Optimal Loss Reduction and Increases in Risk Aversion

Abstract

Optimal Loss Reduction and Increases in Risk Aversion ABSTRACT The impact of increases in risk aversion on the optimal level of loss reduction is analyzed in models in which the magnitude of the prospective loss is uncertain. The results are sensitive to the specification of the source of the uncertainty. Introduction Dionne and Eeckhoudt (1985) show that a more risk averse individual will always choose a higher level of loss reduction (activity which reduces the magnitude of a potential loss) but may not always choose a higher level of loss prevention (activity which reduces the probability of loss).(1) However, following Ehrlich and Becker (1972), they consider only a simple two state model in which the severity of the possible loss is fixed. A crucial property of a two state model is that the productivity of loss reduction is not subject to uncertainty. This article extends the previous analysis by examining the impact of increases in risk aversion on optimal loss reduction within a more general model which allows the productivity of loss reduction to be uncertain. It is shown that the results for the simple model used by Dionne and Eeckhoudt do not carry over to more general models. It is assumed that uncertainly surrounding the loss reduction decision can arise form either of two distinct sources. In the first case, the severity of the potential loss is random, but there is no uncertainty about the effectiveness of loss reduction measures in reducing this loss. In the second case, the size of the potential loss is nonrandom, but the effectiveness of any loss reduction measure depends on conditions which happen to prevail at the time a loss actually occurs. In both cases the productivity of loss reduction is uncertain since the actual size of the loss avoided is unknown when the level of loss reduction is chosen. It is shown that an increase in risk aversion unambiguously increases the optimal level of loss reduction in the first case but not in the second case. Hence the results are critically dependent on the specification of the source of the uncertainty.(2) This contrast in the results for the two alternative specifications has a simple explanation. If uncertainty arises because the potential loss is random, then an increase in loss reduction activity also reduces the variance of the (conditional) loss. Hence, a more risk averse individual chooses a less risky position by selecting a higher level of loss reduction.(3) On the other hand, if the effectiveness of loss reduction measures is uncertain, then an increase in loss reduction activity increases the variance of the (conditional) loss. Consequently, a more risk averse individual may actually prefer a lower level of loss reduction because the risk increasing character of this activity lowers its value to the individual. Uncertainty about the Potential Loss Consider an individual with an initial wealth, W, which is subject to the risk of partial loss. The individual can undertake loss reduction, activity which reduces the size of the actual loss but which leaves the probability of a loss unchanged. In this section it is assumed that the effectiveness of this loss reduction is nonrandom. The magnitude of the loss is written as L(1 - h(x)), where L is a positive random variable representing the size of the potential loss and represents the proportion of the loss avoided by the loss reducing activity, x. It is assumed that there are diminishing returns to loss reduction (h'(x) > 0, h(x) The objective is to maximize (1 - p)U(W - C(x)) + pE[U(W - L(1 - h(x)) - C(x))] (1) where U is a strictly concave utility function and p is the probability that a loss occurs. …