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 \noindent
  JUne 26, 1993

 
  Feb. 12, 1996
\section*{ 7 MORAL HAZARD: HIDDEN ACTIONS OCERHEDS}  
  
 \subsection{Categories of Asymmetric Information Models}  

 \noindent
    We will divide games of asymmetric
information into five categories, to be studied in four chapters.  

  \noindent
 (1)
  {\bf Moral hazard with hidden actions} (Chapter
7)\\
 Smith and Jones begin with symmetric information and agree to a
contract, but then Smith takes an action unobserved by Jones.
Information is complete.

  \noindent (2) {\bf Moral hazard with hidden knowledge } (or {\bf hidden information}) (Chapter 8)
\\
  Smith and Jones begin with symmetric information and agree to a
contract.  Nature then makes a move observed by Smith but not
Jones, and Smith takes some action, which may be simply a report of Nature's move.    Information is complete.

\noindent
(3)
  {\bf Adverse selection} (Chapter 9)\\
 Nature begins the game by choosing Smith's type (his payoff and
strategies), unobserved by Jones. Smith and Jones then agree to a
contract.  Information is incomplete.



\noindent
  (4, 5) {\bf Signalling } and {\bf Screening} (Chapter 10)\\
 Nature begins the game by choosing Smith's type, unobserved by
Jones.  To demonstrate his type, Smith takes actions that Jones can
observe. If Smith takes the action before they agree to a contract,
 he is signalling; if he takes it afterwards, he is being
screened.  Information is incomplete.

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   \newpage


    We will make heavy use of the ``principal-agent model'' to analyze
asymmetric information.  Usually this  term is  applied to moral hazard models, since the problems studied   in the law of agency usually involve an employee who disobeys orders by choosing the wrong actions, but the paradigm will be useful in all of these contexts.  The two players are the principal and the
agent, who are usually representative individuals.  The principal
hires an agent to perform a task, and the agent acquires an
informational advantage about his type, his actions, or the outside
world at some point in the game.  It is usually assumed that the players can make a binding {\bf contract} at some point in the game, which is to say that the principal can commit to paying the agent an agreed sum if   he observes a certain outcome. In the implicit background of such models are courts which will  punish any player who breaks a contract in  a way that can be proven with public information.  

 \noindent
  {\it The {\bf principal} (or {\bf uninformed player}) is
the player who has the coarser information partition.} 

 \noindent
 {\it The {\bf agent} (or {\bf informed player}) is the player who
has the finer information partition.}

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   \newpage
   \begin{tiny}
  \begin{center} {\bf Table 7.1 Applications of the principal-agent
model}  

 \begin{tabular}{|llll|}
\hline
  & & & \\
 & {\bf Principal} & {\bf Agent}  & {\bf Effort or Type and  Signal} \\
\hline
  & & & \\
  {\bf Moral hazard with} &Insurance company & Policyholder & Care to avoid theft\\
 {\bf hidden actions} &  Insurance company & Policyholder &  Drinking and Smoking  \\
 & Plantation owner & Sharecropper & Farming Effort\\
 & Bondholders & Stockholders & Riskiness of corporate projects\\
 & Tenant  & Landlord & Upkeep of the building\\
 & Landlord & Tenant & Upkeep of the building\\
 & Society  & Criminal & Number of robberies\\
 & & & \\
\hline
 & & & \\
  {\bf Moral hazard with} & Shareholders & Company president & Investment decision\\
  {\bf hidden  knowledge} &  FDIC & Bank & Safey of Loans\\
 & & & \\
\hline
& & & \\
 {\bf Adverse selection} & Insurance Company & Policyholder & Infection with the HIV virus\\
 & Employer & Worker & Skill\\
& & & \\
\hline
& & & \\
{\bf  Signalling and } &  Employer & Worker & Skill  and Education\\ 
{\bf screening} &  Buyer & Seller & Durability and Warranty\\
 & Investor & Stock-issuer & Stock value and Percentage Retained\\
 & & & \\
   \hline
\end{tabular}                
\end{center}
 \end{tiny}
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   \newpage
    

 
  \begin{center}
{\bf  ``The Production Game'' }
 \end{center}
 {\bf Players}\\ 
The principal and the agent.
  
\noindent
  {\bf Order of Play} \\
 (1)  The principal offers  the agent a wage $w$. \\
    (2)   The agent decides whether to accept or reject the contract.\\
 (3)  If the agent accepts, he exerts
effort $e$. \\
(4)   Output equals $q(e)$, where $q'>0$. 
  
\noindent
 {\bf Payoffs}\\
 If the agent rejects the contract, then $\pi_{agent} = \bar{U}$ and
$ \pi_{principal} = 0$.\\
 If the agent accepts the contract, then $\pi_{agent} = U(e,w)$ and $
\pi_{principal} = V(q - w).$
 

 
  Denote the monetary value of output by $q(e)$, which is increasing
in effort, $e$.  The agent's utility function $U(e,w)$ is decreasing
in effort and increasing in the wage $w$, while the principal's
utility $V(q-w)$ is increasing in the difference between output and
the wage. 

%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   \newpage
 
 
\noindent
{\bf ``Production Game I'': Full Information.} In the first version of the game,   every move is common knowledge and the contract is a function $w(e)$. 

    
 The agent must be paid some amount $\tilde{w}(e)$ to exert effort
$e$,  where $\tilde{w}(e)$ is defined to be the $w$ that
solves the participation constraint 
  \begin{equation} \label{e1}
 U(e,  w(e))  = \overline{U}. 
 \end{equation}
 Thus, the principal's problem is  
   \begin{equation} \label{e1} 
\begin{array}{cl}
 Maximize&  V(q(e) - \tilde{w}(e ))\\
 e & \\
  \end{array}
 \end{equation}
    The first order condition for this problem is 
\begin{equation} \label{e7.1}
   V'(q(e) - \tilde{w}(e )) \left(\frac{\partial q }{\partial e} -  \frac{\partial  \tilde{w} }{\partial e} \right)=0,
 \end{equation}
 which implies that
 \begin{equation} \label{e7.1a}
     \frac{\partial q }{\partial e} =  \frac{\partial  \tilde{w} }{\partial e}.   
 \end{equation}
  From the implicit function theorem (see Section 13.4) and the participation constraint,  
\begin{equation} \label{e7.2}
 \frac{\partial  \tilde{w} }{\partial e}   = -  \left( \frac{\frac{\partial  U }{\partial e} }{ \frac{\partial  U }{\partial  \tilde{w}}}\right). 
 \end{equation}
 Combining   equations  (7.\ref{e7.1a}) and (7.\ref{e7.2}) yields 
\begin{equation} \label{e7.3}
 \frac{\partial U}{\partial  \tilde{w}}  \frac{\partial q }{\partial e}   = -   \frac{ \partial  U }{\partial e}.
   \end{equation}
 
  
Under   perfect competition  among the principals the  profits are zero, so the reservation utility
$\overline{U}$ is chosen so that at the profit-maximizing effort $e^*$,
$\tilde{w}(e^*) = q(e^*)$, or 
 \begin{equation} \label{e1}
  U(e^*, q(e^*)) = \overline{U}.  
   \end{equation}
 The principal  
  selects the
point on the $U= \overline{U}$ indifference curve that maximizes his profits, which is at the effort $e^*$ and wage $w^*$. He must then design a contract that will induce the agent to choose this effort level. 


%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   \newpage


A number of contracts are equally effective under full information.  
 

(1) The  {\bf forcing contract}  sets $w(e^*) = w^*$ and $w(e \neq e^*) =0$.  This is certainly a strong incentive for the agent to choose exactly $e=e^*$.  

(2) The  {\bf threshold contract}  sets $w(e \geq e^*) = w^*$ and $w(e <e^*) =0$.    This   can be viewed as a flat wage  for low effort levels, equal to 0 in the previous sentence,  plus a bonus if effort reaches $e^*$. Since the agent dislikes effort, the agent will choose exactly $e=e^*$. 
 

 (3) The  {\bf  linear contract}  shown in Figure 7.2  sets $w(e) = \alpha + \beta e$, where  $\alpha$ and $\beta$ are chosen so that $ w^* = \alpha + \beta e^*$ and   the contract line is tangent to the indifference curve $U=\bar{U}$ at $e^*$.  The most northwesterly of the agent's indifference curves that touch  this contract line touches it at $e^*$.    

 
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   \newpage
\noindent
{\bf ``Production Game II'': Full Information and the Agent Moves First.} In this  version  every move is common knowledge and the contract is a function $w(e)$.  The order of play, however, becomes

\noindent
  {\bf Order of Play} \\
 (1)  The  agent offers the principal a contract $w(e)$. \\
    (2)   The principal decides whether to accept or reject the contract.\\
 (3)  If the principal accepts, the agent exerts
effort $e$. \\
(4)   Output equals $q(e)$, where $q'>0$. 

  
   In this game, the agent has all the bargaining
power, not the principal. The participation constraint is now that the principal must
earn zero profits, so $q(e) - w(e) \geq 0$. The agent will maximize his own payoff by driving the principal to exactly zero profits, so  $w(e)= q(e)$. Substituting  $q(e)$ for $w(e)$ to account for the participation constraint, the    maximization problem
for the agent  in proposing an effort level $e$ at a wage $w(e)$   can therefore be written as  
   \begin{equation} \label{e1} 
\begin{array}{cl}
 Maximize&  U(e, q(e))\\
 e & \\
  \end{array}
 \end{equation}
  The first order condition is 
\begin{equation} \label{e7.100}
   \frac{\partial U }{\partial e} + \left(\frac{\partial U }{\partial q} \right) \left(\frac{\partial q }{\partial e} \right) =0.
 \end{equation}
 Since   $\frac{\partial U }{\partial q}=\frac{\partial U }{\partial w}$ when the wages equals output, equation (7.\ref{e7.100}) implies that 
 \begin{equation} \label{e1}
 \frac{\partial U}{\partial w } \frac{\partial q }{\partial e} = -
\frac{ \partial U }{\partial e}.
   \end{equation}
 


%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   \newpage



\bigskip
\noindent
{\bf ``Production Game III'': A Flat Wage under Certainty}. In this version of the game, the principal can condition the wage on neither effort nor output. This is modelled as a principal who observes neither effort nor output, so information is asymmetric.   

 
  
     The outcome of ``Production Game III'' is simple and inefficient.
If the wage is non-negative, the agent accepts the job and exerts
zero effort, so the principal offers  a wage of
zero. 

  
\bigskip
\noindent
{\bf ``Production Game IV'': An Output-Based Wage under Certainty}. In this version, the principal cannot observe effort but can observe output and specify the contract to be $w(q)$.   


    
Now  the principal  picks not a number $w$   but a function $w(q)$. His problem is not quite so  straightforward as in  ``Production Game  I'', where he picked the function $w(e)$, but here too it is possible to achieve the efficient    effort level, $e^*$,  despite the unobservability of effort. 
  The principal starts by 
finding the optimal  effort level $e^*$, as in ``Production Game  I''.  That effort yields  the efficient output level   $q^* = q(e^*)$.  To give the agent the proper incentives, the contract must reward him when output is $q^*$. Again, a variety of contracts could be used. The forcing contract, for example,  would  
 be any wage function such that
$U(e^*,w(q^*)) = \bar{U}$ and $U(e,w(q)) < \bar{U}$ for $e \neq e^*$.
 

 


%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   \newpage


\bigskip
\noindent
{\bf ``Production Game  V'': An Output-Based Wage under Uncertainty}.
  In this version, the principal cannot observe effort but can observe output and specify the contract to be $w(q)$.   Output, however, is a function $q(e,\theta)$  both of effort and   the  state of the world $\theta \in {\bf R}$, which is  chosen by Nature  according to the
probability density $f(\theta)$ as the new move (3.5) of the game.  Move (3.5) comes just  after the agent chooses effort, so the agent cannot  choose a low effort knowing that nature will take up the slack. (If the agent can observe Nature's move before his own, the game becomes moral hazard with hidden knowledge as well as hidden actions). 




 Because of the uncertainty of the state of the world,    effort does not map cleanly onto the
observed output in ``Production Game  V''. A given output   might have been produced by any of several
different effort levels, so a forcing contract will not necessarily achieve the desired effort.  
Unlike in ``Production Game
IV''  the principal cannot deduce that $e = e^*$  from the fact that $q=q^*$. Moreover, even if the contract does induce the agent to choose $e^*$, if it does so by penalizing him heavily when $ q \neq q^*$ it will be expensive for the principal. The  agent's expected utility must be kept equal to $\bar{U}$ because of the participation constraint, and if the agent is sometimes paid a low wage because output happens not to equal $q^*$, he must be paid more when output does equal $q^*$ to make up for it. If the agent is risk averse, this variability in his wage requires  that his  expected wage be higher than the $w^*$ found earlier, because he must be compensated for the extra risk. There is a tradeoff between    incentives and
insurance.

 
 Moral hazard  is a problem when the
functions $q$ and $V$ are not invertible.  Often if the principal could observe the state $\theta$,
he could deduce effort from observed output $q(e, \theta)$, or from
his own observed utility $V(q-w)$.  That $q(e, \theta)$ is {\bf
invertible} means that every point in the space in which $(e, \theta)$ lies
is mapped onto exactly one point in the space in which $q$ lies, and
every point in $q$-space is mapped onto by some point in
$(e, \theta)$-space.  Knowing $q$, you know $(e,\theta)$, and vice versa.
If $q$ and $V$ are not invertible,  the principal can neither observe the
agent's effort level nor deduce it without assuming equilibrium
behavior.    

 The combination of unobservable effort and lack of invertibility in 
  ``Production Game  V''  means that no contract can induce the agent to put forth the efficient effort level without incurring extra costs, which usually take the form of imposing extra risk on the agent.   We will  still try to find a contract that is  efficient  in the sense of maximizing welfare given the informational constraints. 
The terms ``first-best'' and ``second-best''  are used to distinguish these  two kinds of optimality.

 \noindent
{\it A {\bf first-best contract} achieves the same allocation as the
contract optimal when the principal and the agent have the same
information set and all variables are contractible.  }

 \noindent {\it A {\bf second-best contract} is Pareto-optimal given
information asymmetry and constraints on writing contracts.} 

    The difference in welfare between the first-best world and the second-best world is the cost of  the agency problem. 



%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   \newpage


The first four production games were easier because the principal could find a first-best contract without searching very far. 
 Even defining the strategy space in a game like ``Production Game  V''  is tricky, because  the principal is choosing a  function $w(q)$  and the agent is choosing effort as a function of that function: $e(w(\cdot))$.  Finding the optimal contract  when a forcing contract cannot be used   becomes a difficult   problem,  without  general answers  because of the tremendous  variety of possible contracts. 
  

  The principal's problem in ``Production Game V'' is to maximize
his utility knowing that the agent is free to reject the contract entirely and that the contract must give him an incentive to choose the desired effort.       These two constraints come up in every moral hazard problem, and they are named the   {\bf participation constraint } and   the {\bf incentive compatibility contraint}. Mathematically, the principal's  problem is  
   \begin{equation} \label{ebig} 
\begin{array}{cl}
 Maximize& EV(q(\tilde{e},\theta) - w(q(\tilde{e},\theta)))\\
 w(\cdot) & \\
  \end{array}
 \end{equation}
 subject to\\
 \begin{tabular}{lll}
 & \\
 & $ \tilde{ e} = \stackrel{argmax}{e}\; EU(e, w(q(e,\theta)))$&
({\bf incentive compatibility constraint}) \\
 & \\
 & $  EU(\tilde{e}, w(q(\tilde{e},\theta))) \geq \bar{U}$ & ({\bf
participation constraint})\\
 & \\
\end{tabular}


%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   \newpage

  
     
 
%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   \newpage
 

\begin{center}
{\bf   ``Broadway Game I''}
\end{center}
 {\bf Players}\\
  The producer and the investors.

 \noindent
 {\bf Order of Play}\\
 (1) The investors offer a wage contract $w(q)$ as a function of
revenue $q$.\\
 (2) The producer accepts or rejects the contract.\\
   (3) The producer chooses to $Embezzle$ or {\it Do not embezzle}.\\
  (4) Nature picks the state of the world to be $Success$ or
$Failure$ with equal probability.  Table 7.2 shows the resulting
revenue, $q$.
  
\noindent
 {\bf Payoffs.}\\
 The producer is risk averse and the investors are risk  neutral.  The producer's payoff is $U(100)$ if he rejects the contract, where
$U' >0$ and $U''<0$, and the investors' payoff is 0.  Otherwise, 

$$
\begin{array}{l}
  \pi_{producer} = \left\{
\begin{array}{ll}   U( w(q) + 50)   & if \;he\; embezzles \\
  U(w(q))  & if\;he\; is\;honest \\
 \end{array} 
\right.\\
  \pi_{investors} = q - w(q) \\
\end{array}
 $$

\bigskip

 \begin{center}
{\bf Table 7.2   ``Broadway Game I'': Profits} 

 \begin{tabular}{lllccc}
  &       &             &\multicolumn{3}{c}{\bf State of the World}\\
  &       &             &  {\it Failure} (0.5)  &   &  {\it Success}  (0.5)  \\
  &   &  $ Embezzle$     &    $-100$ &   &  $+100$ \\
 & {\bf  Effort} && & &   \\
&  &    {\it  Do not embezzle }     &      $-100$  &    & +500 \\  
 \end{tabular}                
\end{center}


\newpage
 
    Another way to tabulate outputs  is shown in Table 7.3.
    \begin{center}
{\bf Table 7.3   ``Broadway Game I'':  Probabilities of Profits  } 


 \begin{tabular}{lllcccc}
  &       &             &\multicolumn{4}{c}{\bf Profit}\\
  &       &             &  {\bf -100}   &   {\bf +100} &   {\bf +500}  & {\bf Total}  \\
  &   &  $Embezzle$   &     0.5 & 0.5  &  0   & 1\\
 & {\bf Effort} && & &  &      \\
&  &    {\it Do not embezzle }       &     0.5  & 0  &  0.5  & 1\\  
 \end{tabular}                
\end{center}

  The investors will observe $q$ to equal either $-100, +100$, or $+500$, so the producer's contract will specify at most three different wages:  $w(-100), w(+100)$,  and $w(+500)$. 
 The producer's expected payoffs from his two possible actions are  
  \begin{equation} \label{e7.2a}
  \pi(Do\; not\; embezzle) =   0.5 U ( w(-100)) +  0.5 U (  w(+500))    \end{equation}
 and
\begin{equation} \label{e7.2b}
  \pi(Embezzle) =    0.5 U ( w(-100)+50) +  0.5 U (  w(+100)+50).  
 \end{equation}
  The incentive compatibility constraint is    $\pi(Do\; not\; embezzle)  \geq  \pi(Embezzle)$, so 
 \begin{equation} \label{e7.2c}
           0.5 U ( w(-100)) +  0.5 U (  w(+500))  \geq 0.5 U ( w(-100)+50) +  0.5 U (  w(+100)+50), 
 \end{equation}
 and the participation constraint is that
 \begin{equation} \label{e7.2d}
  \pi(Do\; not\; embezzle) =   0.5 U ( w(-100)) +  0.5 U (  w(+500)) \geq U(100).
    \end{equation}

   The following 
 {\bf boiling-in-oil contract}   provides both riskless wages and effective incentives. 

\noindent
 $w(+500) = 100$.\\
 $w(-100) = 100$.\\
 $w(+100) = -\infty$.

    
\newpage
   
   Milder  contracts than  this would also    be effective.  Two wages will be used in equilibrium, a low wage $\underline{w}$ for an output of $q=100$ and a high wage  $\overline{w}$ for any other output. The participation  and incentive compatibility constraints provide two equations to solve for these two unknowns.  To find the mildest possible contract,  the modeller must also specify a function for $U(w)$  which, interestingly enough, was unnecessary for finding the first  boiling-in-oil contract.   Let us specify that 
 \begin{equation} \label{e7.2e}
   U(w) =  100 w - 0.1w^2.  
   \end{equation}
 A quadratic utility function like this is only increasing if its argument is not too large, but since the wage will not exceed $w=1000$, it is a reasonable utility function for this model. 
 Substituting  (7.\ref{e7.2a}) into the participation constraint (7.\ref{e7.2d}) and solving for  the high wage $\overline{w}= w(-100) = w(+500)$ yields 
$\overline{w}=100$ and a reservation utility of 9000. Substituting 
into the incentive compatibility constraint,  (7.\ref{e7.2c}), yields
  \begin{equation} \label{e7.2f}
          9000 \geq 0.5 U (  100+50) +  0.5 U ( \underline{w}  +50).  
 \end{equation}
 When $(7.\ref{e7.2f})$ is solved using the quadratic equation, the answer is (with rounding error), $\underline{w} \leq 5.6$.  A low wage of $-\infty$ is much more severe than is needed.  

   \newpage


   The conditions favoring
boiling-in-oil contracts are


 (1)  The agent is not very risk averse.

(2)  There are outcomes with high probability under shirking that
have low probability under optimal effort. 

(3)
  The agent can be severely punished. 

(4)   It is credible that the principal will carry out the severe
punishment. 
 

   Another first-best contract that can sometimes be used is {\bf
selling the store.} Under this arrangement, the agent buys the entire
output for a flat fee paid to the principal, becoming the {\bf
residual claimant}, since he keeps every additional dollar of output
that his extra effort produces. This is equivalent to fully insuring
the principal, since his payoff becomes independent of the agent's and Nature's moves.  

 In ``Broadway Game I'', selling the store takes the form of the
producer paying the investors 100 ($= 0.5[-100] + 0.5[+500]-100$) and
keeping all the profits for himself. The drawbacks are that (1) the
producer might not be able to afford to pay the investors the flat
price of 100; and (2) the producer might be risk averse and incur a
heavy utility cost in bearing the entire risk.  These two drawbacks
are why producers go to investors in the first place.




 
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