Optimal Risk-Sharing When Risk Preferences are Uncertain

Abstract

In many cases, a firm or agency needs a product that only one vendor can supply and for which the final cost is uncertain. An optimal risk-sharing arrangement is sought when the buyer and contractor agree on the probability distribution of cost but the buyer is uncertain of the contractor's risk-preferences. We find that when the buyer and contractor have exponential utilities, the optimal profit arrangement for the higher risk-averse contractor is no longer linear but concave in the costs. The degree of concavity is affected by the probabilistic beliefs on the contractor's risk-preferences. As the more risk-averse contractor becomes more likely, her chosen profit arrangement becomes less concave approaching the ideal, linear arrangement. The less risk-averse contractor is provided a profit arrangement with a certainty equivalent above her reservation price. This is the price the buyer must pay in order to entice a less risk-averse contractor into agreeing to accept a more risky profit arrangement.

Another formulation is considered that assumes the buyer and contractor maximize approximations to their certainty equivalents in order to provide a more practical and possibly viable approach to sole-source contracting.

Keywords:

• buying
• contract
• risk
• utility