}$ because $D^*(\sigma)$ is close to $\bar{x}$ in that case. The variations in the loading factor then affect self-insurance through the risk-aversion effect only, hence the complementarity between self-insurance and market insurance when the loading factor is large enough. Proposition 1 extends this conclusion to a setting where self-insurance is locally weak for the largest losses, i.e. where $F_a(x|a)$ may be positive but not too large when $x$ is close to $\bar{x}$.

The next Proposition considers the other polar case and shows that market-insurance and self-insurance are substitutable when the loading factor is small.

**Proposition 2** If the individual displays constant absolute risk aversion, $D^*(\sigma)$ and $a^*(\sigma)$ are increasing when $\sigma$ is close to zero.

The intuition of Proposition 2 is the following. We have $D^*(0) = 0$ and $D^*(\sigma) > 0$ when $\sigma > 0$, hence $D^*(\sigma)$ is locally increasing when $\sigma$ is small. Since the policyholder faces no risk when $\sigma = 0$, his self-insurance level is the same as if he were risk-neutral, i.e. $a^*(0) = a_N$, where $a_N$ solves $\partial \mathbb{E}[x|a_N]/\partial a = -1$. In other words, $a_N$ equalizes the marginal decrease in expected loss with the marginal cost of self-insurance. When $\sigma$ is close to 0, $D^*(\sigma)$ is also close to zero. In that case, the individual is almost fully covered and reducing the insurance premium is the main reason for increasing self-insurance. In other words, the direct insurance premium effect dominates the indirect effects, and consequently an increase in $\sigma$ triggers an increase in $a^*(\sigma)$ above $a_N$ because of this cost-cutting objective. This mechanism may be related to the marginal willingness to pay for self insurance, since

$$ \frac{dMWP}{d\sigma} \rightarrow 1 = -\frac{\partial \mathbb{E}[x|a_N]}{\partial a}, $$

when $\sigma \rightarrow 0$. Hence, at the first order, an increase in $\sigma$ in the neighborhood of 0 leads to an increase in the marginal willingness to pay for self-insurance from 1 (its equilibrium value when $D = 0$ and $a = a_N$) to $1+\sigma$. This leads to simultaneous increases in $D^*(\sigma)$ and $a^*(\sigma)$ when $\sigma$ increases, which corresponds to the substitutability between market insurance and self-insurance established in Proposition 2.

## 3.2 Ehrlich and Becker (1972) put in perspective

Ehrlich and Becker (1972) assume that losses are single-valued, and a short digression
allows us to reframe their analysis as a particular case of our framework. Indeed, when the possible losses have a single value, under pure self-insurance we may write,

$$ \mathbb{E}u = \bar{F}u(w - P - a) + (1 - \bar{F})u(w - \ell(a) - P - a + I), $$

where $\ell(a)$ and $I$ are the loss and the insurance payout, respectively, with $\ell'(a) < 0$, $\ell''(a) > 0$, and the insurance-pricing constraint becomes

$$ P = (1 + \sigma)(1 - \bar{F})I = (1 + \sigma)(1 - \bar{F})[\ell(a) - D] \equiv \hat{P}(D, a, \sigma), $$

where $D = \ell(a) - I$ is the loss retained by the policyholder. This yields

$$ \mathbb{E}u = \bar{F}u(w - \hat{P}(D, a, \sigma) - a) + (1 - \bar{F})u(w - D - \hat{P}(D, a, \sigma) - a), $$

which is an adaptation of equation (1) to the case of single-valued losses.<fnref n="19" /> As before, consider the policyholder’s expected utility

$$ \bar{F}u(w - \hat{P}(D, a + \varepsilon, \sigma) - a - \delta) + (1 - \bar{F})u(w - D - \hat{P}(D, a + \varepsilon, \sigma) - a - \delta), $$

when status quo choices $a, D$ are perturbed by an improvement in the risk exposure from $\ell(a)$ to $\ell(a + \varepsilon)$, with additional expenses $\delta$ in return. Let $\delta(\varepsilon, D, a, \sigma)$ be the maximum expenses that the policyholder would accept for such an improvement, with $\delta(0, D, a, \sigma) = 0$. The marginal willingness to pay for this improvement in risk exposure is given by

$$ MWP \equiv \left. \frac{\partial \delta}{\partial \varepsilon} \right|_{\varepsilon=0} = - \frac{\partial \hat{P}(D, a, \sigma)}{\partial a} = -(1 + \sigma)(1 - \bar{F})\ell'(a) > 0 $$

for all $\sigma, a$ and $D$. In other words, when losses are single-valued, the marginal willingness to pay for self-insurance is nothing but the induced discount in insurance premium, for $D$ unchanged. This means that, contrary to the case where losses are multi-valued, this marginal willingness to pay is fully captured by the direct insurance premium effect, and the indirect effects on $D$ do not play any role. The larger the loading factor $\sigma$, the larger the marginal willingness to pay for self-insurance, which explains straightforwardly why self-insurance and market insurance are substitutable when losses are

<footnote n="19">¹⁹In Ehrlich and Becker (1972), the insurance level is measured by the lump-sum payment $I$ and not by the retained loss $D$, but of course this change of variable does not affect the conclusion about function $a^*(\sigma)$.</footnote>
single-valued. This may be easily checked by observing that the maximization of $E_u$ implies that $\hat{P}(D^*(\sigma), a, \sigma) + a$ is minimized when $a = a^*(\sigma)$. This gives

$$ \left. \frac{\partial \hat{P}(D^*(\sigma), a, \sigma)}{\partial a} \right|_{a=a^*(\sigma)} = -1, $$

which corresponds to the equality between the marginal willingness to pay and the marginal cost of self insurance, as in the general model with a continuum of loss. We deduce

$$ \ell'(a^*(\sigma)) = -\frac{1}{(1 + \sigma)(1 - \bar{F})}, $$

and

$$ a^*(\sigma) = \frac{1}{(1 + \sigma)^2(1 - \bar{F})\ell''(a^*(\sigma))} > 0. $$

The larger the loading factor, the larger the optimal self-insurance expenses, hence the substitutability between self-insurance and market insurance in the Ehrlich-Becker self-insurance model.<fnref n="20" />

## 3.3 Full scope for the loading factor

In what follows, we stick to the setting where losses are multi-valued and distributed in $(0, \bar{x}]$. The next two propositions provide sufficient conditions on the distribution of losses for $D^*(\sigma)$ to be increasing and $a^*(\sigma)$ to be bell-shaped when $\sigma$ goes from 0 to $\bar{\sigma}$.

**Proposition 3** Assume $f_{aa}(x|a) - f_{ax}(x|a) \ge 0$ for all $x$ such that $f_a(x|a) < 0$. Then, if the individual displays constant absolute risk aversion, $D^*(\sigma)$ is increasing for all $\sigma \in [0, \bar{\sigma}]$.

Proposition 3 gives a sufficient condition for the deductible to increase with the loading factor $\sigma$ in the CARA case. We know that this property is true in the pure market insurance problem without self-insurance. However, it is also known that, in this problem, a change in the probability distribution of losses affects the optimal deductible in an ambiguous way (see, e.g., Eeckhoudt et al., 1991). Here, when $\sigma$ changes, in addition to the standard price effect on insurance demand, the optimal deductible

<footnote n="20">²⁰Interestingly, in the Ehrlich and Becker (1972) two-state setting, optimal self-insurance expenditures are independent of the individual's risk preferences. In other words, changing the utility function $u(\cdot)$ does not affect $a^*(\sigma)$. This property no longer holds when accident losses are multi-valued. Indeed, the impact of self-insurance on losses below the deductible affects the individual's overall well-being, thereby making risk preferences relevant.</footnote>
$D^*(\sigma)$ is also affected by the change in the risk exposure induced by the variation in the optimal self-insurance level $a^*(\sigma)$. It is shown in the proof of the proposition that the price effect dominates, i.e. that $D^*(\sigma)$ is increasing for all $\sigma \in [0, \bar{\sigma}]$, when $f_{aa}(x|a) > f_{ax}(x|a)$ for all $x$ such that $f_a(x|a) < 0$. This condition corresponds to a limit on the marginal efficiency of self-insurance, in the sense that it is more and more costly to reduce the probability of incurring large losses (i.e., loss values $x$ such that $f_a(x|a) < 0$) when $a$ is increasing.<fnref n="21" /> If this condition were not satisfied, then an increase in $a^*(\sigma)$ induced by a decrease in $\sigma$ could reduce the risk exposure so strongly that the insured would choose a higher deductible. The condition prevents to have such a Giffen effect on market-insurance.

**Proposition 4** Assume that (i) $C(a) = \{x|F_a(x|a) - f(x|a) \ge 0\}$ is a convex subset of $[0, \bar{x})$, and (ii) $C(a') \subseteq C(a)$ for all relevant $(a, a')$, $a < a'$. Then, if $D^*(\sigma)$ is an increasing function and if the individual displays constant absolute risk aversion, $a^*(\sigma)$ is a unimodal function when $\sigma \in [0, \bar{\sigma}]$.

As in the previous propositions, the results are obtained under constant absolute risk aversion. Assumption (i) introduces $C(a)$, the subset of $[0, \bar{x})$ in which $F_a(x|a)$ is larger or equal to $f(x|a)$, i.e. the subset of losses where self-insurance is locally strong. As in Proposition 1, $C(a)$ does not include $\bar{x}$ for all relevant values of $a$. More restrictively, according to assumptions (i) and (ii), $C(a)$ is a sub-interval of $[0, \bar{x})$ that shrinks when $a$ increases. In particular, $C(a') \subseteq C(a)$ when $a' > a$ conveys the idea that the marginal efficiency of self-insurance is decreasing when $a$ is increasing.<fnref n="22" />

We already know from Propositions 1 and 2 that $a^*(\sigma)$ is increasing when $\sigma$ is close to 0, and that it is decreasing when $\sigma$ is close to $\bar{\sigma}$ if $F_a(x|a) < f(x|a)$ when $x$ is close to its upper limit $\bar{x}$. Under the conditions postulated in Propositions 3 and 4, $a^*(\sigma)$ is bell-shaped over the whole interval $[0, \bar{\sigma}]$. In other words, because of the interaction between the direct and indirect effects that determine the marginal willingness to pay

$$df_a = f_{ax}(x_0|a_0)dx + f_{aa}(x_0|a_0)da = [f_{aa}(x_0|a_0) - f_{ax}(x_0|a_0)]da > 0,$$

<footnote n="21">²¹To be more specific, consider a status quo situation where $a = a_0$ and $x = x_0$ with $f_a(x_0|a_0) < 0$. Hence, $-f_a(x_0|a_0)$ is a measure of the marginal efficiency of self-insurance on losses around $x_0$ (whose probability decreases when self-insurance increases). Starting from this status quo, consider a small increase in self-insurance $da > 0$ compensated by a decrease in the loss level $dx = -da$ so that the global cost $x + a = x_0 + a_0$ is unchanged. This compensated increase in self-insurance yields</footnote>

<footnote>which means that, for a global cost $x + a = x_0 + a_0$, the marginal efficiency of self-insurance is decreasing. One may check that the condition is met in the simulation below.</footnote>

<footnote n="22">²²$C(a)$ is a sub-interval of $[0, \bar{x})$ if $F_a(x|a) - f(x|a)$ is quasi-convex in $x$. As shown in the proof of Proposition 3, the condition postulated in this proposition ensures that this function is non-increasing w.r.t. $a$, and thus $C(a') \subseteq C(a)$ if $a' > a$.</footnote>
for self-insurance when the insurance price is changing, there exists $\hat{\sigma} \in (0, \bar{\sigma})$ such that $a^*(\sigma)$ is monotonically increasing when $\sigma < \hat{\sigma}$ and monotonically decreasing when $\sigma > \hat{\sigma}$.

Propositions 1 to 4 may be illustrated by simulating functions $D^*(\sigma)$ and $a^*(\sigma)$. To do so, consider the case where $f(x|a) = (1 - \bar{F})h(x|a)$ where $\bar{F} > 0$ is a constant (pure self-insurance), and

$$h(x|a) = \frac{1}{\bar{x}} \left( 1 - \sqrt{\frac{a}{\bar{a}}} \frac{3\sqrt{3}}{2} (2x/\hat{x} - 1)(1 - (2x/\hat{x} - 1)^2) \mathbf{1}_{\{x \in [0, \hat{x}]\}} \right),$$

with $a \in [0, \bar{a}]$ and $\hat{x} \le \bar{x}$. Hence, $h(x|a)$ is the conditional density of the loss, with a sinusoidal shape when $x \le \hat{x}$, and it is flat above $\hat{x}$, with $h(x|a) = 1/\bar{x}$ if $x \in [\hat{x}, \bar{x})$, $h(x|a) > 1/\bar{x}$ if $x \in (0, \hat{x}/2)$ and $h(x|a) < 1/\bar{x}$ if $x \in (\hat{x}/2, \hat{x})$. Function $h(x|a)$ is illustrated by Fig. 2a in the cases $a = \bar{a}$ (the blue curve), and $a = \bar{a}/2$ (the red curve), with $\bar{a} = 10$, $\hat{x} = 70$ and $\bar{x} = 100$.<fnref n="23" /> Fig. 2b depicts the c.d.f. of the loss when $a$ is at its maximum $\bar{a}$ (the blue curve) and at its minimum $a = 0$ (the red line).

<figure><img src="image_4.png" /><figcaption>Figure 2: Conditional density and probability distributions of the loss ($\bar{x} = 100, \bar{a} = 10$ and $\hat{x} = 70$).</figcaption></figure>

The parameter values used in the simulations are presented in Table 1,<fnref n="24" /> and under these assumptions, we have computed the optimal deductible and self-insurance levels

<footnote n="23">²³Conditional density $h(x|a)$ reaches a maximum when $x = \hat{x}(\sqrt{3} - 1)/2\sqrt{3}$ and a minimum when $x = \hat{x}(\sqrt{3} + 1)/2\sqrt{3}$.</footnote>

<footnote n="24">²⁴Normalization $\bar{a} = 10$ yields $a_N = 1.152$. Furthermore, assuming $\bar{F} = 0.20$ means that there is a 80% chance of incurring a loss, which would be uniformly distributed over $[0, \bar{x}]$ if the individual does not take any self-insurance measure. The value of the initial wealth level $w$ is only relevant to derive the expected utility level reached in each case.</footnote>
<table><caption>Table 1: Calibration values</caption><thead><tr><th>α</th><th>w</th><th>F&#773;</th><th>x&#773;</th><th>a&#773;</th><th>x&#770;</th><th>a<sub>N</sub></th></tr></thead><tbody><tr><td>0.05</td><td>20</td><td>0.2</td><td>100</td><td>10</td><td>70</td><td>1.152</td></tr></tbody></table>

Parameters: α is the ARA coefficient, w the initial wealth, F the probability that no accident occurs, x the maximal loss, a the maximum self-insurance level and x the maximum loss affected by self-insurance measures. a_N is the self-insurance level under risk neutrality.

when the loading factor σ goes from 0 to 150%.<fnref n="25" />

Optimal self-insurance $a^*(\sigma)$ rises at first to reach 1.850 when $\sigma = 50\%$, and then decreases.<fnref n="26" /> Hence, market insurance and self-insurance are substitutable when $\sigma$ is smaller than 50%, and they are complementary for larger values of the loading factor.

<figure><img src="image_5.png" /><figcaption>Figure 3: Deductible and self-insurance as functions of the loading factor.</figcaption></figure>

## 3.4 Large-risk self-insurance

So far, we have presented conditions on the probability distribution of the loss that lead to the complementarity between self-insurance and market insurance, at least for high values of the loading factor. For the sake of completeness, we now present conditions that make Ehrlich and Becker (1972)'s substitutability result between market and self-

<footnote n="25">²⁵The simulations were performed with Mathematica.</footnote>

<footnote n="26">²⁶One can easily verify that the conditions of Proposition 4 hold, i.e. that $C(a) \equiv \{x|F_a(x|a) - f(x|a) \ge 0\}$ is an interval that shrinks when $a$ varies from 0.4 to 2.</footnote>
insurance valid in a continuum-of-loss framework. As shown in Proposition 5, this is the case if self-insurance is locally strong in the upper tier of losses.

**Proposition 5** Assume that $\mathcal{C}(a) \equiv \{x|F_a(x|a) - f(x|a) \ge 0\} = [\tilde{x}(a), \bar{x}]$ with $\tilde{x}'(a) \ge 0$. Then, if $D^*(\sigma)$ is an increasing function and if the individual displays constant absolute risk aversion, market insurance and self-insurance are substitutes for all $\sigma \in [0, \bar{\sigma}]$.

Under the conditions of Proposition 5, the set of loss levels for which self-insurance is locally strong, $\mathcal{C}(a)$, is an interval that includes the largest losses for all $a$, i.e. losses close to $\bar{x}$, contrary to Propositions 1 and 4. The condition $\tilde{x}'(a) \ge 0$ indicates that $\mathcal{C}(a)$ shrinks when $a$ increases, due to a decreasing marginal efficiency of self-insurance. An example of such a case is given by proportional self-insurance, i.e. when loss $x$ is the realization of the random variable $X(a) = X_0\gamma(a)$, with $\gamma(0) = 1$ and $\gamma'(a) < 0$, where $X_0$ is the random variable of the loss without any self-insurance (hence with c.d.f. $F(x|0)$) and $\gamma(a)$ is a loss reduction factor that is the same for all potential losses that would occur if there were no self-insurance.<fnref n="27" /> We then have $F(x|a) \equiv \Pr\{X(a) \le x\} = \Pr\{X_0 \le x/\gamma(a)\} = F(x/\gamma(a)|0)$ for all $x \in (0, \bar{x}\gamma(a)]$, which gives $\tilde{x}(a) = -\gamma(a)/\gamma'(a)$ with $\tilde{x}'(a) \ge 0$ if $\gamma''(a) \ge \gamma'(a)^2/\gamma(a)$, which corresponds to the log-convexity of function $\gamma(a)$.<fnref n="28" />

## 3.5 Self-insurance as a residual loss function

We have assumed that the random loss $x$ is distributed in $[0, \bar{x}]$ according to c.d.f. $F(x|a)$ for all $a \ge 0$. Interestingly, the effect of self-insurance on the individual's risk exposure may also be characterized by a function $L(\cdot|a)$ from $[0, \bar{x}]$ to $[0, \bar{x}]$ called *residual loss function* (RLF) such that $L(x|a) \le x$ and $L(x|0) = x$. By definition of the RLF, the random loss under self-insurance expenses $a$ has the same probability distribution as $L(X_0|a)$ where $X_0$ is the random variable that corresponds to the no-self-insurance case, i.e. $a = 0$. In other words, function $L(\cdot|a)$ is defined by

$$F(L(x|a)|a) = F(x|0) \text{ for all } x \text{ in } [0, \bar{x}],$$

with $L_x > 0, L_a \le 0$.

<footnote n="27">²⁷This is the case considered by Quaas and Baumgärtner (2008) in a model with mean-variance preferences.</footnote>

<footnote n="28">²⁸Under proportional self-insurance, the upper bound of possible losses depends on self-insurance expenses. The proof of Proposition 5 covers this case.</footnote>
Fig. 4 illustrates this definition in the pure self-insurance case (i.e., $\bar{F}'(a) = 0$ for all $a$), and for simplicity, we restrict ourselves to this case in what follows. The black curve depicts the c.d.f. of the loss without self-insurance $F(x|0)$, and the blue curve the distribution of the loss with $a > 0$, $F(x|a)$. There is the same probability that the loss is lower than $x_0$ in the absence of self-insurance and lower than $L(x_0|a)$ under self-insurance at level $a > 0$.<fnref n="29" />

<figure><img src="image_6.png" /><figcaption>Figure 4: Derivation of the residual loss function.</figcaption></figure>

Proposition 1 has shown that self-insurance and market insurance are complementary when $\sigma$ is large, under the assumption $F_a(x|a) - f(x|a) < 0$ when $x$ is close to $\bar{x}$, a case that we have called “locally weak self-insurance”. Furthermore, Proposition 4 has extended this result under the assumption that $F_a(x|a) - f(x|a)$ is positive when

<footnote n="29">²⁹Of course, this does not imply that the random loss under self-insurance is a deterministic function of the random loss in the absence of self-insurance. For the sake of clarity, consider $\Omega$ the underlying set of states of nature (associated with a set of events and a probability measure) and let $X_a : \Omega \to [0, \bar{x}]$ be the random variable such that $X_a(\omega)$ is the loss size in state $\omega \in \Omega$ under self-insurance expenses $a$. Hence, $X_a$ and $L(X_0|a)$ are identically distributed in $[0, \bar{x}]$, which does not imply that $X_a$ is a deterministic function of $X_0$. There is such a deterministic relationship in the model of Li and Peter (2021). They assume $X_a = \Phi(a, \kappa) \in [0, \bar{x}]$, where $\kappa$ is the realization of a one-dimensional random variable $\tilde{\kappa}$ that stands for the uncertain factors that affect the effectiveness of self-insurance, with $\Phi_a < 0$, $\Phi_{aa} \ge 0$ and $\Phi_\kappa < 0$. Hence, higher values of $\kappa$ represent more efficient self-insurance. Equivalently, we have $X_a = \Phi(a, \tilde{\kappa}(X_0))$, where $\tilde{\kappa}(\cdot)$ is given by $x = \Phi(0, \tilde{\kappa}(x))$ for all $x \in [0, \bar{x}]$. In other words $\tilde{\kappa}(X_0)$ and $\tilde{\kappa}$ are identically distributed, with $L(\cdot|a) = \Phi(a, \tilde{\kappa}(\cdot))$ in this model. A simple calculation shows that $C(a) = [0, \hat{x}(a))$ with $\hat{x}'(a) < 0$ (as in Proposition 4) when $\Phi_{a\kappa} < 0$. Li and Peter (2021) say that self-insurance exhibits “increasing difference” in that case, because an increase in self-insurance reduces small losses by more than large losses, a terminology introduced by Hoy (1989) in the case of self-protection.</footnote>
$x$ is in an interval $C(a)$ whose upper and lower bounds are lower than $\bar{x}$ and possibly larger than 0, respectively. The following proposition highlights the close link between this characterization and the RLF.

**Proposition 6** For all $(x, a)$, we have $F_a(x|a) - f(x|a) < 0$ iff $L_a(L^{-1}(x|a)|a) > -1$.

Using Proposition 6 allows us to reformulate Proposition 1 in an intuitive way. Since $L^{-1}(x|a)$ goes to $\bar{x}$ when $x$ goes to $\bar{x}$, the condition in Proposition 1 means that a marginal increase in self-insurance expenses $da$ is equivalent to a residual loss decrease $-L_a(x|a)da$ smaller than $da$. This is why self-insurance was said to be locally weak in that case, although Proposition 6 now shows that “ex post inefficient” would have been a better terminology. Hence, the condition in Proposition 1 means that increasing self-insurance would be ex post inefficient if only the largest losses were considered. The condition $C(a') \subseteq C(a)$ if $a' > a$ in Proposition 4 may also be reinterpreted in the same perspective: when self-insurance expenses increase, there is a widening of the lowest and highest parts of interval $[0, \bar{x}]$ where increasing self-insurance is ex post inefficient, which corresponds to a decreasing marginal efficiency of self-insurance.

A possible interpretation of condition $L_a(L^{-1}(x|a)|a) > -1$ is in terms of regret the individual may feel when he suffers a loss $x$ after spending $a$ on self-insurance. Indeed, assume that the individual's rationality reflects an underlying causal model in which he faces an exogenous loss (i.e. independent of his decisions), distributed according to c.d.f. $F(\cdot|0)$, which can be reduced through self-insurance, the final loss being distributed according to c.d.f. $F(\cdot|a)$. The condition $L_a(L^{-1}(x|a)|a) > -1$ then means that after suffering a loss $x$ and realizing that this loss would have been $L^{-1}(x|a)$ without self-insurance, this individual regrets in retrospect having spent too much money on self-insurance.<fnref n="30" />

# 4 Beyond the CARA case

When absolute risk aversion is not constant, the optimal choices about market insurance and self insurance are affected by wealth effects that have been investigated for each instrument separately in the literature. In the following, we untangle their interplay within our framework. In section 4.1, we highlight sufficient conditions under which the complementarity between market insurance and self-insurance remains valid

<footnote n="30">³⁰The condition in Proposition 1 then means that he experiences such regret after suffering large losses. Similarly, when loss $x$ is in $C(a)$ introduced in proposition 4, he does not regret his self-insurance expenditures.</footnote>
when the loading factor is high. One key condition is that self-insurance is an inferior good. Section 4.2 describes the conditions on the distribution of the global cost of risk that enable this property to hold for individuals who are risk-averse and prudent. Section 4.3 details the impact of self-insurance on this distribution, which can be decomposed into an expected value effect and a volatility effect, and then returns to the notion of local weakness or strength of self-insurance and its interpretation in terms of regret.

## 4.1 Complementarity of self-insurance and market insurance

Wealth effects are at work when we extend our problem beyond the CARA case. In particular, a decrease in the loading factor $\sigma$ under its upper bound $\bar{\sigma}$ creates a negative wealth effect, because of the increase in the insurance premium from 0 to $\hat{P}(D^*(\sigma), a^*(\sigma), \sigma) > 0$. When self-insurance is an inferior good (as market insurance under DARA preferences), this wealth effect reinforces the demand for self-insurance induced by a decrease in $\sigma$, which confirms the complementarity between market insurance and self-insurance when $\sigma$ is large. More formally, we obtain the following result where $A(w_f) = -u''(w_f)/u'(w_f)$ denotes the index of absolute risk aversion as a function of the final wealth $w_f$:

**Proposition 7** If $\lim_{x \to \bar{x}} \partial \ln f(x|a)/\partial a > -A(w - \bar{x} - a^*(\bar{\sigma}))$ and self-insurance is an inferior good in the absence of market insurance, then self-insurance and market insurance are complementary when $\sigma$ is close to $\bar{\sigma}$.

The first assumption in Proposition 7 differs from that in Proposition 1, although both assumptions have in common that an increase in $a$ should not strongly affect the probability distribution of the largest losses.<fnref n="31" /> The other assumption stipulates that self-insurance is an inferior good in the absence of market insurance, and we now examine why this may be the case.

## 4.2 Self-insurance as an inferior good

In the Ehrlich and Becker (1972)'s two-state model without market insurance, Dionne and Eeckhoudt (1985) and Briys and Schlesinger (1990) show that self-insurance is an

<footnote n="31">³¹The condition in Proposition 7 is a sufficient one that is more restrictive than the one in Proposition 1. It is compatible with $f_a(x|a) < 0$ when $x$ is close to $\bar{x}$. In other words, similarly to the condition in Proposition 1, an increase in self-insurance may reduce the probability of the largest losses, but “not too much”.</footnote>
inferior good when the individual displays decreasing absolute risk aversion (DARA). ? shows that this result is not robust in a continuum-of-loss framework and shows that self-insurance can be either an inferior good or a normal good in the DARA case. We detail in the following the effects at work in the continuum-of-loss setup and we discuss the characteristics of loss distributions that make self-insurance an inferior good for individuals who are risk-averse and prudent (i.e. assuming $u' > 0$, $u'' < 0$ and $u''' > 0$).

Considering the global cost of risk $z = x+a$, the individual's program in the absence of market insurance is written as

$$ \max_{a \ge 0} \int_a^{a+\bar{x}} u(w-z)dG(z|a) \equiv U(w,a), $$

where $G(z|a) = F(z-a|a)$. Define $a^*(w)$ the optimal self-insurance level as a function of initial wealth $w$ in this problem without market insurance, with $\partial U(w, a^*)/\partial a = 0$ at an interior solution $a^*(w) > 0$. The implicit function theorem yields

$$ \frac{da^*}{dw} = -\frac{\partial^2 U(w, a)}{\partial w \partial a} $$

where $a = a^*(w)$ and $\partial^2 U(w, a)/\partial a^2 < 0$. Hence $da^*/dw$ and $\partial^2 U(w, a)/\partial w \partial a$ have the same sign, and self-insurance is an inferior good in the absence of market insurance if $\partial^2 U(w, a)/\partial w \partial a < 0$. We have

$$ \frac{\partial^2 U(w, a)}{\partial w \partial a} = \frac{\partial}{\partial a} \left[ \int_a^{a+\bar{x}} u'(w-z)dG(z|a) \right] = \frac{\partial}{\partial a} \left[ \int_a^{a+\bar{x}} v(z-w)dG(z|a) \right] $$

where $v(y) \equiv u'(-y)$, and thus $v'(y) = -u''(-y) > 0$ and $v''(y) = u'''(-y)$. Consequently, if $u''' > 0$ (i.e., the individual is prudent), then $\partial^2 U(w, a)/\partial w \partial a < 0$ if an increase in $a$ reduces the expected utility of an individual with wealth $z$ distributed according to c.d.f. $G(z|a)$ and with utility function $v$ such that $v' > 0$ and $v'' > 0$ (i.e. a risk-lover individual).

As detailed in the following section, self-insurance produces a change in the distribution of the loss that can be described as the sum of a change in the expected value of the global cost of risk and a mean-preserving change in this global cost (hereafter called the mean-value effect and the volatility effect, respectively). The mean-value effect is detrimental to the individual with utility $v(z-w)$ if the expected value of his final wealth decreases with $a$ because $v' > 0$. As regard to the volatility effect, it is also detrimental to this risk-loving individual if the increase in $a$ reduces the risk of his wealth (in the sense of Rothschild and Stiglitz, 1970). Self-insurance is an inferior good
under the cumulated actions of these two effects as shown in the following section.

## 4.3 Self-insurance and the distribution of risk

Consider two self-insurance levels $a_0$ and $a_1 > a_0$, and let $Z_i$ be the random variable corresponding to the global cost $z$ when $a = a_i$, for $i = 0$ and $1$, with c.d.f. $G_i(z) = G(z|a_i)$ and expected value $m_i = \mathbb{E}Z_i$. The expected value effect associated with the change from $Z_0$ to $Z_1$ is the increase in expected cost given by

$$m_1 - m_0 = a_1 - a_0 - (\mathbb{E}X_0 - \mathbb{E}X_1),$$

where $X_i$ is the random loss under self-insurance $a_i$, for $i \in \{0, 1\}$. When $m_1 > m_0$, by analogy with market insurance, this expected value effect may be interpreted as the *implicit loading of self-insurance*. In other words, this is the decrease in expected wealth that results from additional self-insurance expenses.

Let us define random variable $Z_2 = Z_0 + m_1 - m_0$ and $G_2$ and $m_2 = \mathbb{E}Z_2$, its c.d.f. and expected value. Hence, $Z_2$ is the residual global cost of risk when $a = a_0$ after compensating for the decrease in expected wealth when $a$ changes from $a_0$ to $a_1$. By construction, $m_2 = m_1$. The change from $Z_0$ to $Z_1$ when $a$ increases from $a_0$ to $a_1$ may then be decomposed into two steps as follows:

$$Z_0 \xrightarrow[\text{step 1}]{+m_1-m_0} Z_2 \xrightarrow[\text{step 2}]{+Z_1-Z_2} Z_1,$$

where steps 1 and 2 correspond to the expected value effect and to the volatility effect, respectively.

Fig. 5 illustrates this two-step decomposition when $m_0 < m_1$ (i.e., when there is a positive implicit loading of self-insurance). $G_2(z)$ corresponds to a rightward shift of $G_0(z)$. Fig. 5 is drawn in the case where $Z_1$ is a mean-preserving contraction of $Z_2$, with only one crossing point between $G_1$ and $G_2$. In this case, the expected value effect and the volatility effect cumulate, and self-insurance is an inferior good.

In Fig. 5, $G_1(z) > G_0(z)$ if $z \in (\hat{z}_0, \hat{z}_1)$.<fnref n="32" /> We thus have $F(z - a_1|a_1) > F(z - a_0|a_0)$ for all $z \in (\hat{z}_0, \hat{z}_1)$, or equivalently, using the RLF and defining $\Delta a \equiv a_1 - a_0$, $\hat{x}_0 \equiv \hat{z}_0 - a_1$ and $\hat{x}_1 \equiv \hat{z}_1 - a_1$,

$$F(x|a_1) = F(L^{-1}(x|a_1)|0) > F(x + \Delta a|a_0) = F(L^{-1}(x + \Delta a|a_0)|0)$$

<footnote n="32">³²If $G_1(z) < G_0(z)$ for all $z$, then $a_0$ would be obviously a better choice than $a_1$ whatever the individual's preferences.</footnote>
<figure><img src="image_7.png" /><figcaption>Figure 5: Two-step decomposition of an increase in $a$ from $a_0$ to $a_1$. $\bar{z}_i \equiv a_i + \bar{x}$ denotes the upper bound of the support of $Z_i$, $i = 0, 1$. Curve $G_2(z)$ depicts the c.d.f. of $Z_2$ and corresponds to a rightward shift of the c.d.f. of $Z_0$, $G_0(z)$. Curve $G_1(z)$ depicts the c.d.f. of $Z_1$ which is a mean-preserving contraction of $Z_2$.</figcaption></figure>

and thus $L^{-1}(x|a_1) > L^{-1}(x + \Delta a|a_0)$ implying $L(L^{-1}(x|a_1)|a_0) + a_0 > x + a_1$ for all $x \in (\hat{x}_0, \hat{x}_1)$. Following the interpretation in terms of regret introduced in Section 3.5, the individual who suffers a loss $x \in (\hat{x}_0, \hat{x}_1)$ after spending $a_1$ on self-insurance, considers that he would have suffered a loss larger than $x + a_1 - a_0$ with only $a_0$ spent on self-insurance, and thus doesn't regret in retrospect having spent $a_1$ rather than $a_0$.

# 5 Broader insurance contracts scope

So far, we have relied on several simplifying assumptions, most notably symmetric information, which permits self-insurance expenses to be contractible, and a fully diversifiable risk, so that the insurer's cost is proportional to the insurance expected indemnities. Under these conditions, the optimal insurance contract takes the form of a straight deductible policy. When these assumptions do not hold, or when additional frictions are present, some degree of coinsurance, possibly in combination with a deductible, may become optimal. We now turn to an examination of the robustness of our results in such settings, beginning with the implications of (ex ante) moral hazard. Next, we will consider the case in which coinsurance contracts are employed as a
second-best insurance mechanism to mitigate the frictions affecting the insured-insurer relationship.

## 5.1 Non verifiable self-insurance

When self-insurance expenditures are not verifiable, the optimal contractual agreement {$P, I(\cdot), a$} maximizes the policyholder's expected utility subject to the insurer's pricing constraint and to the following incentive constraint:

$$a \in \arg \max_{a'} \int_{0}^{x} u(w - P - a' + I(x) - x) dF(x|a'). \quad (4)$$

Let {$P^{**}(\sigma), I^{**}(\cdot, \sigma), a^{**}(\sigma)$} denote the optimal solution to this expected utility maximization under moral hazard, and let $\bar{\sigma} = \sup\{\sigma \in \mathbb{R}_+ | P^{**}(\sigma) > 0\} \le \bar{\sigma}$ be the value of the loading factor above which the individual does not purchase any insurance. To understand how our conclusions may be affected by the incentive constraint, assume first that $F_a(x|a) \equiv 0$ when $x$ is close to $\bar{x}$, as illustrated in Figure 1. In that case, the optimal deductible without moral hazard $D^*(\sigma)$ is close to $\bar{x}$ when $\sigma$ is close to $\bar{\sigma}$, and thus the policyholder's self-insurance choice $a$ only affects the loss probability distribution in the lower part of its support, which is not covered by this optimal perfect-information insurance policy. In other words, self-insurance has no common value to the insured and to the insurer in that case. Consequently, when $\sigma$ is large enough, the optimal insurance contract under perfect information remains optimal when $a$ is non-contractible, with unchanged self-insurance $a^{**}(\sigma) = a^*(\sigma)$ and a straight deductible contract $I^{**}(x, \sigma) = \max\{0, x - D^*(\sigma)\}$. Hence, Proposition 1 still applies in that case, and market insurance and self-insurance remain complements when $\sigma$ is close to $\bar{\sigma}$. When $F_a(x|a) \ne 0$ for all $x > 0$, then a straight deductible policy is usually no more optimal under moral hazard.<fnref n="33" /> However, our conclusion about the complementarity of self-insurance and market insurance may be extended to this case if self-insurance exerts a sufficiently small effect on the probability distribution of large losses. These conclusions are stated in Proposition 8 and Corollary 1, respectively.

**Proposition 8** If $F_a(x|a) = 0$ when $\hat{x}(a) < x < \bar{x}$, then $P^{**}(\sigma) = P^*(\sigma)$, $I^{**}(\bar{x}, \sigma) = \max\{0, x - D^*(\sigma)\}$ and $a^{**}(\sigma) = a^*(\sigma)$ when $\sigma$ is close to $\bar{\sigma} = \bar{\sigma}$, and self-insurance and market insurance are complements under moral hazard if $\sigma$ is large enough.

<footnote n="33">³³As shown by Holmström (1979), a straight-deductible policy is optimal under moral hazard only in the case of pure self-protection as defined by Ehrlich and Becker (1972), i.e., when the prevention expenditures do not affect the probability distribution of losses conditionally on the occurrence of an accident.</footnote>
**Corollary 1** If $F(\cdot|a)$ is sufficiently close to a c.d.f. $\tilde{F}(\cdot|a)$ such that $\tilde{F}_a(\cdot|a) = 0$ when $\tilde{x}(a) < x < \bar{x}$, then $P^{**}(\sigma)$ and $a^{**}(\sigma)$ are both decreasing when $\sigma$ is close to $\bar{\sigma}$ and self-insurance and market insurance are complements under moral hazard if $\sigma$ is large enough.

Proposition 8 just reflects the fact that self-insurance has no common value when $\sigma$ is large and $F_a(x|a) = 0$ if $x$ is close to $\bar{x}$. Consequently, the optimal solutions under contractible and non-contractible self-insurance coincide when $\sigma$ is large, and the complementarity between self-insurance and market insurance still holds in this setting with moral hazard. Corollary 1 considers the set of loss random variables $\tilde{X}_a$ with c.d.f. $\tilde{F}(\cdot|a)$ such that $\tilde{F}_a(x|a) = 0$ when $x$ is close to $\bar{x}$ and it extends the validity of Proposition 8 to all random losses that are sufficiently close (in the sense of convergence in probability) to a random variable in this set. The intuition of this corollary is as follows. Consider $\tilde{a}^{**}(\sigma)$, the self-insurance level under moral hazard and loss c.d.f. $\tilde{F}(\cdot|a)$. We know from Proposition 8 that $\tilde{a}^{**}(\sigma)$ is decreasing when $\sigma$ is large enough. The proof of Corollary 1 considers a sequence of random variables $X_{na}, n \in \mathbb{N}$, with optimal self-insurance under moral hazard $a_n^{**}(\sigma)$, that converges to $\tilde{X}_a$ in probability. A continuity argument shows that $a_n^{**}(\sigma) \to \tilde{a}^{**}(\sigma)$ for all $\sigma$, when $n \to \infty$. Since $\tilde{a}^{**}(\sigma)$ is decreasing when $\sigma$ is large, this is also the case for $a_n^{**}(\sigma)$. In words, for $\sigma$ large enough, self-insurance and market insurance are complementary for any c.d.f. $F(\cdot|a)$ sufficiently close to a c.d.f. where self-insurance does not affect the probability distribution of large losses.<fnref n="34" />

Consider now the opposite polar case where $\sigma$ is small. According to Proposition 2, self-insurance and market insurance are substitutes in the case where self-insurance expenditures are verifiable. This result stemmed from the fact that the optimal insurance contract tends toward full coverage as $\sigma$ approaches zero, with the indirect effects, including the risk coverage effect, becoming negligible. Self-insurance expenditures were then driven by the direct and indirect insurance coverage effects, which explains the substitutability between market insurance and self-insurance. This no longer holds in a moral hazard context, since the optimal insurance contract provides partial coverage even if there is no loading. Consequently, market insurance and self-insurance may become complements when $\sigma$ is small, as the following illustrative example shows.

<footnote n="34">³⁴Corollary 1 is based on a stronger condition than Proposition 1. Indeed, consider a set of random variables $X_{an}, n \in \mathbb{N}$, with c.d.f. $F_n(\cdot|a) = 1 - e^{-(\lambda_n a + \mu_n x)}$ when $x$ is close to $\bar{x}$, and density $f_n(\cdot|a)$, with $0 < \lambda_n < \mu_n$ for all $n$. We have $F_{na}(x|a) - f_n(x|a) = (\lambda_n - \mu_n)e^{-(\lambda_n a + \mu_n x)} < 0$, and thus all random variable $X_{na}$ satisfies the condition of Proposition 1. However, we have $F_{na}(x|a) > \lambda_n e^{-(\lambda_n a + \mu_n \bar{x})} > 0$ for all $n$, and thus there is no sequence in this set that converges in probability to a random variable $\tilde{X}_a$ with c.d.f. $\tilde{F}(\cdot|a)$ such that $\tilde{F}_a(x|a) = 0$ when $x$ is close to $\bar{x}$.</footnote>
Assume that an accident occurs with probability $1 - \bar{F}$, with only two possible levels of loss $x_1$ and $x_2$, with $x_1 < x_2$, and probabilities $\pi_1 + a$ and $\pi_2 - a$, respectively, conditionally on a loss occurring. Parameters $\pi_1$ and $\pi_2 = 1 - \pi_1$ correspond to the loss conditional probabilities in the absence of self-insurance, and self-insurance expenditures $a \in [0, \pi_2]$ shift probability weight from the large-loss state to the low-loss state, while the probability of an accident remains unchanged. An insurance contract corresponds to an indemnity $I_i$ paid when the loss is $x_i$, and to a premium $P = (1 + \sigma)\mathbb{E}[I]$ where $\mathbb{E}[I] = (1 - \bar{F})[(\pi_1 + a)I_1 + (\pi_2 - a)I_2]$. With CARA utility and denoting $Y_i \equiv e^{\alpha(x_i - I_i)}$, the policyholder's expected utility is $\mathbb{E}u = -e^{\alpha(P+a)}\mathbb{E}[Y]$ where $\alpha$ is the index of absolute risk aversion and $\mathbb{E}[Y] = \bar{F} + (1 - \bar{F})[(\pi_1 + a)Y_1 + (\pi_2 - a)Y_2]$. The first-order condition of the incentive constraint is written as

$$ \alpha \mathbb{E}[Y] = (1 - \bar{F})(Y_2 - Y_1), \qquad (\text{IC}) $$

at an interior solution $0 < a < \pi_2$. Hence, we must have $Y_2 > Y_1$, and thus $x_2 - I_2 > x_1 - I_1 \ge 0$, in order to provide incentives. In other words, the retained loss $x_i - I_i$ should be larger when the loss is large than when it is small, for the policyholder to be incentivized to provide self-insurance effort. For the sake of simplicity and realism, we simplify this example by excluding over-insurance, and thus by assuming $x_1 - I_1 \ge 0$. The insurer's program is then written as $\max_{I_1, I_2, a}\{\mathbb{E}u : (IC), I_i \le x_i\}$ and the optimal solution is denoted $I_1^{**}(\sigma)$, $I_2^{**}(\sigma)$, $a^{**}(\sigma)$. The calculations reported in the appendix show that an interior optimal solution to this problem, i.e. with $\pi_2 > a^{**}(\sigma) > 0$, is such that $I_1^{**}(\sigma) = x_1$, $I_2^{**}(\sigma) < x_2$. Hence, unsurprisingly, there is full coverage of small losses, and partial coverage of large losses. It is also shown that self-insurance and market insurance are complementary for all $\sigma$, and thus $a^{**}(\sigma)$ and $I_2^{**}(\sigma)$ move in the same direction as $\sigma$ increases. When there is no Giffen effect, a case illustrated in Fig. 6, then $I_2^{**}(\sigma)$ and $a^{**}(\sigma)$ are both decreasing functions: the larger the loading factor, the lower the coverage of large losses and the lower the self-insurance expenditures.

## 5.2 Coinsurance contracts

We now consider a setting in which insurers offer coinsurance contracts even though self-insurance efforts are verifiable. In this context, coinsurance is implicitly viewed as a second-best solution that helps mitigate frictions in the insured-insurer relationship, whether these arise from ex post moral hazard, high solvency capital requirements, or other market imperfections. We do not attempt to specify the precise origins of
<figure><img src="image_8.png" /><figcaption>Figure 6: Optimal self-insurance level $a^*(\sigma)$ and insurance indemnity $I_2^*(\sigma)$ under moral hazard. Parameter values $\bar{F} = 3/4$, $\pi_1 = 1/2$, $\pi_2 = 1/2$, $x_1 = 1$, $x_2 = 2.6$, $\alpha = 1$. $\sigma$ varies from 0 to 12%.</figcaption></figure>

these frictions; instead, our focus remains on the interaction between self-insurance and market insurance under such conditions.

A coinsurance contract with a rate $\gamma \in [0, 1]$ specifies a premium $P$ and an indemnity schedule $I(x) = \gamma x$ for all $x \in [0, \bar{x}]$.<fnref n="35" /> Maximizing the policyholder's expected utility with respect to $a$ and $\gamma$, subject to the insurer's break-even constraint is written as

$$ \max_{a, \gamma} \mathbb{E}u = \int_{0}^{\bar{x}} u(w - \tilde{P}(\gamma, a, \sigma) - a - x(1 - \gamma))dF(x|a) \quad (5) $$

where

$$ \tilde{P}(\gamma, a, \sigma) = (1 + \sigma)\gamma \int_{0}^{\bar{x}} x dF(x|a), $$

and we denote $a^*(\sigma)$ and $\gamma^*(\sigma)$ the optimal solution as a function of $\sigma$. Let $\bar{\sigma} = \sup\{\sigma | \gamma^*(\sigma) > 0\}$, where $\gamma^*(\bar{\sigma}) = 0$ and $a^*(\bar{\sigma})$ is the optimal self-insurance when the individual does not purchase market insurance.

**Proposition 9** Assume that the insurer offers co-insurance contracts $I(x) = \gamma x$. Then, when the insured displays constant absolute risk aversion,

$$ -\frac{\partial}{\partial a} \frac{\mathbb{E}[x^2 e^{\alpha x} | a^*(\bar{\sigma})]}{\mathbb{E}[x e^{\alpha x} | a^*(\bar{\sigma})]} < 1 \quad (6) $$

is a sufficient condition for $\gamma^*(\sigma)$ and $a^*(\sigma)$ to be decreasing functions when $\sigma$ is close to $\bar{\sigma}$.

<footnote n="35">³⁵In the terminology of health insurance, $1-\gamma$ is the coinsurance percentage, i.e. the share of health care costs paid by the policyholder.</footnote>
To interpret the condition in Proposition 9, observe that with a CARA utility function $u(w) = -e^{-\alpha w}$, we have

$$ \frac{\mathbb{E}[x^2 e^{\alpha x} | a]}{\mathbb{E}[xe^{\alpha x} | a]} = \frac{\mathbb{E}[x^2 u'(-x) | a]}{\mathbb{E}[x u'(-x) | a]}. $$

Let $g_\alpha(x|a) \equiv \eta e^{\alpha x} f(x|a)$ be a modified density function, with $\eta = 1/\int_0^{\bar{x}} e^{\alpha x} f(x|a)dx$, deduced from the actual density function $f(x|a)$ by weighting it with the marginal disutility of the loss. This yields

$$ \frac{\mathbb{E}[x^2 e^{\alpha x} | a]}{\mathbb{E}[xe^{\alpha x} | a]} = \frac{\int_0^{\bar{x}} x^2 g_{\alpha}(x|a) dx}{\int_0^{\bar{x}} x g_{\alpha}(x|a) dx} = \frac{V_{\alpha}[x|a]}{\mathbb{E}_{\alpha}[x|a]} + \mathbb{E}_{\alpha}[x|a] = \mathbb{E}_{\alpha}[x|a](1 + \mathbb{C}V_{\alpha}[x|a]^2) $$

where $\mathbb{E}_\alpha[x|a]$, $V_\alpha[x|a]$ and $\mathbb{C}V_\alpha[x|a]$ denote the expected value, the variance and the coefficient of variation of the loss under the modified density $g_\alpha(\cdot|a)$, respectively. The condition in Proposition 9 can thus be written as

$$ \frac{\partial}{\partial a} \left[ \mathbb{E}_{\alpha}[x|a] (1 + \mathbb{C}V_{\alpha}[x|a]^2) \right] > -1 \quad \text{when } a = a^*(\bar{\sigma}). $$

Two opposite forces affect this derivative. First, an increase in $a$ leads to a negative first-order shift in the modified probability distribution of the loss, reducing $\mathbb{E}_\alpha[x|a]$. However, an increase in the modified coefficient of variation $\mathbb{C}V_\alpha[x|a]$ may exert a countervailing effect. This is likely to occur when the increase in self-insurance tends to convert medium-sized losses to small losses without significantly affecting the largest losses. The condition in proposition 9 will be satisfied if this compensatory effect is sufficiently large. To illustrate how this can happen, consider the case where the loss is distributed over $[0, +\infty)$ according to a Gamma distribution. It is shown in the appendix that the condition in Proposition 9 holds if

$$ \frac{d}{da} \ln(1 + SK(a)^2/4) \geq -\frac{d}{da} \ln \mathbb{E}[x|a] \quad \text{when } a = a^*(\bar{\sigma}) \qquad (7) $$

where $SK(a) = 2\mathbb{C}V[x|a]$ is the skewness of the gamma distribution, a measure of its asymmetry. The larger the skewness, the more right-tailed is the distribution, i.e. the more the distribution appears as a left-leaning curve. Condition (7) expresses that, in relative terms, the increase in self-insurance must have a sufficiently large impact on the skewness of the distribution of the loss to overcome the decrease of its expected value. Graphically, this implies that the reduction in the expected value should not appear as a simple leftward shift in the distribution but as an increase in its asymmetry due to
<figure><img src="image_9.png" /><figcaption>Figure 7: P.d.f and c.d.f. of losses following a Gamma distribution. The blue curves are characterized by $E[x|a] = 10$ and $SK(a) = 1$ and the red curves by $E[x|a'] = 6$ and $SK(a') = 1.1$.</figcaption></figure>

the mass of the distribution being more concentrated on the left. Fig. 7 illustrates the variation of the density and c.d.f. of the loss due to a change in self-insurance from $a$ to $a' > a$, which increases the skewness and decreases the mean so that (the discrete version of) condition (7) is satisfied.

The following proposition shows that, like in the case where straight-deductible contracts are optimal, co-insurance contracts result in market insurance and self-insurance that are substitutable when the loading factor is small.

**Proposition 10** If the individual displays constant absolute risk aversion, $\gamma^*(\sigma)$ is decreasing and $a^*(\sigma)$ is increasing when $\sigma$ is close to zero.

The intuition of Proposition 10 is the same as that of Proposition 2: When $\sigma$ is close to 0, the individual is almost fully covered and reducing the insurance premium is the main reason for increasing self-insurance. The direct insurance premium effect outweighs the indirect effects when $\sigma$ increases, leading to an increase in $a^*(\sigma)$ above the risk neutral level $a_N$.

# 6 Conclusion

When the price of market insurance is high because of transaction costs, taxes or solvency capital costs, subsidizing insurance may be a matter of governmental decision in order to improve risk-sharing in the economy. The resulting change in insurance price
is likely to affect self-insurance, and this may be a concern for public decision-making. This is particularly the case in agriculture, when self-insurance creates detrimental externalities, for instance when farmers use more pesticides to protect their crops against pests attacks or when they replace rain-fed farming by irrigation and water resources are scarce. Conversely, externalities would be favorable when self-insurance takes the form of biodiversity preservation and ecosystem protection. When such important matters are at stake, understanding how self-insurance choices react to changes in the insurance price becomes particularly important.

In their seminal paper, Ehrlich and Becker (1972) conclude that self-insurance and market insurance are substitutes, and consequently, subsidizing market insurance would tend to discourage policyholders from exerting self-insurance efforts. In the case of agriculture, this means that subsidizing crop insurance would encourage farmers to deviate from less profitable practices based on biodiversity preservation and ecosystem protection, insofar as these technologies play a self-insurance role. In other words, this suggests that favoring farmers’ access to market insurance could undermine the consideration of environmental issues in public policy-making.

As shown in this paper, this would be an over-hasty conclusion. When the cost of accidents is random, contrary to the model of Ehrlich and Becker (1972), then the combination of market insurance and self-insurance goes through the design of the optimal insurance policy. We have considered a model in which this optimal policy corresponds to a straight-deductible contract, which opens the way to an adequate combination of market insurance and self-insurance. Beyond the mere fact that self-insurance shifts the probability distribution of losses leftwards, it has appeared that the substitutability or complementarity between self-insurance and market insurance depends on which levels of loss are less likely to occur when self-insurance expenses increase. When the largest gains from self-insurance correspond to the upper tier of losses, as in the case of proportional self-insurance, then self-insurance exerts risk coverage effects similar to those of an insurance contract covering losses above a deductible, and it is therefore substitutable to market insurance in that case. On the contrary, when self-insurance is more effective in decreasing the probability of medium-sized losses and turning them into small losses, then it is a natural complement to market insurance. In that case, when the high price of insurance leads to a large deductible, reducing the deductible by lowering the loading factor motivates the policyholder to self-insure more intensely. More insurance then goes hand in hand with more self-insurance.

However, the optimality of the straight-deductible contract relies on several simplifying assumptions, most notably the verifiability of self-insurance efforts and the
full diversification of risk in insurers’ portfolios. In the presence of moral hazard, we have shown that if self-insurance does not significantly reduce the probability of catastrophic losses, the optimal contract remains close to a deductible policy, and our main conclusions still apply: self-insurance and market insurance act as complements when insurance prices are high. More generally, as demonstrated in a stylized two-loss setting, because moral hazard modifies individual incentives, an optimal contract that provides partial coverage for large losses can generate complementarity even at lower insurance prices. Similarly, complementarity emerges when insurers offer coinsurance contracts and self-insurance simultaneously reduces expected losses and increases the asymmetry of the loss distribution.

These results highlight that the relationship between self-insurance and market
insurance depends critically on the distribution of losses and the presence of informa-
tional or market frictions. Contrary to the traditional view, complementarity is often
the norm when insurance is expensive and prevention affects only parts of the loss
distribution. This has important implications for public policy. Subsidizing insurance,
especially in agriculture, disaster risk, or health, need not discourage preventive behav-
iors. On the contrary, well-designed insurance programs may foster greater investment
in self-insurance, particularly when they reduce the financial consequences of worst-
case scenarios. Recognizing this nuanced interaction is key to designing effective and
sustainable insurance interventions.
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# Appendix

## A Lemma 1

**Lemma 1** The optimal solution to the individual's problem $a^*, D^*$ satisfies

$$ \int_{0}^{D^{*}} u'(w - x - \hat{P}(D^{*}, a^{*}, \sigma) - a^{*})dF(x|a^{*}) = H(D^{*}, a^{*}, \sigma)u'(w - D^{*} - \hat{P}(D^{*}, a^{*}, \sigma) - a^{*}) \quad (8) $$

$$ \int_{0_{+}}^{D^{*}} u'(w - x - \hat{P}(D^{*}, a^{*}, \sigma) - a^{*}) F_{a^{*}}(x|a^{*})dx = K(D^{*}, a^{*}, \sigma)u'(w - D^{*} - \hat{P}(D^{*}, a^{*}, \sigma) - a^{*}). \quad (9) $$

where

$$ K(D, a, \sigma) \equiv \frac{1}{1 + \sigma} - \int_{D}^{\bar{x}} F_{a}(x|a)dx, \quad (10) $$

$$ H(D, a, \sigma) \equiv \frac{1}{1 + \sigma} - [1 - F(D|a)]. \quad (11) $$

## Proof

Maximizing $\mathbb{E}u$ given by (1) w.r.t. $D$ and $a$ leads to the following first-order optimality conditions

$$ -[1 - F(D^*|a^*)]u'(w - D^* - \hat{P} - a^*) - \frac{\partial \hat{P}}{\partial D} \frac{\partial \mathbb{E}u}{\partial w} = 0 \quad (12) $$

and

$$ \bar{F}'(a^*)u(w - \hat{P}(D^*, a^*, \sigma) - a^*) + \int_{0_+}^{D^*} u(w - x - \hat{P} - a^*) f_a(x|a^*) dx \\ - F_a(D^*|a^*)u(w - D^* - \hat{P} - a^*) - \left(1 + \frac{\partial \hat{P}}{\partial a}\right) \frac{\partial \mathbb{E}u}{\partial w} = 0 \quad (13) $$

respectively, where

$$ \frac{\partial \mathbb{E}u}{\partial w} = \bar{F}(a^*)u'(w - \hat{P}(D^*, a^*, \sigma) - a^*) + \int_{0^+}^{D^*} u'(w - x - \hat{P} - a^*)dF(x|a^*) \\ + [1 - F(D^*|a^*)]u'(w - D^* - \hat{P} - a^*), \quad (14) $$
$$ \frac{\partial \hat{P}(D^*, a^*, \sigma)}{\partial D} = -(1 + \sigma)[1 - F(D^*|a^*)]. \quad (15) $$

Integrating by parts and observing that $F(\bar{x}|a) = 1$ for all $a$ gives $F_a(\bar{x}|a) = 0$, it comes

$$ \frac{\partial \hat{P}(D^*, a^*, \sigma)}{\partial a} = (1 + \sigma) \int_{D^*}^{\bar{x}} (x - D^*) f_a(x|a^*) dx = -(1 + \sigma) \int_{D^*}^{\bar{x}} F_a(x|a^*) dx. \quad (16) $$

Using (12), (14) and (15) gives

$$ (1+\sigma) \int_{0}^{D^*} u'(w-x-\hat{P}-a^*) dF(x|a^*) = u'(w-D^*-\hat{P}-a^*) (1-(1+\sigma)[1-F(D^*|a^*)]). \quad (17) $$

Dividing by $1 + \sigma$ gives (8).

Integrating by parts the integral in (13) gives

$$ \int_{0_+}^{D^*} u(w - x - \hat{P} - a^*) f_a(x|a^*) dx = u(w - D^* - \hat{P} - a^*) F_a(D^*|a^*) - u(w - \hat{P} - a^*) \bar{F}'(a^*) + \int_{0_+}^{D^*} u'(w - x - \hat{P} - a^*) F_a(x|a^*) dx. $$

Substituting in (13) and using (16) yields

$$ \int_{0_+}^{D^*} u'(w - x - \hat{P} - a^*) F_a(x|a^*) dx = \left[ 1 - (1 + \sigma) \int_{D^*}^{\bar{x}} F_a(x|a^*) dx \right] \frac{\partial \mathbb{E}u}{\partial w}. \quad (18) $$

Furthermore, (12) and (15) give

$$ \frac{\partial \mathbb{E}u}{\partial w} = \frac{u'(w - D^* - \hat{P} - a^*)}{1 + \sigma}, \quad (19) $$

and (18) and (19) yield

$$ (1+\sigma) \int_{0_+}^{D^*} u'(w-x-\hat{P}-a^*) F_a(x|a^*) dx = u'(w-D^*-\hat{P}-a^*) \left[ 1 - (1+\sigma) \int_{D^*}^{\bar{x}} F_a(x|a^*) dx \right]. \quad (20) $$

Dividing by $1 + \sigma$ gives (9).
B Proof of Proposition 1

In the CARA case, we may write $u(w) = -e^{-\alpha w}$ where $\alpha > 0$ is the index of absolute risk aversion. Using $u'(w) = -\alpha u(w)$ and factorizing out $u(w - D - \hat{P}(D, a, \sigma) - a)$, equations (8) and (9) can be rewritten as

$$ \int_{0}^{D^{*}} e^{-\alpha(D^{*}-x)} dF(x|a^{*}) = H(D^{*}, a^{*}, \sigma), \qquad (21) $$

$$ \int_{0^{+}}^{D^{*}} e^{-\alpha(D^{*}-x)} F_{a}(x|a^{*}) dx = K(D^{*}, a^{*}, \sigma), \qquad (22) $$

respectively. Totally differentiating these equations, using $H_D = f(D^*|a^*)$, $H_a = F_a(D^*|a^*) = K_D$, $K_a = -\int_{D^*}^{\bar{x}} F_{aa}(x|a^*)dx$, $H_\sigma = K_\sigma = 1/(1+\sigma)^2$, and collecting terms give

$$ \alpha H \frac{dD^*}{d\sigma} - \left[ \int_{0}^{D^*} e^{-\alpha(D^*-x)} dF_a(x|a^*) - F_a(D^*|a^*) \right] \frac{da^*}{d\sigma} = \frac{1}{(1+\sigma)^2}, $$

$$ \alpha K \frac{dD^*}{d\sigma} - \left[ \int_{0}^{D^*} e^{-\alpha(D^*-x)} F_{aa}(x|a^*) dx + \int_{D^*}^{\bar{x}} F_{aa}(x|a^*) dx \right] \frac{da^*}{d\sigma} = \frac{1}{(1+\sigma)^2}, $$

which can be written as

$$ \begin{bmatrix} \alpha H & -A \\ \alpha K & -B \end{bmatrix} \begin{bmatrix} dD^*/d\sigma \\ da^*/d\sigma \end{bmatrix} = \frac{1}{(1+\sigma)^2} \begin{bmatrix} 1 \\ 1 \end{bmatrix} \qquad (23) $$

where

$$ A = \int_{0}^{D^{*}} e^{-\alpha(D^{*}-x)} dF_{a}(x|a^{*}) - F_{a}(D^{*}|a^{*}), $$

$$ B = \int_{0}^{D^{*}} e^{-\alpha(D^{*}-x)} F_{aa}(x|a^{*}) dx + \int_{D^{*}}^{\bar{x}} F_{aa}(x|a^{*}) dx. $$

We have

$$ \begin{vmatrix} \alpha H & -A \\ \alpha K & -B \end{vmatrix} = \alpha(AK - HB) = \alpha\Delta $$

where

$$ \Delta \equiv K \left[ \int_{0}^{D^{*}} e^{-\alpha(D^{*}-x)} dF_{a}(x|a^{*}) - F_{a}(D^{*}|a^{*}) \right] - H \left[ \int_{0}^{D^{*}} e^{-\alpha(D^{*}-x)} F_{aa}(x|a^{*}) dx + \int_{D^{*}}^{\bar{x}} F_{aa}(x|a^{*}) dx \right] \qquad (24) $$

is the determinant of the Hessian matrix of $Eu$ evaluated at a maximum of this function
(i.e. at $D = D^*(\sigma)$ and $a = a^*(\sigma)$). We thus have $\Delta > 0$ and solving (23) yields

$$ 
\begin{bmatrix} dD^*/d\sigma \\ da^*/d\sigma \end{bmatrix} = \frac{1}{\alpha\Delta(1+\sigma)^2} \begin{bmatrix} -B & A \\ -\alpha K & \alpha H \end{bmatrix} \begin{bmatrix} 1 \\ 1 \end{bmatrix} \\ = \frac{1}{\alpha\Delta(1+\sigma)^2} \begin{bmatrix} A-B \\ \alpha(H-K) \end{bmatrix},
$$

hence

$$ \frac{dD^*}{d\sigma} = \frac{\int_0^{D^*} e^{-\alpha(D^*-x)} dF_a(x|a^*) - F_a(D^*|a^*) - \left[ \int_0^{D^*} e^{-\alpha(D^*-x)} F_{aa}(x|a^*) dx + \int_{D^*}^{\bar{x}} F_{aa}(x|a^*) dx \right]}{\alpha\Delta(1+\sigma)^2} \quad (25) $$

and

$$ \frac{da^*}{d\sigma} = \frac{\int_{D^*}^{\bar{x}} [F_a(x|a^*) - f(x|a^*)] dx}{(1+\sigma)^2 \Delta}. \quad (26) $$

As $D^*(\sigma) > \hat{x}(a)$ when $\sigma$ is large enough, which implies $F_a(x|a) < f(x|a)$ for all $x \in [D, \bar{x}]$, we have

$$ \int_{D^*(\sigma)}^{\bar{x}} [F_a(x|a^*(\sigma)) - f(x|a^*(\sigma))] dx < 0 $$

and it comes from (26) that $da^*/d\sigma < 0$ when $\sigma < \bar{\sigma}$, $\sigma$ close to $\bar{\sigma}$. We also have $dD^*/d\sigma > 0$ when $\sigma$ is close to $\bar{\sigma}$ since $D^*(\bar{\sigma}) = \bar{x}$ and $D^*(\sigma) < \bar{x}$ when $\sigma < \bar{\sigma}$.

## C Proof of Proposition 2

We have $D^*(\sigma) > 0$ when $\sigma$ is close to 0 since $D^*(0) = 0$ and $D^*(\sigma) > 0$ when $\sigma > 0$. We may write $F(x|a) = \bar{F}(a) + [1 - \bar{F}(a)]H(x|a)$, where $H(x|a)$ is the c.d.f. of the loss conditionally on a loss occurring, with $H(0|a) = 0$ and $H(\bar{x}|a) = 1$. Hence, we have $F_a(x|a) = \bar{F}'(a)[1 - H(x|a)] + [1 - \bar{F}(a)]H_a(x|a)$ and $f(x|a) = [1 - \bar{F}(a)]h(x|a)$ where $h(x|a) = H_x(x|a)$. We may write

$$ \int_{0_+}^{\bar{x}} [F_a(x|a) - f(x|a)] dx = \bar{F}'(a) \int_{0}^{\bar{x}} [1 - H(x|a)] dx \\ + [1 - \bar{F}(a)] \int_{0}^{\bar{x}} H_a(x|a) dx - [1 - \bar{F}(a)], \quad (27) $$
using $\int_0^{\bar{x}} h(x|a)dx = 1$, and

$$ 
\begin{aligned}
a^*(0) &\in \arg\min_a[1 - \bar{F}(a)] \int_0^{\bar{x}} xh(x|a)dx + a \\
&= \arg\min_a[1 - \bar{F}(a)] \int_0^{\bar{x}} [1 - H(x|a)]dx + a.
\end{aligned}
$$

The corresponding FOC gives

$$ -\bar{F}'(a^*(0)) \int_0^{\bar{x}} [1 - H(x|a^*(0))]dx - [1 - \bar{F}(a^*(0))] \int_0^{\bar{x}} H_a(x|a^*(0))dx + 1 = 0. \quad (28) $$

When $\sigma \to 0$, $D^*(\sigma) \to 0$ and thus, using (27) and (28),

$$ \int_{D^*(\sigma)}^{\bar{x}} [F_a(x|a^*(\sigma)) - f(x, a^*(\sigma))] dx \to \int_{0_+}^{\bar{x}} [F_a(x|a^*(0)) - f(x|a^*(0))] dx = \bar{F}(a^*(0)). $$

When $\bar{F}(a^*(0)) > 0$, using (26) and $\Delta > 0$ then gives $a^*(\sigma) > 0$ when $\sigma$ is close to 0.

When $\bar{F}(a) = 0$ for all $a$, we also obtain $a^*(\sigma) = 0$ when $\sigma \to 0$. Indeed, suppose the contrary, i.e. $a^*(\sigma) \le 0$ when $\sigma \to 0$, $\sigma > 0$. Let $\psi^*(\sigma) = \psi(D^*(\sigma), a^*(\sigma))$ where

$$ \psi(D, a) \equiv \int_D^{\bar{x}} [F_a(x|a) - f(x|a)] dx. \quad (29) $$

We know from (26) in the proof of Proposition 1 that $a^*(\sigma)$ and $\psi^*(\sigma)$ have the same sign. We have $\psi^*(0) = \bar{F}(a^*(0)) = 0$ and

$$ 
\begin{aligned}
\psi^*(\sigma) = & -[F_a(D^*(\sigma)|a^*(\sigma)) - f(D^*(\sigma)|a^*(\sigma))]D^*(\sigma) \\
& + a^*(\sigma) \int_{D^*(\sigma)}^{\bar{x}} [F_{aa}(x|a^*(\sigma)) - f_a(x|a^*(\sigma))]dx.
\end{aligned}
\quad (30) $$

$\bar{F}(a) = F(0|a) = 0$ for all $a$ gives $F_a(0|a) = 0$ implying $F_a(x|a) \to 0$ when $x \to 0$ since $x \mapsto F_a(x|a)$ is right-continuous. Also, as $F(\bar{x}|a) = 1$ for all $a$, we have $F_a(\bar{x}|a) = 0$ for all $a$ and thus $\int_0^{\bar{x}} f_a(x|a)dx = F_a(\bar{x}|a) - F_a(0|a) = 0$. Consequently

$$ \int_{D^*(\sigma)}^{\bar{x}} [F_{aa}(x|a^*(\sigma)) - f_a(x|a^*(\sigma))]dx \to \int_{0}^{\bar{x}} F_{aa}(x|a^*(0))dx \le 0 $$

when $\sigma \to 0$ since $F_{aa}(x|a) \le 0$. As $a^*(\sigma) \le 0$ when $\sigma \to 0$, the second term of (30) is thus non-negative. Furthermore, $D^*(\sigma) \to +\infty$ when $\sigma \to 0$ because the numerator
in (25) goes to a positive limit whereas the numerator goes to zero. We thus have

$$-[F_a(D^*(\sigma)|a^*(\sigma)) - f(D^*(\sigma)|a^*(\sigma))]D^{\prime\prime}(\sigma) \to f(0|a^*(0))D^{\prime\prime}(0)$$

when $\sigma \to 0$, and thus $\psi^{\prime\prime}(\sigma) \to +\infty$ when $\sigma \to 0$. Using $\psi^*(0) = 0$ yields $\psi^*(\sigma) > 0$ when $\sigma > 0$, $\sigma$ close to 0, hence $a^{\prime\prime}(\sigma) > 0$ when $\sigma \to 0$, $\sigma > 0$, a contradiction.

# D MWP for self-insurance

The fact that MWP($D^*(\sigma)$, $a^*(\sigma)$, $\sigma$) = 1 for all $\sigma \in [0, \bar{\sigma}]$ follows directly from (13). In the CARA case, we have

$$MWP(D, a, \sigma) = \frac{\int_{0}^{D} (1 - e^{-\alpha(D-x)}) dF_{a}(x|a)}{\alpha \left( \int_{0^{+}}^{D} e^{-\alpha(D-x)} dF(x|a) + 1 - F(D|a) \right)} + (1 + \sigma) \int_{D}^{\bar{x}} F_{a}(x|a) dx.$$

where

$$\int_{0}^{D} (1 - e^{-\alpha(D-x)}) dF_{a}(x|a) = \bar{F}'(a)(1 - e^{-\alpha D}) + \int_{0^{+}}^{D} (1 - e^{-\alpha(D-x)}) f_{a}(x|a) dx.$$

Integrating by parts gives,

$$\int_{0^+}^{D} (1 - e^{-\alpha(D-x)}) f_a(x|a) dx = [(1 - e^{-\alpha(D-x)}) F_a(x|a)]_0^D + \alpha \int_{0^+}^{D} e^{-\alpha(D-x)} F_a(x|a) dx \\ = -\bar{F}'(a)(1 - e^{-\alpha D}) + \alpha \int_{0^+}^{D} e^{-\alpha(D-x)} F_a(x|a) dx,$$

hence

$$MWP(D, a, \sigma) = \frac{\int_{0^+}^{D} e^{-\alpha(D-x)} F_a(x|a) dx}{\int_{0^+}^{D} e^{-\alpha(D-x)} dF(x|a) + 1 - F(D|a)} + (1 + \sigma) \int_{D}^{\bar{x}} F_a(x|a) dx. \quad (31)$$

Using (11), (10), (22) and (21) give

$$\int_{0}^{D^{*}(\sigma)} e^{-\alpha(D^{*}(\sigma)-x)} dF(x|a^{*}(\sigma)) = \frac{1}{1+\sigma} - [1 - F(D^{*}(\sigma)|a^{*}(\sigma))] \quad (32)$$

and

$$\int_{0^+}^{D^*(\sigma)} e^{-\alpha(D^*(\sigma)-x)} F_a(x|a^*(\sigma)) dx = \frac{1}{1+\sigma} - \int_{D^*(\sigma)}^{\bar{x}} F_a(x|a^*(\sigma)) dx, \quad (33)$$
respectively.

Differentiating (31) with respect to D and using (32) and (33) give

$$ \frac{\partial MWP(D^*(\sigma), a^*(\sigma), \sigma)}{\partial D} = -\alpha(1+\sigma)[1-F(D^*(\sigma)|a^*(\sigma))] \left(1-(1+\sigma) \int_{D^*(\sigma)}^{\bar{x}} F_a(x|a^*(\sigma))dx\right), $$

and thus

$$ \frac{dMWP(D^*(\sigma), a^*(\sigma), \sigma)}{d\sigma} = \int_{D^*(\sigma)}^{\bar{x}} F_a(x|a^*(\sigma))dx - \alpha(1+\sigma)[1-F(D^*(\sigma)|a^*(\sigma))] \left(1-(1+\sigma) \int_{D^*(\sigma)}^{\bar{x}} F_a(x|a^*(\sigma))dx\right) D^{\prime\prime}(\sigma). $$

As $D^*(\sigma) \to \bar{x}$ when $\sigma \to \bar{\sigma}$, we have $\lim_{\sigma \to \bar{\sigma}} F(D^*(\sigma)|a) = F(\bar{x}|a) = 1$ and thus $dMWP/d\sigma \to 0$ when $\sigma \to \bar{\sigma}$. Differentiating w.r.t. $\sigma$ and using $\lim_{\sigma \to \bar{\sigma}} F(D^*(\sigma)|a) = 1$ and $\lim_{\sigma \to \bar{\sigma}} a^{\prime\prime}(\sigma) = 0$ gives

$$ \frac{d^2 MWP}{d\sigma^2} \approx [-F_a(D^*(\sigma)|a^*(\sigma)) + \alpha(1+\sigma)f(D^*(\sigma)|a^*(\sigma))D^{\prime\prime}(\sigma)]D^{\prime\prime}(\sigma), $$

when $\sigma \to \bar{\sigma}$. Since

$$ D^{\prime\prime}(\sigma) \to \frac{1}{\alpha(1+\bar{\sigma})}, $$

when $\sigma \to \bar{\sigma}$, we deduce $d^2MWP/d\sigma^2 > 0$ when $\sigma$ is close to $\bar{\sigma}$ if $f(x|a^*(\bar{\sigma})) - F_a(x|a^*(\bar{\sigma})) > 0$ when $x$ is close to $\bar{x}$. Under this condition, we have $dMWP/d\sigma < 0$ when $\sigma$ is close to $\bar{\sigma}$.

As $D^*(\sigma) \to 0$ when $\sigma \to 0$, (33) gives $\int_0^{\bar{x}} F_a(x|a^*(0))dx = 1$, i.e. $\partial\mathbb{E}[x|a^*(0)]/\partial a = 1$. We thus have $a^*(0) = a_N$. From (10), (11) and (24) it comes $H \to \bar{F}(a_N)$, $K \to 0$ and $\Delta \to -\bar{F}(a_N) \int_0^{\bar{x}} F_{aa}(x|a_N)dx > 0$ when $\sigma \to 0$, and from (25),

$$ \lim_{\sigma \to 0} \frac{dD^*}{d\sigma} = \frac{1}{\alpha \bar{F}(a_N)} > 0. $$

We thus get

$$ \frac{dMWP}{d\sigma} \to 1 \text{ when } \sigma \to 0. $$

## E Proof of Proposition 3

(25) can be written as

$$ \frac{dD^*}{d\sigma} = \frac{A(D, a) - B(D, a)}{\alpha\Delta(1 + \sigma)^2} $$
where $D = D^*(\sigma)$, $a = a^*(\sigma)$, and

$$A(D, a) \equiv \int_{0}^{D} e^{-\alpha(D-x)} [f_{a}(x|a) - F_{aa}(x|a)]dx, \\ B(D, a) \equiv F_{a}(D|a) + \int_{D}^{\bar{x}} F_{aa}(x|a)dx.$$

Let

$$\phi(x|a) \equiv f_a(x|a) - F_{aa}(x|a). \qquad (34)$$

We have $f(x|a) = 0$ for all $a$ when $x \ge \bar{x}$ and thus $f_a(\bar{x}|a) = 0$ because $f(x|a)$ is twice-differentiable at $x = \bar{x}$. Furthermore, we have $F(x|a) = 1$ for all $a$ and $x \ge \bar{x}$, which implies $F_{aa}(\bar{x}|a) = 0$. Hence, we have $\phi(\bar{x}|a) = 0$ for all $a$. For any $x, a$ such that $\phi(x|a) < 0$, we have $f_a(x|a) < 0$ (because $F_{aa}(x|a) \le 0$ for all $x$), and thus $\phi_x(x|a) = f_{ax}(x|a) - f_{aa}(x|a) \le 0$ under the condition postulated in the Proposition. This implies $\phi(x'|a) < 0$ for all $x' > x$, which contradicts $\phi(\bar{x}|a) = 0$. Hence $\phi(x|a) \ge 0$ for all $x \in (0, \bar{x})$, which gives

$$A(D, a) \equiv \int_{0}^{D} e^{-\alpha(D-x)} \phi(x|a)dx \ge 0.$$

Furthermore, we have $B(0, a) = \int_{0}^{\bar{x}} F_{aa}(x|a)dx < 0$, $B(\bar{x}, a) = 0$ and $\partial B(D, a)/\partial D = f_a(D|a) - F_{aa}(D|a) = \phi(D|a) \ge 0$, which implies $B(D, a) < 0$ for all $D \le \bar{x}$. We deduce $A(D, a) - B(D, a) > 0$ and thus $dD^*/d\sigma > 0$ for all $\sigma \in (0, \bar{\sigma})$.

# F Proof of Proposition 4

We know from (26) and (29) that $a^*(\sigma)$ and $\psi^*(\sigma) \equiv \psi(D^*(\sigma), a^*(\sigma))$ have the same sign, and from Propositions 1 and 2 that $\psi^*(0_+) > 0$, $\psi^*(\bar{\sigma}) = 0$ and $\psi^*(\bar{\sigma}_-) < 0$. Let $\hat{\sigma} = \inf\{\sigma : \forall \sigma' \in [\sigma, \bar{\sigma}], \psi^*(\sigma') \le 0\}$ so that $a^*(\hat{\sigma})$ is a local maximum, and suppose that $a^*(\cdot)$ reaches another local maximum at $\tilde{\sigma} < \hat{\sigma}$. Then, it must also reach a local minimum at $\sigma_0 \in (\tilde{\sigma}, \hat{\sigma})$, with $a^*(\sigma_0) < a^*(\tilde{\sigma})$ and $a^*(\sigma_0) < a^*(\hat{\sigma})$, and we must have $\psi^*(\hat{\sigma}) = \psi^*(\tilde{\sigma}) = \psi^*(\sigma_0) = 0$ with $\psi^{*'}(\hat{\sigma}) < 0$, $\psi^{*'}(\tilde{\sigma}) < 0$ and $\psi^{*'}(\sigma_0) > 0$ where, using (30),

$$\psi^{*'}(\sigma) = -[F_a(D^*(\sigma)|a^*(\sigma)) - f(D^*(\sigma)|a^*(\sigma))]D^{*'}(\sigma)$$

for $\sigma \in \{\tilde{\sigma}, \sigma_0, \hat{\sigma}\}$. As $D^{*'}(\sigma) > 0$, $\psi^{*'}(\sigma)$ and $F_a(D^*(\sigma)|a^*(\sigma)) - f(D^*(\sigma)|a^*(\sigma))$ have opposite signs. From (ii), $a^*(\sigma_0) < a^*(\tilde{\sigma})$, $a^*(\sigma_0) < a^*(\hat{\sigma})$, and $\psi^{*'}(\hat{\sigma}) < 0$ and $\psi^{*'}(\tilde{\sigma}) < 0$, we have $D^*(\hat{\sigma}) \in C(a^*(\hat{\sigma})) \subseteq C(a^*(\sigma_0))$ and $D^*(\tilde{\sigma}) \in C(a^*(\tilde{\sigma})) \subseteq C(a^*(\sigma_0))$. Also, as
$\tilde{\sigma} < \sigma_0 < \hat{\sigma}$ and $D^*(\sigma)$ is non-decreasing, we have $D^*(\tilde{\sigma}) \le D^*(\sigma_0) \le D^*(\hat{\sigma})$ and since $C(a^*(\sigma_0))$ is convex, $D^*(\sigma_0) \in C(a^*(\sigma_0))$, implying $\psi^{\prime\prime}(\sigma_0) < 0$, a contradiction.

## G Proof of Proposition 5

In this section, we consider the case where the upper limit of the loss may depend on the self-insurance effort, i.e. $\bar{x}(a)$, with $\bar{x}'(a) \le 0$. We thus have $C(a) = [\tilde{x}(a), \bar{x}(a)]$, with $\tilde{x}'(a) \ge 0$, $\bar{x}'(a) \le 0$, and $D^*(\bar{\sigma}) = \bar{x}(a^*(\bar{\sigma}))$ by definition of $\bar{\sigma}$, and $D^*(\sigma) < \bar{x}(a^*(\sigma))$ for all $\sigma < \bar{\sigma}$.

We know from (26) and (29) that $a^{\prime\prime}(\sigma)$ and $\psi^*(\sigma) \equiv \psi(D^*(\sigma), a^*(\sigma))$ have the same sign, and from Proposition 2 that $\psi^*(0_+) > 0$. Suppose that $a^*(\cdot)$ is not monotonic, i.e. that it reaches a local maximum at $\tilde{\sigma} < \bar{\sigma}$. Then, either it reaches a local minimum at $\sigma_0 \in (\tilde{\sigma}, \bar{\sigma})$, or it is monotonically decreasing over $[\tilde{\sigma}, \bar{\sigma}]$. In the first case, we must have $\psi^*(\tilde{\sigma}) = \psi^*(\sigma_0) = 0$ with $\psi^{\prime\prime}(\tilde{\sigma}) < 0$ and $\psi^{\prime\prime}(\sigma_0) > 0$ where, using (30),

$$ \psi^{\prime\prime}(\sigma) = -[F_a(D^*(\sigma)|a^*(\sigma)) - f(D^*(\sigma)|a^*(\sigma))]D^{\prime\prime}(\sigma) $$

for $\sigma \in \{\tilde{\sigma}, \sigma_0\}$. As $D^{\prime\prime}(\sigma) > 0$, $\psi^{\prime\prime}(\sigma)$ and $F_a(D^*(\sigma)|a^*(\sigma)) - f(D^*(\sigma)|a^*(\sigma))$ have opposite signs. As $\psi^{\prime\prime}(\tilde{\sigma}) < 0$ and $\psi^{\prime\prime}(\sigma_0) > 0$, we have $D^*(\tilde{\sigma}) \in C(a^*(\tilde{\sigma}))$ and $D^*(\sigma_0) \notin C(a^*(\sigma_0))$, hence $D^*(\tilde{\sigma}) \ge \tilde{x}(a^*(\tilde{\sigma}))$ and $D^*(\sigma_0) < \tilde{x}(a^*(\sigma_0))$. From $\tilde{x}'(a) \ge 0$ and $a^*(\tilde{\sigma}) > a^*(\sigma_0)$, we have $\tilde{x}(a^*(\tilde{\sigma})) \ge \tilde{x}(a^*(\sigma_0))$, and thus $D^*(\tilde{\sigma}) > D^*(\sigma_0)$, which contradicts $\tilde{\sigma} < \sigma_0$ and $D^{\prime\prime}(\sigma) > 0$.

In the second case, we have $D^*(\tilde{\sigma}) \ge \tilde{x}(a^*(\tilde{\sigma}))$ as in the first case, and using $D^*(\bar{\sigma}) = \bar{x}(a^*(\bar{\sigma}))$, (21), (22) and (24), we get $H(D^*(\bar{\sigma}), a^*(\bar{\sigma}), \bar{\sigma}) = K(D^*(\bar{\sigma}), a^*(\bar{\sigma}), \bar{\sigma}) = 1/(1+\bar{\sigma})$ and

$$ \Delta \rightarrow \left[ \int_{0}^{\bar{x}(a^*(\bar{\sigma}))} e^{-\alpha[\bar{x}(a^*(\bar{\sigma})) - x]} [f_a(x|a^*(\bar{\sigma})) - F_{aa}(x|a^*(\bar{\sigma}))] dx \right] / (1 + \bar{\sigma}) \neq 0 $$

when $\sigma \rightarrow \bar{\sigma}$, and thus, from (25) and (26), $D^{\prime\prime}(\sigma) \rightarrow 1/\alpha(1+\bar{\sigma}) > 0$ and $a^{\prime\prime}(\sigma) \rightarrow 0$ when $\sigma \rightarrow \bar{\sigma}$. As $a^*(\cdot)$ is monotonically decreasing over $[\tilde{\sigma}, \bar{\sigma}]$, we have $\psi^*(\sigma) < 0$ for all $\sigma \in (\tilde{\sigma}, \bar{\sigma})$, and from (29) and (30), $\psi^*(\bar{\sigma}) = \psi(\bar{x}(a^*(\bar{\sigma})), a^*(\bar{\sigma})) = 0$ and

$$ \psi^{\prime\prime}(\bar{\sigma}) = -[F_a(\bar{x}(a^*(\bar{\sigma})))|a^*(\bar{\sigma})) - f(\bar{x}(a^*(\bar{\sigma})))|a^*(\bar{\sigma}))]D^{\prime\prime}(\bar{\sigma}) = 0. $$

Since $\psi^*(\sigma) < 0$ for $\sigma$ close to $\bar{\sigma}$ and $\psi^{\prime\prime}(\bar{\sigma}) = \psi^*(\bar{\sigma}) = 0$, we thus must have $\psi^{\prime\prime\prime}(\bar{\sigma}) < 0$. Furthermore, using (30), $D^*(\bar{\sigma}) = \bar{x}(a^*(\bar{\sigma}))$ and $F_a(\bar{x}(a^*(\bar{\sigma})))|a^*(\bar{\sigma})) =$
$f(\bar{x}(a^*(\bar{\sigma})))|a^*(\bar{\sigma})) = 0$ yields

$$ \psi^{\star\prime\prime}(\bar{\sigma}) = -D^{\star\prime}(\bar{\sigma})^2 \frac{\partial}{\partial x} [F_a(x|a^*(\bar{\sigma})) - f(x|a^*(\bar{\sigma}))]|_{x=\bar{x}(a^*(\bar{\sigma}))} \\ - D^{\star\prime}(\bar{\sigma})a^{\star\prime}(\bar{\sigma}) \frac{\partial}{\partial a} [F_a(x|a^*(\bar{\sigma})) - f(x|a^*(\bar{\sigma}))]|_{a=a^*(\bar{\sigma})}. $$

Using $a^*(\bar{\sigma}) = 0$ and $\psi^{\star\prime\prime}(\bar{\sigma}) < 0$ gives $\frac{\partial}{\partial x}[F_a(x|a^*(\bar{\sigma})) - f(x|a^*(\bar{\sigma}))]|_{x=\bar{x}(a^*(\bar{\sigma}))} > 0$. As $F_a(D^*(\bar{\sigma})|a^*(\bar{\sigma})) - f(D^*(\bar{\sigma})|a^*(\bar{\sigma})) = 0$, we deduce $F_a(x|a^*(\bar{\sigma})) - f(x|a^*(\bar{\sigma})) < 0$ if $x < D^*(\bar{\sigma})$, and thus $C(a^*(\bar{\sigma})) = \{D^*(\bar{\sigma})\} = \{\bar{x}(a^*(\bar{\sigma}))\}$, i.e. $\tilde{x}(a^*(\bar{\sigma})) = \bar{x}(a^*(\bar{\sigma}))$. As $D^{\star\prime}(\bar{\sigma}) > 0$, we have $D^*(\bar{\sigma}) < D^*(\bar{\sigma})$. Furthermore, $\tilde{x}'(a) \ge 0$ and $a^*(\bar{\sigma}) > a^*(\bar{\sigma})$ gives $\tilde{x}(a^*(\bar{\sigma})) \le \tilde{x}(a^*(\tilde{\sigma})), \text{ hence } D^*(\bar{\sigma}) < \tilde{x}(a^*(\tilde{\sigma})), \text{ a contradiction.}$

## H Proof of Proposition 6

Using $F(L(x|a), a) = F(x|0)$ and $F_a(x|a) = f(x|a) > 0$ for all $x \in (0, \bar{x})$ shows that $L(x|a)$ is increasing, with $L(x|a) \to \bar{x}$ when $x \to \bar{x}$ and $L(x|a) \to 0$ when $x \to 0$. Differentiating $F(x|a) = F(L^{-1}(x|a)|0)$ yields

$$ F_a(x|a) = f(L^{-1}(x|a)|0)L_a^{-1}(x|a), $$

$$ f(x|a) = f(L^{-1}(x|a)|0)L_x^{-1}(x|a). $$

Differentiating the identity $L^{-1}(L(x|a)|a) = x$ yields

$$ L_a^{-1}(L(x|a)|a) = -L_a(x|a)/L_x(x|a), $$

$$ L_x^{-1}(L(x|a)|a) = -1/L_x(x|a) $$

that hold for all $x \in [0, \bar{x}]$, implying

$$ L_a^{-1}(x|a) = -L_a(L^{-1}(x|a)|a)/L_x(L^{-1}(x|a)|a), $$

$$ L_x^{-1}(x|a) = -1/L_x(L^{-1}(x|a)|a). $$

Consequently, we have

$$ F_a(x|a) - f(x|a) = -\frac{f(L^{-1}(x|a)|0)}{L_x(L^{-1}(x|a)|a)}[L_a(L^{-1}(x|a)|a) + 1]. $$
We have $L^{-1}(x|a) \in [x, \bar{x}]$ and thus $L^{-1}(x|a) \to \bar{x}$ when $x \to \bar{x}$. Similarly, $L^{-1}(x|a) \to 0$ when $x \to 0$. Consequently, when $x$ is close to 0 or $\bar{x}$, we have $F_a(x|a) - f(x|a) < 0$ iff $L_a(x|a) > -1$.

# I Proof of Proposition 7

Let $P^*(\sigma) \equiv \hat{P}(D^*(\sigma), a^*(\sigma), \sigma)$ and define $\hat{a}(\sigma)$ as the solution to

$$ \max_a \int_0^{\bar{x}} u(w - x - P^*(\sigma) - a) dF(x|a). $$

In words, $\hat{a}(\sigma)$ is the optimal self-insurance expense when the individual pays $P^*(\sigma) > 0$ and is not covered by the insurance contract (with $P^*(\sigma) > 0$ if $\sigma < \bar{\sigma}$ and $P^*(\bar{\sigma}) = 0$ since $D^*(\bar{\sigma}) = \bar{x}$). Let $\sigma < \bar{\sigma}$. If self-insurance is an inferior good in the absence of market-insurance, we have $\hat{a}(\sigma) > a^*(\bar{\sigma})$. Hence, we have $a^*(\sigma) > a^*(\bar{\sigma})$ if $a^*(\sigma) > \hat{a}(\sigma)$ with $a^*(\bar{\sigma}) = \hat{a}(\bar{\sigma})$. Let

$$ V(D, a, \sigma) = \int_{0}^{D} u(w - x - P^{*}(\sigma) - a) dF(x|a) + [1 - F(D|a)]u(w - D - P^{*}(\sigma) - a), $$

and let $a^{**}(D, \sigma) = \arg\max_a V(D, a, \sigma)$. We have $a^{**}(\bar{x}, \sigma) = \hat{a}(\sigma)$, and since

$$ (\partial \mathbb{E}u / \partial a)|_{D=D^*(\sigma)} - (\partial V / \partial a)|_{D=D^*(\sigma)} = - \hat{P}_a(D^*(\sigma), a, \sigma) (\partial \mathbb{E}u / \partial w)|_{D=D^*(\sigma)} \geq 0 $$

when $a = a^*(\sigma)$, we have $a^*(\sigma) > a^{**}(D^*(\sigma), \sigma)$. Hence, a sufficient condition for $a^*(\sigma) > \hat{a}(\sigma) = a^{**}(\bar{x}, \sigma)$ is given by $a^{**}(D^*(\sigma), \sigma) > \hat{a}(\sigma)$. This is the case if $D \to a^{**}(D, \sigma)$ is decreasing. The F.O.C for $a^{**}(D, \sigma)$ is

$$ \begin{aligned} \frac{\partial V}{\partial a} &= - \int_{0}^{D} u'(w - x - P^{*}(\sigma) - a) dF(x|a) - [1 - F(D|a)]u'(w - D - P^{*}(\sigma) - a) \\ & \quad + \int_{0}^{D} u(w - x - P^{*}(\sigma) - a) f_a(x|a) dx - F_a(D|a)u(w - D - P^{*}(\sigma) - a) \\ &= 0, \end{aligned} $$

and using the implicit function theorem yields

$$ \frac{\partial a^{**}(D, \sigma)}{\partial D} = - \left. \frac{\partial^2 V(D, a, \sigma)}{\partial a \partial D} \right|_{a=a^{**}(D, \sigma)}. $$
From the S.O.C, we have $\partial^2 V/\partial a^2|_{a=a^{**}(D,\sigma)} < 0$, and thus $D \to a^{**}(D, \sigma)$ is decreasing if $(\partial^2 V/\partial a \partial D)|_{a=a^{**}(D,\sigma)} < 0$ where

$$ \partial^2 V / \partial a \partial D = [1 - F(D|a)]u''(w - D - P^*(\sigma) - a) + F_a(D|a)u'(w - D - P^*(\sigma) - a). $$

We have $\partial^2 V/\partial a \partial D < 0$ if $F_a(D|a)/[1 - F(D|a)] < A(w - D - P^*(\sigma) - a)$. Using l'Hospital rule, this condition holds when $D \to \bar{x}$ if

$$ \lim_{x \to \bar{x}} - \frac{f_a(x|a)}{f(x|a)} = - \lim_{x \to \bar{x}} \frac{\partial \ln f(x|a)}{\partial a} < A(w - \bar{x} - a^*(\bar{\sigma})). $$

Under this condition, we have $a^*(\bar{\sigma}) < \hat{a}(\sigma) < a^{**}(D^*(\sigma), \sigma) < a^*(\sigma)$ if $\sigma < \bar{\sigma}$ when self-insurance is an inferior good in the absence of market insurance. In that case, self-insurance and market insurance are complementary when $\sigma$ is close to $\bar{\sigma}$.

# J Proof of Proposition 8

Suppose there exists $\sigma < \bar{\sigma}$ such that the incentive compatibility constraint does not hold when $a = a^*(\sigma)$, $P = P^*(\sigma)$ and $D = D^*(\sigma)$. Hence, there exists $a'$ such that

$$ \int_{0}^{\bar{x}} u(w - P^{*}(\sigma) - a' - x + \max\{0, x - D^{*}(\sigma)\}) dF(x|a') \\ > \int_{0}^{\bar{x}} u(w - P^{*}(\sigma) - a^{*}(\sigma) - x + \max\{0, x - D^{*}(\sigma)\}) dF(x|a^{*}(\sigma)). $$

We have

$$ P^{*}(\sigma) - (1+\sigma) \int_{D^{*}(\sigma)}^{\bar{x}} [x - D^{*}(\sigma)] dF(x|a') = P^{*}(\sigma) - (1+\sigma) \int_{D^{*}(\sigma)}^{\bar{x}} [x - \tilde{D}(\sigma)] d\tilde{F}(x|a^{*}(\sigma)) $$

when $\sigma$ is close to $\bar{\sigma}$, because $D^*(\sigma) \to \bar{x}$ when $\sigma \to \bar{\sigma}$ and $F_a(x|a) = 0$ when $x$ is close to $\bar{x}$. Consequently, $a = a' \neq a^*(\sigma)$, $P = P^*(\sigma)$, $D = D(\sigma)$ dominates the optimal solution of the problem where self-insurance is contractible, hence a contradiction. The proposition follows straightforwardly.

# K Proof of Corollary 1

Let $\tilde{F}(\cdot|a)$ be a c.d.f. with support $[0, \bar{x}]$ and such that $\tilde{F}_a(x|a) = 0$ when $x$ is close to $\bar{x}$. Let $\{P^*(\sigma), D^*(\sigma), \tilde{a}^*(\sigma)\}$ and $\{\tilde{P}^{**}(\sigma), \tilde{I}^{**}(\cdot, \sigma), \tilde{a}^{**}(\sigma)\}$ be the optimal solution
when self-insurance is contractible and not contractible, respectively, and the loss is distributed according to $\tilde{F}(\cdot|a)$. We know from Proposition 1 that $\tilde{a}^{**}(\sigma) = \tilde{a}^{*}(\sigma)$ is decreasing when $\sigma$ is close to $\tilde{\sigma} = \sup\{\sigma \in \mathbb{R}_{+}|D^{*}(\sigma) < \bar{x}\} = \sup\{\sigma \in \mathbb{R}_{+}|P^{*}(\sigma) > 0\}$.

Let $\tilde{X}_a$ be a random variable that corresponds to c.d.f. $\tilde{F}(\cdot|a)$ and consider $\{X_{a,n}, n \in \mathbb{N}\}$ a sequence of random variables with c.d.f. $F_n(\cdot|a)$ such $X_{a,n} \xrightarrow{p} \tilde{X}_a$ when $n \to +\infty$. Let $\{P_n^{**}(\sigma), I_n^{**}(\cdot, \sigma), a_n^{**}(\sigma)\}$ be an optimal solution to the expected utility maximization problem under non-verifiable self-insurance and loss c.d.f $F_n(\cdot|a)$, with $\bar{\sigma}_n^* = \sup\{\sigma \in \mathbb{R}_{+}|\tilde{P}_n^{**}(\sigma) > 0\}$. Let us show that $a_n^{**}(\sigma)$ is decreasing when $\sigma$ is close to $\bar{\sigma}_n^*$ and $n$ is large enough.

Suppose in contrast that for all $N \in \mathbb{N}$ there exists $n > N$ such that $a_n^{**}(\sigma)$ is not decreasing in the neighborhood of $\bar{\sigma}_n^*$ and thus that there exists $\sigma$ lower than $\bar{\sigma}_n^*$ and arbitrarily close to $\bar{\sigma}_n^*$, such that $a_n^{**}(\sigma) \le a_n^{**}(\bar{\sigma}_n^*)$. Denote $a_{\infty}^{**}(\sigma) = \lim_{n \to +\infty} a_n^{**}(\sigma)$, $P_{\infty}^{**}(\sigma) = \lim_{n \to +\infty} P_n^{**}(\sigma)$ and $\bar{\sigma}_{\infty}^* = \lim_{n \to +\infty} \bar{\sigma}_n^*$. We have $a_{\infty}^{**}(\sigma) = \tilde{a}^{**}(\sigma)$ and $\bar{\sigma}_{\infty}^* = \tilde{\sigma}$ because $X_{a,n} \xrightarrow{p} \tilde{X}_a$ for all $a$, when $n \to +\infty$.<fnref n="36" /> Since $\tilde{a}^{**}(\sigma) = \tilde{a}^{*}(\sigma) > \tilde{a}^{*}(\tilde{\sigma})$ when $\sigma$ is close to $\tilde{\sigma}$, we deduce $a_n^{**}(\sigma) > a_n^{**}(\bar{\sigma}_n^*)$ for $n$ large enough, hence a contradiction.

# L Three-state example with moral hazard

The Lagrangian of the insurer's program can be written as

$$-e^{\alpha(P+a)}\mathbb{E}[Y] + \mu(\alpha\mathbb{E}[Y] + (1 - \bar{F})(Y_1 - Y_2))$$

where $\mu$ is the multiplier associated to (IC), and $P = (1 + \sigma)\mathbb{E}[I]$. The f.o.c. w.r.t. $I_1$ and $I_2$ yield

$$-e^{\alpha(P+a^*)}(\pi_1 + a) \left(1 + \sigma - \frac{Y_1}{\mathbb{E}[Y]}\right) - \mu \frac{Y_1}{\mathbb{E}[Y]}[1 + \alpha(\pi_1 + a)] \geq 0, \quad (35)$$

$$-e^{\alpha(P+a^*)}(\pi_2 - a) \left(1 + \sigma - \frac{Y_2}{\mathbb{E}[Y]}\right) + \mu \frac{Y_2}{\mathbb{E}[Y]}[1 - \alpha(\pi_2 - a)] \geq 0, \quad (36)$$

respectively, which are binding if $I_i < x_i$ at the optimal solution. Using (IC), the f.o.c. w.r.t. $a$ gives

$$e^{\alpha(P+a)}(1 + \sigma)(I_2 - I_1) - \alpha\mu = 0 \quad (37)$$

<footnote n="36">³⁶The optimal solution $a_n^{**}(\sigma), P_n^{**}(\sigma)$ may not be univalued. However, by an argument similar to the one used to establish the upper hemi-continuity of the demand correspondence in consumer theory, the limit of the sequence of optimal self-insurance $a_{\infty}^{**}(\sigma)$ is an optimal solution in the problem with the limit random loss variable $\tilde{X}_a$. Since this optimal solution $\tilde{a}^{**}(\sigma)$ is unique, we have $a_{\infty}^{**}(\sigma) = \tilde{a}^{**}(\sigma) = \tilde{a}^{*}(\sigma)$.</footnote>
when $a \in (0, \pi_2)$. As we must have $x_2 - I_2 > x_1 - I_1$, we cannot have both (35) and (36) binding. We thus have $x_2 - I_2 > x_1 - I_1 = 0$. From (IC), using $Y_1 = 1$, it comes

$$Y_2 = 1 + \frac{\alpha}{(1 - \bar{F})[1 - \alpha(\pi_2 - a)]},$$

and

$$\mathbb{E}[Y] = \frac{1}{1 - \alpha(\pi_2 - a)},$$

which are defined for all $a \in [0, \pi_2]$ if $\pi_2 < 1/\alpha$. From (36), and (37), we get

$$I_2^{**} = x_1 + \frac{\alpha(\pi_2 - a^{**})}{1 - \alpha(\pi_2 - a^{**})} \left( \frac{1 - \bar{F}}{\alpha \bar{F} + (1 - \bar{F})[1 - \alpha(\pi_2 - a^{**})]} - \frac{1}{1 + \sigma} \right), \quad (38)$$

at the optimal indemnity level. Furthermore, the optimal self-insurance expenditures $a^{**}$ are such that

$$Y_2 = e^{\alpha(x_2 - I_2^{**})} = 1 + \frac{\alpha}{(1 - \bar{F})[1 - \alpha(\pi_2 - a^{**})]},$$

which can be written as

$$\phi(1 - \alpha(\pi_2 - a^{**}), \sigma) = x_2 - x_1 \quad (39)$$

where

$$\phi(y, \sigma) \equiv \frac{1}{\alpha} \ln \left( 1 + \frac{\alpha}{(1 - \bar{F})y} \right) - \frac{1 - y}{y} \left( \frac{1}{1 + \sigma} - \frac{1 - \bar{F}}{\alpha \bar{F} + (1 - \bar{F})y} \right),$$

with $y \in [1 - \alpha\pi_2, 1]$. By the implicit function theorem, we have

$$\frac{da^*}{d\sigma} = -\frac{1}{\alpha} \left. \frac{\partial \phi(y, \sigma)}{\partial \sigma} \right|_{y=1-\alpha(\pi_2-a^{**})} . \quad (40)$$

From (38) we have $I_2^{**} = \Psi(1 - \alpha(\pi_2 - a^{**}), \sigma)$ where

$$\Psi(y, \sigma) \equiv x_1 + \phi(y, \sigma) - \frac{1}{\alpha} \ln \left( 1 + \frac{\alpha}{(1 - \bar{F})y} \right),$$

and it comes

$$\frac{dI_2^*}{d\sigma} = \left[ \frac{\partial \Psi(y, \sigma)}{\partial y} \alpha \frac{da^*}{d\sigma} + \frac{\partial \Psi(y, \sigma)}{\partial \sigma} \right] \Big|_{y=1-\alpha(\pi_2-a^{**})},$$

where $\partial\Psi(y, \sigma)/\partial\sigma = \partial\phi(y, \sigma)/\partial\sigma$ and $\partial\Psi(y, \sigma)/\partial y = \partial\phi(y, \sigma)/\partial y + 1/(y((1 - \bar{F})y +$
α)). Using (40), we thus have

$$
\begin{aligned}
\frac{dI_2^{**}}{d\sigma} &= \left[ \frac{\partial\phi(y, \sigma)}{\partial y}\alpha\frac{da^{**}}{d\sigma} + \frac{\partial\phi(y, \sigma)}{\partial\sigma} + \frac{1}{y[(1-\bar{F})y + \alpha]}\alpha\frac{da^{**}}{d\sigma} \right] \Bigg|_{y=1-\alpha(\pi_2-a^{**})} \\
&= \frac{1}{y[(1-\bar{F})y + \alpha]} \Bigg|_{y=1-\alpha(\pi_2-a^{**})} \alpha\frac{da^{**}}{d\sigma},
\end{aligned}
$$

and thus $dI_2^*/d\sigma$ and $da^{**}/d\sigma > 0$ have the same sign, which means that market insurance and self-insurance are complements.

M Proof of Propositions 9 and 10

Maximizing $\mathbb{E}u$ given by (5) w.r.t. $\gamma$ and $a$ leads to the following first-order optimality conditions for $\gamma = \gamma^*(\sigma)$ and $a = a^*(\sigma)$:

$$ \frac{\partial \mathbb{E}u}{\partial \gamma} = \int_{0}^{\bar{x}} (x - (1 + \sigma)\mathbb{E}[x|a])u'(w - \hat{P}(\gamma, a, \sigma) - a - x(1 - \gamma))dF(x|a) = 0 \quad (41) $$

and

$$
\begin{aligned}
\frac{\partial \mathbb{E}u}{\partial a} = 0 &= \int_0^{\bar{x}} \left[ 1 + (1 + \sigma)\gamma \frac{\partial \mathbb{E}[x|a]}{\partial a} \right] u'(w - \hat{P}(\gamma, a, \sigma) - a - x(1 - \gamma)) dF(x|a) \\
&\quad + \int_0^{\bar{x}} u(w - \hat{P}(\gamma, a, \sigma) - a - x(1 - \gamma)) f_a(x|a) dx \quad (42)
\end{aligned}
$$

respectively. In the CARA case, using $u(w) = -e^{-\alpha w}$, $u'(w) = -\alpha u(w)$ and factorizing out $u(w - \hat{P}(D, a, \sigma) - a)$, these equations can be rewritten as

$$ \int_{0}^{\bar{x}} (x - (1 + \sigma)\mathbb{E}[x|a^*(\sigma)])e^{\alpha x(1 - \gamma^*(\sigma))}dF(x|a^*(\sigma)) = 0 $$

and

$$ - \int_{0}^{\bar{x}} \left[ \alpha + \alpha(1 + \sigma)\gamma^*(\sigma) \frac{\partial \mathbb{E}[x|a^*(\sigma)]}{\partial a} + \frac{f_a(x|a^*(\sigma))}{f(x|a^*(\sigma))} \right] e^{\alpha x(1 - \gamma^*(\sigma))} dF(x|a^*(\sigma)) = 0. \quad (43) $$

Totally differentiating these equations gives

$$
\begin{aligned}
0 = -\mathbb{E}[x|a] \int_0^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a) - \int_0^{\bar{x}} \alpha x(x - (1+\sigma)\mathbb{E}[x|a]) e^{\alpha x(1-\gamma)} dF(x|a) \frac{d\gamma^*}{d\sigma} \\
+ \left\{ \int_0^{\bar{x}} (x - (1+\sigma)\mathbb{E}[x|a]) e^{\alpha x(1-\gamma)} f_a(x|a) dx - (1+\sigma) \frac{\partial \mathbb{E}[x|a]}{\partial a} \int_0^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a) \right\} \frac{da^*}{d\sigma}
\end{aligned}
$$
and

$$0 = -\alpha\gamma \frac{\partial \mathbb{E}[x|a]}{\partial a} \int_0^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a) - \left\{ \alpha(1+\sigma) \frac{\partial \mathbb{E}[x|a]}{\partial a} \int_0^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a) - \int_0^{\bar{x}} \left[ \alpha + \alpha(1+\sigma)\gamma \frac{\partial \mathbb{E}[x|a]}{\partial a} + \frac{f_a(x|a)}{f(x|a)} \right] \alpha x e^{\alpha x(1-\gamma)} dF(x|a) \right\} \frac{d\gamma^*}{d\sigma} + \frac{\partial^2 \mathbb{E}u}{\partial a^2} \frac{da^*}{d\sigma}$$

where $\gamma = \gamma^*(\sigma)$ and $a = a^*(\sigma)$. This may be rewritten in matrix form as

$$\begin{bmatrix} \frac{\partial^2 \mathbb{E}u}{\partial \gamma^2} & B \\ C & \frac{\partial^2 \mathbb{E}u}{\partial a^2} \end{bmatrix} \begin{bmatrix} d\gamma^*/d\sigma \\ da^*/d\sigma \end{bmatrix} = \begin{bmatrix} \mathbb{E}[x|a] \int_0^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a) \\ \alpha \gamma \frac{\partial \mathbb{E}[x|a]}{\partial a} \int_0^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a) \end{bmatrix} \quad (44)$$

where

$$B = \int_{0}^{\bar{x}} (x - (1 + \sigma)\mathbb{E}[x|a])e^{\alpha x(1-\gamma)} f_a(x|a)dx - (1 + \sigma)\frac{\partial \mathbb{E}[x|a]}{\partial a} \int_{0}^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a),$$

$$C = -\alpha(1 + \sigma) \frac{\partial \mathbb{E}[x|a]}{\partial a} \int_0^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a) + \int_0^{\bar{x}} \left[ \alpha + \alpha(1 + \sigma)\gamma \frac{\partial \mathbb{E}[x|a]}{\partial a} + \frac{f_a(x|a)}{f(x|a)} \right] \alpha x e^{\alpha x(1-\gamma)} dF(x|a),$$

$$\frac{\partial^2 \mathbb{E}u}{\partial \gamma^2} = - \int_0^{\bar{x}} \alpha x (x - (1 + \sigma)\mathbb{E}[x|a]) e^{\alpha x(1-\gamma)} dF(x|a),$$

and $\gamma = \gamma^*(\sigma)$ and $a = a^*(\sigma)$. We deduce

$$\begin{bmatrix} d\gamma^*/d\sigma \\ da^*/d\sigma \end{bmatrix} = \frac{1}{\Delta} \begin{bmatrix} \frac{\partial^2 \mathbb{E}u}{\partial a^2} & -B \\ -C & \frac{\partial^2 \mathbb{E}u}{\partial \gamma^2} \end{bmatrix} \begin{bmatrix} \mathbb{E}[x|a] \int_0^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a) \\ \alpha \gamma \frac{\partial \mathbb{E}[x|a]}{\partial a} \int_0^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a) \end{bmatrix} \\ = \frac{1}{\Delta} \begin{bmatrix} \frac{\partial^2 \mathbb{E}u}{\partial a^2} \mathbb{E}[x|a] \int_0^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a) - \alpha B \gamma \frac{\partial \mathbb{E}[x|a]}{\partial a} \int_0^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a) \\ -C \mathbb{E}[x|a] \int_0^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a) + \frac{\partial^2 \mathbb{E}u}{\partial \gamma^2} \alpha \gamma \frac{\partial \mathbb{E}[x|a]}{\partial a} \int_0^{\bar{x}} e^{\alpha x(1-\gamma)} dF(x|a) \end{bmatrix},$$

where $\gamma = \gamma^*(\sigma)$ and $a = a^*(\sigma)$ and $\Delta > 0$ is the determinant of the Hessian matrix of $\mathbb{E}u$ evaluated at a maximum of this function.

Consider first the case $\sigma \to 0$ of Proposition 10. Using $\gamma^*(\sigma) \to 1$ when $\sigma \to 0$, we
get, denoting $a_0^* = a^*(\sigma)$,

$$B \rightarrow \int_{0}^{\bar{x}} (x - \mathbb{E}[x|a_0^*]) f_a(x|a_0^*) dx - \frac{\partial \mathbb{E}[x|a_0^*]}{\partial a} \int_{0}^{\bar{x}} dF(x|a_0^*) \\ = -\mathbb{E}[x|a_0^*] \int_{0}^{\bar{x}} f_a(x|a_0^*) dx = 0,$$

$$C \rightarrow -\alpha \frac{\partial \mathbb{E}[x|a_0^*]}{\partial a} \int_{0}^{\bar{x}} dF(x|a_0^*) + \int_{0}^{\bar{x}} \left[ \left( \alpha + \alpha \frac{\partial \mathbb{E}[x|a_0^*]}{\partial a} \right) f(x|a_0^*) + f_a(x|a_0^*) \right] \alpha x dx \\ = \alpha^2 \left( 1 + \frac{\partial \mathbb{E}[x|a_0^*]}{\partial a} \right) \mathbb{E}[x|a_0^*],$$

$$\frac{\partial^2 \mathbb{E}u}{\partial \gamma^2} \rightarrow - \int_{0}^{\bar{x}} \alpha x (x - \mathbb{E}[x|a_0^*]) dF(x|a_0^*) = -\alpha \mathbb{V}[x|a_0^*],$$

when $\sigma \to 0$, which gives

$$\lim_{\sigma \to 0} \frac{d\gamma^*}{d\sigma} \rightarrow \frac{\frac{\partial^2 \mathbb{E}u}{\partial a^2} \mathbb{E}[x|a_0^*]}{\Delta} < 0$$

and

$$\lim_{\sigma \to 0} \frac{da^*}{d\sigma} \rightarrow \frac{-\alpha^2 \left(1 + \frac{\partial \mathbb{E}[x|a_0^*]}{\partial a}\right) \mathbb{E}[x|a_0^*]^2 - \alpha^2 \mathbb{V}[x|a_0^*] \frac{\partial \mathbb{E}[x|a_0^*]}{\partial a}}{\Delta}.$$

When $\sigma = 0$, the insured faces no risk and his self-insurance effort corresponds to the risk neutral level, i.e. $a^*(0) = a_N$ satisfying $\partial \mathbb{E}[x|a_N]/\partial a = -1$. Replacing gives

$$\lim_{\sigma \to 0} \frac{da^*}{d\sigma} \rightarrow \alpha^2 \frac{\mathbb{V}[x|a^*(0)]}{\Delta} > 0.$$

Consider now the case $\sigma \to \bar{\sigma}$ of Proposition 9. Using $\gamma^*(\sigma) \to 0$ when $\sigma \to \bar{\sigma}$, we get

$$\lim_{\sigma \to \bar{\sigma}} \frac{d\gamma^*}{d\sigma} \rightarrow \frac{\frac{\partial^2 \mathbb{E}u}{\partial a^2} \mathbb{E}[x|a] \int_{0}^{\bar{x}} e^{\alpha x} dF(x|\bar{a}^*)}{\Delta} < 0$$

and

$$\lim_{\sigma \to \bar{\sigma}} \frac{da^*}{d\sigma} \rightarrow \frac{\alpha \left\{ \left(1 + \bar{\sigma}\right) \frac{\partial \mathbb{E}[x|\bar{a}^*]}{\partial a} \int_{0}^{\bar{x}} e^{\alpha x} dF(x|\bar{a}^*) - \int_{0}^{\bar{x}} \left[ \alpha + \frac{f_a(x|\bar{a}^*)}{f(x|\bar{a}^*)} \right] xe^{\alpha x} dF(x|\bar{a}^*) \right\} \mathbb{E}[x|\bar{a}^*] \int_{0}^{\bar{x}} e^{\alpha x} dF(x|\bar{a}^*)}{\Delta} < 0$$
where $\bar{a}^* = a^*(\bar{\sigma})$. Hence, when $\sigma$ is close to $\bar{\sigma}$, the sign of $da^*/d\sigma$ is the opposite of

$$ 
\begin{aligned}
A &\equiv -\int_0^{\bar{x}} \left\{ (1 + \bar{\sigma}) \frac{\partial \mathbb{E}[x|\bar{a}^*]}{\partial a} - \left[ \alpha + \frac{f_a(x|\bar{a}^*)}{f(x|\bar{a}^*)} \right] x \right\} e^{\alpha x} dF(x|\bar{a}^*) \\
&= -(1 + \bar{\sigma}) \frac{\partial \mathbb{E}[x|\bar{a}^*]}{\partial a} \int_0^{\bar{x}} e^{\alpha x} dF(x|\bar{a}^*) + \int_0^{\bar{x}} \left[ \alpha + \frac{f_a(x|\bar{a}^*)}{f(x|\bar{a}^*)} \right] x e^{\alpha x} dF(x|\bar{a}^*).
\end{aligned}
$$

Using (41) when $\sigma \to \bar{\sigma}$ (and thus when $\gamma^*(\sigma) \to 0$) yields

$$ (1 + \bar{\sigma}) \int_0^{\bar{x}} e^{\alpha x} dF(x|\bar{a}^*) = \frac{\int_0^{\bar{x}} x e^{\alpha x} dF(x|\bar{a}^*)}{\mathbb{E}[x|\bar{a}^*]} \quad (45) $$

Substituting gives

$$ 
\begin{aligned}
A &= \int_0^{\bar{x}} \left[ \frac{f_a(x|\bar{a}^*)}{f(x|\bar{a}^*)} + \alpha - \frac{\partial \mathbb{E}[x|\bar{a}^*]/\partial a}{\mathbb{E}[x|\bar{a}^*]} \right] x e^{\alpha x} dF(x|\bar{a}^*) \\
&= \int_0^{\bar{x}} x e^{\alpha x} f_a(x|\bar{a}^*) dx + \left[ \alpha - \frac{\partial \mathbb{E}[x|\bar{a}^*]/\partial a}{\mathbb{E}[x|\bar{a}^*]} \right] \int_0^{\bar{x}} x e^{\alpha x} dF(x|\bar{a}^*) \\
&= \frac{\partial \mathbb{E}[x e^{\alpha x} | \bar{a}^*]}{\partial a} + \left[ \alpha - \frac{\partial \mathbb{E}[x|\bar{a}^*]/\partial a}{\mathbb{E}[x|\bar{a}^*]} \right] \mathbb{E}[x e^{\alpha x} | \bar{a}^*]
\end{aligned}
$$

We have $A > 0$ iff $G(\alpha) > H(\alpha)$ where

$$ 
\begin{aligned}
G(\alpha) &\equiv \alpha - \frac{\partial \mathbb{E}[x|\bar{a}^*]/\partial a}{\mathbb{E}[x|\bar{a}^*]} \\
H(\alpha) &\equiv -\frac{\partial \mathbb{E}[x e^{\alpha x} | \bar{a}^*]/\partial a}{\mathbb{E}[x e^{\alpha x} | \bar{a}^*]},
\end{aligned}
$$

As $H(0) = G(0)$ and $G'(\alpha) = 1$, $H'(\alpha) < 1$ is a sufficient condition for $A > 0$. We have

$$ 
\begin{aligned}
H'(\alpha) &= -\frac{\partial}{\partial \alpha} \left[ \frac{\partial \ln \mathbb{E}[x e^{\alpha x} | \bar{a}^*]}{\partial a} \right] = -\frac{\partial}{\partial a} \left[ \frac{\partial \ln \mathbb{E}[x e^{\alpha x} | \bar{a}^*]}{\partial \alpha} \right] \\
&= -\frac{\partial}{\partial a} \frac{\mathbb{E}[x^2 e^{\alpha x} | \bar{a}^*]}{\mathbb{E}[x e^{\alpha x} | \bar{a}^*]},
\end{aligned}
$$

which completes the proof.

When the loss is distributed according to a Gamma distribution of parameters $k(a) > 0$ and $\theta(a) > 0$, we have $\mathbb{E}[x|a] = k(a)\theta(a)$, $\mathbb{V}[x|a] = k(a)\theta(a)^2$ and $\mathbb{CV}[x|a] = 1/\sqrt{k(a)} = SK(a)/2$ where $SK(a)$ denote the skewness of the distribution. Its moment-generating function is $\xi(\alpha|a) \equiv \mathbb{E}[e^{\alpha x}|a] = (1-\alpha\theta(a))^{-k(a)}$ (where $\alpha < 1/\theta(a)$),
and it comes

$$ \frac{\mathbb{E}[x^2 e^{\alpha x} | a]}{\mathbb{E}[x e^{\alpha x} | a]} = \frac{\xi''(\alpha | a)}{\xi'(\alpha | a)} = \frac{(k(a) + 1)\theta(a)}{1 - \alpha\theta(a)} = \frac{1 + \mathrm{CV}(x|a)^2}{1/\mathbb{E}[x|a] - \alpha \mathrm{CV}(x|a)^2} $$

Denoting $y(a) = \mathrm{CV}(x|a)$ and $z(a) = \mathbb{E}[x|a]$, the condition in Proposition 9 holds if

$$ \frac{\partial}{\partial a} \left[ \frac{1 + y(a)^2}{1/z(a) - \alpha y(a)^2} \right] \geq 0 $$

where

$$ \frac{\partial}{\partial a} \left[ \frac{1 + y(a)^2}{1/z(a) - \alpha y(a)^2} \right] = \frac{2y(a)y'(a)(1/z(a) + \alpha) + (1 + y(a)^2)z'(a)/z(a)^2}{(1/z(a) - \alpha y(a)^2)^2}, $$

hence if

$$ \frac{2y(a)y'(a)}{1 + y(a)^2} \geq - \frac{z'(a)/z(a)}{1 + \alpha z(a)}. $$

This condition is satisfied if

$$ \frac{2y(a)y'(a)}{1 + y(a)^2} \geq - \frac{z'(a)}{z(a)}, $$

or equivalently

$$ \frac{d}{da} \ln(1 + y(a)^2) \geq - \frac{d}{da} \ln z(a) $$

hence (7).