  We prove the interior $C^{1,1}$ regularity of the indirect utilities which
solve a subclass of principal-agent problems originally considered by Figalli,
Kim, and McCann. Our approach is based on construction of a suitable comparison
function which, essentially, allows one to pinch the solution between
parabolas. The original ideas for this proof arise from an earlier,
unpublished, result of Caffarelli and Lions for bilinear preferences which we
extend here to general quasilinear benefit functions. We give a simple example
which shows the $C^{1,1}$ regularity is optimal.
