Subjective Expected Utility Theory with Costly Actions - Economics ...

3 Discussion
To begin with consider the application of the models developed in the preceding
section to agency theory. A technology, t; is a mapping from the set A£S to the
set of consequences. The agent’s choice of action a 2 A and nature’s “choice” of
astates 2 S jointly determine the outcome t(a; s) 2 R: Thus given a technology,
t; andanaction,a; t (¢; a) is an act in F . In other words, ta := t (a; ¢) 2 F is
a random variable whose value ta (s) is the monetary payo¤s associated with
the action a in the state s: The full support assumption often invoked in the
formulation of agency problem requires that the range of possible payo¤s be
uniform across actions, that is, T = ta (S) for all a 2 A: 5
An incentive contract is a function w : T ! R, wherew (x) is the payo¤ to
the agent as a function of the realized outcome x: Let W denote the set of all
incentive contracts. Note that, given a technology t; each action a and incentive
contract w correspond to the act ^w (a;w) 2 F de…ned by ^w (a;w)(s) =w (ta(s))
for all s 2 S: Similarly, the residual payo¤ ta ¡ ^w (a;w) is also an act in F:
Suppose that the principal’s preference relation, < P ; on F satis…es Savage’s
(1954) axioms and that the agent has a preference relation, < A ; on M (A)£F jAj
satisfying the axioms in either Theorems 1, 2, or 3.
Assume that the production technology is common knowledge, the level of
output is veri…able, and the choice of action is private information of the agent.
Under these assumptions the information regarding the state becomes a focal
issue in the formulation of the agency problem. In decision theory it is generally
assumed that the true state of nature becomes known ex post. Presumably in
order to verify whether he is correctly rewarded according to the act that he
chose, the decision maker must be able to observe the state. However, if the state
and the consequences are observable and the technology is known, each action
the agent may choose can be inferred on some events (subset of states). Hence
the principal is in a position to implement the …rst-best solution by imposing
a su¢ciently large penalty on all the events-output combinations that, under
the known technology, indicate that the agent chose an action other than that
required to attain the …rst-best solution. Thus for the agency problem to be
nontrivial it is necessary to assume that the states are not directly observable. 6
Within this framework the principal’s problem may be stated as follows:
Choose an action-contract pair (a ¤ ;w ¤ ) such that
¡ ta ¤ ¡ ^w (a ¤ ;w ¤ )
¢ P
< ¡ ¢
ta ¡ ^w (a;w) for all (a; w) 2 A £ W (28)
subject to the incentive compatibility constraints
¡ ¢ ¤ A
a ; ^w(a ¤ ;w ¤ ) < ¡ ¢
a; ^w (a;w ¤ ) for all a 2 A (29)
and the participation constraint
¡ ¢ ¤ A
a ; ^w(a ¤ ;w ¤ ) < (¹a; ¹w) ; (30)
5 See Shavel (1979).
6 See, for example, Quiggin and Chambers (1998).
12