Insurance and Precautionary Capital Accumulation in a Continuous-Time Model

Abstract

Introduction The theory of optimal response to is one of the most lively areas of research in economics. Two types of response to have been studied intensively. The first type of response is the transfer of through insurance. A premium is paid today to receive an additional revenue in case of a future This behavior is related to risk aversion, which measures how much one dislikes uncertainty and how much one would be willing to pay to escape risk. In the expected utility framework, aversion originates from the negativity of the second derivative of the utility function. It explains why averse people are willing to buy insurance, even at unfavorable prices. The second type of response to is capital accumulation in the form of deposits. Money is saved today for consumption tomorrow regardless of whether an accident occurred. This is called a prudent behavior. Following Kimball (1990), prudence is meant to suggest the propensity to prepare and forearm oneself in the face of uncertainty. When future incomes are uncertain, prudent consumers will save more. Leland (1968), Sandmo (1970), and Dreze and Modigliani (1972) showed that prudent behaviors are generated by the nonnegativity of the third derivative of the utility function. Prudence (u'|is greater than or equal to~ 0) and aversion (u |is less than or equal to~ 0) have different nature. Most articles in the insurance literature provide static models, thereby excluding precautionary capital accumulation. Conversely, it is generally assumed in the literature on precautionary saving that the exogenous under consideration is not transferable, thereby excluding insurance strategies. The goal of this article is to determine the optimal strategy when both capital accumulation (deposits) and insurance are allowed. Consider an agent facing at every instant a of liability generating some loss in wealth due to an accident. In the short run, the agent is expected to transfer most of the to an insurer because he has no financial reserve. If he is sufficiently lucky and if his rate of consumption is not too large, he will be able to accumulate reserves in the form of deposits that will allow him to retain a larger proportion of the in the future. This is desirable for two reasons: First, financial reserves generate a positive return, and, second, reducing insurance coverage is profitable because insurance is costly. Two important problems should be solved with respect to this strategy: First, what is the optimal level of reserves and the demand for insurance in the long run? Second, we have to reconcile two conflicting objectives in the short run--protecting the agent against large losses and raising reserves to reduce the cost of the in the future (by reducing insurance coverage). These objectives conflict because the first is attained through spending enough money for insurance and the second saves on insurance costs today. As already observed by Dionne and Eeckhoudt (1984), for nonactuarial premia, it is not true that insurance is always more efficient than capital accumulation for the purpose of protection. Using stochastic continuous-time control techniques, a necessary and sufficient condition is derived that guarantees a positive demand for insurance in the long run. The effect of transitory and permanent changes in the price of insurance is also examined either in the short run or in the long run. Contrary to the well-known result obtained in the standard static model, insurance may not be a Giffen good in the sense that a transitory increase in the loading factor of insurance always reduces the demand for insurance in the short run, under aversion alone.(1) The same result holds for the effect of a permanent change on the long-term demand for insurance. Ambiguity is found for the transitory effect of a permanent increase on the price of insurance, with the wealth and substitution effects being contradictory. …