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Principal-agent problems - Applications of game theory 3

Page 1
Principal- agent problems
Geir B. Asheim
Introduction Hidden action Hidden information
Principal-agent problems
Applications of game theory 3
Geir B. Asheim
Department of Economics, University of Oslo
ECON5200 Fall 2009
Principal- agent problems
Geir B. Asheim
Introduction
Hidden action Hidden information
Introduction
How the this topic differs from Adverse selection Adverse selection: Asymmetry of information before time of contract Principal-agent problems: Asymmetry of information after time of contract
Hidden action: About effort put into a job Hidden information: About how the job should be done
Hidden action and hidden information are often combined. Here we will consider each of them in isolation.

Page 2
Principal- agent problems
Geir B. Asheim
Introduction
Hidden action Hidden information
Principal-agent problems Examples and outline
Examples: Owner Manager Firm Workers Investor Entrepreneur Society Criminal Insurance company Insuree Manufacturer Distributor Outline: Hidden action Hidden information
Principal- agent problems
Geir B. Asheim
Introduction
Hidden action
Observable effort Unobservable effort
Hidden information
Hidden action The owner��s profit
The owner of a firm (the principal) wishes to hire a manager (the agent) for a one time project. Firm��s profit ��, where �� �� [��, ¯��], depends on the manager��s effort e, where e �� E ⊆ R, and �� is stochastically related to e, with conditional pdf f (�� | e), where f (�� | e) > 0 for all �� �� [��, ¯��]. Two possible effort choices: eL < eH . The cdf conditional on eH first-order stochastically dominates the cdf conditional on eL : F(�� | eH ) �� F(�� | eL ) for all �� �� [��, ¯��], with < for an open set �� [��, ¯��]. �� ��f (�� | eH )d�� > �� ��f (�� | eL )d��. The owner is risk neutral. Zero profit without contract.

Page 3
Principal- agent problems
Geir B. Asheim
Introduction
Hidden action
Observable effort Unobservable effort
Hidden information
Hidden action The manager��s utility
The manager is an expected utility maximizer with a Bernoulli utility function: u(w,e) uw(w,e) > 0 and uww(w,e) �� 0 for all (w,e) u(w,eL ) > u(w,eH ) for all w Special case: u(w,e) = v(w) − g(e), where v (w) > 0, v (w) �� 0, and g(eL ) < g(eH ). The manager is risk averse, except if uww(w,e) = 0 for all (w,e) (which in the special case translates to v (w) = 0). Without contract, the manager receives his reservation utility ¯u.
Principal- agent problems
Geir B. Asheim
Introduction Hidden action
Observable effort
Unobservable effort
Hidden information
The optimal contract when effort is observable (1)
Step 1: For each e, find cheapest w(��) for which E[u] �� ¯u. min
w(��)
�� w(��)f (�� | e)d�� s.t. �� v(w(��))f (�� | e)d�� − g(e) �� ¯u FOC: −f (�� | e) + ��v (w(��))f (�� | e)=0
1 v (w(��))
= �� If v (w) < 0, then fixed wage: v(w
e
) − g(e)=¯u or w
e
= v
1
u + g(e)) Step 2: Choose eL or eH to maximize E[�� − w(��)]. max
e��{eL,eH }
�� ��f (�� | e)d�� − v
1
u + g(e))

Page 4
Principal- agent problems
Geir B. Asheim
Introduction Hidden action
Observable effort
Unobservable effort
Hidden information
The optimal contract when effort is observable (2)
Proposition
In the principal-agent model with observable efiort, an optimal contract specifies that the manager choose the efiort e

that maximizes �� ��f (�� | e)d�� − v
1
u + g(e)) over {eL,eH}, and pays the manager a fixed wage w
e
= v
1
u + g(e

)). This is the uniquely optimal contract if v (w) < 0 for all w. Let ¯�� = �� ��f (�� | e∗ )d�� − v
1
u + g(e

)).
Principal- agent problems
Geir B. Asheim
Introduction Hidden action
Observable effort
Unobservable effort
Hidden information
The optimal contract when effort is unobservable
Two cases:
1 A risk-neutral manager
Sell the firm to the manager: the manager receives the full marginal returns from his effort and faces all risk.
2 A risk-averse manager
Trade-off between incentives for effort and insurance against risk

Page 5
Principal- agent problems
Geir B. Asheim
Introduction Hidden action
Observable effort
Unobservable effort
Hidden information
Risk-neutral manager
Proposition
In the principal-agent model with unobservable efiort and a risk-neutral manager, an optimal contract generates the same efiort choice and utilities for the owner and the manager as when efiort is observable.
Proof.
Part 1: There exists an accepted contract where the principal receives ¯��. Let the principal sell the project for ��∗ = �� ��f (�� | e∗ )d�� − g(e

) ¯u where e

= arg max �� ��f (�� | e)d�� − g(e). This contract is accepted: �� ��f (�� | e∗ )d�� − g(e

) − ��∗ = ¯u. Also, ��∗ = ¯��. Part 2: No accepted contract where p. receives more.
Principal- agent problems
Geir B. Asheim
Introduction Hidden action
Observable effort
Unobservable effort
Hidden information
Risk-averse manager (1)
Step 1: For each e �� {eL,eH}, find cheapest w(��) for which (PC) E[u(w,e)] �� ¯u and (IC) E[u(w,e)] �� E[u(w,˜e)]. min
w(��)
�� w(��)f (�� | e)d�� s.t. (PC) �� v(w(��))f (�� | e)d�� − g(e) �� ¯u (IC) e solves max
˜e
�� v(w(��))f (�� | ˜e)d�� − ge) Step 2: Choose eL or eH to maximize E[�� − w(��)].

Page 6
Principal- agent problems
Geir B. Asheim
Introduction Hidden action
Observable effort
Unobservable effort
Hidden information
Risk-averse manager (2)
Implementing eL Fixed wage equal to v
1
u + g(eL )) satisfies both (PC) and (IC). Yields principal same utility as when eL is observable. Higher payoff is not feasible when manager chooses eL . Implementing eH (IC) becomes �� v(w(��))f (�� | eH )d��−g(eH ) �� �� v(w(��))f (�� | eL )d��−g(eL ) Let �� �� 0 and �� �� 0 be the multipliers for (PC)&(IC).
−f (�� | eH )+��v (w(��))f (�� | eH )+��[f (�� | eH )−f (�� | eL )]v (w(��)) = 0
1 v (w(��)) = �� + �� [ 1 f (�� | eL ) f (�� | eH ) ]
Principal- agent problems
Geir B. Asheim
Introduction Hidden action
Observable effort
Unobservable effort
Hidden information
Risk-averse manager (3)
Proposition
In the principal-agent model with unobservable efiort and a risk-averse manager, the optimal compensation scheme for implementing eH satisfies
1 v (w(��))
= �� + �� [ 1
f (��|eL) f (��|eH )
] , gives the manager ¯u, and involves a larger expected wage payment than is required when efiort is observable. The optimal compensation scheme for implementing eL involves the same fixed wage payment as if efiort were observable. Whenever the optimal efiort level with observable efiort would be eH , nonobservability causes a welfare loss. If
f (��|eL) f (��|eH )
, the likelihood ratio, is monotone, then the optimal compensation scheme for implementing eH is monotone. If eH is optimal with observable effort, then either implement eH at additional cost due to inefficient risk-sharing, or implement eL .

Page 7
Principal- agent problems
Geir B. Asheim
Introduction Hidden action
Hidden information
Observable state Unobservable state
Hidden information
The owner of a firm (the principal) wishes to hire a manager (the agent) for a one time project. The manager��s effort e (�� [0,��)) is fully observable. The manager��s cost of exhorting effort is private information. u(w,e,��) = v(w − g(e,��)) where g(e,��) measures the disutility of effort in monetary terms. g(0,��)=0 ge(e,��) { > 0 for e > 0 = 0 for e = 0 g�� (e,��) { < 0 for e > 0 = 0 for e = 0 gee(e,��) > 0 for all e ge�� (e,��) { < 0 for e > 0 = 0 for e = 0 Two states: 0 < ��L < ��H �� = Prob(�� = ��H ) �� (0,1)
Principal- agent problems
Geir B. Asheim
Introduction Hidden action Hidden information
Observable state
Unobservable state
The state �� is observable
max
wL,eL,wH ,eH ��0
��[��(eH ) − wH ] + (1 − ��)[��(eL ) − wL ] s.t. ��v(wH − g(eH,��H )) + (1 − ��)v(wL − g(eL,��L )) �� ¯u v (w
H
− g(e
H
,��H )) = v (w
L
− g(e
L
,��L )) v(w
H
− g(e
H
,��H )) = v(w
L
− g(e
L
,��L )) = ¯u �� (e
H
) = ge(e
H
,��H ) and �� (e
L
) = ge(e
L
,��L )
Proposition
In the principal-agent model with observable state, an optimal contract involves an efiort level e
i
in state ��i such that �� (e
i
) = ge(e
i
,��i) and fully insures the manager, setting his wage in each state ��i at the level w
i
such that v(w
i
− g(e
i
,��i)) = ¯u.

Page 8
Principal- agent problems
Geir B. Asheim
Introduction Hidden action Hidden information
Observable state
Unobservable state
The state �� is observed only by the manager (1)
If the owner asks the manager to reveal his type, then — with the optimal contract under observable state — in state ��H the manager prefers to report ��L . The manager must be compensated to provide truthful announcement of state ��H .
Proposition (The revelation principle)
��: The set of possible states. The owner can without loss restrict himself to contracts of the following form:
1 After �� is realized, the manager is asked to reveal state. 2 The contract specifies an outcome [w��),e��)] for each
possible announcement ˆ�� �� ��.
3 In every �� �� ��, truthful announcement is optimal.
Principal- agent problems
Geir B. Asheim
Introduction Hidden action Hidden information
Observable state
Unobservable state
The state �� is observed only by the manager (2)
Assume infinite managerial risk averse: u = min{uL,uH}. max
wL,eL,wH ,eH ��0
��[��(eH ) − wH ] + (1 − ��)[��(eL ) − wL ] s.t. (PCL ) wL − g(eL,��L ) �� v−1 (¯u) (PCH ) wH − g(eH,��H ) �� v−1 (¯u) (ICL ) wL − g(eL,��L ) �� wH − g(eH,��L ) (ICH ) wH − g(eH,��H ) �� wL − g(eL,��H )
1 (PCH
) can be ignored.
2 (PCL
) binds (holds with equality).
3 eL �� e∗
L
and eH = e
H
((ICL ) can be ignored).
4 eL < e∗
L
: [�� (eL
) − ge (eL,��L )] =
�� 1−��
[ge (eL,��L ) − ge (eL,��H )]

Page 9
Principal- agent problems
Geir B. Asheim
Introduction Hidden action Hidden information
Observable state
Unobservable state
The state �� is observed only by the manager (3)
Proposition
In the hidden information principal-agent model with an infinitely risk-averse manager, the optimal contract sets the level of efiort in state ��H at its full observability level e
H
. The efiort level in state ��L is distorted downward from its full observability level e
L
. The manager is inefficiently insured, with uH > ¯u and uL = ¯u. The owner��s expected utility is below its full observability level, while the manager��s utility is at its full observability level (= ¯u). Other topics: The hidden information model with infinite risk aversion can be used to study monopolistic screening, in which a single firm screens workers who, at the time of contracting, have different unobservable productivity levels. Applicable also to regulation, e.g. of a monopolist with unknown cost. Hybrid models of hidden action and hidden information.

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