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\begin{document}
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\begin{titlepage}
 
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\begin{center}
{\Large Governmental Failures in Evaluating Programs\footnote
{
We are grateful for comments by Michelle Garfinkel and by members of
the Focussed Research Group at the University of California, Irvine.
}
} \vfill
 
\bigskip
    Amihai Glazer\\
    Department of Economics\\
    University of California, Irvine \\
    Irvine, California 92717 U.S.A. \\
\bigskip
    Refael Hassin\\
    School of Mathematical Sciences\\
    Tel-Aviv University\\
    Tel-Aviv 69978, Israel
\bigskip
\vfill \today \vfill
\bigskip
\begin{abstract}
 
Consider a government that adopts a program, sees a noisy signal about
its success, and decides whether to continue the program.  Suppose
further that the success of a program is greater if people think it
will be continued.  This paper considers the optimal decision rule
for continuing the program, both when government can and cannot
commit.  We find that welfare can be higher when information
is poor, that government should at times commit to continuing
a program it believes had failed, and that a government which fears
losing power may acquire either too much or too little
information.
 
\end{abstract}
 
\end{center} \medskip \vfill \end{titlepage}
 
\pagebreak
 
\section{Introduction}
 
Government often appears to cancel successful policies, and to continue
failed ones.  The first phenomenon, cancellation of policies that in
hindsight would have been successful, is understandable when government
has imperfect information at the time it must make a decision.  The
second phenomenon, however, is more complex. Government often appears to
continue a program even though analysts using existing information would
conclude that the program will likely fail.\footnote {Instances relating
to technology policy are cited in Cohen and Noll (1991).  For example,
Congress continued funding a supersonic transport long after
cost-benefit analyses showed such a plane would be a commercial failure.
} Two explanations are most common:
 
(1) The goals of politicians are not those formally stated.  For
example, legislators may adopt dam projects not to control floods or
to protect the environment, but to spend money in their districts.
Thus, policy fulfills the unstated goals of politicians.
 
(2) Special interest groups deflect attention away from the purpose
of the program originally proposed. The classic example is economic
regulation.
 
Though both explanations say much about the world, we see them as
incomplete in considering only politics.  They do not recognize that in a
world with imperfect information finding a good program can be difficult.
Moreover, we shall see that when the success of a program depends on
peoples' expectation about its continuation, the optimal decision rule may
be to continue programs likely to fail.
 
Some issues we discuss here also apply to private firms.  In that sense,
our analysis is quite general.  But government faces additional
problems.  First, a democratic government may be unable to assure
economic agents that the program will be continued---it has limited
rights to constrain the actions of a future government.  A court may
require a firm to pay damages arising from breach of contract, but will
rarely uphold a claim of damages against a government that changed a
policy.  Government must then often use indirect methods to make
credible commitments.  Second, commitment is difficult when changes in
government reflect changes in the preferences of political
leaders.\footnote {For an example of government inconsistency, consider
the investment tax credit.  Such a credit was enacted in the United
States in 1962, repealed in 1969, reinstituted in 1971, abolished in
1986, and proposed again in 1993.} In contrast, a succeeding chief
executive of a private firm can share the goals of his predecessor of
maximizing profits.
 
Central to our model is the assumption that the success of a program
is greater if people think it will be continued. Expectations can
affect the success of program in at least two ways. First, the
success of a program may require firms or individuals to incur fixed
costs of investment, so that a program expected to continue will
induce more such investment. Second, the success of a program can
depend on the degree of learning-by-doing. A firm engaged in
learning-by-doing will tend to undertake those activities that it
believes will generate the knowledge most useful under future
conditions. Thus, the behavior of firms in a particular period
depends on their expectations of policy in future periods.\footnote
{Sometimes a program will be more successful when economic agents
believe it will not be continued. A prime example is a tax amnesty.
We shall not consider such policies here.}
 
Consider the regulation of automobiles. In the wake of the energy
crisis of 1973, the U.S. Congress adopted in 1975 the Energy Policy
and Conservation Act, which mandated minimum corporate average fuel
economy (CAFE) standards for new passenger vehicles sold in the
United States. Auto manufacturers were in effect required to double
the fuel-efficiency of their vehicles. Meeting these targets required
costly research, development, and retooling. The firms would be
unwilling to incur these expenses if they thought the regulations
would later be relaxed. Indeed, in 1985 Ford and General Motors did
not meet the standards, and faced penalties of up to \$750
million.\footnote{``Should car mileage limits be kept?" {\em New York
Times}, June 17, 1985 I, 16:1.} The Secretary of Transportation
reduced the standards from 27.5 mpg to 26.0 mpg. Furthermore, in June
the Environmental Protection Agency revised its formula for measuring
fuel efficiency, thereby substantially reducing the penalties Ford
and General Motors faced.\footnote{``U.S. ready to cut auto mileage
standards." {\em New York Times}, July 9, 1985 I, 18:3.} Some of the
support for relaxation of the standards came from studies which
suggested that passengers in small cars were likely to suffer
increased injuries in accidents, and from research which claimed that
automobile emissions (which might be increased if fuel economy
increased) could contribute to global warming. That is, one reason
for the imperfect commitment to enforce the standards was the
uncertain knowledge about the effects of the CAFE policy.
 
Similar credibility problems arose with the Clean Air Act Amendments of
1970 in the United States, which required automobile manufacturers to
reduce emissions by 90 percent by 1975.\footnote{See White (1982).} The
Act was explicitly technology-forcing, requiring manufacturers to make
large investments in new equipment.  Under pressure from the automobile
companies the U.S. government delayed imposition of the standards until
1977. In that year the manufacturers could not meet the standards
either, so Congress amended the law to further temporarily relax the
standards.  One argument for delay was concern that catalytic
converters, required to reduce emissions of carbon monoxide,
hydrocarbons, and nitrous oxides, would generate sulfite pollution.
Later studies showed the concern to be misplaced.  But the existence of
some earlier studies cast doubt on the wisdom of the emission standards,
and thus reduced the credibility that they would be enforced.
 
The following sections consider government policy when it can and cannot
commit.  If government can commit, our problem is to find the optimal
actions of government in the second period.  If government cannot
commit, our problem is to find an equilibrium which describes peoples'
beliefs that government will continue the program.\footnote{ Our
discussion of commitment in public policy relates to work by Strotz
(1955-56), Kydland and Prescott (1977), Barro and Gordon (1983), and
Persson (1988).  They show that current decisions of economic agents
depend, in part, on their expectations of future policy.  Cukierman and
Meltzer (1986) show that a government seeking reelection will prefer to
follow a discretionary policy rather than a rule which could lead to
higher social welfare.}
 
\section{Assumptions}
 
We analyze a two-period model. In period 1 government adopts a new
program. In period 2 let government continue the program with
probability $p$. The program succeeds or fails. Success in period 2
yields a benefit of $S(p)$, and failure imposes a cost of $-F(p)$.
In line with the discussion in the introduction, we suppose that a
program will give greater benefits the more likely people believe it
will be continued. That is, $S'(p) > 0$, and $F'(p) < 0$. Where
appropriate we shall consider the results when $F'(p) > 0$. To be
succinct we shall at times omit the argument $p$ in the functions
$S(p)$ and $F(p)$.
 
If adoption of the program requires no fixed cost, and if even under
failure the benefits are positive, then the program will be adopted in the
first period and continued in the second period.  Thus, the model is
interesting only if there exists a fixed cost, or, equivalently, if $F$ is
positive.\footnote{ Let the fixed cost be $K$.  Instead of $S$ and $F$
consider $S-K$ and $F+K$ and ignore fixed costs.  We then require that
$S-K>0$ and $F+K>0$.} We henceforth assume that $F$ is positive.  The
values of $F$ and $S$ in the first period are relevant only in determining
whether government undertakes the program in that period.  The values are
irrelevant for any decisions made in period 2.  We shall thus assume that
$S$ is sufficiently large, or $F$ sufficiently small, to make adoption of
the program in period 1 worthwhile.
 
The prior probability that the program will succeed is $\pi_{0}$; the
prior probability that it will fail is $1-\pi_{0}$. At the end of period
1 government gets information about success of the program in that
period. After a success government sees a signal of success (a positive
signal) with probability $s$; it sees a negative signal with probability
$1-s$. After failure government sees a signal of failure (a negative
signal) with probability $f$; it sees a positive signal with probability
$1-f$. A program which succeeded in period 1 will also succeed in period
2; a program which failed in one period will later also fail. A
government which does not have perfect information about the program
could use its priors and these signals to determine the posterior
probability that the program is a success or a failure.
 
The main question we address is whether the program should be continued in
period 2.  If the program is continued, then the benefits obtained depend
on whether the program succeeds or fails.  If the program is stopped, then
in period 2 no benefits are obtained.  For simplicity we let the
intertemporal discount rate be zero.
 
 
\section{Optimal solution with commitment}
 
This section considers the government's optimal strategy when it can commit
to any action it desires.  In particular, we allow it to continue a program
in period 2 even if maximizing its utility from that point on would call
for stopping the program.  In other words, we do not require government to
follow time-consistent solutions.  Time-inconsistent strategies are
attractive because an increase in the probability of continuing the program
in period 2 can increase the gains from the program---$S$ increases and $F$
decreases.
 
The choice variable of interest is the probability, $p$, that government
will continue a program.  We make the natural assumption that a positive
signal makes it always beneficial to continue the program.  Thus, the
government can be viewed as choosing the probability that it will
continue the program after a negative signal.\footnote
{Another policy is to stop the program with some probability after a
positive signal.  However, either always continuing or always stopping
the program after a positive signal dominates any randomized policy of
this type.  The reason is that an increase in the probability of
continuing after a positive signal increases both the benefits and the
probability of getting those benefits.  In contrast, an increase in the
probability of continuing after a negative signal increases the
probability of a loss.
}
 
Note incidentally that it can be worthwhile to adopt the program in
period 1 even if in that period expected benefits are negative.  For
adopting the program in period 1 yields information about the program,
and thereby leads to a better decision in period 2.  That is, adopting
the program gives an option value, which can be greater than the
expected loss in period 1.  An example will illustrate.  Let the
program have $S=10$ and $F=11$.  Each outcome has probability 1/2.  In
period 1 expected gains are $1/2(10-11) = -1/2$.  But at the end of
period 1, government learns something about the program's success.
Suppose that at the end of period 1 it gets perfect information about
the program.  Government continues only a successful program.  Since
success occurs with probability 1/2, the expected gain from the
program over the two periods is $ -1/2 + (1/2)(10) = 4.5 $.
 
 
Let $p^-$ be the probability that government will continue the program
after a negative signal. For a given $p^-$ the probability of continuing
the program is
\begin{equation}
        p = \pi_{0}s + (1- \pi_0)(1-f) + p^- \pi_0(1-s) +
            p^{-}(1 - \pi_0)f.
\end{equation}
Expected benefits over the two periods are
\begin{equation}
         V(p^-) = \\ (\pi_{0}s)S(p)
        - (1 - \pi_{0})(1-f)F(p) + p^{-}\pi_{0}(1-s)S(p)
        - p^{-}(1-\pi_{0})fF(p).
\end{equation}
The government chooses $p^{-}$ to maximize $V(p^-)$.
 
Let
\begin{equation}
        p' \equiv {\partial p \over \partial p^-} =
        \pi_0(1-s) + (1-\pi_0)f > 0.
\end{equation}
Then
\begin{eqnarray*}
        {\partial V \over \partial p^-} &&
        = \pi_0sS'(p)p'-(1-\pi_0)(1-f)F'(p)p' \\
        & & \mbox{}
        + \pi_0(1-s)S(p) + p^- \pi_0(1-s)S'(p)p' \\
        & & \mbox{}
        - (1-\pi_0)fF(p) - p^-(1-\pi_0)fF'(p)p'.
\end{eqnarray*}
The optimal value of $p^-$ can be $0, \ 1$, or an interior value
satisfying $ \partial V / \partial p^- = 0 $.
 
\section{Equilibrium solution without commitment} \label{SecNoCommit}
 
Consider next a second-best solution, where government must follow a
time-consistent policy.  That is, it will continue the program after
seeing the signal at the end of the first period only if the expected
benefits of continuing are positive.  As before, $S'(p) > 0$, and
$F'(p) < 0$.  Where appropriate we shall consider the results when
$F'(p) > 0$.  The realized values of the signals and the values of
$S$ and $F$ determine whether government will continue the program in
period 2.
 
The inability to commit means that at the end of period 1, after
observing the signal, government makes the choice that maximizes
expected benefits in period 2.  Let the probability that the state
of nature is $x$ when signal $y$ is observed be $P_{xy}$, where $x \in
\{ S,F \} $ and $y \in \{ +,- \}$.  For example,
\[
         P_{S+} = {\pi_{0}s \over \pi_{0}s + (1-\pi_0)(1-f)}.
\]
Let the signal observed at the end of period $1$ be $i$.  Then
maximizing expected utility in period 2 requires government to
\begin{equation}
          \mbox{maximize \ } [0,  S(p)P_{Si}-F(p)P_{Fi}].
\end{equation}
 
Consider an equilibrium probability ($p$), known to both the government
and the public, that the government will continue the program.  When
government does not randomize, three equilibria in pure strategies can
arise; they are described in the appendix.  Of most interest is the
equilibrium which has government continue the program only after a
positive signal.  (As seen in the previous section, this need not be a
first-best solution.)  Continuation of the program only after a positive
signal is an equilibrium if the government's optimal choice is to
continue after a positive signal and to stop after a negative signal.
The probability of continuing is the probability of a positive signal,
$p_+ \equiv \pi_0 s + (1-\pi_0)(1-f)$.  The condition for continuing
only after a positive signal is then
\begin{equation}                                        \label{p+}
     R^+ \equiv  \frac {(1-\pi_0)(1-f)} {\pi_0s} \le
            S(p_+) / F(p_+)
            \le \frac {(1-\pi_0)f} {\pi_0(1-s)} \equiv R^-.
\end{equation}
The expected benefits in period 2 are
\begin{equation}
        V^+ \equiv S(p_+)\pi_0s - F(p_+)(1-\pi_0)(1-f).
\end{equation}
 
The comparative static properties of $V^+$ are then sometimes surprising.
The partial derivative of $V^+$ with respect to $\pi_0$ is
\begin{equation}
        \frac{\partial V^+}{\partial \pi_0} =
        \frac{\partial p_+}{\partial \pi_0}
        [S'(p_+)\pi_0s - F'(p_+)(1-\pi_0)(1-f)]
        + S(p_+)s + F(p_+)(1-f).
\end{equation}
If $F'> 0$ this derivative can be negative: an increase in the prior
probability of success makes it less attractive to continue after a
positive signal. The effect arises because an increase in $\pi_0$
which leads to an increased probability of continuing the program
increases the loss when the program fails.
 
The partial derivative of $V^+$ with respect to $s$ (the probability
that success generates a positive signal) is
\begin{equation}
        \frac{\partial V^+}{\partial s} =
        \{ S'(p_+)\pi_0s+S(p_+)-F'(p_+)(1-\pi_0)(1-f) \} \pi_0.
\end{equation}
If $F'>0$ this derivative can be negative: the more accurate is a signal
when the program succeeds, the lower the benefits from continuing the
program after a positive signal.  This effect arises because even with
high values of $s$ the program may be continued when it fails.  An
increase in the probability of continuing the program can increase the
loss when a failed program is continued.
 
The partial derivative of $V^+$ with respect to $f$ (the probability
that failure generates a negative signal) is
\begin{equation}
        \frac{\partial V^+}{\partial f} =
        \{ -S'(p_+)\pi_0s + F'(p_+)(1-\pi_0)(1-f)+F(p_+) \}
        (1- \pi_0).
\end{equation}
This derivative can be negative both when $F'<0$ and when $F'>0$.  An
increase in $f$ reduces the probability that the program will be
continued.  When $ F'(p)<0 $, the reduced probability of continuing
the program increases the costs of failure.  Since a failed program
may be continued, the expected benefits in period 2 may decline, and
the derivative can be negative.  When $F'(p)>0$ and $S'(p)>0$, the value
of $\partial V^+ / \partial f$ can nevertheless be negative: the
reduced probability of continuing the program reduces the benefits of
continuing a successful program.
 
Recall that an increase in $f$ means that when the program fails the
signal is more reliable.  We can interpret this as improved information.
Suppose $p=p_+$, so that social welfare is $V^+$.  We saw that $\partial
V^+ / \partial f$ can be negative---an increase in $f$ reduces social
welfare.  The paradox arises not because of the improved information.
The problem is that the increased reliability of the signal
makes government less likely to continue the program.  This reduced
probability can reduce benefits when a failed program is continued.
 
\section{Commitment by ignorance}
 
Though commitment is often problematic, it may be easier in our setting.
The government's action in period 2 depends on the informativeness of the
signal (on the values of $s$ and $f$). The design of a program in period 1
which determines the values of $s$ and $f$ thus affects decisions in
period 2. So a requirement that firms report on their activities may
induce high values of $s$ and $f$, causing the government to continue the
program only after a positive signal.  Not imposing such requirements, or
designing an experimental program with poor sampling procedures, induces
low values of $s$ and $f$; the poor information will cause the government
to either always stop or else always continue the program.
 
To see the benefits of poor information, consider an example where
$s=f=1/2$.  In other words, the signal at the end of period 1
contains no information.  Without commitment the possible equilibria
are $p=0$ and $p=1$.  The condition for $p=1$ to be an equilibrium is
$\pi_0S(1) > (1-\pi_0)F(1)$, or $S(1)/F(1) > (1-\pi_0)/(\pi_0)$.
Expected welfare is $V^1 = \pi_0 S(1) - (1-\pi_0)F(1)$.
 
Suppose government can improve information, so that $s,\ f > 1/2$. The
improvement makes $R^- > R^+$; with a proper $S(p)/F(p)$ function,
$p=p_+$ may be an equilibrium. Expected benefits are then
$V^+=S(p_+)\pi_0s - F(p_+)(1-\pi_0)(1-f)$. If $S(1)$ is much larger
than $S(p_+)$, while $F(1) \approx F(p_+)$, then $V^+ < V^1$. The
improved information can reduce welfare. Welfare may be reduced even
if better signals can be obtained costlessly, and even if perfect
information (expressed by $s=f=1$) is attainable. Under perfect
information expected benefits are $\pi_0S(\pi_0)$. This value may be
greater than, smaller than, or equal to $V^1$. Of course, such a
phenomenon cannot appear when government can commit.
 
Suppose instead that $V^+ > V^1$, or that
\begin{equation}
                \pi_0[sS(p_+)-S(1)] - (1-\pi_0)[(1-f)F(p_+)-F(1)] > 0.
\end{equation}
Then government prefers an equilibrium with $p=p_+$ (continue only
after a positive signal) to an equilibrium with $p=1$. This
inequality will be satisfied if one of the following conditions is
satisfied:
\begin{enumerate}
        \item{} $f$ is large;
        \item{} $s$ is large;
        \item{} $S(p)$ increases slowly with $p$;
        \item{} $F(p)$ decreases slowly with $p$.
\end{enumerate}
 
 
\section{Politics}
 
Often one person or agency determines whether to continue a program, but
a different party controls the quality of information that will be
provided.  Thus, firms developing a new technology may make it difficult
or easy to evaluate the effectiveness of their products. They may give
specific or vague estimates of future prices.  They may  report profits
accurately or inaccurately, promptly or with delay.  The question then
arises of what information firms would want to provide.
 
Similar issues arise when the government that establishes the program
in the first period fears losing power to a new government with
different preferences. The first government determines the reliability
of the information that will be revealed at the end of period 1. The
second government decides in period 2 whether to continue the
program.\footnote{
Recent related work shows that expectations of a change in power can
greatly influence the policies the current government will pursue.
Glazer (1989) shows that collective choices will show a bias
towards durable projects.  Other work considers the macroeconomic
implications of coalitional instability (see Alesina (1989) for a
survey).  Alesina and Tabellini (1990) and Tabellini and Alesina
(1990) show that a government may increase the debt to reduce
expenditures by a later government.  McCubbins, Noll, and Weingast
(1989) show that if Congress fears that an agency will have different
preferences from the current congressional majority, then Congress
may limit an agency's freedom of action.}
 
The manipulation of information as described above can be considered
to determine the values of $s$ and $f$.  To see the effects of such
changes, let the government in period 1 evaluate success at $S_1$ and
failure at $-F_1$.  In period 2 a different government will be in
power.  It evaluates success at $S_2$ and failure at $-F_2$.  The
first government chooses $s$ or $f$.  The second government decides
whether to continue the project.
 
An increase in the value of $s$ or of $f$ makes it more attractive
for government to continue the program in period 2 after a positive
signal, and to stop the program after a negative signal. Note that
this need not mean that reliable signals increase the chances that
the program will be continued---if success is unlikely, then an
increase in $f$ can reduce the chances that the program will be
continued. An increase in $s$ or $f$ may also change a government's
criteria of when to continue the program. With unreliable signals
government may continue the program regardless of the signal. With
reliable signals it may choose to continue only after a positive
signal.
 
To determine how the first government can manipulate information to its
benefit, suppose the second government (in power in period 2) is less
favorable towards the program than is the first government  (in power in
period 1).  That is, $S_2 < S_1$ and $F_2 > F_1$. Two opposite
situations are possible:
 
\begin{enumerate}
 
\item{}
The second government would only continue the program if success is
very likely.  The first government then prefers that the program
generate a reliable signal of success, that is a high value of $s$.
 
\item{}
The second government would continue the program even with no
further information (that is, even if $s=f=1/2$). Given, however, a
sufficiently informative negative signal, the second government
would cancel the program while the first government (if it stayed
in power) would not. The first government then has an incentive to
make the signals unreliable.
 
\end{enumerate}
 
Analogous incentives to manipulate signals appear when the second
government benefits more from the program than does the first
government.  The preferences of the second government may also affect
the first government's decision about adopting the program.  Suppose
that $S_2 > S_1$, that $F_2 < F_1$, and that $s$ and $f$ are fixed.
The first government may then adopt the program only if it has the
option of stopping the program once it thinks the program likely
failed.  But if the second government would continue the program even
when the first government would prefer that it not, then the benefits
to the first government of adopting the program are reduced.  We have
the paradoxical result that the fear that a future government greatly
favors the program makes the first government less inclined to adopt
it in the first place.\footnote
{A similar effect may arise when the first government favors the
program more than the second government.  The higher probability that
the program will be stopped reduces the benefits of adopting it in
the first place.  Here, however, it is not surprising that the
existence of a future government that does not favor the program
reduces the benefits of to the first government of adopting the
program.
}
 
Finally, the discussion in the previous section implies that a
possible change in power may increase the expected benefits of the
program as measured by the first government. A belief by the public
that the second government is more favorable to the program (that is,
has higher values of $S$ and lower values of $F$) can generate a
higher value of $p$, and thus higher benefits to the first government.
In other words, a change in government can have the same effects as a
commitment by the first government to either continue or stop the
program. Thus, a government that cares about policy may have higher
utility in the second period if people expect it to lose reelection.
 
\section{Conclusion}
 
President Clinton said in his 1993 State of the Union message that
``Our every effort will reflect what President Franklin Roosevelt
called `bold, persistent experimentation,' a willingness to stay with
things that work, and stop things that don't."  We agree with the
first principle.  But stopping programs that do not work can be bad
policy.
 
\pagebreak
 
\section{Appendix: Equilibria without commitment}
 
When government cannot commit, three equilibria in pure strategies
can arise.
 
\begin{enumerate}
 
\item The program is never continued.
 
For $p=0$ to be an equilibrium, the government's optimal action must be
to stop the program even after a positive signal.  That is
        $$ S(0)P_{S+}-F(0)P_{F+} \le 0, $$
or
\begin{equation}
        S(0)/F(0) \le \frac{(1-\pi_0)(1-f)}{\pi_0 s} \equiv R^{+}.
\end{equation}
The expected benefits in period 2 are 0.
 
\item The program will be continued only after a positive signal.
 
The probability of continuing is the probability of a positive signal,
so that
\begin{equation}
               p = p_+ \equiv \pi_0 s + (1-\pi_0)(1-f).
\end{equation}
 
Continuation of the program only after a positive signal is an
equilibrium if the government's optimal action is to continue after a
positive signal and to stop after a negative signal.  Thus, the
following two conditions must be satisfied:
\begin{eqnarray*}
        S(p_+)P_{S+}-F(p_+)P_{F+} \ge 0, \\
        S(p_+)P_{S-}-F(p_+)P_{F-} \le 0.
\end{eqnarray*}
These conditions can be written as
\begin{equation}                                        \label{R+}
     R^+ \equiv  \frac {(1-\pi_0)(1-f)} {\pi_0s} \le
            S(p_+) / F(p_+)
            \le \frac {(1-\pi_0)f} {\pi_0(1-s)} \equiv R^-.
\end{equation}
The expected benefits in period 2 are
\begin{equation}
        V^+ \equiv S(p_+)\pi_0s - F(p_+)(1-\pi_0)(1-f).
\end{equation}
 
Note that
\begin{equation}
                \frac{\partial p_+}{\partial \pi_0} = s+f-1 > 0.
\end{equation}
The inequality follows from the assumption that $s$ and $f$ are each
greater than 1/2.  We thus find that the equilibrium probability of
continuing the program is greater the greater the prior probability of
success.  We also note that this partial derivative is larger the
greater the accuracy of the signals, that is the more likely success
generates a positive signal, and the more likely failure generates a
negative signal.
 
\item The program is always continued.
 
For $p=1$ to be an equilibrium, the government's optimal choice must
be to continue the program even after a negative signal:
        $$ S(1)P_{S-}-F(1)P_{F-} \ge 0, $$
or
\begin{equation}                                           \label{e1}
     S(1)/F(1) \ge \frac {(1-\pi_0)f} {\pi_0(1-s)} \equiv R^{-}.
\end{equation}
The expected benefits in period 2 are
\begin{equation}
        V^1 \equiv S(1)\pi_0 - F(1)(1-\pi_0).
\end{equation}
 
\end{enumerate}
 
The equilibrium need not be unique.  In particular, suppose that
\begin{equation}                                     \label{assumption}
        S(p) / F(p) {\mbox \em \ is \ monotone \ increasing}.
\end{equation}
That is, an increase in the probability of success increases the relative
benefits of success more than the costs of failure.  Then any combination of
the possible solutions can apply.
 
Equilibria also exist which make government just indifferent between
continuing and stopping the program after a positive or negative
signal. Suppose that some $p$ in $(0,1)$ satisfies $S(p)/F(p) =
R^{+}$; call this value $p_{I}^{+}$. Similarly, suppose that some $p$
in $(0,1)$ satisfies $S(p)/F(p) = R^{-}$; call this value $p_{I}^{-}$.
If the public believes that $p=p_{I}^{+}$, then government is
indifferent between continuing and stopping the program after a
positive signal. If the public believes that $p = p_{I}^{-}$, then
government is indifferent between continuing and stopping the program
after a negative signal. In such cases of indifference we could view
government as randomizing in a way that conforms with $p$.
Specifically, suppose that equation (\ref{p+}) holds (and $p_+$ is an
equilibrium). In this case the assumption that $S(p)/F(p)$ is monotone
increasing implies that $p_{I}^{+} \le p_+$.
 
If government randomizes after a positive signal (and stops the
program after a negative signal), then $p=p_{I}^{+}$ may result.  In
this case, by definition of $p_{I}^{+}$, government is indeed
indifferent about continuing the program after a positive signal.
Hence, this is an equilibrium.  Similarly, $p=p_{I}^{-}$ may be an
equilibrium.  Nevertheless, an equilibrium with $p_{I}^{+}$ or
$p_{I}^{-}$ is unlikely.  Such equilibria require government to be
indifferent between continuing and stopping the program, and require
government to randomize in the particular ways which support these
equilibria.
 
These possibilities are illustrated in Figure 1 with the curve
$S(p)/F(p)$.  We suppose that $S(p_+)/F(p_+)$ lies between $R^+$ and
$R^-$, that $S(1)/F(1)$ is greater than $R^-$, and that $S(0)/F(0)$ is
less than $R^+$.  Thus, $0 <p_{I}^{+} < p_+ <p_{I}^{-} <1$: $p_+$,
$p_{I}^{+}$ and $p_{I}^{-}$ can all be equilibria.  Figure 2 illustrates
a case where $p_{I}^{+}>p_+$ and $p_{I}^{-}$ is not defined (since
$S(p)/F(p) < R^{-}$ for all $p \in [0,1]$).  The only possible
equilibrium in this case is 0.  Other possibilities exist, depending on
the relation between the curve $S(p)/F(p)$ and the critical values $R^+$
and $R^-$.
 
\pagebreak
 
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\pagebreak
 
\section{Notation}
 
\begin{description}
 
\item[$F$] Loss if program fails
 
\item[$f$] Probability signal is negative if program fails
 
\item[$p$] Probability that government continues program
 
\item[$p^-$] Probability government continues program when it observes
a negative signal
 
\item[$p_+$] Probability that government continues program, given that
it continues only after a positive signal
 
\item[$p_{I}^{+}$] Value of $p$ for which $S(p)/F(p) = R^{+}$, so
that government is indifferent about continuing the program following
a positive signal.
 
\item[$p_{I}^{-}$] Value of $p$ for which $S(p)/F(p) =R^{-}$, so that
government is indifferent about continuing the program following
a negative signal.
 
\item[$P_{xy}$] Probability of outcome $x$ (Success or Failure) given
signal $y$ ($+$ or $-$)
 
\item[$R^+$]  $ \frac{(1-\pi_0)(1-f)}{\pi_{0}s} $; critical value for
determining whether to continue after positive signal
 
\item[$R^-$]  $ \frac{(1-\pi_0)f}{\pi_{0}(1-s)} $; critical value for
determining whether to continue after negative signal
 
\item[$S$] Benefit in each period if program succeeds
 
\item[$s$] Probability signal is positive if program succeeds
 
\item[$V^1$] Expected benefit if the program is always continued
 
\item[$V^+$] Expected benefit if the program is continued only after
a positive signal
 
\item[$\pi_0$] Prior probability program will succeed
 
\end{description}
 
\end{document}
 
 
