Copyright

by

Garrett Paul Schaller

2020

The Dissertation Committee for Garrett Paul Schallercertifies that this is the approved version of the following dissertation:

Credibility Cycles

Committee:

James R. Lowery, Supervisor

Jonathan Cohn

John Hatfield

Nathaniel A. Pancost

Vasiliki Skreta

Sheridan Titman

Credibility Cycles

by

Garrett Paul Schaller

DISSERTATION

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN

May 2020

Dedicated to my parents.

Acknowledgments

I am grateful to the members of my dissertation committee: Jonathan

Cohn, John Hatfield, Richard Lowery (chair), Aaron Pancost, Vasiliki Skreta,

and Sheridan Titman. I also thank Aydogan Alti, Andres Donangelo, William

Fuchs, Daniel Neuhann, Michael Sockin, Mindy Xiaolan, and seminar par-

ticipants at the University of Texas at Austin for their useful comments and

suggestions.

v

Credibility Cycles

Garrett Paul Schaller, Ph.D.

The University of Texas at Austin, 2020

Supervisor: James R. Lowery

I model the strategic interactions between the manager of a firm and

an outside investor in a dynamic cheap talk game with two-sided asymmetric

information. Each period, the investor selectively discloses his information to

influence the manager’s capital investment decision. While the manager knows

that she can learn from the investor’s disclosures, she also knows that the in-

vestor is trying to manipulate her; in equilibrium, the investor’s incentives

to mislead the manager constrain the credibility of his disclosures, leading to

a mutually-deleterious loss of information. I compare the set of cheap talk

equilibria with the Bayesian persuasion equilibrium. My model has implica-

tions for short-termism, management guidance, and investor credibility over

the business cycle.

vi

Table of Contents

Acknowledgments v

Abstract vi

List of Figures ix

Chapter 1. Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Related literature . . . . . . . . . . . . . . . . . . . . . . . . . 6

Chapter 2. Two-period model 10

2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 “Cyclicality” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Commitment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.6 Mandatory disclosures . . . . . . . . . . . . . . . . . . . . . . 26

2.7 Short-termism? . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Chapter 3. Infinite-horizon model 29

3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Cyclicality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Commitment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5 Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.6 Mandatory disclosure . . . . . . . . . . . . . . . . . . . . . . . 40

3.7 Short-termism . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.8 Additional implications . . . . . . . . . . . . . . . . . . . . . . 43

vii

Chapter 4. Conclusion 45

Appendix 47

References 76

viii

List of Figures

2.1 Summary of interactions in the two-period game . . . . . . . . 14

2.2 Equilibrium in the two-period game . . . . . . . . . . . . . . . 17

2.3 Summary of interactions in the two-period guidance game . . 25

3.1 Summary of interactions in the infinite-horizon game . . . . . 33

3.2 Equilibrium over the business cycle . . . . . . . . . . . . . . . 36

3.3 Summary of interactions in the infinite-horizon guidance game 40

3.4 Investment loss . . . . . . . . . . . . . . . . . . . . . . . . . . 44

ix

Chapter 1

Introduction

“To see what is in front of one’s nose needs a constant struggle.”

— George Orwell (1946)

1.1 Overview

Suppose you are the CEO of Apple. Carl Icahn, who owns 1% of Apple,

tells you that you should be paying more dividends. You know that Icahn is

a savvy businessman, and you want to learn from him, but you do not trust

him. Is he giving you good advice, or is he simply trying to cash out?

In this dissertation, I show that you can learn from Carl Icahn, but he

will not tell you everything. In fact, he could not tell you everything even if

he wanted to; as long as he has an incentive to manipulate you, his recom-

mendations will not be fully credible. And despite your skepticism, his advice

will induce you to pay more dividends. More broadly, I show that a minority

investor can influence a CEO solely through strategic communications that are

costless, non-binding, and unverifiable. In equilibrium, these communications

pressure firms into sacrificing long-term growth to boost short-term earnings.1

1There is a longstanding empirical literature on the relationship between financial mar-

1

Formally, I develop a dynamic cheap talk model with two-sided asym-

metric information. The players, a manager and an investor, each have private

information about the marginal product of capital. Each period, the investor

strategically discloses his information to influence the manager’s capital in-

vestment decision, while the manager has an incentive to disclose her own

information to incite additional revelations from the investor. When the in-

vestor is less patient than the manager, his equilibrium disclosures lead the

manager to underinvest relative to the full-information benchmark.

Underinvestment is not a success of investor manipulation, but a failure

of investor credibility: if instead the investor were more patient than the man-

ager, his equilibrium disclosures would still lead the manager to underinvest.

Moreover, the more patient the investor is, the more the manager will under-

invest. Indeed, the investor cannot help but induce short-termism, because

his bias damages the credibility of his communications and impairs informa-

tion transmission. What matters is not the investor’s patience, but the fact

that the manager knows she is being manipulated; she fails to capitalize on

investment opportunities because she cannot take the investor’s advice at face

value. Thus, short-termism is not a result of investor impatience per se, but a

consequence of misaligned preferences.

kets and managerial short-termism. See, for example, Poterba and Summers (1995); Gra-ham, Harvey, and Rajgopal (2005); Lerner, Sorensen, and Stromberg (2011); Bernstein(2015); Kini, Shenoy, and Subramaniam (2016); Agarwal, Vashishtha, and Venkatachalam(2017); Edmans, Fang, and Lewellen (2017); Edmans, Goncalves-Pinto, Groen-Xu, andWang (2018); Feldman et al. (2018); and Thomas (2019).

2

If the investor could commit to a signaling strategy, he would opti-

mally disclose either all or none of his information, depending on his level

of patience. More concretely, in a Bayesian persuasion environment, the in-

vestor would commit to revealing no information if he were substantially less

patient than the manager. Otherwise, he would commit to fully revealing his

information, in which case the manager would neither overinvest nor underin-

vest. Thus, commitment can usually solve the underinvestment problem; while

short-termism is borne of preference misalignment, it is only weaponized by

the investor’s credibility constraints.

I show that while the investor’s credibility is limited, the manager’s

credibility is nonexistent. This is in stark contrast to recent warnings against

management guidance, including a highly-publicized report in the Wall Street

Journal (Dimon & Buffett, 2018; Moyer, 2018) arguing that management guid-

ance stokes a culture of short-termism and underinvestment. On the contrary,

I find that voluntary management disclosures cannot be used to reveal the

manager’s private information or to induce different disclosures from the in-

vestor.

Mandated disclosures, by contrast, can improve communication: if the

manager is forced to truthfully reveal her private information, then good news

will loosen the investor’s credibility constraints and he will disclose more in-

formation. Conversely, when the manager reveals bad news, the investor’s

credibility constraints tighten; for sufficiently bad news, the investor’s disclo-

sures may be coarser than they would have been had the manager kept her

3

information private. Overall, this amplification mechanism may be welfare-

improving because it enhances communication precisely when the manager

plans on making substantial capital investments. Public policies on manda-

tory disclosures, which circumvent the manager’s credibility constraints, can

therefore give the manager a powerful commitment device.

The investor’s credibility depends not only on differences in preferences

but also on the state of the business cycle. I show that during recessions, the

divergence between the manager’s and investor’s utilities is compressed, mak-

ing the manager less skeptical of the investor’s disclosures. At the same time,

the manager’s optimal capital investment is less sensitive to the investor’s

information, making it more difficult for the investor to influence the man-

ager. The utility-compression effect will enhance investor credibility, while the

information-sensitivity effect will degrade it. Because recessions are transitory,

the information-sensitivity effect generally dominates the utility-compression

effect, implying that the investor’s credibility is lower during recessions.

My model therefore microfounds countercyclical uncertainty, which is

the focus of a recent literature in macroeconomics (Bachmann & Bayer, 2014;

Bloom, 2009, 2014; Bloom, Floetotto, Jaimovich, Saporta-Eksten, & Terry,

2018). More concretely, uncertainty rises when disclosures fail to be credible,

hence procyclical credibility generates countercyclical uncertainty. Procyclical

credibility is also consistent with the empirical fact that investment rates are

less dispersed during downturns (Bachmann & Bayer, 2014). In the context

of this model, dispersion in the firm’s investment rate will mirror dispersion

4

in the manager’s beliefs; as the investor loses credibility, the manager loses a

source of payoff-relevant information.

I derive the first complete characterization of cheap talk equilibria for a

natural extension to Crawford and Sobel’s (1982) parameterized setting, which

has been the standard for cheap talk models since its inception. In order to

fit Crawford and Sobel’s example, models in this literature employ restrictive

assumptions to ensure that the difference between agent’s preferred actions

is constant; my extension allows for interactions between state variables and

preferences, so that the difference between agents’ preferred actions can be

subject to asymmetric information. This theoretical innovation permits me

to study the equilibrium effects of discount rates and business cycle dynamics

in cheap talk games; in many financial environments, the relevance of these

features cannot be overstated.

My model provides a general framework for dynamic cheap talk games,

which is an under-explored theoretical literature; the dearth of models in this

area is likely due, at least in part, to the limitations inherent in Crawford and

Sobel’s (1982) example. In terms of applied theory, cheap talk is often used in

models of corporate investment, macroeconomic forecasting, stock purchases,

and bank transparency; while dynamic models are abundant in these litera-

tures, dynamic cheap talk models are virtually nonexistent. My characteriza-

tion of the model leads to tractable dynamics, which deliver novel theoretical

insights. Indeed, in my setting the dynamics reveal countercyclical credibil-

ity and pervasive underinvestment; in other settings, this framework could be

5

used to analyze the value and cyclicality of the Federal Reserve’s credibility,

the market distortions caused by stock recommendations and charlatanism, or

the limits of financial transparency.

The rest of this dissertation is organized as follows. The remainder

of this section reviews the related literature on cheap talk, investment, and

managerial disclosures. Chapter 2 derives a two-period version of the model,

which provides some intuition and offers a foundation for later proofs. Chap-

ter 3 presents the infinite-horizon model and discusses equilibrium results.

Chapter 4 concludes.

1.2 Related literature

Crawford and Sobel (1982) wrote the seminal paper on cheap talk.

They showed conditions under which a biased expert could strategically com-

municate information to an unbiased decision-maker using only costless, non-

binding, and unverifiable signals. In contrast to standard signaling environ-

ments, cheap talk messages have no intrinsic meaning, which makes the ex-

istence of a non-trivial equilibrium all the more surprising. The parametric

example which Crawford and Sobel analyze has become the benchmark model

in the cheap talk literature. In that setting, the difference between the expert’s

and decision-maker’s preferred action is constant. My framework relaxes that

assumption, and therefore allows for interactions among asymmetric informa-

tion, discount rates, and the business cycle; more broadly, this relaxation is

crucial for understanding the effects of cheap talk in standard economic en-

6

vironments, where most models have only been able to determine whether or

not cheap talk is fully-revealing.

The preferences I study in my two-period model are similar to those

in Melumad and Shibano (1991) and Gordon (2010, 2011). Melumad and

Shibano (1991) compare equilibria with no communication to equilibria where

disclosure partitions the state space into two intervals. Gordon (2010) shows

that infinitely-fine disclosure may be feasible in settings where the expert is

more sensitive to the state. Gordon (2011) offers an equilibrium refinement for

this class of preferences, and proposes an algorithm which numerically solves

for the maximum number of feasible intervals under additional assumptions.

Harris and Raviv (2006, 2010), as well as Chakraborty and Yılmaz

(2017), consider the optimal delegation of authority between management and

the board of directors in the presence of two-sided asymmetric information.

My paper also relates to a literature on dynamic cheap talk games.

Golosov, Skreta, Tsyvinski, and Wilson (2014) model a privately-informed

sender’s incentives to progressively reveal his private information over time.

Grenadier, Malenko, and Malenko (2016) consider an option-exercise game in

which the expert and decision-maker have differing levels of patience. In my

setting, the information asymmetry is two-sided, and the cheap talk concerns

information which arrives in each period.

Bachmann and Bayer (2014), Bloom (2009), and Bloom et al. (2018)

estimate the macroeconomic effects of countercyclical uncertainty in an econ-

7

omy with both aggregate and firm-specific shocks; Bloom (2009) also includes

intra-firm (unit-level) shocks. Time-varying uncertainty proves to be quite

devastating in these models; for example, Bloom et al. (2018) estimate that

uncertainty shocks can lead to a 2.5% decline in GDP. I microfound counter-

cyclical uncertainty by showing that investor credibility is tied to the state of

the business cycle. In particular, procyclical credibility induces countercyclical

uncertainty.

In other settings, procyclical learning is used to microfound counter-

cyclical uncertainty. Fajgelbaum, Schaal, and Taschereau-Dumouchel (2017),

Van Nieuwerburgh and Veldkamp (2006), and Veldkamp (2005) obtain this

result because agents learn from each other’s investment activities. In my

setting, learning is a tool which the investor uses to influence the manager,

hence the investor’s communications will always strategically distort his own

information.

There has also been a recent interest in using structural estimation tech-

niques to measure the extent and consequences of short-termism. Bertomeu,

Marinovic, Terry, and Varas (2017) estimate a persuasion game in which man-

agers, who have utility over the sequence of stock prices, disclose their earn-

ings forecasts to the market. Terry (2017) estimates a macroeconomic model

in which managers manipulate the firm’s earnings in order to meet market

expectations. I show that short-termism endogenously arises when managers

rely on the information generated by investors.

As noted in Goldstein and Yang’s (2017) review, there is a great deal of

8

theoretical and empirical research on the information content of stock prices,

and the extent to which managers can use this information to make efficient

capital investment decisions. Yet, it is clear that managers can only learn from

stock prices if investors have information. By modeling communication, I am

considering a distinct channel through which managers can learn from agents

outside the firm.

Moreover, while I refer to the outside agent as an investor, my model

can be applied to any setting in which the manager of a firm is learning from an

agent with distinct preferences. For example, Ali, Amiram, Kalay, and Sadka

(2018) and Hutton, Lee, and Shu (2012) empirically show that sell-side ana-

lyst forecasts are, in many cases, more accurate than management forecasts.

T. Chen, Xie, and Zhang (2017) and Choi, Hann, Subasi, and Zheng (2020)

show that accurate analyst forecasts improve managerial investment efficiency.

Finally, consistent with my model’s implications, Derrien and Kecskes (2013)

find that firms invest less when they lose analyst coverage.

9

Chapter 2

Two-period model

I start by introducing a simple, two-period version of my model. This il-

lustrates the basic intuition for the investor’s credibility constraints, the power

of investor commitment, and the value of mandatory disclosures. However, this

simple model does not sufficiently endogenize the trade-offs that govern the

investor’s credibility cycles, and fails to deliver underinvestment. Comparing

the two-period model with the infinite-horizon model is particularly helpful as

it shows where the dynamics are most useful.

2.1 Setup

Consider a capital-investment problem in which two agents, a manager

and an investor, have equity stakes in the same firm. They both have pri-

vate information about the firm’s marginal product of capital. The investor

has an opportunity to communicate his information by costlessly sending any

non-binding and unverifiable signal to the manager, after which the manager

chooses the firm’s capital investment.1 Their preferences are given by

1For simplicity, I assume that the communication is one-way, such that only the investoris allowed to cheap talk; later on, I amend the model to give the manager a chance to speak.

10

Ej [uj] ≡ ζjEj [Π0 + βjΠ1]

where j ∈ i,m indexes the investor or manager, Πt is the firm’s net profits

in period t, ζj denotes agent j’s share of equity, and βj reflects the patience of

agent j.

At time t = 0, the firm’s net profits are given by

Π0 ≡ ZF (K0)− I − C(I,K0),

where Z denotes aggregate productivity, F (·) is a homothetic production func-

tion,2 K0 denotes the firm’s extant capital stock, I is the manager’s capital

investment decision, and C(·) represents capital adjustment costs. These ad-

justment costs take the form:

C(I,K0) ≡ φ

2

(I

K0

− δ)2

K0,

where I assume φ ≥ 1 to ensure that K1 ≥ 0.

At time t = 1, the firm’s capital investment pays off:

Π1 ≡ AF (K1) +G(Z)K1,

where A determines the productivity of the investment and G(·) > 0 is the

per-unit liquidation value of the firm’s capital stock K1. Capital accumulates

2Any production function with constant returns to scale would yield the same results.As I note in the infinite-horizon setting, this includes a Cobb-Douglas production functionwith both labor and capital.

11

according to

K1 ≡ (1− δ)K0 + I. (2.1)

The firm’s future productivity A has three components: aggregate productivity

Z, firm-specific productivity Y , and industry-specific productivity X.3 In logs:

log(A) ≡ ρ log(Z) + log(Y ) + log(X),

where4

X ∼ Uniform[0, 1],

Y ∼ Uniform [0,Ω] .

The manager has superior knowledge about the firm-specific component of pro-

ductivity; at time t = 0, she observes the future realization of Y . The investor,

by contrast, has superior information about the industry-specific component of

productivity; at t = 0, he observes X and selectively discloses his information

to influence the manager’s choice of I.

Formally, the investor’s strategy space permits him to send some mes-

sage L ∈ [0, 1] to the manager. The mapping from supp (L) to E [X|L] will be

determined in equilibrium; in fact, the mapping itself will be fairly arbitrary,

since cheap talk messages have no intrinsic meaning. This is a subtler point

3I use the phrases “firm-specific” and “industry-specific” for exposition only: it is moreexpedient to write “industry-specific productivity” than to write “the productivity compo-nent observed by the investor.”

4In the dynamic game, ρ will correspond to the persistence parameter of an AR(1) processgoverning Z.

12

about cheap talk, which is wholly unnecessary to delve into for the purposes

of this dissertation.5

The game can be summarized as follows:

1. Investor learns the industry-specific productivity X. Manager learns the

firm-specific productivity Y .

2. Investor communicates a message L about the industry-specific produc-

tivity X.

3. Manager chooses the firm’s capital investment I subject to K1 ≥ 0.

4. Short-term payoffs Π0 are realized.

5. Long-term payoffs Π1 are realized.

Figure 2.1 illustrates the information set and action space for each player.

2.2 Equilibrium

The manager’s value function is

Vm ≡ maxIζm (Π0 + βmEm [Π1]) ,

5For example, if there exists an equilibrium such that the message L′ induces the beliefE [X|L′] = X ′ and the message L′′ induces the belief E [X|L′′] = X ′′, where L′ 6= L′′ andX ′ 6= X ′′, then there also exists an outcome-equivalent equilibrium such that the messageL′ induces the belief E [X|L′′] = X ′′ and the message L′′ induces the belief E [X|L′′] = X ′.Indeed, cheap talk is language in its purest form, in the sense that there is no intrinsicreason why we call canis familiaris “dog” and canis lupus “wolf” instead of the other wayaround.

13

Aggregateproductivity

Z

Industry-specific

productivityX

Firm-specificproductivity

Y

Investor Manager

Capitalinvestment

I

Cheap talk

Figure 2.1: Summary of interactions in the two-period game

This figure depicts each agent’s information and actions in the two-period game.

where her only source of uncertainty is the true value of X, hence her ex-

pectation is taken over the conditional distribution of X given the investor’s

message L.

Similarly, the investor’s value function is

Vi ≡ maxL

ζiEi [Π0 + βiΠ1] ,

where his lack of information about firm-specific productivity Y implies that

he takes expectations over both Y itself and the investment decision I.

14

A Bayesian Nash equilibrium of this game consists of the investor’s

(mixed) messaging rule λ(L|X) and the manager’s (mixed) investment rule

ι(I|L) such that:

1. For each X ∈ supp (X),∫

Λλ(L|X)dL = 1 where the Borel set Λ is the

set of feasible signals and if L∗ is in the support of λ(L|X) then L∗ solves

maxL∈Λ

∫∞(δ−1)K0

Ei [ui|I] ι(I|L)dI.

2. For each L ∈ Λ,∫∞

(δ−1)K0ι(I|L)dI = 1 and if I∗ is in the support of ι(I|L)

then I∗ solves maxI∈[(δ−1)K0,∞)

∫ 1

0Em [um|L]χ(X|L)dX where χ(X|L) ≡

λ(L|X)fX(X)/∫ 1

0λ(L|W )fX(W )dW .

Communication is informative if the manager’s posterior χ(X|L) is not

constant in equilibrium. Communication is influential if the manager’s action

ι(I|L) is not constant in equilibrium. Trivially, an uninformative and hence

non-influential equilibrium always exists such that the manager ignores all

messages; however, it is possible to sustain influential equilibria as follows.

Theorem 2.2.1. For n = 0, ..., N and N ≤ N , there exists an equilibrium

(λ(L|X), ι(I|L)), where λ(L|X) is uniform and supported on [cn, cn+1] if X ∈

[cn, cn+1] and ι(I∗(cn, cn+1)|L) = 1.

In particular, the investor partitions the support of X according to the

sequence cnNn=0 defined by

cn =

ω + csc (Nθ) ((1− ω) sin (nθ) + ω sin (nθ −Nθ)) if βi < βm

ω −(

1+ω(ψN−1)ψN−ψ−N

)ψ−n +

(1−ω(1−ψ−N)ψN−ψ−N

)ψn if βi > βm,

15

where

ω ≡ − E [Y ]G (Z)

E [Y 2]ZρF (1),

θ ≡ arg

(2βi − βm

βm+ i

2√βi (βm − βi)βm

),

ψ ≡ 2βi − βmβm

+2√βi (βi − βm)

βm.

The manager’s best response is

I∗(cn, cn+1) ≡

(βm[(

cn+cn+1

2

)Y ZρF (1) +G (Z)

]+ δφ− 1

φ

)K0.

The maximum number of feasible intervals N is given by

N ≡

⌊

12

+arccos

(( ωω−1)

√βiβm

)2 arctan

(√βmβi−1)⌋

if βi < βm⌊1

log(ψ)log

(ω(1+ψ)−1−ψ−

√ω2(ψ−1)2+(ψ+1)2−2ω(ψ+1)2

2ω

)⌋if βi > βm.

Following Y. Chen, Kartik, and Sobel’s (2008) equilibrium refinement, I focus

on the equilibrium with N intervals.

In equilibrium, the investor discloses thatX lies in some interval [cn, cn+1],

where ∪N−1n=0 [cn, cn+1] = [0, 1] forms a partition of supp(X). Given this message,

the manager chooses the firm’s capital investment according to I∗(cn, cn+1).

Figure 2.2 plots an example of an equilibrium with N = 3 intervals.

The x-axis shows the possible realizations of industry-specific productivity X,

while the y-axis shows investment I; all other variables are fixed. The dashed

line represents the manager’s optimal choice of I under full information:

Im ≡ argmaxI

E [um|X, Y ] .

16

X

I

Im

Ii

c0 c1 c2 c3

I∗(c0, c1)

I∗(c1, c2)

I∗(c2, c3)

Figure 2.2: Equilibrium in the two-period game

This figure plots the manager’s equilibrium investment I∗(cn, cn+1) as a functionof the true realization of X. Each agent’s preferred level of investment Ij , wherej ∈ i,m, is shown for comparison.

This is what the manager would choose if she knew X. The dotted line repre-

sents the investor’s optimal choice of I under full information:

Ii ≡ argmaxI

E [ui|X, Y ] .

This is what the investor would choose if he knew Y and could choose I. The

solid line plots the equilibrium action I∗(cn, cn+1) as defined in Theorem 2.2.1.

17

In this example, the investor is impatient, which is visually evident from the

fact that Ii lies below Im for every possible realization of X.

The discontinuous appearance of the solid line shows the partitioned

nature of this equilibrium. For example, when X ∈ [c0, c1], the investor’s mes-

sage induces I∗(c0, c1). In contrast to the discontinuous best-response function

I∗(c0, c1), each agent’s optimal choice of I, denoted by Ij, is a continuous func-

tion of X.

Given the coarse nature of a partition, is clear that any equilibrium will

feature a loss of information. Moreover, this loss can be mutually deleterious.

When both the dashed and dotted lines lie above the solid line, both agents

would prefer a higher level of investment; when the dashed and dotted lines

lie below the solid line, both agents would prefer a lower level of investment.

While full revelation would benefit both the investor and manager in these

regions, such an equilibrium would ultimately unravel. After all, if the manager

ever took the investor’s recommendations at face value, then the investor would

always shade his recommendations to induce his privately-preferred action Ii.

For a concrete example of this unraveling, suppose the investor is im-

patient, βi < βm, such that he always has an incentive to claim that the

realization of X is slightly lower than it actually is. The manager would ra-

tionally discount the investor’s claim and believe that the true state is slightly

higher than what the investor reports; consequently, the investor would have

an incentive to report that the realization of X is substantially lower than it

actually is, and so on. This unwinding does not have a fixed point, hence full

18

revelation is infeasible, even for a subset of X ∈ supp (X).

Any informative equilibrium features a partition specifically because

the partition itself limits what the investor can communicate. Even if the

investor is impatient, he does not want the manager to simply dismantle the

company in its entirety; in figure 2.2, Ii lies below Im, but it is not horizontal.

For realizations of X that just exceed c1, the investor would rather the manager

invest slightly too much (I∗(c1, c2)) than far too little (I∗(c0, c1)). More ab-

stractly, these partitions preclude unraveling by endogenously discretizing the

investor’s information set. Ultimately, the investor’s disclosures are credible

because they are coarse.

2.3 “Cyclicality”

The investor’s credibility is tied to the state of the business cycle. In a

two-period model, this relationship can be characterized by performing com-

parative statics on the aggregate productivity parameter Z.

This exercise is not without irony. By definition, a cycle would require

at least three periods such that an object may deviate from, then return to,

its initial position. Because cycles cannot be represented in this two-period

setting, I instead perform comparative statics on the “business cycle” here,

which itself is not cyclical at all. Cycles do, however, appear in the dynamic

model.

Despite these conceptual issues, this exercise can provide some useful

19

intuition. The following proposition details the relationship between investor

disclosures and the state of the business cycle.

Theorem 2.3.1. Let N denote the maximum number of intervals attainable

in equilibrium. Then

∂N

∂Z≥ 0

if and only if

G′(Z)

G(Z)< ρZ−1.

Intuitively, a negative aggregate shock has two competing effects. First,

it compresses the wedge between the agent’s liquidation values:

∂

∂Z|(βm − βi)G(Z)| > 0,

hence a decline in Z implies a decline in |(βm − βi)G(Z)|. All else equal,

this improves communication, as the agent’s preferences are effectively more-

aligned.

Second, a negative aggregate shock decreases the marginal value of the

investor’s private information: following the shock, Im is less sensitive to the

realization of X, which makes it more difficult for the investor to influence the

manager. Indeed, when Z is low, the manager will not invest much regardless

of what the investor says. All else equal, this degrades communication.

Interestingly, the intuition behind this second effect bears an analogy

to the failure of full revelation. As I previously reasoned, and as theorem 2.3.1

20

shows, full revelation cannot be an equilibrium; put differently, there is no

equilibrium in which the investor truthfully reveals the exact realization of

X. Of course, a fully-revealing equilibrium can be thought of as the limit

of a partitioned equilibrium such that there are infinitely-many, infinitely-

small intervals. If instead we have an equilibrium with a fixed number of

intervals N , and then shrink the support of X, the resulting partition will

start to look like full revelation; that is to say, the partition will start to look

like something which cannot be an equilibrium. To maintain an informative

equilibrium, the number of intervals N must therefore decline. When Z is low,

the investor cannot induce disparate actions, as the manager will be investing

very little regardless of what she believes X to be; a negative aggregate shock

is effectively compressing the support of X. In more-conventional terms, an

investor of type X ′ ∈ [cn, cn+1] will have a harder time separating from an

investor of type X ′′ ∈ [cn+1, cn+2] when their revelations lead to similar actions,

i.e., when I∗ (cn, cn+1) is close to I∗ (cn+1, cn+2). When Z declines, the set of

rationalizable I shrinks.

In figure 2.2, a negative aggregate shock is equivalent to simultaneously

lowering the slope and intercept of both Im and Ii. Since both the slope

intercept of Ij scale with βj, the shock will have a larger effect on the slope

and intercept of the more-patient agent.

As Theorem 2.3.1 demonstrates, the direction of cyclicality of the two-

period model depends on the form of the per-unit liquidation value G(·). With-

out specifying additional features of G(·), it is impossible to discern which of

21

the two effects dominates. The dynamic model disciplines this by consider-

ing an infinite-horizon game where G(·) is effectively replaced by the firm’s

continuation value.

2.4 Commitment

As the previous sections demonstrate, the investor’s credibility con-

straints necessarily engender a loss of information. What would happen if

we resolve the investor’s credibility problem while maintaining the underlying

preference misalignment?

To answer this question, I consider the equilibrium of a game in which

the investor can commit to a signaling strategy before learning the realization

of X. In the words of Kamenica and Gentzkow (2011), this is a Bayesian

persuasion game.6

As before, let χ (X|L) denote the manager’s posterior distribution of

X conditional on receiving some message L. In the cheap talk game, I defined

the investor’s strategy as a distribution over messages λ (L|X). While I could

formulate the commitment problem in a similar manner, I instead write the

investor’s problem as a decision over posterior distributions; this yields a more

familiar representation of the Bayesian persuasion problem. Prior to learning

X, the investor commits to a distribution over posteriors ξ(χ) in order to

6The investor does not need to know any more than what he reveals to the manager,hence this is equivalent to a game in which the investor commits to acquiring a particularsignal which he then conveys to the manager.

22

maximize his ex-ante expected utility. He solves:

maxξ

∫supp(ξ)

Eχ [ζi (Π0 + βiΠ1)] dξ(χ)

subject to Bayes plausibility∫supp(ξ)

χdξ(χ) = fX ,

where fX denotes the prior distribution of X. The following theorem produces

the equilibria of this game.

Theorem 2.4.1. If βi/βm < 1/2, then no revelation is optimal for the in-

vestor.

If βi/βm = 1/2, then any revelation is optimal for the investor.

If βi/βm > 1/2, then full revelation is optimal for the investor.

If the investor is extremely impatient, βi/βm < 1/2, then he will opti-

mally commit to disclose no information at all. Therefore, when the investor is

extremely impatient, giving him commitment power can only make the man-

ager worse off by denying her any access to the investor’s private information.

Indeed, for some βi/βm < 1/2, there may exist an informative and influential

equilibrium in the cheap talk game, hence commitment can make the manager

strictly worse off.

Conversely, if the investor is either patient or moderately impatient,

βi/βm > 1/2, then he will commit to fully revealing his private information. In

this case, commitment makes the manager strictly better off. Despite the fact

23

that the credibility constraints were only symptomatic of preference misalign-

ment, we can nevertheless restore the manager’s first-best level of investment

by giving the investor commitment power.

2.5 Guidance

In the baseline model, the investor learns the industry-specific produc-

tivity X and communicates with the manager by sending message L. The

manager does not disclose any information regarding the firm-specific produc-

tivity Y ; the investor can only infer the realization of Y after observing the

manager’s choice of I. In practice, however, we might think that managers

have an incentive to strategically communicate their own private information

as a way to influence investor disclosures.7 Figure 2.3 illustrates the game with

management guidance.

In the context of this model, this type of communication can be broadly

understood as guidance: discretionary communication from managers to share-

holders about managerial expectations of firm performance. Guidance is rel-

atively common, where “about 20 percent of Business Roundtable members

[give] quarterly guidance and about 60 percent [give] annual targets” (Dimon,

2018, as cited in Moyer, 2018). Proponents of this communication contend that

“guidance...improves communications with Wall Street,” among other things

7Note that I am considering managerial disclosures which occur before the investor sendshis own message. If instead the manager were to disclose information after the investor isfinished communicating, then the manager’s message would trivially have no effect on theequilibrium.

24

Aggregateproductivity

Z

Industry-specific

productivityX

Firm-specificproductivity

Y

Investor Manager

Capitalinvestment

I

Cheap talk

Cheap talk

Figure 2.3: Summary of interactions in the two-period guidance game

This figure depicts each agent’s information and actions in the two-period gamewith managerial guidance.

(Moyer, 2018). Critics, by contrast, argue that guidance induces managerial

myopia by unduly pressuring managers to meet their short-term forecasts at

the expense of long-term investment goals.

Counter to both of these arguments, the following theorem shows that

guidance is completely neutral: if managers are allowed to communicate their

information to the market, their disclosures will generally fail to be credible.

Theorem 2.5.1. Managerial disclosures in the “guidance game” are credible

25

if and only if all types of managers are indifferent among all types of messages.

Such disclosures cannot affect equilibrium outcomes.

In other words, barring an equilibrium in which the manager tells the

truth despite having no incentives to do so, guidance fails to be informative.

Indeed, at best, guidance can only convey information if it does not affect the

equilibrium. Intuitively, the manager discloses information for the sole purpose

of inciting additional revelations from the investor. The investor knows that

the manager would say anything to get more information, hence he cannot

believe a word of what she says.

2.6 Mandatory disclosures

As the previous section demonstrates, the manager is generally inca-

pable of reporting Y through discretionary disclosures. How will the structure

of the equilibrium change if the manager is instead forced to reveal her private

information each period?8

Theorem 2.6.1. Let N(Y ) denote the maximum number of intervals attain-

able when Y is common knowledge. Then

N(Y ) ≥ N (2.2)

8Equivalently, we can think of this equilibrium as one in which the manager has crediblycommitted to revealing her private information.

26

if and only if

Y ≥ E [Y 2]

E [Y ]. (2.3)

Furthermore,

∂N(Y )

∂Y≥ 0. (2.4)

As inequality (2.4) shows, the investor’s communications will be coarser

when the manager reveals negative news about the firm’s prospects. Forced

managerial disclosures would therefore embed another amplification mecha-

nism into this problem: good news brings insight, while bad news begets

ignorance. Nevertheless, as indicated by inequalities (2.2) and (2.3), this am-

plification can be welfare-improving as it gives the manager additional infor-

mation precisely when she intends to make a substantial capital investment

decision.

A mandatory-disclosure policy is especially valuable if the investor

would otherwise babble, i.e., if N = 1. Whether this occurs during booms

or recessions depends on the specification of G(·); once again, an attempt

to glean business-cycle implications from this two-period model is hamstrung

by the need to specify G(·). And, once again, the infinite-horizon setup will

resolve this issue.

2.7 Short-termism?

Thus far, I have used the two-period model to illustrate the economic

intuition behind some of the results in this dissertation. Additional intuition

27

comes from a result which the two-period model cannot deliver.

Theorem 2.7.1. Let I∗(cn, cn+1) denote the manager’s best-response function

as defined in Theorem 2.2.1. Let Im denote the manager’s optimal level of

investment in the full-information benchmark. Then

E [I∗(cn, cn+1)] = E[Im

].

Thus, underinvestment does not occur in this two-period environment.

The manager chooses the the firm’s level of capital investment based on her

expectation of the marginal product of capital. Moreover, the manager has

rational expectations, hence the investor cannot move the manager’s average

beliefs about the firm-specific component of productivity. Indeed, the only

myopia here is in the model itself; underinvestment is a dynamic result.

28

Chapter 3

Infinite-horizon model

The infinite-horizon problem yields the main results for this disserta-

tion. Much of the intuition underpinning the investor’s credibility constraints,

the role of investor commitment, and the limitations of managerial disclosures

can be carried over from the two-period model. Crucially, the dynamics in this

section will discipline the economic forces which engender credibility cycles,

and show that the cost of limited credibility is ultimately underinvestment.

3.1 Setup

Time is discrete and indexed by t. There are two agents, a manager an

investor, who hold equity stakes in the same firm. Each period, the manager

chooses a level of capital investment. Both agents have private information re-

garding the firm’s marginal product of capital; the investor selectively discloses

his information in an attempt to influence the manager’s capital investment

decision. His disclosures are costless, non-binding, and unverifiable; in other

29

words, he communicates through cheap talk.1 Their preferences are given by

Ej,t [uj,t] ≡ Ej,t

[∞∑s=t

βs−tj ζjΠs

]where j ∈ i,m indexes the investor or manager, and βj reflects their pa-

tience. Agent j’s equity stake ζj ∈ (0, 1) entitles them to a fraction of the

firm’s net profits Πt.

Each period, a firm with capital stock Kt generates net profits

Πt ≡ AtF (Kt)− It − C (It, Kt, ) ,

where C(·) denotes capital adjustment costs and F (·) is a homothetic produc-

tion function.2 Adjustment costs take the form

C (It, Kt, ) ≡φ

2

(ItKt

− δ)2

.

Capital takes time to build, hence the firm’s investment at time t determines

the next period’s capital stock. Capital accumulates according to

Kt+1 = It + (1− δ)Kt. (3.1)

The technological process At has three components: aggregate productivity Zt,

firm-specific productivity Yt, and industry-specific productivity Xt. In logs:

log (At) ≡ log (Zt) + log (Yt) + log (Xt) ,

1For now, I only let the investor communicate his private information. As I show later,giving the manager a chance to communicate will generally fail to make a difference.

2Any CRTS production function would yield the same results. For example, we mayhave a Cobb-Douglas production function with both labor and capital, in which case we canthink of this F (·) as a the production function evaluated at the optimal choice of labor.

30

where

Xti.i.d.∼ Uniform[0, 1],

Yti.i.d.∼ Uniform[0,Ω],

log (Zt) = ρ log (Zt−1) + σεt,

εti.i.d.∼ N(0, 1).

Aggregate productivity Zt and the firm’s extant capital stock Kt are common

knowledge at the start of time t; both agents are symmetrically uninformed

about the future realization of Zt+1. The asymmetric information in this model

concerns Xt+1 and Yt+1.

The manager has superior knowledge about the firm-specific compo-

nent of productivity: at time t, she privately observes the realization of Yt+1

The investor, by contrast, has superior information about the industry-specific

component of productivity; at time t, he privately observes the realization of

Xt+1 and selectively discloses his information to influence the manager’s cap-

ital investment decision It.

For a given time t, the game can be summarized as follows:

1. Aggregate productivity Zt is publicly observable at the start of time t.

2. Investor learns the industry-specific productivity Xt+1. Manager learns

the firm-specific productivity Yt+1.

3. Investor communicates a message Lt about the industry-specific produc-

tivity Xt+1.

31

4. Manager chooses the firm’s capital investment It.

5. Profits Πt are realized at the end of time t.

Figure 3.1 illustrates the information set and action space for each player at

each period t. As in the two-period model, both players are symmetrically

informed about the current level of aggregate productivity Zt; however, they

are symmetrically uninformed about the future level of aggregate productivity

Zt+1. For each agent’s optimization problem at time t, the state variable Zt

is relevant to the extent that it contains information regarding the future

realization of Zt+1.

3.2 Equilibrium

The manager’s value function is

Vm,t ≡ maxIt

Em,t

[∞∑s=t

βs−tm ζmΠs

]

= maxIt

ζm

(Πt + Et

[∞∑

s=t+1

βs−tm Πs|Lt, Yt+1

]),

where the expectation taken at time t, Et, accounts for the publicly-observable

state variables Kt and Zt. The investor’s value function is

Vi,t ≡ maxLt

Ei,t

[∞∑s=t

βs−ti ζiΠi

]

= maxLt

ζi

∞∑s=t

βs−ti Et [Πs|Xt+1] ,

32

AggregateproductivityE [Zt+1|Zt]

Industry-specific

productivityXt+1

Firm-specificproductivity

Yt+1

Investor Manager

Capitalinvestment

It

Cheap talk

Figure 3.1: Summary of interactions in the infinite-horizon game

This figure depicts each agent’s information and actions at time t.

where the investor’s uncertainty over Yt+1 implies that he must also form

expectations over It.

Following Grenadier et al. (2016), I consider perfect Bayesian equilibra

in Markov strategies (PBEM).

Theorem 3.2.1. For n = 0, ..., Nt and Nt ≤ Nt, there exists an equilibrium

such that the investor’s messaging rule partitions the support of Xt+1 according

33

to the sequence cn,tNtn=0 defined by

cn,t ≡

ωt + csc (Ntθ) ((1− ωt) sin (nθ) + ωt sin (nθ −Ntθ)) if βi < βm

ωt −(

1+ωt(ψNt−1)ψNt−ψ−Nt

)ψ−n +

(1−ωt(1−ψ−Nt)ψNt−ψ−Nt

)ψn if βi > βm,

where

ωt ≡ −(βmEt

[V 0m,t+1

]− βiEt

[V 0i,t+1

])Et [Yt+1]

(βm − βi)Et[Y 2t+1

]Et [Zt+1]F (1)

,

Et[V 0j,t+1

]≡ Et

[ζ−1j

Vj,t+1

Kt+1

|Xt+1 = 0

],

θ ≡ arg

(2βi − βm

βm+ i

2√βi (βm − βi)βm

),

ψ ≡ 2βi − βmβm

+2√βi (βi − βm)

βm.

The manager’s best response is

I∗t (cn,t, cn+1,t) ≡

(βm[( cn,t+cn+1,t

2

)Yt+1Et [Zt+1]F (1) + Et

[V 0m,t+1

]]+ δφ− 1

φ

)Kt.

The maximum number of feasible intervals Nt is given by

Nt ≡

⌊

12

+arccos

((ωtωt−1

)√βiβm

)2 arctan

(√βmβi−1)⌋

if βi < βm⌊1

log(ψ)log

(ωt(1+ψ)−1−ψ−

√ω2t (ψ−1)2+(ψ+1)2−2ωt(ψ+1)2

2ωt

)⌋if βi > βm.

When the investor is impatient, βi < βm, solving for the set of equi-

libria requires (i) deriving imaginary roots, and (ii) finding the conditions

34

under which an oscillating function is monotonic over a bounded interval of

non-negative integers. These are the features which invoke the trigonometric

functions in Theorem 3.2.1.

In equilibrium, the investor’s messages partition the support of Xt+1.

For a given partition cn,tNn=0, if the investor observes Xt+1 ∈ [cn,t, cn+1,t],

he discloses Xt+1 ∼ Uniform [cn,t, cn+1,t]. After receiving this message, the

manager makes a capital investment decision according to her best response

function I∗(cn,t, cn+1,t). As in the two-period model, the investor’s coarse dis-

closures are a consequence of his credibility constraints; full revelation yields

no fixed point, hence the investor’s information set must be endogenously dis-

cretized.

3.3 Cyclicality

As Theorem 3.2.1 demonstrates, the investor’s credibility is tied to the

state of the business cycle. The investor’s credibility will be procyclical under

the following conditions.

Theorem 3.3.1. Let N denote the maximum number of intervals attainable

in equilibrium. Then

∂Nt

∂Zt≥ 0

if and only if

βmEt[∂V 0

m,t+1

∂Zt

]− βiEt

[∂V 0

i,t+1

∂Zt

]βmEt

[V 0m,t+1

]− βiEt

[V 0i,t+1

] < ρZ−1t .

35

Xt+1

It

(a)

Xt+1

It

(b)

Xt+1

It

(c)

Xt+1

It

(d)

Figure 3.2: Equilibrium over the business cycle

This figure depicts changes in the equilibrium over the business cycle. As in fig-ure (2.2), the step function represents I∗t (cn,t, cn+1,t), while the dashed and dottedlines are the manager’s and investor’s preferred level of investment, respectively.The aggregate Zt is decreasing as we move from panel (a) to panel (d).

While the appearance of Theorem 3.3.1 is far more daunting than that

of Theorem 2.3.1, the intuition is unchanged. A negative aggregate shock has

two competing effects. First, it compresses the wedge between the agent’s

residual continuation values, which makes the manager less skeptical of the

36

investor’s disclosures. Second, it decreases the marginal value of the investor’s

private information: following the shock, the manager’s optimal choice of It

is less sensitive to the realization of Xt+1, which makes it more difficult for

the investor to influence the manager. The first effect enhances the investor’s

credibility, while the second effect degrades it.

In contrast to the two-period model, the infinite-horizon game formal-

izes this trade-off using an endogenous wedge in preferences. In particular, the

dynamics reveal the strength of the information effect relative to the continu-

ation effect; figure 3.2 plots a numerical example. A persistent, impermanent

shock will have a substantial effect on the marginal product of capital, but

only a modest effect on the agent’s continuation values.

3.4 Commitment

In equilibrium, the proximate cause of the information loss is the in-

vestor’s lack of credibility. What would happen if the same biased investor

could commit to his disclosures?

More concretely, suppose that at the start of each period t, the investor

has the ability to commit to a signaling strategy before learning the realization

of Xt+1. In particular, I consider a case in which the investor has the ability

to commit to his disclosures within each period, but not across periods.

At the beginning of each period t, before learning Xt+1, the investor

chooses a distribution ξ (χ) over posteriors χ concerning the future realization

37

of Xt+1.3 He therefore solves:

maxξ

∫supp(ξ)

Eχ [ui,t] dξ (χ)

subject to Bayes plausibility∫supp(ξ)

χdξ (χ) = fX .

As in the two-period model, this problem generically yields a set of starkly-

different equilibria.

Theorem 3.4.1. If βi/βm < 1/2, then no revelation is optimal for the in-

vestor.

If βi/βm = 1/2, then any revelation is optimal for the investor.

If βi/βm > 1/2, then full revelation is optimal for the investor.

Once again, for βi/βm < 1/2, giving the investor commitment power

can make the manager strictly worse off. Conversely, for βi/βm > 1/2, this

form of investor commitment will fully ameliorate the information loss.

From an ex-ante perspective, the equilibrium play is ultimately deter-

mined by the manager’s sensitivity to the investor’s disclosures. If the in-

vestor commits to revealing no information, then he is effectively playing with

a manager who is completely insensitive to his disclosures. A patient investor,

3For a slightly more detailed explanation of this notation, see the corresponding Bayesianpersuasion problem in the two-period model.

38

βi > βm, already believes that a fully-informed manager would be too insensi-

tive to revelations about Xt+1; consequently, he would not want to make the

manager even less sensitive by committing to no revelation. The converse is

true of an exceedingly-impatient investor, βi/βm < 1/2, who would rather play

with an informationally-insensitive manager.

3.5 Guidance

In this infinite-horizon setting, managerial disclosures remain, at best,

an extremely fragile construct. Indeed, suppose that at period t the manager

has an opportunity to disclose information regarding Yt+1 prior to the investor

sending a message concerning Xt+1; figure 3.3 illustrates this game. This

produces a familiar theorem.

Theorem 3.5.1. Managerial disclosures in the “guidance game” are credible

if and only if all types of managers are indifferent among all types of messages.

Such disclosures cannot affect equilibrium outcomes.

The manager’s attempts at communication are ultimately limited by

her single-minded preference to extract additional information out of the in-

vestor. If the manager could send a message to induce the investor to disclose

additional information, then she would always have an incentive to do so.

Moreover, if she had access to a set of messages which could equally incite

additional disclosures from the investor, then each message must be outcome-

equivalent to revealing no information at all; indeed, this is perhaps the more

39

AggregateproductivityE [Zt+1|Zt]

Industry-specific

productivityXt+1

Firm-specificproductivity

Yt+1

Investor Manager

Capitalinvestment

It

Cheap talk

Cheap talk

Figure 3.3: Summary of interactions in the infinite-horizon guidance game

This figure depicts each agent’s information and actions at time t in the game withmanagerial guidance.

ruinous charge from Theorem 3.5.1. Taken together, these results demonstrate

the neutrality of management guidance.

3.6 Mandatory disclosure

If the manager were instead forced to disclose her private information

each period, we would obtain the following result.

Theorem 3.6.1. Let Nt (Yt+1) denote the maximum number of intervals at-

40

tainable at time t when Yt+1 is common knowledge. Then

∂Nt (Yt+1)

∂Yt+1

≥ 0.

As in the two-period model, mandatory disclosures will generate yet

another amplification effect, such that managers bearing good news are re-

warded with additional information, while managers possessing bad news are

left in the dark.

In essence, this is a microcosm of Theorem 3.3.1, whereby the man-

ager’s private information determines how susceptible she is to the investor’s

influence. A manager bearing bad news is prone to pare down her level capi-

tal investment, hence the set of actions which the investor can induce will be

relatively circumscribed.

3.7 Short-termism

Now that we have moved beyond the two-period setting, we can obtain

the following result.

Theorem 3.7.1. Let I∗t (cn,t, cn+1,t) denote the manager’s best-response func-

tion as defined in Theorem 3.2.1. Let Im denote the manager’s optimal level

of investment in the full-information benchmark. Then

E [I∗t (cn,t, cn+1,t)] ≤ E[Im,t

].

In other words, the manager will underinvest.

41

In the two-period model, Theorem 2.7.1 showed that the investor could

not alter the manager’s average level of capital investment. To an extent, that

result still holds in a dynamic setting; in any given period t, the investor cannot

skew the manager’s beliefs about the average level of productivity. What

makes the dynamic result different is the fact that the manager anticipates

that she will continue to play this cheap talk game in future periods. Indeed,

if at time t the investor could commit to fully revealing his information in all

future periods, the manager would no longer underinvest.

To better understand the intuition behind this result, note that the

manager’s value function at time t+ 1 is given by:

Vm,t+1 ≡ maxIt+1

Πt + βmEm,t+1 [Vm,t+2]

≡ maxIt+1

At+1F (Kt+1)︸ ︷︷ ︸Revenue from It

−It+1 − C(It+1, Kt+1) + βmEm,t+1 [Vm,t+2]︸ ︷︷ ︸Continuation value

.

At time t, the manager has rational expectations regarding the future real-

ization of At+1: regardless of the investor’s revelation scheme, the manager’s

beliefs about the overall productivity At+1 will be unbiased. Consequently,

the investor’s disclosures at time t will not influence the manager’s expected

revenue Em,t [At+1F (Kt+1)] from investing It. However, the investment It is

valuable not only for the revenue it generates at t + 1 but also for the con-

tinuation value it offers in periods t+ 1 and beyond. This continuation value

is impinged by the investor’s inability at time t to commit to full revelation

in subsequent periods; it is this impingement that drives down the manager’s

average level of capital investment E [It].

42

Short-termism is therefore an issue of inter-period commitment in the

infinite-horizon model. This result also recontextualizes the insights of Theo-

rem 2.7.1. In particular, for all but the most impatient investors, this inter-

period commitment problem can be resolved with an intra-period commitment

device.

3.8 Additional implications

Figure 3.4 shows the symmetry of the underinvestment problem.4 If the

investor is less patient than the manager, then the manager underinvests; the

less patient the investor is, the more the manager underinvests. If instead the

investor is more patient than the manager, then the manager still underinvests;

however, the more patient the investor is, the more the manager underinvests.

This implies that empirical tests linking investor horizon to underinvestment

must crucially account for the alignment between managerial and investor

preferences, not investor horizons per se. Indeed, while there has long been

a concern that financial markets encourage managerial myopia, the empirical

evidence supporting these fears has been mixed, with some studies studies

substantiating this fear and other studies finding no evidence at all (Lerner et

al., 2011). This model can help to reconcile these mixed results.

The consequences of this information loss will also be related to firm-

4While this function has largely been smoothed, some elements of its piecewise construc-tion remain apparent. Each corner is approximately a point at which Nt|Zt=mode(Z) changesdiscretely; however, because each piece a high-dimensional function of βi/βm, not all of thesepoints are visually conspicuous.

43

βi/βm

Investment gap

0

1

Figure 3.4: Investment loss

This figure shows the average level of investment relative to the full-informationbenchmark. As the agent’s preferences diverge, information is lost, hence the un-derinvestment problem becomes more severe.

level characteristics. As theorem Theorem 2.7.1 shows, the manager’s capital

investment choice will be proportional to the size of the firm. Accordingly,

large firms will suffer the most substantial loss of investment due to this loss

of information.

Finally, firms which rely more heavily on this information will be more

impacted by variation in the information loss. If a firm hinges on learning

from outside agents, then business cycle downturns, which dampen business

opportunities and destroy information transmission, will be particularly dev-

astating. Young firms may therefore be the greatest victims of the credibility

cycle.

44

Chapter 4

Conclusion

When an investor advises a manager on her capital investment deci-

sions, differences in their preferences can impair communication. In equilib-

rium, the investor’s credibility is generally procyclical: during economic down-

turns, his ability to disclose information becomes increasingly constrained.

Procyclical credibility microfounds widely-observed empirical patterns of coun-

tercyclical uncertainty and procyclical investment dispersion (Bachmann &

Bayer, 2014; Bloom, 2009, 2014; Bloom et al., 2018).

Investor impatience leads to underinvestment, despite the fact that

managers retain their preference for long-term investments. Yet, the notion

that impatience is inexorably linked to underinvestment may be right for the

wrong reasons; indeed, even if the investor were more patient than the man-

ager, the firm would still underinvest. Investor commitment can restore the

manager to her first-best level of investment, provided the investor is not

too impatient. Thus, short-termism is borne of preference misalignment and

weaponized by the investor’s credibility constraints.

Voluntary disclosures from the manager to the investor fail to be mean-

ingful. However, if the manager is instead forced to reveal her private informa-

45

tion, she may actually induce the investor to disclose more of his own informa-

tion in equilibrium. Unfortunately, this would also tie the investor’s credibility

constraints to realizations of the manager’s private information. In particular,

the manager may become less informed when she is forced to reveal bad news

regarding the firm’s future prospects. These amplification effects compound

to make the firm’s lowest points even worse; downturns become catastrophes,

and when it rains, it pours.

46

Appendix

47

Proof of Theorem 2.2.1

Manager’s problem

Given the capital accumulation equation (2.1), the manager’s problem can be

cast in terms of a choice over K1 ∈ R+. Observe that

∂2Em [um]

∂K21

= − φ

K0

< 0,

hence the manager will never play mixed strategies in equilibrium. Let K∗1 (L)

denote the manager’s best response to some message L.1 Let L∗ (X) denote

an equilibrium message sent by the investor in state X.

Suppose K ′1, K ′′1 are actions induced in an influential equilibrium such

that K ′1 < K ′′1 . Since ui is continuous in X, there exists X ∈ supp (X) such

that

Ei[ui|K1 = K ′1, X = X

]= Ei

[ui|K1 = K ′′1 , X = X

].

Furthermore, since

∂2Ei [ui]∂K2

1

= − φ

K0

< 0

and

∂2Ei [ui]∂K1∂X

= Ei [Y ]ZρF (1) > 0,

1Implicitly, the manager’s best response K∗1 (L) is also a function of parameters and themanager’s private information Y .

48

we know that

K ′1 < Ki

(X)< K ′′1 ,

Ei [ui|K1 = K ′1, X = X ′] > Ei [ui|K1 = K ′′1 , X = X ′] for all X ′ < X

Ei [K∗1 (L∗ (X))] < Ki

(X)

for all X ′ < X,

Ei [ui|K1 = K ′1, X = X ′′] < Ei [ui|K1 = K ′′1 , X = X ′′] for all X ′′ > X,

Ei [K∗1 (L∗ (X))] > Ki

(X)

for all X ′′ > X,

where

Ki (X) ≡ argmaxK1

Ei [ui] .

In other words, if the investor in state X is indifferent between K ′1, K ′′1, then

he will strictly prefer K ′′1 over K ′1 for all X > X. Consequently, any action

K∗1 (L∗ (X)) will only be induced by an investor who observes X ∈ [cn, cn+1] ⊆

supp (X) such that ∪N−1n=0 [cn, cn+1] = supp (X).

Suppose a message L∗ is supported on X ∈ [cn, cn+1]. The manager’s

posterior is

χ (X|L) ≡ λ (L|X) fX (X)∫ cn+1

cnλ (L|W ) fX (W ) dW

=fX (X)∫ cn+1

cnfX (W ) dW

,

hence her best response is

K∗1 (cn, cn+1) ≡ K∗1 (L∗) =

(βm[(

cn+cn+1

2

)Y ZρF (1) +G (Z)

]+ φ− 1

φ

)K0.

(1)

49

Investor’s problem

At each cutoff cn+1, the investor is indifferent between sending a message in

[cn, cn+1] and sending a message in [cn, cn+1], hence

Ei [ui|K1 = K∗1 (cn, cn+1) , X = cn+1] = Ei [ui|K1 = K∗1 (cn+1, cn+2) , X = cn+1] .

In order to derive the monotonic sequence cnNn=0, I rewrite this equality as

a second-order difference equation:

βmcn+2 + 2 (βm − βi) cn+1 + βmcn + 4 (βm − βi)Ei [Y ]G (Z)

E [Y 2]ZρF (1)= 0.

The Z-transform of this difference equation is given by

∞∑n=0

z−n(βmcn+2 + 2 (βm − βi) cn+1 + βmcn + 4 (βm − βi)

Ei [Y ]G (Z)

E [Y 2]ZρF (1)

)= 0

⇒ βm(C[n]− z−1c1 − c0

)+ 2 (βm − βi) z (C[n]− c0)

+ βmC[n] + 4 (βm − βi)Ei [Y ]G (Z)

E [Y 2]ZρF (1)= 0

⇒ C[n] =z3c0 + z2

(c1 + βm−4βi

βmc0

)− z

(c1 + 2(βm−2βi)

βmc0 + 4

(βm−βiβm

)Ei[Y ]G(Z)

E[Y 2]ZρF (1)

)(z − 1)

(z −

(2βi−βmβm

− 2√βi(βi−βm)

βm

))(z −

(2βi−βmβm

+2√βi(βi−βm)

βm

))(2)

If βi < βm, the denominator of equation (2) contains two complex numbers. I

therefore split the remainder of this proof into cases.

Case 1. βi < βm

50

It is convenient to rewrite the complex portion of equation (2) using Euler’s

formula:

2βi − βmβm

±2√βi (βi − βm)

βm

=2βi − βm

βm± i

2√βi (βm − βi)βm

=

√√√√(2βi − βmβm

)2

+

(2√βi (βm − βi)βm

)2

e±i arg

(2βi−βmβm

+i2√

βi(βm−βi)βm

)

≡ e±iθ,

where θ ≡ arg

(2βi−βmβm

+ i2√βi(βm−βi)βm

)is, without loss of generality, the prin-

cipal value of the arg function. Equation (2) can therefore be rewritten as

C[n] =z3c0 + z2

(c1 + βm−4βi

βmc0

)− z

(c1 + 2(βm−2βi)

βmc0 + 4

(βm−βiβm

)Ei[Y ]G(Z)

E[Y 2]ZρF (1)

)(z − 1) (z − e−iθ) (z − eiθ)

.

Cauchy’s residue theorem applies to both real and complex poles, hence I use

it to calculate the inverse Z-transform such that each zr denotes a pole of the

51

above function:

cn = Z−1C[n]

=3∑r=1

Res(C[n]zn−1, zr

)= Res

(C[n]zn−1, 1

)+ Res

(C[n]zn−1, e−iθ

)+ Res

(C[n]zn−1, eiθ

)=c0 +

(c1 + βm−4βi

βmc0

)−(c1 + 2(βm−2βi)

βmc0 + 4

(βm−βiβm

)Ei[Y ]G(Z)

E[Y 2]ZρF (1)

)(1− e−iθ) (1− eiθ)

+e−(n+2)iθc0 + e−(n+1)iθ

(c1 + βm−4βi

βmc0

)(e−iθ − 1) (e−iθ − eiθ)

−e−niθ

(c1 + 2(βm−2βi)

βmc0 + 4

(βm−βiβm

)Ei[Y ]G(Z)

E[Y 2]ZρF (1)

)(e−iθ − 1) (e−iθ − eiθ)

+e(n+2)iθc0 + e(n+1)iθ

(c1 + βm−4βi

βmc0

)(eiθ − 1) (eiθ − e−iθ)

−eniθ

(c1 + 2(βm−2βi)

βmc0 + 4

(βm−βiβm

)Ei[Y ]G(Z)

E[Y 2]ZρF (1)

)(eiθ − 1) (eiθ − e−iθ)

≡ − E [Y ]G (Z)

E [Y 2]ZρF (1)+ (R1 + iM1) e−inθ + (R2 + iM2) einθ. (3)

Since the coefficients take the form R + iM , the cutoff cn can easily be ex-

pressed using boundary conditions other than c0, c1. In general, it will be

useful to set the boundaries at the endpoints of the state space: c0, cN =

Xmin, Xmax.

I will now use the analytic expression for cn in equation (3) to derive

the number of feasible intervals, i.e., the set of N such that there exists an

equilibrium with N intervals. Consequently, I will need to prove the sequence

52

cn is monotonically increasing in n for n ∈ 0, 1, ..., N. Define

sn ≡ cn +E [Y ]G (Z)

E [Y 2]ZρF (1)

= (R1 + iM1) e−inθ + (R2 + iM2) einθ

= csc (Nθ) (sN sin (nθ)− s0 sin (nθ −Nθ)) , (4)

where the last line uses trigonometric substitutions alongside the boundary

conditions

s0 =E [Y ]G (Z)

E [Y 2]ZρF (1),

sN = 1 +E [Y ]G (Z)

E [Y 2]ZρF (1). (5)

Observe that sn is a linear combination of two sine functions, each with period

2πθ

. By the harmonic addition theorem, sn is a sinusoid with period 2πθ

and

positive amplitude. Furthermore βm > 0 implies θ ∈ (0, π) and 2πθ∈ (2,∞).

For an arbitrary sine function with amplitude a, period 2πb

, and phase

shift − bc, define

Qq,z ≡[− c

b+ z

2π

b+ (q − 1)

π

2b,−c

b+ z

2π

b+ q

π

2b

), (6)

where z ∈ Z. If a > 0, then a sin (bn+ c) is positive an increasing for n ∈ Q1,z,

positive and decreasing for n ∈ Q2,z, negative and decreasing for n ∈ Q3,z,

and negative and increasing for n ∈ Q4,z.2 Thus, when a > 0, Qq,z denotes the

2These statements concern the way in which sn responds to marginal changes in n.However, the goal here is to prove sn is monotonic for n ∈ 0, 1, ..., N; since sn is anoscillating function, these marginal properties have limited use.

53

quadrant q of the sine function. This expression will be useful for investigating

the range in which sn is monotonic.

Let η denote the phase shift of sn. Suppose there exists n′ ∈ Q3,z, i.e.,

an n′ such that sn is negative and decreasing when evaluated at n = n′. By

the definition of Qq,z, it follows that we have

n′ ≥ η + z2π

θ+ 2

π

2θ

⇔ n− 1 ≥ η + (z − 1)2π

θ+ 4

π

2θ+π

θ− 1

> η + (z − 1)2π

θ+ 4

π

2θ

⇒ sn′−1 > sn′ . (7)

Therefore, if sn′ is a valid cutoff such that n′ ∈ Q3,z, then sn′−1 cannot be a

valid cutoff because sn′−1 > sn′ violates monotonicity.

Similarly, if n′′ ∈ Q2,z+1, then

n′′ < η + (z + 1)2π

θ+ 2

π

2θ

⇔ n′′ + 1 < η + (z + 2)2π

θ+ 1− π

θ

< η + (z + 2)2π

θ

⇒ sn′′+1 < sn′′ . (8)

Therefore, if sn′′ is a valid cutoff such that n′′ ∈ Q2,z+1, then sn′′+1 cannot be

a valid cutoff because sn′′+1 < sn′′ violates monotonicity.

Given inequalities (7) and (8), it is sufficient to consider n ∈ 0, 1, ..., N

54

which satisfy

n ∈ Q3,z ∪ Q4,z ∪ Q1,z+1 ∪ Q2,z+1.

In fact, − E[Y ]G(Z)E[Y 2]ZρF (1)

< 0 implies sn > 0, hence we need only consider

n ∈ Q1,z+1 ∪ Q2,z+1. (9)

Note that Q1,z+1∪Q2,z+1 has length πθ, hence the number of sustainable intervals

N is bounded above by πθ; however, this is not a sharp bound, as sn must still

be monotonic for n ∈ 0, 1, ..., N under the boundary conditions (5).

Indeed, this bound can be slightly tightened by recalling that inequal-

ity (8) implies that at most one cutoff can lie in Q2,z+1. Moreover, if N satisfies

sN−1 ∈ Q1,z+1 and sN ∈ Q2,z+1, then sN−1 < sN if and only if the length of the

interval defined by n : sn ≤ sN and n ∈ Q2,z+1 is less than 12. Intuitively,

this is because the peak of the sinusoid sn lies between sN−1 ∈ Q1,z+1 and

sN ∈ Q2,z+1, hence sN−1 < sN will only hold if sN−1 is further from the peak

than sN . Thus, the number of intervals N must satisfy

N ≤⌊

1

2+

π

2θ

⌋, (10)

where the floor function b·c reflects the fact that N must be an integer.

For n and N which satisfy conditions (9) and (10), respectively, the

sequence snNn=0 will be monotonic if and only if s0 < s1 and sN−1 < sN . In

equilibrium, N must therefore satisfy

s1 − s0 = sN csc (Nθ) sin (θ) + s0 (csc (Nθ) sin (θ)− 1) > 0,

sN − sN−1 = −s0 csc (Nθ) sin (θ)− sN (csc (Nθ) sin (θ)− 1) > 0.

55

Since s0, sN , sin (θ) > 0, algebraic manipulation reveals that these conditions

are equivalent to

rN ∈(s0

sN,sNs0

)if sin (Nθ) > 0, (11)

rN ∈(sNs0

,s0

sN

)if sin (Nθ) < 0, (12)

where

rN ≡ (sin (Nθ)− sin (Nθ − θ)) csc (θ) .

Observe that 0 < s0 < sN implies s0sN∈ (0, 1) and sN

s0∈ (1,∞), hence

(sNs0, s0sN

)is not a valid interval. Consequently, N must also satisfy sin (Nθ) > 0. I do

not impose this restriction, as it will be satisfied so long as inequality (10)

holds. Indeed, θ ∈ (0, π) implies[1,

1

2+

π

2θ

)⊂[0,

2π

θ+π

θ

),

where the right-hand corresponds to the function Q1,0, as defined in equa-

tion (6), applied to the sinusoid sin (Nθ). In other words, sin (Nθ) is positive

and increasing for the set of N which satisfy inequality (10).

In order to find the conditions under which rN satisfies the first part of

condition (11), it is useful to observe that

r1 = 1 ∈(s0

sN,sNs0

),

56

hence N = 1 constitutes a valid equilibrium.3 Moreover,[∂rN∂N

]N=1

=

[−θ sec

(θ

2

)cos

(Nθ − θ

2

)]N=1

= −θ tan

(θ

2

)< 0,

hence rN is positive and decreasing in N at N = 1. Furthermore,

rN = 0⇔ N =1

2± π

2θ+ z

2π

θ,

where z ∈ Z. Notably, θ ∈ (0, π) implies that N = 12

+ π2θ

is the smallest

N which satisfies rN = 0, which is at least as large as the upper bound from

condition (10). In summary, rN is a positive and decreasing function of N for

the set of N which satisfy conditions (9), (10), and (11).

This has two important implications. First, since r1 = 1 < sNs0

, the only

additional restriction imposed by condition (11) is

rN >s0

sN, (13)

because rN < r1 <sNs0

and sin (Nθ) > 0 are satisfied by conditions (9) and

(10). Second, if N = N ′ > 2 satisfies conditions (9), (10), and (11), then

N = N ′ − 1 will also satisfy these conditions. Thus, if N denotes the largest

N ∈ Z++ which satisfies these conditions, then the set of feasible N will be

1, ..., N. All that remains is to find N .

3This corresponds to the babbling equilibrium, where no information is transmitted.

57

I start by finding the largest N > 1 which satisfies inequality (13). In

particular, since rN is decreasing in N , I consider the set of N > 1 which solve4

rN ≡ sec

(θ

2

)cos

(Nθ − θ

2

)=

s0

sN.

The general solution to this equation is

N =1

2±

arccos(s0sN

cos(θ2

))+ z2π

θ, (14)

where z ∈ Z. Recall θ ∈ (0, π), therefore

cos

(θ

2

)=

√1 + cos (θ)

2,

where

cos (θ) =

cos

(arctan

(2√βi(βm−βi)βm

(2βi−βmβm

)−1))

if 2βi−βmβm

> 0

cos

(arctan

(2√βi(βm−βi)βm

(2βi−βmβm

)−1)

+ π

)if 2βi−βm

βm< 0

=2βi − βm

βm,

hence

cos

(θ

2

)=

√1 + cos (θ)

2=

√βiβm

.

Equation (14) is therefore equivalent to

N =1

2±

arccos(s0sN

√βiβm

)+ z2π

θ.

4Recall that sN is determined by the boundary conditions (5), hence it is not a function

of N .

58

Since s0sN,√

βiβm∈ (0, 1), the function arccos

(s0sN

√βiβm

)∈(0, π

2

). In order for⌊

N⌋

to simultaneously satisfy N > 0 and inequality (10), it must therefore be

the case that

N =1

2+

arccos(s0sN

√βiβm

)θ

. (15)

In fact, this N also satisfies N > 1, which follows from s0sN,√

βiβm∈ (0, 1) and

1

2+

arccos(s0sN

√βiβm

)θ

>1

2+

arccos(√

βiβm

)θ

=1

2+

1

2

(θ

θ

)= 1.

Moreover, since βi < βm, we can rewrite equation (15) using

θ = 2 arctan

√(

2βi−βmβm

)2

+

(2√βi(βm−βi)βm

)2

−(

2βi−βmβm

)(

2√βi(βm−βi)βm

)

= 2 arctan

(√βmβi− 1

),

hence

N =1

2+

arccos(s0sN

√βiβm

)2 arctan

(√βmβi− 1) . (16)

Thus, N =⌊N⌋≥ 1 is the largest N ∈ Z++ which satisfies conditions (9),

(10), and (11), where N is given by equation (16).

59

Equations (1), (4), (5), and (16) characterize the equilibria. When

βi < βm, there is an equilibrium for N ∈ 1, ..., N such that the investor

partitions the support of X according to the sequence cnNn=0 defined by

cn ≡ ω + csc (Nθ) ((1− ω) sin (nθ) + ω sin (nθ −Nθ)) ,

where

ω ≡ − E [Y ]G (Z)

E [Y 2]ZρF (1),

θ ≡ arg

(2βi − βm

βm+ i

2√βi (βm − βi)βm

).

The manager’s best response is

K∗1 (cn, cn+1) ≡

(βm[(

cn+cn+1

2

)Y ZρF (1) +G (Z)

]+ φ− 1

φ

)K0,

hence

I∗ (cn, cn+1) ≡

(βm[(

cn+cn+1

2

)Y ZρF (1) +G (Z)

]+ δφ− 1

φ

)K0.

The maximum number of feasible intervals N is given by

N ≡

1

2+

arccos((

ωω−1

)√βiβm

)2 arctan

(√βmβi− 1).

Case 2. βi > βm

The inverse Z-transform of equation (2) yields identical results to equation (3),

save for the fact that the poles are now real. In particular, the cutoff function

cn takes the form

cn = − E [Y ]G (Z)

E [Y 2]ZρF (1)+ P1ψ

−n + P2ψn, (17)

60

where

ψ ≡ 2βi − βmβm

+2√βi (βi − βm)

βm.

I will use the analytic expression for cn in equation (17) to derive the

set of N such that there exists an equilibrium with N intervals. For each N , I

will therefore need to prove that the sequence cn is monotonically increasing

in n for n ∈ 0, 1, ..., N. The function sn from equation (4) now takes the

form

sn ≡ cn +E [Y ]G (Z)

E [Y 2]ZρF (1)

= −

(1 + ω

(ψN − 1

)ψN − ψ−N

)ψ−n +

(1− ω

(1− ψ−N

)ψN − ψ−N

)ψn, (18)

where the last line follows from the boundary conditions (5), and

ω ≡ − E [Y ]G (Z)

E [Y 2]ZρF (1).

For marginal increases in n, the function sn satisfies:5

∂sn∂n

=

(1 + ω

(ψN − 1

)ψN − ψ−N

)ψ−n log (ψ) +

(1− ω

(1− ψ−N

)ψN − ψ−N

)ψn log (ψ) .

In this case, βi > βm, hence ψ > 1 and(1− ω

(1− ψ−N

)ψN − ψ−N

)> 0.

5Note that the cutoff function cn is only economically meaningful for n ∈ Z++. Incontrast to the previous case, the infinitesimal properties of cn will nevertheless prove to bequite useful.

61

If

(1+ω(ψN−1)ψN−ψ−N

)> 0, then the function sn will be strictly monotonic in n.

However, since ω < 0, there exists an N ′ > 0 such that

(1+ω

(ψN′−1

)ψN′−ψ−N′

)<

0. For such an N ′, the function sn will be strictly monotonic in n if and only

if

0 <∂sn∂n

⇔ 1

2 log (ψ)log

(−

1 + ω(ψN

′ − 1)

1− ω (1− ψ−N ′)

)< n,

hence ∂sn∂n

crosses zero from below at a single n and remains positive thereafter.

Consequently, sn will be monotonic in n ∈ 0, 1, ..., N if and only if s0 < s1,

which is trivially satisfied for any N ∈ (0, N ′).

In other words, there exists an equilibrium with N intervals if and only

if

s1 − s0 ≡ −

(1 + ω

(ψN − 1

)ψN − ψ−N

)ψ−1 +

(1− ω

(1− ψ−N

)ψN − ψ−N

)ψ + ω > 0. (19)

Note that

[s1 − s0]N=1 = 1,

hence the babbling equilibrium N = 1 is always feasible. Moreover, for N ≥ 1,

the length of the initial interval [s0, s1] satisfies

∂

∂N(s1 − s0) = −

ψN−1 (ψ2 − 1) log (ψ)(ψ2N + 1− ω

(ψN − 1

)2)

(ψ2N − 1)2 < 0. (20)

62

Thus, if an interval of size N + 1 ∈ Z++ is feasible, an interval of size N is also

feasible. Similar to the previous case, all that remains is to find the largest N

which satisfies condition (19).

Given condition (20), finding the maximal N ≥ 1 amounts to finding

the roots of the expression on left-hand side of condition (19). The general

solution to [s1 − s0]N=N = 0 takes the form

N± =1

log (ψ)log

ω(1 + ψ)− 1− ψ ±√ω2 (ψ − 1)2 + (ψ + 1)2 − 2ω (ψ + 1)2

2ω

.

Observe that

1

log (ψ)log

ω(1 + ψ)− 1− ψ −√ω2 (ψ − 1)2 + (ψ + 1)2 − 2ω (ψ + 1)2

2ω

= 1− 1

log (ψ)log

ω(1 + ψ)− 1− ψ +√ω2 (ψ − 1)2 + (ψ + 1)2 − 2ω (ψ + 1)2

2ω

,

(21)

hence only one of these roots will satisfy N ≥ 1. In fact, the satisfactory root

will be the left-hand side of equation (21), as algebraic manipulation reveals:

ω(1 + ψ)− 1− ψ +√ω2 (ψ − 1)2 + (ψ + 1)2 − 2ω (ψ + 1)2

2ω< 1

if and only if

−4ω (ψ + 1) > 0,

which necessarily holds true since ω < 0 and ψ > 1. Thus, the maximal

63

number of feasible intervals is given by the floor of

N =1

log (ψ)log

ω(1 + ψ)− 1− ψ −√ω2 (ψ − 1)2 + (ψ + 1)2 − 2ω (ψ + 1)2

2ω

.

(22)

Equations (1), (18), and (22) characterize the equilibria. When βi >

βm, there is an equilibrium for N ∈ 1, ..., N such that the investor partitions

the support of X according to the sequence cnNn=0 defined by

cn ≡ ω −

(1 + ω

(ψN − 1

)ψN − ψ−N

)ψ−n +

(1− ω

(1− ψ−N

)ψN − ψ−N

)ψn,

where

ω ≡ − E [Y ]G (Z)

E [Y 2]ZρF (1),

ψ ≡ 2βi − βmβm

+2√βi (βi − βm)

βm.

The manager’s best response is

K∗1 (cn, cn+1) ≡

(βm[(

cn+cn+1

2

)Y ZρF (1) +G (Z)

]+ φ− 1

φ

)K0,

hence

I∗ (cn, cn+1) ≡

(βm[(

cn+cn+1

2

)Y ZρF (1) +G (Z)

]+ δφ− 1

φ

)K0.

The maximum number of feasible intervals N is given by

N ≡

1

log (ψ)log

ω(1 + ψ)− 1− ψ −√ω2 (ψ − 1)2 + (ψ + 1)2 − 2ω (ψ + 1)2

2ω

.Q.E.D.

64

Proof of Theorem 2.3.1

Following Theorem 2.2.1, the maximum number of feasible intervals is given

by

N ≡

⌊

12

+arccos

(( ωω−1)

√βiβm

)2 arctan

(√βmβi−1)⌋

if βi < βm⌊1

log(ψ)log

(ω(1+ψ)−1−ψ−

√ω2(ψ−1)2+(ψ+1)2−2ω(ψ+1)2

2ω

)⌋if βi > βm.

The partial derivative of N with respect to any argument will neces-

sarily be a discrete-valued function due to the floor function b·c. To show that

the floor function is nondecreasing in an argument, it is sufficient to show that

the expression inside the floor functions are nondecreasing in an argument. I

therefore take the following partial derivative:

∂N

∂ω=

√

βiβm

(2 (ω − 1)2

√1− βiω2

βm(ω−1)2arctan

(√βmβi− 1))−1

if βi < βm

− (ψ + 1)

(ψ√ω2 (ψ − 1)2 + (ψ + 1)2 − 2ω (ψ + 1)2 log (ψ)

)−1

if βi > βm.

Note that ω < 0, hence ∂N∂ω

> 0. Since Z only affects N through ω,

∂N

∂Z=∂N

∂ω

∂ω

∂Z,

thus sgn(∂N∂Z

)= sgn

(∂ω∂Z

). Furthermore,

∂ω

∂Z=

E [Y ]Z−(ρ+1) (ρG (Z)− ZG′ (Z))

E [Y 2]F (1),

hence ∂N∂Z≥ 0 if and only if

0 < ρG (Z)− ZG′ (Z)

⇔ G′ (Z)

G (Z)< ρZ−1.

65

Q.E.D.

Proof of Theorem 2.4.1

The investor’s expected utility under an arbitrary posterior belief χ is given

by

K0

φ

[β2m

(βiβm− 1

2

)Eχ [X]2 E

[Y 2]Z2ρF (1)2

+

(2β2

m

(βiβm− 1

2

)G (Z) + βi (φ− 1)

)Eχ [X]E [Y ]ZρF (1)

+ β2m

(βiβm− 1

2

)G (Z)2 + βi (φ− 1)G (Z) + (ZF (1)− δ)φ+

1

2

].

If βiβm

< 12, the investor optimizes by minimizing his ex-ante expectation of

Eχ [X]2, hence he reveals no information. Conversely, if βiβm

> 12, the investor

optimizes by maximizing his ex-ante expectation of Eχ [X]2, hence he fully

reveals his private information. In the singular case where βiβm

= 12, the in-

vestor’s ex-ante expected utility is independent of his revelations, therefore

any strategy is optimal.

Q.E.D.

Proof of Theorem 2.5.1

After learning Y , but before communicating with the investor, the manager

would choose her message to maximize her expectation of

K0

φ

[1

2β2mEχ|γ [X]2 Y 2Z2ρF (1)2 + βm (βmG (Z)− 1 + φ)Eχ|γ [X]Y ZρF (1)

+1

2β2mG (Z)2 + βm (φ− 1)G (Z) + (ZF (1)− δ)φ+

1

2

].

66

Reminiscent of Theorem 2.4.1, the manager would tailor her disclosures to

maximize her expectation of Eχ|γ [X]2, where χ|γ denotes a posterior distri-

bution over X induced by an investor whose posterior beliefs over Y are sum-

marized by γ; unsurprisingly, the manager’s utility is maximized when the

investor fully reveals his information.

Note that the manager’s expected utility is monotonically increasing in

E[Eχ|γ [X]2

]for every possible realization of Y . Consequently, the manager’s

disclosures will be credible if and only if she is indifferent among every feasible

posterior γ for all Y ∈ supp (Y ).

Moreover, following Theorem 2.2.1, the manager’s expectation of Eχ|γ [X]2

will be monotone in the ratio

Eγ [Y 2]

Eγ [Y ].

If the manager indifferent among every set of feasible posteriors, then each

pair (γ′, γ′′) of these posteriors must satisfy

Eγ′ [Y 2]

Eγ′ [Y ]=

Eγ′′ [Y 2]

Eγ′′ [Y ]

⇔ Eγ′[Y 2]Eγ′′ [Y ]− Eγ′′

[Y 2]Eγ′ [Y ] = 0

⇔ P (γ′′)Eγ′[Y 2]Eγ′′ [Y ]− P (γ′′)Eγ′′

[Y 2]Eγ′ [Y ] = 0

⇔∑γ′′

P (γ′′)Eγ′[Y 2]Eγ′′ [Y ]−

∑γ′′

P (γ′′)Eγ′′[Y 2]Eγ′ [Y ] = 0

⇔ Eγ′[Y 2]E [Y ]− E

[Y 2]Eγ′ [Y ] = 0

⇔ Eγ′ [Y 2]

Eγ′ [Y ]=

E [Y 2]

E [Y ], (23)

67

where the right-hand side of equation (23) uses the prior distribution of Y .

Thus, the manager’s disclosures cannot affect the set of equilibrium outcomes.

Q.E.D.

Proof of Theorem 2.6.1

Suppose Y is common knowledge. Then for each Y ∈ supp (Y ), Theorem 2.2.1

yields

N(Y ) ≡

⌊

12

+arccos

(( ω(Y )ω(Y )−1)

√βiβm

)2 arctan

(√βmβi−1)⌋

if βi < βm⌊1

log(ψ)log

(ω(Y )(1+ψ)−1−ψ−

√ω(Y )2(ψ−1)2+(ψ+1)2−2ω(Y )(ψ+1)2

2ω(Y )

)⌋if βi > βm.

where

ω(Y ) ≡ − G (Z)

Y ZρF (1).

As the proof of Theorem 2.3.1 demonstrates, ∂N∂ω≥ 0 for ω < 0, hence

N ≤ N(ω) if and only if

ω ≤ ω(Y )

⇔ E [Y ]

E [Y 2]≥ 1

Y

⇔ Y ≥ E [Y 2]

E [Y ].

Q.E.D.

68

Proof of Theorem 2.7.1

Recall that Theorem 2.2.1 produced the manager’s best-response function

I∗ (cn, cn+1) ≡

(βm[(

cn+cn+1

2

)Y ZρF (1) +G (Z)

]+ δφ− 1

φ

)K0.

Given any partition of the state space cnNn=0, the ex-ante expected level of

investment is given by

E [I∗ (cn, cn+1)] =

∫supp(Y )

N−1∑n=0

I∗ (cn, cn+1)

(cn+1 − cncN − c0

)dFY (Y )

=

(βm[(

c0+cN2

)E[Y ]ZρF (1) +G (Z)

]+ δφ− 1

φ

)K0.

If instead the manager were fully informed on on the realization of X, her

ex-ante expected level of investment would be

E[Im

]=

∫supp(Y )

∫supp(X)

ImdFX(X)dFY (Y )

=

(βm [E[X]E[Y ]ZρF (1) +G (Z)] + δφ− 1

φ

)K0

=

(βm[(

c0+cN2

)E[Y ]ZρF (1) +G (Z)

]+ δφ− 1

φ

)K0

= E [I∗ (cn, cn+1)] .

Thus, the manager will not underinvest in the two-period model.

Q.E.D.

69

Proof of Theorem 3.2.1

Given the capital accumulation equation (3.1), the manager’s problem can be

cast in terms of a choice over Kt+1 ∈ R+, hence her value function satisfies

Vm,t = maxKt+1

ζmΠt + βmEm,t [Vm,t+1] .

Under symmetric information, the homotheticity of the production function

F (·) and adjustment cost function C (·) would imply that each agent’s value

function Vj,t is homothetic with respect to the extant capital stock Kt. I

therefore conjecture and later verify that the value functions under asymmetric

information are also homothetic. In this case, the manager’s value function

can be written as

Vm,t = maxKt+1

ζmΠt + βmEm,t[Vm,t+1

Kt+1

]Kt+1

= maxKt+1

ζm(Πt + βmEm,t

[At+1F (1) + V 0

m,t+1

]Kt+1

),

where

Et[V 0m,t+1

]≡ Et

[ζ−1m

Vm,t+1

Kt+1

|Xt+1 = 0

]is not a function of Kt+1.

Following the proof of Theorem 2.2.1, I obtain a familiar set of equi-

libria. In particular, there is an equilibrium for Nt ∈ 1, ..., Nt such that the

investor partitions the support of Xt+1 according to the sequence cn,tNtn=0

defined by

cn,t ≡

ωt + csc (Ntθ) ((1− ωt) sin (nθ) + ωt sin (nθ −Ntθ)) if βi < βm

ωt −(

1+ωt(ψNt−1)ψNt−ψ−Nt

)ψ−n +

(1−ωt(1−ψ−Nt)ψNt−ψ−Nt

)ψn if βi > βm,

70

where

ωt ≡ −(βmEt

[V 0m,t+1

]− βiEt

[V 0i,t+1

])Et [Yt+1]

(βm − βi)Et[Y 2t+1

]Et [Zt+1]F (1)

,

θ ≡ arg

(2βi − βm

βm+ i

2√βi (βm − βi)βm

),

ψ ≡ 2βi − βmβm

+2√βi (βi − βm)

βm.

The manager’s best response is

K∗t+1 (cn,t, cn+1,t) ≡

(βm[( cn,t+cn+1,t

2

)Yt+1Et [Zt+1]F (1) + Et

[V 0m,t+1

]]+ φ− 1

φ

)Kt,

hence

I∗t (cn,t, cn+1,t) ≡

(βm[( cn,t+cn+1,t

2

)Yt+1Et [Zt+1]F (1) + Et

[V 0m,t+1

]]+ δφ− 1

φ

)Kt.

The maximum number of feasible intervals Nt is given by

Nt ≡

⌊

12

+arccos

((ωtωt−1

)√βiβm

)2 arctan

(√βmβi−1)⌋

if βi < βm⌊1

log(ψ)log

(ωt(1+ψ)−1−ψ−

√ω2t (ψ−1)2+(ψ+1)2−2ωt(ψ+1)2

2ωt

)⌋if βi > βm.

Finally, note that ωt, θ, and ψ are unaffected by the extant capital stock Kt,

while the manager’s best-response function K∗t+1 (cn,t, cn+1,t) is linear in Kt.

Taken together, this confirms that Vj,t is homothetic in Kt.

Q.E.D.

71

Proof of Theorem 3.3.1

Following the proof of Theorem 2.3.1,

sgn

(∂Nt

∂Zt

)= sgn

(∂ωt∂Zt

),

where

∂ωt∂Zt

=Et [Yt+1]Z−1

t

(ρ(βmEt

[V 0m,t+1

]− βiEt

[V 0i,t+1

])− Zt

(βmEt

[∂V 0

m,t+1

∂Zt

]− βiEt

[∂V 0

i,t+1

∂Zt

]))(βm − βi)Et

[Y 2t+1

]Et [Zt+1]F (1)

.

Consequently, ∂Nt∂Zt≥ 0 if and only if

0 < ρ(βmEt

[V 0m,t+1

]− βiEt

[V 0i,t+1

])− Zt

(βmEt

[∂V 0

m,t+1

∂Zt

]− βiEt

[∂V 0

i,t+1

∂Zt

])

⇔βmEt

[∂V 0

m,t+1

∂Zt

]− βiEt

[∂V 0

i,t+1

∂Zt

]βmEt

[V 0m,t+1

]− βiEt

[V 0i,t+1

] < ρZ−1t .

Q.E.D.

Proof of Theorem 3.4.1

The investor’s expected utility under an arbitrary posterior belief χ is given

by

Kt

φ

[β2m

(βiβm− 1

2

)Eχ [Xt+1]2 Et

[Y 2t+1

]Et [Zt+1]2 F (1)2

+ 2β2m

(βiβm

(Et[V 0m,t+1

]+ Et

[V 0i,t+1

]2

)−

Et[V 0m,t+1

]2

)Eχ [Xt+1]Et [Yt+1]Et [Zt+1]F (1)

+ β2m

(βiβm

Et[V 0i,t+1

]−

Et[V 0m,t+1

]2

)Et[V 0m,t+1

]+ βi (φ− 1)Et

[V 0i,t+1

]+ (AtF (1)− δ)φ+

1

2

]

72

If βiβm

< 12, the investor optimizes by minimizing his ex-ante expectation of

Eχ [Xt+1]2, hence he reveals no information. Conversely, if βiβm

> 12, the in-

vestor optimizes by maximizing his ex-ante expectation of Eχ [Xt+1]2, hence

he fully reveals his private information. In the singular case where βiβm

= 12, the

investor’s ex-ante expected utility is independent of his revelations, therefore

any strategy is optimal.

Q.E.D.

Proof of Theorem 3.5.1

After learning Yt+1, but before communicating with the investor, the manager

would choose her message to maximize her expectation of

Kt

φ

[1

2β2mEχ|γ [Xt+1]2 Y 2

t+1Et [Zt+1]2 F (1)2

+ βm(βmEt

[V 0m,t+1

]− 1 + φ

)Eχ|γ [Xt+1]Yt+1Et [Zt+1]F (1)

+1

2β2mEt

[V 0m,t+1

]2+ βm (φ− 1)Et

[V 0m,t+1

]+ (ZtF (1)− δ)φ+

1

2

],

where χ|γ denotes a posterior distribution over X induced by an investor

whose posterior beliefs over Y are summarized by γ. As in Theorem 2.5.1,

this function is monotonically increasing in Et[Eχ|γ [Xt+1]2

]for every possible

realization of Yt+1. Thus, the manager’s disclosures will be credible if and only

if she is indifferent among every feasible γ for all Yt+1 ∈ supp (Yt+1), and none

of these disclosures can affect the set of equilibrium outcomes.

Q.E.D.

73

Proof of Theorem 3.6.1

Suppose Yt+1 is common knowledge. Then for each Yt+1 ∈ supp (Yt+1), Theo-

rem 3.2.1 yields

Nt (Yt+1) ≡

⌊

12

+arccos

((ωt(Yt+1)ωt(Yt+1)−1

)√βiβm

)2 arctan

(√βmβi−1)

⌋if βi < βm⌊

1log(ψ)

log

(ωt(Yt+1)(1+ψ)−1−ψ−

√ωt(Yt+1)2(ψ−1)2+(ψ+1)2−2ωt(Yt+1)(ψ+1)2

2ωt(Yt+1)

)⌋if βi > βm,

where

ωt (Yt+1) ≡ −βmEt

[V 0m,t+1

]− βiEt

[V 0i,t+1

](βm − βi)Y 2

t+1Et [Zt+1]F (1).

As in the proof of Theorem 3.3.1, the form of ωt (Yt+1) implies ∂Nt(Yt+1)∂Yt+1

≥ 0.

Q.E.D.

Proof of Theorem 3.7.1

Recall that Theorem 3.2.1 produced the manager’s best-response function

I∗t (cn,t, cn+1,t) ≡

(βm[( cn,t+cn+1,t

2

)Yt+1Et [Zt+1]F (1) + Et

[V 0m,t+1

]]+ δφ− 1

φ

)Kt.

74

The expected difference between the manager’s first-best level of capital in-

vestment and her equilibrium capital investment is therefore given by

E[Im,t − I∗t (cn,t, cn+1,t)

]=

∫ ∫ [∫supp(Xt+1)

Im,tdFX (Xt+1)

−Nt−1∑n=0

I∗t (cn,t, cn+1,t)

(cn+1,t − cn,tcNt,t − c0,t

)]dFY (Yt+1)dFZ (Zt+1|Zt)

=Kt

φβm

(Et[V 0m,t+1

]− Et

[V 0m,t+1

])≥ 0,

where V 0m,t+1 denotes the full-information counterpart of V 0

m,t+1. Thus, given

any equilibrium partition of the state space cn,tNtn=0, the manager will on

average invest less than she would under the full-information benchmark.

Q.E.D.

75

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