We revisit the first-order approach (FOA) in a classical setting of moral hazard model with multi-dimensional signal. After providing formal justification of Lagrangian duality, we reformulate the issue of validiting the FOA as an issue of the existence of a fixed point of the agent's best reaction to the principal's targeted effort level. Therefore, it is unnecessary to show the validity based on the global concavity of the agent's expected under a subclass of monotone contract. The new method allows the relaxation of several requirements of previous approaches. We generalize some results of Sinclair-Desgagne (1994) and Conlon (2009a) to validate the FOA for either the mixture probability model without the likelihood ratio order, or certain exponential family distributions with a bounded likelihood ratio.
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