Optimal-contract characterization without concavity

Determine whether an optimal contract exists and, when it exists, identify it for the finite-horizon continuous-time Principal–Agent problem with deterministic discount rate and a non-decreasing principal utility function whose concave envelope is finite, without imposing a concavity assumption on the principal utility.

Background

Theorem 2.1 gives an explicit formula for the principal’s value through the concave envelope of the principal’s utility and constructs an optimal contract when the relevant argument lies in the contact set where the utility equals its concave envelope. Outside that contact set, the theorem does not provide an optimizer.

The subsequent example demonstrates that an optimal contract can fail to exist when the principal utility is non-concave and the relevant value lies outside the contact set. Thus, the unresolved issue is to determine general existence conditions and characterize optimal contracts beyond the sufficient contact-set condition, potentially identifying what additional form of concavity is necessary.