Abstract
This study develops a theoretical, and experimental analysis addressing the issue of premium variations on the demand for insurance. Accounting for risk attitudes, our contribution disentangles the decision to buy insurance from the conditional demand (the non-null demand for insurance). Partially validating our theoretical predictions, our experimental results show that, when it has an effect, a non-massive increase in the premium (either in the unit price or the fixed cost) exclusively results in an exit from the insurance market (the risk lovers first, then the risk averters). Moreover, our study highlights a key feature of risk-seeking agents' behavior; they exhibit behavior consistent with gambling and opportunism rather than a lack of interest in insurance.
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This mechanism appears highly relevant to the insurance industry. Insurance is conditioned by a large variety of fixed administrative expenses (marketing costs, price formation, fees for administration and audit) that need to be paid regardless of the level of coverage while this latter element is associated with a unit price that varies with the amount purchased. As a rule, insurers assesses, for each category of policyholders, a pure premium reflecting their risk exposure. A loading is applied to this actuarial premium to cover management costs and taxes, but some administrative costs are charged as pure fixed costs. In real life, policyholders may not know the ins and outs of the sharing between the fixed and variable components of the insurance premium.
The analysis of a less-than-actuarial unit price is particularly appropriate in several cases. First, under adverse selection with heterogeneous agents ranked according to their probability of an accident, a “pooling” mean price may confront the riskiest individuals with a less-than-actuarial unit price. Second, a less-than-actuarial unit price is not uncommon in some public health policies. Finally, a less-than-actuarial unit price combined with a positive fixed cost may achieve and exceed the break-even point.
Since p > q, the FOC implies that U′(W1) < U′(W2). It follows that W1 > W2 and I < x.
In this optimization problem, optimal insurance demand I is an implicit function of the parameters (W0, p, C, q). Differentiating the 1st order condition, denoted \( H\left( I \right) = \frac{\partial EU}{\partial I} = 0 \), we get: \( \frac{dI}{dC} = - \frac{\partial H}{\partial C}/\frac{\partial H}{\partial I}. \) As \( \frac{\partial H}{\partial I} = \frac{{\partial^{2} EU}}{{\partial I^{2} }} < 0, \) the sign of this impact is determined by the sign of \( \frac{\partial H}{\partial C} = p\left( {1 - q} \right)U^{{\prime \prime }} \left( {W_{1} } \right) - \left( {1 - p} \right)qU^{{\prime \prime }} \left( {W_{2} } \right) \), which eventually depends on the difference between the two coefficients of absolute risk aversion, evaluated respectively for W1 and W2: \( - A\left( {W_{1} } \right) + A\left( {W_{2} } \right)\quad {\text{where}}\quad A\left( W \right) = - U^{{\prime \prime }} \left( W \right)/U^{{\prime }} \left( W \right). \)
When wealth increases, aversion to any given risk decreases. The marginal benefit of insurance declines with wealth and so does the demand for insurance.
Relying on the 1st order condition, the impact of a change in p is given by: \( \frac{dI}{dp} = - \frac{\partial H}{\partial p}/\frac{\partial H}{\partial I} \). Again, since \( \frac{\partial H}{\partial I} < 0, \) we find that the sign of \( \frac{dI}{dp} \) depends on the sign of \( \frac{\partial H}{\partial p} \). We then have \( \frac{\partial H}{\partial p} = - EU^{{\prime }} + I\frac{\partial H}{\partial C},\quad {\text{where}} EU^{{\prime }} = \left( {1 - q} \right)U^{{\prime }} \left( {W_{1} } \right) + qU^{{\prime }} \left( {W_{2} } \right) \). The 1st term refers to a negative substitution effect and the 2nd term is the wealth effect previously identified.
Where t = q u′ (W0 – x − C)/EU(0).
Obviously, I is not independent of p but the individual’s welfare EU cannot increase with p, even if the demand for insurance is simultaneously adjusted.
The demand for insurance of a risk neutral agent is trivial. Indeed, a risk neutral agent will find it profitable to get insured if the mathematical expectation of I is higher than P, so that pI + C ≤ qI. Especially when I = x, we obtain C ≤ (q − p)x. We do not detail the behavior of risk-neutral individuals since they were pooled with risk averters in our experimental data analysis.
Since W0 > W0 − x, U′(W0) > U(W0 − x), because the marginal utility is increasing for a RL.
The global demand for insurance (GD with GD \( \in \left[ {0;x} \right] \)) is the genuine demand for insurance. At an individual level i, GDi = DIi*CDi, where DIi stands for the decision of individual i to buy insurance (1 or 0) and CDi for her conditional demand for insurance (the amount of insurance bought if the decision is positive (DIi = 1). At an individual level, CDi does not exist if DIi = 0 and CDi = GDi otherwise.
At the aggregate mean level, \( GD = \frac{1}{n}\mathop \sum \nolimits_{i = 1}^{n} GD_{i} \). If we denote k the number of times DIi = 0, GD can be rewritten as the product of the propensity to buy insurance (proportion of individuals buying a strictly positive amount of insurance coverage) by the mean conditional demand (the mean demand of those buying a strictly positive amount of insurance coverage):
$$ GD = \frac{1}{n}\mathop \sum \limits_{i = 1}^{n} GD_{i} = \frac{1}{n}\mathop \sum \limits_{{GD_{i} \ne 0}} GD_{i} = \frac{1}{n}\mathop \sum \limits_{{GD_{i} \ne 0}} CD_{i} = \left( {\frac{n - k}{n}} \right)\left( {\frac{{\mathop \sum \nolimits_{{GD_{i} \ne 0}} CD_{i} }}{n - k}} \right) $$Therefore, at an aggregate level, GD = PI*CD and PI is between 0 and 1.
Where −6 (final loss with insurance) = −10 (loss without insurance) − 4 (premium) + 8 (indemnity).
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We refer to situations where, for example, one choice A appears within a long series of B choices (and inversely).
Table 5 presents the demand for insurance, by contract.
However, the same Wilcoxon signed rank tests were also carried out accounting for these intermittent coverage situations and provided the same results. Also, the results hold when using parametric tests such as the Student's test or the Pearson correlation coefficient.
Namely, individuals who switch several times from one option (safe or risky) to the other; subjects who choose option A in line 10 (i.e. who prefer less money to more with certainty).
Especially, as seen in Table 1, when the unit price is more than actuarial, RAs can only partially cover, while the RLs’ theoretical insurance choices are binary, 0 or 1000.
Computed in the order the subjects faced the different contracts.
The difference, though important (73 vs 66%), is not significant (p value = 0.4804) most likely due to the small number of periods an accident occurs.
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Acknowledgements
We are grateful to the editor as well as two anonymous referees for their highly valuable comments and suggestions. We also thank Nathalie Etchart-Vincent for her remarks. Remaining errors or omissions are ours. This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the «Investissement d’Avenir» program, through the “iCODE Institute project” funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02.
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Appendix
Appendix
The Holt and Laury (H&L) method to elicit risk preference is widely used in literature. All experimentalists are aware of noise in decision-making with H&L and other methods to elicit risk preference. In Lévy-Garboua et al. (2012), the authors found 31.3% inconsistent subjects in the standard H&L low payoff treatment. Our adapted H&L in the loss domain carries the imperfection of the standard H&L.
In Table 9, we present the same Wilcoxon signed rank tests as Table 6 on a high-quality sample of RLs where inconsistent behaviors have been removed: multiple switches from one option (safe or risky) to the other and the three subjects who prefered less money to more with certainty in choosing option A in line 10.
Compared to the full sample, all statistically significant results remain with the high-quality sample of RAs, but some p values differ, however.
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Corcos, A., Pannequin, F. & Montmarquette, C. Leaving the market or reducing the coverage? A model-based experimental analysis of the demand for insurance. Exp Econ 20, 836–859 (2017). https://doi.org/10.1007/s10683-017-9513-8
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DOI: https://doi.org/10.1007/s10683-017-9513-8