The Mirrlees-Rogerson conditions for the first-order approach to principal-agent problems to be valid do not work if the principal can observe more than one relevant statistic. This is a problem since much of the literature since Holmstrom (1979) deals with precisely such a case. We also argue that the Mirrlees-Rogerson assumption that the distribution function of output is convex in the agents action is unsatisfactory even in the context of the basic model; it is too restrictive, being satisfied by a very few of the distributions occurring in statistics textbooks. The present paper replaces this assumption and provides conditions which justify the first-order approach in the multi-statistic case.