复旦微观经济学Hidden actions

Junji Xiao (Lecture 8)
Micro
3 / 42
Hidden actions
Introduction
Hidden action v.s. hidden information
Solution to hidden information problems is self-selection menus or screening. Solution to hidden action problems is contracts. Presence of incentive constraints (ICs) and individual rationality (IRs) constraints in both problems.
A two-step procedure for solving for optimal contract
Step 1: For each recommended e¤ort e, choose a wage function w (π ) R in order to maximize (π w (π ))f (π je )d π subject to (IC) and (IR), yielding the expected pro…t as a function of e. Step 2: maximize the expected pro…t function by choosing e.
where γ is the Lagrangian multiplier.
Junji Xiao (Lecture 8)
Micro
13 / 42
Hidden actions
Optimal contract design
Benchmark: e¤ort is observable (cont’ d)
Case 1: the manager is strictly risk averse or v 00 < 0
e
Z
v (w (π ))f (π je )d π
g (e ).
Manager
accepts contract w (π ), if max
e
Z
v (w (π ))f (π je )d π
g (e )
u, (IR )
where u is the manager’ reservation utility level, and s reject o¤er otherwise.
We may think of the choice e in the maximization problem as “recommended e¤ort.”
(IC) is then interpreted as “obedience,” or “following recommendation.”
Junji Xiao (Lecture 8)
Micro
8 / 42
Hidden actions
Optimal contract design
Optimal contract design
Observation: For any wage contract w (π ), manager chooses e to maximize payo¤, given by max
This observation is akin to revelation principle argument.
Junji Xiao (Lecture 8) Micro 9 / 42
Hidden actions
Optimal contract design
Optimal contract design
Can solve for each w (π ) in terms of γ and substitute in (IR) to get γ and then each w (π ), as functions of e. w (π ) is a constant we (γ). risk-sharing (full insurance for manager): (IR) v (we ) g (e ) = u ) we = v 1 (g (e ) + u ) In second step, owner chooses e to maximize max
Junji Xiao (Lecture 8)
Micro
6 / 42
Hidden actions
Model
General setup (cont’ d)
Timing of game
Principal makes a take-it-or-leave-it o¤er (a contract) to the manager. Game ends if the manager rejects the o¤er. Manager chooses e¤ort e if accepting the o¤er. Output π is realized, and wage is paid according to contract and game ends.
Owner chooses wage function w (π ) and e¤ort e to maximize max
Z
e,w (π )
(π
w (π ))f (π je )d π
s.t. (ICs)
Z
v (w (π ))f (π je )d π
Z
g (e )
Z
v (w (π ))f (π je 0 )d π
Junji Xiao (Lecture 8)
Micro
11 / 42
Hidden actions
Optimal contract design
Benchmark: e¤ort is observable
Assume that manager’ e¤ort, e, is observable to owner. s Suppose contracts can be written on actions directly.
Junji Xiao (Lecture 8)
Micro
2 / 42
Hidden actions
Introduction
Origin of “moral hazard”
Insurance problems: insuree can take unobservable actions that a¤ect the value of insured asset. General characteristics of moral hazard
There are two parties that can voluntarily choose to enter a contractual relationship. Unobserved actions of one party, called agent, exert externality on the payo¤ of the other, called principal. Principal can write an enforceable contract on veri…able outcomes that depend on agent’ action. s Other economic examples: plainti¤-attorney, employer-employee, homeowner-contractor, landlord-tenant, banker-borrower.
Junji Xiao (Lecture 8)
Micro
4 / 42
Hidden actions
Model
General setup
Imagine that the owner of a …rm (the principal) wishes to hire a manager (the agent) for a one-time project. Manager’ payo¤ is s u (w , e ) = v (w ) g (e ).
w (π ) is wage, and π w (π ) is pro…t when output is π.
Байду номын сангаас
Assume output π 2 [π, π ] is stochastically related to manager’ s e¤ort e
conditional density function f (π je ) for all e 2 E and all π 2 [π, π ];
Junji Xiao (Lecture 8)
Micro
7 / 42
Hidden actions
Model
General setup (cont’ d)
Implicit assumptions:
Symmetric information at contracting stage. E¤ort e is not observable to owner. Output is π observable and veri…able. No renegotiation of contract–both manager and owner can commit to contract.
g (e 0 )
for all e 0 , and (IRs) v (w (π ))f (π je )d π g (e ) u.
Junji Xiao (Lecture 8)
Micro
10 / 42
Hidden actions