CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM action that affects the outcome of the project. The revenue from the venture is assumed to be a random function of the agents action. Let a denote the set of actions available to the agent with generic element a. Typically, A is either a finite set or an interval of real numbers. Let s denote a set of states with generic element s. For simplicity, we assume that the set S is finite. The probability of the state s conditional on the action a is denoted by p(a, s). The revenue in state s is denoted by R(s)20 The agent's utility depends on both the action chosen and the consump- tion he derives from his share of the revenue. The principals utility depend only on his consumption. We maintain the following assumptions about references · The agent' s utility function u:A×R+→ R is additively separable: u(a, c)=U(c-v(a) Further, the function U: R+-+R is C and satisfies U(c>0 and U"(c)≤0. The principals utility function V: R-R is C and satisfies V(c)>0 andW"(c)≤0. Notice that the agents consumption is assumed to be non-negative. This is interpreted as a liquidity constraint or limited liability. The principals consumption is not bounded below; in some contexts this is equivalent to assuming that the principal has large but finite wealth and non-negative consumption 3.2 Pareto efficiency The principal and the agent jointly choose a contract that specifies an action and a division of the revenue. A contract is an ordered pair(a, w()E Axw, where w is the set of incentive schemes and w(s)20 is the payment to the agent in state s Suppose that all variables are observable and verifiable. The principal and the agent will presumably choose a contract that is Pareto-efficient This leads us to consider the following decision problem(DP1) a∑>p(sV(F)-()2 CHAPTER 3. THE PRINCIPAL-AGENT PROBLEM action that affects the outcome of the project. The revenue from the venture is assumed to be a random function of the agent’s action. Let A denote the set of actions available to the agent with generic element a. Typically, A is either a finite set or an interval of real numbers. Let S denote a set of states with generic element s. For simplicity, we assume that the set S is finite. The probability of the state s conditional on the action a is denoted by p(a, s). The revenue in state s is denoted by R(s) ≥ 0. The agent’s utility depends on both the action chosen and the consump￾tion he derives from his share of the revenue. The principal’s utility depends only on his consumption. We maintain the following assumptions about preferences: • The agent’s utility function u : A × R+ → R is additively separable: u(a, c) = U(c) − ψ(a). Further, the function U : R+ → R is C2 and satisfies U0 (c) > 0 and U00(c) ≤ 0. • The principal’s utility function V : R → R is C2 and satisfies V 0 (c) > 0 and V 00(c) ≤ 0. Notice that the agent’s consumption is assumed to be non-negative. This is interpreted as a liquidity constraint or limited liability. The principal’s consumption is not bounded below; in some contexts this is equivalent to assuming that the principal has large but finite wealth and non-negative consumption. 3.2 Pareto efficiency The principal and the agent jointly choose a contract that specifies an action and a division of the revenue. A contract is an ordered pair (a, w(·)) ∈ A×W, where W = {w : S → R+} is the set of incentive schemes and w(s) ≥ 0 is the payment to the agent in state s. Suppose that all variables are observable and verifiable. The principal and the agent will presumably choose a contract that is Pareto-efficient. This leads us to consider the following decision problem (DP1): max (a,w(·)) X s∈S p(a, s)V (R(s) − w(s))