In this paper, we examine second-best efficient allocations of risk when some forms of incompleteness are introduced in risk- sharing contracts. In the first model, there are two independent sources of risk, but risk-sharing contracts can be made contingent to only one of the two sources. We examine the condition under which those who bear the non-transferable risk should bear relatively less of the transferable risk in the economy. Decreasing absolute prudence, i.e. -u'''/u'')'< 0, is shown to be necessary and sufficient for such a property to hold. In the second model, there is a complete set of contingent markets, but some agents have no direct access to them. We examine efficient allocation of risk in a pool gathering a trader and a non-trader. The contract incompleteness comes from the fact that it can be made contingent only upon the risk initially borne by the pool. It is generally not true that a larger share of the pool's risk should be borne by the trader, despite his ability to diversify risk in his portfolio. Decreasing absolute prudence provides an upper bound to the share of the risk that should be borne by the trader.
