Mathematics in Contract Theory (preliminary Version)

1.1. Problem formulation. To understand the flavor of questions in contract theory, we start with an one-period model in this section. Consider an agent whose utility is modelled by an exponential utility with the risk aversion γA, i.e., UA(c) = − 1 γA e −γAc. If hired at time 0, this agent manages a project, which produces an output at time 1. The output is random, and it is modelled by a normal random variable X with the mean α and the standard deviation σ, i.e., X ∼ N(α, σ). The mean α is determined by the agent’s effort. If the agent works hard, α is high, otherwise α is low. The randomness of the output is characterized by σ which cannot be controlled by the agent. The effort is costly to the agent. The cost is given by a cost function g(α) which is the monetary cost to the agent if he exerts effort α. We consider the following cost function