A convex mapping for the first order approach: A note

The first order approach to solving the standard one-dimensional principal-agent model is conditional upon the relevant stochastic production function obeying two noteworthy restrictions: that the Likelihood Ratio be monotonically increasing in output, and that the distribution function be convex in effort. It is usually claimed that such conditions are very restrictive, as very few of the standard probability distributions satisfy both properties. The purpose of this note is to show that some simple transformations or parametrizations are available, that enable one to work out convenient distributions with the required properties