 % regmean.tex.
 %  28 DEcember 1991.April 8, 2000, contact and cite info.

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         \titlepage
          
         \begin{center}
\begin{large}
    {\bf Managerial Conservatism and   Rational Information
Acquisition }\\
             \end{large}
                
                      
                    \bigskip
                    Eric Rasmusen \\


Published:{\it  Journal of Economics and Management Strategy} (Spring
1992), 1: 175-202. \\

                     
                    {\it Abstract}\\
                    \end{center}
                    \par\noindent
 Conservative managerial behavior can be  rational and profit-
maximizing.
If the valuation of innovations  contains white noise   and the status
quo would be preferred to random innovation, then any innovation that
does not appear to be
substantially better than the status quo should
be rejected.
The  more successful the firm,
 the higher the threshold for accepting innovation should be, and
the greater the conservative bias. Other things equal, more
successful firms will  spend less on research, adopt fewer
innovations, and be less likely to advance
the industry's best practice.
\begin{small}
       
\noindent
   \hspace*{.2in}
 Yale Law
 School, Box 401A Yale Station, New Haven, Ct. 06520. (203) 432-4022.
 UCLA AGSM Business Economics Working Paper \#89-31.
File:/papers/regmean/99regmean.tex. Draft: 9.1 (Draft 1.1, October
1989.)

 \begin{small}
               \noindent
\hspace*{20pt} 2000: Eric Rasmusen, 	Professor of Business
Economics and Public Policy and Sanjay Subhedar Faculty Fellow,
Indiana University,
Kelley School of Business, BU 456,
  1309 E 10th Street,
  Bloomington, Indiana, 47405-1701.
  Office: (812) 855-9219.   Fax: 812-855-3354. Erasmuse@indiana.edu.
Php.indiana.edu/$\sim$erasmuse.
 \end{small}


 

 I would like to thank David Hirshleifer, Steven Lippman, George
Lowenstein, Edward Miller, Emmanuel Petrakis, Ivan Png, Steve
Postrel, Ivo Welch, two anonymous referees, and participants in the
Chicago GSB Uncertainty Workshop for helpful comments, and Michael
Kim and George Michaelides for research assistance.

            \end{small}

%---------------------------------------------------------------

\newpage
 \noindent
 {\bf Introduction}

 American managers, it is commonly said, are  conservative and
short-sighted, passing up new ideas and avoiding risks.
 Hayes
\& Garvin (1982) reflect a common  opinion when they
complain in the {\it Harvard Business Review} of excessive managerial
conservatism in the form of high hurdle rates:
 
 \begin{small}
 \begin{quotation}
 Such hurdle rates often bear little resemblance either to a
company's real cost of capital (even after appropriate adjustment for
differences in risk) or to the actual rates of return (net of
deterioration replenishment) that the company can reasonably expect
to earn from alternative investments. Again and again, we have
observed the use of pretax hurdle rates of 30\% or more in companies
whose actual pretax returns on investment were less than 20\%.
(Hayes \& Garvin, p.  76)
 \end{quotation}
\end{small}

  
 

One should keep in mind that managerial conservatism and myopia may
just be popular myth. A recent Wall Street Journal/NBC poll found
that 51\% of Americans thought that Japan had more of a long-term
perspective, but only 10\% of the Japanese did, suggesting that
people everywhere like to decry their compatriots'
shortsightedness.\footnote{
``Which Land is Fairest of Them All? Japanese, Americans
Talk to Pollsters,'' {\it Wall Street Journal}, 15 June 1990, p. A6.}
Some evidence, however, does seem to support the idea.
 Ross (1986) interviewed twelve manufacturers about their
decisionmaking process regarding investments in energy conservation
and examined their internal records.  He concluded that the hurdle
rate for large projects was near the companies' cost of capital, but
for small projects the hurdle rate was higher, and ``Decisions are
then based on the primary quantitative measure from the analysis
supplemented by informal adjustments made in the minds of
decisionmakers.''  Pruitt \& Gitman (1987) sent questionnaires on
project evaluation to financial officers in Fortune 500 firms, and
found they believed that forecasts were consistently overoptimistic,
implying a need for conservative use of forecasts.  80\% of
respondents felt that revenue forecasts of capital-budgeting
proposals were overstated. 37\% of them thought that this was
intentional and 36\% attributed it to inexperience.  59\% of them
agreed, and 20\% disagreed with the statement: ``In general,
decisionmakers who evaluate forecasts consider them to be optimistic
in their estimates and adjust forecasts to correct them.''

 The perception is common enough that economists have devoted
considerable effort trying to explain it. Recent articles include
Narayanan (1985), Stein (1988), Shleifer \& Vishny
(1989), and Hirshleifer \& Thakor (1991).
 A variety of explanations for conservative bias have been developed
based on incomplete rationality or rational but not profit-maximizing
behavior. People, including managers, might simply make systematic
mistakes, as discussed in Libby \& Fishburn (1977).  Or, bounded
rationality might generate conservative behavior rules, as in Day
(1987), Heiner (1983), and Kuran (1988). Even if managers are
conventionally rational, principal-agent problems can generate
conservative bias in a variety of ways.  Most simply, agents may
overestimate the value of new projects because they are rewarded more
highly when projects are adopted. The principal then imposes a
conservative bias to undo the overestimate. Agency problems can also
have more complicated effects, of which I will describe just two to
give the flavor of them. In Holmstrom \& Ricart (1986), the agent
initially is ignorant of his ability, but he and the rest of the
world can discover it by undertaking projects.  The agent can either
ignore a potential project or truthfully report a signal of its
quality to the principal to obtain approval for undertaking it. If
the agent is simply paid his estimated marginal product, he will veto
too many projects, because he is risk-averse and fears discovering
that he has low ability.  But if he is given a downward-rigid wage,
which is part of the second-best optimal contract, then he
overinvests unless the principal engages in conservative capital
rationing. In Lambert (1986), a risk-averse agent must be motivated
to expend effort in measuring the value of an innovation and to adopt
it if it is superior to the status quo. The principal does not
observe effort or the agent's report, only the cash flow from the
decision.  Lambert finds a conservative bias if the average
innovation is superior to the status quo, because the agent must be
given sufficient incentive to expend effort.

 It is possible, however, to explain conservatism without either
irrationality or agency problems. Most simply, those who see
conservatism might be ignoring adjustment costs, especially the cost
of management time. This explanation would not explain, however, why
financial officers think that project revenue forecasts are
consistent overestimates.  The present article explores a different
argument, valid even in the absence of adjustment costs, which
suggests that managers process information rationally, but in a way
that seems overcautious to outsiders, and that the borderline refused
project, which seems clearly profitable, in fact would yield zero
profits. The argument will be based on regression to the mean: if
profitability estimates are made with error, even if the error is
unbiased, then a project measured to be unusually profitable is
probably not as profitable as it appears.  Regression to the mean is
symmetric with respect to unusually unprofitable projects, but such
projects would be rejected anyway.  Put somewhat differently, the
distribution of profitabilities can be treated as a Bayesian prior,
and the measurement of a particular new project's profitability is
one data point of information. Under conditions specified below, the
mean of the posterior distribution will lie between the prior mean
and the information, and if the information is greater than the prior
mean, the posterior mean is less than the information, giving rise to
conservatism.  The model follows a line of inquiry begun in the
corporate finance literature, where
 Brown (1974, 1978) and Smidt (1979) noted that
 measurements of adopted projects will appear overoptimistic because
projects with low measured values will not be adopted.  Miller (1978,
1987) expanded the argument using examples which suggest that
managers should apply a conservative bias in capital budgeting.  The
present article will focus on the assumptions behind the argument,
and the implications for which firms will collect information and
innovate.



 Section 1 will lay out the model and establish that a conservative
bias is rational (Proposition 1). It will also give an example with
normal distributions to show the magnitude of the bias, and compare
its size with the effect of risk aversion. Section 2 will show that as
firms
progress they should become increasingly conservative (Proposition
2), and that even with conservative decisionmaking most projects will
prove disappointing (Proposition 3).  Section 3 will discuss
empirical implications, and
 Section  4  will  suggest applications to
 a  variety of other contexts.  Section  5 concludes.

%---------------------------------------------------------------
\pagebreak
 \begin{center}
{\bf 1. The Model}
  \end{center}


A risk-neutral manager whom we shall call ``the boss'' chooses a
single ``policy'' for each of two periods to maximize his firm's
profits.  A policy is a method of using the resources available to
the firm, and it may be an investment project, a technological
innovation, or an organizational form. Many policies are possible,
but the per-period profitability of policy $i$, $\theta_i$, is
unknown to the boss. He does know that $\theta_i$ is drawn from a
continuous distribution with a symmetric density $f_\theta(\theta)$
which has mean $\overline{\theta}>0$ and is either unimodal or
uniform.  Once a policy is adopted, $\theta_i$ becomes known, but the
policy cannot be reversed until the following period.\footnote{
\label{footnote} Otherwise, the boss might adopt a policy simply to
discover its profitability, which is not purely an adoption decision,
but rather a way to acquire information. The problem of when to
switch policies to acquire information is the ``multi-armed bandit''
problem discussed in Weitzman (1979).} At the end of the first
period, the policy can be changed.  The policy chosen for the first
period will be called the ``status quo,'' with profitability denoted
$\theta_0$, and our focus will be on the boss's incentive to innovate
in the second period.

 In each period, the boss may commission $n$ staffers to estimate the
profitability of $n$ different policies, at a cost of $c$ per policy
estimated.\footnote{This is distinct from the problem analyzed by Sah
\& Stiglitz (1988) of combining the reports of $n$ staffers on {\it
one} policy.} Estimates from previous periods grow stale and are no
longer available.\footnote{This assumption would become relevant if a
firm adopted a bad policy in the first period which would, ex post,
be abandoned for a different policy whose value had been estimated in
the first period. Such a possibility does not affect Proposition 1,
but it would require extra care in Section 2's analysis of the
optimal $n$.}
Staffers do not know $f_\theta$, and staffer $i$ reports his
measurement of $\theta_i$, denoted $y_i$, where $y_i = \theta_i +
u_i$.  The variable $u_i$ is a random error with mean zero,
independent of $\theta_i$ and $u_j$ for $j \neq i$ and distributed
according to a symmetric continuous density $f_u$ which is unimodal
or uniform.  Assume also, for reasons explained below, that the
support of $f_u$ is at least as wide as the support of $f_\theta$.
For convenience, let $f(x)$ denote the marginal density for any
variable $x$; $f(x|z)$, the conditional density; $f(x,z)$, the joint
density; and $E(x)$ the expected value; where $f$ and $E$ are derived
from whatever functions are appropriate to their arguments. The
following lemma will play a central role:

{\it LEMMA 1: (i) $E (\theta|y)$ is increasing in $y$ and (ii)
$\overline{\theta} < E (\theta|y) <y$ for $y> \overline{\theta}$,
 $E (\theta|y) =y$ for  $y= \overline{\theta}$, and
  $ y < E (\theta|y) <\overline{\theta}$ for  $y< \overline{\theta}$.}

 {\it Proof:} See Appendix.

Lemma 1 says that the expected value of the new policy conditional on
the estimate is higher if the estimated value is higher, and that the
expected value lies between the estimate and the value of the average
new policy.

  Initially, the firm has no
status quo, and the boss must decide how to choose a policy. He
could blindly accept an uninvestigated policy, which has expected
profitability $\overline{\theta}$, or accept one of the investigated
policies, which has measured profitability $y$ and expected
profitability
 $E(\theta|y)$. The optimal decision rule is to choose an
uninvestigated policy if all the investigated policies have estimated
profitabilities less than $\overline{\theta}$, and to choose the
policy with the greatest estimated profitability otherwise. If we
define $y_m =Max \{y_i\}_{i=1}^n$, the firm adopts policy $m$ if $
y_m \geq \overline{\theta}$, and an uninvestigated policy otherwise.

    Let us assume that $c$ is low
enough that it is worth investigating at least one policy:
   \begin{equation} \label{e1}
 E[(\theta -\overline{\theta})|(y \geq \overline{\theta})]
\int_{\overline{\theta}}^{\infty} f(y) dy \geq c.
 \end{equation}
 Equation (\ref{e1}) says that the expected gain in profitability
from having the option to accept the investigated policy if its
estimated value exceeds $\overline{\theta}$ equals at least $c$.



  If the status quo were chosen blindly, it would be true that $E
(\theta_0) = \overline{\theta}$.  But the expected value can only
increase,  given the option of adopting an investigated policy. It
takes the value
  \begin{equation} \label{e2}
E (\theta_0) = \overline{\theta} +  E[(\theta_m -\overline{\theta})
|(y_m \geq \overline{\theta})] \int_{\overline{\theta}}^{\infty}
f(y_m) dy_m.
 \end{equation}
   The second term of equation (\ref{e2}) is positive because there
is a positive probability that $ \theta_m> \overline{\theta}$, and
Lemma 1 then tells us that $E(\theta_m|y_m) > \overline{\theta}$.
  Thus we have

{\it LEMMA 2: For the average new firm, the status quo is superior to
blind innovation: $\theta_0 >\overline{\theta}$.}

 The question of what adoption rule the boss should follow regarding
innovations can now be addressed. For the average firm, Lemma 2 says
that the status quo is $\theta_0 >\overline{\theta}$.  We will see
that for this average firm, the optimal decision procedure is to
adopt the new policy if $y$ is greater than some threshold. Let
$\overline{y}$ denote the adoption threshold and let $y^*$ denote the
optimal value of $\overline{y}$. In the absence of estimation error,
the boss's optimal adoption rule would be $\overline{y}=\theta_0$:
``Accept
the new policy if and only if $y \geq \theta_0$.''  With estimation
error, on the other hand, the optimal rule is  described by
Proposition 1.


{\it PROPOSITION 1: The average firm uses a threshold rule with a
conservative bias: $y^* > \theta_0$.}

{\it Proof:}   Lemma 2 says that for the average firm, $\theta_0 >
\overline{\theta}$. Thus, the
 boss will reject innovations with $E(\theta|y) <\overline{\theta}$,
and since
 Lemma 1 says that if $y > \overline{\theta}$ then $E(\theta|y)$
increases in $y$, a threshold rule is optimal.  The boss will choose
the threshold $y^*$ so that $E(\theta|y \geq y^*) \geq \theta_0$, so
the threshold is such that
  \begin{equation} \label{e201}
     E(\theta|y^*) = \theta_0.
      \end{equation}
       Lemma 1 says that if $y > \overline{\theta}$ then $E(\theta|y)
< y$. It follows that $E(\theta|y^*) < y^* $, and if equation
(\ref{e201}) is to be satisfied, it must be that $y^* > \theta_0$.
$\Box$


\bigskip

 As an example, let us assume normality of the
distributions of new policies and  measurement errors, $f_\theta$
and $f_u$.  The boss is trying to use the observed  variable
$  y = \theta + u$
 to estimate the unobserved  variable
$ \theta = \overline{\theta} + \epsilon$,
  where $u$ and $\epsilon$ are independent random variables with zero
mean, and $\overline{\theta}$ is a constant.  If $u \sim N(0,
\sigma^2_u)$ and $\epsilon \sim N(0, \sigma^2_\epsilon)$, then
$\theta \sim N(\overline{\theta}, \sigma^2_\epsilon)$ and $y \sim
N(\overline{\theta}, \sigma^2_u +\sigma^2_\epsilon$).  What the boss
cares about is the conditional distribution $f(\theta|y)$, which is
also normal, with parameters that can be calculated (see, e.g.,
Casella [1985]). The mean is
 \begin{equation} \label {e19}
 E (\theta|y)= \left( \frac{\sigma^2_u}{\sigma^2_u
+\sigma^2_\epsilon} \right) \overline{\theta} +
\left(\frac{\sigma^2_\epsilon}{\sigma^2_u +\sigma^2_\epsilon} \right)
y,
 \end {equation}
 in which case $ E (\theta|y)<y$ if $y > \theta_0$.
The variance is
 \begin{equation} \label {e20}
 Var(\theta|y)=
 \frac{\sigma^2_u\sigma^2_\epsilon}{\sigma^2_u +\sigma^2_\epsilon}.
 \end {equation}
  To find the optimal threshold, $y^*$, the boss  solves for $y$ in
the equation
 \begin{equation} \label {e21}
 E (\theta|y)= \theta_0.
 \end{equation}
 Using equation (\ref{e19}), equation (\ref{e21}) becomes
 \begin{equation} \label {e22}
  \left( \frac{\sigma^2_u}{\sigma^2_u +\sigma^2_\epsilon} \right)
\overline{\theta}
+ \left( \frac{\sigma^2_\epsilon}{\sigma^2_u +\sigma^2_\epsilon}
\right) y = \theta_0.
 \end {equation}
Solving (\ref{e22}) for $y$ gives
 \begin{equation} \label {e23}
 y^*  = \left( \frac{\sigma^2_u
+\sigma^2_\epsilon}{\sigma^2_\epsilon} \right) \theta_0 -
\left( \frac{\sigma^2_u}{\sigma^2_\epsilon} \right) \overline{\theta},
 \end {equation}
 or, equivalently,
 \begin{equation} \label {e23.5}
  y^* = \theta_0 + \frac{\sigma^2_u}{\sigma^2_\epsilon} (\theta_0 -
\overline{\theta}).
 \end {equation}
  Equation (\ref{e23.5}) confirms Proposition 1.  If a random draw is
likely to be worse than the status quo because
$\theta_0>\overline{\theta}$, then equation (\ref{e23}) says that
$y^*>\theta_0$. The conservative bias increases in the variance of
the measurement error, $\sigma^2_u$, and decreases in the variance of
the possible new-policy values, $\sigma^2_\epsilon$.\footnote{The
effect of the variance here is not general. With the normal density,
mean
and variance fully characterize the distribution, but what really
matters is how much of the new-policy density is for policy values
superior to the status quo. Probability mass between the new-policy
mean and the status quo increases variance, but not the
attractiveness of new policies. This is a little like the value of an
option: it is not exactly variance of the underlying asset price that
gives an option value; it is the probability that the asset value
will be beyond the strike price.}


Table 1 shows different values calculated from equation
(\ref{e23}), which may give some idea of how empirically relevant the
conservative bias might be. The maintained assumptions are that
$\sigma_\epsilon=15$ and $\theta_0 = 100$, while $\sigma_u$ and
$\overline{\theta}$ take various values.


\pagebreak
 \begin{center} {Table 1}\\
  {Acceptance Thresholds for Normal Distributions}\\
 $y^* = \theta_0 + \frac{\sigma^2_u}{\sigma^2_\epsilon} (\theta_0 -
\overline{\theta})$

\begin{tabular}{lr| cccc}
\hline
\hline
 & & \multicolumn{4}{c}{ Standard Error of Measurement
($\sigma_u$)}\\
 \hline
   & & 5 &10 &15 &40\\
\hline
 & &  & & &\\
 & 50 &  106 &122 &150 &456\\
 New & 80 &102 & 109 & 120 & 242\\
Policy& 90 & 101 &104 & \underline{{\bf 110}} &171 \\
Mean & 95 & 100.6 &102 &105 &136\\
 ($\overline{\theta}$) & 98 & 100.2 &101 &102 &114\\
  & 100.0& 100.0 &100.0 &100.0 & 100.0 \\
 & 110  &99 &96 &90 &29 \\
 & &  & & &\\
\hline
 \multicolumn{4}{l}{Assumed: $\theta_0 = 100$, $\sigma_\epsilon =
15$ } & \multicolumn{2}{c}{ (fractions rounded)}\\
\hline
 \hline
  \end{tabular}
 \end{center}

 Consider, for example, the boldfaced entry in Table 1,
$y^*(90,15)=110$.  This means that if the average new policy has a
value of 90, and about 1/3 of the staffer's measurements are wrong by
more than 15, the new policy needs to look about 10\% better than the
status quo to be acceptable to the boss. Thus, the effect is sizeable
for plausible parameter values.




\bigskip
\noindent
 {\bf Discussion of Proposition 1}

 The most intuitive interpretation of the conservative bias is as
regression towards the mean. The staffer
might measure the new policy's value to be high for either of two
reasons:
 (1) the true value $\theta$ is above average, or (2) the
error $u$ is positive. Having observed a high measured value, the
subjective probabilities of both (1) and (2)   should rise.  Ex
ante, the probability of a positive error is no greater than that of
a negative error. But positive errors tend to push the measured value
above the average, so, ex post, positive errors are more common for
above-average observations of $y$.  In that limited range, the errors
have a positive mean, not a zero mean, and the observed value
overestimates the true value.  Since positive errors are more likely
given an observed value above average, if the staffer made a second
measurement, on the same policy but with  independent error, he
would most likely make a less positive or a negative error, and
the second measured value would be less than the first.  The
measurement ``regresses towards the mean.''  Because the high
first measurement might have been produced by a high true value, the
expected value of the second measurement is still above
$\overline{\theta}$: the measurement only regresses {\it towards} the
mean, not all the way.  Thus, for a particular $y$ it might be true
that $
y > \theta_0 > E(\theta|y)> \overline{\theta}$.\footnote{
 All of this has disregarded adoption costs, another difference
between a status quo and an innovation. Adoption costs very obviously
skew the conclusions towards conservatism. But in fact they skew them
even more than might be obvious.  Suppose that $\theta_0=100$ and
$\overline{\theta}=100$, but the implementation cost of $c=5$ is not
included in the definition of $\theta$.  The obvious conservative
bias is that the boss should refuse the new policy if $\theta=104$.
But if $\theta$ is not known, and the staffer reports that $y=106$,
so that $y-c>\theta_0$, the boss should still refuse the new policy.
Because $(y-c)$ equals 101, which is uncomfortably close to 100, $(E
(\theta|y) - c)$ is less than 100.  Thus, adoption costs create more
conservatism than one might think.}

The language of Bayesian statistics provides a terser interpretation
of Proposition 1.  The boss has a prior distribution, $f(\theta)$,
which he updates using the information $y$ to obtain the posterior
distribution $f(\theta|y)$. Given the assumptions of the model, the
posterior mean lies between the prior mean and the information, and
hence has a lower value than the information, which induces a
conservative bias.


  Two elements of the model are key to the result. The first
is Lemma 2's statement that the status quo is superior to a blind draw
from the pool of policies. In the present model, Lemma 2 was generated
by
the assumption that the status quo was itself generated by
investigation of possible policies.  Other ``front-ends'' to
Proposition 1 are also possible and plausible; for example, that new
firms are randomly assigned policies, but only new firms with
policies of more-than-average profitability survive to the second
period.  The second key element is the assumption that the policy's
profitability becomes known once it is adopted. The
status quo has the advantage of being a known value, and hence not
subject to regression to the mean. If the true profitability of the
status quo were unknown,  the conservative bias would disappear,
because the status quo's estimated profitability would be completely
symmetric to the estimated profitability of the newly investigated
policies.

It should also be noted that the conservative bias may not be the
only effect of poor information about alternative policies. In
particular, if the firm can discover the value of a policy relatively
quickly and reverse the adoption decision, then an innovative bias
may be appropriate, as was mentioned in footnote \ref{footnote}. In
the extreme, the firm could briefly adopt each possible policy in
turn, and then choose the best one for a permanent policy. In such a
case, adoption has two purposes---information collection, for the
initial adoptions, and direct profitmaking for the serious adoption.
The initial adoptions replace the staffer's estimate, and if the
information acquired by adoption was imperfect like the staffer's,
then a conservative bias would still be applied in making the final
adoption decision.

 Proposition 1 applies to the average firm,  not to every firm.
  A minority of firms will have been unlucky in the first period, and
they will not have a conservative bias. They will innovate, but not
due to a bias in the sense used here. Rather, their adoption
procedure will be the same as that of a new firm: adopt the best
investigated policy if its estimated profitability is greater than
$\overline{\theta}$, and adopt a random uninvestigated policy
otherwise. The unlucky firm will certainly abandon its status quo,
but it will never adopt an investigated policy whose estimated value
is less than the status quo--- it would prefer to adopt an
uninvestigated policy if the investigation proved disappointing.
Although if $y < \theta_0< \overline{\theta}$ it is true that
$E(\theta|y)>y$, by Lemma 1, the lemma also says that if $y <
\overline{\theta}$ then $E(\theta|y) < \overline{\theta}$, so the
blind choice is preferable to adjusting an investigated policy's
valuation upwards and accepting it. Thus, unlucky firms do not so
much impose an innovative bias as simply start over. This extends to
their choice of how many policies to investigate. Except that the
profits from a policy can only last one period instead of two (which
is a modelling artifact), the unlucky firm faces the same fallback
position as the new firm, an expected profitability of
$\overline{\theta}$, so status quo is irrelevant, because it is sure
to be rejected.

In the case of either firm, average or unlucky, a virtue of the
regression argument is that managers need not understand it to behave
according to it: they can learn to be optimally conservative as a
rule of thumb.  The boss might well be conservative for the wrong
reasons, thinking in terms of agency problems or
irrationality---``My staff get emotionally attached to the
projects they research.''  Even though he misunderstands the process,
he could grasp that the conservative bias should be less for more
accurate measurements: ``Anderson is smarter than Brown, so he avoids
exaggeration.'' Or, he might reach his conclusion by blind
empiricism, adjusting the adoption threshold until he finds he is
adopting only policies that are ex post superior to the status quo.

 On the other hand, one might ask whether the regression argument is
simple enough that the subordinate would apply it himself. This is a
matter of interpreting the model, which divides the decision process
into two stages: first a staffer measures the value of $\theta$ and
delivers a report $y$, and then a boss calculates $(\theta|y)$ and
makes the decision. This distinguishes two kinds of information
processing.  What business schools teach MBA students is how to use
theory to measure the value of things.  What they spend less time
teaching, perhaps because it is harder to teach, perhaps because it
is easier to learn, is a vague knowledge of how things actually are.
Such ``soft'' knowledge comes with experience, and senior
decisionmakers might positively prefer that their juniors not add
noise to staff estimates by subjective adjustments from their tiny
personal experience. Splitting the decision process divides hard
knowledge from soft knowledge, measuring from deciding. The staffer
watches the trees; and the boss, the forest.  Were there no division
of labor of this kind, the firm could dispense with the boss and
simply have the staffer make the decisions.  But even if there is
just one person involved, the distinction between measurement and
decision is useful.  It could be the boss himself, not the staffer,
who estimates the value. In one part of his mind, he believes his
measurement of $y$ to be an unbiased estimate of $\theta$.  If he is
sophisticated, he steps back and realizes that he is probably
overestimating.


\bigskip
\noindent
{\bf  The Technical Assumptions}

    Proposition 1 relied on the finding in Lemma 1 that $E
(\theta|y)$ lies in the open interval $(E\theta,y)$; that is, the
posterior mean lies between the prior mean and the data. This may
seem obvious, but it is false for certain distributions which violate
the assumptions of the model.
  The counterexample in Figure 1 shows why
 the assumptions exclude bimodal or skewed densities.  In Figure 1,
$f_\theta$ is
weakly unimodal but right-skewed, and $f_u$ has a small variance.
Since $f_u$ has a small variance, the observation of $y$ was more
likely generated by $\theta = m(\theta)>y$, where $m(\theta)>y$, than
by smaller values of $\theta$, so $\overline{\theta}< y< E (\theta|y)
$.  Symmetry
without unimodality allows a similar perversity: if a probability
peak were added at $\theta_1$   to make $f_\theta$
symmetric, it would remain true that $\overline{\theta}< y< E
(\theta|y) $.



\epsfysize=3in

\epsffile{regmean1.eps}



The assumption that the support of the error density is wider than
the support of the new-policy density is important only when the
new-policy density is uniform.  Consider the following example.
 Let
$\theta_0 = 100$,  let $f_\theta$ be   uniform   over
 $[93,103]$ with mean $\overline{\theta}=98$, and let
$f_u$ be  uniform
over $[-2,+2]$.   This violates the support assumption because the
$\theta$'s have a wider support than the $u$'s.  As a result,  the
outcome of
$y=100$ can only be generated by $\theta \in [98,102]$. Hence,
$E(\theta|(y=100)) = 100$, and the new policy should be adopted if $y
> 100$. The
optimal threshold is $y^*=100$, because values of $y$ near $\theta_0$
are as likely to be the result of negative as of positive measurement
errors. Proposition 1 fails.

If, on the other hand, $f_u$ is uniform, not over $[-2,+2]$, but over
$[-10,+10]$, then the support assumption is satisfied. Then an outcome
of $y=100$ is generated by $\theta \in [90,103]$, and $E(\theta|y=100)
= 96.5$. Thus, $E(y|\theta)< y$, and a conservative bias must be
applied.\footnote{If $f_\theta$ is not uniform, the support assumption
is unnecessary. If, for example, $f_\theta$ is  triangular over
$[93,103]$ and $f_u$ is   uniform
over $[-2,+2]$, the support assumption is violated, and
  $y=100$ could only be generated by $\theta \in [98,102]$. The
 probability that $y=100$  was generated by  $\theta \in
[98,100]$, however, is greater than the probability  of $\theta \in
[100,102]$ (which is farther from the mean of 98).
Therefore, $y^*>100$, and there is a conservative bias.}

\bigskip
\noindent
{\bf  Risk Aversion as an Altenative Explanation}

 The model assumes that the boss is risk-neutral, but risk aversion
would also make a known status quo preferable to an uncertain
alternative. This is true whether the uncertainty arises from
exogenous events in the world or from the measurement error of the
staffer, since in either case the boss cannot perfectly predict
profitability.   Which argument, regression or risk aversion, more
reasonably explains the conservative boss?

 Table 2 illustrates a variant model in which the
decisionmaker is risk-averse.  The status quo is $\theta_0=100$, and
$\overline{\theta}=100$ also, so Proposition 1 fails to apply and
$E(\theta|y)= y$. As in Table 1, the standard deviation of the new
policy population is $\sigma_\epsilon=15$.  The acceptance threshold
is $100 + P \left(
\frac{\sigma_\epsilon^2\sigma_u^2}{\sigma_\epsilon^2
+\sigma_u^2}\right)$, because each unit of variance requires a
premium of $P$.\footnote{Note that the value of the status quo and
the measured value of the new policy are assumed to be certainty
equivalents, except for measurement error. If the status quo has a
non-stochastic return and the new policy is risky (even beyond
measurement error), then the problem is simply that the staffer has
not
used the proper discount rate.  The problem analyzed in this section
is different: it asks whether the risk due to measurement error is
important relative to the regression effect.}

 \begin{center}
{Table 2}\\
{Acceptance Thresholds for a Risk-Averse Decisionmaker}\\
  ($100 + P \left(
\frac{\sigma_\epsilon^2\sigma_u^2}{\sigma_\epsilon^2
+\sigma_u^2}\right)$)

\begin{tabular}{lr|cccc}
\hline
\hline
 & & \multicolumn{4}{c}{ Standard Error of Measurement
($\sigma_u$)}\\
 \hline
  & & 5 &10 &15 &40\\
\hline
  & &  & & &\\
  & 0.104 & 102.3&107.2 & {\bf 112} & 121  \\
Price  & 0.052 & 101.2 &103.6 & 106 & 110\\
 of    & 0.026 & 100.6&101.8 & 102.9   & 105 \\
Risk     & 0.013 &  100.3  &100.9 & 101.5&102.6\\
 ($P$) & 0.0& 100.0 &100.0 &100.0 & 100.0 \\
  & &  & & &\\
\hline
 \multicolumn{4}{l}{ Assumed: $\theta_0=100$, $\overline{\theta} =
100$, $\sigma_\epsilon=15$  } &\multicolumn{2}{c}{ (fractions
rounded)}\\
\hline
\hline
  \end{tabular}
 \end{center}

 What price of risk is reasonable?  The situation being modelled is a
boss
facing idiosyncratic risk, rather than an investor facing market
risk, so one might think that the price should be negligible and risk
aversion is unimportant, but let us suppose that idiosyncratic risk
does matter for some reason such as the necessity of tying managerial
compensation to firm value. The stock market's risk premium
provides a benchmark for the price of risk.  The
mean real return on the stock market from 1889 to 1978 was 7.0\% and
the standard deviation was 16.5, compared with a mean of 0.8 and a
standard deviation of 5.7 for low-risk securities.  An increase of
240 percent-squared in variance thus requires an increase of 6.2\% in
return, a price of $P=0.026$ percent-squared.\footnote{The returns
are from p.  147 of Mehra \& Prescott (1985), who constructed them
from the annual average Standard and Poor's Composite Stock Price
Index, the consumption deflator, and various low-risk short-term
securities.  The point of their article is that the amount of implied
relative risk aversion is implausibly high, so 0.026 is most likely
an overestimate.}

 Consider again the boss who refuses a new policy with measured value
of 110 in favor of a status quo with value 100. Suppose that
$\sigma_u=15$.  From Table 1, the regression argument explains the
boss's behavior if $P=0$ and $\overline{\theta}=90$.  From Table 2,
risk aversion explains it if $P = 0.104$ and $\overline{\theta}=100$.
The amount of risk aversion needed seems high---four times as high as
the market price of risk. Moreover, since the project risk is
idiosyncratic, it should actually be priced lower.

  The regression argument has different empirical implications than
simple risk aversion.  One difference is that if the distribution of
new policies has greater variance ($\sigma^2_\epsilon$ increases),
regression has a weaker effect, but risk aversion,  a stronger.
\footnote{Regarding risk aversion, note that in the example
with normal densities, equation (\ref{e20}) says that $
Var(\theta|y)= \frac{\sigma^2_u\sigma^2_\epsilon}{\sigma^2_u
+\sigma^2_\epsilon}$.  $\sigma^2_\epsilon$ is $
\frac{dVar(\theta|y)}{d \sigma^2_\epsilon} =
\frac{\sigma^2_u}{\sigma^2_u +\sigma^2_\epsilon}-
\frac{\sigma^2_u\sigma^2_\epsilon}{(\sigma^2_u
+\sigma^2_\epsilon)^2}>0$.} A second difference is that the
regression effect does not depend on the covariance of the random
terms with other assets in the economy, unlike risk aversion.  Thus,
the conservatism due to the regression argument should not depend on
the beta of the measurement error or the degree to which the
decisionmaker is diversified.



%---------------------------------------------------------------
\bigskip

 \pagebreak
  \begin{center}
      {\bf 2. Progress and Disappointment}
   \end{center}

It was possible to prove Proposition 1 without reference to the way
in which the boss discovered $y$. The  proposition applies whether $y$
is the best of $n$ values or not, and whether $n$ is chosen optimally
or not.  But it is also interesting to look at the effect of
conservatism on industry progress when research levels are optimized.

Let the industry consist of  a leading firm with a status quo of
$\alpha$ and a lagging firm with a status quo of $\beta < \alpha$.
To avoid the strategic
considerations that are  the subject of the large literature
surveyed in Reinganum (1989), assume that the policy of one
firm does not affect the profits of the other. As before, each firm
must choose a research level $n$ and an acceptance threshold  $y^*$.
Which firm is more conservative?

\noindent
 {\it PROPOSITION 2: Progress instills conservatism.  The leading firm
\\
  \hspace*{ .1in} (a)   has a greater conservative bias,\\
 \hspace*{ .1in}  (b)  has a higher threshold for
adoption,\\
  \hspace*{ .1in} (c)  does less research,\\
   \hspace*{ .1in} (d) is less likely to advance its policy,\\
     \hspace*{ .1in} (e) is less likely to advance the industry's
best practice.\\
 All of these points except  (a) remain true even if research
discovers a policy's value with perfect accuracy.
  }

\noindent
{\it Proof:}
See Appendix.

  Points (a) and (b) of Proposition 2 say that the leading firm will
require a greater apparent advantage of the new policy over its
status quo as well as having a higher threshold. As the policy in use
improves, it becomes less and less likely that a new policy is
genuinely better. Unless there is an exogenous shock to the system
that improves the new-policy pool, firms should become more and more
skeptical of apparently superior new policies. Points (c), (d) and
(e) concern the amount of research done by the leading firm. Point
(c) is the most difficult point to prove.  The simple intuition is
that for a given level of research the leading firm is more likely to
reject every investigated policy, and so it is less willing to spend
on research. The complications arise because the lagging firm is more
likely to find an improvement with even a small amount of research,
so additional research might be redundant and it is not clear without
careful analysis which firm has the higher return from research at
the margin. Since it does turn out that the leader does less
research,
in addition to rejecting a greater fraction of investigated policies,
the leader is less likely to improve. Since
the lagging firm adopts innovations more often, it is not only more
likely to improve over its own current policy, but also more likely
to improve over the leader's  current policy.

 Proposition 2 has implications for the life cycle of the industry.
 Even if the industry starts off only mildly conservative, it will
become more conservative as time passes. New policies are adopted
only if they are expected to be superior to the status quo, which
becomes less probable as the status quo improves.  Moreover, since
firms give less thought to changing their policies as they improve,
the lagging firm is both more likely to improve and more likely to
advance the best practice of the industry. On the industry level,
this generates the familiar notion that firms in a young industry
will be bolder and innovate more than firms in a mature industry, and
that struggling firms are more likely to take risks than successful
firms.

 Two caveats should be made. First, interactions between firms have
been ignored, since such interactions have tremendous variety and can
favor research by either the leader or the
laggard.\footnote{Examples: The interaction favors the leader's
research if consumer switching costs induce once-and-for-all switches
when one firm acquires a large enough lead. The interaction favors the
lagging firm's research if there are leakages from innovation so the
lagging firm shares in the leader's progress.  } Second, it
was implicitly assumed that the two firms are equally good at the
technology of innovation. If one firm has a superior ability to find
good policies or to estimate policy values, that firm will more
likely become the leader and  will have a tendency to do
more research which must be balanced against the conservative bias
arising from its advanced technology.

\bigskip

  The regression argument implies another form of apparently
irrational behavior consistently adopting innovations that turn out to
be disappointing. Let us define ``disappointment'' to be the
decisionmaker's state of mind when he discovers that the new policy
is less valuable than predicted by the forecast (i.e.,
 $\theta$ is less than $y$).  This definition is appropriate if one
believes that a person can rationally appreciate his own bounded
rationality and expect to be disappointed. A decision
can lead to disappointment ($\theta <y$), but still be correct
($\theta > \theta_0$).
 From Proposition 1 and the fact that all policy changes accepted have
$y>\theta_0$, it is easy to see why adoption will on average be
followed by disappointment: that is simply another way to state that
$E(\theta|y) < y.$ This idea, which can essentially be found in Brown
(1978) and Harrison \& March (1984), is listed here as Proposition 3.


{\it PROPOSITION 3: A rational decisionmaker will be disappointed on
average when he adopts a  new policy: $E(\theta|y) < y$.}



%---------------------------------------------------------------

\bigskip

\begin{center}
 {\bf  3.  Empirical Implications}
 \end{center}

 Proposition 1 has strong empirical implications, because it cannot
explain genuine managerial risk aversion, but it can explain {\it
apparent} risk aversion, reconciling our usual belief that managers
maximize profits with the common perception that in this particular
aspect of decisionmaking they do not. The model explains this
perception as the result of outsiders observing managers rejecting
projects which  unbiased analysts have stated are superior to
the status quo.  The opinion of analysts and outsiders may never be
contradicted, because the true value of a rejected project is never
discovered by the firm that rejects it. In the case of
 unused technological innovations, it may never be discovered by
anybody, and the outsider will strongly suspect  that the firm has
done its research to acquire   ``sleeping patents'' for strategic
purposes. If the project is not patentable, then it might be adopted
by another firm, but this too might serve to confirm the outside
observer's suspicions, since sometime the rejected project will turn
out to be profitable after all.
  Using the metric ``number of successful
adoptions,'' to compare  managers,  the naive manager who takes
estimates at face value, or the manager of a firm that begins with an
inferior status quo would win, because the
 sophisticated manager of an already-successful firm would indeed
reject more truly
profitable projects---as a simple consequence
of having higher standards.  The sophisticated manager  also rejects
more unprofitable projects, but this  could easily be overlooked.

 The regression argument can also explain why many managers share the
mistrust of Hayes \& Garvin (1982)  concerning academically
uncontroversial capital budgeting methods such as discounted cash
flow. If estimation of the discounted value is done without regard to
the regression argument, the recommendations will be wrong. Schnall \&
Sundem
(1980) expected to find that a survey of companies' capital budgeting
techniques would find greater use of formal methods in firms with
riskier environments. Instead, they found the opposite: risky firms
are more informal. The
regression argument has an explanation: where measurement error is
greater, the naive use of discounted cash flow leads to more
frequent unprofitable innovation. Firms with high measurement
error might find that trained intuition works better than naive
discounting, although a sophisticated use of discounted cash flow
might work better still.

 The regression argument explains perceptions of myopia in the same
way as perceptions of risk aversion,  if one adds the assumption that
projects with more distant returns are measured less accurately.  If
that is true, rational managers will impose a greater conservative
bias for long-term projects, and  it's estimated return will have
to pass a higher hurdle rate. Managers will also
reject a greater percentage of genuinely good long-term projects,
because a greater proportion of the measured high values will be due
to bad projects.  This increased conservatism towards long-term
projects might be interpreted as irrational myopia.

Some readers may be troubled by the existence of innovative
industries,
because the regression model does not seem to apply there. In the
computer
industry, after all, the firm that does not innovate does not
survive. It does not follow, however, that computer firms are not
conservative in the sense of this article, turning down a multitude
of projects with apparently positive present discounted values.
Nothing in the model says that a firm with a conservative bias will
not innovate; only that it will be careful when it innovates. If a
single staffer is sent out, a conservative firm will most likely
retain the status quo. But it might be optimal to investigate 200 new
policies with the expectation that 10 will appear superior to the
status quo, that only 3 will pass the threshold for acceptance, and
that if none pass it, the best alternative is to exit the industry. A
firm can be both extremely conservative in its decisionmaking
  and   extremely likely to innovate.  Whether the computer industry
is conservative in this sense is an interesting empirical question.



 The regression argument can explain the more specific empirical
observations mentioned in the introduction.  Hayes \& Garvin (1982)
found hurdle rates of 30\% or more, compared to actual returns of
20\%.  This is exactly the finding that $y^* > \theta_0$ on average.
Pruitt \& Gitman (1987) found that 80\% of responding financial
officers thought that measurements were overoptimistic, 59\% thought
that decisionmakers scaled them back, 37\% blamed intentional
overestimation and 36\% blamed inexperience. This is evidence that
$E(\theta|y)<y$ in the opinion of decisionmakers, and that they
attribute this to both agency problems and to staffer ignorance (not
knowing $f_\theta$).  Ross (1986) found that the hurdle rate for
large energy conservation projects was near the cost of capital, that
for small projects the hurdle rate was higher, and that for small
projects the decisionmakers made informal adjustments to the
estimates.  The distinction between large and small projects is
 interesting, and might be due to more precise staffers being assigned
to larger projects, or to the staffer for the large project being the
same person as the decisionmaker, and thus able to add the
conservative bias.


 Two additional studies are relevant. Beardsley
\& Mansfield (1978) looked at data on the forecasted and actual
success
of new products and processes developed between 1960 and 1965 by an
anonymous multibillion-dollar company. Of the 57 new projects, six
  were within 10\% of the forecast, 25 had pessimistic
forecasts, and 26 had optimistic forecasts, with a tendency to
pessimism for large and optimism for small projects.  This is the
same relative pattern observed by Ross.\footnote{The  inaccurate
forecasts for large projects are puzzling. Classification bias could
explain this (if a project with a small forecast is successful
enough, it becomes a large project),  or  the unusual macroeconomic
growth of the 1960's, which for most industries would have made
rational forecasts look pessimistic.}


 Little and Mirrlees (1991) report on results from an internal World
Bank report by Pohl \& Dabrarko (1989) which compared ex ante project
appraisals to ex post estimates of the returns. The appraisals
averaged 17 percent in 1968 and rose in a clear trend to 29 percent
in 1980, possibly driven by the ``McNamara effect'' of an executive
who favored increased lending. What is more relevant to the
regression argument is that the average ex post estimate, showing no
similar trend, ranged between 13 percent and 17 percent, averaging 16
percent, and that this was well above the cost of capital.  This is
consistent with the regression argument, since the ex post estimate
is still, in the case of these public, LDC projects, an estimate with
a possibly large error.\footnote{Little and Mirrlees suggest another
reason why estimates would be overoptimistic on average: If the
hurdle rate is 10 percent, and it becomes clear to the staff that the
estimate will be well above 10 percent, then the staff will be less
careful to avoid large positive errors. If the estimate seems to be
close, on the other hand, the staff will measure more carefully. On
average, this leads to overestimates, but harmless ones.}

\bigskip

%---------------------------------------------------------------
\bigskip
\begin{center}
 {\bf  4.  A Miscellany of  Applications}
 \end{center}

The regression argument applies to a variety of situations in which
the value of a status quo is known precisely and innovation would not
be made to a random alternative. This section suggests a few
speculative applications.

  {\it The Fallacy of Sunk Cost.} Suppose that the status quo is 100
dollars that might be invested safely at a return of 5\% or spent on
the new project, that $y^* =150$, and that upon observing $y = 200$,
the manager adopts the project.  Suppose further that the manager
observes the true value $\theta$ after 20 dollars is spent, and he
discovers that it equals 100.  The manager will continue with the
project, paying the additional 80 to get the value of 100 (a return
of 25\%) .  He has received bad news, but his information is now
precise.  An outside observer, however, would see that the
measurement of the total return has dropped from 100\% to 0\%, and
the return from continuing the project has sunk to 25\%, which is
below the original 50\% threshold for acceptance, yet the manager, an
apparent victim of the fallacy of sunk cost, refuses to abandon the
project. If, in general, the information about a project's value
becomes more accurate as more is spent on the project, the threshold
for continuing the project will fall over time, giving rise to
apparent ignorance of the fallacy of sunk cost.

  {\it Loss Aversion.}
 Libby \& Fishburn (1977) note that various studies find experimental
subjects to be more averse to losses than to risk {\it per se}.  The
regression argument explains this if potential loss is measured more
accurately than potential gain. If the experimental subject begins
with a known 100 dollars, and is offered, in exchange, a fair gamble
with a known bad outcome of 0 and an unknown good outcome estimated
at 201, the regression argument applies, and the subject should
behave conservatively.  If, on the other hand, the subject starts
with 0 dollars and faces a choice between a nonstochastic outcome
estimated at 100 dollars and the risky gamble between 0 and 201, the
effect of the regression argument is not clear, since it applies to
both alternatives. But the first tradeoff, with the known 100, is
more usual, because in business  and personal decisions alike,
the potential loss---the resources invested---is usually known with
more accuracy than  the potential gain.


 {\it The Ellsberg Paradox}. Suppose that two urns are
each filled with 100 black and white balls. Urn X has 50 white and 50
black balls, whereas urn Y has  an unknown number of each color.
Experimental subjects should be indifferent between betting on the
draw of a black ball from urn X and betting on the draw of a black
ball from urn Y, but they generally prefer urn X, which has the known
probabilities.

Bordley \& Hazen (forthcoming) explain this using the idea that
players have pessimistic priors over unknown distributions. The
subject of the experiment fears that the experimenter will fill
urn Y with fewer black balls, and that even if the subject were
offered a choice between white and black, the experimenter would
somehow rig the game.

  The regression argument elaborates on this slightly. Assume that
the subjects know that most strangers who offer bets have stacked the
odds in their favor.  In the experiment with black and white balls,
the experimenter may do his best to convince the subjects that he is
offering a fair bet. But the rational subject should be aware that he
can be fooled, particularly in a situation contrived by a person
clever enough to have a doctorate in mathematics. The subject does
not trust himself to have truly figured out all the angles of the
experiment, and so he picks urn X, which is easier to understand. Not
because of measurement error, but because of recognizedly bounded
rationality, the subject regresses the odds of urn Y towards the
unfavorable odds of the typical tricky bet.


%---------------------------------------------------------------




  {\it Scientific Conservatism.} Under the plausible assumption that
the average randomly conceived explanation for a phenomenon is
inferior to the current explanation, the regression argument predicts
that scholars would be slow to accept a new idea, even if it seems
correct to them. (In fact, if the regression argument is correct,
scholars should be slow to accept the regression argument!) The way
classical hypothesis testing is used may be an example of this.
Classical hypothesis testing imposes a strong bias towards not
rejecting the null hypothesis.  That might be justified as a rule of
thumb, if the regression argument is applicable because a randomly
selected alternative hypothesis is likely to be further from the
truth than the null hypothesis.


\bigskip

 {\it Marriage.} If a woman requires that a man be 20\% better
than a randomly drawn man if she is to abandon the status quo of
single life, the regression argument says that the man's measured
value must be more than 20\% better than average.  She will
hesitate to marry even if the potential husband seems to meet her
standards. Of course, unless women are naive---which is perhaps
plausible in this context---it is not possible for every woman's
husband to be above average in the same characteristic. If tastes
differ over characteristics, however, this becomes a matching
problem, such as has been analyzed by various authors (see Mortensen
[1988] for references), and every woman could expect her husband to
be above average in the characteristics she values.  Empirically, one
could work backwards from the regression argument to check its
assumptions in this application. If, for example, most wives lower
their estimates of their husbands after marriage, one might conclude
that they had attempted to marry above-average men.  If, on the other
hand, your wife is not disappointed in you, she probably wasn't very
selective in her choice of a husband.

\bigskip
  Despite this variety of applications, the regression
argument is not capable of being twisted to explain every kind of
conservative behavior. Below are listed a number of situations to
which it does {\it not} apply.

 \noindent
  {\it Not the High Risk Premium on Equity.} The regression argument
might have the implication that the market value of an asset would be
less than its measured value.  But it cannot explain the equity
premium puzzle of Mehra \& Prescott (1985).  If ex ante stock returns
were very high relative to the riskless rate of return, the reason
might be the regression argument rather than risk aversion.  But what
Mehra and Prescott point out is that {\it ex post} stock returns are
peculiarly high, whereas the regression argument says that ex post
there is no conservative bias.


 \noindent
  {\it Not Takeovers.} Roll (1986) proposes, in his ``hubris
hypothesis,'' that the lack of statistically significantly positive
returns to bidders in takeovers might be explained by naive optimism.
There is an asymmetry because pessimistic bidders never attempt a
takeover. Hubris is not formally modelled, but it might arise from
lack of recognition of the regression argument. Naive decisionmaking
is unusually foolish in this context, however: in a takeover, the
bidder knows that other managers have looked at the same project, and
rejected it, and that the target managers are likely to have
information superior to his own. Miller (1977) follows a similar line
of reasoning, but suggests that the winning bidder is not alone in
his optimism, which is what forces up the takeover price. If one
believes that bidders are irrational, the regression argument can be
used to explain the precise nature of their mistakes, but the story
needs  irrationality as well as the regression argument.

 \noindent
 {\it Not Conservative Accounting Practices.} Employees sometimes
steal a company's goods, but they never surreptitiously add to them.
Hence, when those goods are audited, the auditor should expect the
amount of goods to be less than the nominal amount. Moreover, there is
measurement error in the audit, particularly if it is using random
sampling.  One might conclude that if the audit shows the amount of
goods to be close to the nominal amount, the decisionmaker should
adjust his estimate downwards, applying what might be called a
conservative bias. This is an application of  regression towards
the mean, but  not the same application as in this paper.  The
choice is not between a status quo and a new project; the audit is a
pure measurement problem. Moreover, if the first step of the audit
reveals an unusually low measured value, the value should be adjusted
{\it upwards} (which will happen in a full fifty percent of cases),
which does not fit our notion of conservative accounting practices.
Hence, the regression argument does not seem to explain such
practices.

%---------------------------------------------------------------


\begin{center}
 {\bf 5. Concluding Remarks}
 \end{center}

  Using the idea of regression towards the mean, this article offers
a normative reason for conservatism in decisionmaking and a positive
reason for perceptions of excess conservatism. The theory is
simple--- simpler than the complex decision trees and refinements of
risk aversion that one finds in textbooks on decision theory and
finance, and simpler than the agency arguments used to explain
conservatism in the economics literature.  The key assumptions are
that the value of the status quo is known better than the value of
innovations, and that random uninvestigated innovation is
unprofitable. With the addition of mild technical assumptions, these
assumptions imply that raw measurements of innovation values, even if
made with unbiased error, will lead to overinnovation if taken
literally as the basis for decisions. Managers should be
conservative, and their conservatism should increase with the success
of their policies.

%---------------------------------------------------------------
\newpage
\noindent
 {\bf Appendix: Proofs.}


{\it LEMMA 1: (i) $E (\theta|y)$ is increasing in $y$ and (ii)
$\overline{\theta} < E (\theta|y) <y$ for $y> \overline{\theta}$,
 $E (\theta|y) =y$ for  $y= \overline{\theta}$, and
  $ y < E (\theta|y) <\overline{\theta}$ for  $y< \overline{\theta}$.}


 {\it Proof:}
 What must be shown is that the posterior mean is (i) increasing in
the observed datum, and (ii) lies in the open interval between the
prior mean
and the observed datum.  This posterior mean, the conditional
expectation of $\theta$, is
 \begin{equation} \label{e10a}
 E(\theta|y) = \int \theta  f(\theta|y) d\theta.
 \end{equation}
 Using
Bayes Rule:
 \begin{equation} \label{e10}
  f(\theta|y) = \frac{f(y|\theta) f(\theta)}{ f(y)}.
 \end{equation}
   Substituting for $f(\theta|y)$ from  (\ref{e10}) into (\ref{e10a})
gives
 \begin{equation} \label{e12}
 E(\theta|y) = \int \theta \left( \frac{f(y|\theta) f(\theta)}{ f(y)}
\right) d\theta = \left( \frac{1}{ f(y)} \right) \int \theta
f(y|\theta) f(\theta) d\theta.
 \end{equation}

(i) The first part of (\ref{e12}) can be written as
 \begin{equation} \label{e12a}
 E(\theta|y) = \int \theta f(\theta) \left( \frac{f(y|\theta) }{
f(y)} \right) d\theta.
 \end{equation}
 Since $\overline{\theta} = \int \theta f(\theta) d\theta$, the
question is what effect multiplying each $\theta$ in the integral by
the weights $
 \frac{f(y|\theta) }{ f(y)}$ has. For given $y$, this weight is
greatest for $\theta=y$ and is equal for $\theta=y+\delta$ and
$\theta = y - \delta$ given the assumption that $f_u$ is symmetric
and weakly unimodal.  Hence, if $y$ increases, the weights on all the
values of $\theta$ less than $y$ decrease and those on the greater
values increase, so $E(\theta|y$) will increase in $y$.

(ii)
 Fix $y$ at some level greater than $\overline{\theta}$.  It will
be shown that $E(\theta|y) < y$, and the other results will follow
easily.
  Because $f(y|\theta)$ is symmetric around $\theta=y$,
 \begin{equation} \label{e13}
  \int \theta f(y|\theta) d\theta = y.
 \end{equation}

 It  is sufficient to show that multiplying $\theta$ by $f(y|\theta)
f(\theta)$ in the integral in (\ref{e12}), instead of by
$f(y|\theta)$ as in (\ref{e13}), puts relatively greater weight on
smaller values of $\theta$.  (Dividing by $f(y)$ just makes the
density integrate to one.) The symmetry of $f_u$ implies that
$f(y|(\theta=y-\delta)) = f(y|(\theta=y+\delta))$ for any $\delta$,
so what needs to be shown is that
 \begin{equation} \label{e14}
 f_\theta(y-\delta) \geq f_\theta (y+\delta),
 \end{equation}
 with strict inequality for at least one value of $\theta$ such that
$f(y|\theta) > 0$ (otherwise, $f(\theta) f(y|\theta) = 0$, and the
strictness of (\ref{e14}) would be irrelevant). Given (\ref{e14}),
multiplication by $f(\theta)$ in expression
(\ref{e12}) weights the values of $\theta$ less
than $y$ more heavily than those greater than $y$.

Inequality (\ref{e14}) is true for $\delta<y-\overline{\theta}$
because $f'_\theta \leq 0$ in that range, by the assumption of weak
unimodality of $f_\theta$.  Inequality (\ref{e14}) is true for
$\delta>y-\overline{\theta}$ because $y-\delta$ is nearer to
$\overline{\theta}$ than is $y+\delta$, and the symmetry and weak
unimodality of $f_\theta$ together imply that $f_\theta(\theta)$ is
weakly greater for values of $\theta$ nearer $\overline{\theta}$.

Inequality (\ref{e14}) is strict for at least one  $\theta$ such that
$f(y|\theta)>0$,
either because $f'_\theta <0$ for some $\delta< y-\overline{\theta}$
(if $f_\theta$ is sloping) , or because there is some $\delta$ such
that $f_\theta(y-\delta) > f_\theta (y+\delta) = 0$ (if $f_\theta$ is
flat).    This is true for flat $f_\theta$  under the assumption that
$f_u$
has a larger support than $f_\theta$.   The support
of $f(y|\theta)$ overlaps with at least the most positive half of the
support of $f_\theta$, so $f(y|\theta)>0$ for at least one $\delta$
such that inequality (\ref{e14}) is strictly satisfied.  Hence,
equation (\ref{e12}) does have heavier weight than (\ref{e13}) for
some $\theta$ such that $f(y|\theta)>0$.

 Given that equation (\ref{e13}) shows that the unweighted function
integrates to $y$, the heavy weights on low values of $\theta$ in
equation (\ref{e12}) imply that the weighted function integrates to
less than $y$. Hence, $E(\theta|y) < y$ if $ y > \overline{\theta}$.

Parallel arguments show that $E (\theta|y) =y$ for $y=
\overline{\theta}$, and
  $E (\theta|y) >y$ for  $y< \overline{\theta}$.
 Given  that
$E (\theta|y=overline{\theta}) =\overline{\theta}$ and that
 $E (\theta|y)$ is increasing in $y$ from part (i), it follows that
$E (\theta|y) < \overline{\theta}$ for  $y< \overline{\theta}$ and
$E (\theta|y) > \overline{\theta}$
for  $y> \overline{\theta}$.
This completes the description of the intervals in which the
posterior means lie.  $\Box$

\bigskip

\noindent
 {\it PROPOSITION 2: Progress instills conservatism.  The leading firm
\\
  \hspace*{ .1in} (a)   has a greater conservative bias,\\
 \hspace*{ .1in}  (b)  has a higher threshold for
adoption,\\
  \hspace*{ .1in} (c)  does less research,\\
   \hspace*{ .1in} (d) is less likely to advance its policy,\\
     \hspace*{ .1in} (e) is less likely to advance the industry's
best practice.\\
 All of these points except for (a) remain true even if research
discovers a policy's value with perfect accuracy.
  }

\noindent
{\it Proof:}
 Part (a) says that $y^*- \theta_0$ increases in $\theta_0$.  This is
equivalent to $ y^* - E(\theta|y^*) $ increasing in $\theta_0$.
 To see this, refer to the proof of Lemma 1, starting with
inequality (\ref{e14}).  Assume that $y > \theta_0$.  The proof showed
that
if $f_\theta (y-\delta) > f_\theta(y+\delta)$ for some $\theta$ such
that $f(y|\theta) >0$, then $y - E(\theta|y) >0$.  For the present
proof, the size of $y - E(\theta|y)$ matters. The size of $y -
E(\theta|y)$ increases with the difference $f_\theta (y-\delta) -
f_\theta(y+\delta)$ and with the range of $\theta$ for which the
difference is positive.  If $y$ is greater, then since $f_\theta$ is
unimodal, either or both of the difference and range increase.
Hence, $y- E(\theta|y)$ increases with $y$, and since $y^*$ increases
with $\theta_0$, part (a) is proved.


 Part (b) says that $y^*$ rises with $\theta_0$.  $y^*$ is chosen so
that $E(\theta|y^*) = \theta_0$, and since Lemma 1 established that
$E(\theta|y)$ increases in $y$, it follows that for larger
$\theta_0$, $y^*$ is larger.


Parts (c), (d), and (e)  pertain to optimal choice of $n$. Define
$y_m(n) \equiv Max \{y_i \}_{i=1,\ldots n}$.  The boss will adopt
the innovation if and only if $y_m(n) > y^*$; otherwise, the
investigation was useless, {\it ex post}.

  The advantage of searching is that $Ey_m(n)$ increases,
so that $E(\theta|y_m(n))$ increases also. Values of $y_m(n)$ below
$y^*$ are useless, and assigning zero weight to these, the expected
value of a research level of $n$ is
 \begin{equation} \label{e200}
 V(n) = \theta_0 F(y^*)^n + \int_{y^*}^\infty E(\theta|y) n f(y)
F(y)^{n-1} dy - cn.
 \end{equation}
  The reasoning behind this is as follows.  If $y_m(n) < y^*$ , then
all new policies are rejected, and the firm's value remains the
status quo value of $\theta_0$.  The probability of any one new
policy being worse than the status quo is $F(y^*)$, so the
probability of all of them being worse is $F(y^*)^n$ and the first
term of (\ref{e200}) is $\theta_0 F(y^*)^n$.  The second
term represents the values and probabilities of accepted policies.
Only values greater than $y^*$ are relevant, giving the integration
bounds of $y^*$ and $\infty$. Consider a measurement $y$, with
expected true value $E(\theta|y)$,  probability density
$f(y)$ for each of the $n$ measurements, and probability mass $n
f(y)$ over all of them. With probability $F(y)$, each of the $n-1$
other measurements is less than $y$, so with probability $F(y)^{n-1}$
they are all less.  The third term arises from the research cost $cn$.

 It is convenient to  rewrite $V(n)$ as follows:
  \begin{equation} \label{e3}
 V(n) = \theta_0+ \int _{-\infty}^ \infty \int_{y^*}^\infty \left[
1-F(y)^n \right] f (\theta|y) dy d \theta - cn.
 \end{equation}
   This expression equals equation (\ref{e200}) because it can be
integrated by parts to get
    \begin{equation} \label{e4}
     \begin{array}{ll}
 V(n) & = \theta_0 + \left. E(\theta_y)y \left[ 1- F(y) \right]
\right|^\infty_{y^*} - \int_{y^*}^\infty \left[ -nF(y)^{n -1} f(y)
E(\theta|y) \right] dy - cn\\
 & \\
  & = \theta_0 + 0 - E(\theta|y^* )+ E(\theta|y^*)F(y^*)^n +
\int_{y^*}^\infty \left[nF(y)^{n -1} f(y) y \right] dy - cn\\
   & \\
    & = \theta_0 F(y^*)^n + \int_{y^*}^\infty E(\theta|y) n f(y)
F(y)^{n-1} dy - cn, \\
 \end{array}
 \end{equation}
 which is  (\ref{e200}).

 Using (\ref{e3}), it can be seen that there are diminishing returns
to research. $n$ is a discrete variable, but one may differentiate
with respect to it and ignore all but discrete values.  This gives an
expression for the marginal value of search:
  \begin{equation} \label{e5}
 \frac{ \partial{V}}{\partial n} = - \int_{-\infty}^\infty
\int_{y^*}^\infty F(y)^n log (F(y))f(\theta|y)dy d \theta- c
  \end{equation}
  Expression (\ref{e5}) is positive for small $c$, because $F$ is
less than one.  But search has diminishing returns:
    \begin{equation} \label{e6}
 \frac{ \partial^2{V}}{\partial n^2}  = - \int_{-\infty}^\infty
\int_{y^*}^\infty F(y)^n log^2 (F(y)) f(\theta|y) dy d\theta  <0.
   \end{equation}
   Also, the marginal benefit of search falls with $y^*$, because
        \begin{equation} \label{e7}
 \frac{ \partial^2{V}}{\partial n \partial y^* } =
\int_{-\infty}^\infty F(y^*)^n log (F(y^*)) d\theta <0.
   \end{equation}
   These inequalities can be used to prove part (c), which says that
$n^*$, the optimal level of research, falls in $\theta_0$. Using the
implicit function theorem, $\frac{d n} {d y^*} <0$.  As the status
quo improves (for if $\theta_0$ increases, so does $y^*$), $n^*$ does
not increase, and may decrease. (The inequality is not strict,
because $n$ is a discrete variable and for small changes in $y^*$,
$n^*$
might not change.)


 Part (d) says that $Prob( y_m> y^*(\theta_0))$ falls with
$\theta_0$.  The leading firm  does less research and has a
higher threshold, both of which reduce the probability that $y_m >
y^*$, so this is clearly true.

 Part (e) says that $Prob(y_m > \alpha)$ falls with $\theta_0$.  Let
the leading firm's optimal acceptance threshold be $y^*_\alpha$.
Although both firms adopt policies with $y > y^*_\alpha$, the
lagging firm also adopts some other policies. These other policies
are most likely inferior, but with positive probability their true
value is greater than $\alpha$. Hence, the lagging firm has a higher
probability than the leading firm of choosing a policy with true
value better than the best existing policy in the industry.

 If research is perfectly accurate, $u=0$ and $y=\theta$. This
implies that $y^* = \theta_0$.  Part (b) obviously is still true,
because the leading firm has a higher $\theta_0$. None of the
proof of parts (c), (d), and (e) used the fact that $y\neq \theta$,
so perfectly accurate research is a special case covered by those
parts of the proposition. Only part (a) is invalid, which it is
because with accurate research the conservative bias equals zero for
both firms.   $\Box$


%---------------------------------------------------------------

\newpage

\noindent
 {\bf References.}


 Beardsley, George \& Edwin Mansfield (1978), ``A Note on the
Accuracy of Industrial Forecasts of the Profitability of New Products
and Processes,'' {\it Journal of Business}, 51: 127-135.


 Bordley, Robert \& Gordon Hazen (forthcoming) ``SSB and Weighted
Linear Utility as Expected Utility with Suspicion,''{\it Management
Science}.

 Brown, Keith (1974), ``A Note on the Apparent Bias of Net Revenue
Estimates for Capital Investment Projects'' {\it Journal of Finance},
September 1974, 29, 4: 1215-16.

 Brown, Keith (1978), ``The Rate of Return of Selected Investment
Projects,'' {\it Journal of Finance}, September 1978, 33,4:
1250-1253.

 Casella, George (1985), ``An Introduction to Empirical Bayes Data
Analysis,'' {\it American Statistician}, May 1985, 39: 83-87.


 Day,  Richard (1987), ``The General Theory of Disequilibrium and
Economic Evolution,'' in D. Batten, J. Casti, and B. Johansson, eds.,
{\it Economic Evolution and Structural Adjustment}, Berlin:
Springer-Verlag, 1987.

 Harrison, J. \& J. March (1984), ``Decision Making and Postdecision
Surprises,'' {\it Administrative Science Quarterly}, March 1984,
26-42.

Hayes, Robert \& David Garvin (1982), ``Managing as if Tomorrow
Mattered,'' {\it Harvard Business Review}, May-June 1982, Vol 60, pp.
71-79

 Heiner, Ronald (1983), ``The Origin of Predictable Behavior,'' {\it
American Economic Review}, 83: 560-595.

 Hirshleifer, David \& Anjan Thakor (1991), ``Managerial
Conservatism, Project Choice, and Debt,'' UCLA AGSM Finance Working
Paper \#14-89, January 1991.

 Holmstrom, Bengt \& Joan Ricart (1986), ``Managerial Incentives and
Capital Management,'' {\it Quarterly Journal of Economics}, November
1986, 835-860.

 Kuran, Timur (1988), ``The Tenacious Past: Theories of Personal and
Collective Conservatism.'' {\it Journal of Economic Behavior and
Organization}, 10: 143-171.

 Lambert, R. (1986), ``Executive Effort and Selection of Risky
Projects,'' {\it Rand Journal of Economics}, Spring 1986, 17: 77-88.

 Libby, Marc \& Peter Fishburn (1977), ``Behavioral Models of Risk
Taking
in Business Decisions: A Survey and Evaluation,'' {\it Journal of
Accounting Research}, 15: 272-292.


Little, Ian \& James Mirrlees (1991), ``Project Appraisal and
Planning Twenty Years On,'' {\it Proceedings of the World Bank Annual
Conference on Development Economics, 1990}, Washington: World Bank,
1991.


 Mehra, Rajnish \& Edward Prescott (1985), ``The Equity Premium: A
Puzzle,'' {\it Journal of Monetary Economics}, 15: 145-61.

 Miller, Edward (1977), ``Risk, Uncertainty, and Divergence of
Opinion,'' {\it Journal of Finance}, September 1977, pp. 1151-67.

Miller, Edward (1978), ``Uncertainty Induced Bias in Capital
Budgeting,'' {\it Financial Management}, Autumn 1978, pp. 12-18.

Miller, Edward (1987), ``The Competitive Market Assumption and
Capital Budgeting Criteria,'' {\it Financial Management}, Winter
1987, pp. 22-28.


Mortensen, Dale (1988), ``Matching: Finding a Partner for Life or
Otherwise,'' {\it American Journal of Sociology}, 94 (supplement):
S215-S240.


Narayanan, M. (1985), ``Managerial Incentives for Short-Term
Results,'' {\it Journal of Finance}, December 1985, 1465-1484.

Pohl, Gerhard \& Mihaljek Dabrarko (1989), ``Project Evaluation in
Practice: Uncertainty at the World Bank,'' Economic Advisory Staff,
World Bank.

Pruitt, Stephen \& Lawrence Gitman (1987), ``Capital Budgeting
Forecast Bias: Evidence from the Fortune 500'' {\it Financial
Management}, Spring 1987, 46-51.

Reinganum, Jennifer (1989) ``The Timing of Innovation: Research,
Development and Diffusion'' In Schmalensee, Richard \& Robert Willig,
eds.  {\it The Handbook of Industrial Organization.} New York:
North-Holland, 1989.

 Roll, Richard (1986), ``The Hubris Hypothesis of Corporate
Takeovers,'' {\it Journal of Business}, 59: 197-216.

 Ross, Marc (1986), ``Capital Budgeting Practices of Twelve Large
Manufacturers'' {\it Financial Management}, Winter 1986, 15,4: 15-22.

 Sah, Raaj \& Joseph Stiglitz (1988) ``Committees, Hierarchies, and
Polyarchies,'' {\it Economic Journal} (June 1988) 98: 451-70.


Schnall, Lawrence \& Gary Sundem (1980), ``Capital Budgeting Methods
and Risk: A Further Analysis,'' {\it Financial Management}, (Spring
1980) 9: 7-11.

Shleifer, Andrei \& Robert Vishny (1989), ``Equilibrium Short
Horizons of Investors and Firms,'' mimeo, University of Chicago
Graduate School of Business, November 1989.

 Smidt, S. (1979), ``A Bayesian Analysis of Post-Audits,'' {\it
Journal of Finance}, June 1979, 34,3: 675-688.

Stein, Jeremy (1988), ``Takeover Threats and Managerial Myopia,''
{\it Journal of Political Economy}, February 1988, 96:61-80.

Weitzman, Martin (1979), ``Optimal Search for the Best Alternative,''
{\it Econometrica}, May 1979, 47: 641-54.

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