# Ignorance Is Bliss:
The Screening Effect of (Noisy) Information††thanks: Previously titled
“Looking the Other Way: The Screening Role of (Weak) Internal Monitoring”. We
are indebted to Phil Dybvig, Paolo Fulghieri, Kebin Ma, Beatrice Michaeli,
Sebastian Pfeil, Richard Saouma, Chao Tang, Curtis Taylor, Alexei Tchistyi,
Andrew Winton, Huaxiang Yin, and participants at the MIT Asia Conference in
Accounting, Summer Institute of Finance, Global Management Accounting Research
Symposium, Econometric Society NASM, Stony Brook International Conference on
Game Theory, the Conference on Markets and Economies with Information
Frictions, the Financial Markets and Corporate Governance Conference, RSM
Corporate Finance Day, CFAR, TJAR, and EEA-ESMS for helpful comments and
suggestions. Robin Luo provided excellent research assistance.

Felix Zhiyu Feng Wenyu Wang Yufeng Wu Gaoqing Zhang Michael G. Foster School
of Business, University of Washington. Email<EMAIL_ADDRESS>School of
Business, Indiana University. Email<EMAIL_ADDRESS>College of
Business, University of Illinois Urbana-Champaign. Email:
<EMAIL_ADDRESS>School of Management, University of Minnesota.
Email<EMAIL_ADDRESS>
(February, 2023)

###### Abstract

This paper studies how the firm designs its internal information system when
facing an adverse selection problem arising from unobservable managerial
abilities. While more precise information allows the firm to make ex-post more
efficient investment decisions, noisier information has an ex-ante screening
effect that allows the firm to attract on-average better managers. The
tradeoff between more effective screening of managers and more informed
investment implies a non-monotonic relationship between firm value and
information quality, and a marginal improvement of information quality does
not necessarily lead to an overall improvement of firm value.

JEL Classification: M41, D82, G34
Key Words: information system design, screening, corporate governance,
accounting noise

##  1 Introduction

Internal information system and managerial talent both play a critical role in
firm operations in practice. Firms rely on high-quality information to assess
the prospect of their investment opportunities, and on high-quality managers
to safeguard the success of any investments undertaken. While the literature
has long recognized the importance of internal information system design and
managerial screening, these two are usually studied separately and in distinct
settings.

In this paper, we propose and analyze a model in which the two decisions
interact with each other. Consistent with conventional wisdom, the firm makes
more efficient investment decisions when it adopts a system that generates
more precise information. However, we demonstrate the existence of a novel
channel in which information quality also affects the screening of managers
whose ability is unobservable. Moreover, we find that surprisingly, the firm
achieves a more effective screening result by adopting a system that generates
noisier information. The tradeoff between better screening and more informed
investment decisions results in a non-monotonic relationship between the
quality of a firm’s information system and its overall value: an improvement
in the former does not always lead to an increase in the latter.

In the model, a firm needs to hire a manager from a pool of potential
candidates to supervise a risky investment project. Each candidate’s innate
ability, which we refer to as his type, determines the probability that the
project will succeed if he is hired. The firm is also able to obtain a binary
signal about whether the project will succeed or not from its internal
information system. The more precise the signal is, the more likely a good
(bad) signal indicates eventual success (failure). The firm’s information
quality is publicly observable and cannot be changed once it is chosen.

To hire a manager, the firm offers a compensation contract to all candidates,
but faces an adverse selection problem because it does not observe each
candidate’s type. As a result, the contract must offer the managers a certain
amount of utility, known as the managers’ rent, in exchange for the truthful
reporting of their types. Seeing the contract and the firm’s information
quality, each manager forms rational expectations of the rent he can extract
based on his own type. Those who are willing to accept the firm’s contract
form a managerial pool, from which a randomly selected manager will be hired.
We normalize the value of all managers’ outside options to zero, and assume
they prefer unemployed when they are indifferent. Therefore, the managerial
pool consists of only the candidates who can expect to receive positive rent
from the contract.

After the manager is hired but before the investment is made, the firm
receives a signal from its internal information system and makes the decision
whether to continue or scrap the project. Continuation incurs costs for both
the firm and the manager, but generates some output only if the project
succeeds. If the project is scrapped, both the firm and the manager receive
zero payoff. Thus, the managerial pool is equivalent to the set of managers
who expect to receive investment under some conditions.

Crucially, while the investment outcome is assumed to be observable and
contractible, the firm’s signal is not. This reflects the fact that internal
information regarding a firm’s operation is usually soft, subjective, and
difficult to fully characterize in a formal contract. Consequently, the firm
cannot ex-ante commit to any investment policy contingent on the realization
of the signal. Instead, by calculating the ex-post conditional probability of
success based on the manager’s (reported) type and the actual signal received,
the firm makes one of three investment decisions: never invest, always invest,
and invest only when the good signal is received. This divides the space of
the manager candidates into three investment zones. Low-type managers with
insufficient conditional probability of success regardless of the signal are
in the no investment zone. High-type managers with sufficient conditional
probability of success even under the bad signal are in the unconditional
investment zone. The rest, intermediate-type managers are in the conditional
investment zone, within which investment is made if and only if the firm
receives a good signal.

The range of the conditional investment zone depends on the quality of the
signal. When the signal is more precise, the conditional probability of
success given the good signal is higher for any type of manager. Consequently,
the conditional investment zone expands, and the minimal type of manager that
can expect to receive investment becomes lower. Meanwhile, because the
manager’s rent comes from his compensation, which he can receive only if
investment is made, the minimal type of manager in the conditional investment
zone is also the minimal type in the managerial pool. This implies our main
result that a more precise signal reduces the average quality of the manager
the firm can hire, which we refer to as the screening effect of information
quality. When information is perfect, i.e., when a good (bad) signal leads to
success (failure) for sure, the investment decision is made based on the
signal only. All managers are in the conditional investment zone, and no
manager will be excluded from the managerial pool. In contrast, when the
signal is noisier, the firm relies less on the signal to make the investment
decision. The conditional investment zone shrinks while the average quality of
the managerial pool increases. In the extreme case when the signal is
completely uninformative, the conditional investment zone disappears, and the
investment decision is solely based on the type of the manager. Only the
managers with sufficiently high types (i.e., sufficiently high unconditional
likelihood of success) will receive investment. All other managers are
excluded from the managerial pool, allowing the firm to achieve the best
screening result.

Next, we analyze the overall effect of information quality on the ex-ante firm
value, i.e., the expected firm profit before hiring the manager. While a
noisier signal achieves more effective screening, it does not necessarily
imply that the firm is always better off. In the model, the firm has two
objectives: hiring better managers, and investing only when it is likely going
to succeed. Noisier information helps the firm achieve the first objective,
but better information helps the firm achieve the second objective. We show
this tradeoff of the dual roles of information leads to a non-monotonic
relationship between information quality and firm value. For firms with a
strong existing information system, the effect of more informed investment
dominates, and firm value increases in information quality. However, for firms
with a weak existing information system, the screening effect of noisier
information dominates, and firm value actually decreases in information
quality.

Our model produces several implications for the design of internal information
system in practice. In particular, our results suggest that marginal
improvements of information quality are not always beneficial, and the overall
impact of small or gradual adjustments of a firm’s internal information system
on firm value should be evaluated with caution. Our results also provide a
possible explanation for the observed heterogeneity among firms’ internal
governance practices, such as the manners and intensities at which firms
conduct internal monitoring or auditing, or the various degrees of latitude
the managers enjoy in making operational decisions. These diverse policies
could be the result of firms balancing the direct effect of more informed
decisions and the indirect effect of screening more capable managers. In
particular, a lack of frequent monitoring and auditing or a large degree of
discretion for the managers is not necessarily the result of failure in
internal governance.

We also extend the benchmark model to differentiate the role of information
quality from the role of accounting conservatism, a downward bias under which
bad signals are more likely to be generated than good signals (e.g., as in
Gigler, Kanodia, Sapra, and Venugopalan, 2009, Caskey and Laux, 2017, etc.).
We show that, opposite to information noisiness, accounting conservatism has a
negative screening effect, whereby more conservative accounting leads to a
decline of the quality of the managerial pool. This is because while more
conservative accounting and noisier information both lower the likelihood of a
good signal, conservatism also results in the good signal becoming more
informative, in which case the project is more likely to succeed if the good
signal is observed. This expands the lower bound of the conditional investment
zone, attracting more low-type managers to the pool, and reducing the efficacy
of screening.

Our paper is closely related to the literature that studies firms’ ex-ante
design of information systems. For instance, Bertomeu and Cheynel (2015) study
the optimal asset measurement rules for assets used as collateral and show
that the optimal measurement is asymmetric. Jiang and Yang (2017) study the
design of accounting rules to reduce firms’ signaling cost via excessive share
retention, and show that the optimal accounting rule is consistent with
conservative reporting. Friedman, Hughes, and Michaeli (2020, 2022) consider
the ex-ante optimal design of information systems when managers can acquire
and/or disclose additional private information. Bertomeu, Cheynel, and
Cianciaruso (2021) study the design of measurement systems when there is
uncertainty regarding whether verifiable information exists. Perhaps more
related are the studies that examine the signaling role of firms’ ex-ante
accounting choices.111In comparison, a few other studies consider the
signaling role of firms’ ex-post reporting discretion, e.g., Stocken and
Verrecchia (2004), Fan (2007), Baldenius and Meng (2010), etc. In particular,
Bagnoli and Watts (2005) examine a firm’s choice of conservative reporting
policies, and derive the conditions under which such choice serves as a signal
for the investors to infer the firm’s private information. Dordzhieva, Laux,
and Zheng (2022) show that firms may choose to adopt a liberal accounting
system in order to signal favorable private information and attract external
financing. Different from these studies, our paper argues that the design of
information system also has a screening effect when the firm faces an adverse
selection problem arising from unobservable abilities, thus highlighting a
novel mechanism in the literature of the ex-ante design of information system.

Our paper is also related to the accounting literature that considers the role
of accounting information in firms’ investment decisions, particularly whether
the information system should be made more or less conservative. For example,
Nan and Wen (2014) show that conservative (liberal) reporting helps to
mitigate investment inefficiency in less (more) profitable industries. Caskey
and Laux (2017) find that conservative accounting improves the oversight of
managers’ investment decisions, but at the same time, provides managers with
incentives to distort the accounting system. Laux and Ray (2020) show that
more conservative reporting can improve investment efficiency and stimulate
innovation, when managers’ incentive contracts are adjusted optimally in
response to changes in the accounting system. Laux and Laux (2021) show that
the false alarm cost of conservative information systems can be alleviated
when managers are allowed to gather additional information upon receiving
unfavorable accounting reports. In this literature, the optimality of
conservative or liberal systems often results from the tradeoff between the
costs of false negative and false positive errors. In our model, the costs and
benefits of noisy or biased information systems are not driven by these two
errors. Instead, the tradeoff emerges from the ex-ante more effective
screening of managers and the ex-post more informed investment decisions.

Finally, our paper is related to the broad literature that explores the impact
of corporate governance and board monitoring on firms’ real and financial
decisions, and their implications on firm value, e.g., Mehran (1992); Hermalin
(2005); Burns, Kedia, and Lipson (2010); Baldenius, Melumad, and Meng (2014),
etc. These studies in general focus on the positive value of more information,
both in terms of quantity and quality, and how that contributes to curbing
managers’ empire-building incentives, facilitating external financing, or
lowering excessive compensation. In contrast, our paper highlights a potential
cost associated with more precise information. Admittedly, our paper is not
the first to argue that better information is not always beneficial (e.g.,
Arya, Glover, and Sivaramakrishnan, 1997; Kanodia, Singh, and Spero, 2005;
Gao, 2010; Kanodia and Sapra, 2016; etc.). However, the mechanism is different
from existing studies that arrive at similar conclusions, which can be broadly
categorized into two groups. The first group of studies, represented by Raheja
(2005), Adams and Ferreira (2007), and Harris and Raviv (2008), typically
assume the board has multiple roles (e.g., as monitors and advisors of the
managers), and over-committing to one role (e.g., intensity of monitoring) can
impair the effectiveness of the other. The second group, represented by
Burkart, Gromb, and Panunzi (1997), Goldman and Slezak (2006), and Banerjee
and Szydlowski (2021), assume the informed agent has multiple actions (e.g.,
maintaining productivity and directing investments), and overly-stringent
monitoring can mitigate the agency friction in one action but exacerbates the
friction in the other action. Our paper shows that even though the board
(representing the firm) has a single role of collecting information, and the
manager has a single role of supervising investment, internal information
itself has multiple effects. In particular, noisier information can facilitate
the ex-ante screening of managers when the managers’ ability is not directly
observable.

##  2 Model

This section introduces the model. Section 2.1 lays out the timing of the
events and the basic model structure. Section 2.2 discusses and motivates the
key assumptions. Section 2.3 describes the optimization objective of each
party of the model and defines the equilibrium.

###  2.1 Timing of Events

A firm needs to hire a manager from a pool of potential candidates to
supervise a risky investment project. Both the firm and the managers are risk-
neutral, but the latter ones are protected by limited liability. Figure 1
summarizes the timing of events, which are described in detail below:

$t=0$$t=1$$t=2$Firm chooses informationquality $q$.Firm offers a
compensationcontract. A manageraccepts the contractand reports type
$\hat{\theta}$.Firm observes signal $s$and chooses whether toinvest. The
investmentoutcome is realized. Figure 1: Timeline of the model.

At $t=0$, the firm is endowed with an opportunity to invest in a risky
project. The payoff of the project $\pi\in\left\\{0,1\right\\}$ is binary:
either the project succeeds and returns a unit of output or it fails and
yields $0$. The success of the project requires the firm to hire a manager to
supervise the operation of the project, as the success probability
$\theta\in[0,1]$ depends on the manager’s ability (type). The pool of
potential manager candidates is a continuum indexed by each manager’s type
$\theta$, which is privately known only to the manager himself. All the other
parties share a common belief that $\theta$ follows a distribution
characterized by a cumulative density function $F(\theta)$.

To explore how the firm’s design of its internal information system can
endogenously alter the pool of managerial candidates, we assume that at $t=0$,
the firm makes an observable choice of the quality (or more precisely, the
noisiness) of its internal information system, which is denoted by $q$. The
system then generates a signal $s\in\\{0,1\\}$ about the outcome $\pi$ of its
project at $t=2$ that follows the following distribution:

$\Pr(s=0|\pi=1)=\Pr(s=1|\pi=0)=q.$ (1)

In other words, $q$ represents the probability that the signal is the opposite
of the underlying state. A higher $q$ thus represents a noisier information
system. We assume that $0<q<1/2$, so that $s=1$ can be appropriately labeled
as the good signal and $s=0$ as the bad signal based on the implied likelihood
of the high and low state, respectively.222In the main model, we assume, for
simplicity, a symmetric information system in which $q$ captures the noisiness
of both the good the bad signals. In Section 4, we extend the model to
consider an asymmetric information system and examine the effect of accounting
conservatism on the screening of managers. We assume that the signal $s$ is
the firm’s private information and unverifiable.

At $t=1$, the firm offers a compensation contract in order to hire a manager
from the candidate pool. For simplicity, we assume that the firm has no excess
cash and cannot make payments to the manager unless the project succeeds and
generates cash flows. Accordingly, the firm’s compensation contract to the
manager is a wage $w\geq 0$ payable if and only if the project
succeeds.333Appendix A.1 considers an extension that allows the manager to be
compensated even when the project fails. Our main results regarding the
screening effect of information quality remain intact. We allow the firm to
optimally tailor its wage structure. In particular, it can be made contingent
on the manager’s reported type.

We normalize the managers’ outside option to zero, and assume that they prefer
to remain unemployed when they are indifferent. Therefore, managers are
willing to accept the firm’s contract if their expected net payoff from
working in the firm is strictly positive.444Alternatively, we can assume the
managers prefer to be employed when indifferent, but the value of their
outside option is strictly positive. We present the analysis of this extension
in Appendix A.2. The results are qualitatively the same. For simplicity, we
assume a one-time random matching between the firm and the pool of managers
willing to accept the firm’s contract. Upon his employment, the manager
reports his type $\hat{\theta}$. We place no restrictions on the manager’s
reporting choice.

At $t=2$, the firm receives the private signal $s$ from its internal
information system. Upon observing the signal and conditional on the manager’s
reported type $\hat{\theta}$, the firm chooses whether to invest in its risky
project. If the firm chooses not to invest, both the firm and its manager
receive zero payoffs. Alternatively, if the firm chooses to invest, it incurs
an investment cost $k>0$. Investment also incurs a cost $c>0$ from the manager
regardless of whether it eventually succeeds. This can be interpreted as the
direct personal cost the manager needs to bear as long as investment is
made.555In other words, to maintain focused on the screening mechanism in the
presence of adverse selection, we do not model hidden effort choices by the
manager or any resulting moral hazard problem. We assume that $k+c<1$ to
ensure that the net surplus from the project is positive if it succeeds.

###  2.2 Discussion of Assumptions

We now discuss and motivate several key assumptions of our model.

First, we assume that the firm’s signal $s$ is private and unverifiable so
that the firm cannot ex-ante commit to an investment strategy based on the
realization of the signal. Similar assumptions are often made in the
incomplete contracting literature (e.g., Dewatripont and Tirole, 1994). In
practice, much of the information known privately to the firm’s board of
directors often reflects the directors’ own subjective assessments of the
project’s prospect; accordingly, specifying the information content of the
signals and how the firm should respond to them ex-ante is too costly, making
the signals non-contractible (Laux and Laux, 2021).

Second, we assume that the firm’s choice of information quality $q$ is made
ex-ante and observable before the firm hires the manager. Similar assumptions
are commonly made in the Bayesian persuasion literature.666See, for example,
Kamenica and Gentzkow (2011), Michaeli (2017), Bertomeu, Cheynel, and
Cianciaruso (2021), Friedman, Hughes, and Michaeli (2020, 2022), Dordzhieva,
Laux, and Zheng (2022), among others. This assumption is essential as it
allows us to study how the firm’s informational choice affects its screening
of potential manager candidates. We interpret such informational choices
broadly, such as the composition, dedication, and industry experience of the
firm’s board of directors. These reflect persistent and fundamental structures
of the firm that are unlikely to fluctuate in the short run.

Finally, we assume a one-time random matching between the firm and the pool of
manager candidates willing to accept the firm’s contract. This is mainly for
simplifying the structure of the managerial labor market. It rules out the
competition among managers and any potential repeated signaling or bargaining
games. However, the fact that each candidate can decide whether to join the
managerial pool after seeing the contract is crucial. As we explain later,
this allows the firm to use the contract terms to shape the distribution of
managers that it can potentially be matched with, i.e., the contract serves as
a screening device for the firm. The assumption can also be relaxed, for
example, by allowing some sorting between the manager and the firm, or by
allowing the firm to take more than one draw from the managerial pool. As long
as there is some randomness in the matching process or some costs every time
the firm samples its managerial pool, our mechanism remains qualitatively
intact.

###  2.3 Optimization Objectives and Equilibrium

Because type is the manager’s private information, the firm faces an adverse
selection problem. The firm’s compensation contract and investment policy must
therefore provide each manager with the incentives to report his type
truthfully. Formally, let $p_{s}(\hat{\theta})\equiv\Pr(\pi=1|s,\hat{\theta})$
denote the conditional probability that the project will succeed given the
signal $s$ and the manager’s reported type $\hat{\theta}$, and
$w(\hat{\theta})$ denote the wage to the manager at success. Because the firm
cannot ex-ante commit, its investment decision, denoted by
$x(s,\hat{\theta})$, must be ex-post optimal, i.e.,

$x(s,\hat{\theta})=\begin{cases}1,\text{ if
}p_{s}(\hat{\theta})[1-w(\hat{\theta})]>k,\\\ 0,\text{ otherwise.}\end{cases}$
(2)

Given this investment policy, each manager forms rational expectations of the
likelihood of receiving investment and chooses his reporting strategy
$\hat{\theta}$ to maximize his expected utility from the contract, denoted by

$R_{\hat{\theta}}(\theta)=\mathrm{E}[x(s,\hat{\theta})(w(\hat{\theta})-c)]\;,$
(3)

Following the convention, we will focus on incentive-compatible (IC) contracts
under which each manager finds it optimal to report his type truthfully, i.e.,

$\theta=\hat{\theta}^{\ast}\equiv\arg\max_{\hat{\theta}}R_{\hat{\theta}}(\theta),\text{
for all }\theta.$ (4)

We will then drop the notation $\hat{\theta}$ in the subsequent analyses as
long as the IC condition is met or imposed. For example, the manager’s
expected utility from the contract in the equilibrium is simply denoted by
$R(\theta)\equiv R_{\hat{\theta}^{\ast}=\theta}(\theta)$. This is commonly
known as the managerial rent, which refers to the utility that a manager needs
to receive in exchange for revealing his private information truthfully.

The manager will accept the contract if and only if he receives positive rent
from the firm’s contract. That is, the firm’s managerial pool $\Theta$ is
defined as

$\displaystyle\Theta=\\{\theta:R(\theta)>0\\}.$ (5)

The firm’s objective is to maximize its ex-ante expected payoff before
matching with a manager and has three instruments (controls) to do so: wage
$w$, the investment policy $x$, and the information quality $q$. Because our
focus is the impact of information quality on firm value, we write the firm’s
objective as $\max_{q}V(q)$, where

$\displaystyle
V(q)=\max_{x,w}\int_{\Theta}\mathrm{E}\left[x(s,\theta)(\pi-w(\theta)-k)\right]dF(\theta),$
(6)

denotes the expected firm value under a given level of information
quality.777To minimize notations, we use $F(\theta)$ as the cumulative density
function for the overall, unconditional distribution of $\theta$ and use
$\int_{\Theta}v(\theta)dF(\theta)$ to represent the conditional expectation of
any function $v(\theta)$ given $\theta\in\Theta$. As it will become clear
later, $\Theta=\\{\theta:\theta>\theta_{1}\\}$ for some $\theta_{1}$ in the
equilibrium. In that case,
$\int_{\Theta}v(\theta)dF(\theta)=\frac{1}{1-F(\theta_{1})}\int_{\theta_{1}}^{1}v(\theta)dF(\theta)$
Consequently, we define a firm’s contract as optimal under a given $q$, and
refer to the policies and payoffs under such contract as the equilibrium
policies and payoffs, if the contract solves (6) subject to constraints (2),
(4), and (5). That is, the investment decisions $x$ must be ex-post optimal;
the contract must be incentive compatible; and the managerial pool consists of
only those with strictly positive rents.

##  3 Model Solution

This section presents the solution to the model. We derive the equilibrium
managerial rent under a given information system in Section 3.1 and present
our main result, the screening effect and how it varies with the firm’s
information quality, in Section 3.2. Section 3.3 offers an alternative
explanation for the intuition of the screening effect for interested readers.
We explore the overall impact of information quality on firm value in Section
3.4 and discuss some comparative statics in Section 3.5.

###  3.1 Managerial Rent

We begin with deriving the firm’s optimal screening contract (i.e., the wage
and investment policy) and the implied equilibrium managerial rent for a given
level of information quality. For ease of exposition, we will derive the
equilibrium heuristically first, before collecting all the results to present
formally. In particular, we will assume that there is an incentive compatible
contract under which the wage for success is a constant, i.e., $w(\theta)=w$
for all $\theta$. We verify later that this is indeed a property of the
optimal contract.

Given the firm’s information quality $q$ and the type of the manager hired
$\theta$, the probability of all events can be summarized in the following
matrix:

| Success $\pi=1$ | Failure $\pi=0$
---|---|---
Good signal $s=1$ | $\theta(1-q)$ | $(1-\theta)q$
Bad signal $s=0$ | $\theta q$ | $(1-\theta)(1-q)$

We can define $p_{s}(\theta)$, the conditional probability of success given
the signal $s$ and type $\theta$, as:

$\displaystyle p_{1}(\theta)\equiv\Pr(\pi=1|s=1)$
$\displaystyle=\frac{\theta(1-q)}{\theta(1-q)+(1-\theta)q},$ (7)
$\displaystyle p_{0}(\theta)\equiv\Pr(\pi=1|s=0)$ $\displaystyle=\frac{\theta
q}{\theta q+(1-\theta)(1-q)}.$ (8)

Conditional on each signal, the firm makes the investment if and only if
$p_{s}(\theta)(1-w)>k$. This implies two cutoff types: $\theta_{0}^{f}$ and
$\theta_{1}^{f}$, which represents the point at which the firm is indifferent
between investment or not conditional on a bad signal ($s=0$) and on a good
signal ($s=1$), respectively. The superscript $f$ denotes these are cutoffs
based on the firm’s investment decision. These two cutoffs solve the two
corresponding indifferent conditions:

$\displaystyle p_{1}(\theta_{1}^{f})(1-w)$ $\displaystyle=k,$ (9)
$\displaystyle p_{0}(\theta_{0}^{f})(1-w)$ $\displaystyle=k.$ (10)

It is straightforward to show that $\theta_{1}^{f}<\theta_{0}^{f}$ when
$q<1/2$. Intuitively, for any given $\theta$, the good signal is more likely
to be associated with success than the bad signal is. Therefore, the minimal
type of manager that the firm is willing to invest in is lower with the good
signal than with the bad signal. Moreover,

$\displaystyle\lim_{\theta\rightarrow 0}p_{0}(\theta)$
$\displaystyle=\lim_{\theta\rightarrow 0}p_{1}(\theta)=0,$ (11)
$\displaystyle\lim_{\theta\rightarrow 1}p_{0}(\theta)$
$\displaystyle=\lim_{\theta\rightarrow 1}p_{1}(\theta)=1,$ (12)

which means that as the manager’s type approaches zero (one), the conditional
likelihood of success approaches zero (one) regardless of the signal. Since
the return from investment is strictly negative (positive) when the success
likelihood is zero (one), the space of managerial type is divided into three
“zones” based on whether the manager will receive the investment or not if he
reports his type truthfully:

  * •

Low-type managers with $\theta\leq\theta_{1}^{f}$ belong to the “no-
investment” zone in which the project will be scrapped regardless of the
signal, and thus receive no rent, i.e., $R(\theta)=0$.

  * •

High-type managers with $\theta>\theta_{0}^{f}$ belong to the “unconditional
investment” zone, in which investment will be made regardless of the signal.
The rent for managers in this zone, denoted by $R_{u}(\theta)$, is

$R_{u}(\theta)=\theta w-c.$ (13)

  * •

Intermediate-type managers with $\theta_{1}^{f}<\theta\leq\theta_{0}^{f}$
belong to the “conditional investment” zone, in which investment will be made
if and only if the firm receives the good signal. The rent for managers in
this zone, denoted by $R_{c}(\theta)$, is

$R_{c}(\theta)=\theta(1-q)(w-c)-(1-\theta)qc.$ (14)

The left panel of Figure 2 illustrates these three functions: $R=0$,
$R_{u}(\theta)$, and $R_{s}(\theta)$ for the entire space of $\theta\in(0,1)$.
Note that both $R_{u}$ and $R_{s}$ can be lower than zero if $\theta$ is
sufficiently low. This is because investment is risky not only to the firm but
also to the manager: he has to bear a cost $c$ and only receives the wage $w$
if the project succeeds. This leads to an important observation regarding
incentive compatibility: once placed in each zone, the manager’s rent is a
function of his true type only. Thus, the only potential benefit from
misreporting his type is to distort the firm’s investment decision by
manipulating the firm’s belief about the manager’s success likelihood. Since
the manager can freely choose which investment zone he wishes to be in through
his report, a contract is incentive compatible if and only if the manager’s
equilibrium rent is the upper segments of all three functions ($R=0$, $R_{u}$,
and $R_{s}$), shown in the right panel of Figure 2:

Figure 2: This figure illustrates the construction of the equilibrium
managerial rent function $R(\theta)$. The left panel plots the managerial rent
for the three investment zones for all $\theta\in(0,1)$: $R(\theta)=0$ (no-
investment, dotted line), $R_{c}(\theta)$ (conditional investment, equation
(14), dashed line), and $R_{u}(\theta)$ (unconditional investment, equation
(13), dashed-dotted line). The right panel plots the equilibrium rent (blue
solid line), which is the upper segment of the individual rent functions shown
in the left panel. $\theta_{1}$ and $\theta_{0}$ correspond to the left and
right boundaries of the conditional investment zone in the equilibrium defined
in (17) and (18), respectively.

The equilibrium managerial rent has two kink points. The left kink, denoted by
$\theta_{1}^{m}$, represents the manager that receives equal rents from being
in the no-investment zone or in the conditional investment zone, and solves

$\displaystyle\theta_{1}^{m}(1-q)(w-c)-(1-\theta_{1}^{m})qc=0,$ (15)

where the superscript $m$ indicates that this is a cutoff based on the
manager’s reporting strategy. The right kink, denoted by $\theta_{0}^{m}$,
represents the manager that receives equal rents from being in the conditional
investment or the unconditional investment zones, and solves

$\displaystyle\theta_{0}^{m}(1-q)(w-c)-(1-\theta_{0}^{m})qc=\theta_{0}^{m}w-c.$
(16)

Incentive compatibility thus requires that all managers form the correct
rational expectation regarding the boundaries of each investment zone, and no
manager is strictly better off relocating himself into a different zone by
misreporting his type. That is, a contract is incentive compatible only if

$\displaystyle\theta_{1}^{f}$ $\displaystyle=\theta_{1}^{m}\equiv\theta_{1},$
(17) $\displaystyle\theta_{0}^{f}$
$\displaystyle=\theta_{0}^{m}\equiv\theta_{0}.$ (18)

If one of these two conditions is not met, e.g., if
$\theta_{0}^{f}<\theta_{0}^{m}$, then managers with type
$\theta\in\left(\theta_{0}^{f},\theta_{0}^{m}\right)$ will receive
unconditional investment if they report their types truthfully. However, their
rents are higher if they are in the conditional investment zone, which they
can achieve by under-reporting their type, violating incentive compatibility.
Intuitively, these are managers with intermediate levels of $\theta$, which
represent a modest unconditional probability of success. They are better off
being in the conditional investment zone, where they can receive higher rents
by free-riding the firm’s signal regarding their likelihood of success.
Similarly, if $\theta_{0}^{m}<\theta_{0}^{f}$, some managers with type
$\theta\in\left(\theta_{0}^{m},\theta_{0}^{f}\right)$ are better off over-
reporting their type in order to receive unconditional investment. These are
managers with sufficiently high unconditional success probability and can
obtain more rents from maximizing the likelihood of receiving the wage
payment, which is possible only if the investment is made.

Finally, we prove that although wage is allowed to be made contingent on the
manager’s reported type, the firm will optimally choose not to do so and
instead offer a flat wage $w^{*}$ to all managers. The complete argument is in
the appendix but the intuition is as follows. First, wage must be independent
of type within each investment zone, otherwise every manager in that zone will
always report his type to be the one that earns the highest wage.
Consequently, there can be at most two levels of wage, one for intermediate-
type managers in the conditional investment zone, and one for high-type
managers in the unconditional investment zone. We show that if the former is
strictly smaller than the latter, the IC condition (18) will be violated; if
the former is strictly larger than the latter, the contract is incentive
compatible but not optimal: the firm can do better by lowering the higher wage
for the intermediate-type managers in the conditional investment zone, which
improves both the screening (i.e., higher $\theta_{1}$, discussed in details
in Section 3.2) and the firm’s profit when the project does succeed.
Therefore, the wage must be a constant under the optimal contract.

Collecting all the results, we now formally summarize the optimal screening
contract and the implied equilibrium managerial rent as follows:

###### Proposition 1

Given the firm’s information quality (noise) $q$, the optimal incentive
compatible screening contract that satisfies (17) and (18) has the following
properties:

  * i)

The wage for success is a constant given by

$\displaystyle w$ $\displaystyle=w^{*}\equiv\frac{c}{k+c}.$ (19)

  * ii)

The implied boundaries of the investment zones are:

$\displaystyle\theta_{1}(q)$ $\displaystyle=\frac{q(k+c)}{(1-q)-(1-2q)(k+c)},$
(20) $\displaystyle\theta_{0}(q)$
$\displaystyle=\frac{(1-q)(k+c)}{q+(1-2q)(k+c)}.$ (21)

  * iii)

The equilibrium managerial rent $R(\theta)$ is given by

$R(\theta)=\begin{cases}0\;,\text{ if }\theta\in(0,\theta_{1}(q)],\\\
\theta(1-q)(w^{\ast}-c)-(1-\theta)qc\;,\text{ if
}\theta\in(\theta_{1}(q),\theta_{0}(q)],\\\ \theta w^{\ast}-c\;,\text{ if
}\theta\in(\theta_{0}(q),1).\end{cases}$

###  3.2 The Screening Effect of Information Quality

Proposition 1 leads to our main result of this section: the managerial pool
and how it varies with the firm’s information quality $q$.

###### Proposition 2

Under the optimal screening contract characterized in Proposition 1, the
firm’s managerial pool consists of all the managers with type higher than a
cutoff level: i.e., $\Theta=\\{\theta:\theta>\theta_{1}(q)\\}$, where
$\theta_{1}(q)$ is given by (20). The cutoff is increasing in the noisiness of
the firm’s signal, i.e., $\theta_{1}^{\prime}(q)>0$ for all $q$.

Proposition 2 follows naturally from the properties of the optimal screening
contract summarized in Proposition 1. Because only managers with strictly
positive rent in the equilibrium will join the managerial pool, the pool
consists of all managers with types higher than $\theta_{1}(q)$, i.e., all
managers in the conditional and unconditional investment zones, and the
minimal type of the managerial pool is also the left boundary of the
conditional investment zone.

Importantly, $\theta_{1}(q)$ is an increasing function in $q$, meaning that a
noisier signal achieves more effective screening of managers by excluding more
low-type candidates from the managerial pool. We refer to this as the
(positive) screening effect of noisier information. Figure 3 (left panel)
visualizes such screening effect by plotting two rent functions under
different levels of $q$ and their corresponding $\theta_{1}(q)$. Intuitively,
given that the equilibrium wage is constant, $\theta_{1}$ solves

$p_{1}(\theta_{1})=\frac{k}{1-w^{\ast}}=k+c,$ (22)

where $p_{1}(\theta)$, given in (7), is an increasing function of $\theta$ and
a decreasing function of $q$. The more precise the signal, the higher
$p_{1}(\theta)$ is for any $\theta$, once a good signal is observed.
Consequently, higher-quality information (i.e. lower $q$) lowers the minimal
type of manager with whom the firm is willing to invest under a good signal.
This expands the left boundary of the conditional investment zone, allowing
more managers with lower $\theta$ to have the opportunity to trigger
investment. These managers in turn can expect to receive positive rents and
thus are willing to join the managerial pool. In contrast, when information
quality decreases (i.e., $q$ increases), $p_{1}(\theta)$ is less informative
about the likelihood of success. As a result, only managers with sufficiently-
high types can trigger the investment even under the good signal, and more
low-type managers are left in the no-investment zone and thus out of the
managerial pool.

Figure 3: The left panel plots the equilibrium rent functions and the
resulting $\theta_{1}(q)$, the minimal type in the firm’s managerial pool,
under different levels of information quality $q_{1}$ and $q_{2}$, where
$q_{1}<q_{2}$. The right panel plots the rent function and the resulting
$\theta_{commit}$, the minimal type in the firm’s managerial pool, for the
“with-commitment” benchmark discussed in Section 3.3.

Put differently, the firm in our model has two objectives: screening for high-
ability managers, and investing only when they know the project will succeed.
The benefit of a better signal is reflected in the latter objective by
improving the firm’s assessment of the likelihood of success from the
unconditional probability $\theta$ to the conditional probability
$p_{1}(\theta)$. However, this improvement of the success likelihood also
benefits the intermediate-type managers with modest unconditional success
probability, who otherwise would not be given the investment without the
(good) signal. Thus, a better signal also bears the cost of drawing more of
those managers to the managerial pool, undermining the first objective of
screening. In comparison, lower information quality reduces the accuracy of
the firm’s assessment of the success likelihood, which hurts not only the firm
but also the intermediate-type managers, thus allowing the firm to achieve
more effective screening by excluding more of those managers from the
managerial pool.

###  3.3 The Role of Commitment

We have so far established how the screening of managers depends on the
quality of the firm’s information. Given this is the central result of the
paper, we offer in this section an alternative understanding of its underlying
intuition based on its connection with the firm’s (lack of) commitment power.
Readers who are more interested in the implications of our model can skip this
section and move on to the next.

Consider the case in which the firm can ex-ante commit to any investment
policy regardless of the signal. Suppose the firm’s objective is to achieve
the best screening result (i.e., maximizing the minimal type in the managerial
pool) while still offering the equilibrium wage $w^{*}=c/(k+c)$ found in (19)
for success.888Technically speaking, with commitment, the firm can offer any
wage $w>c$ because the IC conditions (17) and (18) are no longer relevant when
the firm can commit to any $\theta^{f}_{s}$. We assume the firm still offer
the equilibrium wage $w^{*}$ here to simplify the illustration of the role of
commitment. The firm’s optimal policy is to offer $w^{*}$ to all managers and
commit to scrapping (continuing) all investments if the manager’s reported
type is less (more) than $\theta_{commit}=c/w^{*}$. In other words, the space
of managers is partitioned into two zones only: the “no investment” zone for
low-type managers (i.e., $\theta<\theta_{commit}$) who also earn zero rent,
and the “unconditional investment” zone for high-type managers (i.e.,
$\theta>\theta_{commit}$) who earn positive rent. The right panel of Figure 3
illustrates the managerial rent and the resulting managerial pool under this
contract. The firm achieves the best possible screening result because
$\theta_{commit}=c/w^{\ast}>\theta_{1}(q)$ (given in equation (20)) for any
$q\in(0,1/2)$. Intuitively, the ability to commit to investment policies
eliminates the “conditional investment” zone, where the firm’s private signal
is used. This lowers the rent for intermediate-type managers (to zero) and
thus excludes them from the managerial pool.

Without commitment, the firm is unable to sustain this screening result. The
lack of commitment creates a conditional investment zone for intermediate-type
managers who rationally expect some rents in the equilibrium. This is because
$p_{1}(\theta)$, the conditional probability of success, is sufficiently high
for these managers. Consequently, while the firm would like to exclude most of
them from the managerial pool by committing never to invest, it is not sub-
game optimal to do so once a good signal is observed. A more precise signal
(i.e., lower $q$) increases $p_{1}(\theta)$ for all $\theta$, thus lowering
the minimal type of managers that expects to receive investment (and thus
positive rents) in the equilibrium, drawing them to the managerial pool.999In
addition, the lack of commitment implies that the firm cannot offer any wage
$w<w^{*}$ while maintaining incentive compatibility. It can be shown that
$\theta_{0}^{f}<\theta_{0}^{m}$ if $w<w^{*}$. Therefore, managers with type
$\theta\in(\theta_{0}^{f},\theta_{0}^{m})$ will have the incentive to
misreport their type and be placed into the conditional investment zone, which
yields them higher rents than reporting truthfully and being placed into the
unconditional investment zone. Therefore, compared to the “with-commitment”
benchmark, the lack of commitment raises the wage needed to maintain incentive
compatibility and creates the conditional investment zone that attracts some
intermediate-type managers to the managerial pool.

In contrast, a noisier signal improves the firm’s screening result by lowering
the conditional probability of success given the good signal, causing the firm
to scrap more often for the intermediate-type managers. This intuition can be
best illustrated in the extreme case in which the signals are completely
uninformative (i.e., $q=1/2$). This is equivalent to no signal being
available, and the firm only makes investment decisions based on the reported
managerial type. Consequently, the equilibrium coincides with the “with-
commitment” equilibrium described above, where the firm achieves the best
screening result.101010Of course, the “with-commitment” equilibrium is not
necessarily the optimal for the firm overall, due to the same tradeoff between
better screening and more informed investment described in the previous
section. In short, the effect of the precision of the signal on the firm’s
screening result is related to the friction generated by the lack of
commitment. Noisier information mitigates the firm’s lack of commitment issue
by lowering the usefulness of the signal, thus increasing the likelihood of
the scenarios in which the firm can commit to (i.e., no investment as long as
the manager’s type is low and invest only when type is sufficiently high).

The role of commitment in our model shares some similarity with that in Zhu
(2022) which, unlike us, studies a dynamic moral hazard problem. In that
model, the firm can gather information about the performance of the employees
through continuous monitoring, but cannot commit not to use the information
for excessive punishment. Anticipating this, employees ex-ante demand a
contract with low pay-performance-sensitivity. This reduces the likelihood of
ex-post punishment but also the incentives for the employees to exert effort,
leading to an overall decline of the workforce productivity. Different from
Zhu (2022), we study an adverse selection problem arising from unobservable
managerial type. The firm cannot commit not to use the private signal to make
investment decision, which attracts rent-seeking low-type manager and lowers
the average quality of the managerial pool. Moreover, high-quality information
in Zhu (2022) is always beneficial to the firm, but the production of such
information is costly (e.g., requiring impatient firm investors to wait
longer). In our model, the quality of information is intrinsically costless
but carries two endogenous effects (screening versus informed investment) that
manifest in opposite directions.

###  3.4 Information Quality and Firm Value

We are now ready to examine the overall relationship between information
quality and the ex-ante firm value $V(q)$, defined in (6). Given the contract
described in Proposition 1, $V(q)$ can be written as

$\displaystyle V(q)$
$\displaystyle=\left(\frac{1}{1-F(\theta_{1})}\right)\left[\int^{\theta_{0}}_{\theta_{1}}v_{c}(\theta)dF(\theta)+\int^{1}_{\theta_{0}}v_{u}(\theta)dF(\theta)\right],$
(23)

where

$\displaystyle v_{c}(\theta)$
$\displaystyle=\theta(1-q)(1-w^{*}-k)-(1-\theta)qk,$ (24) $\displaystyle
v_{u}(\theta)$ $\displaystyle=\theta(1-w^{*})-k,$ (25)

$w^{*}$, $\theta_{1}$, and $\theta_{0}$ are given in (19), (20), and (21).
$v_{c}(\theta)$ and $v_{u}(\theta)$ represents respectively the firm’s
expected payoff from hiring a $\theta$-type manager in the conditional and
unconditional investment zones.

The firm’s information quality affects the firm value through three channels:
$\theta_{1}$, the lowest type in the managerial pool; $\theta_{0}$, the
boundary between the conditional and unconditional investment zones; and the
slope of $v_{c}(\theta)$ in the conditional investment zone. The first channel
corresponds to the screening effect: the firm’s objective of hiring the best
manager. The second and third channels reflect the firm’s objective of
investing when the project is likely going to succeed. To better illustrate
the impact of information quality on firm value through these two objectives,
differentiating (23) with respect to $q$ and observing that
$v_{c}(\theta_{1})=0$ and $v_{c}(\theta_{0})=v_{u}(\theta_{0})$ in the
equilibrium, we have:

$\displaystyle V^{\prime}(q)$ $\displaystyle=\frac{\partial
V}{\partial\theta_{1}}\frac{d\theta_{1}}{dq}+\frac{\partial
V}{\partial\theta_{0}}\frac{d\theta_{0}}{dq}+\left(\frac{1}{1-F(\theta_{1})}\right)\left[\int^{\theta_{0}}_{\theta_{1}}\frac{\partial
v_{c}(\theta)}{\partial q}dF(\theta)\right]$ (26)
$\displaystyle=\underbrace{V(q)\left(\frac{f(\theta_{1})}{1-F(\theta_{1})}\right)\theta_{1}^{\prime}(q)}_{\text{screening
effect,
}>0}+\underbrace{\left(\frac{1}{1-F(\theta_{1})}\right)\left[\int^{\theta_{0}}_{\theta_{1}}\frac{\partial
v_{c}(\theta)}{\partial q}dF(\theta)\right]}_{\text{change in investment
profit, }<0}.$ (27)

The first term in (27) reflects the marginal screening effect, which is
positive because $\theta^{\prime}_{1}(q)>0$ (Proposition 2). The second term
in (27) reflects the marginal change in the firm’s investment profit in the
conditional investment zone, which is negative because $\partial
v_{c}(\theta)/\partial q<0$. That is, a noisier signal erodes the firm’s
expected profit in the conditional investment zone by reducing the
informativeness of the good signal. Altogether, (27) demonstrates the tradeoff
the firm faces when designing an optimal information system.

Figure 4: The left panels plot $\theta_{1}(q)$, the minimal type in the firm’s
managerial pool, given by (20). The right panels plot $V(q)$, the ex-ante firm
value, given by (23)-(25). In the top two panels, $c=0.15$. In the bottom two
panels, $k=0.15$. $\theta$ follows uniform distribution.

Figure 4 depicts several numerical examples for $\theta_{1}(q)$ and the firm
value $V(q)$. In all the examples, $\theta_{1}$ decreases in $q$, consistent
with the prediction in Proposition 2. The U-shaped firm value $V(q)$ also
echoes other studies (e.g., Morris and Shin, 2002; Goldstein and Yang, 2019;
etc.) showing that the value of information quality is non-monotonic. However,
unlike the prior studies, our result is driven by a novel tradeoff between
more effective screening and more informed investment. While proving all the
properties of $V(q)$ (such as its global convexity) illustrated by these
numerical examples turns out to be analytically challenging, we can
nonetheless prove the key takeaway—that neither more nor less precise
information is always necessarily better for the firm—under certain
conditions:

###### Proposition 3

If $\theta$ follows a uniform distribution, then there exists
$0<\underline{q}<\overline{q}<1/2$ such that $V^{\prime}(q)>0$ for all
$q>\overline{q}$ and $V^{\prime}(q)<0$ for all $q<\underline{q}$.

Proposition 3 states that the screening effect dominates when the firm’s
information quality is sufficiently low, and an improvement of information
quality (i.e., lowering $q$) for these firms may actually decrease their
value. This can be easily seen algebraically in (27): when $q=1/2$ (meaning
that the signal is useless), $\theta_{0}=\theta_{1}$, and only the first term
that corresponds to the screening effect exits. Therefore,
$V^{\prime}(1/2)>0$, and by continuity, $V^{\prime}(q)$ must be positive for
$q$ that is sufficiently high. Intuitively, the benefit from the signal is
more informed investment in the conditional investment zone. When the signal
is not very informative, the firm relies less on the signal for investment.
The conditional investment zone shrinks, manifesting mostly the screening
effect. In contrast, when the signal is very precise (i.e., $q$ is close to
zero), the firm makes most of its investment decision based on the signal. The
conditional investment zone is wide, and any deterioration of the precision of
the signal has a substantial negative effect on the return to investment in
that zone and thus, the overall firm value (i.e., $V^{\prime}(q)<0$).

Results of this section have important implications for the firm’s design of
its internal information system. While a perfect information system (i.e., $q$
near zero) is valuable, marginal improvements of information quality are not
always beneficial. In practice, producing useful information for operational
decisions such as investment is often costly and requires a substantial amount
of subjective judgment, rendering a perfect information system infeasible and
any significant improvement of an existing system challenging. Given these
constraints, the overall impact of gradual adjustments of information system
should be evaluated with caution. Our results also provide a possible
explanation for the observed heterogeneity in both the quality of internal
information system and how information is utilized. In practice, firms vary
substantially in their manners and intensities of conducting internal
monitoring or auditing, and managers are allowed various degrees of latitude
in making operational decisions. Our results suggest that such diversity could
be the result of firms optimally balancing the direct effect of more informed
decisions and the indirect screening effect of information. In particular, a
lack of frequent monitoring and auditing or a large degree of liberty for the
managers are not necessarily the result of failure in internal governance.

###  3.5 Comparative Statistics

This section derives some comparative statics of our main results. To maximize
transparency, our model is intentionally made simple with only two parameters,
both pertaining to the cost of investment: $k$, the firm’s physical cost, and
$c$, the manager’s personal cost. The proposition below summarizes how they
affect the screening of managers and the overall firm value:

###### Proposition 4

Fixing any signal quality $q>0$, $\theta_{1}(q)$, the minimal type in the
managerial pool, increases with the firm’s investment cost $k$ and the
manager’s investment cost $c$. That is, $\partial\theta_{1}/\partial k>0$, and
$\partial\theta_{1}/\partial c>0$.

Suppose $\theta$ follows a uniform distribution. Firm value is decreasing in
the manager’s investment cost $c$ when the signal is either perfectly
informative (i.e., $q=0$) or completely uninformative (i.e., $q=1/2$). That
is, $\partial V(0)/\partial c<0$ and $\partial V(1/2)/\partial c<0$. Firm
values in those states are increasing in the firm’s investment cost $k$ when
$k$ is low and decreasing in $k$ when $k$ is high. That is, there exists
$k^{*}$ such that $\partial V(0)/\partial k>0$ and $\partial V(1/2)/\partial
k>0$ when $k<k^{*}$, and $\partial V(0)/\partial k<0$ and $\partial
V(1/2)/\partial k<0$ when $k>k^{*}$.

Proposition 4 entails two sets of results. First, both the firm’s investment
cost $k$ and the manager’s investment cost $c$ have the same positive
screening effect by increasing $\theta_{1}$. Intuitively, $\theta_{1}$
represents the lowest type the firm needs in order to invest when receiving
the good signal. Because both $k$ and $c$ represent the cost of investment but
do not affect the gross return or the probability of success when signal
quality $q$ is given, the higher those costs, the higher the managerial type
needed for investment to take place.111111Based on the alternative intuition
discussed in Section 3.3, a higher cost of investment mitigates the firm’s
lack of commitment problem by forcing the firm to withhold investment from a
larger set of low-type managers even under the good signal. The left panels of
Figure 4 illustrate these results.

However, while a higher $k$ or $c$ improves the screening, both erode the
firm’s net return (profit) from investment. The second set of results of
Proposition 4 says that in the extreme cases of perfectly informative or
uninformative signals, the erosion of the firm’s profit from investment
dominates for all values of $c$. Given that firm value $V(q)$ is mostly
U-shaped in $q$, a higher $c$ lowers the firm value for all $q$, as
illustrated by the lower right panel of Figure 4.121212While we are unable to
analytically prove these results other than at $q=0$ or $q=1/2$, we have not
found any exceptions for any $q\in(0,1/2)$ in our numerical exercises.
Interestingly, the effect of $k$ on the firm value is non-monotonic: when $k$
is low, the screening effect dominates, and firm value is increasing when $k$
becomes larger. When $k$ is high, the erosion of investment profit dominates,
and firm value is lower when $k$ becomes larger. In other words, given $q$ and
$c$, there exists an interior optimal level of $k$ that maximizes the firm
value, as illustrated by the upper right panel of Figure 4.

Proposition 4 generates some interesting implications. If the firm has some
control over the magnitude of the investment costs, the firm is better off
driving down the manager’s investment cost $c$ as low as possible. Meanwhile,
the firm does not always benefit from lowering the physical investment cost
$k$. When such cost is low, a marginal increase of such cost may improve
overall firm value by attracting on-average more capable managers through a
positive screening effect.

##  4 Information Quality and Accounting Conservatism

The baseline model so far assumes a symmetric information structure in which
the good and bad signals are equally informative. The extant literature has
also explored asymmetric information structures especially that under
conservative accounting, which entails a downward bias such that bad signals
are more likely to be disclosed than good signals (e.g., as in Gigler,
Kanodia, Sapra, and Venugopalan, 2009, Caskey and Laux, 2017, etc.). This
section extends the baseline model by introducing conservative bias in the
information system and compares the screening effect of accounting
conservatism and that of information noise.

Specifically, we modify the model to allow the firm to choose the noisiness of
the signal and the degree of conservatism separately. Let $\lambda$ represents
the firm’s choice of conservatism, we assume

$\displaystyle\Pr(s=0|\pi=1)=q+\lambda,$ (28)
$\displaystyle\Pr(s=1|\pi=0)=q-\lambda.$ (29)

That is, while information noise $q$ affects the likelihood that the signal is
the opposite of the underlying state, conservatism $\lambda$ always increases
the likelihood of the bad signal. We assume $\lambda<q<1/2$ so that both
probabilities above are well defined. Put differently, given $\theta$, the
probability of all events can be summarized as follows:

| Success $\pi=1$ | Failure $\pi=0$
---|---|---
Good signal $s=1$ | $\theta(1-q-\lambda)$ | $(1-\theta)(q-\lambda)$
Bad signal $s=0$ | $\theta(q+\lambda)$ | $(1-\theta)(1-q+\lambda)$

Applying the same solution method used for deriving Proposition 1, we can
summarize the screening effects of information noise and conservatism as
follows:

###### Proposition 5

Given the firm’s information quality (noise) $q$ and the level of conservatism
$\lambda$, the optimal incentive compatible screening contract has the
following properties:

  1. 1.

Wage $w=w^{*}$, where $w^{*}$ satisfies (19).

  2. 2.

The space of $\theta$ is divided into the same three regions as described in
Proposition 1, where the boundaries of the conditional investment zones are:

$\displaystyle\theta_{1}(q,\lambda)$
$\displaystyle=\frac{(q-\lambda)(k+c)}{(1-q-\lambda)(1-k-c)+(q-\lambda)},$
(30) $\displaystyle\theta_{0}(q,\lambda)$
$\displaystyle=\frac{(1-q+\lambda)(k+c)}{q+\lambda+(1-2q)(k+c)}.$ (31)

  3. 3.

The manager’s equilibrium rent is given by

$\displaystyle R(\theta)=\begin{cases}0,\text{ if
}\theta\in(0,\theta_{1}(q,\lambda)]\\\
\theta\left[(1-q-\lambda)w^{*}-(1-2q)c\right]-(q-\lambda)c,\text{ if
}\theta\in(\theta_{1}(q,\lambda),\theta_{0}(q,\lambda)]\\\ \theta
w^{*}-c,\text{ if }\theta\in(\theta_{0}(q,\lambda),1)\end{cases}$

The managerial pool consists of all the managers with types higher than the
cutoff level $\theta_{1}(q,\lambda)$, i.e.,
$\Theta=\\{\theta:\theta>\theta_{1}(q,\lambda)\\}$. The cutoff is increasing
in the noisiness of the firm’s signal while decreasing the firm’s
conservatism, i.e.,

$\displaystyle\frac{\partial\theta_{1}(q,\lambda)}{\partial q}>0,\quad\text{
and }\quad\frac{\partial\theta_{1}(q,\lambda)}{\partial\lambda}<0$ (32)

for all $0\leq\lambda<q<1/2$.

Figure 5 illustrates the different screening effects of information noise and
conservationism. The effect of the former is depicted in the left panel (which
is qualitatively analogous to the left panel of Figure 3). An increase in the
information noisiness means the signal is less valuable. The firm thus relies
less on the signal for the investment decision, which leads to a smaller
conditional investment zone and lower managerial rent in that zone, generating
a positive screening effect.

Figure 5: This figure illustrates the different screening effects of
information noise $q$ and accounting conservatism $\lambda$. The left panel
shows the effect of an increase in $q$ (i.e., $q_{2}>q_{1}$) while holding
$\lambda$ constant. The right panel shows the effect of an increase in
$\lambda$ (i.e., $\lambda_{2}>\lambda_{1}$) while holding $q$ constant. In
both panels, the blue solid line corresponds to a smaller $q$ or $\lambda$
while the red dashed line corresponds to a larger $q$ or $\lambda$.
$\theta_{1}$ corresponds to the minimal type in the managerial pool as given
in (30).

In contrast, the effect of conservatism, depicted in the right panel of Figure
5, suggests the opposite screening effect. On the one hand, more conservative
accounting implies that the bad signal is noisier, which increases the
conditional probability of success when the firm receives such a signal. As a
result, the firm is more willing to continue investment for lower type
manager, which means that $\theta_{0}$, the right boundary of the conditional
investment zone, decreases as $\lambda$ increases. However, when the firm is
more conservative, the good signal becomes more informative of actual success.
As a result, the firm is also more willing to continue investment conditional
on the good signal even for a low-type manager. This expands the left boundary
of the conditional investment region as $\theta_{1}$ is lower. This effect is
particularly strong for the low-type managers who are far away from the right-
boundary of the conditional investment region (where an increase in type leads
to investment even under the bad signal). Altogether, the conditional
investment region shifts more to the left when the firm is more conservative.
Since the screening effect is determined by the left boundary of the
conditional investment zone, more conservatism leads to a decline in
$\theta_{1}$.

In short, while both information noisiness and accounting conservatism affect
the degree to which the firm relies on the signal for the investment decision,
the two have contrasting impact on the screening of managers. Noisier
information leads to positive screening (i.e., an improvement of the minimal
type in the managerial pool) while more conservative accounting leads to the
opposite screening result.

##  5 Conclusion

Internal information is vital for the success of modern businesses.
Conventional wisdom suggests that firms should design their internal
information systems to be as precise as possible in order to make the most
efficient operational decisions. In this paper, we highlight a novel mechanism
through which a firm, when facing an adverse selection problem arising from
unobservable managerial ability, may benefit from designing a noisy
information system in order to facilitate the screening of managers. The
tradeoff between more effective screening and more informed investment results
in the ex-ante firm value to be non-monotonic in information quality, and a
marginal improvement of information quality does not necessarily lead to an
overall improvement of firm value. Accounting conservatism has the opposite
screening effect as information noise, because more conservative accounting
reduces the expected quality of managers.

Our model can be extended in several directions. For simplicity, we consider a
single representative firm facing a pool of managerial candidates. Introducing
additional firms and allowing competition among them could produce new,
insightful predictions. We also focus on adverse selection as the primary
agency friction and consider a static model in which information is generated
and used only once. Exploring the role of internal information in disciplining
undesirable managerial actions could be fruitful, especially in a dynamic
environment following the literature of dynamic moral hazard and monitoring
(e.g., Piskorski and Westerfield, 2016; Varas, Marinovic, and Skrzypacz, 2020;
Marinovic and Szydlowski, 2021; etc.). We leave topics like these for future
research.

Appendix

##  Appendix A Extensions

###  A.1 Payment for Failure

This section expands the baseline model to allow the firm to also make
payments to the manager when the project fails. Let $b$ denote such payment,
and let $w$ denote the extra payment if the project succeeds, i.e., the total
compensation for success is $b+w$. Then, we can re-write the firm’s investment
conditions (9) and (10) as

$\displaystyle p_{1}(\theta^{f}_{1})(1-w)=\hat{k},$ (33) $\displaystyle
p_{0}(\theta^{f}_{1})(1-w)=\hat{k},$ (34)

where $\hat{k}=k+b$. Similarly, (15) and (16) can be re-written as

$\displaystyle\theta_{1}^{m}(1-q)(w-\hat{c})-(1-\theta_{1}^{m})q\hat{c}$
$\displaystyle=0,$ (35)
$\displaystyle\theta_{0}^{m}(1-q)(w-\hat{c})-(1-\theta_{0}^{m})q\hat{c}$
$\displaystyle=\theta_{0}^{m}w-\hat{c},$ (36)

where $\hat{c}=c-b$. It is straightforward to see that these conditions are
identical to (9), (10), (15) and (16) with a mere change of variables:
$\hat{k}$ and $\hat{c}$ instead of $k$ and $c$. Thus, the exact same analysis
used in Section 3 applies. The optimal extra wage for success is
$w=\hat{c}/(\hat{k}+\hat{c})$. The contract is incentive compatible: all
managerial candidates will report their types truthfully and are divided into
the same three investment zones with the following boundaries:

$\displaystyle\theta_{1}(q)$
$\displaystyle=\frac{q(\hat{k}+\hat{c})}{(1-q)-(1-2q)(\hat{k}+\hat{c})}=\frac{q(k+c)}{(1-q)-(1-2q)(k+c)},$
(37) $\displaystyle\theta_{0}(q)$
$\displaystyle=\frac{(1-q)(\hat{k}+\hat{c})}{q+(1-2q)(\hat{k}+\hat{c})}=\frac{(1-q)(k+c)}{q+(1-2q)(k+c)}.$
(38)

These are identical to the boundaries derived in the baseline model (20 and
21) and are not functions of $b$. In other words, the screening effect of
information quality, which is reflected by how $\theta_{1}$ varies with $q$
(i.e., Proposition 2), is not affected by the firm’s ability to make payments
for failure.

###  A.2 Positive Outside Option Value

This section explores a variation of the baseline model in which each manager
has an outside option worth $R_{\min}>0$. We first presents the case in which
$R_{\min}$ is a constant. Then, we demonstrate the robustness of the model
predictions when $R_{\min}$ is an increasing function of managerial type
$\theta$. We also assume that when indifferent, managers prefer to be
employed. This is without the loss of generality, as the results are virtually
identical if we assume indifference implies unemployment instead.

Consider first the case in which $R_{\min}$ is a constant. For ease of
exposition, we make the following technical assumption:

###### Assumption 1

The firm’s choice of information quality $q$ is bounded by $\bar{q}$ (i.e.,
$q\leq\bar{q}$), where $\bar{q}<1/2$. The manager’s outside option $R_{\min}$
is a constant satisfying $R_{\min}<c\Delta$ where

$\displaystyle\Delta$
$\displaystyle\equiv\frac{(1-2\bar{q})(1-k-c)}{(k+c)(1-2\bar{q})+\bar{q}}>0$
(39)

In other words, we assume that the firm’s signal is not completely
uninformative,131313However, its maximal noisiness $\bar{q}$ can be
arbitrarily close to the absolute maximum (1/2). and the manager’s outside
option value is not too high. As it will become clear later, this ensures that
the minimal type of the managerial pool is within the conditional investment
zone, where the screening effect of information quality manifests.

Since the analysis in Section 3.1 does not depend on the value of outside
options, all results in Proposition 1 still apply. That is, the optimal wage
is a constant given by (19). The contract is incentive compatible: managers
will report their types truthfully, and are divided into three investment
zones with boundaries given by (20) and (21). However, because $R_{\min}>0$,
the minimal type in the managerial pool, denoted by $\theta_{\min}$, no longer
coincides with the left boundary of the conditional investment zone
($\theta_{1}(q)$). Instead, $\theta_{\min}$ corresponds to the type of the
manager whose rent from employment equals his outside option, or
$R(\theta_{\min})=R_{\min}$. The following proposition summarizes the
properties of $\theta_{\min}$:

###### Proposition 6

Suppose each manager’s outside option is worth a constant value $R_{\min}$.
Given the firm’s information quality $q$, the contract derived in Proposition
1 is optimal and incentive compatible. The firm’s managerial pool is
$\Theta=\\{\theta:\theta\geq\theta_{\min}(q)\\}$, where

$\displaystyle\theta_{\min}(q)=\frac{(R_{\min}+qc)(k+c)}{\left((1-q)(1-k-c)+q(k+c)\right)c}$
(40)

If Assumptions 1 is satisfied, then $\theta_{\min}<\theta_{0}$, and
$\theta_{\min}^{\prime}(q)>0$ for all $q\leq\bar{q}$. That is, $\theta_{\min}$
is increasing in the noisiness of the firm’s signal.

The intuition behind this result is the same as that behind Proposition 2 in
the baseline model and can be illustrated using the left panel of Figure 6.
When $R_{\min}$ is a constant and not too high, the managerial pool consists
of all the managers in the unconditional investment zone and some managers in
the conditional investment zone. An increase in the noisiness of the signal
narrows the conditional investment zone and lowers the marginal managerial
rent in that zone. When information is more noisy, only managers with
sufficiently high types can expect to receive rents commensurate with their
outside options. Thus, a noisier signal achieves a better screening result by
excluding more low-type candidates from the managerial pool.141414If
Assumption 1 is not met, then $\theta_{\min}^{\prime}(q)>0$ when $q$ is small,
and $\theta_{\min}^{\prime}(q)=0$ otherwise. In other words, the screening
effect of noisier information still exists, just not for all levels of
information quality.

Figure 6: The left panel plots $\theta_{\min}(q)$, the minimal type in the
firm’s managerial pool, given by (40), under different levels of information
quality $q_{1}$ and $q_{2}$ (where $q_{1}<q_{2}$) when $R_{\min}$, the
manager’s outside option (black dotted line), is a positive constant. The
right panel plots plot $\theta_{\min}(q)$ under different levels of
information quality $q_{1}$ and $q_{2}$ (where $q_{1}<q_{2}$) when $R_{\min}$
is a linearly increasing function of $\theta$.

Our results are also qualitatively robust if $R_{\min}$ is an increasing
function of $\theta$. That is, if the value of the manager’s outside option
increases with manager’s type. Because the analysis and the intuition of this
case are very similar to that when $R_{\min}$ is constant, we omit the
analysis in the interest of space and present the intuition graphically, using
the right panel of Figure 6. In this example, we assume that outside option
value increases linearly in type, i.e., $R_{\min}=\rho\theta$ for some
$\rho>0$. As the graph shows, $\theta_{\min}(q)$ is still increasing in $q$.
That is, noisier information still achieves better screening.

Finally, because the distribution of managerial type is continuous, the
results of both extensions illustrated in Figure 6 are virtually identical
whether the managers prefer employment or not when they are indifferent (i.e.,
when their rent from employment equals the value of their outside options).

In short, the assumptions in the baseline model—that all managers have outside
options worth zero and they favor unemployment when indifferent—are made for
ease of exposition only. They do not materially affect the main result that
noisier information can facilitate the screening for better managers.

##  Appendix B Proofs

###  B.1 Proof of Propositions 1 and 2

The proof of Proposition 1 is divided into two parts. First, we solve the
equilibrium wage and the boundaries of the conditional investment zones. Then,
we show that a constant wage is indeed optimal.

Part I: Define

$\displaystyle\mu_{0}(\theta)$ $\displaystyle=\frac{(1-\theta)(1-q)}{\theta
q}$ (41)

as the likelihood ratio between failure and success given a bad signal, and

$\displaystyle\mu_{1}(\theta)$ $\displaystyle=\frac{(1-\theta)q}{\theta(1-q)}$
(42)

the likelihood ratio between failure and success given a good signal. Then,

$\displaystyle p_{0}(\theta)$ $\displaystyle=\frac{\theta q}{\theta
q+(1-\theta)(1-q)}=\frac{1}{1+\mu_{0}(\theta)},$ (43) $\displaystyle
p_{1}(\theta)$
$\displaystyle=\frac{\theta(1-q)}{\theta(1-q)+(1-\theta)q}=\frac{1}{1+\mu_{1}(\theta)}.$
(44)

Using these notations, (9) and (10) imply

$\displaystyle w$
$\displaystyle=1-(1+\mu_{1}(\theta^{f}_{1}))k=1-(1+\mu_{0}(\theta^{f}_{0}))k.$
(45)

Combining them with (15), and (16), and imposing the incentive compatibility
condition $\theta^{f}_{s}=\theta^{m}_{s}$ ($s=0,1$) yield

$\displaystyle w$ $\displaystyle=w^{*}\equiv\frac{c}{k+c},$ (46)

which, substituted into (45), implies

$\displaystyle 1+\mu_{1}(\theta_{1})=1+\mu_{0}(\theta_{0})=\frac{1}{k+c},$
(47)

which yields (20) and (21). It is easy to verify that $\theta_{1}(q)>0$,
$\theta_{0}(q)<1$, and $\theta_{1}(q)<\theta_{0}(q)$ always when $0<q<1/2$.
i.e., the three investment zones exist for all $q\in(0,1/2)$. When $q=0$,
$\theta_{1}(0)=0$ and $\theta_{0}(0)=1$, i.e., the entire space of $\theta$ is
the conditional investment zone. When $q=1/2$,
$\theta_{1}(1/2)=\theta_{0}(1/2)=k+c<1$, i.e., the conditional investment zone
disappears. $\square$

Part II: $w^{*}$ must be a constant under the optimal screening contract.
First, wage must be independent of type within each investment zone. If not,
then a manager can report his type to be the one that earns him the highest
wage while not altering his investment zone. Thus there can be at most two
wages: $w(\hat{\theta})=w_{c}$ if $\hat{\theta}<\theta_{0}$, and
$w(\hat{\theta})=w_{u}$ if $\hat{\theta}\geq\theta_{0}$, where $\theta_{0}$ is
the boundary between the conditional investment zone and the unconditional
investment zone satisfying the firm’s ex-post optimal investment condition:
$p_{0}(\theta_{0})(1-w_{u})=k$. Suppose $w_{c}<w_{u}$. Then, because
$p_{0}(\theta)$ decreases smoothly in $\theta$, there must exist
$\dot{\theta}<\theta_{0}$ such that $p_{0}(\theta)(1-w_{c})>k$ for all
$\theta\in(\dot{\theta},\theta_{0})$, contradictory to the fact that
$\theta_{0}$ is the boundary of the conditional investment zone. Therefore it
must be that $w_{c}\geq w_{u}$. However, given $w_{u}$, let $R^{i}(\theta)$,
$\theta^{i}_{0}$ and $\theta^{i}_{1}$ ($i=1,2$) be the managerial rent and the
boundaries of the conditional investment zones for $w^{i}_{c}$, with
$w^{1}_{c}<w^{2}_{c}$. Then $\theta^{1}_{1}>\theta^{2}_{1}$ (i.e., a better
screening result), and $R^{1}(\theta)<R^{2}(\theta)$ for all
$\theta\in(\theta^{1}_{1},\theta^{2}_{0})$ (i.e., lower rents for managers in
the conditional investment zone). Thus $w^{1}_{c}$ is strictly preferred to
$w^{2}_{c}$ by the firm. Therefore, it must be that $w_{c}=w_{u}=w$ for all
managers under the optimal screening contract.

The proof of Proposition 2 is straightforward from differentiating
$\theta_{1}(q)$ with respect to $q$:

$\displaystyle\theta_{1}^{\prime}(q)$
$\displaystyle=\frac{k+c}{(1-q)-(1-2q)(k+c)}+\frac{q(k+c)(1-2(k+c))}{[(1-q)-(1-2q)(k+c)]^{2}}$
(48) $\displaystyle=\frac{(k+c)(1-k-c)}{[(1-q)-(1-2q)(k+c)]^{2}}>0,$ (49)

because $1>k+c>0$. $\square$

###  B.2 Proof of Propositions 3 and 4

The proof is organized into three parts.

Part I: Substituting (20) into $v_{c}(\theta)$ implies that
$v_{c}(\theta_{1})=0$. Then, $v_{c}(\theta)>0$ for all
$\theta\in(\theta_{1},\theta_{0})$ because

$v^{\prime}_{c}(\theta)=(1-q)(1-w^{*}-k)+qk>0.$

Substituting (21) into $v_{u}(\theta)$ implies that
$v_{u}(\theta_{0})=v_{c}(\theta_{0})>0$. Then,
$v^{\prime}_{u}(\theta)=1-w^{*}>0$ implies that $v_{u}(\theta)>0$ for all
$\theta\in(\theta_{0},1)$. Therefore, $V(q)>0$ for all $q$.

Part II: Differentiating (20) and setting $q=1/2$ implies that

$\theta_{1}^{\prime}\left(\frac{1}{2}\right)=4(k+c)(1-k-c)>0.$

Therefore,

$V^{\prime}\left(\frac{1}{2}\right)=V\left(\frac{1}{2}\right)\left(\frac{f(\theta_{1})}{1-F(\theta_{1})}\right)\theta_{1}^{\prime}\left(\frac{1}{2}\right)>0.$

Since $V^{\prime}(q)$ is continuous, there must exist $\bar{q}<1/2$ such that
$V^{\prime}(q)>0$ for all $q>\bar{q}$.

Let $\mu={\rm E}(\theta)$ be the unconditional expectation of $\theta$, then,

$\displaystyle V(0)$
$\displaystyle=\int_{0}^{1}\theta(1-w^{*}-k)dF(\theta)=\mu
k\left(\frac{1}{k+c}-1\right),$ (50) $\displaystyle V^{\prime}(0)$
$\displaystyle=V(0)f(0)\theta_{1}^{\prime}(0)-\int_{0}^{1}\left[\theta(1-w^{*}-k)+(1-\theta)k\right]dF(\theta)$
(51)
$\displaystyle=V(0)\left[f(0)\left(\frac{k+c}{1-k-c}\right)-1\right]-\int_{0}^{1}(1-\theta)kdF(\theta)$
(52) $\displaystyle=\mu
k\left[\left(\frac{1}{k+c}-1\right)\left[f(0)\left(\frac{k+c}{1-k-c}\right)-1\right]+1\right]-k.$
(53)

If $\theta$ follows a uniform distribution, then $\mu=1/2$, $f(0)=1$, and

$\displaystyle V^{\prime}(0)$
$\displaystyle=\frac{k}{2}\left(3-\frac{1}{k+c}\right)-k<0,$ (54)

because $3-1/(k+c)<2$. Since $V^{\prime}(q)$ is continuous, there must exists
$\underline{q}>0$ such that $V^{\prime}(q)<0$ for all $q<\underline{q}$.

Part III: The results that $\partial\theta_{1}/\partial k>0$ and
$\partial\theta_{1}/\partial c>0$ are obvious. If $\theta$ follows a uniform
distribution, then

$\displaystyle V(0)$
$\displaystyle=\int_{0}^{1}\theta(1-w^{*}-k)d\theta=\frac{k}{2}\left(\frac{1-k-c}{k+c}\right)=\frac{1}{1-k-c}\int^{1}_{k+c}(\theta
w^{*}-k)d\theta=V\left(\frac{1}{2}\right).$

Clearly, $\partial V(0)/\partial c<0$ and $\partial V(1/2)/\partial c<0$.
Define

$h(k)=\ln\left[k\left(\frac{1-k-c}{k+c}\right)\right]=\ln
k+\ln(1-k-c)-\ln(k+c).$

Then

$\displaystyle h^{\prime}(k)$
$\displaystyle=\frac{1}{k}-\frac{1}{1-k-c}-\frac{1}{k+c},$ (55)

with $\lim_{k\rightarrow 0}h^{\prime}(k)>0$ and $\lim_{k\rightarrow
1-c}h^{\prime}(k)<0$. Since

$\displaystyle h^{\prime\prime}(k)$
$\displaystyle=-\frac{1}{k^{2}}-\frac{1}{(1-k-c)^{2}}+\frac{1}{(k+c)^{2}}<0,$
(56)

there exists $k^{*}$ such that $h^{\prime}(k)>0$ for all $0<k<k^{*}$ and
$h^{\prime}(k)<0$ for all $k^{*}<k<1-c$. Consequently, $\partial V(0)/\partial
k>0$ and $\partial V(1/2)/\partial k>0$ for all $0<k<k^{*}$, while $\partial
V(0)/\partial k<0$ and $\partial V(1/2)/\partial k<0$ for all $k^{*}<k<1-c$.
$\square$

###  B.3 Proof of Proposition 5

Following the proof of Proposition 1, define

$\displaystyle\mu_{0}(\theta)$
$\displaystyle=\frac{(1-\theta)(1-q+\lambda)}{\theta(q+\lambda)}$ (57)

as the likelihood ratio between failure and success given a bad signal, and

$\displaystyle\mu_{1}(\theta)$
$\displaystyle=\frac{(1-\theta)(q-\lambda)}{\theta(1-q-\lambda)}$ (58)

the likelihood ratio between failure and success given a good signal. Then,

$\displaystyle p_{0}(\theta)$
$\displaystyle\equiv\Pr(\pi=1|s=0)=\frac{1}{1+\mu_{0}(\theta)},$ (59)
$\displaystyle p_{1}(\theta)$
$\displaystyle\equiv\Pr(\pi=1|s=1)=\frac{1}{1+\mu_{1}(\theta)}.$ (60)

Then, the firm’s ex-post optimal investment decision implies that

$\displaystyle w$
$\displaystyle=1-(1+\mu_{1}(\theta^{f}_{1}))k=1-(1+\mu_{0}(\theta^{f}_{0}))k,$
(61)

where $\theta^{f}_{1},\theta^{f}_{0}$ are defined analogously to (9) and (10),
respectively. Similarly, the managerial rent function satisfies

$\displaystyle\theta^{m}_{1}(1-q-\lambda)(w-c)+(1-\theta^{m}_{1})(q-\lambda)$
$\displaystyle=0,$ (62)
$\displaystyle\theta^{m}_{0}(q+\lambda)(w-c)+(1-\theta^{m}_{0})(1-q+\lambda)$
$\displaystyle=\theta^{m}_{0}w-c,$ (63)

where $\theta^{m}_{1},\theta^{m}_{0}$ are defined analogously to (15) and
(16), respectively. Combining (61) with (62) and (63) and imposing the
incentive compatibility condition $\theta^{f}_{s}=\theta^{m}_{s}=\theta_{s}$
($s=\\{0,1\\}$) imply that $w=w^{*}\equiv k/(k+c)$, and $\theta_{1}$ and
$\theta_{0}$ given by (30) and (31), respectively.

Finally, differentiating $\theta_{1}(q,\lambda)$ with respect to $q$ and
$\lambda$ yields:

$\displaystyle\frac{\partial\theta_{1}(q,\lambda)}{\partial q}$
$\displaystyle=\frac{k+c}{(1-q-\lambda)(1-k-c)+(q-\lambda)}-\frac{(q-\lambda)(k+c)^{2}}{[(1-q-\lambda)(1-k-c)+(q-\lambda)]^{2}}$
$\displaystyle=\frac{(k+c)[(1-2q)(1-k-c)+(q-\lambda)]}{[(1-q-\lambda)(1-k-c)+(q-\lambda)]^{2}}>0.$
(64) $\displaystyle\frac{\partial\theta_{1}(q,\lambda)}{\partial\lambda}$
$\displaystyle=-\frac{k+c}{(1-q-\lambda)(1-k-c)+(q-\lambda)}+\frac{(q-\lambda)(k+c)(2-k-c)}{[(1-q-\lambda)(1-k-c)+(q-\lambda)]^{2}}$
$\displaystyle=-\frac{(k+c)(1-2q)(1-k-c)}{[(1-q-\lambda)(1-k-c)+(q-\lambda)]^{2}}<0.$
(65)

because $1-2q>0$ and $q>\lambda$. $\square$

###  B.4 Proof of Proposition 6

The proof of Proposition 1 does not involve the value of $R_{\min}$ and thus
still applies. That is, the contract characterized in Proposition 1 is still
optimal and incentive compatible. Given that $R(\theta)$ is increasing in
$\theta$, $\Theta=\\{\theta:\theta\geq\theta_{\min}(q)\\}$, where
$R(\theta_{\min}(q))=R_{\min}$ yields (40). If Assumption 1 holds, then

$\displaystyle\theta_{\min}(q)<\frac{(\Delta+q)(k+c)}{(1-q)(1-k-c)+q(k+c)}<\theta_{0},$
(66)

for all $q<\bar{q}$. Differentiating $\theta_{\min}(q)$ with respect to $q$
yields

$\displaystyle\theta_{\min}^{\prime}(q)>\theta_{\min}(q)[1-k-c+\Delta(1-2(k+c))]>0.$
(67)

## References

  * Adams and Ferreira (2007) Adams, R. B. and D. Ferreira (2007).  A theory of friendly boards.  Journal of Finance 62(1), 217–250.
  * Arya et al. (1997) Arya, A., J. C. Glover, and K. Sivaramakrishnan (1997).  The interaction between decision and control problems and the value of information.  The Accounting Review, 561–574.
  * Bagnoli and Watts (2005) Bagnoli, M. and S. G. Watts (2005).  Conservative accounting choices.  Management Science 51(5), 786–801.
  * Baldenius et al. (2014) Baldenius, T., N. Melumad, and X. Meng (2014).  Board composition and CEO power.  Journal of Financial Economics 112(1), 53–68.
  * Baldenius and Meng (2010) Baldenius, T. and X. Meng (2010).  Signaling firm value to active investors.  Review of Accounting Studies 15(3), 584–619.
  * Banerjee and Szydlowski (2021) Banerjee, S. and M. Szydlowski (2021).  Friends don’t lie: Monitoring and communication with risky investments.  Working paper.
  * Bertomeu and Cheynel (2015) Bertomeu, J. and E. Cheynel (2015).  Asset measurement in imperfect credit markets.  Journal of Accounting Research 53(5), 965–984.
  * Bertomeu et al. (2021) Bertomeu, J., E. Cheynel, and D. Cianciaruso (2021).  Strategic withholding and imprecision in asset measurement.  Journal of Accounting Research 59(5), 1523–1571.
  * Burkart et al. (1997) Burkart, M., D. Gromb, and F. Panunzi (1997).  Large shareholders, monitoring, and the value of the firm.  Quarterly Journal of Economics 112(3), 693–728.
  * Burns et al. (2010) Burns, N., S. Kedia, and M. Lipson (2010).  Institutional ownership and monitoring: Evidence from financial misreporting.  Journal of Corporate Finance 16(4), 443–455.
  * Caskey and Laux (2017) Caskey, J. and V. Laux (2017).  Corporate governance, accounting conservatism, and manipulation.  Management Science 63(2), 424–437.
  * Dewatripont and Tirole (1994) Dewatripont, M. and J. Tirole (1994).  A theory of debt and equity: Diversity of securities and manager-shareholder congruence.  Quarterly Journal of Economics 109(4), 1027–1054.
  * Dordzhieva et al. (2022) Dordzhieva, A., V. Laux, and R. Zheng (2022).  Signaling private information via accounting system design.  Journal of Accounting and Economics 74(1), 101494.
  * Fan (2007) Fan, Q. (2007).  Earnings management and ownership retention for initial public offering firms: Theory and evidence.  The Accounting Review 82(1), 27–64.
  * Friedman et al. (2020) Friedman, H. L., J. S. Hughes, and B. Michaeli (2020).  Optimal reporting when additional information might arrive.  Journal of Accounting and Economics 69(2-3), 101276.
  * Friedman et al. (2022) Friedman, H. L., J. S. Hughes, and B. Michaeli (2022).  A rationale for imperfect reporting standards.  Management Science 68(3), 2028–2046.
  * Gao (2010) Gao, P. (2010).  Disclosure quality, cost of capital, and investor welfare.  The Accounting Review 85(1), 1–29.
  * Gigler et al. (2009) Gigler, F., C. Kanodia, H. Sapra, and R. Venugopalan (2009).  Accounting conservatism and the efficiency of debt contracts.  Journal of Accounting Research 47(3), 767–797.
  * Goldman and Slezak (2006) Goldman, E. and S. L. Slezak (2006).  An equilibrium model of incentive contracts in the presence of information manipulation.  Journal of Financial Economics 80(3), 603–626.
  * Goldstein and Yang (2019) Goldstein, I. and L. Yang (2019).  Good disclosure, bad disclosure.  Journal of Financial Economics 131(1), 118–138.
  * Harris and Raviv (2008) Harris, M. and A. Raviv (2008).  A theory of board control and size.  Review of Financial Studies 21(4), 1797–1832.
  * Hermalin (2005) Hermalin, B. E. (2005).  Trends in corporate governance.  Journal of Finance 60(5), 2351–2384.
  * Jiang and Yang (2017) Jiang, X. and M. Yang (2017).  Properties of optimal accounting rules in a signaling game.  Journal of Accounting and Economics 63(2-3), 499–512.
  * Kamenica and Gentzkow (2011) Kamenica, E. and M. Gentzkow (2011).  Bayesian persuasion.  American Economic Review 101(6), 2590–2615.
  * Kanodia and Sapra (2016) Kanodia, C. and H. Sapra (2016).  A real effects perspective to accounting measurement and disclosure: Implications and insights for future research.  Journal of Accounting Research 54(2), 623–676.
  * Kanodia et al. (2005) Kanodia, C., R. Singh, and A. E. Spero (2005).  Imprecision in accounting measurement: Can it be value enhancing?  Journal of Accounting Research 43(3), 487–519.
  * Laux and Laux (2021) Laux, C. and V. Laux (2021).  Accounting conservatism and managerial information acquisition.  Available at SSRN 3642806.
  * Laux and Ray (2020) Laux, V. and K. Ray (2020).  Effects of accounting conservatism on investment efficiency and innovation.  Journal of Accounting and Economics 70(1), 101319.
  * Marinovic and Szydlowski (2021) Marinovic, I. and M. Szydlowski (2021).  Monitoring with career concerns.  RAND Journal of Economics.  Forthcoming.
  * Mehran (1992) Mehran, H. (1992).  Executive incentive plans, corporate control, and capital structure.  Journal of Financial and Quantitative Analysis, 539–560.
  * Michaeli (2017) Michaeli, B. (2017).  Divide and inform: Rationing information to facilitate persuasion.  The Accounting Review 92(5), 167–199.
  * Morris and Shin (2002) Morris, S. and H. S. Shin (2002).  Social value of public information.  American Economic Review 92(5), 1521–1534.
  * Nan and Wen (2014) Nan, L. and X. Wen (2014).  Financing and investment efficiency, information quality, and accounting biases.  Management Science 60(9), 2308–2323.
  * Piskorski and Westerfield (2016) Piskorski, T. and M. M. Westerfield (2016).  Optimal dynamic contracts with moral hazard and costly monitoring.  Journal of Economic Theory 166, 242–281.
  * Raheja (2005) Raheja, C. G. (2005).  Determinants of board size and composition: A theory of corporate boards.  Journal of Financial and Quantitative Analysis, 283–306.
  * Stocken and Verrecchia (2004) Stocken, P. C. and R. E. Verrecchia (2004).  Financial reporting system choice and disclosure management.  The Accounting Review 79(4), 1181–1203.
  * Varas et al. (2020) Varas, F., I. Marinovic, and A. Skrzypacz (2020).  Random inspections and periodic reviews: Optimal dynamic monitoring.  Review of Economic Studies 87(6), 2893–2937.
  * Zhu (2022) Zhu, J. Y. (2022).  Better monitoring…worse productivity?  Working paper. University of Kansas.