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This is designed for one 75-minute lecture using {\it Games and Information}.

 October 3, 2006

 

\newpage


 \begin{LARGE}
\begin{center}
{\bf  7 Moral Hazard: Hidden Actions}
 \end{center}


  PRINCIPAL-AGENT MODELS

 \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.}

  
 

  \begin{center}
  {\bf  Production Game I: Full Information }
\end{center}
 {\bf Players}\\
The principal and the agent.

\noindent
  {\bf The 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 $\pi_{agent} = U(e,w), U_e<0, U_w>0$ and $
\pi_{principal} = V(q - w), V'>0.$

 
  {\bf reservation utility}, $\bar{U}$ 

\newpage

\begin{center}
  {\bf  Production Game I: Full Information }
\end{center}
 
\noindent
 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   the function   that  makes him just willing to accept the
contract,   so
 \begin{equation} \label{e7.1}
 U(e,  w(e))  = \overline{U}.
 \end{equation}
  Thus, the principal's problem is
 \begin{equation} \label{e7.2}
 \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{e7.3}
 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{e7.4}
 \frac{\partial q }{\partial e} =  \frac{\partial  \tilde{w} } {\partial e}.
\end{equation}
  From  condition (\ref{e7.1}), using the implicit function theorem  (see
section 13.4), we get
  \begin{equation} \label{e7.5}
  \frac{\partial  \tilde{w} }{\partial e}   = -  \left( \frac{\frac{\partial  U
}{\partial e} }{ \frac{\partial  U }{\partial \tilde{w}}}\right).
 \end{equation}
  Combining   equations  (\ref{e7.4}) and (\ref{e7.5}) yields
\begin{equation} \label{e7.6}
 \left(  \frac{\partial U}{\partial  \tilde{w}} \right)  \left(  \frac{\partial
q }{\partial e} \right)  = -  \left(  \frac{ \partial  U }{\partial e} \right).
  \end{equation}
  

\newpage

 
\includegraphics[width=180mm]{fig07-02.jpg}


\begin{center} {\bf Figure 2:   The Efficient Effort Level in Production Game I}
\end{center}

Under   perfect competition  among the principals the  profits are zero, so the
reservation utility, $\overline{U}$, will be at  the level such that at the
profit-maximizing effort $e^*$, $\tilde{w}(e^*) = q(e^*)$, or
\begin{equation} \label{e7.7}
 U(e^*, q(e^*)) = \overline{U}.
 \end{equation}
 The principal selects the point on the $U= \overline{U}$ indifference curve
that maximizes his profits,   at effort $e^*$ and wage $w^*$.

\newpage


\includegraphics[width=100mm]{fig07-03.jpg}



The principal must then
design a contract that will induce the agent to choose this effort level. The
following three contracts  are equally effective under full
information.

\noindent
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 $e=e^*$.

\noindent
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 this
contract,  plus a bonus if effort reaches $e^*$. Since the agent dislikes
effort, the agent will choose exactly $e=e^*$.

\noindent
3 The  {\bf  linear contract}   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 $e^*$.   

\newpage

 Let's now fit out Production Game I with specific functional forms.  Suppose
the agent   exerts  effort $e \in [0, \infty]$, and  output equals
\begin{equation} \label{e7.7a}
  q(e)= 100*log(1+e),
 \end{equation}
 so $q'= \frac{100}{1+e} >0$ and  $q''= \frac{-100}{(1+e)^2} <0$.  If  the agent
rejects the contract, let $\pi_{agent} = \bar{U}=3$ and $ \pi_{principal} = 0$,
whereas if  the agent accepts the contract, let $\pi_{agent} = U(e,w)= log(w) -
e^2$ and $ \pi_{principal} = q(e) - 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 wage that makes the agent willing to
participate, i.e., as in equation (\ref{e7.1}),
\begin{equation} \label{e1q}
 U(e,  w(e))  = \overline{U},  \;\;\;   so \;\;log (\tilde{w}(e))  - e^2 = 3.
\end{equation}
 Knowing  the particular functional form as we do, we can solve (\ref{e1q}) for
the wage function:
\begin{equation} \label{e1qq}
 \tilde{w}(e) =   Exp(3+e^2),
  \end{equation}
 where we use $Exp(x)$ to mean Euler's constant (about 2.718) to the power $x$,
since the conventional notation of $e^x$ would be confused with $e$ as effort.


 Equation (\ref{e1qq}) makes sense.  As effort rises, the wage must rise to
compensate, and rise more than exponentially if utility is to be kept equal to
3.

\newpage

\noindent
 Now that we have a  necessary-wage function $\tilde{w}(e )$, we can attack the
principal's problem, which  is
 \begin{equation} \label{e1qqq}
 \begin{array}{cl}
 Maximize&  V(q(e) - \tilde{w}(e ))=  100 *log (1+e) - Exp(3+e^2)\\
 e & \\
\end{array}
 \end{equation} The first-order condition for this problem is \begin{equation}
\label{e7.1s}
 V'(q(e) - \tilde{w}(e )) \left(\frac{\partial q }{\partial e} - \frac{\partial
\tilde{w} }{\partial e} \right)=0,
\end{equation}
 so for our problem,
 \begin{equation} \label{e7.1q}
 \left(  \frac{100}{1+e} \right)  - 2e(Exp(3+e^2)) =0,
 \end{equation}
which cannot  be solved analytically.   Using the computer program Mathematica,
I found that    $e^* \approx 0.77$, from which, using the formulas above, we get
$q^*  \approx   57$    and $w^* \approx  37$. The payoffs are $\pi_{agent} = 3$
and $\pi_{principal} \approx 20.$

If $\overline{U}$ were high enough, the principal's payoff would be zero.  If
the market for agents were competitive, this is what would happen, since the
agent's reservation  payoff would be the utility of  working for another
principal instead of $\overline{U}=3$.

\newpage

\includegraphics[width=200mm]{fig07-03.jpg}


\begin{center}
 {\bf Figure 3:  Three Contracts that Induce Effort $e^*$ for Wage $w^*$}
\end{center}


 To obtain $e^*=0.77$, a number of styles of contract  could be used.

1 The  {\bf forcing contract}  sets $w(e^*) = w^*$ and $w(e \neq 0.77) = 0$.
Here,  $w(0.77) = 37$ (rounding up) and $w(e \neq e^*) =0$.

2  The  {\bf threshold contract}  sets $w(e \geq e^*) = w^*$ and $w(e <e^*) =0$.
Here,  $w(e \geq 0.77) = 37$ and $w(e <0.77) =0$.


\newpage

\includegraphics[width=200mm]{fig07-03.jpg}


\begin{center}
 {\bf Figure 3:  Three Contracts that Induce Effort $e^*$ for Wage $w^*$}
\end{center}


3 The  {\bf  linear contract}     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 slope of
that indifference curve is the derivative of the  $\tilde{w}(e)$ function, which
is
 \begin{equation} \label{e1sss}
 \frac{\partial  \tilde{w}(e)}{\partial e}  =    2e* Exp(3+e^2).
 \end{equation}
 At   $e^*=0.77$, this takes the value 56 (which only coincidentally is near the
value of $q^*=57$).   That is the $\beta$ for the linear contract. The $\alpha$
must solve $w(e^*) = 37 = \alpha  + 56  (0.77)$, so $\alpha = -7$.

 We ought to be a little concerned as to whether the agent will choose the
effort we hope for if he is given the   linear contract.  We constructed it so
that he would be willing to accept the contract, because if he chooses $e=
0.77$, his
utility will be  3. But might he prefer to choose some larger or smaller $e$ and
get even more utility? No, because his utility is concave.   That  makes the
indifference curve convex, so its slope is always increasing  and  no preferable
indifference curve touches the equilibrium contract line.

\newpage

\noindent
{\bf Quasilinearity  and Alternative Functional Forms for the Production Game}

Consider the following three functional forms for utility:
\begin{equation} \label {e400}
\begin{array}{ll}
  U(e, w) & = log(w) - e^2 \;\;\;\;\;\;\;  \;\;\;\;\;\;\; \;\;\;\;\;\;\;(a) \\
 & \\
   U(e, w) & = w - e^2 \;\;\;\;\;\;\;  \;\;\;\;\;\;\;  \;\;\;\;\;\;\;
\;\;\;\;\;\;\;(b) \\
 & \\
 U(e, w) & = log(w  - e^2) \;\;\;\;\;\;\;  \;\;\;\;\;\;\; \;\;\;\;\;\;\;(c) \\
\end{array}
\end{equation}
  Utility function (a) is what we just used in Production Game I. Utility
function (b) is an example of {\bf quasilinear preferences}, because   utility
is separable in  one good--- money, here---  and  linear in that good. This kind
of utility function is commonly used to avoid wealth effects that would
otherwise occur in the interactions among the various goods in the utility
function. Separability  means that giving an agent a higher wage does not, for
example,  increase his marginal disutility of effort.  Linearity means
furthermore that giving an agent a higher wage does not change his tradeoff
between money and effort, his marginal rate of substitution, as it would in
function (a), where a richer agent is less willing to accept money for higher
effort. In effort-wage diagrams, quasilinearity implies that the indifference
curves are parallel along the effort axis (which they are {\it not} in Figure
2).

\newpage

 

  If utility is quasilinear, the  efficient effort level is independent of which
side has the bargaining power   because the gains from efficient production are
independent of how those gains are distributed so long as each party has no
incentive to abandon the relationship. This as the same lesson as   the Coase
Theorem's:,   under   general conditions  the activities undertaken will be
efficient and  independent of the distribution of property rights (Coase
[1960]). This property of the efficient-effort level means that the modeller is
free to make the assumptions on bargaining power that  help  to focus attention
on the information problems he is studying.

   There are thus three reasons why modellers so often use take-it-or-leave-it
offers. The first  two reasons were discussed earlier in the context of
Production Game I: (1) such offers are  a good way to model competitive markets,
and (2) if the reservation payoff of the player without the bargaining power is
set high enough, such offers lead to the same outcome as would be reached if
that player had more bargaining power. Quasi-linear utility provides a third
reason: (3) if utility is quasi-linear, the optimal effort level   does not
depend on who has the bargaining power, so the modeller is justified in choosing
the simplest model of bargaining.



\newpage

 
\begin{equation} \label {e400}
\begin{array}{ll}
  U(e, w) & = log(w) - e^2 \;\;\;\;\;\;\;  \;\;\;\;\;\;\; \;\;\;\;\;\;\;(a) \\
 & \\
   U(e, w) & = w - e^2 \;\;\;\;\;\;\;  \;\;\;\;\;\;\;  \;\;\;\;\;\;\;
\;\;\;\;\;\;\;(b) \\
 & \\
 U(e, w) & = log(w  - e^2) \;\;\;\;\;\;\;  \;\;\;\;\;\;\; \;\;\;\;\;\;\;(c) \\
\end{array}
\end{equation}

  Quasilinear utility functions most often are chosen to look like (b), but what quasilinearity really requires is
just linearity in the special good ($w$ here)  for some monotonic transformation
of the utility function.

 Utility function (c) is   a logarithmic transformation of (b), which is a
monotonic transformation, so it too is quasilinear. 
 That is because marginal
rates of substitution, which is what matter  here,  are a feature of general
utility functions, not the Von Neumann-Morgenstern functions we typically use.


Thus, utility function (c) is also a quasi-linear function, because it is just a
monotonic function of (b).  



\newpage



\begin{equation} \label {e400}
\begin{array}{ll}
  U(e, w) & = log(w) - e^2 \;\;\;\;\;\;\;  \;\;\;\;\;\;\; \;\;\;\;\;\;\;(a) \\
 & \\
   U(e, w) & = w - e^2 \;\;\;\;\;\;\;  \;\;\;\;\;\;\;  \;\;\;\;\;\;\;
\;\;\;\;\;\;\;(b) \\
 & \\
 U(e, w) & = log(w  - e^2) \;\;\;\;\;\;\;  \;\;\;\;\;\;\; \;\;\;\;\;\;\;(c) \\
\end{array}
\end{equation}



  Returning to the solution of  Production Game I, let us now use a different
approach to get to the same answer as we did using the principal's maximization
problem   (\ref{e1qqq}).  Instead, we will return to the general optimality
condition(\ref{e7.6}), here repeated.
$$
 \left(  \frac{\partial U}{\partial  \tilde{w}} \right)  \left(  \frac{\partial
q }{\partial e} \right)  = -   \frac{ \partial  U }{\partial e}
\;\;\;\;\;\;\;\;\;\;\;\;   (\ref{e7.6})
$$
  For any of our three utility functions we will continue using the same output
function   $q(e)= 100*log(1+e)$ from  (\ref{e7.7a}), which has the first
derivative
$q'= \frac{100}{1+e}$.


\noindent
Using utility function (a),  $\frac{\partial U}{\partial  \tilde{w}}= 1/w$.  and
$\frac{\partial U}{\partial e}= -2e$, so equation (\ref{e7.6}) becomes
\begin{equation}   \label{e401}
 \left(  \frac{1}{w} \right)  \left(  \frac{100}{1+e}  \right)  = -    (-2e).
  \end{equation}
  If we  substitute  for $w$ using the function  $\tilde{w}(e) =   Exp(3+e^2)$
that we found in equation (\ref{e1qq}), we get essentially the same equation as
(\ref{e7.1q}), and so outcomes are the same---
  $e^* \approx 0.77$,   $q^*  \approx   57$ ,    and $w^* \approx  37$,
$\pi_{agent} = 3$,  and $\pi_{principal} \approx 20.$




\newpage



\begin{equation} \label {e400}
\begin{array}{ll}
  U(e, w) & = log(w) - e^2 \;\;\;\;\;\;\;  \;\;\;\;\;\;\; \;\;\;\;\;\;\;(a) \\
 & \\
   U(e, w) & = w - e^2 \;\;\;\;\;\;\;  \;\;\;\;\;\;\;  \;\;\;\;\;\;\;
\;\;\;\;\;\;\;(b) \\
 & \\
 U(e, w) & = log(w  - e^2) \;\;\;\;\;\;\;  \;\;\;\;\;\;\; \;\;\;\;\;\;\;(c) \\
\end{array}
\end{equation}



  Returning to the solution of  Production Game I, let us now use a different
approach to get to the same answer as we did using the principal's maximization
problem   (\ref{e1qqq}).  Instead, we will return to the general optimality
condition(\ref{e7.6}), here repeated.
$$
 \left(  \frac{\partial U}{\partial  \tilde{w}} \right)  \left(  \frac{\partial
q }{\partial e} \right)  = -   \frac{ \partial  U }{\partial e}
\;\;\;\;\;\;\;\;\;\;\;\;   (\ref{e7.6})
$$
  For any of our three utility functions we will continue using the same output
function   $q(e)= 100*log(1+e)$ from  (\ref{e7.7a}), which has the first
derivative
$q'= \frac{100}{1+e}$.



Using utility function (b),  $\frac{\partial U}{\partial  \tilde{w}}= 1 $   and
$\frac{\partial U}{\partial e}= -2e$, so equation (\ref{e7.6}) becomes
\begin{equation}   \label{e404}
 \left(   1 \right)  \left(  \frac{100}{1+e}  \right)  = -    \left(  -2e
\right)
  \end{equation}
  Notice that $w$ has disappeared. The optimal effort no longer depends on the
agent's wealth. Thus, we don't need to use the wage function to solve for the
optimal effort. Solving  directly, we get
 $e^* \approx 6.59$ and    $q^*  \approx  203$.   The wage function will be
different now, solving $ w-e^2 =3$, so $w^* \approx  43$,  $\pi_{agent} = 3$,
and $\pi_{principal} \approx 160.$   (These numbers are not really comparable to
when we used utility function (a), but they will be useful in Production Game
II.)



\newpage



\begin{equation} \label {e400}
\begin{array}{ll}
  U(e, w) & = log(w) - e^2 \;\;\;\;\;\;\;  \;\;\;\;\;\;\; \;\;\;\;\;\;\;(a) \\
 & \\
   U(e, w) & = w - e^2 \;\;\;\;\;\;\;  \;\;\;\;\;\;\;  \;\;\;\;\;\;\;
\;\;\;\;\;\;\;(b) \\
 & \\
 U(e, w) & = log(w  - e^2) \;\;\;\;\;\;\;  \;\;\;\;\;\;\; \;\;\;\;\;\;\;(c) \\
\end{array}
\end{equation}



  Returning to the solution of  Production Game I, let us now use a different
approach to get to the same answer as we did using the principal's maximization
problem   (\ref{e1qqq}).  Instead, we will return to the general optimality
condition(\ref{e7.6}), here repeated.
$$
 \left(  \frac{\partial U}{\partial  \tilde{w}} \right)  \left(  \frac{\partial
q }{\partial e} \right)  = -   \frac{ \partial  U }{\partial e}
\;\;\;\;\;\;\;\;\;\;\;\;   (\ref{e7.6})
$$
  For any of our three utility functions we will continue using the same output
function   $q(e)= 100*log(1+e)$ from  (\ref{e7.7a}), which has the first
derivative
$q'= \frac{100}{1+e}$.


Using utility function (c),  $\frac{\partial U}{\partial  \tilde{w}}= 1/(w-e^2)
$  and  $\frac{\partial U}{\partial e}= -2e/(w-e^2)$, so equation (\ref{e7.6})
becomes
\begin{equation}   \label{e402}
 \left(  \frac{1}{w-e^2} \right)  \left(  \frac{100}{1+e}  \right)  = -
\left(  \frac{-2e}{w-e^2} \right)
  \end{equation}
 and with a little simplification,
\begin{equation}   \label{e403}
   \frac{100}{1+e}     =    2e.
  \end{equation}
The variable  $w$  has again disappeared, so   as with utility function (b)  the
optimal effort does not  depend on the agent's wealth.   Solving  for the
optimal effort yields
 $e^* \approx 6.59$  and    $q^*  \approx  203$, the same as with utility
function (b).   The wage function is different, however. Now it   solves $log
(w-e^2)=3$, so $w =e^2+ exp(3)$  and $w^* \approx  63$,  $\pi_{agent} = 3$,  and
$\pi_{principal} \approx 140.$



\newpage

\noindent
 {\bf Production Game III: A Flat Wage Under Certainty}

 In this version of the game, the principal can condition the wage neither on
effort nor on output. This is modelled as a principal who observes neither
effort nor output, so information is asymmetric.
  In Production Game III, we have finally reached ``moral hazard''.

 

\bigskip
  \noindent
 {\bf Production Game IV: An Output-Based Wage under Certainty}

In this version, the principal cannot observe effort but he can observe output
and specify the contract to be $w(q)$.

Unlike in  Production Game III,   the principal now  picks not a number $w$
but a function $w(q)$. 

 

 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

 
 \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 $f(\theta)$.
 


 Because of the uncertainty about the state of the world, effort does not map
cleanly onto   observed output in Production Game  V. A given output   might
have been produced by any of several different effort levels, so a forcing
contract based on output will not necessarily achieve the desired effort.
Unlike   in Production Game IV, here  the principal cannot deduce   $e  \neq
e^*$  from   $q \neq q^*$. 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 $\bar{U}$ so he will accept the contract, and if he  is
sometimes paid a low wage because output happens not to equal $q^*$ despite his
correct effort, 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 against risk.

 Put more technically, moral hazard  is a problem when $q(e)$ is not a one-to-
one function and  a single value of $e$ might result in any of a number of
values of $q$, depending on the value of $\theta$.  In this case the output
function is not invertible; knowing $q$, the principal cannot deduce the value
of $e$ perfectly without assuming equilibrium behavior on the part of the agent.

  
 \noindent {\it A {\bf first-best contract} achieves the same allocation as the
contract that is 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.}

    
 \newpage

\noindent
  {\bf 7.3 The Incentive Compatibility and  Participation 
Constraints  }

  The principal's objective 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 the agent an incentive to choose the desired effort.
These two constraints arise 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{e7.11}
 \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)))$& (incentive
compatibility constraint)  (\ref{e7.11}a) \\
  & \\ & $  EU(\tilde{e}, w(q(\tilde{e},\theta))) \geq \bar{U}$ & (participation
constraint)  (\ref{e7.11}b)\\
  & &  \\
  \end{tabular}

   

  To support the effort level $e$, the wage contract $w(q)$ must satisfy the
incentive compatibility and participation constraints. Mathematically,  the
problem of finding the least cost $C(\tilde{e})$ of supporting the effort level
$\tilde{e}$ combines steps one and two.
 \begin{equation} \label{e7.12}
 \begin{array}{lcl}
  C(\tilde{e}) =& Minimum& Ew(q(\tilde{e},\theta))\\
 & w(\cdot) &
 \end{array}
  \end{equation}
 subject to constraints (\ref{e7.11}a) and (\ref{e7.11}b).

 Step three takes the principal's problem of maximizing his payoff, expression
(\ref{e7.11}), and restates it as
 \begin{equation} \label{e5}
  \begin{array}{cl}
 Maximize& E V( q(\tilde{e},\theta) - C(\tilde{e})).\\
  \tilde{e}
& \end{array}
  \end{equation}
  After finding which contract most cheaply induces each effort, the principal
discovers the optimal effort by solving problem (\ref{e5}).

  

\newpage

\noindent
 {\bf 7.1 Categories of Asymmetric Information Models}

 \noindent
It used to be that the economist's   first response to  peculiar behavior which
seemed to contradict basic  price theory was ``It must be some kind of price
discrimination.''  Today, we have a new answer: ``It must be some kind of
asymmetric information.'' In a game of asymmetric information, player Smith
knows something that player Jones does not. This covers a broad range of models
(including price discrimination itself), so  it is not surprising that so many
situations come under its rubric. We will look at them in five chapters.

  \noindent
  {\bf Moral hazard with hidden actions} (Chapters 7 and 8)  \\
 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
  {\bf Adverse selection} (Chapter 9) \\
Nature begins the game by choosing Smith's type, unobserved by Jones. Smith and
Jones then agree to a contract.  Information is incomplete.

 \noindent   {\bf  Mechanism design} in adverse selection and  {\bf post-
contractual hidden knowledge }) (Chapter 10)  \\
 Jones is designing a contract for Smith designed to elicit Smith's private
information.  This may happen under adverse selection--- in which case Smith
knows the information prior to contracting--- or post-contractual hidden
knowledge (also called moral hazard with hidden information)---in which case
Smith will  learn it  after contracting.

\noindent   {\bf Signalling } and {\bf Screening} (Chapter 11) \\
  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.

\includegraphics[width=100mm]{fig07-01.jpg}



\begin{center}
{\bf Figure 1:   Categories of Asymmetric Information Models }\end{center}

 \end{LARGE}


\newpage

\noindent
  {\bf Production Game II: Full Information. Agent Moves First.}

 In this  version,   every move is common knowledge and the contract is a
function $w(e)$.  The order of play, however, is now as follows

\noindent
{\bf The 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 $q'>0$.

   Now  the agent has all the bargaining power, not the principal. Thus, instead
of requiring that the contract be at least barely acceptable to the agent, our
concern is that the contract be at least barely acceptable to the  principal,
who must   earn zero profits  so $q(e) - w(e) \geq 0$. 


\newpage

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 this 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{e7.8}
\begin{array}{cl}
 Maximize&  U(e, q(e))\\
e & \\
 \end{array}
 \end{equation}
 The first-order condition is
  \begin{equation} \label{e7.9}
 \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 (\ref{e7.9}) implies that
  \begin{equation} \label{e7.10}
 \left( \frac{\partial U}{\partial w } \right)  \left( \frac{\partial q }
{\partial e}\right)  = -  \left( \frac{ \partial U }{\partial e} \right).
\end{equation}

   \newpage

We can see the wealth effect by solving out optimality equation (\ref{e7.10})
for  the specific functional forms of Production Game I from
 expression  (\ref{e400}).

\noindent
Using utility function (a) from   expression  (\ref{e400})
\begin{equation}   \label{e401a}
 \left(  \frac{1}{w} \right)  \left(  \frac{100}{1+e}  \right)  = -    (-2e).
  \end{equation}
  That is the same as in Production Game I, equation (\ref{e401}), but now $w$
is different. It is not found by driving the agent to his reservation payoff,
but by driving the principal to zero profits: $ w=q$.  Since $q= 100 *log(1+e)$,
we can  substitute that in for $w$ to get
 \begin{equation}   \label{e401b}
 \left(  \frac{1}{ 100* log(1+e) } \right)  \left(  \frac{100}{1+e}  \right)  =
2e.
  \end{equation}
  When solved numerically, this yields $e^* \approx 0.63$, and thus $q=w \approx
49$, and $\pi_{principal} = 0$ and $\pi_{agent}  \approx3.49$. In Production
Game I, the optimal effort using this utility function was 0.77 and the agent's
payoff was 3. The difference arises because there the agent's wealth was lower
because the principal had the bargaining power. In Production Game II the agent
is, in effect, wealthier, and   since his marginal utility of money is lower, he
chooses to convert some (but not all) of that  extra wealth into what we might
call leisure--- working less hard.


Using  the quasilinear utility functions (b)  and  (c) from   expression
(\ref{e400}), recall that both have the same optimality condition, the one we
found in equations (\ref{e404}) and (\ref{e403}):
$$
   \frac{100}{1+e}     =    2e  \;\;\;\;\;\;\;\;\;\;\;\;   (\ref{e403})
$$
As we observed before, $w$ does not appear in equation (\ref{e403}), so the wage
equation does not matter to $e^*$. But that means that in Production Game II,
$e^* \approx 6.59$ and $q^*   \approx  203$, just as  in Production Game I. With
quasilinear utility, the efficient action does not depend on bargaining power.
Of course, the wage and payoffs do depend on who has the bargaining power. In
Production Game II,   $w^*=q^*   \approx  203$,    and $\pi_{principal} =0.$
The agent's payoff  is higher than in Production Game I, but it differs, of
course, depending on the payoff function. For utility function (b) it is
$\pi_{agent}  \approx 160$  and for utility function (c) it is $\pi_{agent}
\approx 5.08$.

\end{document}



