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\begin{LARGE}
 \begin{center}
 {\bf  8 Further Topics in Moral Hazard }

 \end{center}
 \end{LARGE}


This is designed for one 75-minute lecture using {\it Games and Information}. Probably I
have more material than I will end up covering.

These slides just cover sections 8.1 (efficiency wage), 8.6 (teams) and 8.7 (multi-
tasking).

 October 4, 2006



\newpage


 \begin{LARGE}


 \noindent
 { \bf 8.1 Efficiency Wages }

Is the  aim of an incentive contract    to punish the agent if he chooses the
wrong action?

 Not exactly.

  Rather, it  is to create a difference between the
agent's expected payoff from right and wrong actions.

That can be
done either with  the  stick  of punishment or the carrot of reward.

\newpage

\begin{center}
{\bf  The  Lucky  Executive  Game}
 \end{center}
  {\bf Players}:
  A corporation and an executive.

\noindent
{\bf The  Order of play}\\
  1   The corporation offers the executive a contract which pays   $w(q) \geq 0$
depending  on profit, $q$. \\
2  The executive accepts  the contract, or rejects it and receives his
reservation utility of $\overline{U} = 5$\\
 3  The executive exerts effort $e$ of either 0 or 10. \\
 4 Nature  chooses profit according to Table 1. \\

\noindent
 {\bf Payoffs}:
Both players   are risk neutral. The corporation's payoff is  $q-w$. The
executive's payoff is $(w-e)$ if he accepts the contract.

   \begin{center}
{\bf  Table 1:  Output in the Lucky Executive Game}

  \begin{tabular}{ l| cc|c  }
 &  \multicolumn{2}{c|}{\bf Probability of Outputs} & \\
{\bf Effort}  &  0 & 400 & Total \\
 & & & \\
  \hline
 & & & \\
 $Low$ ($e=0$) & 0.5 & 0.5 & 1\\
 & & & \\
 $High$ ($e=10$) &0.1 & 0.9 & 1\\
 & & & \\
 \hline
 \end{tabular}
\end{center}


\newpage

{\bf  Table 1:  Output in the Lucky Executive Game}

  \begin{tabular}{ l| cc|c  }
 &  \multicolumn{2}{c|}{\bf Probability of Outputs} & \\
{\bf Effort}  &  0 & 400 & Total \\
 & & & \\
  \hline
 & & & \\
 $Low$ ($e=0$) & 0.5 & 0.5 & 1\\
 & & & \\
 $High$ ($e=10$) &0.1 & 0.9 & 1\\
 & & & \\
 \hline
 \end{tabular}
 


    Since both players are risk neutral, you might think that  the first-best
can be achieved by selling the store, putting the entire risk on the agent.  The
participation constraint  if the executive exerts high effort is
\begin{equation} \label{e1}
0.1[w(0) -10]   +0.9[w(400)-10]  \geq 5,
  \end{equation}
 so his  expected wage must equal 15.

 The incentive compatility constraint is
\begin{equation} \label{e87}
  0.5 w(0) + 0.5 w(400) \leq   0.1 w(0) + 0.9w(400)-10,
\end{equation}
 which can be rewritten as $w(400) - w(0) \geq 25,$  so the gap between the
executive's wage for high output and low output must equal at least 25.

  A contract that satisfies both constraints is $\{w(0)= -345, w(400)= 55\}$.


But this contract is not feasible, because the game requires $w(q) \geq 0$: the 
  {\bf bankruptcy constraint}.

\newpage

  The
participation constraint  if the executive exerts high effort is
\begin{equation} \label{e1}
0.1[w(0) -10]   +0.9[w(400)-10]  \geq 5,
  \end{equation}
 so his  expected wage must equal 15.

The incentive compatility constraint is
\begin{equation} \label{e87}
  0.5 w(0) + 0.5 w(400) \leq   0.1 w(0) + 0.9w(400)-10,
\end{equation}

  What can be done is to  use the carrot instead of the stick and abandon
satisfying the  participation constraint as an equality.

 All that is needed
for constraint (\ref{e87}) is a  gap  of 25 between the high wage and the low
wage.

  Setting  the low wage as low as is feasible,  the corporation can use
the contract  $\{w(0)= 0, w(400)= 25\}$  and   induce high effort.

 The
executive's expected  utility, however, will   be  $0.1(0)+ 0.9(25) - 10 =
12.5$, more than double his reservation utility of 5.

 He is very happy in this
equilibrium-- but the corporation is reasonably happy, too. The corporation's
payoff is  $337.5   (= 0.1 (0-0) + 0.9 (400-25)$, compared with the $195  (= 0.5
(0-5) + 0.5 (400-5))$ it would get if it paid a lower expected wage.




  \newpage



   This discussion should remind you of Section 5.4's Product Quality Game.


There too, purchasers   paid more than the reservation price in order to give
the seller an incentive to behave properly, because a seller who misbehaved
could be punished by termination of the relationship.

 The key characteristics
of  such models are  a constraint on the amount of  contractual punishment for
misbehavior and a participation constraint that  is not binding in equilibrium.


   Repetition allows for a situation in which the agent could
considerably increase his payoff  in one period by misbehavior such as stealing
or low quality  but refrains because  he would lose his position and  lose all
the future efficiency wage payments.


\newpage




 \noindent
 {\bf  *8.6 Joint Production by Many Agents: The Holmstrom Teams Model}



\noindent
{\it A {\bf team} is a group of agents who independently choose effort levels
that result in a single output for the entire group.}





\begin{center}
{\bf  Teams}\\
(Holmstrom [1982])
 \end{center}
 {\bf Players}\\
 A principal and $n$ agents.

\noindent
 {\bf The order of play}\\
 1 The principal offers a contract to each agent $i$ of the form $w_i(q) $,
where $q$ is total output.\\
2 The agents decide whether or not to accept the contract.\\
 3 The agents simultaneously pick effort levels $e_i$, ($i = 1,\dots,n$).\\
 4   Output is $q(e_1,\ldots e_n)$.

 \noindent
 {\bf Payoffs}\\
 If any agent rejects the contract, all payoffs equal zero. Otherwise,
 \begin{center}
\begin{tabular}{ll} $ \pi_{principal}$ &  $   = q - \sum_{i=1}^n w_i; $\\ & \\
$\pi_{i}$ & $=  w_i - v_i(e_i)$, where $v'_i> 0$ and $v''_i > 0$.\\
\end{tabular}
 \end{center}

\newpage



 Despite the risk neutrality of the agents, ``selling the store'' fails to work
here, because the team of agents still has the same problem as the employer had.
The team's problem is cooperation between agents, and the principal is
peripheral.

 \includegraphics[width=150mm]{fig08-04.jpg}

\begin{center}
 {\bf Figure 4:   Contracts in the Holmstrom Teams Model }
 \end{center}


\noindent
 Denote the efficient vector of actions by $e^*$. An efficient contract,
illustrated in Figure 4(a),  is
 \begin{equation} \label{e10}
w_i(q) = \left\{
  \begin{array}{ll} b_i& {\rm if}\; q \geq q(e^*)\\
 & \\
 0 & {\rm if} \;q < q(e^*)\\
  \end{array} \right.
\end{equation}
 where $\sum_{i=1}^n b_i = q(e^*)$ and $b_i > v_i(e^*_i)$.




 Contract (\ref{e10}) gives agent $i$ the wage $b_i$ if all agents pick the
efficient effort,   and nothing if any of them shirks.

 \newpage

\noindent
 {\bf Proposition  1.} { \it If there is a budget-balancing constraint, no
differentiable wage contract $w_i(q)$ generates an efficient Nash equilibrium.}



\noindent
 Agent $i$'s problem is
 \begin{equation}\label{e11}
 \stackrel{Maximize}{e_i} \;\;\;  w_i(q(e)) - v_i(e_i).
 \end{equation}
His first-order condition is
 \begin{equation}\label{e12}
 \left( \frac{dw_i}{dq} \right) \left( \frac{dq}{de_i} \right) - \frac{dv_i}
{de_i} = 0.
\end{equation}
 With budget balancing and a linear utility function, the pareto optimum
maximizes the sum of utilities (something not generally true), so the optimum
solves
 \begin{equation} \label{e13}
 \begin{array}{cl} Maximize & {\displaystyle q(e) - \sum_{i=1}^n v_i(e_i)}\\
  e_1,\ldots, e_n  & \\
\end{array}
\end{equation}



  The first-order condition is that the marginal dollar contribution to output
equal the marginal disutility of effort:
 \begin{equation} \label{e14}
\frac{dq}{d{e_i}} - \frac{d{v_i}}{d{e_i}} = 0.
\end{equation}

Equation (\ref{e14}) contradicts equation (\ref{e12}), the agent's first-order
condition,   because $\frac{dw_i}{dq}$ is not equal to one.



\newpage



 \noindent
 { \bf *8.7  The Multitask Agency Problem}
:
{\bf  Multitasking I: Two Tasks, No Leisure} Holmstrom \& Milgrom (1991)
 
 

\noindent
 {\bf The Order of Play}\\
 1 The principal offers  the agent either an incentive contract of the form
$w(q_1)$ or a monitoring contract  that pays $m$ under which he pays the agent a
base wage of $\overline{m}$ plus $ m_1$   if he observes him working on Task 1
and $ m_2$ if he observes him working on Task 2  (the  $\overline{m}$ base is
superfluous notation in Multitasking I,    but is used in Multitasking II). \\
2 The agent  decides whether or not to accept the contract.\\
 3  The agent picks efforts $e_1$ and $e_2$ for the two tasks such that $e_1+e_2
=1$, where 1 denotes the total time available.\\
 4 Outputs are $q_1(e_1)$ and $q_2(e_2)$, where $\frac{dq_1}{de_1}>0$ and
$\frac{dq_2}{de_2}>0$ but we do not require decreasing returns to effort.



 \noindent
 {\bf Payoffs}:
 If any agent rejects the contract, all payoffs equal zero. Otherwise,
\begin{equation} \label{e17a}
\begin{array}{ll}
  \pi_{principal}  &  =   q_1 + \beta q_2 - m  - w - C   ;\\
 & \\
  \pi_{agent}   &  = m + w - e_1^2 -e_2^2,     \\
  \end{array}
\end{equation}
where $C$,  the cost of monitoring,  is     $\overline{C}$ if a monitoring
contract is used and zero otherwise.

\newpage



 The first best  can be found by  choosing $e_1$ and
$e_2$ (subject to $e_1+e_2 = 1$) and $C$  to maximize the sum of the payoffs,
\begin{equation} \label{e100}
  \pi_{principal}+   \pi_{agent}  =    q_1(e_1) + \beta q_2(e_2)    - C
  - e_1^2 -e_2^2,
\end{equation}

 In the first-best, $C=0$ of course-- no costly  monitoring is needed.

Substituting $e_2= 1-e_1$ and using the first-order condition  for $e_1$
yields
 \begin{equation} \label{e101}
  C^*=0  \;\;\;\;\;  e_1^* = \frac{1}{2} + \left(  \frac{\frac{dq_1}{de_1}  -
\beta \left( \frac{dq_2}{de_2} \right)}{4}  \right)   
  \end{equation}
\begin{equation} \label{e101}
   e_2^* = \frac{1}
{2} -  \left( \frac{\frac{dq_1}{de_1}  -  \beta \left( \frac{dq_2}{de_2} \right)
}{4}   \right).
  \end{equation}

  Thus, which effort should be bigger depends on $\beta$ (a measure of the
relative value of Task 2) and the diminishing returns to effort  in each task.

If, for example, $\beta >1$ so Task 2's output is more valuable and the
functions $q_1(e_1)$ and $q_2(e_2)$ produce the same output for the same effort,
then  from (\ref{e101}) we can see that $e_1^*< e_2^*$, as one would expect.

\newpage

  Can an  incentive contract   achieve  the first best?

Define $q_1^*, q_2^*, e_1^*$ and $e_2^*$ as the first-best levels of those
variables  and  define the minimum wage payment that would induce the agent to
accept a contract requiring the first-best effort as
\begin{equation} \label{e101a}
     w^* \equiv (e_1^*)^2 + (e_2^*)^2
  \end{equation}


What happens with  the profit-maximizing  flat-wage
contract, which could be either the incentive contract $w(q_1) = w^* $  or  the
monitoring contract $\{w^*, w^* \}$?  The agent's effort choice would   be to
split his effort equally between the two tasks, so $e_1=e_2=0.5$.  To satisfy
the participation constraint it  would be necessary that  $\pi_{agent}   = w^*
+w - e_1^2 -e_2^2 \geq 0,$ so $\pi_{agent}   = w^*  - 0.25 -0.25=0$ and $w^*  =
0.5$.

 

What about a sharing-rule incentive contract, in which the wage rises with
output (that is, $ \frac{dw}{dq_1}>0$)?

The principal
must worry about an externality of sorts: the greater the agent's effort on Task
1, the less will be his effort on Task 2.
 Even if extra $e_1$ were
 free, the principal might not want it-- and  might be willing to
pay to stop it.

\newpage

 Consider the simplest sharing-rule contract, the linear one with $ \frac{dw}
{dq_1}=b,$ so $w(q_1) = a + bq_1$.  The agent will pick $e_1$ and $e_2$ to
maximize
\begin{equation} \label{e103}
 \pi_{agent} =  a + bq_1(e_1)- e_1^2 -e_2^2,
\end{equation}
      subject to $e_1+e_2 =1$ (which allows us to rewrite the maximand in terms
of just $e_1$, since $e_2= 1-e_1$).   The first-order condition is
\begin{equation} \label{e104}
     \frac{d\pi_{agent}}{de_1} = b  \left( \frac{dq_1}{de_1} \right) - 2e_1^* -
2(1-e_1^*)  (-1)=0,
\end{equation}
so
\begin{equation} \label{e105}
     e_1^*  = \frac{1}{2}  + \left( \frac{b}{4} \right)  \left( \frac{dq_1}
{de_1} \right).
\end{equation}

  If $e_1^* \geq 0.5$,  the linear contract will work just fine.  

 If  $e_1^* <0.5$,   the linear contract cannot achieve the first best
with a positive value for $b$. Even under a flat wage ($b=0$), the agent will
choose $e_1=0.5$, which is too high.

 If the principal rewards the agent for more
of the observable output $q_1$, the principal will get too little of the
unobservable output $q_2$.  Instead, the contract must actually punish the agent
for high output!

 It must have at least a slightly negative value for $b$, so as
to defeat the agent's preferred allocation of effort evenly across the tasks.

\newpage


     Chapter 7 compared three contracts: linear, threshold, and forcing
contracts.  The threshold contract will work  as well or better than the linear
contract in Multitasking I. It at least does not provide  incentive to go above
the threshold, which is positively bad in this model.

 The forcing contract is
even better, because the principal positively dislikes having $e_1$ be too
great.



  Thus, in equilibrium the principal chooses some contract that elicits the
first-best effort $e^*$, such as the forcing contract,
\begin{equation}  \label{e105a}
\begin{array}{l}
      w(q_1=q_1^*)  = w^*, \\
  \\
  w(q_1  \neq q_1^*) =0.\\
 \end{array}
\end{equation}


\newpage

  A monitoring contract, which would incur monitoring cost $\overline{C}$, is
suboptimal, since an incentive contract can achieve the first-best  anyway, but
let's see how the optimal  monitoring contract would work.

 Let us set
$\overline{m}=0$ in Multitasking I.

   The agent will choose his effort to maximize
\begin{equation} \label{e106}
\begin{array}{ll}
 \pi_{agent}& =   e_1 m_1 + e_2 m_2 - e_1^2 -e_2^2  \\
 & \\
 & =  e_1  m_1 + (1-e_1)  m_2 -  e_1^2 - (1-e_1)^2,  \\
\end{array}
\end{equation}
 since with probability $e_1$ the monitoring  finds him working on Task 1 and
with probability $e_2$ it  finds him on Task 2. Thus,
\begin{equation} \label{e107}
\frac{d \pi_{agent}}{de_1} =     m_1 -m_2 - 2e_1  - 2(-1)(1-e_1) =0
\end{equation}
so  if the principal wants the agent to pick the particular effort $e_1=e_1^*$
that we found in equation (\ref{e101})  he should choose $m_1^*$ and $m_2^*$ so
that
\begin{equation} \label{e108}
    m_1^*  =    4e_1^* +m_2^*  - 2
\end{equation}


If $e_1^*> e_2^*$, which means that $e_1^*>0.5$, equation
(\ref{e108}) tells us that $m_1^* > m_2^*$, just as we would expect.


\newpage

 We have one equation for  the two unknowns  of $m_1^*$ and $m_2^*$ in
(\ref{e108}), so we need to add some information.  Let us  use the fact that if
the participation constraint is satisfied exactly  then we can set the agent's
payoff from (\ref{e106}) equal to zero,
 which is a second equation for our two unknowns.

 After going through the
algebra to solve (\ref{e108})   together with the binding participation
constraint, we get
\begin{equation} \label{e109}
    m_1^*  =   4e_1^* -  2(e_1^*)^2 -1
\end{equation}

 \begin{equation} \label{e110}
\begin{array}{ll}
    m_2^* & =      [4e_1^* -  2(e_1^*)^2 -1] +2 - 4e_1^*\\
 & \\
 & = 1- 2(e_1^*)^2\\
 \end{array}
\end{equation}
 These have the expected property that $\frac{d m_1^*}{de_1^*} = -4e_1^* +4 >0$
and  $\frac{d m_2^*}{de_1^*} =    -4e_1^* <0$.



\newpage

\begin{center}
{\bf  Multitasking II: Two Tasks Plus Leisure}
  \end{center}
 This game is the same as Multitasking I, except that now the agent's effort
budget constraint is  not $e_1+e_2 =1$, but $e_1+e_2 \leq 1$.
 

 Again let us begin with the first best. This can be found by  choosing $e_1$
and    $e_2$ and $C$  to maximize the sum of the payoffs:
\begin{equation} \label{e111}
    q_1(e_1) + \beta q_2(e_2)    - C    - e_1^2 -e_2^2,
\end{equation}
     subject to $e_1+e_2 \leq 1$, the only change in the optimization problem
from Multitasking I.

  We now cannot use the trick of substituting for $e_2$ using the constraint
$e_2=1-e_1$, since it might happen that the   effort budget  constraint is not
binding at the optimum.

 
Maybe $e_1^*+ e_2^*=1$, as in Multitasking I, so that   the
first-best  effort levels are the same as in that game.


But positive leisure for the agent in the first-best, i.e., the   effort budget
constraint being non-binding,  is a realistic case.

\newpage

  In Multitasking I,  a flat wage
led to $e_1=e_2=0.5$.

In Multitasking II, it would lead  to $e_1=e_2=0$, quite a
different result.

   A low-powered incentive contract is disastrous,
because pulling the agent  away from high effort on Task 1 does not leave him
working harder on Task 2.

  A high-powered sharing-rule incentive contract  in which the wage rises with
output performs much better, even though  we cannot reach the first best as we
did in  Multitasking I.

 Since the flat wage leads to $e_2=0$ anyway, adding
incentives for the agent to increase $e_1$ cannot do any harm.

 Effort on Task 2
will remain zero-- so the first-best is unreachable-- but a suitable sharing
rule can lead to $e_1=e_1^*$.

   The combination $(e_1=e_1^*, e_2=0)$ is   the second-best incentive-contract
solution  in Multitasking II, since at $e_1^*$ the  marginal disutility of
effort equals the
marginal utility of the marginal product of effort.

\newpage

 The combination $(e_1=e_1^*, e_2=0)$ is   the second-best incentive-contract
solution  in Multitasking II.

 That conclusion  might be
misleading, though. We have assumed that the disutility of effort on Task I is
separable from the disutility of effort on Task II.

That is why   even if the
agent is devoting no effort to Task II he should not work any harder on Task I.

More realistically, the disutility of effort would be some nonseparable function
$f(e_1, e_2)$ such that the efforts are ``substitute  bads'' and $\frac{d^2f }
{de_1 de_2 } >0$.

In that case, in the second-best the principal, unable to
induce $e_2$ to be positive, would push $e_1$ above the first-best level, since
the agent's marginal disutility of $e_1$ would be less at $(e_1^*,0)$ than at
$(e_1^*, e_2^*)$.



Thus, one lesson of Multitasking II is that if
  an agent has a strong temptation to spend his time on tasks which have no
benefit for the principal,  the situation is much closer to the conventional
agency models than to Multitasking I.

\newpage

  The
 first-best effort levels {\it can} be attained, but it requires a
monitoring contract .



   The  agent will choose his effort to maximize
\begin{equation} \label{e112}
 \pi_{agent} =  \overline{m} +  e_1 m_1 + e_2 m_2    - e_1^2 -e_2^2,
\end{equation}
subject to    $e_1+e_2 \leq 1$.

 Unlike in Multitasking I,  the base wage
$\overline{m}$ matters, since it may happen that the principal monitors the
agent and finds him working on neither Task 1 nor Task 2. The base wage may even
be negative, which can be interpreted as  a bond   for good effort posted by the
agent or as  a fee he pays for the privilege of  filling the job and possibly
earning $m_1$ or $m_2$.


    The principal  will pick      $m_1$ and $m_2$ to induce the agent to choose
$e_1^*$ and $e_2^*$, so he will pick them to  solve the first-order conditions
of the agent's problem for $e_1^*$ and $e_2^*$:
\begin{equation} \label{e113}
\begin{array}{ll}
\frac{\partial \pi_{agent}}{\partial e_1} &= m_1 - 2e_1=0\\
 & \\
 \frac{\partial \pi_{agent}}{\partial e_2} &= m_2 - 2e_2=0\\
 & \\
\end{array}
\end{equation}
 These can be solved to yield  $ m_1 = \frac{e_1^*}{2}$ and   $m_2 =
\frac{e_2^*}{2}$.

\newpage

  We still need to determine the base wage,  $ \overline{m} $.
Substituting into the participation constraint, which will be binding, and
recalling that  we defined the agent's  reservation expected wage as $w^* =
e_1^2 +e_2^2$,
\begin{equation} \label{e113a}
\begin{array}{ll}
\pi_{agent} &=   \overline{m} +  e_1 m_1 + e_2 m_2  - e_1^2 -e_2^2 =0  \\
 & \\
&  \displaystyle{=   \overline{m} +  e_1^* \left( \frac{e_1^*}{2} \right) +
e_2^*\left( \frac{e_2^*}{2}  \right) - w^*  =0} \\
 & \\
&=       \overline{m} +   \left( \frac{1}{2} \right) w^*   - w^*  =0 \\
\end{array}
\end{equation}
 so $ \overline{m} =\frac{w^*}{2}$.

\newpage

$ \overline{m} =\frac{w^*}{2}$

   The base wage is thus positive; even if the principal finds the agent
shirking when he monitors, he will pay him more than zero.

 That is intuitive
when $e_1^* +e_2^* <1$, because then the principal wants the agent to take some
leisure in equilibrium, rather than have to pay him more for a leisureless job.


It is more surprising that  the base wage is positive when $e_1^* +e_2^* =1$;
that is, when  efficiency requires zero leisure. Why pay the agent anything at
all for inefficient behavior?

 The answer is that the base wage is important only for inducing the agent to
take the job and has no influence whatsoever on the agent's choice of effort.

Increasing the base wage does not make the agent more likely to take leisure,
because he gets the base wage regardless of how much time he spends on each
activity.

 If $e_1^* +e_2^* =1$, then the agent chooses zero leisure despite
knowing that he would still receive his base pay for doing nothing, because the
incentive   of $m_1$ and $m_2$ is great enough that he does not want to waste
any opportunity  to get that incentive pay.





\end{LARGE}

\end{document}




