A Verification Theorem for the First-Order Approach in Continuous-Time Principal–Agent Problems with Hidden Savings

In continuous-time principal–agent models the agent can usually save privately, or divert the returns on delegated capital, and the recursive formulation of the principal’s problem then imposes only on-path incentive constraints. Whether the resulting relaxed problem coincides with the true one — the validity of the first-order approach — is a verification question those constraints leave open, and one the literature has settled only in special cases. This paper settles it in general. Characterizing the agent’s non-Markovian problem by its stochastic Hamilton–Jacobi–Bellman equation, we show that concavity of the agent’s value function in hidden wealth is sufficient for the first-order approach, and we reduce concavity to a single checkable condition. Under hidden effort the condition bounds the endogenous volatility of the agent’s marginal value of wealth by the interest rate times the geometric mean of the curvatures of consumption utility and effort cost; derived from a backward stochastic Riccati equation and signed by a comparison theorem, it specializes at the optimum to a primitive bound on absolute risk aversion and reduces under constant relative risk aversion to the volatility restriction of Di Tella and Sannikov (2021). Under hidden returns, where diversion is linear, consumption-side concavity is automatic and the entire content of the first-order approach is an off-path incentive constraint — deterrence of diversion robust to hidden savings. In both models the identification of the relaxed problem with the truth, previously asserted in this literature, becomes a proved consequence of a transparent condition.