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Author:

• Rongzhu Ke
  Chinese University of Hong Kong

This paper develops a new method to justify the validity of the first order approach (FOA). The main idea is to search for a This paper develops a new method to justify the validity of the first order approach (FOA). The main idea is to search for a fixed point of the agent's best reponse against a Mirrlees- Holmstrom (MH) class contract offered by the principal, instead of using the double relaxed approach (Rogerson, 1985). We first prove a necessary and sufficient condition for the validity of FOA. The necessary parts suggests whenever the FOA is valid, the solution to the principal agent model exist, while the sufficient part also suffices for the existence of the solution. The condition thus can be used to develop a numerical algorithm. Second, based on the necessary and sufficient condition, we also propose a sequence of sufficient conditions for the validity of FOA and existence of the solution, which still are weaker than the global concavity. By these sufficient conditions, we unify and improve Jewitt's (1988) and Rogerson's (1985) proofs of validity of FOA. In addition, when the agent's utility is separable, we show if the log likelihood ratio is monotonically increasing in output but decreasing in effort, the best response correspondence agrainst a MH contract has at most one fixed point, and the fixed point will be monotonically increasing as the shadow price of incentive compatible constraint increases. Finally, our characterization of the .fixed-point approach also sheds lights on solving a general non-linear bi-level optimization problem with non-convex constrained set fixed point of the agent's best reponse against a Mirrlees- Holmstrom (MH) class contract offered by the principal, instead of using the double relaxed approach (Rogerson, 1985). We first prove a necessary and sufficient condition for the validity of FOA. The necessary parts suggests whenever the FOA is valid, the solution to the principal agent model exist, while the sufficient part also suffices for the existence of the solution. The condition thus can be used to develop a numerical algorithm. Second, based on the necessary and sufficient condition, we also propose a sequence of sufficient conditions for the validity of FOA and existence of the solution, which still are weaker than the global concavity. By these sufficient conditions, we unify and improve Jewitt's (1988) and Rogerson's (1985) proofs of validity of FOA. In addition, when the agent's utility is separable, we show if the log likelihood ratio is monotonically increasing in output but decreasing in effort, the best response correspondence agrainst a MH contract has at most one .fixed point, and the .fixed point will be monotonically increasing as the shadow price of incentive compatible constraint increases. Finally, our characterization of the .fixed-point approach also sheds lights on solving a general non-linear bi-level optimization problem with non-convex constrained set.