Dynamic Multitasking and

Managerial Investment Incentives∗

Florian Hoffmann† Sebastian Pfeil‡

February 2017

Abstract

We study long-term investment in a dynamic agency model with multitasking.

The manager‘s short-term task determines current performance which deteriorates if

he invests in the firm‘s future profitability, his long-term task. The optimal contract

dynamically balances incentives for short- and long-term performance such that in-

vestment is distorted upwards (downwards) relative to first-best in firms with high

(low) technological returns to investment. These distortions decrease as good perfor-

mance relaxes endogenous financial constraints arising from the agency problem, im-

plying negative (positive) investment-cash flow sensitivities. Investment distortions

and cash flow sensitivities increase in absolute terms with short-term performance

pay and external financing costs.

Keywords : Continuous time contracting, multiple tasks, delegated investment, man-

agerial compensation, endogenous financing frictions, investment dynamics.

JEL Classification : D86 (Economics of Contract: Theory), D92 (Intertemporal Firm

Choice, Investment, Capacity, and Financing), J33 (Compensation Packages, Pay-

ment Methods).

∗We thank Patrick Bolton, Wei Cui (CICF discussant), Guido Friebel, Roman Inderst, Peter Kondor,Christian Leuz, Stephan Luck, Thomas Mariotti, Konstantin Milbradt (AFA discussant), Jean-CharlesRochet, Paul Schempp, Vladimir Vladimirov, Neng Wang, John Zhu, and conference and seminar par-ticipants at AFA 2015 in Boston, CICF 2013 in Shanghai, Columbia Business School, Erasmus Schoolof Economics, Frankfurt School of Finance and Management, Stockholm School of Economics, TilburgUniversity, University of Amsterdam, University of Bonn, and University of Zurich for helpful comments.An earlier version of this paper has been circulated under the title “A Dynamic Agency Theory of Del-egated Investment”. Pfeil acknowledges support by the University Research Priority Program FinReg ofthe University of Zurich.

†University of Bonn. E-mail: fhoffmann@uni-bonn.de.‡University of Bonn. E-mail: pfeil@uni-bonn.de.

1 Introduction

A manager responsible for a firm‘s operations usually has some form of discretion in

running the day-to-day business, which relies on his specific skills or private information.

Due to the separation of ownership and control, this gives rise to an agency problem, which

has been the focus of much of the recent dynamic financial contracting literature. 1 Yet, in a

dynamic world, firms also have to take strategic decisions and invest in order to maintain

long-term profitability. Typically, this investment process also relies on information of

the same manager or is even delegated to him with discretion. In fact, many strategic

investment decisions, such as investment in process innovation, product development or

R&D, share the following features: i) investment expenditures are not easily verifiable by

firm owners and therefore cannot be contracted upon,2 ii) their outcome is uncertain, and

iii) they have persistent effects on the firm‘s profitability. The manager‘s hidden actions,

thus, affect both the firm‘s current period payoffs as well as its long-term profitability,

giving rise to a dynamic multitask problem.

To capture these ideas, we introduce delegated non-contractible investment in the firm‘s

future profitability, into a continuous time cash flow diversion model. We analyze how the

need to incentivize the manager to meet both short-term as well as long-term targets af-

fects the optimal compensation scheme as well as the efficiency of the investment process.

Under the optimal long-term contract, investment will depend on the entire history of

past performance and be distorted away from the first-best level. We further show that

both the sign of the investment distortion as well as the comparative statics of investment

(with respect to realized cash flows, corporate governance or financial frictions) crucially

depend on the extend to which a firm‘s profitability depends on investment expenditures,

i.e., the technological “returns to investment.” Investment is distorted upwards relative to

the first-best benchmark when returns to investment are high and it is distorted down-

wards when they are low. These distortions decrease in absolute terms with financial slack

1E.g., DeMarzo and Sannikov (2006), DeMarzo and Fishman (2007), Biais et al. (2007), or Hoffmannand Pfeil (2010). See also Biais et al. (2013) for an excellent survey.

2More broadly this also applies to managerial effort and time, entrepreneurial talent, investment ininternal personnel resources, or the development of new styles and products that must be kept secret fromcompetitors. Some soft investment expenditures further are - unlike hard investment in plants, propertyor other equipment - notoriously hard to measure based on a firms accounting figures. Likewise, it is oftendifficult to separate investment expenditures from ordinary operating expenses (c.f., Bebchuk and Stole1993, or Dutta and Reichelstein 2003).

1

and, thus, past performance, implying a negative relation between investment and real-

ized cash flows in sectors with high, and a positive relation in sectors with low returns

to investment. Further, if the costs of raising external funds increase, so will both the

distortions in investment as well as its sensitivity to past performance. Our multitask

model of investment, thus, provides a new perspective on investment-cash flow sensitivi-

ties as a measure of financial constraints (c.f., Fazzari et al. 1988 and Kaplan and Zingales

1997). In particular, in line with empirical evidence (Peters and Taylor 2016), it predicts

negative investment-cash flow sensitivities in firms or sectors with high non-contractible

investment expenditures (such as many forms of intangible investment, e.g., in R&D or

process innovation).

The key agency frictions in our dynamic multitask theory of investment are due to

i) unobservable cash flows and ii) non-contractible investment expenditures. An agency

problem with respect to investment then arises endogenously as a result of compensation

for short-term performance which is necessary to deter the manager from diverting cash

flows for private consumption: Since his compensation is tied to realized cash flows and

investment expenditures are not observable to firm owners, the manager has an incentive

to cut profitable investment in order to boost short-term earnings. Our model, thus, shares

the notion that short-term incentive schemes can induce managers to focus excessively on

current period outcomes with the literature on managerial short-sightedness. 3 However, in

our model these unintended consequences of short-term incentive pay can be mitigated by

tying the manager‘s compensation also to indicators of investment success/failure such as

project milestones or the number of patents granted (see Balkin et al. 2006 for evidence).

Still, as is well known from related (single-task) dynamic financial contracting models

(cf. DeMarzo and Sannikov 2006), exposing the manager to compensation risk in order

to provide incentives is costly. In our model with bilateral risk neutrality this formally

is a consequence of limited liability, which requires to terminate the contract if the man-

ager‘s discounted expected payoff, his continuation value, hits a lower bound in which case

the manager must be replaced, which is costly ex-post, but part of the ex-ante optimal

long-term contract. Hence, the need to tie the manager‘s pay to (imperfect) signals of

3The short-term bias has been attributed to career concerns (Narayanan 1985), takeover threats (Stein1988), concerns about the firm‘s stock prices over a near-term horizon (Stein 1989), mispricing of long-term assets (Shleifer and Vishny 1990), reputational herding (Zwiebel 1995) and short-term financing (vonThadden 1995). In fact, there is ample empirical evidence that managers cut, e.g., R&D spending in orderto meet short term earnings targets (Baber et al. 1991, Bushee 1998, Cheng 2004, Brown and Krull 2008).

2

investment success or failure such as to incentivize investment creates additional agency

costs. The optimal investment schedule is then determined by trading-off the agency

costs of investment with the potential efficiency gains as captured, in particular, by the

respective returns to investment. We show that the agency costs of investment are non-

monotonic in the implemented investment level. In particular, marginal agency costs are

positive for low and negative for high levels of investment. As a consequence, investment

is distorted downwards relative to the first-best (contractible investment) benchmark in

industries with low returns to investment, i.e., industries in which first-best investment is

low. By contrast, investment is distorted upwards relative to first-best in industries where

returns to investment and therefore also first-best investment is high.

To understand this result consider, first, the compensation policy that incentivizes a

given investment level at minimal agency costs, which arise from firm owners’ effective risk

aversion with respect to variation in the manager‘s expected compensation. When the im-

plemented investment level and therefore also the probability of success is rather low, it is

optimal to provide incentives mainly through rewards for investment success, while a high

level of investment is optimally incentivized mainly through punishment for a then un-

likely investment failure. Intuitively, the performance signal (investment success/failure)

which is least likely to occur given the implemented investment level is most informative

about the manager‘s hidden investment. Hence, under the optimal compensation policy,

incentives are mainly provided following the least likely signal which, as a result, is most

costly in terms of incentive provision as it induces most variation in the manager‘s ex-

pected compensation. Thus, when choosing the optimal level of investment, the firm can

reduce agency costs by reducing the probability of the more costly signal realization. In

particular, if investment is high and, hence, optimally incentivized with harsh punishment

following an investment failure, the probability of the more costly signal realization (fail-

ure) can be reduced by incentivizing even higher investment expenditures, which tends to

decrease agency costs. We show that, as a result, delegated investment will be optimally

distorted upwards relative to the first-best benchmark when the returns to investment,

and, accordingly, also first-best investment, are sufficiently high. Similarly, if the returns

to investment are low, delegated investment will be distorted downwards relative to the

(low) first-best investment benchmark.

When the manager‘s track record improves, his stake in the firm under the optimal

3

contract increases. This mitigates the agency problem and, thus, reduces investment dis-

tortions. Accordingly, investment expenditures are negatively related to past performance

(i.e., the entire history of cash flows) in overinvesting firms with high returns to invest-

ment, while in underinvesting firms with low returns to investment the relation is positive.

Note in particular, that in our dynamic optimal contracting model it is the manager‘s con-

tinuation value, which summarizes the entire performance history under the incumbent

manager and can therefore be directly related to the firm‘s financial slack, rather than

current period performance that determines investment (see also DeMarzo et al. (2012)).

We provide an implementation of the optimal contract, in which the firm retains earn-

ings to build up a cash buffer, that is used to cover investment expenditures, potential oper-

ating losses, and payments to investors. If these cash holdings (”financial slack”) increase,

financial constraints decrease and so do investment distortions, implying higher/lower in-

vestment in under-/overinvesting firms. Eventually, financial constraints are sufficiently

relaxed and the firm starts issuing dividends; on the other hand, when cash holdings fall

to zero, the firm is no longer able to honor its payments to investors, triggering a costly

restructuring of the firm, which involves replacing the incumbent manager and raising

new capital. We find that if the costs to raise external funds increase, so do investment

distortions and investment-cash flow sensitivities, for any given value of financial slack.

Note, however, that this amplification implies that investment increases and the relation

between financial slack and investment becomes more negative in firms with high returns

to investment, while investment decreases and its sensitivity to financial slack becomes

more positive in firms with low returns to investment.

Our multitask theory of investment, thus, provides a novel perspective on the rela-

tionship between investment-cash flow sensitivities and financial constraints, which arise

endogenously from the agency problem. By focussing on investment in future profitabil-

ity per unit of (physical) capital instead of standard capital investment, our model can be

viewed as a first step towards accounting for the increasing importance of alternative forms

of investment such as R&D, process innovation or intangible investment more generally

(c.f., Brown and Petersen 2009). In this sense, recent empirical evidence in Peters and

Taylor (2016) indeed confirms our model prediction that investment-cash flow sensitivities

should be negative in high intangible investment firms. Accordingly, our model could also

contribute to explain the puzzling findings by Chen and Chen (2012) that on average (i.e.,

4

not controlling for intangible investment intensity) investment-cash flow sensitivities have

disappeared despite the fact that firms are still financially constrained.

The multitask nature of our dynamic optimal contracting model further generates new

testable implications regarding the relation between pay for short-term performance and

long-term investment. If, in our model, the cash flow diversion problem becomes more

severe, e.g. due to worse corporate governance, then i) the optimal contract has to be

more sensitive with respect to short-term performance. This exacerbates the agency prob-

lem with respect to long-term investment, such that, ii) investment distortions increase in

absolute terms. That is, steeper short-term incentives are associated with higher invest-

ment expenditures in firms with high returns to investment and with lower investment

expenditures in firms with low returns to investment.

From a methodological perspective, our model is most closely related to Hoffmann and

Pfeil (2010) who introduce exogenous (Poisson) shocks to firm profitability in a dynamic

cash flow diversion model a la DeMarzo and Sannikov (2006). While the firm‘s long-term

profitability in their model is purely determined by exogenous ”lucky” shocks, it can be

controlled by the manager in the present analysis via investment expenditures, giving rise

to a fully dynamic multitask problem (c.f., Holmstrom and Milgrom 1991). In that respect

our analysis is related to Zhu (2016) who considers a model of persistent moral hazard

in which the agent can choose between a ”short-term” or a ”long-term” action in every

period. He characterizes the optimal contract that always induces the ”long-term” action.

Our focus is instead on determining the optimal interior investment level at every possible

history while abstracting from problems of persistent private information.

Similarly, our approach is related to Varas (2015) who studies a multitasking model of

project completion in which the agent can complete a project faster by reducing its quality

which is a one-time action. In contrast, we consider a fully dynamic and repeated problem

in which the agent can divert cash flow for private consumption as well as move funds

within the firm in order to meet either short- or long-term targets, where his discretion

is restricted by the optimal contract. In this sense, our model is related to the capital

budgeting literature starting with Harris and Raviv (1996) and extended to a dynamic

setting in Malenko (2016).

Optimal investment with dynamic agency is also analyzed in the two-period model

of Gertler (1992) as well as the multiperiod discrete time models in Quadrini (2004),

5

Clementi and Hopenhayn (2006), and DeMarzo and Fishman (2007). Further, DeMarzo

et al. (2012) analyze dynamic agency in a neoclassical investment setting using continuous

time methods. All of these contributions consider capital investment which can be verified

by firm owners. In our model, by contrast, the manager privately controls investment

and, thus, has to be incentivized to invest in firm owners’ interest. Further, investment

in our model does not scale cash flows by increasing the capital stock as in neoclassical

investment settings, but rather affects mean cash flows per unit of (constant) firm size.

The remainder of the paper is organized as follows. We introduce the model in Section

2. In Section 3 we derive the optimal dynamic contract and provide a detailed discussion

of optimal compensation and investment. Section 4 provides an implementation of the

optimal contract and derives empirical implications. Section 5 concludes. All proofs are

in Appendix A. Appendix B contains some additional material.

2 The Model

We consider an infinite horizon continuous time principal-agent model of a firm whose

owners (the principal) hire a manager (the agent) to operate the business, which includes

investment in the firm‘s future profitability. First, we present the firm‘s cash flow process

and investment technology. Second, we introduce the agency problem between firm owners

and the manager, and formulate the optimal contracting problem.

2.1 Cash Flow Process and Investment Technology

The firm‘s cash flows, Y = {Yt : t ≥ 0}, net of investment expenditures, I = {It ∈ [0, I ] :

t ≥ 0}, evolve according to4

dYt = (μt − It) dt + σdZt, (1)

where μt ∈{μl, μh

}, with 0 < μl < μh, denotes the drift rate of cash flows, which we

refer to as the firm‘s “profitability”, σ the instantaneous volatility, and Z is a standard

Brownian motion on a complete probability space.

Investment affects the firm‘s future profitability by determining its ability to adopt and

4Throughout the text we will denote processes by bold letters.

6

μt

Notechn.shock

μt+ = μt

1 − νdt

Techn.shock

μt+ = μl (investment failure)1 − p(It)

μt+ = μh (investment success)p(It)

νdt

Figure 1: At each t, a new technology is available with probability νdt, which the firmis able to adopt with probability p(It), leading to high future profitability (μt+ = μh).With probability 1 − p(It), the firms fails to adopt the new technology and has low futureprofitability (μt+ = μl). If no new technology is available profitability remains unchanged.

commercialize new technologies as they become available.5 As illustrated in Figure 1, the

industry is subject to (rare) exogenous technology shocks governed by a Poisson process N

with intensity ν, indicating the availability of a new technology. If there is a technology

shock (dNt = 1), the firm is able to adopt the new technology with probability p(It) and

its future profitability will be high (μt+ := lims→t− μs = μh). We refer to this event as an

investment success, which can be interpreted as moving to or staying at the research frontier

(depending on whether μt = μl or μt = μh). With probability 1−p(It) the firm is, however,

unable to adopt the new technology sufficiently quickly to stay competitive, and its future

profitability will, thus, be low (μt+ = μl). We refer to this as an investment failure, which

can be interpreted as falling or staying behind the research frontier (depending on whether

μt = μh or μt = μl). For future reference, note that the occurrence of an investment success

is, thus, governed by a Poisson process Ng with arrival rate νp (It) and the occurrence of an

investment failure by a Poisson process Nb with arrival rate ν (1 − p(It)).6 In between two

technology shocks, profitability remains unchanged such that investment has persistent

effects. Finally, we stipulate that the success probability p(∙) ∈ [0, 1] is an increasing and

5Concretely, investment expenditures can be interpreted as a firm‘s choice of absorptive capacity in thesense of Cohen and Levinthal (1990), describing its capability of “assimilating new, external informationand apply it to commercial ends”. See also Board and Meyer-ter-Vehn (2013) for an application of thisconcept in a model of firm reputation where investment determines future product quality.

6Investment in our model, thus, depreciates instantly, reflecting the need to keep up with fast tech-nological progress. Indeed, there is evidence that R&D investment depreciates much faster than physicalcapital (see e.g. Bernstein and Mamuneas 2006). Nevertheless, the main drivers of our results do notdepend on this assumption and, hence, our insights should extend also to the case where investmentdepreciates gradually over time, which however significantly complicates the analysis.

7

strictly concave function of the investment amount I, satisfying p(0) = 0 and p(I) = 1.7

2.2 Agency Problem

Firm owners have to hire a manager to (profitably) run the business which relies on the

manager‘s specific skills or private information. The manager is protected by limited

liability and has limited wealth. Hence, owners have to bear the costs of setting up the

firm (normalized to zero) and cover operating losses. Both parties are risk neutral and the

manager is more impatient, i.e., he discounts future consumption at rate γ > r.

Running the firm requires some form of discretion over the firm‘s cash flows, which

we model as follows: Firm owners do not observe cash flows Y directly but only the

manager‘s contractible reports Y. The difference between actual and reported cash flows

is determined by the manager‘s hidden action which is the source of the agency problem.

In particular, we assume that the manager can divert cash flows for private consumption.

To capture possible costs of concealing and taking funds out of the firm we stipulate

that the manager can consume fraction λ ∈ (0, 1] of each unit diverted. Furthermore, as

investment similarly requires a considerable understanding of the firm‘s internal processes

as well as the market it is operating in, it is also delegated to the manager, and investment

expenditures I are not directly observable to firm owners. Therefore, the manager can

manipulate net cash flows Y by deviating from the prescribed investment schedule (cf.,

(1)). Still, in order to provide incentives for investment, the contract can condition on the

(observable) investment outcome, as governed by the Poisson processes Ng and Nb.8

The principal can commit to a long-term compensation contract (U, τ ) specifying –

based on reported cash flows and observed investment outcomes – transfers dUt to the

agent, as well as the (random) time τ , when the agent is fired and replaced, which incurs the

principal fixed costs k > 0. Although replacing the agent is costly, it is needed and indeed

7While we frame this investment technology within the context of technology adoption it is more widelyapplicable. E.g., one could similarly interpret the process N as unpredictable demand shocks, to whichthe firm can react quickly enough only if it has invested sufficient resources, e.g., into building capacityin market research or product design. What is key to our analysis is that future profitability dependsstochastically on past investment via unpredictable shocks and that investment has persistent effects.

8Technology shocks in our model should be thought of as capturing important technological break-throughs in the respective industries and are, hence, observable, as is whether the firm is able to adopt thenewly available technology or not. That contracts condition on such signals is also in line with empiricalevidence: Managerial compensation in research intensive industries is often linked to signals indicatingsuccessful innovation such as meeting project milestones or filing new patents (cf., e.g., Balkin et al. 2000).As current profitability is thus known to the principal we can abstract from issues of learning.

8

part of the ex-ante optimal contract in order to incentivize the agent who is protected by

limited liability, which requires the cumulative wage process U = {Ut : 0 ≤ t ≤ τ} to be

non-decreasing. We further assume that the agent cannot save privately, implying that

dYt − dYt ≥ 0, i.e., he can only underreport cash flows.9 Hence, the agent‘s consumption

flow at time t is given by

dCt = dUt + λ[dYt − dYtdt

].

Upon being replaced, the agent receives his outside option, which is normalized to zero.

The incumbent agent‘s total expected wealth at the start of his employment, hence, is

w0 = ES0

[∫ τ

0

e−γtdCt

]

, (2)

where ES denotes the expectation under the probability measure QS induced by the agent‘s

strategy S ={

Yt, It : 0 ≤ t ≤ τ}

.10 The actual value of w0 > 0 in (2) is determined by the

two parties’ relative bargaining power. In what follows, we will assume that the principal

enjoys all bargaining power and the agent accepts any contract with w0 ≥ 0.

If the agent is replaced, the principal‘s value is given by Lτ , which denotes the (en-

dogenous) expected profit from the relationship with a new agent, net of replacement costs

k. Hence, at t = 0, the principal‘s total expected profit, delivering the agent an expected

payoff of w0, and given μ0 ∈{μl, μh

}, is

f0 = E0

[∫ τ

0

e−rt(dYt − dUt

)+ e−rτLτ

]

. (3)

Given a compensation contract (U, τ ), the agent chooses a feasible strategy S to max-

imize his initial expected payoff w0. A strategy S is called incentive compatible if it

maximizes w0 given (U, τ ). Hence, an incentive compatible contract can be described by

the triple (S∗,U, τ ), where S∗ is the (incentive compatible) strategy that the principal

wants to induce. The associated (global) incentive constraint is given by

ES∗

0

[∫ τ

0

e−γtdCt

]

≥ ES0

[∫ τ

0

e−γtdCt

]

, for any S 6= S∗. (4)

We can simplify the analysis considerably by relying on a version of the revelation principle,

9This is without loss of generality, given that the agent is risk-neutral and relatively impatient.10If obvious, we will not state the measure associated with the expectation operator in the following.

9

which allows us to restrict attention to truth-telling contracts, implementing Y = Y.11

The contracting problem then is to find an incentive compatible truth-telling contract

maximizing the principal‘s expected profit f0 for given μ0, delivering w0 to the agent and

satisfying limited liability.12 We refer to the solution of this constrained maximization

problem, which includes the optimal investment profile, as the optimal contract.

To complete the description of the agency problem, let us give a short and heuristic

outline of the timing of events taking place in any infinitesimal time interval [ t, t+dt] prior

to replacement of the incumbent manager: i) The agent decides on investment (It), current

cash flows are realized (Yt) and reported to the principal (Yt), ii) there is an investment

success (failure) with probability νp (It) dt (with probability ν [1 − p (It)] dt), iii) the agent

gets compensated (dUt) and the continuation decision is taken (I{τ>t}). Note in particular

that the agent chooses investment prior to observing whether a technology shock occurred

or not, which is the main important “sequentiality” in our continuous time model, while

compensation and possible replacement of the manager occur thereafter.13

3 Model Solution

In this section we solve for the optimal contract using the dynamic programming approach.

As we will show, the optimal contract can be written in terms of the agent‘s continuation

payoff as the single state variable. We derive the dynamics of this key state variable along

with local incentive compatibility constraints in Section 3.1. In Section 3.2 we set up the

recursive formulation of the contracting problem and characterize the optimal contract.

Finally, in Section 3.3, we discuss in detail the distortions in investment relative to the

contractible investment benchmark as well as its dynamics under the optimal contract.

11For a formal argument compare Lemma 1 and Proposition 2 in DeMarzo and Sannikov (2006).12There is no relevant participation constraint, as the agent enjoys a positive rent, once he is hired.13Formally, I is predictable with respect to F = {Ft, t ≥ 0}, the filtration generated by

(Z,Ng,Nb

),

while U is adapted to F = {Ft, t ≥ 0}, the filtration generated by (Y,Ng,Nb) and τ is an F -measurablestopping time.

10

3.1 Continuation Payoff and Local Incentive Compatibility

For any truth-telling contract, define the agent‘s continuation payoff, wt, as his future

expected discounted payoff at time t, given he will follow strategy S∗ from t onwards, i.e.,

wt = ES∗

t

[∫ τ

t

e−γ(s−t)dCs

]

.

As the agent‘s investment decision has to be predictable with respect to investment out-

comes (dN gt and dN b

t ), it is convenient to work with the left-limit of his continuation payoff

wt which we denote by wt− := lims↑t ws. Intuitively, while wt is the continuation payoff

after observing whether a technology shock occurs in t or not, wt− denotes the respective

value before this uncertainty is resolved. This variable will serve, together with μt, as the

state variable in the recursive formulation of the optimal contracting problem. Applying

standard martingale techniques, the evolution of wt− can be characterized as follows:14

Lemma 1. For any contract, there exist some predictable processes α = {αt : 0 ≤ t ≤ τ}

and βj ={βj

t : 0 ≤ t ≤ τ}, j ∈ {g, b}, such that, if the manager follows the recommended

strategy S∗ = {Yt, I∗t : 0 ≤ t ≤ τ}, his continuation payoff at any moment of time evolves

according to:

dwt− = γwt−dt − dUt + αt

(dYt − (μt − I∗

t ) dt)

(5)

+ βgt

(dN g

t − νp(I∗t )dt

)+ βb

t

(dN b

t − ν (1 − p(I∗t )) dt

),

where dYt − (μt − I∗t ) dt is the increment of a Brownian motion and dN g

t , dN bt are incre-

ments of standard Poisson processes with arrival rates νp(It) and ν (1 − p(It)) respectively.

To build some intuition let us discuss in more detail the evolution of wt− in (5). Due

to promise keeping, the agent‘s promised wealth, wt− , has to grow at his discount rate,

γ, while it must decrease with direct payments, dUt. It also depends on his reporting

strategy via the sensitivity αt and on the investment outcome via the sensitivities βgt and

βbt . So, if the agent underreports cash flows, he can immediately consume λ

(dYt − dYt

),

while his continuation payoff is reduced by αt

(dYt − dYt

). Incentive compatibility, thus,

requires that αt ≥ λ. As for deviations from the recommended investment level, if the

14Cf. Theorem III 4.34 in Jacod and Shiryaev (2003) for the relevant martingale representation theorem.See also e.g. Piskorski and Tchistyi (2010, Proposition 1) for an application to dynamic contracts.

11

agent raises It marginally above I∗t , his continuation payoff changes (in expectation) by

[νp′(I∗

t )(βg

t − βbt

)− αt

]dt. That is, the probability of an investment success – triggering

“reward” βgt – increases by νp′(I∗

t )dt, while the probability of a failure – triggering “pun-

ishment” βbt – decreases by νp′(I∗

t )dt. At the same time, wt− is reduced by αtdt in response

to the reduction in net current cash flows dYt (recall that the additional investment expen-

ditures are taken out of current cash flows). Hence, the agent has no incentives to invest

more than the recommended level I∗t if νp′(I∗

t )(βg

t − βbt

)≤ αt. In a similar manner, if the

agent reduces It marginally to inflate net current cash flows dYt, his continuation payoff

changes (in expectation) by −[νp′(I∗

t )(βg

t − βbt

)− αt

]dt, implying that the agent has no

incentives to invest less than I∗t if νp′(I∗

t )(βg

t − βbt

)≥ αt. These observations lead to the

local incentive compatibility conditions in Lemma 2 below.

Lemma 2. The truth-telling contract {S∗,U, τ} with I∗t ∈

(0, I)

is incentive compatible

if and only if

αt ≥ λ (6)

and

βgt − βb

t =αt

νp′(I∗t )

(7)

holds for all t ∈ [0, τ ), almost surely. Further, the limited liability constraint implies for

all t ∈ [0, τ ) that

wt− + βjt ≥ 0, j ∈ {g, b} . (8)

Incentive constraint (7) reflects the interaction between the problem of non-contractible

investment and the cash flow diversion problem: To provide incentives for investment, the

agent‘s continuation payoff has to increase following an investment success and decrease

following a failure. More precisely, according to (7), the sensitivity of the agent‘s contin-

uation payoff with respect to the investment outcome, βgt − βb

t , has to increase with the

sensitivity to reported cash flows, αt, for any given level of investment. Thus, a more

severe cash flow diversion problem, reflected by greater diversion benefits λ, would require

to impose more risk on the agent‘s income in two respects: First, from (6), the sensitivity

αt has to increase, because with higher diversion benefits, the agent‘s income has to be

linked more closely to reported cash flows in order to induce truthful reporting. Second,

with his income being more sensitive to cash flows, the agent has an incentive to inflate

cash flows by deviating from the recommended investment level. Hence, according to (7),

12

his income has to be more responsive to the investment outcome as well.

3.2 Optimal Contract

The optimal contract can now be derived using the dynamic programming approach. De-

note by f i(w), i ∈ {l, h} the principal‘s value function, that is, the highest profit the

principal can attain under any incentive compatible contract delivering w to the agent

and given the prevailing drift rate of cash flows, μi, where we drop time subscripts for

notational convenience.

The contracting problem can be greatly simplified by noting that, following any history,

the optimal (continuation) contract is independent of the current profitability state μi,

i ∈ {l, h}. Intuitively, this result follows from the fact that both the basic cash flow

diversion problem as well as the agency problem with respect to investment are independent

of μi. In particular, for a given level of investment, the probability of an investment success

or an investment failure does not depend on the current level of μi, implying that also the

optimal investment policy must be independent of the prevailing profitability level. To

see this more formally, consider the process κ counting the number of replacements and

the associated time points τ(κ). For convenience, set τ(0) = 0. Then we can write the

principal‘s value function as

f i(w) = maxI,U,τ

E

∞∫

0

e−rs (dYs − dUs) − k∞∑

κ=1

e−rτ(κ)

∣∣∣∣∣w, μi

,

where maximization is subject to the incentive constraint in (4), the agent‘s limited liability

as well as promise-keeping w0 = w. For any given point in time, denote the (stochastic)

time of the next technology shock by T and recall that technology shocks arrive according

to a Poisson process with exogenous arrival rate, such that T is clearly beyond both the

agent‘s and the principal‘s control. Thus, substituting from (1), we have

f i(w) =μi

(r + ν)+max

I,U,τ

E

∞∫

T

e−rsμsds −

∞∫

0

e−rs (dUs + Isds) − k∞∑

κ=1

e−rτ(κ)

∣∣∣∣∣w

, (9)

where the term in curly brackets is clearly independent of μ0 = μi for any w and so are

the relevant constraints. Hence, the optimal contract does not depend on μ0, and, from 9,

13

the value functions in the two profitability states only differ by an additive constant.

Lemma 3. The principal‘s value functions in the high and in the low profitability state

satisfy

f(w) := f l(w) = fh(w) − Δ, (10)

with

Δ :=1

r + ν

(μh − μl

). (11)

As the optimal contract is independent of the prevailing profitability state, the differ-

ence in the respective value functions, Δ, is a purely technological parameter, capturing

the direct gain from moving to (or staying at) the research frontier, μh − μl, properly

accounting for the Markov switching structure.

From Lemma 3 we can now characterize the optimal contract based on the single state

variable w. Consider, first, the optimal compensation policy. Clearly, the principal can

always compensate the agent directly by paying him a lump-sum of dU > 0 (at marginal

costs of −1) and then move to the optimal contract with reduced continuation payoff

(w − dU). However, deferring compensation may be valuable: The higher w, the smaller

is the probability that it falls to zero, which causes an inefficiency as the agent has to be

replaced at costs k > 0. In contrast to the costs of deferring compensation, which are

due to the wedge in discount rates (γ > r), this benefit declines, however, as w increases.

This is reflected in the concavity of f(w), which will be shown formally in the proof

of Proposition 2 below. As a consequence compensation is optimally deferred until the

threshold w is reached, where f ′(w) = −1 and the agent is paid cash.

Next, consider f(w) for w ∈ [0, w]. Since the principal discounts at rate r, his expected

flow of value at time t must be rf i(w)dt. This has to be equal to the expected instantaneous

net cash flow (μi − I) dt plus the expected change in the principal‘s value function. Hence,

using Ito‘s lemma, the change of variables formula for jump processes, and (10), we find

that the principal‘s value function must satisfy the following HJB equation on w ∈ [0, w]:

(r + ν) f(w) = maxα,β,I

μl − I + νp(I)Δ + 12σ2α2f ′′(w)

+[γw − ν

[βgp(I) + βb (1 − p(I))

]]f ′(w)

+νp(I)f(w + βg) + ν (1 − p(I)) f(w + βb)

(12)

s.t. (6), (7), (8),

14

while it extends linearly to the right of the compensation threshold, i.e., f(w) = f(w) −

(w − w) for w ≥ w. To solve for the optimal contract we have to pin down a solution to

(12) and the compensation threshold w. For this we require three boundary conditions:

The first (“value matching”) condition

f(0) = f(w∗) − k, (13)

with w∗ ∈ arg maxw {f(w)}, reflects the fact that upon firing the incumbent agent at w =

0, the principal receives f(w∗) from the relation with a new agent and bears replacement

costs k.15 The second is a standard “smooth pasting” condition at the compensation

threshold

f ′(w) = −1, (14)

while the optimal choice of w is guaranteed by the third boundary condition (“super

contact”)

f ′′(w) = 0. (15)

Before we characterize the solution to the boundary value problem in (12)-(15), the

optimal contract with non-contractible investment, it is useful to study the benchmark

case when investment expenditures are contractible.

Contractible Investment Benchmark. When investment expenditures are observ-

able, the problem reduces to a standard single-task cash flow diversion problem in which

the principal directly controls investment. Formally, the principal then also solves the

boundary value problem in (12)-(15), but the incentive constraint for investment (7) is

irrelevant. To avoid confusion we denote the respective value function with contractible

investment by fCI(w) and index contractual parameters, such as the compensation bound-

ary wCI , where necessary. The solution to this benchmark case is summarized in the

following Proposition.

Proposition 1. Assume investment expenditures are contractible, then, under the optimal

truth-telling contract, investment is given by It = IFB, ∀t, where IFB denotes the constant

15That the new agent starts employment with promised wealth w∗ maximizing the principal‘s value isnot crucial for our results. We just require that the principal‘s bargaining power stays constant over time.

15

first-best investment level solving

IFB ∈ arg maxI∈[0,I]

{νp(I)Δ − I} . (16)

The incumbent agent‘s continuation payoff evolves according to (5) with αt = λ, βgt =

βbt = 0 and It = IFB, ∀t. When wt− ∈ [0, wCI), dUt = 0; when wt− ≥ wCI payments

dUt cause wt− to reflect at wCI . The incumbent agent is replaced when wt− = 0. The

principal‘s expected payoff at any point in time is given by f iCI(wt), i ∈ {h, l}, which

satisfies fCI(w) := f lCI(w) = fh

CI(w) − Δ, where fCI (w) is concave, strictly so for 0 ≤

w < wCI and solves, for w ∈ [0, wCI ], the HJB equation

rfCI(w) = μl − IFB + νp(IFB)Δ + γwf ′CI(w) +

1

2σ2λ2f ′′

CI(w) (17)

with boundary conditions fCI(0) = fCI(w∗CI)−k, where w∗

CI ∈ arg maxw {fCI(w)}, f ′CI(wCI) =

−1 and f ′′CI(wCI) = 0.

As is standard in dynamic cash-flow diversion models (cf., e.g., DeMarzo and Sannikov

2006), the agent‘s incentive constraint with respect to cash flow diversion (6) binds under

the optimal contract. This is intuitive, because it is costly to provide incentives. Formally,

the result follows from the concavity of fCI (w) and the observation that increasing α would

increase the instantaneous volatility of w. A similar argument also implies that, when

investment expenditures are contractible and there is, thus, no need to provide incentives

based on the investment outcome, it is optimal to choose βgCI = βb

CI = 0. Formally, this

follows from the fact that the value functions in the two profitability states differ only by

an additive constant such that the costs of compensating the agent, as reflected in the

slope f ′CI(w), are independent of current profitability.16 As a consequence, the agent‘s

continuation value is insensitive to the investment outcome and the principal‘s investment

problem is, thus, independent of the cash-flow diversion problem. Hence, the optimal

investment policy is equal to first-best, i.e. the investment level that a profit-maximizing

risk-neutral owner-manager with discount rate r would choose.17 From (16), this first-

16 In our setting, non-zero sensitivities with respect to the investment outcome can, thus, be optimalonly for incentive reasons as in our main model with non-contractible investment solved below. If thecosts of compensating the agent differed across states i ∈ {h, l}, e.g., due to state-dependent hiring costs,it would also be efficiency enhancing to specify βg

CI , βbCI 6= 0 (cf., Hoffmann and Pfeil (2010) for a formal

model of this “reward for luck” effect).17Notably, this is different in the neoclassical investment model considered by DeMarzo et al. (2012),

16

best investment schedule is constant and satisfies, if interior, the first-order condition

νp′(I)Δ = 1, which trades off the potential for higher profitability with the costs of

increasing investment.

Non-Contractible Investment. In the remainder of the paper we now consider the full

model where investment is non-contractible and, hence, the manager has to be incentivized

in order to invest in the interest of firm owners. The following Proposition characterizes the

optimal contract in this case, i.e., the solution to the boundary value problem in (12)-(15).

Proposition 2. The optimal truth-telling contract with non-contractible investment takes

the following form:

Optimal investment I(w) as well as the sensitivities α(w) = λ and βj(w), j ∈ {g, b},

are independent of μi, i ∈ {l, h} and chosen as maximizers in (12). The incumbent agent‘s

continuation payoff evolves according to (5) with αt = λ, βjt = βj(wt−), j ∈ {g, b} and

It = I(wt−), ∀t. When wt− ∈ [0, w), dUt = 0; when wt− ≥ w payments dUt cause wt−

to reflect at w. The incumbent agent is replaced when wt− = 0. The principal‘s expected

payoff at any point in time is given by f i(wt), i ∈ {h, l}, which satisfies f(w) := f l(w) =

fh(w) − Δ, where f (w) is concave, strictly so for 0 ≤ w < w and solves, for w ∈ [0, w],

the HJB equation in (12) subject to the boundary conditions (13) to (15).

As the basic cash-flow diversion model is by now well understood (cf. e.g. DeMarzo

and Sannikov (2006) and the discussion following Proposition 1 above), in the following

Section 3.3 we focus on the investment task and discuss in detail the optimal investment

schedule, as well as the optimal choice of reward and punishment used to incentivize these

non-contractible investment expenditures.

3.3 Optimal Investment

When investment expenditures are non-contractible, incentive compatibility requires that

the optimal contract contains a certain degree of reward following an investment success,

βg(w), and/or punishment after a failure, βb(w), creating a direct link between the in-

vestment and cash-flow diversion problems. In particular, the need to deviate from the

where the returns to investment (Tobin‘s Q) are reduced by a cash flow diversion problem. Thus, eventhough investment is contractible in their model, it is always distorted below first-best. In contrast, inour benchmark model with contractible investment, both the return to investment, reflected by Δ, as wellas its costs, I, are independent of the agency problem and investment is equal to first-best.

17

first-best choices of these sensitivities (cf. Proposition 1) causes additional agency costs

relative to the contractible investment benchmark, as it is no longer possible to perfectly

smooth the costs of compensating the agent across investment outcomes. Intuitively, since

firm owners’ are effectively risk-averse with respect to variation in the agent‘s continuation

value w, as reflected in the concavity of f(w), making the manager‘s compensation contin-

gent on the investment outcome is costly. In determining the optimal level of investment,

the principal, thus, trades off the potential for higher profitability with these agency costs

of investment.18

To see this formally, note that from (12) we can write the principal‘s problem of finding

the optimal investment level as follows:

I(w) ∈ arg maxI∈[0,I]

{νp(I)Δ − I − Φ(w, I)} , (18)

with Φ(w, I) =

{νp(I)

[f(w) + βg(w)f ′(w) − f(w + βg(w))

]

+ν(1 − p(I)

)[f(w) + βb(w)f ′(w) − f(w + βb(w))

]

}

(19)

where βg(w) and βb(w) are chosen optimally, minimizing Φ(w, I). The first term on the

right-hand side in (18) gives the technological benefit and cost of investment, while Φ(w, I)

captures the agency costs of investment. In particular, since f(w) is concave, both terms

in square brackets in (19) are strictly positive whenever non-zero βg(w) and βb(w), as are

required by incentive compatibility, induce additional variation in the agent‘s continuation

value w.

In the following, we now first study the optimal incentives for investment, and the

wedge between first-best investment IFB and optimal investment I(w) for a given value

of the agent‘s continuation utility w. Then, we analyze the implied compensation and

investment dynamics when w evolves as described in Proposition 2.

Optimal Incentives for Investment. Recall that to provide incentives for a given

value of investment the sum of reward and punishment, βg − βb, has to satisfy incentive

compatibility constraint (7). Optimizing the principal‘s value subject to this restriction

18Note that in our model investment will usually be distorted away from its first-best value, which canalso be viewed as a component of agency costs. Still, when referring to the “agency costs of investment”we mean the agency costs for a given level of investment, arising from the fact that the principal can nolonger perfectly smooth the costs of compensating the agent across profitability states.

18

implies from (12) that interior values of βg(w) and βb(w) minimize the agency costs of

investment Φ(w, I(w)) and, thus, have to satisfy the following first-order condition:

p (I(w)) [f ′(w) − f ′(w + βg(w))] = (1 − p (I(w)))[f ′(w + βb(w)) − f ′(w)

], (20)

equalizing the expected costs of providing investment incentives through reward (left hand

side) with the expected costs of providing incentives through punishment (right hand

side). Intuitively, while the principal would like to keep marginal compensation costs,

f ′(w), constant across investment outcomes (which would require βg = βb = 0 as in the

benchmark case with contractible investment characterized in Proposition 1), the best that

he can do without violating incentive compatibility is to keep them constant in expectation.

From (20) together with the (strict) concavity of f(w) it is then immediate that, for

w ∈ (0, w), both punishment as well as reward are used to incentivize interior investment

levels.

The relative importance of punishment and reward, however, depends on the imple-

mented level of investment I(w): Assume that investment and, thus, the probability of

success, p(I) increases. Then, (20) implies that the relative use of reward and punishment

optimally adjusts such that the costs of providing incentives through reward decrease rel-

ative to the costs of providing incentives through punishment. That is, to implement a

given investment level, reward and punishment are chosen such that the investment out-

come that is more likely to occur is less costly in terms of incentive provision. As optimal

investment and the respective incentive compensation are jointly determined in equilib-

rium, to formalize this result for a given value of the state variable w, we consider different

sets of parameters ψ ∈ {μh, μl, ν, λ, r, γ, σ, k} which map into different values of I(w).19

Lemma 4. Fix w ∈ (0, w) and consider two sets of parameter values ψ′ and ψ′′, such

that I(w)|ψ′ < I(w)|ψ′′ under the optimal contract of Proposition 2. Then, as long as the

limited liability constraint does not bind, rewards for investment success and punishments

for investment failure are chosen such that the ratio of the costs of providing incentives

through punishment relative to the costs of providing incentives through reward is strictly

19As will be shown below, optimal investment I(w) also varies with w such that the following compar-ative statics similarly hold fixing ψ and considering different levels of w.

19

increasing in I(w)|ψ, i.e.,

f ′(w + βb(w)) − f ′ (w)

f ′(w) − f ′(w + βg(w))

∣∣∣∣ψ′

<f ′(w + βb(w)) − f ′ (w)

f ′(w) − f ′(w + βg(w))

∣∣∣∣ψ′′

.

Further, βg(w) → 0 as I(w)|ψ → I, while for I(w)|ψ → 0, as the agency problem vanishes,

−βb(w)/βg(w) → 0, i.e., the first unit of investment is incentivized with rewards only. For

all I(w)|ψ /∈{0, I}, both reward and punishment are used, i.e., βg(w)|ψ > 0 > βb(w)|ψ.

Intuitively, the less likely signal realization is more informative about the agent‘s action

and, thus, predominantly used for incentive provision. Hence, the principal‘s optimal

compensation policy implies that, if he wants to implement a high level of investment, an

investment failure is particularly costly in terms of incentive provision, while for low values

of investment an investment success is the more costly investment outcome. This relation

between the implemented level of investment and the costs of providing incentives through

punishment relative to reward will be key in understanding the distortions in investment

which we will analyze next.

Investment Distortions. In order to determine whether the firm invests too much or

too little relative to the (first-best) benchmark with contractible investment expenditures,

it is crucial to understand how the agency costs of investment in (19) change in I, i.e.,

∂Φ(w, I)/∂I . If these marginal agency costs of investment are positive, investment will be

distorted downwards relative to first-best, while it will be distorted upwards if marginal

agency costs are negative. We will show that the interaction between cash-flow diversion

and investment gives rise to agency costs which are non-monotonic in I such that both

under- as well as overinvestment can arise in equilibrium.

It is instructive to look, first, at the cases where the problem in (16) has a corner

solution. Clearly, when the investment technology is very unprofitable, such that, in the

extreme case, IFB = 0, the principal can trivially implement the first-best investment level

without incurring any agency costs as there is no need to provide incentives. However,

also IFB = I can be implemented without inducing any additional volatility in w. In fact,

as the probability of success then is p(I) = 1, from Lemma 4, all incentives are optimally

provided “off-equilibrium” by relying on punishment only, as long as this does not violate

limited liability. But agency costs, as given in (19), are zero for βg(w) = 0 and p(I) = 1,

20

such that optimal investment is equal to the first-best level.

Lemma 5. Fix w ∈ [0, w]. Then, as long as the limited liability constraint does not

bind, investment under the optimal contract of Proposition 2 is equal to first-best whenever

first-best investment attains a corner solution, i.e., I(w) = IFB for IFB ∈ {0, I}.

While implementing first-best investment does not cause any agency costs if IFB = 0,

or if IFB = I, agency costs to incentivize IFB are strictly positive for interior values which,

from Lemma 4, call for both reward, βg > 0, as well as punishment, βb < 0. Since

agency costs to incentivize first-best investment are strictly positive if IFB is interior, and

zero when IFB attains a corner solution, marginal agency costs are positive if first-best

investment is sufficiently low and they are negative if first-best investment is sufficiently

high. As a result, investment under the optimal contract will be distorted downwards in

the former and upwards in the latter case.

Proposition 3. Fix w ∈ (0, w). Then, as long as the limited liability constraint does

not bind, investment is distorted downwards, I(w) < IFB, for sufficiently low values of

IFB > 0, while it is distorted upwards, I(w) > IFB, for sufficiently high values of IFB < I.

Investment is, thus, distorted in order to reduce the probability of the more costly

signal realization. To see this, recall from Lemma 4 that the cost of providing incentives

through punishment relative to the cost of providing incentives through reward is increasing

in the implemented level of investment. Therefore, if the investment technology is very

profitable, first-best investment is very high and, accordingly, an investment failure very

unlikely. Since punishment for failure would be very costly in this case, it is optimal to

increase I(w) above the first-best value in order to further reduce the probability of failure.

The opposite case applies if the investment technology is rather unprofitable such that

first-best investment and the probability of success is low. The costs due to rewarding

the manager for success would then be particularly high, investment I(w) is optimally

distorted downwards below first-best investment in order to further reduce the probability

of having to bear these high incentive costs.

Compensation and Investment Dynamics. So far, we studied how investment dis-

tortions and the incentive scheme used to implement investment depend on the profitability

of the investment technology as captured by IFB. For this analysis we kept the agent‘s

21

continuation value fixed. In the following we will now analyze the resulting dynamics of

investment and compensation when w evolves as described in Proposition 2. In order to

do so, it is useful to look, first, at extreme values of the state space, i.e., w = w and w = 0.

As the agent‘s continuation value reaches the compensation boundary (w = w), he has

accumulated so much wealth inside the firm that the agency problem is relaxed sufficiently

for firm owners’ effective risk-aversion to disappear. In fact, rewards are taken out as cash

payments and do not induce costly variation in w (cf. Proposition 2). Hence, it is optimal

not to punish the agent and to rely exclusively on rewards to incentivize investment. As

a result, agency costs of investment are zero and investment equals first-best. 20

For w < w the principal‘s value function is strictly concave such that it is optimal

to use both rewards and punishments according to the optimal compensation policy in

(20). However, for low values of the agent‘s continuation payoff, punishment is eventually

restricted by the binding limited liability constraint (8). In fact, in the limiting case, as we

approach the termination boundary (w = 0), only rewards can be used to incentivize any

I > 0, as the agent is “too poor to be punished.” Hence, incentives have to be provided

mainly through rewards such that an investment success is the investment outcome that

is particularly costly in terms of incentive provision. As a result, investment will be

optimally distorted downwards for all IFB ∈ (0, I). These observations are summarized in

the following Lemma.

Lemma 6. Fix IFB ∈ (0, I). Then under the optimal contract of Proposition 2

i) There exists a threshold wβ ∈ (0, w) such that the limited liability constraint for βb(w)

binds on w ∈ [0, wβ] and the agent is instantly fired following a failure. Further,

investment is distorted downwards I(w) < IFB for w sufficiently small.

ii) For w ∈ (wβ, w) incentives are provided through punishment and reward, which are

optimally chosen according to (20). Investment may be distorted upwards or down-

wards as shown in Proposition 3.

iii) For w = w, incentives are provided through rewards only and investment is equal to

first-best, I(w) = IFB.

20To see this formally, substitute w in (18) and note that f (w) extends linearly to the right of w.

22

Figure 2 illustrates the compensation and investment dynamics implied by Lemma 6 in

an equilibrium with overinvestment (high IFB in left panels), and in one with underinvest-

ment (low IFB in right panels). The two scenarios are generated by varying the “returns

to investment” Δμ := μh − μl, while all other parameters and functional forms are kept

constant. Note that Δμ is a purely technological parameter in that it does not affect the

agency problem for a given investment schedule.21

As shown in Lemma 6, there is no punishment (βb(w) = 0) if the agent‘s track record

is very poor (w → 0) or if it is very good (w → w), while βb(w) is strictly negative

in-between. The upper panels of Figure 2 illustrate a case where the punishment policy

βb(w) ≤ 0 is in fact U-shaped in the agent‘s continuation value w. The lower the degree

of punishment, keeping all else constant, the more generous rewards are needed to satisfy

incentive compatibility. Notably, this is true independently of whether the lack of punish-

ment is imposed by the binding limited liability constraint (for low w), or if it is in fact

optimal not to punish (for high w). As a result, also the reward policy βg(w) ≥ 0 in our

numerical example is U-shaped in w.

Turning to the optimal investment policy I(w) as depicted in the lower panels of Figure

2, note first that investment is distorted downwards for low Δμ (right panel) and upwards

for high Δμ (left panel) given that w is sufficiently large, which corresponds to a slack

limited liability constraint. In both cases I(w) approaches the respective first-best value for

w → w. Hence, in the equilibrium with overinvestment (left panel), investment decreases

in w, provided the limited liability constraint is not binding, while, in the equilibrium with

underinvestment (right panel) investment increases with the agent‘s continuation value.

Since changes in w track the firm‘s realized cash flows dY , this implies that investment-

cash flow sensitivities can be positive as well as negative, depending only on technological

parameters, in particular the returns to investment. These findings will be analyzed more

systematically and related to empirical evidence in Section 4.2.

4 Implementation and Empirical Implications

In this section, we will first discuss the implementation of the optimal contract, in partic-

ular, its implications for the firm‘s financial slack. Then, we use this implementation to

21I.e., if one had to implement in both cases the same (exogenously given) investment schedule, thenthe optimal contract would be identical, despite the difference in Δμ.

23

5 15 -1

-0.5

0

0.5

1

5 15 -0.2

0

0.2

0.4

0.6

5 15 0.6

0.7

0.8

0.9

1

1.1

5 15 0.19

0.2

0.21

0.22

0.23

0.24

Figure 2: The upper panels illustrate the punishment and reward dynamics and the lowerpanels the investment dynamics under the optimal contract for low Δμ = 1.1 and for highΔμ = 2.2. We stipulate p(I) = φ

√I and the parameter values used for calibration are:

ν = 1.2, σ = 10, λ = 0.5, k = 15, γ = 0.15, r = 0.1, φ = 0.95.

derive empirical implications regarding the relation between investment expenditures and

available internal funds, how this relation is affected by financial frictions, the impact of

corporate governance on investment, as well as the relation of investment to the structure

of executive compensation.

4.1 Implementation of the Optimal Contract

Building on ideas that are well established in the literature on dynamic financial contract-

ing, the optimal contract as characterized in Section 3 can be be implemented in several

ways (see, e.g., DeMarzo et al. (2012) for a discussion). The robust feature that these im-

plementations have in common is that they interpret the key state variable w as a measure

of the firm‘s financial slack. To see this relation, recall that the dynamics of our optimal

contract, as characterized in Proposition 2, are governed by the evolution of the agent‘s

promised utility w. When w falls to zero, this triggers a restructuring of the firm, which

is costly for its owners and involves the replacement of the incumbent management. As,

24

for incentive reasons, w moves with cash flows with sensitivity λ, the maximum loss that

a firm can sustain without triggering a restructuring, i.e., its financial slack, is equal to

w/λ. While this insight can be formalized in a variety of ways, for illustrative purposes,

in the following we consider one particular implementation of the optimal contract based

on cash reserves as a measure of financial slack, together with equity and a portfolio of

derivative securities or insurance contracts.

More precisely, as in DeMarzo et al. (2012), the firm is equity financed and uses cash

reserves to cover its short-term liquidity needs. Let Mt denote the level of cash reserves,

earning interest r. Cash reserves grow if cash flows are positive and they are used to cover

operating losses, which corresponds to negative cash flows. Further, equity holders’ require

a minimum dividend, given by

dDt = [μt − It − (γ − r) Mt] dt, (21)

which is paid out of the cash reserves Mt. This minimum dividend comprises of the

expected free cash flow μt − It minus an adjustment factor that reflects the discounting

difference γ − r. If the firm fails to meet the minimum payout rate (21), or the cash

holdings are exhausted, the manager is laid off, which is critical in providing incentives

for the manager not to divert cash flows. Incentives for investment can then be provided

by creating exposure of the firm’s financial slack Mt to the investment outcome, which

we formalize through a portfolio of derivative securities contingent on the investment

outcome.22 Derivatives are fairly priced given investors’ beliefs on the level of investment

It. Concretely, we stipulate that holding a state-price security that pays one unit in case

of an investment success (failure) incurs flow costs of P g = νp (It) (P b = ν (1 − p (It))), so

that the instantaneous net payoff from holding such a security is given by dSjt = dN j

t −P jt dt,

for j ∈ {g, b}. We denote the number of securities of type j ∈ {g, b} held by the firm at

22Note that we stipulate that the investment outcome is a verifiable event. In our interpretation ofinvestment into absorptive capacity (c.f., footnote 5) an investment success could, e.g., be a patent grantedto the firm and an investment failure a patent granted to a competitor. In this case, successes and failureswould be reflected in the firm‘s stock price, implying that the required exposure could be created byderivatives based on this underlying. For an alternative implementation in a setting with only downsiderisk see Biais et al. (2010). There the firm is requested to maintain an insurance contract against accidentcosts which, for incentive reasons, entails only partial coverage and a down-sizing covenant.

25

time t by njt , and require the firm to hold at any time a portfolio of size

njt =

βj (λMt)

λ, j ∈ {g, b} , (22)

of the respective security.23 Else, the manager is replaced. Other than that, the manager

is free to choose investment and to distribute cash in form a special dividend X at any

time. The manager receives compensation in form of fraction λ of this special dividend.

Overall, the firm‘s cash reserves, thus, follow

dMt = rMtdt + dYt + ngt dSg

t + nbtdSb

t − dDt − dXt. (23)

When Mt hits zero for the first time, the firm can no longer pay the minimum dividend

dD and, thus, goes into restructuring. In this process the incumbent manager is fired and

equity holders realize a payoff corresponding to Lτ , where τ denotes the first time at which

Mt falls to zero. The value of the firm’s equity claim is then given by

P (Mt, μt) = Et

[∫ τ

t

e−r(s−t) (dDs + (1 − λ) dXs) + e−r(τ−t)Lτ

]

, (24)

and we have the following result:

Proposition 4. Suppose the firm has initial cash reserves M0 and can operate as long as

Mt ≥ 0. When the manager is fired unless he maintains the minimum payout rate dDt and

holds a derivative security portfolio ngt , nb

t , it is optimal for him to refrain from diverting

funds and to implement the optimal investment profile as characterized in Proposition 2.

The firm accumulates cash reserves Mt until Mt = w/λ, and pays out all cash in excess

of this amount. Given this policy, the manager‘s payoff is wt = λMt, which coincides

with the continuation value of Proposition 2, and the equity value satisfies P (Mt, μt) =

f(λMt, μt) + Mt.

Note that in the implementation given in Proposition 4, the state-price securities are

not used to hedge against investment failure. By contrast, although firm value is a concave

function of financial slack, the firm’s derivative position deliberately creates exposure of its

financial position to the uncertain investment outcome in order to provide the appropriate

23Note from βb < 0 (c.f., (20) and the subsequent discussion), the firm holds a short position in securityb, i.e., nb

t < 0.

26

incentives for investment. Let us also comment a bit more on the interpretation of the

restructuring process triggered when Mt = 0. In this event, which could be interpreted as

insolvency, the firm needs to raise cash from the capital market, which involves a fixed cost

of k. With the equity value net of the required cash injection given by f(λMt, μt), the firm

continues to operate with a new manager and initial cash reserves of M∗ = w∗/λ. Hence,

we have P (0, μ) = P (M∗, μ)−k, where we interpret k as the cost of raising external funds,

which captures the key financing friction in our model.

4.2 Empirical Implications

In this section, we compute cross-sectional averages for different corporate figures in the

context of the implementation studied in Section 4.1 and derive a number of novel testable

implications. We employ a Monte Carlo simulation to compute the cross-sectional averages

of equilibrium investment, INVt := I(λMt) = I(wt), and the elasticity of investment to

cash holdings,

ICSt :=∂ ln I(λMt)

∂ ln Mt

,

the “investment-cash sensitivity.”24

Since the firm in our implementation operates forever, cross-sectional (“ensemble”)

averages of INV and ICS can be obtained by computing the corresponding path-wise

long-term averages25

INV :=1

T

T∑

t=0

It, (25)

ICS :=1

T

T∑

t=0

ICSt, (26)

with T sufficiently large. For the numerical analysis below we use a total number of T = 107

periods with period length h = 10−3. We further specify p(I) = φ√

I, with φ > 0 such

that I ∈[0, 1/φ2

]. For more details and an outline of the employed algorithm, we refer to

24ICS, thus, refers to the percentage change in investment induced by a percentage change in cashholdings, whereas the empirical literature usually considers investment sensitivities with respect to cashflows (i.e., the absolute change in cash holdings). While elasticities better capture the fact that under theoptimal contract investment (and its derivative) depends on financial slack, i.e., cash holdings, and notonly on cash flows, our qualitative results still hold true if cash flows would be considered instead.

25This ergodicity property is exploited in a similar context also by Brunnermeier and Sannikov (2014)or Klimenko et al. (2016).

27

Appendix B. Based on the analysis in Section 3, we consider the cross-sectional averages

for subsets of firms (e.g. industries), which differ in the returns to investment as measured

by the increase in expected cash flows due to successful investment, Δμ = μh − μl. We

vary Δμ between 0 and an upper limit that guarantees an interior solution to (18) for all

w ∈ (0, w).

0 0.5 1 1.5 2

-0.02

-0.01

0

0.01

0.02

0.03

0.04=.1=.5=.9

0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.510-3

k=5k=15k=30

Figure 3: Plots average investment distortions (left panel), and average elasticity of invest-ment with respect to cash (right panel) as a function of the industry returns to investmentΔμ. We stipulate p(I) = φ

√I and the parameter values used for calibration are: ν = 1.2,

σ = 10, λ = 0.5, k = 15, γ = 0.15, r = 0.1, φ = 0.95.

Investment distortions. The left panel of Figure 3 shows how average industry in-

vestment varies with the returns to investment in the industry, Δμ. In particular, the

investment distortion relative to the contractible investment benchmark is negative for the

average firm in a low returns to investment industry while it is positive when returns to

investment in the respective industry are high.

Implication 1. Compared to average investment in otherwise similar owner-manager

firms, average (non-contractible) investment is

i) distorted downwards in industries with low returns to investment, and

ii) distorted upwards in industries with high returns to investment.

A common presumption in much of the managerial short-termism literature is that

the manager is biased towards short-term performance and, thus, neglects (hard to ob-

serve) long-term investment. Varying the diversion benefits λ, the left panel of Figure

28

3 further illustrates that, under a long-term contract that acknowledges such distorted

incentives, the relation between compensation for short-term performance and long-term

investment can also be positive, in line with empirical evidence.26 As the agency problem

becomes more severe, more compensation for short-term performance is needed to induce

truth-telling (α = λ). In our multitask setting, then also the exposure to the investment

outcome needs to increase to incentivize a given level of investment. As a consequence,

investment distortions go up, which implies either more severe overinvestment or more

severe underinvestment, depending on the returns to investment.

Implication 2. The relation between compensation for short-term performance and long-

term investment is

i) negative in industries with low returns to investment, and

ii) positive in industries with high returns to investment.

Given that better corporate governance reduces the manager‘s potential to divert funds

for private consumption (lower λ), Implication 2 can be interpreted also as a statement on

the relation between corporate governance and long-term investment.27 Our model there-

fore describes a novel channel relating the impact of corporate governance on investment to

industry characteristics, which does not assume any direct costs or benefits of investment

on the manager’ s side.

Investment-Cash (Flow) Sensitivities. The right panel of Figure 3 plots the average

investment-cash sensitivity ICS implied by the optimal contract as a function of the

returns to investment Δμ. Since the agency problem is mitigated as the manager‘s “stake”

in the firm grows, investment distortions relative to the constant first-best level decrease

with positive firm performance and, thus, the level of financial slack. As a consequence,

the sign of ICS is determined by the sign of the average investment distortion as reported

in Implication 1.

26While some empirical studies report a positive relation between compensation schemes such as stock,stock options or bonuses and long-term investment such as R&D (Barker and Mueller, 2002; Coles et al.,2006), others find a negative relationship (e.g., Holthausen et al., 1995).

27In line with this implication, Brav et al. (2016) find a negative relation between hedge fund activismand R&D expenditures in their sample of high-tech firms, while Aghion et al. (2013) report a strongpositive relation between institutional ownership and R&D. The role of incentive pay as a substitute forcorporate governance is documented, e.g., in Core et al. (1999) or, more recently, Fahlenbrach (2009).

29

Implication 3. The relation between investment and internal funds (cash) is

i) positive in industries with low returns to investment, and

ii) negative in industries with high returns to investment.

That investment in our model changes with financial slack and, thus, also with cash

flows, is a consequence of endogenous financial frictions arising from the agency problem.

It is therefore useful to relate Implication 3 to the large empirical literature on investment-

cash flow sensitivities as a measure of financial constraints. The question whether a (pos-

itive) investment-cash flow sensitivity is a good measure of financial constraints, as put

forward by Fazzari et al. (1988), has been repeatedly challenged in the literature start-

ing with Kaplan and Zingales (1997). Empirical studies that analyze the time-series

dimension of investment-cash flow sensitivities have found them to decline steadily over

the last decades, thus, casting further doubts on their suitability as a measure of finan-

cial constraints.28 Some authors argue that for physical investment, the decline can be

explained by a shift to intangible investment, such as R&D, which has been excluded from

most existing studies (c.f., Brown and Petersen 2009). However, the fact that also for

intangible investment such as R&D the cash flow sensitivities are insignificant or negative,

leads Chen and Chen (2012) to conclude that “(i)f one believes that financial constraints

have not disappeared, then investment-cash flow sensitivity cannot be a good measure

of financial constraints. The decline and disappearance are robust to considerations of

R&D ... and remain a puzzle.” According to Peters and Taylor (2016), who show that

investment-cash flow sensitivities are negative in high-intangible investment firms, the rise

of intangible investment may explain the disappearance of average sensitivities. 29 How-

ever, there does not exist a theory explaining why one should expect sensitivities to be

negative in high-intangible investment firms. Our model can be viewed as a first step

towards such a theory and shows that the empirical results can be consistent with optimal

dynamic financial contracting and endogenous financial constraints.

28See, e.g., Agca and Mozumdar (2008) for evidence on declining investment-cash flow sensitivities.Campello et al. (2010) provide survey evidence that financial constraints were still a first order issue forAmerican firms at the end of 2008. Similar results can be found in Almeida et al. (2012).

29Similarly, Hovakimian (2009) finds that firms with negative investment-cash flow sensitivities havehigher R&D expenditures, higher market-to-book ratios, and lower asset tangibility than firms with posi-tive investment-cash flow sensitivities.

30

In our model, a firm‘s financial slack reduces the probability with which the firm has

to raise costly external funds and, thus, relaxes financial constraints relative to the given

costs of external financing k. Varying k then allows us to conduct comparative statics with

respect to the severity of financial for any given level of cash holdings. We find that, if the

costs of external financing increase, investment becomes more sensitive to cash flows. As

illustrated in the right panel of Figure 3, investment-cash sensitivities increase in absolute

terms with financing frictions, but the sign and strength of this relation depends on the

industry returns to investment Δμ.

Implication 4. The relation between the costs of external financing and investment-cash

sensitivities is

i) positive in industries with low returns to investment, and

ii) negative in industries with high returns to investment.

Implication 4 provides a novel perspective on the precise sense in which investment-

cash flow sensitivities should be expected to increase with the degree of a firm‘s financial

constraints. In particular, our model predicts that investment should indeed become more

sensitive to cash flows as financial constraints become more severe, however, in absolute

terms. Hence, since investment-cash flow sensitivities are negative when the returns to

investment are high, an increase in the costs of external funds leads to even lower sensi-

tivities in such industries (see Peters and Taylor 2016 for empirical evidence for the case

of intangible investment). Using sensitivities as a measure for financial constraints, thus,

requires, in light of our model, to control for this dependence on the returns to investment.

5 Conclusion

In this paper, we analyze the dynamics of corporate investment under endogenous financial

constraints in a dynamic agency model with multitasking. The manager privately observes

the firm‘s cash flows which he can divert for private consumption and, in addition, use to

finance non-contractible investment in the firm‘s future profitability. Thus, when reported

cash flows are low (or even negative), the principal does not know whether this is because

of stealing or because of investment. To ensure truth-telling and investment in the interest

of firm owners the optimal compensation scheme ties the manager‘s compensation both

31

to reported cash flows as well as to imperfect performance signals indicating investment

success or failure. Exposing the manager to compensation risk is, however, costly such

that incentivizing higher investment causes additional agency costs.

The optimal investment profile trades off these agency costs of investment with the

potential efficiency gains. As a result, investment will be history dependent and distorted

away from its first-best level. Investment distortions decrease (in absolute terms) with

good performance, as the manager‘s stake in the firm increases mitigating the agency

problem. Our model predicts overinvestment and negative investment-cash flow sensitivi-

ties in industries where returns to non-contractible “intangible” investment such as R&D

are high (e.g., as being in a position of technological leadership is important like in the

pharmaceutical industry). By contrast, in industries where this is not the case, investment

will be distorted downwards and increasing in cash flows.

We provide an implementation of the optimal contract, in which the history dependence

of the optimal contract is captured by the firm‘s financial slack. The firm retains earnings

to build up a cash buffer, that is used to cover investment expenditures, potential operating

losses, and payments to investors. If cash holdings fall to zero, the firm is no longer able to

honor its payments to investors, triggering a costly restructuring of the firm, which involves

replacing the incumbent manager and raising new capital. As cash holdings increase,

financial constraints are relaxed and investment distortions decrease. If raising external

funds is more expensive, investment will be more severely distorted and more sensitive to

past performance, for any given level of financial slack; i.e., in firms with high returns to

investment, investment increases and the relation between financial slack and investment

becomes more negative, while investment decreases and its sensitivity to financial slack

becomes more positive in firms with low returns to investment. Our multitask theory,

thus, offers a new perspective on the interpretation of investment cash-flow sensitivities

as a measure of financial constraints, taking into account the increased importance of

alternative forms of investment such as process innovation, R&D, or intangible investment

in general.

32

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36

Appendix A Omitted Proofs

Proof of Lemma 1. For S = S∗, we define the manager’s expected lifetime utility

evaluated conditional on time t information by:

vt =

∫ t

0

e−γsdUs + e−γtwt, (A.1)

which, by construction, is a martingale with respect to the filtration generated by (Z,N j),

j ∈ {g, b} under the probability measure QS∗. By the martingale representation theorem

(cf. Theorem III 4.34 in Jacod and Shiryaev 2003), we can express vt for some predictable

processes α and βj , j ∈ {g, b} as

vt = v0 +

∫ t

0

e−γsαs

(dYs − (μs − I∗

s ) ds)

+

∫ t

0

e−γsβgs (dN g

s − νp(I∗s )ds)

+

∫ t

0

e−γsβbs

(dN b

s − ν (1 − p(I∗s )) ds

),

where, under QS∗, Yt−

∫ t

0(μs − I∗

s ) ds is a standard Brownian motion and N gt −∫ t

0νp(I∗

s )ds

as well as N bt −∫ t

0ν (1 − p(I∗

s )) ds are compensated Poisson processes. Differentiating both

this representation and the definition of vt in (A.1) yields (5). Q.E.D.

Proof of Lemma 2. Consider any feasible policy of the agent, S ={

Yt, It : 0 ≤ t ≤ τ}

,

with dYt − dYt ≥ 0 and It ≥ 0. The associated expected lifetime utility is given by

vt = v0 +

∫ t

0

e−γsαs

(dYs − (μs − I∗

s ) ds)

+

∫ t

0

e−γsλ(dYs − dYs

)(A.2)

+

∫ t

0

e−γsβgs (dN g

s − νp(I∗s )ds) +

∫ t

0

e−γsβbs

(dN b

s − ν (1 − p(I∗s )) ds

),

where dYt − (μt − I∗t ) dt = σdZt for S = S∗, and dN g

t (dN bt ) is a Poisson process with

intensity νp(It) (ν (1 − p(It))). Differentiating (A.2) and taking expectations gives

eγtE [dvt] = (λ − αt) E[dYt − dYt

]+ αt (I∗

t − It) dt + ν(βg

t − βbt

)(p(It) − p(I∗

t )) dt.

Clearly, vt is a martingale for S = S∗. For S = S∗ to be incentive compatible, the drift of

vt has to be non-positive for all possible deviations, i.e., vt has to be a supermartingale for

any feasible S 6= S∗. Consider, first, a deviation from truth-telling, i.e., underreporting for

consumption dYt − dYt > 0. This deviation is suboptimal for the agent if αt ≥ λ. Second,

increasing It marginally above I∗t , is suboptimal for the agent if αt ≥ νp′(It)

(βg

t − βbt

),

37

while, third, decreasing It marginally below I∗t , is suboptimal if αt ≤ νp′(It)

(βg

t − βbt

). So,

incentive compatibility requires (6) and (7) to hold. Q.E.D.

Proof of Lemma 3. See main text. Q.E.D.

Proof of Proposition 1. This result is a straightforward extension of Hoffmann and

Pfeil (2010), who study a dynamic cash-flow diversion models with exogenous shocks to

profitability; we therefore will be brief. Note, first, that fCI(w) is strictly concave, which

follows from the same arguments as in Hoffmann and Pfeil (2010). Further, the incentive

constraint binds, i.e., α = λ. Interior solutions for I are then given by

Δ +[fCI(w + βg) − fCI(w + βb)

]−(βg − βb

)f ′

CI(w) =1

νp′(I), (A.3)

while the βj , j ∈ {g, b}, are determined from f ′CI(w+βg) = f ′

CI(w) = f ′CI(w+βb). Due to

the strict concavity of fCI this immediately implies that βg = 0 and βb = 0, and, plugging

into (A.3), we find that optimal investment is equal to the first-best level as characterized

in (16), such that the proposed solution achieves efficient investment. Verification is then

standard (c.f., Hoffmann and Pfeil 2010) and therefore omitted. Q.E.D.

Proof of Proposition 2. We first show that the function f(w) is concave in w and

then verify that the contract of Proposition 2 maximizes the principal‘s expected payoff.

Concavity. Consider the function f(w) that, for a given w and μ, solves

(r + ν) f(w) =

μ + νp(I (w))Δ − I (w) + 12(α (w))2σ2f ′′(w)

+[γw − ν

(βg (w) p(I (w)) + βb (w) (1 − p(I (w)))

)]f ′(w)

+νp(I (w))f (w + βg (w)) + ν (1 − p(I (w))) f(w + βb (w))

(A.4)

for w ∈ [0, w] and f (w) = f(w) − (w − w) for w > w, with boundary conditions f (0) =

maxw {f (w) − k}, f ′ (w) = −1, and f ′′ (w) = 0.

Investment I (w) satisfies the first order condition

1 =

νp′(I (w))(βg (w) − βb (w)

)fw(w)

+νp′(I (w))[f(w + βg (w)) + Δ − f(w + βb (w))

]

+α (w) p(I (w)) p′′(I(w))p′(I(w))2

[f ′(w) − f ′(w + βg (w))]

, (A.5)

and βi (w) , i ∈ {g, b} is determined by

f ′(w) = p(I (w))f ′(w + βg (w)) + (1 − p (I (w))) f ′(w + βb (w)) (A.6)

38

for βi (w) ≥ −w while βi (w) = −w otherwise.

Differentiating (A.4) and using (A.5) and (A.6) yields

− (γ − r) f ′ (w) =

[

γw − ν

(α (w) p(I (w))

νp′(I (w))+ βb (w)

)]

f ′′ (w) +1

2σ2 (α (w))2 f ′′′ (w) .

From the boundary conditions at w we getf ′′′(w) = 2 γ−r

σ2α(w)2> 0, such that ∃ε > 0 with

f ′′′(w − ε) > 0 and f ′′(w − ε) < 0.

The proof then is by contradiction. So assume that ∃w := sup{w < w : f ′′(w) ≥ 0},

where it holds by continuity that f ′′(w) = 0 and f ′′′(w) < 0, implying that f ′(w) =

−f ′′′(w)12

σ2α(w)2

γ−r> 0. Now, consider two points w1 < w < w2 close to w, such that

f ′′(w1) > 0 > f ′′(w2) and w1f ′(w1) = w2f ′(w2) and observe that f(w) can be written as

rf(w) = γwf ′(w) +1

2σ2 (α (w))

2

f ′′(w) + g(w),

with

g (w) = μ − I (w) + ν

p(I (w))(f(w + βb (w) + α(w)

νp′(I(w))) + Δ

)

+ (1 − p(I (w))) f(w + βb (w)) − f(w)

−(

α(w)p(I(w))νp′(I(w))

+ βb (w))

f ′(w)

.

Next, compute the differential of g (w) around w, dg (w)|w=w = g′(w)dw, and observe that

g′(w) = Iw (w, μ)

−1 + νp′(I (w))[f(w + βg (w)) + Δ − f(w + βb (w))

]

−α (w) f ′(w)

+α(w)p(I(w))p′′(I(w))p′(I(w))2

[f ′(w) − f ′(w + βg (w))]

+ν(1 + βb

w (w))[

p(I (w))f ′(w + βg (w))

+ (1 − p(I (w))) f ′(w + βb (w)) − f ′(w)

]

−ν

(α (w) p(I (w))

νp′(I (w))+ βb (w)

)

f ′′(w)

= 0,

which follows from f ′′(w) = 0 together with (A.5) and either (A.6), or, in case the limited

liability constraint binds, βbw (w) = ∂βb (w) /∂w = −1. Thus, evaluating f(w) in w1 and

w2, we get

r[f(w1) − f(w2)

]=

1

2σ2α (w)2 [f ′′(w1) − f ′′(w2)

]> 0,

where we have used that the effect of a change in g(w) around w is of second order and will

thus be dominated by the change in f ′′(w). However, this directly contradicts f ′(w) > 0.

39

Verification. For any incentive compatible contract (S, U , τ ) , define

Gt =

∫ t

0

e−rs (dYs − dUs) + e−rtf (wt− , μt)

and recall that wt− evolves according to

dwt− = γwt−dt − dUt + αtσdZt + βgt (dN g

t − νp(It)dt) + βbt

(dN b

t − ν (1 − p(It)) dt),

where αt ≥ λ and βgt − βb

t = αt

νp′(It). Recall that f (w) = f

(w, μl

)= f

(w, μh

)− Δ so that

from differentiating G using Ito‘s lemma and the change in variables formula for point

processes as well as (A.4), we get

ertdGt = ν

(p(It) [f (wt− + βg

t ) + Δ] + (1 − p(It)) f(wt− + βbt)

−[βg

t p (It) + βbt (1 − p (It))

]f ′ (wt−) − It

)

dt (A.7)

−ν

(p(I (w)) [f (w + βg (w)) + Δ] + (1 − p(I (w))) f(w + βb (w))

−[βg (w) p(I (w)) + βb (w) (1 − p(I (w)))

]f ′ (w) − I (w)

)

dt

+1

2

(α2

t − λ2)σ2f ′′ (wt−) dt − [1 + f ′ (wt−)] dUt + [σ + αtσf ′ (wt−)] dZt

+ [f (wt− + βgt ) + Δ] dM g

t + f(wt− + βb

t

)dM b

t − f (wt−)(dM g

t + dM bt

)

for w = wt− ∈ [0, w], where M i denote the compensated point processes associated with

N i, i ∈ {g, b}. Now observe that the sum of the first two lines is less or equal to zero,

because βi (w) , i ∈ {g, b} and I (w) are the solution to

maxβi≥−w,I

[p(I) [f (w + βg) + Δ] + (1 − p(I)) f(w + βb)

−(βgp (I) + βb (1 − p (I))

)f ′ (w) − I

]

.

Now turn to the third line of (A.7). The first term is non positive as f ′′ ≤ 0 and αt ≥ λ

for any t ≥ 0. The second term is non positive as f ′ ≥ −1 and dU ≥ 0 while Z and M i

are martingales. Hence, Gt is a supermartingale and a martingale if and only if for t > 0,

βit = βi (w), It = I (w), αt = λ and dUt > 0 only when w > w.

Now consider the principal‘s expected payoff under any incentive compatible contract

(S, U , τ ) :

E

[∫ τ

0

e−rs (dYs − dUs) + erτLτ ,

]

where τ denotes the time when the incumbent manager is replaced and Lτ the principal‘s

expected profits from restarting with a new agent, net of replacement costs k. We then

40

have that

E

[∫ τ

0

e−rt (dYs − dUs) + erτLτ ,

]

= E [Gt∧τ ] + E

[

1t≤τ

(∫ τ

t

e−rs (dYs − dUs) + e−rτLτ − e−rtf (wt−)

)]

≤ f (w0−) + E

[

1t≤τ

(∫ τ

t

e−rs (dYs − dUs) + e−rτLτ − e−rtf (wt−)

)]

= f (w0−) + e−rtE

[

1t≤τ

(

E

[∫ τ

t

e−r(s−t) (dYs − dUs) + e−r(τ−t)Lτ |Ft

]

− f (wt−)

)]

,

where the inequality follows since Gt∧τ is a supermartingale and G0 = f (w0−). Now note

that

E

[∫ τ

t

e−r(s−t) (dYs − dUs) + e−r(τ−t)Lτ |Ft

]

<μh

r− wt− ,

and, as f ′ ≥ −1, we have that f (wt−) + wt ≥ Lτ , which yields

E

[∫ τ

0

e−rt (dYs − dUs) + erτLτ ,

]

≤ f (w0−) + e−rtE

[

1t≤τ

(μh

r− Lτ

)]

.

Taking t → ∞ yields

E

[∫ τ

0

e−rt (dYs − dUs) + erτLτ ,

]

≤ f (w0−) . (A.8)

Finally, under the contract stated in Proposition 2, Gt∧τ is a martingale and, hence, (A.8)

holds with equality. Q.E.D.

Proof of Lemma 4. Whenever the limited liability constraint (8) does not bind, for

which it is sufficient that w ≥ λνp′(I)

, first order condition (20) holds. The first claim then

follows immediately from rewriting (20) as

f ′(w + βb(w)) − f ′(w)

f ′(w) − f ′(w + βg(w))=

p(I(w))

1 − p(I(w)), (A.9)

and observing that p(I(w))|ψ′ < p(I(w))|ψ′′ since I(w)|ψ′ < I(w)|ψ′′ and p′(∙) > 0. Next,

when I(w)|ψ → I and the limited liability constraint is slack, strict concavity of f(w)

implies that (20) can only be satisfied for βg(w) → 0. In case p(I) → 0, (20) similarly

requires that βb(w) → 0, while, from incentive compatibility, βg(w) → λνp′(0)

. For bounded

p′(0), it then trivially follows that −βb(w)/βg(w) goes to zero as I(w) → 0. Now, if

p′(I) → ∞ as I → 0, both βg(w) and βb(w) go to zero as I → 0. Then, up to a first-order

41

approximation, we have f ′(w+βb(w)) = f ′(w)+βb(w)f ′′(w), and f ′(w+βg(w)) = f ′(w)+

βg(w)f ′′(w). Hence, there exists an ε > 0 such that, from (A.9), we have p(ε)1−p(ε)

= −βb(w)βg(w)

.

Finally, the last claim follows from strict concavity of f(w) for w < w, implying that

(20) can only be satisfied for βg > 0 > βb. Q.E.D.

Proof of Lemma 5. See main text. Q.E.D.

Proof of Proposition 3. Denote the right-hand side of the HJB in (12) by L and take

the first derivative with respect to I, noting that βg = βb + λνp′(I)

from incentive constraint

(7), to obtain

∂L∂I

= νp′(I (w))Δ − 1 −

[(λ

ν

−p′′(I(w))

[p′(I(w))]2

)

δgb (w) + νp′ (I (w))(δg (w) − δb (w)

)]

,

(A.10)

with

δgb (w) := p(I(w)) (1 − p(I(w))) ν[f ′(w + βb(w)) − f ′(w + βg (w))

]≥ 0, (A.11)

δj (w) := f(w) + βj (w) f ′(w) − f(w + βj (w)) ≥ 0, j ∈ {g, b} , (A.12)

where we have used (20), which holds whenever the limited liability constraint does not

bind, and the inequalities in (A.11) and (A.12) follow from strict concavity of f(w) for

w < w. The term in square brackets in (A.10) are the marginal agency costs of investment

which we denote by φ(w).

To show the overinvestment result for sufficiently high levels of IFB it is sufficient to

show that the marginal agency costs of investment are strictly negative if I(w) = IFB = I,

such that ∂L/∂I > 0 for IFB = I. The result then follows from continuity and the fact

that I(w) = IFB is the uniquely optimal investment level for IFB = I. So, note that

p(I) = 1 implies that δgb(w) = 0 and, by Lemma 5, βg(w) = δg(w) = 0, such that

φ(w)|IFB=I = νp′(I)δb(w) < 0, where the inequality follows again from the concavity of

f(w) and the fact that βb(w) > 0 by incentive constraint (7).30

As for the remaining case, underinvestment if IFB is sufficiently low, we will show

that ∂L/∂I |I(w)=IFB→ 0 from below as IFB ↘ 0. To see this note that, in the limit, δg

dominates δb from Lemma 5, such that φ(w) is strictly positive. The result then follows

from continuity and the fact that I(w) = IFB is the uniquely optimal investment level for

IFB = 0. Q.E.D.

30Note also that IFB = I implies from νp′(IFB)Δ − 1 = 0 that p′(I) > 0, which is also a necessarycondition for limited liability not to bind.

42

Proof of Lemma 6. For part i) we first show that there exists wβ ∈ (0, w) such that

βb(w) = −w for w ∈ [0, wβ]. To see this, note that, for any I(w) > 0, we must have

βg(w)− βb(w) > 0, which together with the strict concavity of f(w) for w ∈ [0, w) implies

that the first-order condition in (20) cannot be satisfied in a neighborhood of w = 0. Thus,

we get the stated result, with βb(w) = −w and βg (w) = βb + λνp′(I(w))

> 0 on w ∈ [0, wβ],

where

wβ := min

{

w > 0 : p(I(w))f ′

(λ

νp′(I(w))

)

+(1 − p(I(w))

)f ′(0) = f ′(w)

}

.

Next, let us show the underinvestment result for small w. With the limited liability

constraint binding, marginal agency costs of investment φ(w) in (A.10) can be rewritten

to obtain

φ(w) = νp′(I(w))

[

f(λ

νp′(I(w))) −

(

f(0) +λ

νp′(I(w))f ′(w)

)]

+ λp(I(w))p′′(I(w))

(p′(I(w)))2 [f ′(w) − f ′(w + βg(w))] .

It remains to show that this expression is negative for small w. To see this, note first, that

the expression in the second line is negative for all w by strict concavity of f(w). As for

the remaining right-hand-side expression, it also follows from concavity of f(w) that this

is negative for w small enough.

Part ii) is immediate as for w ∈ (wβ, w) the strict concavity of f(w) together with (20)

directly implies βg(w) > 0 and βb(w) < 0 and, since limited liability constraint (8) does

not bind, Proposition 3 applies.

For part iii), note that as f(w) extends linearly for w > w, we have that f ′(w+βg(w)) =

f ′(w) and (20) can only be satisfied if βb(w) = 0. It remains to show that I(w) = IFB.

Since βg(w) > 0, but f ′(w + βg(w)) = f ′(w), and βb(w) = 0, the marginal agency costs of

investment φ(w) in (A.10) are equal to zero, implying first-best investment. Q.E.D.

Proof of Proposition 4. Under the proposed implementation, cash reserves evolve

according to

dMt = γMtdt +(dYt − (μt − It) dt

)+

βg (λMt)

λ(dN g

t − νp (It) dt)

+βb (λMt)

λ

(dN b

t − ν (1 − p (It)) dt)− dXt.

43

Now, define wt = λMt to get

dwt = λdMt = γwtdt + λ(dYt − (μt − It) dt

)+ βg (wt) (dN g

t − νp (It) dt)

+βb (wt)(dN b

t − ν (1 − p (It)) dt)− λdXt.

Letting dUt = λdXt, incentive compatibility under the proposed implementation then

follows from incentive compatibility of the optimal contract characterized in Proposition 2

and the agent’s value is given by wt. Note further, that the agent is indifferent as to when

to issue the special dividend.

Next, consider the valuation of the equity claim, which follows from arguments similar

to those in DeMarzo et al. (2012), in particular their Proposition 2. Substituting from

(23), (24) can be written as

P (Mt, μt) = Et

[∫ τ

t

e−r(s−t) ((μs − Is) ds − dUs) + e−r(τ−t)Lτ

]

+Et

[∫ τ

t

e−r(s−t) (rMsds − dMs)

]

= f(Wt, μt) + rEt

[∫ τ

t

e−r(s−t)Msds

]

− Et

[∫ τ

t

e−r(s−t)dMs

]

= f(Wt, μt) + Mt = f i(λMt) + Mt,

where we have used integration by parts. Q.E.D.

Appendix B Numerical Implementation

To solve numerically for the optimal contract of Proposition 2, we take the following

iteration steps.

1. Solve for the principal’s value function f (0) and the free boundary w(0) without

technology shocks. That is, we solve the ODE in (12) with ν = 0 (thus the initial

investment is I(0) = 0 and the initial rewards and punishments are βg(0)

= βb(0) = 0).

2. Given f (0), w(0), and βb(0) , update the optimal investment scheme I(1) according to

(18) and subject to incentive compatibility, i.e., βg(I(1))

= βb(0) + λ/(νp′(I(1))

).

3. Given βg(0)

, βb(0) , and I(1), update the principal’s value function f (1) and the free

boundary w(1).

44

4. Given f (1), w(1), and I(1), update the optimal rewards and punishments βg(1)

and

βb(1) according to the first order condition (20), subject to the (binding) incentive

constraint (7) and the limited liability constraint (8). That is, we solve (20) for βb(1) ,

such that βg(1)

= βb(1) + λ/(νp′(I(1))

)and βb(1) ≥ w.

5. Repeat steps 2 to 4 until the problem converges. The convergence criterion is

max

[

supw

∣∣I(i+1) − I(i)

∣∣ , sup

w

∣∣f (i+1) − f (i)

∣∣]

< 10−5.

45