The first-order approach when the cost of effort is money

We provide sufficient conditions for the first-order approach in the principal-agent problem when the agent’s utility has the nonseparable form u(y−c(a))$u(y-c\left(a\right))$ where y$y$ is the contractual payoff and c(a)$c\left(a\right)$ is the money cost of effort. We first consider a decision-maker facing prospects which cost c(a)$c\left(a\right)$ and with distributions of returns y$y$ that depend on a$a$. The decision problem is shown to be concave if the primitive of the cdf of returns is jointly convex in a$a$ and y$y$, a condition we call Concavity of the Cumulative Quantile (CCQ) and which is satisfied by many common distributions. Next we apply CCQ to the distribution of outcomes (or their likelihood-ratio transforms) in the principal-agent problem and derive restrictions on the utility function that validate the first-order approach. We also discuss another condition, log-convexity of the distribution, and show that it allows binding limited liability constraints, which CCQ does not.