Security design with flexible moral hazard and limited liability

I study security design with moral hazard and two-sided limited liability when an entrepreneur can choose any distribution of project returns subject to a cost. I first characterize which return distributions can be induced by a security and when the first-best can be financed. Whenever the first-best can be financed, it is optimal and can be implemented by debt; more generally, any security that implements it pays the investor a constant amount on the support of the first-best distribution. The agency costs that may prevent first-best financing are related to the convexity of the cost function. For a given return distribution, greater convexity of the cost function increases the rent that the entrepreneur has to retain and can therefore make first-best financing more difficult. When the first-best cannot be financed, I derive second-best distributions for cost functions that are monotone in risk or depend on generalized moments. Under suitable conditions, the second-best is first-order stochastically dominated by the first-best. The optimal return distributions I characterize have finite support. Since an optimal security is uniquely determined only on, but not off, the support of the optimal return distribution, optimal securities are not unique.