Problems of calculating variations from contract theory.

Summary This thesis is devoted to some optimization problems arising in contract theory. We first consider some one-dimensional problems. In this simple case, the principal's program consists in minimizing a certain cost on the cone of increasing functions. We consider non-convex cases and use a variant of the Hardy-Littlewood inequality. We then consider variational problems under convexity constraints. After giving existence results for non-convex Lagrangians, we introduce a penalty to recover the Euler equation due to Pierre-Louis Lions. In a collaborative work with Thomas Lachand-Robert, we then establish C1 regularity results for minimizers in various situations (Dirichlet, Neumann, Choné and Rochet model. . . ). Finally, in a paper co-authored with Thomas Lachand-Robert and Bertrand Maury, we present a method for numerical approximation of quadratic problems under convexity constraints. The presented algorithm is, to our knowledge, the only one whose convergence is established. In the second part, we characterize incentive contracts in a general way by means of abstract convexity and subdifferentiability notions used in the theory of mass transportation and we prove the existence of optimal incentive contracts without any particular functional specification on the agents' preferences. We then use the link between the characterization of incentive contracts and a class of optimal transportation problems. First, existence, uniqueness, and duality results are established for mass transportation problems where the cost verifies a generalizing Spence-Mirrlees hypothesis, which is classical in the economic literature in dimension 1. This allows us, returning to incentive problems, to de��monstrate a reallocation principle: any measurable allocation profile can be uniquely rearranged into an implementable profile, via a suitable optimal transfer problem. The final section is devoted to two specific economic problems. The first one is motivated by insurance fraud issues that give rise to a non-convex problem and consists of a paper co-authored with Rose-Anne Dana and Maxime Renaudin. The second one is related to the design of labor contracts in the presence of two-dimensional adverse selection. It is a paper co-authored with Damien Gaumont.