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\begin{document}

\title{Topics in Contracts and Organizations:\\
Lecture Notes}
\author{Steve Tadelis \\
%EndAName
UC Berkeley}
\date{September 2007}
\maketitle
\tableofcontents

\chapter{Reputational Incentives}

\section{Introduction}

\begin{itemize}
\item Explicit contracts are a useful tool to generate incentives, but they
must be enforced for them to be effective. When such enforcement
institutions are not available, these contracts cannot work.

\item A substitute to formal explicit contracts is the concern individuals
have for their \emph{reputation}.

\item The ``economics of reputation'' can be summarized by: \emph{perform
well today and receive more tomorrow, perform poorly today and receive less
tomorrow}.

\item There are many ways one can model this simple idea, and we will
discuss several modelling technologies, and the applications that motivate
them.
\end{itemize}

\section{Career Concerns}

\subsection{Introduction}

An important question regarding incentives is whether explicit incentives
are at all needed to induce agents to work hard. Fama (1980) argues that
market forces will cause agents to exert high levels of effort because their
future prospects will depend on their past performance. In particular,
agents will work hard earlier in their careers to get a ``reputation'' for
being very productive, which in turn will yield higher future wages through
the competitive market for (managerial) labor. The bottom line of Fama's
argument is that market forces create strong incentives to perform, and thus
explicit contracts may not be needed, or at least may not be too important.

An attempt to model Fama's informal argument was made by Holmstrom (1999)
who investigates more carefully when Fama's conjectures would indeed hold,
and how general they may be. The first model sets out to see how the
competitive labor market induces reputational (career) concerns that in turn
provide effort incentives. Conditions under which optimal (first best)
incentives arise in equilibrium are provided, as well as conditions for
departures from first best. Comparative statics and effort dynamics are also
analyzed. The second model is based on the last section of Holmstrom (1999)
and is further developed in Holmstrom and Ricart-I-Costa (1986) and looks at
how career concerns distort the choices of managers who have to choose
investment projects for the firms they work in. This is a very interesting
take on the dissonance between the real value of investments (the value to
the firm) and the reputational value of investments (that the manager cares
about).

\subsection{Work Incentives through Reputational Concerns}

Consider an infinitely lived agent (manager) who is endowed with labor, and
can choose in every period $t=1,2,...$ to provide effort $a_{t}\geq 0$ at a
private cost $g(a_{t})$. Assume that $g^{\prime }(\cdot )>0$ and $g^{\prime
\prime }(\cdot )>0.$

Assume that the manager is risk neutral, and that his utility is separable
in effort and wages, and is also time separable so that there is no role for
consumption smoothing. Thus, given a stream of wages $w=\{w_{1},w_{2},...\}$
and of effort levels $a=\{a_{1},a_{2},...\},$ the manager's utility is given
by: 
\begin{equation*}
U(w,a)=\sum_{t=1}^{\infty }\beta ^{t-1}[w_{t}-g(a_{t})]\,.
\end{equation*}

To model the production technology assume that the output a manager can
produce in period $t$ is given by 
\begin{equation}
y_{t}=\eta +a_{t}+\varepsilon _{t\;},  \label{y}
\end{equation}%
where $\varepsilon _{t}\sim N(0,\frac{1}{h_{\varepsilon }})$ is a normally
distributed noise parameter with mean 0 and precision $h_{\varepsilon }$
(precision is defined as the inverse of variance). The manager's ability (or
talent) is $\eta \sim N(m_{1},\frac{1}{h_{1}}),$ and it is determined ex
ante and fixed (at this stage of the model) over time. It is also normally
distributed with mean $m_{1}$ and precision $h_{1}$. This technology
captures the idea that output is random, and depends both on a manager's
innate talent as well as on the effort he exerts.

We assume that contingent contracts are not feasible. That is, the manager
cannot be employed using a contract that ties his wages to performance. We
can therefore without loss restrict attention to markets in which managers
are paid in advance for their services. In particular, the manager's wage in
period $t$ is $w_{t}$ and can depend \emph{only implicitly} on past
performance. Our object of analysis will be the choice of the manager's
stream of actions given his beliefs about how his current actions affect his
future prospects.

The labor market is modeled as a competitive market in which spot wages are
bid up to the expected product that a manager will provide. As mentioned
earlier, the manager will choose his efforts taking into account how wages
will respond to effort. In turn, wages themselves will be determined given
the belief of \textquotedblleft the market\textquotedblright\ about the
manager's effort levels. Thus, an equilibrium must have \textit{rational
expectations} of the market about the manager's effort levels, and these
effort levels must be a best response of the manager to the market's correct
beliefs and the belief updating process. In particular, at time $t,$ given a
past sequence of outcomes $y^{t-1}=(y_{1},y_{2},...,y_{t-1}),$ and given the
(correct) beliefs about the effort choice $a_{t}(y^{t-1}),$ the market will
set wages as follows: 
\begin{equation}
w_{t}(y^{t-1})=E[y_{t}|y^{t-1}]=E[\eta |y^{t-1}]+a_{t}(y^{t-1})\;.
\label{wages}
\end{equation}%
Then, in equilibrium, the actions of the manager will be a best response to
the wage process, which means that the sequence $\{a_{t}(y^{t-1})\}_{t=1}^{%
\infty }$ will solve, 
\begin{equation}
\underset{\{a_{t}(y^{t-1})\}_{t=1}^{\infty }}{\max }\;\sum_{t=1}^{\infty
}\beta ^{t-1}[Ew_{t}(y^{t-1})-Eg(a_{t}(y^{t-1}))]\;,  \label{effort}
\end{equation}%
where the expectations over the cost of effort is due to the fact that $%
y^{t-1}$ is a random variable ex ante.

Thus, an equilibrium to this infinite horizon model is a sequence of path
dependent efforts $\{a_{t}^{*}(y^{t-1})\}_{t=1}^{\infty }$ and a sequence of
path dependent wages $\{w_{t}(y^{t-1})\}_{t=1}^{\infty }$ such that given $%
\{a_{t}^{*}(y^{t-1})\}_{t=1}^{\infty }$, the sequence of wages is correctly
calculated, and the sequence $\{a_{t}^{*}(y^{t-1})\}_{t=1}^{\infty }$ is a
best response to the sequence of wages. To solve for the equilibrium it is
instructive to start with a two period model.

\subsubsection{A Two Period Model}

Clearly, since wages are paid in advance, the manager will have no incentive
to exert any effort in the second period. What, then, induces the manager to
exert effort in the first period? To see this consider the stochastic
process of output. In equilibrium, $a_{1}^{\ast }$ is correctly anticipated,
so that \emph{in equilibrium} the market infers 
\begin{equation*}
z_{1}\equiv \eta +\varepsilon _{1}=y_{1}-a_{1}^{\ast }\;,
\end{equation*}%
and through this inference the market learns about the talent parameter $%
\eta $ by calculating the conditional (posterior) expectation $E[\eta
|z_{1}] $. Since both $\eta $ and $\varepsilon _{1}$ are Normal, it is well
known that the posterior of $\eta $ is Normal with $[\eta |z_{1}]\sim
N(m_{2},\frac{1}{h_{2}})$, where\footnote{%
This is a well known derivation that appears in dynamic rational
expectations models. It seems to have been introduced into economics by
Jovanovic (1979) \textquotedblleft Job Matching and the Theory of
Turnover,\textquotedblright\ \textit{JPE} \textbf{87(5)}:972-989, and a
closer model to the one in this paper without moral hazard was developed by
Harris and Holmstrom (1982) \textquotedblleft A Theory of Wage
Dynamics,\textquotedblright\ \textit{ReStud} \textbf{49}:315-333. For a
reference to the actual derivation of this Bayes updating see M. DeGroot
(1984) \textit{Probability and Statistics}, Addison-Wesley, Proposition 3 on
page 324.} 
\begin{equation}
m_{2}(z_{1})=\frac{h_{1}}{h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{%
h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\cdot z_{1}\;,  \label{m2}
\end{equation}%
and 
\begin{equation}
h_{2}=h_{1}+h_{\varepsilon }\;.  \label{h2}
\end{equation}

Clearly, given the equilibrium effort $a_{1}^{*}$, the first period wage
must be, 
\begin{equation*}
w_{1}=Ey_{1}=E[\eta +a_{1}^{*}+\varepsilon _{1}]=m_{1}+a_{1}^{*}\;.
\end{equation*}
To solve for the equilibrium effort $a_{1}^{*},$ we can use (\ref{m2}) to
rewrite (\ref{wages}) for the second period, 
\begin{eqnarray*}
w_{2}(y_{1}) &=&m_{2}(z_{1})+a_{2}^{*} \\
&=&m_{2}(z_{1})
\end{eqnarray*}
where the second equality follows trivially from the fact that $a_{2}^{*}=0$%
. Thus, given the market beliefs about first period effort, $a_{1}^{*}$, the
expected second period wage the manager expects to receive ex ante is, 
\begin{eqnarray}
E[w_{2}(y_{1})] &=&E[m_{2}(z_{1})]  \notag \\
&=&\frac{h_{1}}{h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }}{%
h_{1}+h_{\varepsilon }}\cdot \underset{Ez_{1}}{\underbrace{(\overset{Ey_{1}}{%
\overbrace{m_{1}+a_{1}}}-a_{1}^{*})}}\,.  \label{Ew2}
\end{eqnarray}

We can now insert (\ref{Ew2}) into the agent's problem (\ref{effort}) for
the two period case, and we obtain: 
\begin{equation}
\underset{a_{1}}{\max }\;m_{1}+a_{1}^{\ast }-g(a_{1})+\beta \cdot \left[ 
\underset{E[w_{2}(y_{1})]}{\underbrace{\frac{h_{1}}{h_{1}+h_{\varepsilon }}%
\cdot m_{1}+\frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\cdot
(m_{1}+a_{1}-a_{1}^{\ast })}}\right] \;,  \label{Eu2}
\end{equation}%
which yields the first order condition (FOC), 
\begin{equation}
g^{\prime }(a_{1}^{\ast })=\beta \cdot \frac{h_{\varepsilon }}{%
h_{1}+h_{\varepsilon }}\in (0,1)\;.  \label{FOC1}
\end{equation}%
Under standard Inada assumptions on $g(\cdot )$, this FOC implies that some
effort will be exerted in the first period. Furthermore, from the convexity
of $g(\cdot ),$ there are simple comparative statics regarding the
equilibrium effort level: it is increasing in $\beta $ (the future is more
important), increasing in $h_{\varepsilon }$ (the noise has less influence),
and decreasing in $h_{1}$ (ability is more noisy).

Intuitively, the incentives are provided through reputational concerns: if
everyone believed that the manager is not exerting effort in the first
period, then the outcome of the first period would be a direct signal of the
manager's ability. With these market beliefs, engaging in some effort will
cause the market to pay a higher wage in the second period since the added
output will be attributed to ability. This in turn gives incentives to
engage in effort, and equilibrium considerations must make the effort
correctly inferred by the market. Thus, the manager is using effort to bias
his ability upwards, but this is not fooling the market --- the rational
expectations equilibrium solution means that this effort is anticipated, and
lack of effort will (in expectations) cause the market to infer poorly about
the manager's ability.

As for the comparative statics, the results are also intuitive. Discounting
obviously makes the reputational returns to effort less effective. The more
uncertainty there is about ability (which in a way is saying that there is
more ignorance about the manager's ability), the more influential effort is
on beliefs, which is what happens when there is high variance in $\eta $ (a
low $h_{1}$). On the other hand, a high variance in $\varepsilon $ makes the
returns to effort low since the noise in $\varepsilon $ swamps the influence
of effort on output (the production process is noisy).\footnote{%
Note that for the two period model $\varepsilon _{t}$ is not needed to
generate the career concerns - the noise on $\eta $ is enough and we can
have $h_{\varepsilon }\rightarrow \infty $. For longer horizons, if there is
no $\varepsilon $ noise then ability is perfectly learned after one period
and therefore $a_{t}^{*}=0$ for all $t>1$.}

How does the equilibrium effort level compare to the first best effort
level? In this simple two period model it is clear that there is
underinvestment in both periods. First note from (\ref{y}) that the marginal
returns to effort are always 1, so that it is first best optimal to set
effort so that $g^{\prime }(a^{FB})=1$ for all $t$. In the second period
there will be no effort, and in the first period we have from (\ref{FOC1})
that $g^{\prime }(a_{1}^{*})<1,$ and $g^{\prime \prime }(\cdot )>0$ implies
that $a_{1}^{*}<a^{FB}$.

If the manager has a longer, but finite lifetime of length $T$, it is easy
to see that $a_{T}^{*}=0$, and that incentives decline over time (see the
next subsection for a more formal argument). However, it is also easy to see
that with more periods there can be too much effort in the first periods. To
illustrate this point assume that the model lasts for three periods, but
that the manager can only exert effort in his first period so that $%
a_{2}=a_{3}=0$. Then, the returns to effort are (almost) doubled: $%
E[w_{2}(y_{1})]$ is determined as in (\ref{Ew2}) above, and the third period
wage is given by,\footnote{%
This is derived by the fact that 
\begin{equation*}
m_{3}(z_{2})=\frac{h_{2}}{h_{2}+h_{\varepsilon }}\cdot m_{2}(z_{1})+\frac{%
h_{\varepsilon }}{h_{2}+h_{\varepsilon }}\cdot Ez_{2}\,,
\end{equation*}
and substituting for $m_{2}(z_{1}),\,Ez_{2}$, and recalling that $%
h_{2}=h_{1}+h_{\varepsilon }$ will yield this equation.} 
\begin{eqnarray}
E[w_{3}(y_{1},y_{2})] &=&E[m_{3}(z_{1},z_{2})]  \notag \\
&=&\frac{h_{1}}{h_{1}+2h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }}{%
h_{1}+2h_{\varepsilon }}\cdot [\underset{Ez_{1}}{\underbrace{%
(m_{1}+a_{1}-a_{1}^{*})}}+\underset{Ez_{2}}{\underbrace{m_{1}}}]\,.
\label{Ew3}
\end{eqnarray}

The manager's utility is now the utility given in (\ref{Eu2}) plus $\beta
^{2}\cdot E[w_{3}(y_{1},y_{2})],$ which gives the FOC, 
\begin{equation}
g^{\prime }(a_{1}^{*})=\beta \cdot \frac{h_{\varepsilon }}{%
h_{1}+h_{\varepsilon }}+\beta ^{2}\cdot \frac{h_{\varepsilon }}{%
h_{1}+2h_{\varepsilon }}\;.  \label{FOC2}
\end{equation}
Now observe that as $\beta \rightarrow 1$ (important future), then either $%
h_{1}\rightarrow 0$ (noisy ability) or $h_{\varepsilon }\rightarrow \infty $
(low noise in production) will cause the FOC to yield $g^{\prime
}(a_{1}^{*})>1$ which implies that $a_{1}^{*}>a^{FB}$. Intuitively, the
effects of the first period effort last for more than just one period, so
that the cumulative effects of this effort will add up and cause the manager
to work harder than is first-best optimal. Again, the market is not fooled
(rational expectations), but, as Holmstrom (1999) nicely puts it, ``[The
manager] is trapped in supplying the equilibrium level that is expected of
him, because, as in a rat race, a lower supply of labor will bias the
evaluation procedure against him.''

\subsubsection{The infinite Horizon Model with fixed ability}

Now consider the original model with an infinite horizon, so that (\ref%
{effort}) and (\ref{wages}) apply. We now define analogous to the above, 
\begin{equation*}
z_{t}\equiv \eta +\varepsilon _{t}=y_{t}-a_{t}^{*}(y^{t-1})\;,
\end{equation*}
so that observing the sequence $z_{t}$, the market learns about the talent
parameter $\eta $ by calculating the conditional (posterior) expectation $%
E[\eta |z^{t-1}],$ where $z^{t-1}=(z_{1},z_{2},...,z_{t-1})$. As before,
since both $\eta $ and $\varepsilon _{t}$ are Normal, it is well known that
the posterior of $\eta $ is Normal with $[\eta |z^{t-1}]\sim N(m_{t},\frac{1%
}{h_{t}})$, where 
\begin{equation*}
m_{t}(z^{t-1})=\frac{h_{1}}{h_{1}+(t-1)h_{\varepsilon }}\cdot m_{1}+\frac{%
h_{\varepsilon }}{h_{1}+(t-1)h_{\varepsilon }}\cdot \sum_{s=1}^{t-1}z_{s}\;,
\end{equation*}
and 
\begin{equation*}
h_{t}=h_{1}+(t-1)h_{\varepsilon }\rightarrow \infty \;.
\end{equation*}

As before, given the equilibrium decision rule $a_{t}^{*}(y^{t-1})$, the
expected wages the manager expects to receive ex ante is, 
\begin{equation}
E[w_{t}(y^{t-1})]=\frac{h_{1}}{h_{t}}\cdot m_{1}+\frac{h_{\varepsilon }}{%
h_{t}}\cdot \sum_{s=1}^{t-1}\underset{Ez_{s}}{\underbrace{[\overset{Ey_{s}}{%
\overbrace{m_{1}+a_{s}}}-Ea_{s}^{*}(y^{s-1})]}+Ea_{t}^{*}(y^{t-1})}\,.
\label{Ewt}
\end{equation}
By substituting (\ref{Ewt}) into (\ref{effort}) we can explicitly solve the
agent's problem. The marginal benefit of effort $a_{t}$ in future period $%
s>t $ is $\alpha _{s}\equiv \frac{h_{\varepsilon }}{h_{s}}$ , which implies
that the manager's FOCs are, 
\begin{equation*}
g^{\prime }(a_{t}^{*})=\sum_{s=t}^{\infty }\beta ^{s-t}\cdot \alpha
_{s}\equiv \gamma _{t}\;.
\end{equation*}
Note that $h_{t}\rightarrow \infty $, which implies that the marginal
returns in the far future disappear: $\alpha _{s}\rightarrow 0$. This
implies that as time goes by the whole marginal returns to effort disappear: 
$\gamma _{t}\rightarrow 0,$ which in turn implies that $a_{t}^{*}(y^{t-1})%
\rightarrow 0$. This is purely because as time goes by, everyone has a
better estimate of the manager's ability $\eta $, and the influence of
effort is less effective (the market is more informed about the manager's
ability). The comparative statics given for the two period case still apply
to this case, and clearly, effort in the first periods may be above or below
the first best level for the same reasons as mentioned above.

A conclusion of this model is that Fama's original argument needs some
qualifications: moral hazard problems are not fully solved by the market,
and in fact effort will be almost always not optimal: it may be too high in
the beginning of one's career and too low towards to end. Therefore, an
interesting question is when would the market really work? For incentives
not to disappear, there has to be some perpetual uncertainty about the
manager's ability, so that effort can continue to influence beliefs (the
market cannot become ``too convinced'' as to the manager's true ability).
This is explored in the following subsection which is based on section 2.2
in Holmstrom (1999).

\subsubsection{The infinite Horizon Model with varying ability}

Now assume that ability follows a random walk as follows, 
\begin{equation}
\eta _{t+1}=\eta _{t}+\delta _{t}\;,  \label{variable}
\end{equation}
where $\delta _{t}\sim N(0,\frac{1}{h_{\delta }}).$ The Bayes updating will
be similar to before, with 
\begin{equation}
m_{t+1}=\mu _{t}m_{t}+(1-\mu _{t})z_{t}\;,\text{ where \ }\mu _{t}=\frac{%
h_{t}}{h_{t}+h_{\varepsilon }}\;.  \label{mt}
\end{equation}
However, $h_{t}$ will proceed differently than before because of the changes
in $\eta _{t}$. Namely, the $\delta $-shocks prevent the market from
learning the value of $\eta $, and in particular we get\footnote{%
To see this, let $\widehat{h}_{t}$ be the precision of $\eta _{t+1}$ before
observing $y_{t+1}$. As before, $\widehat{h}_{t}=h_{t}+h_{\varepsilon }$ .
Note that $\widehat{h}_{t}$ is the same as the precision of $\eta _{t}$
after observing $y_{t},$ and from the independence of $\eta _{t}$ and $%
\delta _{t}$ it follows from the variable process of $\eta _{t+1}=\eta
_{t}+\delta _{t}$ that $\frac{1}{h_{t+1}}=\frac{1}{\widehat{h}_{t}}+\frac{1}{%
h_{\delta }}\,.$ Substituting for $\widehat{h}_{t}=h_{t}+h_{\varepsilon }$
we obtain the desired equality. Note that if $Var(\delta _{t})\rightarrow 0$%
, this is just $h_{\delta }\rightarrow \infty $ and the equality approached
the one shown for the model without variability of $\eta $.} 
\begin{equation*}
h_{t+1}=\frac{(h_{t}+h_{\varepsilon })h_{\delta }}{h_{t}+h_{\varepsilon
}+h_{\delta }}\;.
\end{equation*}
Differentiating $h_{t+1}$ with respect to $h_{t}$ yields, 
\begin{eqnarray*}
\frac{\partial h_{t+1}}{\partial h_{t}} &=&\frac{h_{\delta
}(h_{t}+h_{\varepsilon }+h_{\delta })-(h_{t}+h_{\varepsilon })h_{\delta }}{%
(h_{t}+h_{\varepsilon }+h_{\delta })^{2}} \\
&=&\frac{h_{\delta }^{2}}{(h_{t}+h_{\varepsilon }+h_{\delta })^{2}}\in (0,1)
\end{eqnarray*}
which implies that there is some $h^{*}$ such that as $t\rightarrow \infty ,$
$h_{t}\rightarrow h^{*}$. This also implies that $\mu _{t}\rightarrow \mu
^{*}=\frac{h^{*}}{h^{*}+h_{\varepsilon }}$ (see Holmstrom (1999) for a
closed form solution).

Thus, we can consider behavior in a stationary steady state of this model,
in which case the effort level will be some $a^{*}.$ To solve for the value
of a steady state effort level, we must have that the marginal benefit from
changing the steady state effort level in a specific period is equal to the
marginal cost, or, 
\begin{eqnarray*}
g^{\prime }(a^{*}) &=&\beta (1-\mu ^{*})+\beta ^{2}\mu ^{*}(1-\mu
^{*})+\beta ^{3}(\mu ^{*})^{2}(1-\mu ^{*})+\cdot \cdot \cdot \\
&=&\frac{\beta (1-\mu ^{*})}{1-\beta \mu ^{*}}
\end{eqnarray*}
and we obtain,

\begin{description}
\item[Proposition 1:] (Holmstrom (1999)) The stationary level is $a^{*}\leq
a^{FB}$. If $\beta =1,\,\frac{1}{h_{\varepsilon }}>0$, and $\frac{1}{%
h_{\delta }}>0$ then $a^{*}=a^{FB}$. Furthermore, $a^{*}$ is increasing in $%
\beta $ and in $h_{\varepsilon }$, and decreasing in $h_{\delta }\,.$
\end{description}

Intuitively, the comparative statics for the model without variability also
apply here: the more patient the manager, or the more accurate are the
observations of output (higher $h_{\varepsilon }$), the higher the
reputational concerns. Similarly, the more stochastic is the ability process
(lower $h_{\delta }$), the higher the reputational concerns.

We can also look at the dynamics of the system before a steady state is
reached. The following (intuitive) proposition follows from the detailed
analysis in Holmstrom (1999):

\begin{description}
\item[Proposition 2:] (Holmstrom (1999)) Starting at $t=0$ with some $h_{1}, 
$ the optimal effort sequence $\{a_{t}^{*}\}$ will converge monotonically to 
$a^{*}$. If $h_{1}<h^{*}$ then $a_{t}^{*}\searrow a^{*}$, and if $%
h_{1}>h^{*} $ then $a_{t}^{*}\nearrow a^{*}$.
\end{description}

If one accepts the hypothesis that information is less accurate at the
beginning of time, then $h_{1}<h^{*}$ and $\{a_{t}^{*}\}$ will converge
monotonically to $a^{*}$ from above, implying that initially there will be
several (maybe many) periods where more than optimal effort will be exerted.
If $\beta <1$ then as time goes by less than optimal effort will be exerted:
managers work too hard when young, and too less when old, a prediction that
makes sense for the finite horizon case (but is more complicated to solve
for!). If $\beta =1$ (no discounting) then except for the limit, there will
always be too much effort!

\begin{description}
\item[Remark:] One observation is that Fama's conjecture is true only for
the case where there is no discounting, and where there is some persistent
noise about output and ability. Holmstrom shows in section 2.4 of the paper
that this is true only for the linear model (in which (\ref{y}) and (\ref%
{variable}) are linear). Of course, it is not surprising that Fama's
conjecture would be true under very special conditions. Still, the
conjecture that market forces will affect incentives without implicit
contracts is rather general, and will be seen again later.
\end{description}

\clearpage 

\subsection{Explicit Contracts and Career Concerns}

An interesting question is, how will the presence of career concerns affect
explicit contracts if these were allowed in the model? Namely, how will the
presence of career concerns affect the strength of explicit incentives? As
we have demonstrated, career concerns are strongest early in a manager's
career, and deteriorate thereafter. This intuitively suggests that there is
more scope for providing explicit incentives later in a manager's career.

In the model above the manager is assumed to be risk neutral. If the manager
lived in isolation (no reputational concerns), and had access to the
technology, then he would always choose the first-best effort level. An
immediate implication is that if firms could write contingent contracts with
managers so that $w_{t}$ can depend on $y_{t}$, then by setting $w_{t}=y_{t}$
we again get first best effort levels and no interesting dynamics.

Thus, to introduce contracts into the model one also has to introduce some
trade-off, a natural one being risk smoothing between the firms and the
managers. This is the framework of Gibbons and Murphy (1992), in which the
intuition above is formalized and derived.

To proceed, assume that the technology is as in (\ref{y}) above, 
\begin{equation*}
y_{t}=\eta +a_{t}+\varepsilon _{t}\,,
\end{equation*}
an that the information structure and timing of events is the same as
before. The two differences are as follows:

\begin{enumerate}
\item Firms can write explicit incentives contracts of the form 
\begin{equation*}
w_{t}(y_{t})=c_{t}+b_{t}(y_{t})\,.
\end{equation*}

\item The manager is risk averse with CARA utility 
\begin{equation}
U(w,a)=-e^{-r\sum_{t=1}^{\infty }\beta ^{t-1}[w_{t}-g(a_{t})]}  \label{CARA}
\end{equation}
\end{enumerate}

The restriction to linear contracts is for technical tractability, and for a
clear measure of explicit incentives: a higher $b$ means higher returns to
output, which is interpreted as \emph{stronger explicit incentives}.%
\footnote{%
In general linear contracts are not optimal, but as Holmstrom and Milgrom
(1987) show, in a dynamic model that is interpreted as a manager choosing
the mean of a normal distribution, and with this type of utility function, a
linear contract is optimal.} Another restriction is that the wages at period 
$t$ depend on the output of that period alone, so that long term contracts
are ignored. This assumption is justified as an alternative to renegotiation
proof contracts and is formally done in the appendix of Gibbons and Murphy
(1992).

We will analyze the two period model, but the qualitative results generalize
to longer horizons. At the beginning of period 1, firms will offer wages $%
w_{1}(y_{1})=c_{1}+b_{1}y_{1}$ where $c_{1}$ and $b_{1}$ are not contingent
on anything since no information has been revealed. As before, the
observation of $y_{1}$ contains information, and thus, second period
compensation will \textit{implicitly} depend on $y_{1}$ so that 
\begin{equation*}
w_{2}(y_{2}|y_{1})=c_{2}(y_{1})+b_{2}(y_{1})y_{2}\,.
\end{equation*}

As for the first career concern model, we solve for the equilibrium using
backward induction starting from period 2 after the events of period 1 were
realized. In equilibrium, $a_{1}^{*}$ is correctly anticipated, so that it
is \emph{as if} the market observes 
\begin{equation*}
z_{1}\equiv \eta +\varepsilon _{1}=y_{1}-a_{1}^{*}\;,
\end{equation*}
and through this observation the market learns about the talent parameter $%
\eta $ by calculating the conditional (posterior) expectation $E[\eta
|z_{1}] $. As before, the posterior of $\eta $ is Normal with $[\eta
|z_{1}]\sim N(m_{2},\frac{1}{h_{2}})$, where 
\begin{equation}
m_{2}(z_{1})=\frac{h_{1}}{h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{%
h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\cdot z_{1}\;,  \label{m22}
\end{equation}
and 
\begin{equation}
h_{2}=h_{1}+h_{\varepsilon }\;.  \label{h22}
\end{equation}

To continue, given any $c_{1},\,b_{1},\,c_{2}$ and $b_{2},$ and the utility
specified in (\ref{CARA}) above, after $a_{1}$ has been chosen, and $y_{1}$
has been observed, the agent's second period problem is given by, 
\begin{equation}
\max_{a_{2}}\;-\overset{\text{constant}}{\overbrace{e^{-\frac{r}{\beta }%
[c_{1}+b_{1}y_{1}-g(a_{1})]}}}\cdot E\left[ e^{-r[c_{2}+b_{2}(\eta
+a_{2}+\varepsilon _{2})-g(a_{2})]}|z_{1}\right]  \label{max2}
\end{equation}%
Now using (\ref{m22}) and (\ref{h22}), and ignoring the constant term in (%
\ref{max2}), we can write the manager's problem as,\footnote{%
This follows from the identity 
\begin{equation*}
E\left[ e^{-rx}\right] =e^{-rE[x]+\frac{1}{2}r^{2}var(x)}
\end{equation*}%
where in our case the random variable is $x=b([\eta
|z_{1}]+a_{2}+\varepsilon _{2}),$ which has mean $m_{2}+a_{2}$ and variance $%
b_{2}^{2}(\frac{1}{h_{2}}+\frac{1}{h_{\varepsilon }})$.} 
\begin{equation}
\max_{a_{2}}\;-e^{-r[c_{2}+b_{2}(m_{2}+a_{2})-g(a_{2})]+\frac{1}{2}%
(rb_{2})^{2}(\frac{1}{h_{2}}+\frac{1}{h_{\varepsilon }})}  \label{max22}
\end{equation}%
and since after $y_{1}$ was realized, the agent takes $b_{2}$ and $c_{2}$ as
given, the necessary FOC for determining $a_{2}^{\ast }(b_{2})$ is given by 
\begin{equation}
g^{\prime }(a_{2})=b_{2}\,.  \label{FOC3}
\end{equation}

The firm's problem is to choose a compensation scheme that maximizes
expected profits, subject to the agent's best response given by (\ref{FOC3})
and to the zero profit condition. The zero profit condition implies that
given any $b_{2}$ and the manager's best response $a_{2}^{*}(b_{2})$, we
must have that 
\begin{eqnarray}
c_{2}(b_{2}) &=&(1-b_{2})E[y_{2}|y_{1}]  \notag \\
&=&(1-b_{2})[m_{2}+a_{2}^{*}(b_{2})]  \label{c2}
\end{eqnarray}
so that the firm's problem is reduced to choosing $b_{2}$, and then $c_{2}$
will be calculated using (\ref{c2}) above.

There is a convenient way, however, to solve this problem. Two facts are
known: first, the firm is maximizing expected profits, and second, these
profits are transferred to the manager due to labor market competition. This
implies that we can solve for the competitive equilibrium by \emph{%
maximizing the manager's expected utility,} subject to the following two
constraints: first, the manager is behaving according to his best response $%
a_{2}^{\ast }(b_{2})$ as given by (\ref{FOC3}), and second that $c_{2}$ must
satisfy the zero profit condition in (\ref{c2}). Substituting these
constraints in the manager's utility, the equilibrium is solved by choosing $%
b_{2}$ to maximize, 
\begin{equation}
\max_{b_{2}}\;-e^{-r[m_{2}+a_{2}^{\ast }(b_{2})-g(a_{2}^{\ast }(b_{2}))]+%
\frac{1}{2}(rb_{2})^{2}(\frac{1}{h_{2}}+\frac{1}{h_{\varepsilon }})}
\label{max23}
\end{equation}%
and differentiating (\ref{max23}) with respect to $b_{2}$ gives the
necessary FOC, 
\begin{equation}
\frac{da_{2}^{\ast }(b_{2})}{db}-g^{\prime }(a_{2}^{\ast }(b_{2}))\cdot 
\frac{da_{2}^{\ast }(b_{2})}{db}-r(\frac{1}{h_{2}}+\frac{1}{h_{\varepsilon }}%
)b_{2}=0\,.  \label{FOC4}
\end{equation}

By implicitly differentiating the managers FOC (\ref{FOC3}) with respect to $%
b_{2}$ we obtain that 
\begin{equation*}
\frac{da_{2}^{*}(b_{2})}{db}=\frac{1}{g^{\prime \prime }(a_{2}^{*}(b_{2}))}
\end{equation*}
and we can then implicitly solve for (\ref{FOC4}) to obtain the slope of the
second period compensation scheme, 
\begin{equation}
b_{2}^{*}=\frac{1}{1+r(\frac{1}{h_{2}}+\frac{1}{h_{\varepsilon }})g^{\prime
\prime }(a_{2}^{*}(b_{2}^{*}))}\;.  \label{b2}
\end{equation}

Recall that we assumed $g^{\prime \prime }(\cdot )>0,$ which implies that
the strength of incentives decreases when either the manager is more risk
averse ($r$ increases) or when there is more uncertainty (either $h_{2}$ or $%
h_{\varepsilon }$ decreases), which is what we would expect from standard
moral hazard theory. To guarantee that this FOC (\ref{FOC3}) is sufficient,
it is enough to assume that $g^{\prime \prime \prime }(\cdot )\geq 0$.

A nice feature of (\ref{b2}) is that the optimal level of incentives, $%
b_{2}^{\ast }$, does not depend on $y_{1}$. This follows from the CARA
assumption on utility (there are no wealth effects), and by the fact that
the variance of $[\eta |z_{1}]$ is deterministic and independent of $y_{1}$.
This means that the impact of career concerns will be on $c_{2}$, not on $%
b_{2}$, which simplifies things dramatically. This is captured by, 
\begin{equation}
c_{2}^{\ast }(b_{2}^{\ast })=(1-b_{2}^{\ast })\left[ \frac{h_{1}}{%
h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }}{%
h_{1}+h_{\varepsilon }}\cdot \underset{z_{1}}{\underbrace{(\overset{y_{1}}{%
\overbrace{\eta +a_{1}+\varepsilon _{1}}}-a_{1}^{\ast })}}\right] \,
\label{c22}
\end{equation}

We can now turn to the first period problem taking the solution to the
second period problem as given. The worker takes $b_{1}$ and $c_{1}$ as
given, and solves 
\begin{equation*}
\max_{a_{1}}\;-E\left[ e^{-r[c_{1}+b_{1}(\eta +a_{1}+\varepsilon
_{1})-g(a_{1})]}\cdot e^{-r\beta [c_{2}^{*}(b_{2}^{*})+b_{2}^{*}(\eta
+a_{2}^{*}(b_{2}^{*})+\varepsilon _{2})-g(a_{2}^{*}(b_{2}^{*}))]}\right]
\end{equation*}
Substituting (\ref{c22}) and taking the market conjecture for first period
effort to be $a_{1}^{*}$, the manager's objective function is, 
\begin{equation*}
-E\left[ e^{-r\left[ c_{1}+b_{1}(\eta +a_{1}+\varepsilon
_{1})-g(a_{1})+\beta \left( (1-b_{2}^{*})\left[ \frac{h_{1}}{%
h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }}{%
h_{1}+h_{\varepsilon }}\cdot (\eta +a_{1}+\varepsilon _{1}-a_{1}^{*})\right]
+b_{2}^{*}(\eta +a_{2}^{*}(b_{2}^{*})+\varepsilon
_{2})-g(a_{2}^{*}(b_{2}^{*}))\right) \right] }\right]
\end{equation*}
which can be rewritten using the certainty equivalent representation as, 
\begin{eqnarray}
&&e^{-r\left[ c_{1}+b_{1}(m_{1}+a_{1})-g(a_{1})+\beta \left( (1-b_{2}^{*})%
\left[ \frac{h_{1}}{h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }%
}{h_{1}+h_{\varepsilon }}\cdot (m_{1}+a_{1}-a_{1}^{*})\right]
+b_{2}^{*}(m_{1}+a_{2}^{*}(b_{2}^{*}))-g(a_{2}^{*}(b_{2}^{*}))\right) \right]
}  \notag \\
&&\cdot e^{\frac{1}{2}r^{2}\left[ \left( b_{1}+\beta \cdot
(1-b_{2}^{*})\cdot \frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\,+\beta
b_{2}^{*}\right) ^{2}\left( \frac{1}{h_{1}}+\frac{1}{h_{\varepsilon }}%
\right) -2\left( b_{1}+\beta \cdot (1-b_{2}^{*})\cdot \frac{h_{\varepsilon }%
}{h_{1}+h_{\varepsilon }}\right) \beta b_{2}^{*}\frac{1}{h_{e}}\right] }\,.
\label{max26}
\end{eqnarray}

Notice that the manager's actual first period effort shows up twice. First,
as an \emph{explicit incentive} since he is rewarded in the first period
with an explicit incentive slope of $b_{1}$. Second, as an \emph{implicit
incentive} is provided by the career concerns since there is Bayes updating
about the manager's ability in the second period. This can be seen clearly
in the following FOC of the manager's problem that solves for $%
a_{1}^{*}(b_{1})$, 
\begin{equation}
g^{\prime }(a_{1})=\underset{\text{incentive}}{\underset{\text{explicit}}{%
\underbrace{\underset{}{b_{1}}}}}+\underset{\text{implicit incentive}}{%
\underbrace{\beta \cdot (1-b_{2}^{*})\cdot \frac{h_{\varepsilon }}{%
h_{1}+h_{\varepsilon }}}}\,.  \label{a1}
\end{equation}
The implicit incentive is positive since from (\ref{b2}) we have $%
b_{2}^{*}\in (0,1)$. The implicit career concern component has the same
comparative statics as in the model without explicit contracts: effort is
increasing in patience ($\beta $) and in the precision of the output process
($h_{\varepsilon }$) and decreasing in the precision of prior ability ($%
h_{1} $).

In equilibrium we must have the market's conjecture being correct, and
therefore the zero profit condition implies that 
\begin{eqnarray}
c_{1}(b_{1}) &=&(1-b_{1})E[y_{1}]  \notag \\
&=&(1-b_{1})[m_{1}+a_{1}^{*}(b_{1})]\;.  \label{c1}
\end{eqnarray}
As before, we can solve for the competitive equilibrium by maximizing the
manager's expected utility in (\ref{max26}) subject to his best response $%
a_{1}^{*}(b_{1})$ as given by (\ref{a1}), and by taking $c_{1}$ as given by (%
\ref{c1}). This implicitly solves for $b_{1}^{*}$ as follows, 
\begin{eqnarray}
b_{1} &=&\frac{1}{1+r(\frac{1}{h_{1}}+\frac{1}{h_{\varepsilon }})g^{\prime
\prime }(a_{1}^{*}(b_{2}^{*}))}  \label{b1} \\
&&-\beta (1-b_{2}^{*})\frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}-\frac{%
r\beta b_{2}^{*}\frac{1}{h_{1}}g^{\prime \prime }(a_{1}^{*}(b_{2}^{*}))}{1+r(%
\frac{1}{h_{1}}+\frac{1}{h_{\varepsilon }})g^{\prime \prime
}(a_{1}^{*}(b_{2}^{*}))}  \notag
\end{eqnarray}
Comparing (\ref{b1}) with (\ref{b2}) shows that $b_{1}^{*}<b_{2}^{*}$ for
three reasons. The first is the \emph{output risk effect. }The first term in
(\ref{b1}) is itself smaller than the term in (\ref{b2}) because $(\frac{1}{%
h_{1}}+\frac{1}{h_{\varepsilon }})>(\frac{1}{h2}+\frac{1}{h_{\varepsilon }})$%
. This is a direct consequence of the learning that occurs about the
manager's ability, which reduces the ex post variance. Due to risk aversion,
this implies that there should be less explicit incentives in the first
period since there is more risk. The second is the \emph{career concerns
effect,} captured by the second term in (\ref{b1}) -- these are the
incentives provided by career concerns, so they need not be provided by
explicit incentives. The third is the \emph{insurance effect,} captured by
the third term in (\ref{b1}). This is due to the fact that future wages
depend on human capital, which ex ante is a random variable itself (the
manager and the firms are learning about his type). Since firm's are assumed
to be risk neutral, there is room to insure the manager for uncertainty with
respect to his human capital realizations, which puts extra pressure to
reduce explicit incentives.

\subsection{Reputational Concerns and Investment Decisions}

The previous two sections show that reputational career concerns can provide
self interested agents with an incentive to exert effort. An interesting
related question that is raised in Holmstrom (1999) is how reputational
career concerns will affect the decision of managers to invest in risky
projects. This question is addressed in more detail in Holmstrom and Ricart
I Costa (1986), and we will go over their introductory example to illustrate
the main points of the paper.

Consider a manager who can be employed through a competitive market as
before, and assume that there can be no contingent contracts. Therefore, his
wage in every period will be bid up to his expected marginal product. The
economy lasts for two periods, $t\in \{1,2\}$, and in each period the
manager observes a signal $s_{t}\in \{-1,1\}$, where $\Pr \{s_{t}=1\}=\frac{1%
}{2}$. The manager's ability is given by $\varepsilon _{t}\in \{-1,1\}$ and
is conditional on his ``talent'': the manager can be either ``good'' or
``bad'', and the distribution of $\varepsilon _{t}$ is time independent and
is given by, 
\begin{equation*}
\Pr \{\varepsilon _{t}=1\}=\left\{ 
\begin{array}{ll}
\frac{3}{4} & \text{if manager is good} \\ 
&  \\ 
\frac{1}{4} & \text{ if manager is bad}%
\end{array}
\right. \,
\end{equation*}
That is, a good manager is more likely than a bad manager to have high
ability.

The technology in each period is given by the time invariant relationship, 
\begin{equation*}
y_{t}=s_{t}+\varepsilon _{t}\text{ ,}
\end{equation*}
so that for every period $y_{t}\in \{-2,0,2\}.$

The information structure is as follows: Both the manager and the firms who
can employ him do not know the manager's talent, but it is common knowledge
ex ante that $\Pr \{$good$\}=p_{1}$. At the beginning of each period only
the manager observes the signal associated with his project, and he can
decide whether to invest in the project or not. If investment does not occur
in the first period, then nothing is observed and the updated beliefs of the
manager and the firms about the manager's talent are just $p_{2}^{0}=p_{1}.$
If however, investment occurs, we assume that both output, $y_{t},$ and the
signal, $s_{t},$ are observed by the manager and the firms. This means that
everyone can infer $\varepsilon _{t}$, and both the manager and the firms
will use the same posterior updating rule. In particular, beliefs are then
updated to $p_{2}^{S}$ if high ability was realized and the projected
succeeded ($\varepsilon _{t}=1$), and to $p_{2}^{F}$ if low ability was
realized and the projected failed ($\varepsilon _{t}=0$). This implies that $%
p_{2}\in \{p_{2}^{F},p_{2}^{0},p_{2}^{S}\},$ and since beliefs form a
martingale then, 
\begin{equation}
E[p_{2}|p_{1}]=p_{1}\,.  \label{martingale}
\end{equation}

The equilibrium will have a first period non-contingent wage $w_{1}$, and a
second period contingent wage $w_{2}(p_{2})$. Given a posterior belief $%
p_{2},$ the expected returns to a firm from investing in period 2
conditional on $s_{2}$ are, 
\begin{equation*}
E[y_{2}|s_{2}=1\}=1+p_{2}(\frac{3}{4}-\frac{1}{4})+(1-p_{2})(\frac{1}{4}-%
\frac{3}{4})=p_{2}+\frac{1}{2}>0\;,
\end{equation*}
and 
\begin{equation*}
E[y_{2}|s_{2}=-1\}=-1+p_{2}(\frac{3}{4}-\frac{1}{4})+(1-p_{2})(\frac{1}{4}-%
\frac{3}{4})=p_{2}-\frac{3}{2}<0\;.
\end{equation*}
This implies that it is efficient to invest if and only if $s_{2}=1$.

Since the second period is the last, the manager would be indifferent
between following the efficient rule and any other rule, so assume that he
does follow it, which implies that the expected value to a firm of hiring a
manager is $E[y_{2}|p_{2}]=\frac{1}{2}(p_{2}+\frac{1}{2})\,,$and from the
competitive labor market assumption this is also $w_{2}(p_{2}),$ so 
\begin{equation}
w_{2}(p_{2})=\frac{1}{2}p_{2}+\frac{1}{4}\,.  \label{w2}
\end{equation}%
Now in the first period $w_{1}$ is not contingent, and the manager has to
choose whether to invest after observing $s_{1}$. By not investing he
guarantees himself $w_{2}(p_{2}^{0})=\frac{1}{2}p_{1}+\frac{1}{4}$ for sure.
If he invests, he faces a lottery that depends on the inferred value of $%
\varepsilon _{1},$ so that success implies a wage $w_{2}(p_{2}^{S})=\frac{1}{%
2}p_{2}^{S}+\frac{1}{4}$ and failure implies a wage $w_{2}(p_{2}^{F})=\frac{1%
}{2}p_{2}^{F}+\frac{1}{4}$. But ex ante, regardless of the signal value $%
s_{1}$, the martingale property (\ref{martingale}) implies that the expected
value of this lottery is exactly $\frac{1}{2}p_{2}+\frac{1}{4}$, which he
can get for sure by not investing!

This argument implies that if the manager has the slightest amount of risk
aversion, he will choose not to invest even when $s_{1}=1$, and this is
socially not optimal. The reason is that the expected financial value of
such an investment is positive, but the effect on the manager's future human
capital is negative since risk is created. It is easy to see that any finite
repetition of this technology will yield investment only in the last period
for any amount of risk aversion.

This is a very special model and the results are rather extreme. Holmstrom
and Ricart I Costa (1986) develop this further with more general information
structures on the signal and the manager's ability. In general, high risk
aversion causes underinvestment in projects, and low risk aversion creates
over investment (where optimal investment is some sort of cutoff rule given
the value of the signal $s_{t}$). This last observation is used as a
rationale for the need to centrally ``ration'' investments in firms. Namely,
assuming that a manager who proposes an investments reveals $s_{t}$, then a
central authority (the firm's owner) can make the right financial decision
and will veto projects with low values of $s_{t}$. This alleviates the
problem of over investment in projects that are not attractive from a risk
perspective.

\clearpage

\section{Signaling Models of Reputation}

\subsection{Introduction}

The models of career concerns were developed using symmetric uncertainty
over the ability, or ``type'' of the managers. When the managers know their
own types, the analysis becomes much more complex since different types may
choose different actions, thus ``signalling'' something about their type.%
\footnote{%
A quick observation of the Holmstrom (1999) model suggests that for a two
period economy, the equilibrium with symmetric uncertainty over types is
also an equilibrium when managers know their own type. There are, however,
potentially many more equilibria, and with longer horizons the analysis
becomes quite cumbersome.} When we consider the provision of dynamic
incentives in a world where managers know their own types, we would expect
that the simple ``economics of reputation'' argument will go through: when a
manager has a short horizon, without explicit contracts he will have little
or no incentives to perform. With longer horizons, however, it is tempting
to conjecture that long-term reputational concerns will kick in.

\subsection{Reputation Formation and Dynamic Incentives}

To demonstrate that this intuition can be made precise, and offer some
interesting results on the dynamics of reputations, we consider a model
based on Diamond (1989) who explores reputation formation and dynamic
incentives in debt markets. Part of the model uses a competitive assumption
on the market for loans (that lenders are competitive and earn zero profits)
in a similar way that Holmstrom models the labor market. On the other side,
however, there is a difference: Diamond's managers (who select investment
projects) know something about their ``ability'' and can choose how to
behave based on this knowledge. When their ``bad'' behavior precisely
signals who they are, the ramifications are loss of credit, which is the
incentive mechanism to keep managers in line.

Formally, consider a market for loans in which lenders are plentiful (a
continuum of measure $m>1$), and can invest their endowments in some outside
option. If they chose this option in a certain period, this yields a return
of $r>1$ at the end of a period for each unit of input they invested at the
beginning of that period. Assume that every period a new cohort of lenders
is born, each with an endowment of 1 unit of input, while the previous
cohort retires. This prevents lenders from writing long term contracts with
borrowers, and ``credit histories'' are the sole source of information upon
which debt contracts will be contingent.

Borrowers are born at the beginning of time, and live for $T$ periods which
is the length of the economy. Borrowers have no endowments but they have
access to technologies, and each technology requires one unit of input and
offers a return at the end of the period. Borrowers differ in their type,
which is described as follows.

\begin{itemize}
\item A proportion $f_{G}$ of borrowers are ``Good'' ($G$) types who have
access to only one technology that guarantees a return $G>r$ (so these are
profitable from a lender's perspective).

\item A proportion $f_{B}$ of borrowers are ``Bad'' ($B$) types who have
access to only one technology that yields a return of $B>G$ with probability 
$\pi $, and $0$ with probability $1-\pi $, where $\pi B<r$ (so these are not
profitable from a lender's perspective).

\item A proportion $f_{GB}=1-f_{G}-f_{B}$ of borrowers are ``Bad-or-Good'' ($%
BG$) types who have access to both technologies described above, but can
choose only one in each period.
\end{itemize}

Thus, we have a model that combines adverse selection (the $G$ and $B$
types) with moral hazard (the $BG$ type). Assume that the measure of (the
continuum of) borrowers is $1,$ so that competition pushes the returns of
investors down to $r$. Also, assume that the type of a borrower is his
private information, but the proportions of types and the technologies are
common knowledge. Furthermore, the following informational assumptions are
made:

\begin{itemize}
\item The realized output of a project is the borrower's private information.

\item The chosen payments made by borrowers are public information.

\item If a borrower defaults, the project is liquidated and neither the
borrower, nor the lender, receive any returns from it.
\end{itemize}

These informational assumptions will play an important role. The last of
these will guarantee that no direct information is learned from default, but
prevents ``take-the-money-and-run'' strategies which can always support
equilibria where the debt market is inactive, and would be the unique
equilibrium with a finite horizon.

Borrowers and lenders are risk neutral, and maximize their discounted
expected consumption. Given a sequence of random consumptions $\widetilde{c}%
=\{\widetilde{c}_{1},\widetilde{c}_{2},...,\widetilde{c}_{T}\}$ and discount
factor $\beta <1$ for a borrower, his utility is given by\footnote{%
Notice that here we have $\beta ^{t}$ instead of $\beta ^{t-1}$. This is of
no importance but we are loyal to Diamond's formalization.} 
\begin{equation*}
U(\widetilde{c})=\sum_{t=1}^{T}\beta ^{t}E[\widetilde{c}_{t}]\,.\,
\end{equation*}

The informational assumptions together with a limited liability constraint
on borrowers ($c_{t}\geq 0$) imply that optimal contracts will be simple
debt contracts of the form:\footnote{%
Without limited liability each borrower can consume negative amounts early
on, invest in $G$ projects, and then be self sufficient to continue to
invest, rendering the model uninteresting.}

\begin{itemize}
\item \textbf{``you get }$1$\textbf{\ today and you repay }$r_{t}$\textbf{\
at the end of this period.''}
\end{itemize}

We continue by treating each period $t$ as having four distinct stages:

\begin{enumerate}
\item Borrowers offer debt contracts with face value $r_{t}$ (can differ
across borrowers)

\item Lenders accept some contract or reject and use their outside option

\item BG borrowers choose projects

\item borrowers observe output and decide how much to repay (with
liquidation following repayments below $r_{t}$)
\end{enumerate}

\subsubsection{A 1-Period Economy}

If this economy were to last for just one period, $T=1$, the behavior of $BG$
types would depend on the parameters as follows. There are four obvious
facts about the equilibrium behavior:

\begin{enumerate}
\item If the debt repayment is equal to some $r_{T}$, then it is obvious
that no borrower would repay more than $r_{T}$ (this implies nothing about
the type of borrower).

\item If returns are above $r_{T}$ then no borrower would pay less than $%
r_{T}$ since this results in liquidation and zero utility.

\item In equilibrium, it cannot be that $r_{T}>G$ since then no $G$ type
would repay, and this implies that no investments would be profitable.

\item In equilibrium, $r_{T}\geq r$, since the probability of repayment is
no greater than 1, and lenders must receive an expected return of $r$.
\end{enumerate}

To continue, 1-4 above imply that the utility of a borrower from a $G$
project is $G-r_{T}$, while the utility from a $B$ project is $\pi (B-r_{T})$%
. Thus, a $BG$ type borrower will select a risky $B$ project if and only if $%
\pi (B-r_{T})>G-r_{T}$, or, 
\begin{equation}
r_{T}>\frac{G-\pi B}{1-\pi }\,.  \label{risky}
\end{equation}

This implies that the action of $BG$ types depends on the face value of
projects. However, given the competitive assumption on the debt market, a
rational expectations equilibrium will have $r_{T}$ depending on the
(correct) beliefs of lenders about the action of $BG$ types to guarantee a
return of $r$ to the lenders. Define, 
\begin{equation*}
r_{T}^{b}\equiv \frac{r}{\pi f_{B}+\pi f_{BG}+f_{G}}\;,
\end{equation*}
and, 
\begin{equation*}
r_{T}^{g}\equiv \frac{r}{\pi f_{B}+f_{BG}+f_{G}}\;.
\end{equation*}
That is, $r_{T}^{b}$ would be the equilibrium face value that yields a
return of $r$ if lenders conjecture that \textit{all} $BG$ types chose the $%
B $ project, and $r_{T}^{g}$ would be the equilibrium face value that yields
a return of $r$ if lenders conjecture that \textit{all} $BG$ types chose the 
$G $ project. It is easy to see that $r<r_{T}^{g}<r_{T}^{b}$, which implies,

\begin{description}
\item[claim:] There are four cases for the one-period economy:

\begin{enumerate}
\item If $r_{T}^{g}>G,$ or if $r_{T}^{g}>\frac{G-\pi B}{1-\pi }$ and $%
r_{T}^{b}>G,$ then no loans are made

\item If $r_{T}^{g}>\frac{G-\pi B}{1-\pi }$ and $r_{T}^{b}\leq G,$ then the
unique equilibrium has $r_{T}=r_{T}^{b}$, and $BG$ types choose $B$ projects

\item If $r_{T}^{b}<\frac{G-\pi B}{1-\pi }$ then the unique equilibrium has $%
r_{T}=r_{T}^{g}$, and $BG$ types choose $G$ projects

\item If $r_{T}^{g}<\frac{G-\pi B}{1-\pi }$ and $\frac{G-\pi B}{1-\pi }%
<r_{T}^{b}\leq G,$ then there are three equilibria: ($i$) $r_{T}=r_{T}^{b}$,
and $BG$ types choose $B$ projects, ($ii$) $r_{T}=r_{T}^{g}$, and $BG$ types
choose $G$ projects, and ($iii$) $r_{T}=\frac{G-\pi B}{1-\pi },$ and $BG$
types choose $G$ projects with probability $p=\frac{(1-\pi )r-(G-\pi B)[\pi
(f_{B}+f_{BG})+f_{G}]}{(G-\pi B)(1-\pi )f_{BG}}$
\end{enumerate}
\end{description}

The proof is rather straightforward. Case 1 describes two conditions that
prevent lending in equilibrium. The first condition, $r_{T}^{g}>G,$ implies
that there cannot be an equilibrium interest rate (with beliefs that all $BG$
types choose a safe project) that is less than $G,$ and from our previous
analysis this implies that no loans can be made. The second condition, $%
r_{T}^{g}>\frac{G-\pi B}{1-\pi }$ (if lenders believe that $BG$ types choose
safe projects, then these borrowers would actually choose risky ones) and $%
r_{T}^{b}>G$ imply that with pessimistic beliefs the interest is too high.
In case 2, $r_{T}=r_{T}^{g}$ is not consistent with equilibrium because this
would cause $BG$ types to choose $B$ projects, but $r_{T}^{b}$ is consistent
with equilibrium and promises expected returns of $r$. Similarly, in case 3, 
$r_{T}=r_{T}^{b}$ is not consistent with equilibrium because this would
cause $BG$ types to choose $G$ projects, but $r_{T}^{g}$ is consistent with
equilibrium and promises expected returns of $r$. Finally, in case 4, both
of these face values are consistent with equilibrium, and there is a third
equilibrium with mixed strategies.\footnote{%
In this equilibrium $p$ is calculated to guarantee that the correct face
value causes $BG$ types to be indifferent, that is, $p$ is chosen to solve, 
\begin{equation*}
\frac{G-\pi B}{1-\pi }=\frac{r}{\pi f_{B}+[p+(1-p)\pi ]f_{BG}+f_{G}}\;.
\end{equation*}%
Note that there is a fifth case which is similar to 4: If $r_{T}^{g}<\frac{%
G-\pi B}{1-\pi }$ and $r_{T}^{b}>G,$ then there are three equilibria: ($i$) $%
r_{T}=r_{T}^{b}$, and no loans are granted, ($ii$) $r_{T}=r_{T}^{g}$, and $%
BG $ types choose $G$ projects, and ($iii$) $r_{T}=\frac{G-\pi B}{1-\pi },$
and $BG$ types choose $G$ projects with probability $p=\frac{(1-\pi
)r-(G-\pi B)[\pi (f_{B}+f_{BG})+f_{G}]}{(G-\pi B)(1-\pi )f_{BG}}\,.$}

To make the problem interesting, the following assumption is made:

\begin{description}
\item[Assumption:] For all values of $f_{B},\,f_{BG},\,$and $f_{G}$ for
which the market does not fail, if $T=1$ then $BG$ types will choose the
risky project, that is, 
\begin{equation}
G>r>\frac{G-\pi B}{1-\pi }\,.  \label{A1}
\end{equation}
\end{description}

Note that since in equilibrium we must have $r_{T}\geq r$, then (\ref{A1})
implies (\ref{risky}) and thus risky projects are chosen. Thus, our analysis
results in the following lemma:

\begin{lemma}
(Diamond, 1989) At the final period $T$, (a) all borrowers offer a debt
contract with the lowest interest rate that provides an expected return of $%
r $; (b) all borrowers who can repay do so; and (c) Only borrowers for whom
beliefs suggest a high enough probability of being a $G$ type will get a
loan in period $T$.
\end{lemma}

\subsection{$T>1$ Periods}

\subsubsection{Lending Decisions, Repayment Decisions, and Rates:\newline
}

We will solve for a sequential equilibrium for the $T>1$ period model. Note
that there may be more than one sequential equilibrium, but we will
characterize general properties of any equilibrium.

From (\ref{A1}) above, it follows that if a borrower is revealed at time $T$
to be either a $B$ or a $BG$ type, then he will not receive a loan. Backward
induction implies,

\begin{lemma}
(Diamond, 1989) If at time $t\leq T$ a borrower is revealed to be a type $B$
or $BG$, then he will get no loans afterward
\end{lemma}

The following two lemmas follow (almost) immediately from the one period
analysis

\begin{lemma}
(Diamond, 1989) If a loan is made at date $t$ then $r_{t}\in [r,G]$
\end{lemma}

\begin{lemma}
(Diamond, 1989) If returns are above $r_{t}$ then borrowers repay exactly $%
r_{t},$ while if returns are below $r_{t}$ then the project is liquidated.
\end{lemma}

Lemma 3 follows the same logic as fact 3 in the characterization of the
1-period model above. Lemma 4 follows a similar logic to fact 2: If a
borrower repays less than $r_{t},$ then the project is liquidated giving him
zero, and he is revealed as not being a $G$ type, guaranteeing a future
utility of $V_{t+1}=0$. If he repays $r_{t}$ then he gets some positive
value this period, and a future value of $V_{t+1}\geq 0$. Paying more than $%
r_{t}$ just reduces this period's payoff since it cannot credibly signal his
type or his future payoffs. Lemmas 2 and 4 imply,

\begin{lemma}
(Diamond, 1989) Any payment less than $r_{t}$ at time $t$ leads to no loans
for all $t^{\prime }\geq t$.
\end{lemma}

Lemma 5 implies that in any equilibrium of the $T$ period model, interest
rates (face values) evolve as follows: Any default by a borrower $i$ leads
to termination of loans to that borrower. Any borrower who has not defaulted
until some time $t,$ will face a rate of $r_{t}$, which must be correct
using Bayes' rule on the remaining pool of borrowers and beliefs over the
actions of $BG$ types. Furthermore, all borrowers who ``survived'' up till
date $t$ will pool at the interest rate $r_{t}$ that gives lenders an
expected return of $r$.\footnote{%
As before, there is no way to credibly signal one's type by offering a menu
of rates.} Thus,

\begin{lemma}
(Diamond, 1989) All surviving borrowers at time $t$ offer the same rate $%
r_{t}$ that gives lenders an expected return of $r$.
\end{lemma}

To complete the characterization of equilibrium we need to specify the
action of $BG$ types, and show that there are rate sequences and action
sequences that together form an equilibrium.

\subsubsection{Project Choice and Rates:\newline
}

Let $V_{t}$ be the equilibrium value to a $BG$ type of getting a loan at
time $t$ and making optimal project and repayment decisions from period $t$
onward. From (\ref{A1}), $BG$ types choose $B$ in period $T,$ which yields a
value in this period of 
\begin{equation*}
V_{T}=\beta \pi (B-r_{T})\,.
\end{equation*}
Working backward, define for each $t<T$ the values $V_{t}^{b}$ and $%
V_{t}^{g} $ as the values from choosing a $B$ or $G$ project in period $t$,
and continuing with optimal choices from period $t+1$ onward. That is, 
\begin{equation}
V_{t}^{b}=\beta \pi (B-r_{t}+V_{t+1})\,,  \label{Vtb}
\end{equation}
and, 
\begin{equation}
V_{t}^{g}=\beta (G-r_{t}+V_{t+1})\,,  \label{Vtg}
\end{equation}
and we must have in equilibrium that 
\begin{equation*}
V_{t}=\max \{V_{t}^{b},V_{t}^{g}\}\,.
\end{equation*}
This implies that a safe project is chosen if and only if $\beta
(G-r_{t}+V_{t+1})\geq \beta \pi (B-r_{t}+V_{t+1})$, or, 
\begin{equation}
r_{t}-V_{t+1}\leq \frac{G-\pi B}{1-\pi }\;,  \label{safe-t}
\end{equation}
a condition similar to the reverse of (\ref{risky}). Notice here that the
value from choosing a safe project takes into account the cost, which is the
repayment $r_{t}$ \emph{less} the value of continuing as an active borrower.
This continuation value is a benefit from repaying, which happens for sure
if the safe project is chosen. This implies the following Lemma,

\begin{lemma}
(Diamond, 1989) Safe projects are optimal at time $t$ if and only if (\ref%
{safe-t}) holds, or equivalently, 
\begin{equation}
(1-\pi )V_{t}\geq \beta \pi (B-G)\,.  \label{Gt}
\end{equation}
Risky projects are optimal if and only if the opposite of (\ref{Gt}) holds.
\end{lemma}

Condition (\ref{Gt}) is derived from having (\ref{safe-t}) hold together
with $V_{t}=V_{t}^{g}$ as given by (\ref{Vtg}) above. The opposite is
derived from having (\ref{safe-t}) (weakly) violated together with $%
V_{t}=V_{t}^{b}$ as given by (\ref{Vtb}) above. Loosely speaking, the LHS of
(\ref{Gt}) is the loss from choosing a risky project, and the RHS is the
gain.

Interest rates are crucial in determining the relative attractiveness of
safe projects. Before determining these rates endogenously in equilibrium,
it is instructive to notice the following. Reducing $r_{t}$ at some $t$
makes safe projects more attractive at time $t$, and also increases $V_{t}$
(since both $V_{t}^{g}$ and $V_{t}^{b}$ are decreasing in $r_{t}$).
Increasing $V_{t}$ causes an increase in $V_{t-1}$, which makes safe
projects more attractive at time $t-1$.

This argument implies that a reduction of $r_{t}$ for all $t$ by the same
amount, makes the choice of safe projects increasingly attractive as the
date of the period goes back. This is at the heart of the intuition for
reputational concerns: if there is a stream of benefits that follows
\textquotedblleft good\textquotedblright\ behavior, then the earlier one is
in his career, the stringer are the incentives to pursue that stream.
Clearly, $r$ is a lower bound on $r_{t}$ for all $t$, and also, the more
periods there are in the future, the more attractive a safe project will be
(there is more to lose by choosing a risky project). Thus, if reputation
effects will not kick in with the lowest interest rate and the largest
number of periods, then there cannot be reputation effects for any
parameters. This intuition is captured in the following Lemma which offers a
necessary condition for reputation to work:

\begin{lemma}
(Diamond, 1989) Type BG borrowers will select the safe project on some date $%
t$ \underline{only if} 
\begin{equation}
\frac{\beta (G-r)}{1-\beta }\geq \frac{\beta \pi (B-r)}{1-\beta \pi }\;.
\label{infin}
\end{equation}
\end{lemma}

\begin{description}
\item[proof:] By (\ref{A1}) a risky project will be chosen at $T$. If a safe
project is sometimes optimal then there exists some last period $t^{\prime
}<T$ at which a safe project is chosen, and risky projects are chosen for
all $t>t^{\prime }$. Choosing safe at $t^{\prime }$ implies that (\ref%
{safe-t}) is satisfied, 
\begin{equation}
r_{t^{\prime }}-V_{t^{\prime }+1}\leq \frac{G-\pi B}{1-\pi }\;.
\label{proof1}
\end{equation}
and choosing risky projects thereafter implies, 
\begin{eqnarray*}
V_{t^{\prime }+1} &=&\sum_{t=1}^{T-t^{\prime }}(\beta \pi )^{t}(B-r_{t})\leq
\sum_{t=1}^{T-t^{\prime }}(\beta \pi )^{t}(B-r) \\
&<&\sum_{t=1}^{\infty }(\beta \pi )^{t}(B-r)=\frac{\beta \pi (B-r)}{1-\beta
\pi }\;,
\end{eqnarray*}
which together with (\ref{proof1}) and $r_{t}\geq r$ for all $t$ implies 
\begin{equation}
r-\frac{\beta \pi (B-r)}{1-\beta \pi }<\frac{G-\pi B}{1-\pi }\;,
\label{proof3}
\end{equation}
which is equivalent to (\ref{infin}). $Q.E.D.$
\end{description}

Note that the condition in (\ref{infin}) is very straightforward, and
implies that choosing the safe project and repaying $r$ in every period of
an infinite horizon economy is better than committing to a strategy of
always choosing a risky project, and repaying when it succeeds, while
terminating when it fails.

If the necessary condition (\ref{infin}) is satisfied, then a sufficient
condition on future rates for reputation to work is given as follows:

\begin{lemma}
(Diamond, 1989) Fix $t^{\prime }$. If for all $t\in \{t^{\prime },t^{\prime
}+1,...,T\},\,$%
\begin{equation}
r_{t}<\beta G+\frac{(1-\beta )(G-\pi B)}{(1-\pi )}\,,  \label{suffic}
\end{equation}%
then there exists $T<\infty $ such that 
\begin{equation*}
r_{t^{\prime }}-V_{t^{\prime }+1}\leq \frac{G-\pi B}{1-\pi }\;,
\end{equation*}%
and a safe project is chosen at $t^{\prime }$. Furthermore, (\ref{suffic})
is feasible ($r_{t}\geq r$) if and only if (\ref{infin}) is satisfied.
\end{lemma}

\begin{description}
\item[proof:] Since the borrower can always choose safe projects for all $%
t\geq t^{\prime }$ it follows that 
\begin{equation*}
V_{t^{\prime }}\geq \sum_{t=1}^{T-t^{\prime }}\beta
^{t}(G-r_{t})>\sum_{t=1}^{T-t^{\prime }}\beta ^{t}\left( G-\left[ \beta G+%
\frac{(1-\beta )(G-\pi B)}{(1-\pi )}\right] \right) \equiv \zeta (T)\;.
\end{equation*}%
But notice that 
\begin{eqnarray*}
\lim_{T\rightarrow \infty }\zeta (T) &=&\frac{\beta }{1-\beta }\left( G-%
\left[ \beta G+\frac{(1-\beta )(G-\pi B)}{(1-\pi )}\right] \right) \\
&=&\frac{\beta \pi (B-G)}{1-\pi }\,,
\end{eqnarray*}%
which implies that for $T$ large enough, $(1-\pi )V_{t^{\prime }}\geq \beta
\pi (B-G)$, and from Lemma 7 (\ref{Gt}) a safe project will be chosen at $%
t^{\prime }$. \newline
To see that this is feasible if and only if (\ref{infin}) holds, notice that
for (\ref{suffic}) to hold, it must be that 
\begin{eqnarray*}
r &<&\beta G+\frac{(1-\beta )(G-\pi B)}{(1-\pi )} \\
&=&\frac{\beta \pi (B-G)}{1-\pi }+\frac{G-\pi B}{1-\pi }\;,
\end{eqnarray*}%
which after rearranging is equivalent to (\ref{infin}). $Q.E.D.$
\end{description}

\subsubsection{Bayesian Rational Interest Rates:\newline
}

As the analysis of the one period model suggests, the interest rate $r_{t}$
is bounded by two rates: the high bound is with beliefs that of the
surviving borrowers, all $BG$ types choose $B$, and the low bound is with
beliefs that of the surviving borrowers, all $BG$ types choose $G$. For $t=1$
these are given by 
\begin{equation*}
r_{1}^{b}=\frac{r}{\pi f_{B}+\pi f_{BG}+f_{G}}\;,
\end{equation*}%
and, 
\begin{equation*}
r_{1}^{g}=\frac{r}{\pi f_{B}+f_{BG}+f_{G}}\;.
\end{equation*}%
These bounds change for surviving borrowers who never defaulted according to
Bayes' rule. Let $f_{\theta t}$ be the fraction of type $\theta \in
\{B,BG,G\}$ borrowers who have never defaulted up till period $t$. Clearly, $%
f_{Gt}=f_{G}$ for all $t$. Using a version of the law of large numbers, $%
f_{Bt}=\pi ^{t-1}f_{B}$. The fraction of $BG$ types depends on their
actions. If in period $t$ they choose $G$ with probability $\sigma \in
\lbrack 0,1]$, then $f_{BGt}=\sigma f_{BGt-1}+(1-\sigma )\pi f_{BGt-1}$.
This implies that the bounds for period $t$ are, 
\begin{equation*}
r_{t}^{b}=r\cdot \frac{f_{Bt}+f_{BGt}+f_{G}}{\pi f_{Bt}+\pi f_{BGt}+f_{G}}\;,
\end{equation*}%
and, 
\begin{equation*}
r_{t}^{g}=r\cdot \frac{f_{Bt}+f_{BGt}+f_{G}}{\pi f_{Bt}+f_{BGt}+f_{G}}\;.
\end{equation*}%
We will soon make use of the fact that if there is a countable number of $BG$
types (measure zero) then $r_{t}^{b}=r_{t}^{g}$ since the decisions of $BG$
types have no impact. This is a trick that is introduced into the model to
loose one factor that can cause multiple equilibria.

\subsubsection{The Endgame with $f_{BG}=0:$\newline
}

Lemmas 8 and 9 derived necessary and sufficient conditions for reputation to
work as $T\rightarrow \infty $ when the interest rate path is \textit{%
exogenous}, but in equilibrium it is \textit{endogenous}. It turns out that
the necessary condition in Lemma 8 is actually necessary and sufficient to
ensure that reputation will work as $T\rightarrow \infty $ if the loan
market is active.

We will say that reputation works if, as $T\rightarrow \infty ,$ the number
of periods for which $BG$ types choose risky projects is bounded. Formally,
the \emph{endgame} begins at a date $t^{\prime }$, where $t^{\prime }$ is
the smallest $t$ with $V_{t}^{g}\leq V_{t}^{b}$ which occurs after $all$
dates $t^{\prime \prime }$ for which $V_{t}^{g}>V_{t}^{b}$. From the fact
that $V_{T}^{g}<V_{T}^{b}$, backward induction implies that such a date $%
t^{\prime }$ exists. If the endgame is bounded then as $T\rightarrow \infty
, $ the number of periods for which reputation works is unbounded.

To see that the endgame is bounded, consider a restricted model with $%
f_{BG}=0,$ and $f_{B}=1-f_{G}\in (0,1).$ This implies $%
r_{t}=r_{t}^{g}=r_{t}^{b}$ is not affected by the behavior of $BG$ types. In
particular, if $r_{1}<G$ (so the market for loans is open at $t=1)$ then
interest rates are given by the following deterministic series, 
\begin{equation}
r_{t}=r\cdot \frac{\pi ^{t-1}f_{B}+1-f_{B}}{\pi ^{t}f_{B}+1-f_{B}}\,.
\label{rt0}
\end{equation}
These interest rates are charged to those borrowers who never defaulted. If $%
r_{1}>G$ then the market never opens. It is easy to see that in this case $%
r_{t}$ is decreasing in $t$, and that $\lim_{T\rightarrow \infty }r_{t}=r$.
This implies that as $T\rightarrow \infty $, there will be an arbitrarily
large number of periods in which for all $\varepsilon >0$, $%
r_{t}<r+\varepsilon $. Lemma 9 then implies,

\begin{lemma}
(Diamond, 1989) If $f_{BG}=0$ and $f_{B}=1-f_{G}\in (0,1),$ then condition (%
\ref{infin}) is necessary and sufficient for a bounded endgame as $%
T\rightarrow \infty $
\end{lemma}

For $f_{BG}>0$ but small, a continuity argument would suffice to prove the
same result, but this would not work for large values of $f_{BG}.$ Diamond
(1989) extends the analysis and proves the following,

\begin{description}
\item[Proposition 1:] (Diamond 1989) If the loan market is active, then the
endgame is of bounded length as $T\rightarrow \infty $ if and only if (\ref%
{infin}) holds.
\end{description}

\subsubsection{The Dynamics of Reputation:\newline
}

Proposition 1 explains what happens near the end of the game as $%
T\rightarrow \infty $. We are interested, of course, in what happens before
that. Assume that (\ref{infin}) holds, so that reputation will have value
early in the horizon. We will describe two special cases that build on the
restricted model with $f_{BG}=0.$

\paragraph{Moral Hazard without Adverse selection ($f_{BG}=$ $f_{B}=0$):}

\smallskip

In this case (\ref{rt0}) implies that $r_{t}=r$ for all $t$. From (\ref{A1})
we have $V_{T}=\beta \pi (B-r)$ and risky projects are selected. If it is
optimal to choose risky projects from period $t^{\prime }$ up till $T,$ then 
\begin{equation*}
V_{t^{\prime }}=\sum_{t=1}^{T-t^{\prime }}(\beta \pi )^{t}(B-r)\,.
\end{equation*}
But notice that this is a decreasing function in $t^{\prime },$ which
implies that if (\ref{infin}) holds, and $T$ is sufficiently large, then
there is some $t^{\prime }<T$ for which 
\begin{equation*}
r_{t^{\prime }}-V_{t^{\prime }+1}\leq \frac{G-\pi B}{1-\pi }\;,
\end{equation*}
and safe projects will be chosen from period 1 until period $t^{\prime }$.
We conclude with the result that is consistent with the classic game
theoretic reputation models\footnote{%
Kreps, Milgrom, Roberts and Wilson (1982), Kreps and Wilson (1982), Milgrom
and Roberts (1982).}:

\begin{description}
\item[Proposition 2:] With no adverse selection interest rates are constant,
and if reputation ever works it works immediately at $t=1$ and stops working
at some $t^{\prime }<T$.
\end{description}

\paragraph{Moral Hazard with Adverse selection ($f_{BG}=0,$ $f_{B}>0$):}

In this case $r_{t}$ follows (\ref{rt0}), and if $T\rightarrow \infty $ then
for some $t^{\prime }$ large enough, $r_{t}$ is close to $r$ for all $%
t>t^{\prime }$ which implies that the analysis for moral hazard with no
adverse selection applies from $t^{\prime }$ onward. However, the beginning
of the economy might be very different. In particular, with enough adverse
selection ($f_{BG}>0$ large enough) then $r_{1}$ may be too high to induce $%
BG$ types to choose safe projects. Therefore, as time goes by, those $BG$
types who did not default will face lower interest rates, and at some point
these may be low enough to induce selection of safe projects. That is, in
this scenario reputation effects are not strong enough initially, but for
those who succeeded to repay by the good fortune of having successful risky
projects, their reputation will at some point be worth maintaining. For the
case in which $f_{BG}=0$, $r_{t}$ falls over time, and we have:

\begin{description}
\item[Proposition 3:] (Diamond 1989) If $r_{t}$ falls over time and a type $%
BG$ borrower optimally selects safe projects at time $t^{\prime \prime }$
and risky projects at some $t^{\prime }<t^{\prime \prime }$, then risky
projects are optimal for all $t<t^{\prime }$. This implies that if safe
projects are optimal at two dates $t_{1}<t_{2},$ then safe projects are
optimal for all $t\in \{t_{1},t_{1}+1,...,t_{2}\}.$
\end{description}

It is interesting to note that proposition 3 does not use the length of the
horizon $T$. There are two reasons to choose risky projects. The first is
having a short horizon ahead (so maintaining a reputation has low returns),
and the second is if interest rates are too high. Thus, since interest rates
are falling (conditional on not defaulting) in this special case, then if a
safe project is selected at time $t$, it follows that the only reason to
switch to risky projects later is a short horizon. This argument implies
that such a switch is irreversible: if at some date $t^{\prime }<t$ there
was a risky project selected, then it must be due to high interest rates,
and therefore risky projects should be chosen for all $t^{\prime \prime
}<t^{\prime }$.

Diamond then shows that Propositions 2 and 3 generalize to the case with $%
f_{BG}>0$. Intuitively, if there are very few $B$ types, and $BG$ types are
assumed to choose safe projects, then $r_{1}$ is close to $r$ and if
condition (\ref{infin}) is satisfied then reputation will work immediately.
It is a larger pool of $B$ types that would cause initial interest rates to
rise, which may not be consistent with $BG$ types choosing safe projects. In
this case, there will be a period of reputation acquisition where both $B$
and $BG$ types will be thrown out of the loans market. When the interest
rate drops enough to create incentives to maintain the reputation
accumulated by those $BG$ borrowers who were lucky enough not to default.
This dynamic of reputation acquisition is a nice contribution to the
literature on signalling models of reputation.

\clearpage

\section{Reputation as an Asset}

\subsection{Introduction}

In the reputation models described above, agents will create a reputation of
being good, and the market will rationally infer this equilibrium behavior.
In this setup, however, incentives to be good disappear as agents approach
the end of their active career. If the concerns to maintain a reputation can
be extended to terminal periods by using an indicator that survives beyond
an agent's active career, then this problem might be mitigated. Indeed,
there is a fundamental difference between an individual's reputation and a
firm's reputation: a firm's reputation is a tradeable asset. If a reputation
is acquired under a firm's name, or \textit{entity}, and it is separated
from the \textit{identity} of the firm's owner, then incentives might
survive throughout the owner's career. This section discusses some models
that capture reputation as a tradeable asset, and analyze its effects on the
incentives of agents who sell off their firm's reputation.

\subsection{Moral Hazard and Repeated Games}

A first attempt to model this idea was made by Kreps (1990). In his model, a
buyer can either ``Trust'' a seller or ``Not trust'' him, The seller can
then ``Honor'' or ``Abuse'' trust. Not trusting implies a payoff of 0 for
both agents. Trust followed by honor results in a payoff of 1 for each
agent, while trust followed by abuse results in a payoff of 2 to the seller,
and -1 to the buyer.

It is easy to see that in the one-period version of this game, the unique
equilibrium has no trust offered, since if trust is offered then it will
clearly be abused. Any finite period version of this game will have the same
property by backward induction.

If, however, this game is repeated infinitely often with some discount
factor $\beta <1$, then a version of the folk theorem would apply: there are
large enough discount factors for which the following ``Nash reversion''
strategies form a (subgame perfect) equilibrium of the game: The buyer
starts by offering trust, and continues to offer trust as long as trust was
never abused. If trust was ever abused then the buyer will cease to offer
trust forever. The seller's strategy is to honor trust as long as he never
abused trust in the past, and to abuse trust otherwise. It is easy to see
that for discount factors $\beta \geq 0.5$ this is indeed an equilibrium.

What happens, however, if there is instead a sequence of buyers, each living
for only one period, who face this long lived seller? Basically, the same
argument would apply: if the seller discounts the future at a rate $\beta
\geq 0.5$ then a simple revision of the strategies is an equilibrium: The
first buyer starts by offering trust, and each subsequent buyer will offer
trust if and only if trust was never abused. The seller's strategy is the
same. Kreps refers to this equilibrium as one in which the seller has a 
\textit{reputation} of honoring trust, and this reputation is a valuable
asset that is not worth destroying by abusing it.

Now, what happens if the seller is too replaced by a sequence of short lived
sellers? Kreps demonstrates that reputation can become a tradeable asset --
and provide incentives -- even when buyers live for only one period. The
argument is simple and appealing: a ``name'' will be chosen for the firm by
the first seller. Each subsequent seller will be trusted by the buyer of
that period if and only if trust was never abused in the past, \emph{and}
the current seller acquired the firm's ``name'' from his predecessor.
Sellers' strategies will be to buy the ``name'' and honor trust if and only
if trust was never abused. Thus, a seller will be able to sell his own good
name if and only if he himself honors the trust of the client. If the loss
from not being able to sell a name outweighs the benefits from abusive
behavior, then sellers will have incentives to honor trust, and cooperative
behavior is sustained in equilibrium.\footnote{%
For example, with $\beta =1$ and a name price $v>1$ this is an equilibrium
since the gain from abuse is 1, while the loss from abuse is $v>1$.}

The appealing feature of Kreps's equilibrium is that short lived agents
become ``ageless'' in the sense that they do not really face a terminal
period. The theory, however, has some shortcomings. First, there are many
equilibria in which the firm is not bought, its name has no value, and thus
no incentives are provided. These ``bad'' equilibria are no more or less
likely than the ``good'' reputational equilibrium. Furthermore, the good
equilibrium is supported by clients coordinating their punishments in
response to previous poor behavior of the firm's owner. The extent to which
their reaction is rational depends on past actions that are irrelevant to
future fundamentals, since past behavior has no \textit{direct} link to
future behavior but is indirectly linked through the players' strategies.
Indeed, if one restricts beliefs and actions to depend only on
payoff-relevant information, then only the non-reputational equilibrium
survives. Second, this theory is mute with respect to how reputations arise
and become valuable assets. In particular, there is no account of how a
firm's reputation, represented by the value of its name, may increase (or
decrease) in value as is commonly observed in reality. This too is due to
the fact that client beliefs are not tied down, but are rather assigned
arbitrarily. Finally, the price of a name is not directly linked to the true
value of a reputation, but rather acts like a bond that prevents abusive
behavior.

\subsection{A Market Approach with Moral Hazard and Adverse Selection}

This section will build on the model presented in Tadelis (2002). The first
main insight provides a rationale for an active market for firm reputations.
The second main insight is that the market for names, which is supported by
the presence of adverse selection, can alleviate the problems associated
with moral hazard despite the finite life horizon of agents. The reputation
concerns provide incentives for opportunistic agents throughout their
career: young agents are concerned with their future income, while old
agents are concerned with the value of their firm's name. It is shown that
the incentives provided by these two reputational concerns are \textit{%
quantitatively} the same; the market for names causes agents to behave as if
they have no terminal period since they internalize the future value of
their good name. Thus, the ``ageless'' feature of Kreps's exogenously chosen
equilibrium arises endogenously in the present model. The third insight is
that the market for names cannot clearly separate between ``good'' and
``bad'' agents. In particular, there is no equilibrium in which good agents
fully separate themselves by buying successful names. A direct implication
of this result is that the model generates sensible reputation dynamics:
reputations increase after good performance and decrease after bad
performance, a crucial characteristic for reputations to provide incentives.

\subsubsection{The Model}

In each period a risk neutral buyer employs a risk neutral seller for that
period only. The service provided is either a success, valued at 1, or a
failure, valued at 0. Clients face adverse selection and moral hazard. There
are two indistinguishable types of $\ $sellers: good $\ $sellers ($G$-types)
in proportion $\gamma $, and opportunistic $\ $sellers ($O$-types) in
proportion $1-\gamma $. $G$-types succeed with probability $P_{G}\in (0,1)$,
while $O$-types choose the probability of success by exerting effort $e\in
[0,1]$ at a private cost $c(e)$, $c^{\prime }(e)>0,$ $c^{\prime \prime
}(\cdot )\geq 0$. An $O$-type's probability of success is given by $%
P_{O}(e)=eP_{G}$.

Assume that sellers are active for two periods, and that wealth is valuable
after they retire. Sellers have overlapping-generations (OLG) demographics,
in which the total size of the population, and the distribution of types is
constant over time. In contrast, clients live for only one period and can
only observe the firms' (names') track records.\footnote{%
Similar to Diamond (1989), the implication of this assumption is that a
firm's reputation, summarized by its past performance, is the only
intertemporal linkage.}

Each $\ $seller is a firm, which is represented by a \textit{name}, and no
two firms can share the same name. A$\ $seller can either \textit{choose} a
new name (with no track record) or he can \textit{buy} a name from a$\ $%
seller who is about to retire, thus inheriting the track record associated
with that name. The value of a firm's service is determined by the perfect
observation of that firm's past performance.

There is a continuum of clients and $\ $sellers, and the price of supplying
a service is determined competitively. To simplify, assume that the clients
are plentiful so that competition causes prices to be bid up to the client's
expected surplus.

\begin{description}
\item[Assumption A1:] \textit{Compensation cannot be based on the
transaction's outcome.}
\end{description}

\noindent That is, problems of verifiability exclude outcome-contingent
contracts. This implies that each client who employs an $\ $seller will pay
up-front for the expected value of the service supplied.

\begin{description}
\item[Assumption A2:] \textit{Shifts of name-ownership are not observable by
clients.}
\end{description}

This implies that a client employing a firm with a history cannot determine
whether the current $\ $seller running the firm is himself responsible for
that history, or whether he has just bought it. Thus, the actual identity of
the $\ $seller who provides the service is separated from the firm's entity.
This introduces noise into the economy: the impact of the current owner on
the firm's past performance is uncertain.

\begin{description}
\item[Assumption A3:] \textit{At the beginning of each period every active $%
\ $seller can either choose to retain his past name or unobservably change
it.}
\end{description}

\noindent This assumption is symmetric to assumption A2. Once the $\ $seller
chooses a new name, then his past performance is erased.\footnote{%
An agent can also abandon his past name and then buy a name from another
agent. In all the equilibria of the model presented it turns out that agents
who wish to abandon their past are indifferent between choosing a new name
or buying a name. Therefore,assume that agents who wish to erase their past
will just choose a new name.}

\begin{description}
\item[Assumption A4:] \textit{With probability }$\varepsilon \geq 0$\textit{%
\ each$\ $seller cannot change his name.}
\end{description}

\noindent This assumption eliminates some ``unreasonable belief'' equilibria
that can arise when all $\ $sellers can change their name costlessly. The
time line that describes a single period appears in Figure 1.

\vskip 60pt

\setlength{\unitlength}{.35in}\hskip 30pt%
\begin{picture}(14,2)
\put(1,2.07){\vector(1,0){12}}
\put(1,1.72){$\uparrow$}
\put(2.3,2.19){$\downarrow$}
\put(6.6,2.19){$\downarrow$}
\put(7.5,1.72){$\uparrow$}
\put(11.5,2.19){$\downarrow$}
\put(11.5,1.72){$\uparrow$}
\put(-.3,.8){\makebox(2,1.1)[l]{\footnotesize New agent chooses}}
\put(-.3,.4){\makebox(2,1.1)[l]{\footnotesize or buys name}}
\put(1,2.7){\makebox(2,1.1)[l]{\footnotesize Client hires firm}}
\put(1,2.3){\makebox(2,1.1)[l]{\footnotesize and pays upfront}}
\put(5.3,2.7){\makebox(2,1.1)[l]{\footnotesize Opportunistic type}}
\put(5.3,2.3){\makebox(2,1.1)[l]{\footnotesize chooses effort}}
\put(6.3,.8){\makebox(2,1.1)[l]{\footnotesize Success or failure}}
\put(6.3,.4){\makebox(2,1.1)[l]{\footnotesize realized}}
\put(10.2,2.7){\makebox(2,1.1)[l]{\footnotesize Retiring agent}}
\put(10.2,2.3){\makebox(2,1.1)[l]{\footnotesize can sell name}}
\put(10.2,.8){\makebox(2,1.1)[l]{\footnotesize Continuing agent}}
\put(10.2,.4){\makebox(2,1.1)[l]{\footnotesize can change name}}
\put(13,1.9){\makebox(2,1.1)[l]{\footnotesize time}}

\end{picture}

\textbf{\centerline{\bf Figure 1: The time line for each period.}}\vskip 20pt

\subsubsection{Benchmark: No Reputation Markets}

This section outlines a simple benchmark that will help relate the model to
the previous reputation models. We restrict attention to two periods, each
period having the sequence of events depicted in Figure 1 above. To capture
the essence of an OLG economy there will be three generations of equal size
as described in Figure 2 below:

\vskip 5pt

\hskip 200pt$t=1$\hskip 57pt$t=2$

\hskip 80ptGeneration 0:\hskip 10pt\TEXTsymbol{\vert}\_\_\_\_\_\_\_\_\_\_%
\TEXTsymbol{\vert}

\vskip 2pt

\hskip 80ptGeneration 1:\hskip 10pt\TEXTsymbol{\vert}\_\_\_\_\_\_\_\_\_\_%
\TEXTsymbol{\vert}\_\_\_\_\_\_\_\_\_\_\TEXTsymbol{\vert}

\vskip 2pt

\hskip 80ptGeneration 2:\hskip 105pt\TEXTsymbol{\vert}\_\_\_\_\_\_\_\_\_\_%
\TEXTsymbol{\vert}

\vskip10pt\textbf{\centerline{\bf Figure 2: A two period economy.}}\vskip %
20pt

Throughout this benchmark section we assume that names cannot be traded.
This implies that all $\ $sellers of generation 2 will have new names (no
histories) at the beginning of period 2. Sellers of generation 1 will have
the choice of sticking to their history or changing it, and $\ $sellers of
generation 0 will retire, and their names will disappear with them.

This model is solved by a dynamic Rational Expectation Equilibrium (REE) as
follows. In period $t=1$ all $\ $sellers have no history, and the wage $%
w_{1} $ will depend on clients' beliefs with respect to the effort level of
the $O$-types. In period $t=2$ there will be three wages depending on
whether a$\ $seller had a past success ($S$), a past failure ($F$), or a new
name ($N$). Denote these wages by $w_{2}(S)$, $w_{2}(F)$ and $w_{2}(N)$
respectively. Two facts are obvious:

\begin{enumerate}
\item All $\ $sellers who failed in the first period will be better off
changing their name and ``mingling'' with the new $\ $sellers (this is
easily verified using Bayes rule). Thus, one can restrict attention to
equilibria in which only $S$ and $N$ histories are observed in the second
period.

\item $O$-types will choose $e=0$ in the second period, so $w_{2}(h)$ only
depends on clients' beliefs about the likelihood of $h$ belonging to a $G$%
-type.
\end{enumerate}

Assume without loss that second period income is not discounted so that the
expected utility of an $O$-type at $t=0$ is, 
\begin{eqnarray*}
Eu_{O} &=&w_{1}+eP_{G}w_{2}(S)+(1-eP_{G})w_{2}(N)-c(e) \\
&=&w_{1}+eP_{G}\Delta w+w_{2}(N)-c(e)\,\,,
\end{eqnarray*}
where $\Delta w\equiv w_{2}(S)-w_{2}(N)$. The $O$-type's choice of effort
will affect second period wages, and these in turn feedback into the
incentives of $O$-types. Thus, an equilibrium will be characterized by the
tuple $\left\langle w_{1},w_{2}(S),w_{2}(N),e\right\rangle $ such that $e$
is a best response given $w_{2}(S)$ and $w_{2}(N)$, and all wages are
correct given rational expectations about $e$. In particular, given an
effort level $e\in [0,1]$ equilibrium beliefs imply that, 
\begin{equation}
\Pr \{G|S\}=\frac{\gamma P_{G}}{\gamma P_{G}+(1-\gamma )eP_{G}}\text{ ,}
\label{PrS}
\end{equation}
and, 
\begin{eqnarray}
\Pr \{G|N\} &=&\frac{\gamma (1-P_{G})+\gamma }{\gamma (1-P_{G})+(1-\gamma
)(1-eP_{G})+1}  \label{PrN} \\
&=&\frac{2\gamma -\gamma P_{G}}{2-P_{G}[\gamma +(1-\gamma )e]}\text{ .} 
\notag
\end{eqnarray}
That is, a firm with a past success is generated by $G$-types and $O$-types
from generation 1 who succeeded, which accounts for (\ref{PrS}) above. A
firm with a new name is generated by all new (generation 2) $\ $sellers, and
by all the $\ $sellers from generation 1 who failed (and then changed their
name), which accounts for (\ref{PrN}) above. Equilibrium wages are $%
w_{2}(h)=\Pr \{G|h\}\cdot P_{G}\,$.

Fixing $\Delta w$, the equilibrium $e$ solves the FOC, 
\begin{equation*}
P_{G}\Delta w=c^{\prime }(e).
\end{equation*}
It is easy to see that in equilibrium the level of effort will be
sub-optimal, which follows from the fact that optimal effort must solve $%
P_{G}\geqslant c^{\prime }(e)$.\footnote{%
This follows from maximizing social surplus: $eP_{G}-c(e),\;e\in [0,1]$.
Since in equilibrium $0<w(N)<w(S)<1$, then $\Delta w<1$ and sub-optimal
effort arises in equilibrium.} It is also easy to see that $\Delta w$ is
decreasing in $e$ (this follows from (\ref{PrS}) and (\ref{PrN}) above), so
that if $O$-types are choosing higher effort in equilibrium, then the wage
differential is smaller. Thus we can calculate the highest wage differential
when all $O$-types are ``bad'' and choose $e=0$, and the lowest wage
differential when all $O$-types behave as $G$-types ($e=1$). Let

\begin{equation*}
\Delta w_{B}=\frac{2(1-\gamma )P_{G}}{2-\gamma P_{G}}\;,\text{ \ and\ \ }%
\Delta w_{G}=0\;,
\end{equation*}
as the \textit{bad} and \textit{good} \textit{wage differentials}
respectively.

\begin{description}
\item[Proposition 1:] \textit{There is a unique equilibrium of the
two-period model with no reputation markets. If }$P_{G}\Delta w_{B}\leqslant
c^{\prime }(0)$ \textit{then }$e=0$\textit{\ in equilibrium. If }$%
P_{G}\Delta w_{B}>c^{\prime }(0)$ \textit{then }$e\in (0,1)$\textit{\ in
equilibrium.}

\item[\protect\underline{{\protect\footnotesize proof:}}] {\footnotesize If $%
c^{\prime }(0)\geqslant \Delta w_{B}$ and all $O$-types choose $e=0$ in the
first period then $\Delta w=\Delta w_{B},$ and no $O$-type has an incentive
to increase $e$ since $c^{\prime }(0)\geqslant \Delta w_{B}$ and $c^{\prime
\prime }(e)>0,$ implying that this is an equilibrium. No other equilibrium
can be sustained since any higher effort level implies a lower $\Delta w\,$%
implying that $c^{\prime }(\widehat{e})>\Delta w$. Since $c^{\prime }(1)>0$
and $\Delta w_{G}=0$, then choosing $e=1$ in the first period cannot be an
equilibrium. If $c^{\prime }(0)<\Delta w_{B},$ $e=0$ can be supported in
equilibrium due to the $O$-type's FOC. Since $\Delta w$ is continuous and
decreasing in $e,$and since $c^{\prime }(e)$ is increasing in $e,$ there
exists a unique $e\in (0,1)\ $such that $c^{\prime }(e)=\Delta w$ and this
is the unique equilibrium. $Q.E.D.$ }
\end{description}

The intuition is similar to the previous reputation models we saw. If
opportunistic sellers are not expected to exert any effort in the first
period, then first period success must be attributed to good types, and the
wage differential is greatest, which can provide incentives if the marginal
cost of effort is not too high. To summarize, career concerns may help solve
the moral hazard problem for young$\ $sellers, but not for old ones.

\subsubsection{The Market for Reputations}

We now depart from the benchmark model and allow trade of names. As before,
clients will pay up-front for services given their expected benefit. In
period $t=1$ no prior information is available to the clients, so they will
pay the same wage to all firms. In period $t=2$ there will be two kinds of
firms: some firms will have a past history while others will not. In turn,
firms with a past history of success can either be operated by a seller who
succeeded and continued, or by a new $\ $seller who bought the name.

An equilibrium will be characterized by the wages that clients pay in $t=1$,
by the strategies of $O$-types of each generation in every period, and by
prices in two markets in $t=2$: the wages clients will pay for hiring firms
with different track records and the prices $\ $sellers will pay for names
with different track records. Since only past histories are observable then
two distinct names with the same history should generate the same
expectations for future success in $t=2$. For this reason let $S$ denote any
name in $t=2$ with a past success, $F$ with a past failure, and $N$ a name
with no past. For simplicity only allow retiring $\ $sellers to possibly
sell their name, and only new $\ $sellers to possibly buy names in $t=2$.
Also, restrict attention to equilibria in which only $S$ names are traded,
so that all $\ $sellers who fail in the first period will change their name.
As before, denote wages by $w_{2}(h)$, $h\in \{S,F,N\}$, and the equilibrium
price of $S$ names in $t=2$ will be denoted $v(S)$. Also, let $\Pr \{S|h\}$
denote the conditional probability of a firm succeeding given that its
history is $h$.

\begin{description}
\item[Proposition 2:] $S$ \textit{names will be traded in all equilibria.}

\item[\protect\underline{{\protect\footnotesize proof:}}] {\footnotesize %
Assume that there exists an equilibrium in which no names are traded. This
implies that the value of a name with a past success must be zero since the
supply of $S$ names is positive with measure $\gamma P_{G}+(1-\gamma -\beta
)eP_{G}>0$. This in turn implies that $w_{2}(S)\leq w_{2}(N),$ since
otherwise $S$ names would be valuable to sellers in $t=2$. $O$-types of
generation 0 earn a wage in $t=1$ that does not depend on their action, and
since names are not traded it is a dominant strategy to choose $e=0$. Now
consider the $O$-types of generation 1 in the first period. Their wage in $%
t=1$ also does not depend on their action, but their performance can affect
their wage in $t=2$. However, since $w_{2}(S)\leq w_{2}(N),$ then choosing $%
e=0$ in the first period is a dominant strategy. Furthermore, if names are
not traded then Assumption A4 implies that $\Pr \{S|S\}=P_{G}$,\footnote{%
{\footnotesize Assumption A4 guarantees that $\Pr \{G|S\}=1$ when no names
are traded. This assumption rules out equilibria of the following form: All
agents abandon their name after the first period, and in the second period
clients believe that a firm with \textit{any} history is worse than the
average firm with no track record. No new agent would choose to buy any
name, and all agents would indeed change their name. But, if a proportion $%
\varepsilon $ of the population will have their names stick then by Bayes
rule, $\Pr \{G|S\}=1$, which restores trade in names.}} and $1-\gamma >0$
implies that $\Pr \{S|N\}<P_{G}$. This in turn implies that $%
w_{2}(S)>w_{2}(N)$, a contradiction. $Q.E.D.$ }
\end{description}

Since at least $G$-types succeed, there is always a supply of $S$ names for
sale at the beginning of the second period. Therefore, if there is no trade
of such names then having a past success must be worthless to sellers at the
beginning of the second period. This implies that opportunistic sellers in
the first period have no incentive to exert effort because it generates no
future benefits. But then, if good names are not traded, then successful
histories \textit{must} belong to $G$-types of generation 1. This would
create expectations of success, which means that new sellers will be willing
to buy these names, and disguise themselves as good types, since trading
names is not observable.

With full observability of ownership shifts trade in names can still be
supported in equilibrium, but other equilibria arise: assigning beliefs to
clients that only $O$-types who will choose $e=0$ buy names will support
equilibria with no trade of names.\footnote{%
If a small proportion of the clients observe name trading, these clients'
beliefs must correspond to the actual buyers of names. This will lower the
value of names, but if the proportion of informed clients is not too large
then proposition 2 still holds.} Thus, the lack of observability guarantees
an active market for names. To characterize equilibrium, the following
result is helpful:

\begin{description}
\item[Lemma 1:] \textit{In any equilibrium, all new $\ $sellers will be
indifferent between buying a }$S$\textit{\ name and not buying one, and the
price of a }$S$\textit{\ name is }$v(S)=w_{2}(S)-w_{2}(N).$
\end{description}

This follows because in any equilibrium $w_{2}(h)$ depends only on clients'
beliefs and not on the outcome of the second period, implying that all types
have the same benefit from buying a $S$ name. Since the supply of $S$ names
is less than (measure) 1, and the potential demand is 1 (the measure of new $%
\ $sellers) then the price of a $S$ name must cause indifference to clear
the market. This yields,

\begin{description}
\item[Proposition 3:] \textit{In any equilibrium, all first period} $O$-%
\textit{types have identical incentives and thus choose the same effort
level.}
\end{description}

The intuition is simple: since the price of a name will be $%
v(S)=w_{2}(S)-w_{2}(N)=\Delta w$ then the wage differential that $O$-types
of generation 1 face when they are young is equal to the \emph{sales premium}
that $O$-types from generation 0 get from selling a successful name. Thus,
the incentives provided to young are identical to those provided to ``old''.

The market equilibrium analysis shows that wages and name prices are tightly
connected, which in turn implies that incentives to work are \textit{%
independent} of an $\ $seller's future horizon. This was generated by a
model in which $\ $sellers are active for only two periods, but the economic
intuition behind the result seems to be quite general.

The only equilibrium parameter that affects the first period wage is the
(correct) beliefs clients have about $e$ in $t=1$. In particular,

\begin{equation*}
w_{1}=\left[ \gamma +(1-\gamma )e\right] P_{G}\,.
\end{equation*}
In the second period all $O$-types will choose $e=0$. Let $\mu $
(respectively $\rho $) denote the proportion of $G$ (respectively $O$) types
who buy $S$ names in $t=2$. An equilibrium for the two period model will be
a tuple $\left\langle \mu ,\rho
,w_{1},w_{2}(S),w_{2}(F),w_{2}(N),v(S),e\right\rangle $. Note that the wages
firms charge clients, and the prices new sellers are willing to pay for
names will be generated by the correct beliefs about $(\mu ,\rho ,e)$, so
that a triplet $(\mu ,\rho ,e)$ will in fact uniquely determine the other
equilibrium parameters.

In equilibrium $(\mu ,\rho ,e)$ must satisfy the market clearing condition, 
\begin{equation}
\gamma P_{G}+(1-\gamma )eP_{G}=\mu \gamma +\rho (1-\gamma )\text{ .}
\label{clear}
\end{equation}
so the supply of $S$ names (the left hand side of (\ref{clear})) is equal to
the demand (the right hand side). Since $O$-types choose $e=0$ in $t=2$
(like in the benchmark case) then for all $h,$ $w_{2}(h)=\Pr \{S|h\}=\Pr
\{G|h\}\cdot P_{G}$. Given $(\mu ,\rho ,e)$ that satisfy (\ref{clear})
above, 
\begin{eqnarray}
\Pr \{G|S\} &=&\frac{\gamma P_{G}+\mu \gamma }{\gamma P_{G}+(1-\gamma
)eP_{G}+\mu \gamma +\rho (1-\gamma )}  \label{PrG;S} \\
&&  \notag \\
&=&\text{ }\frac{\gamma P_{G}+\gamma \mu }{2\gamma P_{G}+2(1-\gamma )eP_{G}}%
\text{ ,}  \notag
\end{eqnarray}
and, 
\begin{eqnarray}
\Pr \{G|N\} &=&\frac{\gamma (1-P_{G})+(1-\mu )\gamma }{%
\begin{array}{c}
\gamma (1-P_{G})+(1-\gamma )(1-eP_{G}) \\ 
+(1-\mu )\gamma +(1-\rho )(1-\gamma )%
\end{array}
}  \label{PrG,Phi} \\
&&  \notag \\
&=&\frac{2\gamma -\gamma P_{G}-\mu \gamma }{2-2\gamma P_{G}-2eP_{G}(1-\gamma
)}\text{ ,}  \notag
\end{eqnarray}
where the second equality in both equations follows from market clearing and
some simple algebra. The following proposition characterizes the set of
equilibria in which only $S$ names are traded.

\begin{description}
\item[Proposition 4:] There exist $\underline{\mu }<\overline{\mu }$ so that 
$(\mu ,\rho ,e)$ is an equilibrium if and only if the following three
conditions hold:\newline
$(i)\;$ $\;\mu \in [\underline{\mu },\overline{\mu }]$\newline
$(ii)$ $\;(\mu ,\rho ,e)$ satisfy market clearing\newline
$(iii)\;c^{\prime }(e)=\Delta wP_{G}$
\end{description}

Proposition 4 implies that there is a continuum of equilibria with respect
to prices: the interval $[\underline{\mu },\overline{\mu }]$ is non-empty,
and any $\mu \in [\underline{\mu },\overline{\mu }]$ can be supported in
equilibrium. This follows from the indifference result established in Lemma
1, that is, as long as the price for names reflects the wage differential
that the name generates, sellers will be indifferent. In particular, the
equilibrium price for $S$ names is increasing in $\mu $ so that $\mu =%
\underline{\mu }$ supports the lowest price equilibrium. In this equilibrium
many bad types of generation 2 are buying the names so that the price of a $%
S $ name is either zero (there are enough good types in the second period so
that having no history is not too harmful), or it is positive (there are too
few good types so that even when all $S$ names are bought by bad types, this
is still better than having no history). Any equilibrium with $\mu >%
\underline{\mu }$ commands a positive price for $S$ names. The price is
bounded by $\mu =\overline{\mu }$, which is when the largest possible number
of new good types are buying these names without violating market clearing.
The effect of the market for names on the incentives of opportunistic $\ $%
sellers in the first period is investigated in the next section.

\subsubsection{Incentive Effects}

As Proposition 2 implies, if a regulator can make the identity of owners
transparent, then pessimistic beliefs will cause the market for names to
shut down, resulting in the benchmark economy with no trade of names. This
provides a useful benchmark to see how the market for names affects the $O$%
-types' equilibrium effort level. By comparing the two models, with and
without a market for names, one can evaluate the welfare implications of a
regulators actions with respect to imposing disclosure of identities.

It is somewhat cumbersome to perform a welfare comparison of the two models
since there are multiple equilibria when names are traded. A way to proceed
is to identify a reasonable equilibrium selection and then compare the two
models. Since the multiplicity arises because sellers who buy names are
indifferent, it is reasonable to argue that the matching between buyers and
sellers of names is random,\footnote{%
In Tadelis (2002) this is derived by adding another source of heterogeneity
that is $iid$ across types.} which in turn implies that the composition of
name buyers is equal to the composition of the population of sellers. That
is, $\mu $ and $\rho $ are given by, 
\begin{equation*}
\mu ^{*}=\rho ^{*}=\gamma P_{G}+(1-\gamma )eP_{G}.
\end{equation*}

In equilibrium $\mu ^{*}$ will be simultaneously determined with $e$ so that 
$e$ is a best response of the $O$-types. Fixing $\mu ^{*}$ we can compute
the bad and good wage differentials for this equilibrium (as we did for the
benchmark model), 
\begin{equation*}
\Delta w_{B}(\mu ^{*})=\frac{P_{G}(1-\gamma )}{2(1-\gamma P_{G})}\,,\text{
and }\Delta w_{G}(\mu ^{*})=0\,\,.
\end{equation*}

As mentioned earlier, both with and without an active market for names there
is under-provision of incentives, so social surplus is therefore higher in
the setup that generates more total effort. The effects of the market for
names on the incentives of young$\ $sellers of generation 1 it is ambiguous
-- it may be that the introduction of a market for names will cause the wage
differential to decrease, thus lowering incentives. However, incentives may
be created for the ``old'' sellers of generation 0.

To see how parameters affect the welfare conclusions, it is again
illustrative to consider the extreme equilibria associated with the good and
bad wage differentials. Since $\Delta w_{G}=0$ is the same, consider the
change in $\Delta w_{B}.\ $If $\Delta w_{B}(\mu ^{*})<\Delta w_{B},$ then
there exists cost functions $c(\cdot )$ for which $\Delta w_{B}(\mu
^{*})<c^{\prime }(0)<\Delta w_{B}$. Thus, for this case young $\ $sellers
exerted some effort in the benchmark model, but with trade of names neither
young nor old sellers will exert effort in the first period. This is clearly
a decrease in social surplus. If, however, $\Delta w_{B}(\mu ^{*})>\Delta
w_{B}$ then the reverse is true: there exist cost functions for which no
sellers exert effort in the benchmark model, but when trade of names is
considered then both young and old $\ $sellers will exert some effort in the
first period. Thus, the market for names is bad if $\Delta w_{B}(\mu
^{*})<\Delta w_{B}$, which reduces to, 
\begin{equation}
P_{G}<\frac{2}{3\gamma }\,\text{.}  \label{dif1}
\end{equation}
In Tadelis (2002) there is a more detailed analysis that shows that this
ambiguity is true for interior solutions as well. Comparative statics show
that the social surplus when the market for names is active (relative to the
benchmark model) increases (that is, $e^{*}$ increases for both types) when $%
P_{G}$ or $\gamma $ increase. The intuition follows quite simply: As either $%
P_{G}$ or $\gamma $ increase, the wage differential and the proportion of
good types buying names increase. The latter effect does not appear in the
benchmark model.

Note that the analogy of generation 1 sellers with ``young'' $\ $sellers and
generation 0 $\ $sellers with ``old'' $\ $sellers is clear. The next section
considers the infinite horizon economy in which there is no terminal period
and shows that the results obtained above are robust.

\subsubsection{Longer Horizons and Sorting}

To analyze the incentive effects of a continuing market for names, it is
natural to consider the infinite horizon version of the model described
above. The following Proposition is the parallel of proposition 1 for the
infinite horizon model:

\begin{description}
\item[Proposition 6:] \textit{Names with a history consisting of only one
success and no failures must be traded in all equilibria.}
\end{description}

The proof of this proposition follows from the proof of Proposition 2. The
intuition is the same: these names have a positive supply, and under the
assumption of no trade it must be that they convey positive information and
have value.

Since names are associated with histories, this introduces a complicated
problem for the analysis of the infinite horizon model. Formally, the set of
histories $H$ is the set of all histories consisting of successes and
failures (including $N,$ no history). Then, even when concentrating \textit{%
only} on Steady State Equilibria (SSE), the analysis becomes quite
intractable.

A way to simplify the analysis is to distinguish between histories and
reputations (see Tadelis (2001)). Appendix B introduces a formal reduction
of histories into equivalence classes using a well defined reputation
reduction. This reduction is rather straightforward and it uses the model's
stationarity.

Using this reduction, Appendix B analyzes a stationary SSE that is similar
qualitatively to the equilibrium of the two period model: names last as long
as they do not fail, and once they fail they are rationally discarded. The
wage differentials for the derived SSE are exactly those derived for the two
period model, but since there is no terminal period then incentives are
provided in every period. Note that the equilibrium identified in the finite
horizon benchmark model is the unique stationary SSE for the infinite
horizon model without trade of names.

An intuitive conjecture is that if the economy lasts for more than 2
periods, then the value of a good reputation should be higher for a good
type who is more likely to maintain it, than for an opportunistic type who
has to exert effort to do so. This reasoning would be consistent with Kreps
(1990), if his ideas were incorporated into the current incomplete
information framework. That is, if good types find it easier to maintain a
reputation, then they should be able to outbid opportunistic types who are
more likely to ruin a reputation. Notice, however, that in the equilibrium
analyzed above this separation did not occur. It turns out that this is no
coincidence, as the following proposition states:

\begin{description}
\item[Proposition 7:] \textit{For the infinite horizon model there is no
equilibrium in which only }$S$\textit{\ names are traded, and these are
bought only by good types and by opportunistic types who choose to be good.}
\end{description}

To see this, consider a candidate sorting equilibrium in which only good
types buy $S$ names. This means that clients must (correctly) predict that
any $S$ name is bought by a good type, so that a name with a history that
starts with a success is more likely to belong to a good type \textit{no
matter what the continuation history will be}. This may seem somewhat
counterintuitive, but the intuition is quite simple: If a name had a success
in period $t$, then the person who continues with this name at period $t+1$
is more likely to be good. Thus, failures will not be ``punished'' with low
wages since clients cannot update their beliefs too strongly, and the value
of the name will remain high.

This, in turn, causes $O$-types to value a $S$ name more than $G$-types
because the value of buying a $S$ name depends on the \textit{alternative
option} of not buying one. $O$-types face a less attractive future if they
start with a new name because it is harder for them to build a good
reputation. Therefore, if the stream of payments from having a $S$ name is
the same (or close) for both types, then having a poor alternative will make 
$O$-types value a $S$ name more than $G$-types. Thus, in any equilibrium
enough $O$-types (choosing $e<1$) buy success names so that clients will
sufficiently update their beliefs after a name with a good reputation starts
failing.

In Tadelis (1999) two reputational effects arise in a pure adverse selection
model. The \textit{reputation maintenance effect} captures the idea that
more able types are more likely to maintain a good name. This allows good
types to reap benefits over a longer horizon (on average), which in turn
gives them a higher willingness to pay for a good name. The \textit{%
reputation start-up effect} captures the idea that good types can more
easily build a good name. Therefore, if a firm's reputation does not
depreciate then good types will have a lower willingness to pay than bad
types. The introduction of moral hazard in this paper shows the generality
of these effects. It is the market equilibrium approach of both models that
identifies the alternative of buying a good name, which is building one.%
\footnote{%
Using terminology from the signaling literature, the ``single crossing''
condition in my model is endogenous, and depends on who buys names. Mailath
and Samuelson (2000) develop a repeated game model with incomplete
information and imperfect monitoring and show that when a reputation is
``too good'' it is more likely to be bought by a bad type. Given their
partial equilibrium approach, however, they need to \textit{exogenously}
assume that good types have a better outside option than bad types. In my
models the outside option is derived \textit{endogenously} using a market
equilibrium analysis.}

To summarize, an important difference between the model in this paper and
standard models of reputation is in the general competitive equilibrium
approach employed here. This is key in deriving two results. First, young
and old $\ $sellers face the same incentives created by the market. The two
markets in this model -- one for services and one for names -- are linked:
good names are scarce, and thus their price captures their full value, which
is the wage differential they generate in the market for services. Second,
reputations cannot fully sort good $\ $sellers from bad ones.

\subsubsection{Appendix A: Proofs}

\setlength{\baselineskip}{.15in}

\begin{description}
\item[\protect\underline{{\protect\footnotesize proof of Proposition 4:}}] 
{\footnotesize Market clearing must be satisfied in any equilibrium, thus $%
(\mu ,\rho ,e)$ must satisfy (\ref{clear}). Also, it must be the case that
in equilibrium $v(S)\geqslant 0$, or equivalently, $w_{2}(S)-w_{2}(N)%
\geqslant 0$ (otherwise no $\ $seller would by a $S$ name)$.$ Since $%
w_{2}(h)=\Pr \{G|h\}\cdot P_{G}, $ then using (\ref{PrG;S}) and (\ref%
{PrG,Phi}) above this inequality can be written as, 
\begin{equation*}
\frac{\gamma P_{G}+\gamma \mu }{2\gamma P_{G}+2(1-\gamma )eP_{G}}\geqslant 
\frac{2\gamma -\gamma P_{G}-\mu \gamma }{2-2\gamma P_{G}-2eP_{G}(1-\gamma )}%
\text{ ,}
\end{equation*}
which after rearranging becomes, 
\begin{equation}
\mu \geqslant (2\gamma -1)P_{G}+2eP_{G}(1-\gamma )\;.  \label{delta_tilde}
\end{equation}
Denote the RHS of (\ref{delta_tilde}) by $\widetilde{\mu }.$ Observe that, 
\begin{eqnarray*}
\widetilde{\mu } &\equiv &P_{G}(2\gamma -1+2e(1-\gamma )) \\
&<&P_{G}(2\gamma -1+2(1-\gamma )) \\
&=&P_{G}\;,
\end{eqnarray*}
Where the first (weak) inequality follows from $\widetilde{\mu }$ increasing
in $e$, and from $e<1.$ Define $\underline{\mu }\equiv $ Max$\{0,\widetilde{%
\mu }\}$, and note that to satisfy $v(S)\geqslant 0$ it must be that $\mu
\geq \underline{\mu }$. Now, let $\widehat{\mu }$ be the proportion of $G$%
-types that are needed to clear the market with no bad types buying $S$
names at $t=1$, which is derived from market clearing as follows, 
\begin{equation*}
\widehat{\mu }\equiv P_{G}\left[ 1+e\left( \frac{1}{\gamma }-1\right) \right]
\;.
\end{equation*}
Notice that $\widehat{\mu }\geq P_{G}$ since $\frac{1}{\gamma }\geq 1$.
Also, for certain parameter values (e.g., $\gamma <<1$) we can have $%
\widehat{\mu }>1$, which means that only good types buying $S$ names cannot
clear the market. Define $\overline{\mu }\equiv $ Min$\{\widehat{\mu },1\}$.
The interval $[\underline{\mu },\overline{\mu }]$ is non-empty which follows
from $\widetilde{\mu }<P_{G}$ and $\widehat{\mu }\geq P_{G}$. Thus, if $(\mu
,\rho ,e)$ is an equilibrium then it must be that $\mu \in [\underline{\mu },%
\overline{\mu }]$, that $(\mu ,\rho )$ satisfy market clearing, and that $e$
is a best response of the $O$-types in the first period which satisfies $%
c^{\prime }(e)=\Delta wP_{G}$. The converse follows immediately: if $\mu \in
[\underline{\mu },\overline{\mu }],$ $(\mu ,\rho ,e)$ satisfy market
clearing and $e$ satisfies $c^{\prime }(e)=\Delta wP_{G}$ for the $O$-types
in the first period, then the wages and prices generated by correct beliefs
will constitute an equilibrium, and $e$ is a best response. $Q.E.D$}
\end{description}

\subsubsection{Appendix B: Infinite Horizon Analysis}

{\footnotesize Let $H$ denote the set of all possible histories and let $%
h\in H$ be a generic history. To reduce the set $H$ to a smaller set of
reputations, consider four equivalence classes of histories. That is,
clients will treat any two histories in the same equivalence class with the
same beliefs. In particular, }

\begin{description}
\item[{\protect\small Definition 1:}] {\footnotesize Let $r:H\rightarrow
\{N,S$ $,M,F\}$ be a reputation reduction satisfying:\newline
$r(N)=N\,;$ \newline
$r(h)=S$ for all $h\in H_{S}\equiv \{NS,NSS,NSSS,...\};$ $\vspace{0in}$%
\newline
$r(h)=M$ for all $h\in H_{M}\equiv \{NSF,NSSF,NSSSF,...\};$\newline
$r(h)=F$ for all $h\in H\backslash (N\cup H_{S}\cup H_{M})$. }
\end{description}

{\footnotesize This reduction implies that clients take any one of the
infinite number of histories and map them into one of four classes: $\{N,S$ $%
,M,F\}.\,$The analysis that follows will focus on the unique stationary SSE
consistent with the equilibrium selection of random matching, and consistent
with restricting attention to the case where only $S$ names are traded (it
will be confirmed that $F$ or $M$ names will not be traded, and agents with $%
M$ or $F$ names would prefer to change them to $N$ names). That is, the
names from the equivalence class $H_{s}$ are randomly assigned to the young
agents who enter the economy. It will also be confirmed that the restriction
to these four classes is consistent with equilibrium beliefs, so that no
irrationality is involved. }

{\footnotesize Denote the SSE wages as $w(N),$ $w(S),$ $w(M)$ and $w(F)$
respectively. In each period there are ``young'' and ``old'' agents as in
any standard OLG model. The following result is parallel to Proposition 3: }

\begin{description}
\item[{\protect\small Proposition B1:}] {\footnotesize In any SSE, all young
and old $O$-types have identical incentives and thus choose the same effort
level. }

\item[{\protect\small proof:}] {\footnotesize If in equilibrium only $S$
names are traded then all other names are never better than a new name. This
implies that incentives do not depend on the name an agent has since any
failure will cause a change of name. Thus, the utility an agent $i$ who
succeeds with probability $P_{i}$ gets from buying an $S$ name is $%
u_{i}(S)=w(S)+P_{i}w(S)+(1-P_{i})w(N)\,,$whereas his utility from not buying
a name is, $u_{i}(N)=w(N)+P_{i}w(S)+(1-P_{i})w(N)\,,$ and the value from
having a $S$ name is $\Delta u_{i}=w(S)-w(N)$. Since all agents --
regardless of their type -- get this benefit, then there is scarcity of
names and the price of a $S$ name must be $v(S)=w(S)-w(N)$. \thinspace $%
\,Q.E.D.$ }
\end{description}

{\footnotesize It follows that the supply of $S$ names has the same
structure as for the two-period model because old agents who sell a $S$ name
are those who succeeded in their last period, regardless of what happened in
their first period. Thus, stationarity is used to obtain the market clearing
condition, 
\begin{equation}
\gamma P_{G}+(1-\gamma )eP_{G}=\mu ^{*}.  \label{clearing}
\end{equation}
The SSE is therefore characterized by the tuple $\{\mu
^{*},e,w(S),w(N),w(F),w(M),v\}.$ Computing the SSE wages using Bayes' rule
as before yields, 
\begin{eqnarray}
w(S) &=&\frac{\gamma (P_{G}+\mu ^{*})+e(1-\gamma )(eP_{G}+\mu ^{*})}{\gamma
P_{G}+(1-\gamma )eP_{G}+\mu ^{*}}\cdot P_{G}  \label{s} \\
&=&\frac{\left( 
\begin{array}{c}
\gamma (P_{G}+\gamma P_{G}+e(1-\gamma )P_{G}) \\ 
+e(1-\gamma )(eP_{G}+\gamma P_{G}+e(1-\gamma )P_{G})%
\end{array}
\right) }{2(\gamma P_{G}+(1-\gamma )eP_{G})}\,\cdot P_{G}\text{. }  \notag
\end{eqnarray}
and, 
\begin{eqnarray}
w(N) &=&\frac{\gamma (1-P_{G}+1-\mu ^{*})+e(1-\gamma )(1-eP_{G}+1-\mu ^{*})}{%
\gamma (1-P_{G})+(1-\gamma )(1-eP_{G})+1-\mu ^{*}}\cdot P_{G}  \label{n} \\
&=&\frac{\left( 
\begin{array}{c}
\gamma (2-\gamma P_{G}-e(1-\gamma )P_{G}-P_{G}) \\ 
+e(1-\gamma )(2-\gamma P_{G}-e(1-\gamma )P_{G}-eP_{G})%
\end{array}
\right) }{2(1-\gamma P_{G}-(1-\gamma )eP_{G})}\cdot P_{G}\text{ .}  \notag
\end{eqnarray}
Where the second equality in both cases follows from substituting $\mu ^{*}$
from (\ref{clearing}) above. Using Bayes rule, a $M$ name must be generated
by a $S$ name that was followed by a failure and then continues to be
active. Such a name can only be generated by young agents who bought a $S$
name, and then failed and are stuck with their name. According to the
equilibrium under construction, no mater how these names evolve (success or
failure) they will not be sold, and hence once an opportunistic type
continues with such a name he will choose $e=0.$ Thus, 
\begin{eqnarray}
w(M) &=&\frac{\mu ^{*}\gamma (1-P_{G})}{\mu ^{*}(\gamma +e(1-\gamma
))(1-P_{G})+\mu ^{*}(1-e)(1-\gamma )}\cdot P_{G}  \notag \\
&=&\frac{\gamma (1-P_{G})}{1-\gamma P_{G}-(1-\gamma )eP_{G}}\text{ }\cdot
P_{G}\,\text{.}  \notag
\end{eqnarray}
Similarly, in equilibrium a $F$ name is generated by a $N$ name that was
followed by a failure and then continues to be active. (From A4, any other
history in $H_{F}$ should not be observed in equilibrium, which will be
supported by the appropriate beliefs.) Such a name can only be generated by
young agents who did not buy a $S$ name, and then failed and are stuck with
their name. As with $M$ names, opportunistic types who continues with such a
name will choose $e=0.$Thus, 
\begin{eqnarray}
w(F) &=&\frac{(1-\mu ^{*})\gamma (1-P_{G})}{(1-\mu ^{*}))(\gamma +e(1-\gamma
))(1-P_{G})+(1-\mu ^{*})(1-e)(1-\gamma )}\cdot P_{G}  \label{f} \\
&=&\frac{\gamma (1-P_{G})}{1-\gamma P_{G}-(1-\gamma )eP_{G}}\cdot P_{G}\text{
.}  \notag
\end{eqnarray}
Any other name that is considered to be a $F$ name, $h\in H_{F}\backslash
\{NF\}$ (by the definition of the reputation reduction) occurs with zero
probability and it is possible to assign any beliefs by clients to any such
name, in particular to have all such histories command a wage of $w(F)$ as
calculated above. }

{\footnotesize The analysis establishes that in our candidate equilibrium, $%
w(F)=w(M),$ and to verify that an agent who failed will prefer to change
names over sticking to a $F$ or $M$ name it must be shown that $%
w(N)\geqslant w(F)$. Let $\Delta \equiv [\Pr \{G|N\}-\Pr \{G|F\}]\frac{1}{%
P_{G}}$. Using (\ref{n}) and (\ref{f}) above we have, 
\begin{eqnarray*}
\Delta &=&\frac{\left( 
\begin{array}{l}
\gamma (2-\gamma P_{G}-e(1-\gamma )P_{G}-P_{G}) \\ 
+e(1-\gamma )(2-\gamma P_{G}-e(1-\gamma )P_{G}-eP_{G})%
\end{array}
\right) }{2(1-\gamma P_{G}-(1-\gamma )eP_{G})}-\frac{\gamma (1-P_{G})}{%
1-\gamma P_{G}-(1-\gamma )eP_{G}}\text{ } \\
&&\text{ } \\
&>&\frac{\left( 
\begin{array}{l}
\gamma (2-\gamma P_{G}-e(1-\gamma )P_{G}-P_{G}) \\ 
+e(1-\gamma )(2-\gamma P_{G}-e(1-\gamma )P_{G} \\ 
-eP_{G})%
\end{array}
\right) }{2(1-\gamma P_{G}-(1-\gamma )eP_{G})}-\frac{\left( 
\begin{array}{l}
\gamma (1-P_{G}) \\ 
+\frac{1}{2}e(1-\gamma )(2-\gamma P_{G}-e(1-\gamma )P_{G} \\ 
-eP_{G})%
\end{array}
\right) }{1-\gamma P_{G}-(1-\gamma )eP_{G}} \\
&&\text{ } \\
&=&\frac{\gamma P_{G}(1-\gamma -e(1-\gamma ))}{2(1-P_{G}(\gamma +e(1-\gamma
))}>0\text{ .}
\end{eqnarray*}
This completes the characterization of our candidate equilibrium. Now it is
possible to establish the proof of proposition 6: }

\begin{description}
\item[{\protect\small proof of proposition 7:}] {\footnotesize Assume in
negation that only $S$ names are traded and are bought only by $G$-types. If 
$M$ names are not traded then $\Pr \{G|M\}=1$, which follows because such a
history can belong only to a good type. This is true because no trade of $M$
names implies that such a history must belong to an $\ $seller who bought a $%
S$ name and then failed, and by assumption such an $\ $seller must be good.
In this case $w(M)>w(N)$, which in turn implies that $M$ names must be
traded. Thus assume that $w(M)>w(N)$ and that $M$ names are traded in
equilibrium. The utility that a young good type entering the economy gets
out of owning a $S$ name is, 
\begin{equation*}
u_{G}(S)=w(S)+P_{G}w(S)+(1-P_{G})w(M)+P_{G}^{2}v(S)\text{ ,}
\end{equation*}
while his utility from not owning a name is, 
\begin{equation*}
u_{G}(N)=w(N)+P_{G}w(S)+(1-P_{G})w(N)+P_{G}^{2}v(S)\text{ ,}
\end{equation*}
because after failing in his first period such an $\ $seller who started
with a new name will wish to change his bad ($F$) name. The benefit from
owning a name is therefore, 
\begin{equation*}
u_{G}(S)-u_{G}(N)=w(S)-w(N)+(1-P_{G})[w(M)-w(N)]\text{ .}
\end{equation*}
Given that the choice of an opportunistic type depends only on $w(S)-w(N)$
and not on the name he actually has, the preferences (and thus, utility
differences) of opportunistic types who in equilibrium chooses $e<1$ are
derived similarly to the calculations above. The utility that a young
opportunistic type entering the economy gets out of owning a $S$ name is, 
\begin{equation*}
u_{O}(S)=w(S)+eP_{G}w(S)+(1-eP_{G})w(M)+(eP_{G})^{2}v(S)-2c(e)\text{,}
\end{equation*}
while his utility from not owning a name is, 
\begin{equation*}
u_{O}(N)=w(N)+eP_{G}w(S)+(1-eP_{G})w(M)+(eP_{G})^{2}v(S)-2c(e)\text{ ,}
\end{equation*}
and, 
\begin{equation*}
u_{O}(S)-u_{O}(N)=w(S)-w(N)+(1-eP_{G})[w(M)-w(N)].
\end{equation*}
But since $w(M)>w(N)$ and $P_{G}>eP_{G}$ then $%
u_{O}(S)-u_{O}(N)>u_{G}(S)-u_{G}(N)$. That is, $O$-types have a larger
benefit from owning a $S$ name at $t=2$, which contradicts the assumption
that only $G$-types buy $S$ names at $t=2$ in equilibrium. $Q.E.D.$ }
\end{description}

\chapter{Relational Contracting}

\setlength{\baselineskip}{.22in}

\section{Introduction}

Most of the models we saw so far had an element of Bayesian learning which
is meant to capture the idea of reputation formation, and it's effects on
dynamic incentives. An outlier was Kreps' model of firm reputation as an
asset: his model was in the spirit of repeated games, and the so called
``reputational equilibrium'' was one in which parties used an ongoing
relationship to support behavior that could not have been supported in a one
(or finitely many) period setting.

The idea of using a ``bad'' equilibrium as a threat to support a ``good''
equilibrium is the essence of what is often called ``relational''
contracting. That is, behavior is governed by the contracting parties rather
than being enforced by the courts, and the threat of forgoing a valuable
relationship is what keeps parties in line and supports behavior that is not
myopically optimal.

Some of this literature (e.g., Kreps (1990), Klein and Leffler (1981), Baker
Gibbons and Murphy (2001), and others) uses the term ``reputation'' to
describe these equilibria. Namely, the parties want to maintain their
reputation of following the good behavior and are reluctant to loose this
reputation. I would argue that this is a stretch of the term reputation. It
is actually two (or more) parties getting together and realizing that they
can use the ongoing relationship to support behavior that otherwise could
not have been supported. Thus, though we would expect parties to realize
their gains from ongoing relationships, this is not closely related to the
ideas of Bayes learning in the reputational models we have seen so far.

Many of the papers in this line of research show that parties can use their
ongoing relationships to improve upon myopic behavior, but most study
specialized, not optimal contacts. Furthermore, they vary in the information
structure available to the parties. Here we will discuss a recent paper by
Levin (2002) who unifies, generalizes and extends the framework of
relational contracts and studies optimal contracts under different
information structures. Furthermore, Levin establishes a nice link to the
studies of complete (static) contracts that we studied earlier (namely,
static models of moral hazard and hidden information such as Mirrlees (1971)
and Holmstrom (1979)).

\section{A General Model}

Consider a principal and an agent who are both risk neutral.\footnote{%
Since we focus on incentive provision and not on risk sharing, this is a
reasonable assumption. For the trade-off between risk and incentives in a
repeated moral hazard framework see Rogerson (1985), Radner (1985), Spear
and Srivastava (1987), Atkeson (1991), and Pearce and Stachetti (1998).}
They both discount future payoffs at a discount rate $\delta \in (0,1)$. At
each period $t=1,2,...$ the parties decide whether or not to transact. Three
parameters govern their relationship: First, a random cost parameter
observed by the agent, which is $i.i.d.$ and denoted by $\theta _{t}\in
\Theta =[\underline{\theta },\overline{\theta }]$ with density $p(\theta )$,
is realized each period. Second, the agent chooses effort level $e_{t}\in [0,%
\overline{e}]$ at a private cost $c(e_{t},\theta _{t})$ increasing in $e$
and $\theta $, convex in $e$, and supermodular in $(e,\theta )$ ($c_{e\theta
}>0$). Normalize costs so that $c(0,\theta _{t})=0.$ The agent's action
affects output $y_{t}\in Y=[\underline{y},\overline{y}]$ with continuous
density $f(y|e)$. There are several informational setups that can be
analyzed, as described in the following table:

\begin{center}
\begin{tabular}{|l|ccc|}
\hline
& $\mathbf{\theta }$ & $\mathbf{e}$ & $\mathbf{y}$ \\ \hline\hline
Common Monitoring: &  &  &  \\ 
\qquad Perfect Information & Both & Both & Both \\ 
\qquad Hidden Information & agent & Both & Both \\ 
\qquad Hidden Action & Both & agent & Both \\ \hline
Private Monitoring: &  &  &  \\ 
\qquad Hidden Action & Both & agent & principal \\ \hline
\end{tabular}
\end{center}

That is, different assumptions on what information is observed by whom will
change the focus of analysis, and we will study optimal contracts and their
implications in these different informational setups.\footnote{%
Perfect informatio includes Klein-Leffler 1981), Bull (1987), Kreps (1990)
and MacLeod and Malcomson (1989) among others. Hidden Action includes Green
and Porter (1984), Radner (1985) and Radner,Myerson and Maskin (1986).
\noindent Hidden Information includes Green (1987) and Atkeson and Lucas
(1992). Private monitoring includes Compte (1998) and Kandori and Matsushima
(1998).}

Assume that if the parties choose not to collaborate (they are separated)
then per-period payoffs are $\overline{\pi }$ for the principal and $%
\overline{u}$ for the agent, and denote total ``separation surplus'' as $%
\overline{s}=\overline{\pi }+\overline{u}$. Let $W_{t}$ denote the payment
from the principal to the agent in period $t$. We will assume that courts
will enforce \emph{noncontingent payments}, but that courts cannot observe
any of the above parameters. This implies that payments take on the form $%
W_{t}=w_{t}+b_{t}$ where $w_{t}$ will be enforced by the court as long as
the relationship is not voluntarily terminated,\footnote{%
Implicitly it is assumed that courts can verify whether some trade occurred,
but not the characteristics of this trade. Payments to third parties are
also ruled out.} and $b_{t}$ is a voluntary payment from the principal to
the agent that can be contingent on observables and messages ($%
w_{t},b_{t}\in \Re $). Thus, the principal's per period payoff is $%
y_{t}-W_{t}$, the agent's is $W_{t}-c(e_{t},\theta _{t})$, and the total
surplus is $y_{t}-c(e_{t},\theta _{t})$.

Assume that for every $\theta $, there is a unique maximizer $e^{FB}(\theta
) $ that solves, 
\begin{equation*}
e^{FB}(\theta )=\arg \max_{e}E[y|e]-c(e,\theta )\,\text{,}
\end{equation*}
which is decreasing in $\theta $ (this follows from the supermodularity
condition $c_{e\theta }>0)$. We assume that first best effort leads to a
valuable relationship, but that zero effort does not. Namely, $E_{y,\theta
}[y-c|e^{FB}(\theta )]\geq \overline{\pi }+\overline{u},$ and $%
E[y-c|e=0]\leq \overline{\pi }+\overline{u}$.

\subsection{Relational Contracts}

A relational contract is defined as a pair of payment plans $\zeta =(\zeta
^{w},\zeta ^{b})$ and of strategies $\sigma =(\sigma ^{p},\sigma ^{a}).$ The
court enforced plan $\zeta ^{w}$ specifies the sequence $w_{t}$ as a
function of current employment status and of payments $(w_{1},...,w_{t-1})$,
while the discretionary plan $\zeta ^{b}$ specifies the sequence $b_{t}$ as
a function of the mutually observed history up to time $t$, and the mutually
observed outcome at time $t$. The agent's strategy $\sigma ^{a}$ specifies
his decisions at time $t$ (whether to transact, his choice of $e$, and of
payments to the principal) as a function of his observed history of play,
and similarly define $\sigma ^{p}$.\footnote{%
See the appendix in Levin (2000) for a formal definition of outcomes and
strategies.}

\begin{description}
\item[Definition:] We say that a pair of strategies $(\sigma ^{p},\sigma
^{a})$ is a \textit{Perfect Public Equilibrium} (PPE) of the repeated game
if given the pair of payment plans $(\zeta ^{w},\zeta ^{b}),$\newline
($i$) Play following any history is a \textit{Nash Equilibrium}, and\newline
($ii$) Strategies at time $t$ are \textit{public}: they may be based on
private information at time $t$, but must be based \textit{only on public
(mutually observed) information} from time $\tau <t$.
\end{description}

We say that a relational contract is \textit{self enforcing} if it is a PPE
as defined above. It is important to note that as long as one party is using
public strategies, then the other party does not gain from using private
strategies. This follows because past private information is not payoff
relevant.

Since in the repeated game parties care about their expected payoff starting
from any time $t$ onward, denote the principal's normalized expected payoff
as, 
\begin{equation*}
\pi _{t}=(1-\delta )E_{t|(\sigma ,\zeta )}\left\{ \sum_{\tau =t}^{\infty
}\delta ^{\tau -t}\left[ \text{\textbf{1}}_{\{\text{trade at }\tau \}}\left(
y_{t}-W_{t}\right) +\text{\textbf{1}}_{\{\text{no trade at }\tau \}}%
\overline{\pi }\right] \right\} ,
\end{equation*}
where \textbf{1}$_{\{\cdot \}}$ is the indicator function. Similarly we can
define the agent's normalized expected payoff $u_{t},$ and the normalized
expected joint surplus 
\begin{equation*}
s_{t}=u_{t}+\pi _{t}\,.
\end{equation*}

The following two observations are helpful:

\begin{enumerate}
\item \textbf{Individual Rationality:} In any self enforcing contract, $\pi
_{t}\geq \overline{\pi }$ and $u_{t}\geq \overline{u}$.

\item \textbf{Linearity:} If there is a self enforcing contract that
generates $s\geq \overline{s}$, then any division of $s$ subject to 1 above
can be supported by a self enforcing contract (Theorem 1 in Levin (2002)).
\end{enumerate}

Observation 1 above is trivial. As for 2, assume that there is a self
enforcing contract $(\sigma ,\zeta )$ that generates payoffs $\pi $ and $u$
such that $\pi +u=s>\overline{s}$. Consider a pair of payoffs $\pi ^{\prime
}\neq \pi $ and $u^{\prime }$ such that $\pi ^{\prime }+u=s$, $\pi ^{\prime
}\geq \overline{\pi }$ and $u^{\prime }\geq \overline{u}$. Now take a new
contract $(\sigma ^{\prime },\zeta ^{\prime })$ that is the same as $(\sigma
,\zeta )$, except for the first period court enforced payment which is now $%
w_{1}^{\prime }=w_{1}+\frac{u^{\prime }-u}{1-\delta }$ , and the decision
not to accept this new contract is followed by everlasting separation. Since
all actions and expected payoffs are independent of $w_{1}^{\prime }$, and
since $\pi ^{\prime }\geq \overline{\pi }$ and $u^{\prime }\geq \overline{u}$
imply that initiating the new contract is beneficial, then this new contract
is self enforcing and it generates the desired expected payoffs. This 
\textit{linearity result} will be very useful in characterizing contracts.

In what follows we will concentrate on \emph{optimal self enforcing contracts%
}:\footnote{%
In the appendix of Levin (2000) the existence of such an optimal contact for
the relevant cases is assured.}

\begin{description}
\item[Definition:] We say that a self enforcing contract $(\sigma ,\zeta )$
is \textit{optimal} if there is no other self enforcing contract with
greater expected surplus.
\end{description}

\subsection{Common Monitoring: Stationarity}

Let $\varphi _{t}\subseteq \{\theta _{t},e_{t},y_{t}\}$ denote the vector of
commonly observed variables at date $t$. Common monitoring implies that $%
\varphi _{t}$ must include $y_{t}$, but may or may not include $\theta _{t}$
and $e_{t}$ (always observed by the agent).

\begin{description}
\item[Definition:] We say that a contract $(\sigma ,\zeta )$ is \textit{%
stationary} if on the equilibrium path, $e_{t}=e(\theta _{t}),\,w_{t}=w,$
and $b_{t}=b(\varphi _{t})$ for all $t$.
\end{description}

That is, a stationary contract is one in which the enforced payment $w_{t}$
is constant, bonus payments in period $t$ depend only on the realizations of
period $t$, and choices do not depend on the (irrelevant) observed history.

An interesting feature of the model is, as implied by the linearity
observation above, that to provide the agent with incentives, the principal
can either use self enforced discretionary payments, or self enforced
continuation payoffs. However, since parties are risk neutral, these two
instruments are perfect substitutes.

With common monitoring the principal needs no incentives since he has no
private information. Thus, we can use \emph{extreme} continuation payoffs to
provide incentives for the agent, and fine tune the division of surplus
using self enforced discretionary payments. This intuition is made formal:

\begin{description}
\item[Theorem:] (Levin, Theorem 2) \textit{If an optimal contract exists
with common monitoring then there exists a stationary self enforcing
contract that is optimal.}

\item[proof:] First we show that if an optimal self enforcing contract $%
(\sigma ,\zeta )$ exists, then it must be that in every period $t$ the
expected surplus is stationary. To see this let $(\sigma ,\zeta )$ generate
ex ante expected payoffs $u+\pi =s^{*}$. Assume that the first period
outcome $\varphi $ is followed by payments $W(\varphi )=w+b(\varphi )$ with
continuation payments $u(\varphi )+\pi (\varphi )=s(\varphi )$. We have, 
\begin{equation}
s^{*}=(1-\delta )E_{\theta ,y}[y-c\,|\,e(\theta )]+\delta E_{\theta
,y}[s(\varphi )\,|\,e(\theta )]\,.  \label{s*}
\end{equation}
Assume without loss that deviations from payments are followed by
everlasting separation. For $(\sigma ,\zeta )$ to be self enforcing it must
be that ($i$) $u\geq \overline{u}$ and $\pi \geq \overline{\pi },$ ($ii$)
for all $\theta ,$%
\begin{equation}
e(\theta )\in \underset{e}{\arg \max }\,E_{\varphi }\left[ W(\varphi )+\frac{%
\delta }{1-\delta }u(\varphi )\,\,\left| \,e\right. \right] -c(e,\theta )\,,
\label{e*}
\end{equation}
($iii$) that discretionary payments are incentive compatible, 
\begin{equation}
b(\varphi )+\frac{\delta }{1-\delta }u(\varphi )\,\geq \frac{\delta }{%
1-\delta }\overline{u}\;,\text{ and \ }-b(\varphi )+\frac{\delta }{1-\delta }%
\pi (\varphi )\,\geq \frac{\delta }{1-\delta }\overline{\pi }\,\,,
\label{IC1}
\end{equation}
and ($iv$) that continuation payoffs $\pi (\varphi )$ and $u(\varphi )$ are
themselves PPE payoffs (the perfection requirement). To see that $s(\varphi
)=s^{*},$ assume that this were not the case. Then by the fact that the
continuation $s(\varphi )$ is supported by a PPE, and the definition of
optimal contracts, it can't be true that $s(\varphi )>s^{*}$ for any $%
\varphi $ that occurs with positive probability. If $s(\varphi )<s^{*},$
then increasing $\pi (\varphi )$ to $s^{*}-u(\varphi )$ is feasible from
observation 2 above, and this relaxes (\ref{IC1}) without affecting (\ref{e*}%
). But since $(\sigma ,\zeta )$ is optimal, this cannot increase total
surplus, so it must be that $s(\varphi )=s^{*}$ for any $\varphi $ that
occurs with positive probability, and from (\ref{s*}) $E_{\theta
,y}[y-c\,|\,e(\theta )]=s^{*}.$ \newline
We can now construct a stationary contract $(\sigma ,\zeta )$ that
implements $e(\theta )$ from (\ref{e*}) in every period, and gives the agent
some arbitrary utility $u^{*}\in [\overline{u},s^{*}-\overline{\pi }]$.
Given $w$ (the first period payment of $\zeta ^{w}),$ and $u$ (the expected
payoff from $(\sigma ,\zeta )),$ define stationary payoffs to be 
\begin{eqnarray*}
w^{*} &=&w-\frac{1}{1-\delta }u+\frac{1}{1-\delta }u^{*\,}\,, \\
b^{*}(\varphi ) &=&b(\varphi )+\frac{\delta }{1-\delta }u(\varphi )-\frac{%
\delta }{1-\delta }u^{*}\,.
\end{eqnarray*}
As above, any deviation is followed by termination. The agent's expected
payoff is 
\begin{eqnarray}
&&E_{\varphi }[W^{*}(\varphi )-c\,|\,e(\theta )]  \notag \\
&=&u^{*}+\overset{=0\text{ from (\ref{e*})}}{\overbrace{E_{\varphi
}[W(\varphi )+\frac{\delta }{1-\delta }u(\varphi )-c\,|\,e(\theta )]-\frac{1%
}{1-\delta }u}}\, \\
&=&u^{*}\,,  \notag
\end{eqnarray}
while the principal's is $\pi ^{*}=s^{*}-u^{*}$, and both $\pi ^{*}\geq 
\overline{\pi }$ and $u^{*}\geq \overline{u}$. From (\ref{Eu}) it is clear
that the agent will choose $e(\theta )$ in each period. It follows from (\ref%
{e*}) that discretionary payments are incentive compatible (see Levin for
the algebra). $Q.E.D.$
\end{description}

This theorem shows that we must have \noindent \textit{Sequential Efficiency}%
. Since we always have efficient continuation play, and the utility
possibility frontier is linear with slope -1, then we can replace variation
in continuation play with transfers, and thus get stationarity.

The fact that the utility possibility frontier is linear with slope -1, and
that we can support any payoffs of the form $\pi ^{*}=s^{*}-u^{*}$, with $%
\pi ^{*}\geq \overline{\pi }$ and $u^{*}\geq \overline{u}$, means that we
can use extreme punishments ($\overline{u}$ for the agent and $\overline{\pi 
}$ for the principal) on the equilibrium path. This means that we can punish
deviators as harshly as possible while keeping continuation play efficient.

\begin{description}
\item[Definition:] An optimal contract $(\sigma ,\zeta )$ is \textit{%
strongly perfect} if following any sequence of play at $t$, the continuation
contract from $t+1$ is optimal.

\item[Proposition:] (Levin, Proposition 3) \textit{Under common monitoring,
if an optimal contract exists then there are optimal stationary self
enforcing contracts that are strongly perfect.}

\item[proof:] For any division of surplus use the stationary contract
constructed in the previous proof with punishments as follows: following a
deviation of the agent, use the optimal continuation that generates $%
\overline{u}$ for the agent and $s^{*}-\overline{u}$ for the principal, and
the converse for deviations of the principal. $Q.E.D.$
\end{description}

The intuition is clearly obtained in the following diagram:

\vskip 180pt

In the diagram, $N$ is the payoffs from choosing everlasting separation.
Instead of using $N$ to punish a deviator, use $A$ to punish deviations by
the agent, and $P$ to punish deviations by the principal, where the only
possible deviations can be not making discretionary promised payments
(recall that $e$ was a best response since it solved the agent's FOC). Note
that the results of stationarity apply to all informational settings with 
\emph{common monitoring}.

\subsection{Common Monitoring: Optimal Incentive Provision}

We saw that with stationary contracts the parties implement the same effort
schedule and the same payment plan at every date. Thus, the solution to an
optimal stationary contract is maximizing total surplus subject to two
constraints: an incentive compatibility constraint for the choice of effort,
which was (\ref{e*}) above, and dynamic enforcement (incentive) constraints
which were given in (\ref{IC1}) above. Clearly, (\ref{IC1}) implies that
per-period payments cannot be too high, or else the principal will deviate,
and they cannot be too low, or else the agent will deviate. These limits
will have important implications to efficiency properties of effort in
equilibrium. To continue, for a stationary effort schedule $e(\theta )$
define the expected surplus as, 
\begin{equation*}
s(e)=E_{\theta ,y}[y-c\,|\,e(\theta )]\,.
\end{equation*}

\begin{description}
\item[Theorem:] (Levin, Theorem 4) \textit{A stationary self enforcing
contract can implement effort schedule }$e(\theta )$\textit{\ if and only if
there are payments such that for all }$\theta ,$%
\begin{equation}
e(\theta )\in \underset{e}{\arg \max }\,E_{\varphi }\left[ W(\varphi
)\,\left| \,e\right. \right] -c(e,\theta )\,,  \tag{(IC)}
\end{equation}
\textit{\ and,} 
\begin{equation}
\frac{\delta }{1-\delta }(s(e)-\overline{s})\geq \max_{\varphi }W(\varphi
)-\min_{\varphi }W(\varphi )\,,  \tag{(DE)}
\end{equation}
\end{description}

For a formal proof see Levin (2000). It follows from the stationarity, and
from the fact that (\ref{IC1}) has to hold for every $\varphi $, and the Max
and Min become necessary and sufficient conditions.

The feature of (DE) is that there must be a bound on the variability of
payments, which is basically what happens in standard repeated game models.
If the variability is too large, there will be instances where one of the
parties will want to renege. Notice, however, that as $\delta \rightarrow 1$%
, the bounds become very large, and first best outcomes will be possible in
this risk neutral setup. We continue to study what happens when first best
is not attainable.

\subsubsection{Perfect Observability}

As in MacLeod and Malcomson (1989), let $\varphi _{t}=\{\theta
_{t},e_{t},y_{t}\}.$ To implement a schedule $e(\theta )$ we need a payment
plan $W(\theta ,e)=w+b(\theta ,e)$ that satisfy (IC) and (DE). (Since $e$ is
observable, $y$ adds nothing.) We can restrict attention to ``forcing
contract'': $W(\theta ,\widehat{e})=W(\theta ,0)$ for all $\widehat{e}\neq
e(\theta )$, and $W(\theta ,e)=W(\theta ,0)+c(e(\theta ),\theta )$.
Following the previous theorem, this can be done if and only if (DE) is
satisfied, or 
\begin{equation*}
\frac{\delta }{1-\delta }(s(e)-\overline{s})\geq \max_{\theta }c(e(\theta
),\theta )\,,
\end{equation*}
which is the condition obtained by MacLeod and Malcomson, extended to random
cost functions. First Best will not necessarily be obtained for low enough
discount factors.

\subsubsection{Hidden Action (Moral Hazard)}

Assume $e_{t}$ is privately observed by the agent, but that both observe $%
y_{t}$. For simplicity assume $c(e,\theta )=c(e)$\ (i.e., $p(\theta )$\ is
degenerate) so we focus only on moral hazard and $\varphi _{t}=y_{t}$. Using
the first order approach (assuming MLRP and CDFC)\footnote{%
As described earlier, the first order approach is valid if the distribution
of output satisfies the \textit{monotone likelihood ratio property} (MLRP),
that is, $\frac{f_{e}(y|e)}{f(y|e)}$ is increasing in $y,$ and the \textit{%
convexity of the distribution function condition} (CDFC), that is, $%
F(y|e=c^{-1}(x))$ is convex in $x$. See Rogerson (1985).} we can rewrite
(IC) as, 
\begin{equation}
\frac{d}{de}\left( E[W(y)\,|\,e]-c(e)\right) =0\,,  \label{Foc10}
\end{equation}
and the problem is now to choose $e$ and $W(y_{t})$ to maximize $%
s(e)=E[y-c\,|\,e],$ $\,$subject to (\ref{Foc10}) and (DE).

\begin{description}
\item[Proposition:] \textit{Under MLRP and CDFC the optimal contract either
implements }$e^{FB}$\textit{\ or is ``one step'': }$W(y)=\underline{W}$%
\textit{\ for all }$y<\widehat{y}$\textit{\ , and }$W(y)=\overline{W}$%
\textit{\ for all }$y\geq \widehat{y}$\textit{\ where }$\overline{W}-%
\underline{W}=\frac{\delta }{1-\delta }(s(e^{*})-\overline{s})$\textit{, and 
}$\widehat{y}$\textit{\ is given by }$\frac{f_{e}(\widehat{y}|e)}{f(\widehat{%
y}|e)}=0$\textit{.}

\item[proof:] The proof follows from risk neutrality and from the first
order approach. By risk neutrality, incentives are highest when $w=0$. The
problem is then, 
\begin{equation*}
\begin{array}{ll}
\max_{e,b(y)} & s=E[y-c\,|\,e] \\ 
\;\;\;\;\;\;\text{s.t.} & \int_{y}b(y)\frac{f_{e}(y\,|\,e)}{f(y\,|\,e)}%
dy-c^{\prime }(e)=0 \\ 
& \frac{\delta }{1-\delta }(s(e)-\overline{s})\geq \max_{y}b(y)-\min_{y}b(y)%
\end{array}%
\end{equation*}
and differentiating the Lagrangian with respect to $b(y)$ yields the shadow
cost $\mu \cdot \frac{f_{e}(y\,|\,e)}{f(y\,|\,e)}$, with $\mu >0$. This
implies that for all $y<\widehat{y}$, $b(y)$ should be minimal, and for all $%
y>\widehat{y}$, $b(y)$ should be maximal. Whether or not first best is
attainable depends on the discount factor. $Q.E.D.$
\end{description}

We get that optimal contracts are simple ``bang-bang'' contracts. (As in the
Abreu, Pierce and Stacchetti (1990) model, extremal payoffs are optimal.)

\subsubsection{Hidden Information}

Assume now that the agent privately observes $\theta _{t}$, but both observe 
$\varphi _{t}=(y_{t},e_{t})$. As with perfect observability, payments need
not be conditioned on $y_{t}$. Using the revelation principle (recall that
the shocks $\theta _{t}$ are $i.i.d.$ across periods), a stationary
screening contract will be an effort schedule $e(\theta )$ and payments $%
W(e)=w+b(e).$ If a stationary contract implements $e(\theta ),$ then the
expected surplus is 
\begin{equation}
s=\int_{\underline{\theta }}^{\overline{\theta }}\left\{ E[y|e(\theta
)]-c(e(\theta ),\theta )\right\} p(\theta )d\theta \,.  \label{obj}
\end{equation}

\begin{description}
\item[Proposition:] (Levin, Proposition 5) \textit{Under hidden information,
a stationary self enforcing contract can implement }$e(\theta )$\textit{\ if
and only if }$e(\theta )$\textit{\ is non-increasing, and } 
\begin{equation}
\frac{\delta }{1-\delta }(s(e)-\overline{s})\geq c(e(\underline{\theta }),%
\underline{\theta })+\int_{\underline{\theta }}^{\overline{\theta }%
}c_{\theta }(e(\widetilde{\theta }),\widetilde{\theta })p(\widetilde{\theta }%
)d\widetilde{\theta }  \tag{IC-DE}
\end{equation}
\end{description}

The proof is omitted and appears in Levin (2000). The idea follows well
known results from the hidden information literature with monopolistic
screening (e.g., Mirrlees (1971), Mussa and Rosen (1978)). That is, global
incentive constraints are replaced with local incentive constraints and a
monotonicity condition on the choice function $e(\theta )$.

As with (DE), (IC-DE) captures the idea that variation in incentive
provision is bounded for a contract to be self enforcing. If we can induce $%
e^{FB}(\theta )$, then our constraints will not be binding. However, if
(IC-DE) is binding, then we will generally not implement $e^{FB}$, and there
must be types for which incentives are sub-optimal. Furthermore, due to the
bound on payment variation, if we want to increase incentives for these
types, we must reduce incentives for some other types. This leads to
trade-offs that are not present in the standard static hidden information
models.

We proceed by defining $S(e,\theta )=E[y|e]-c(e,\theta )$, and assume that,

\begin{description}
\item[(A1)] $S(e,\theta )$ is concave in $e$ for all $\theta $, and $%
c(e,\theta )$ and $c_{\theta }(e,\theta )$ are convex in $e$.

\item[(A2)] $c(e,\theta )$ and $c_{\theta }(e,\theta )$ are supermodular in $%
(e,\theta ),$ and $p(\theta )$ is nonincreasing in $\theta $.
\end{description}

With static hidden information models, these two conditions would guarantee
full separation of types, and no ``pooling''. Following the standard
techniques of control theory that are used for the static models, Levin
derives the following result:

\begin{description}
\item[Proposition:] (Levin, Proposition 6) \textit{Under hidden information,
an optimal contract either implements }$e^{FB}(\theta ),$\textit{\ or
implements a second best schedule }$e^{*}(\theta ).$\textit{\ Any second
best schedule has }$e^{*}(\theta )<e^{FB}(\theta )$\textit{\ for all }$%
\theta ,$\textit{\ and involves some pooling of cost types. Either (i) }$%
e^{*}(\theta )$\textit{\ is constant, or (ii) there is a unique }$\widehat{%
\theta }>\underline{\theta }$\textit{\ such that }$e^{*}(\theta )$ \textit{%
is constant for }$\theta <\widehat{\theta },$\textit{\ and decreasing above
it.}
\end{description}

The intuition for this proposition goes as follows. If incentives are
provided to cause the highest cost type to choose some effort $e>0$, then
this implies that all lower cost types have incentives as well. That is, the
principal gets a large benefit from providing incentives to the highest cost
type. The reason why there is no ``efficiency at the top'' for the lowest
cost type comes from the (IC-DE) constraint that puts bounds on the average
variation of payments. Assume that the lowest cost type had first best
incentives. Then by lowering the variation of his incentives, there would be
no first order loss, but there will be a first order gain from transferring
the added variation to the highest cost type. The reasoning for pooling at
the top (lowest costs) is the same: the largest benefits from providing
incentives and separation comes from the highest cost types.

To conclude this case, it is optimal to distort the effort choice of all
types, i.e. $e^{*}(\theta )=e^{FB}(\theta )$\ or $\forall \theta $,\ $%
e^{*}(\theta )<e^{FB}(\theta ),$ and there is less screening (more pooling)
than the static case.

\subsection{Private Monitoring/Hidden Action}

\begin{itemize}
\item $A$ privately chooses $e_{t}.$

\item \noindent $P$ privately observes $y_{t}$.

\item $P$ makes announcement $m_{t}.$
\end{itemize}

\textbf{Issue 1.}\ Must provide incentives for agent and induce principal to
monitor honestly $\Rightarrow $ equilibrium contracts can't be sequentially
efficient, joint surplus must vary.

\textbf{Issue 2.}\ How should principal release information?

\begin{itemize}
\item Full report each period (PPE) $\Rightarrow $ under CDFC, MLRP,
```nearly one-step' termination contracts'' are optimal.
\end{itemize}

\subsubsection{Review Strategies}

\begin{itemize}
\item Report every $T$ periods. (T-PPE)

\begin{itemize}
\item Cost:\ More temptation to renege

\item Benefit:\ Better inference.

\item Abreu, Pearce, and Milgrom '91; Compte '98; Kandori and Matsushima '98;
\end{itemize}
\end{itemize}

\begin{description}
\item[Theorem:] (Compte,K-M). For $\delta $ near $1$, can approximate
first-best with $T$ near $\infty $.
\end{description}

\begin{itemize}
\item General $\delta $. Choose finite $T$ termination contract.

\begin{itemize}
\item One-step incentives optimal (CDFC).

\item Optimal statistical test: terminate if and only if:\medskip 
\begin{equation*}
\lambda +\mu _{1}\frac{f_{e}}{f}(y_{1}|e_{1})+...+\mu _{T}\frac{f_{e}}{f}%
(y_{T}|e_{T})<0
\end{equation*}
\bigskip \medskip
\end{itemize}

\item Ongoing review/partial revelation is better.
\end{itemize}

\end{document}
