Christian Gollier

Risk sharing on the labour market and second-best wage rigidities

The starting point of the analysis is that a significant proportion of the labour force is not covered by efficient labour contracts. In particular, the non-existence of forward labour contracts limits the opportunity of efficient risk-sharing through private arrangements among young labour-suppliers and capital-owners. In such a context, state-dependent labour taxes and employment subsidies would be required to achieve ex ante Pareto efficiency. In the absence of employment subsidies, a certain (limited) degree of downward wage rigidity is in general Pareto superior to full wage flexibility. At the second-best optimum so defined, productive efficiency is sacrificed in some states, in order to achieve greater efficiency in risk-sharing. Thus, the unemployment associated with wage rigidity is inefficient, though voluntary due to the levels of benefits, and second-best efficient. A limited degree of wage discrimination by hiring date is typically part of the second-best policy.

Second-Best Insurance Contract Design in an Incomplete Market*

The optimal form of insurance contracts for multiple risks is examined. A well-known result in the literature is that, under fairly general conditions, an insurance policy with a deductible for aggregate losses is optimal. Real-world markets, however, are typically incomplete in that they require separate contracts for separate loss exposures. For instance, insurable damage to ones home is not generally allowed to affect the insurance indemnity for (unrelated) insurable damage to ones automobile. We show that separate deductibles are second-best optima in this setting. We compare the indemnity provided in this second-best setting with first-best solutions. The effect of second-best contracts on the individuals total insurance demand is also examined. Copyright 1995 by The editors of the Scandinavian Journal of Economics.

The Spillover Effect of Compulsory Insurance

The assumption usually made in the insurance literature that risks are always insurable at the desired level does not hold in the real world: some risks are not—or are only partially—insurable, while others, such as civil liability or health and workers injuries, must be fully insured or at least covered for a specific amount. We examine in this paper conditions under which a reduction in the constrained level of insurance for one risk increases the demand of insurance for another independent risk. We show that it is necessary to sign the fourth derivative of the utility function to obtain an unambiguous spillover effect. Three different sufficient conditions are derived if the expected value of the exogenous risk is zero. The first condition is that risk aversion be standard—that is, that absolute risk aversion and absolute prudence be decreasing. The second condition is that absolute risk aversion be decreasing and convex. The third condition is that both the third and the fourth derivatives of the utility function be negative. If the expected value of the exogenous risk is positive, a wealth effect is added to the picture, which goes in the opposite direction if absolute risk aversion is decreasing.

Relatively weak increases in risk and their comparative statics

Abstract We propose in this paper a new condition on increases in risk to obtain the desirable comparative statics properties for non-linear payoffs. Our new concept of relatively weak increases in risk extends the definition of relatively strong increases in risk (Black and Bulkley, International Economic Review , 1989, 30, 119–130).

The Economics of Adding and Subdividing Independent Risks: Some Comparative Statics Results

In this paper we address the problem of determining whether adding independent risks or subdividing them is a good substitute for insurance. Despite the fact that accepting more i.i.d. risks increases total risk, it is shown that some risk-averse decision makers can rationally reduce their demand for insurance by doing so. Similarly, a better diversified portfolio of i.i.d. risky assets can rationally be more insured, even if diversification is a risk-reduction scheme. We derive conditions sufficient to obtain unambiguous comparative statics results. Assuming that absolute risk aversion is decreasing and that the fourth derivative of the utility function is positive, we show that diversification is an exceptionally good substitute for insurance. Under the same conditions, adding independent risks to wealth reduces the demand for insurance on each unit.

On the Inefficiency of Bang-Bang and Stop-Loss Portfolio Strategies

We show in this article that bang-bang portfolio strategies where the investor is alternatively 100% in equity and 100% in cash are dynamically inefficient. Our proof of this result is based on a simple second-order stochastic dominance (SSD) argument. It implies that this is true for any decision criterion that satisfies SSD, not necessarily expected utility. We also examine the stop-loss strategy in which the investor is 100 percent in equity as long as the value of the portfolio exceeds a lower limit where the investor switches to 100 percent in cash. Again, we show that this strategy is inefficient under second-order risk aversion. However, a slight modification of it–in which all wealth exceeding a minimum reserve is invested in equity–is shown to be an efficient dynamic portfolio strategy. This strategy is optimal for investors with a nondifferentiable utility function.

Economics of Radiation Protection: Equity Considerations

In order to implement cost-benefit analysis of protective actions to reduce radiological exposures, one needs to attribute a monetary value to the avoided exposure. Recently, the International Commission on Radiological Protection has stressed the need to take into consideration not only the collective exposure to ionising radiation but also its dispersion in the population. In this paper, by using some well known and some recent results in the economics of uncertainty, we discuss how to integrate these recommendations in the valuation of the benefit of protection.

Deductible insurance and production: A comment

Abstract We consider a risk-averse firm facing output price uncertainty. Contrary to Machnes (1995), we show that a fair deductible insurance does not need to increase the optimal level of output. We provide some sufficient conditions to obtain an unambiguous comparative static effect.

Portfolio selection by mutual insurance companies and optimal participating insurance policies

Abstract The specific problem of portfolio selection by insurance companies is addressed in this paper. Assuming risk aversion on capital markets, it is optimal to transfer part of the investment risk to policyholders (participating policies), as done in life insurance. This remark leads to two results. First, it implies that underwriting profits should be countercyclic. Second, it allows for riskier investments by the insurer. Indeed, optimal risk transfer increases the risk tolerance of the pool. Regulatory constraints on investment behavior in the insurance industry may then be questioned when participating policies are provided.

A Model of Comparative Statics for Changes in Stochastic Returns with Dependent Risky Assets

In this article, we show how the order of Linear Stochastic Dominance proposed by Gollier (1995) can be applied to situations with dependent risky assets. This order was shown to be the least constrained necessary and sufficient condition to guarantee that all risk-averse agents reduce their risky position when an increase in risk is imposed. This was done in a model with only one source of risk, as in the standard portfolio problem with one safe asset and one risky asset. We obtain the necessary and sufficient condition for a change in the joint distribution of returns to yield an unambiguous comparative statics result when the two assets are risky. We show in particular that the concept of Linear Stochastic Dominance is sufficient to generate the desired result. These results are linked to existing sufficient conditions in the one-safe-one-risky-asset model, as the condition of strong increase in risk or the monotone likelihood ratio order. They are also compared to those in models where restrictions are on the set of concave utility functions.