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Liquidity, Innovation, and EndogenousGrowth∗
Semyon Malamud† Francesca Zucchi‡
October 9, 2016
Abstract
We study how financing frictions and corporate cash hoarding
affect firms’ invest-
ment in innovation and shape economic growth. By nesting a
dynamic corporate
finance model with financing frictions into a model of
endogenous growth, we show
that financing frictions have two offsetting effects on growth.
First, financing fric-
tions exacerbate entry barriers and deter new firms from
innovation (the hurdle
effect). Second, financing frictions spur innovation by
incumbents by reducing exit
threats and prompting a substitution from production to
innovation (the haven ef-
fect). Financing frictions can foster growth if the haven effect
prevails, which is the
case if entry barriers are sufficiently large.
Keywords: Innovation; Cash management; Financing frictions;
Endogenous growth
JEL Classification Numbers: G31; G32; O31; O40
∗We thank Jean-Paul Décamps, Michael Fishman, Teodor Godina,
Mikhail Golosov, Sebastian Gry-glewicz, Luigi Guiso, Barney
Hartman-Glaser, Zhiguo He, Dalida Kadyrzhanova, Leonid Kogan,
ArvindKrishnamurthy, Ernst Maug, Konstantin Milbradt, Erwan
Morellec, Boris Nikolov, Dino Palazzo, PaolaSapienza, Lukas Schmid,
Toni Whited, an anonymous referee, and the participants of the 12th
AnnualConference in Financial Economics Research (IDC Herzliya),
the 7th EBC Conference (Tilburg), the2016 AFA Annual Meeting (San
Francisco), the 2016 EFA Annual Meeting (Oslo), and the 2016
NorthAmerican Summer Meeting of the Econometric Society
(Philadelphia) for helpful comments. Any remain-ing errors are our
own. Semyon Malamud gratefully acknowledges the Lamfalussy
Fellowship Programsponsored by the European Central Bank. The views
expressed in the paper are those of the authorsand do not
necessarily represent those of the ECB, the Eurosystem, the Federal
Reserve System, or theirstaff.†Swiss Finance Institute, EPFL, and
CEPR. E-mail: [email protected]‡Federal Reserve Board of
Governors. E-mail: [email protected]
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1 Introduction
Innovation is pivotal to economic growth. The emergence of new
products and improve-
ments in existing goods shape the world in which we live and
drive the dynamics of the
economy. Yet, investment in innovation (or research and
development, R&D) is costly
and difficult to finance. It requires a long gestation period
before becoming productive,
it is not pledgeable, and its outcome is uncertain. To safeguard
financial flexibility, in-
novative firms hoard cash reserves (or liquidity).1 While much
work has been done to
understand how financing frictions affect corporate cash
management and investment,
there is no work studying how financing frictions jointly affect
firms’ production and
innovation decisions and factor into economic growth. This paper
seeks to fill this gap.
To this end, we develop a tractable model of the effects of
financing frictions and
corporate cash hoarding on innovation and economic growth. The
model consists of two
building blocks. The first is a dynamic corporate finance model
with financing frictions
for a continuum of incumbents and entrants. We use the model to
investigate how con-
strained incumbents manage their production-innovation mix as
well as to study entrants’
incentives to invest in innovation. The second building block is
a model of endogenous
growth, in which we nest our firm maximization problem. Similar
to extant growth mod-
els, innovations by incumbents and entrants that increase the
quality of products are the
key drivers of economic growth. We depart from these
contributions by recognizing that
innovative firms face financing frictions and, thus, hoard
precautionary cash reserves.
We start by solving the intertwined optimization problems of
constrained incumbents
and entrants. Each incumbent produces one good and invests in
innovation to improve
the quality of this good. Incumbents are financially constrained
in that they have costly
access to external financing whenever they seek to refinance
current operations or to
market technological breakthroughs. To maintain financial
flexibility, incumbents have
an incentive to retain earnings in cash reserves and adjust
production and innovation rates
in response to operating shocks. Each incumbent faces the threat
of creative destruction—
1 R&D is a major determinant of corporate cash reserves, as
illustrated by Lyandres and Palazzo(2016), Ma, Mello, and Wu
(2013), Falato, Kadyrzhanova, and Sim (2013), Falato and Sim
(2014), andBegenau and Palazzo (2016).
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i.e., losing its market position when an entrant launches a
better product in the same
product line. Entrants only invest in innovation and do not
produce any goods. To single
out the effects of financing frictions, we compare the optimal
policies in the constrained
economy with those associated with an identical, but
unconstrained economy.
We show that constrained incumbents scale down production and
increase markups
in response to negative operating shocks. Negative operating
shocks drain corporate cash
reserves and weaken a firm’s financial strength. In an attempt
to avert a costly refinancing,
an incumbent reduces its production rate to limit cash flow risk
and cut operating costs.
Our model predicts that incumbents produce less and charge
higher markups in the
constrained economy than in the unconstrained economy. A
constrained incumbent’s
production rate increases with cash reserves and equals the
production rate associated
with the unconstrained economy only at its target cash level.
The resulting production
rate is cyclical (and markups countercyclical) to operating
(firm-specific) shocks.
A key result of our model is that incumbents may invest more in
innovation in the
constrained economy than in the unconstrained economy. Despite
the tendency to scale
down production, constrained incumbents may increase their
innovation rate in response
to negative operating shocks, thus substituting production for
innovation. The reason is
the following. Both the marginal gain and cost of investing in
innovation increase as cash
reserves shrink. The marginal gain is the probability-weighted
change in firm value when
attaining a technological breakthrough, in which case the firm
accesses fresh monopoly
rents and raises funds in light of a success (the achievement of
a breakthrough) rather
than a failure (running out of funds because of bad operating
performance). The marginal
cost of innovation is the associated decrease in cash reserves,
which makes the firm more
constrained. If the gain rises faster than the cost as cash
reserves decrease, the optimal
innovation rate decreases with cash reserves. We show that this
is the case for constrained
firms with volatile cash flows and low operating margins. For
these firms, it is less costly
(in terms of foregone profits) to reallocate resources from
production to innovation.
Although this result may be surprising in light of the
documented positive relation
between R&D and corporate cash hoarding (see footnote 1),
one should not confuse the
relation between R&D and target cash reserves with the
relation between R&D and de-
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viations from this target. While extant works have focused on
the first relation, our
contribution is to shed light on the second one. Our results
help rationalize the observa-
tion that some firms have actually increased their R&D
expenditures despite the tighter
constraints during the recent crisis (see Archibugi, Filippetti,
and Frenz, 2013). Moreover,
we illustrate which firm characteristics should prompt this
behavior.
Our model also shows that constrained entrants spend more in
innovation than in the
unconstrained economy if their cash reserves are sufficiently
large. Similar to incumbents,
the entrants’ marginal cost and gain from innovation increase as
cash reserves decrease.
Differently, entrants do not produce any goods and lack regular
cash flows that replenish
cash reserves and smooth R&D expenditures. Thus, the
marginal value of entrants’ cash
is larger than that of incumbents, and the increase in the
marginal cost of innovation
associated with decreasing cash reserves is more likely to
offset the rising gain. When
cash reserves are large (i.e., after a financing round), the
marginal cost of innovation
is relatively low, but the marginal gain is greater than in the
unconstrained economy.
Conversely, when cash reserves are small, the marginal cost of
innovation is large and
causes constrained entrants to innovate less than in the
unconstrained economy.
More generally, our model predicts that financing frictions do
not need to imply lower
firm values.2 Financing frictions inflate entry costs and cause
a decrease in the measure
of active entrants, which lessens incumbents’ exit threats. As a
result, incumbents have
longer horizons, are more valuable, and innovate more in the
constrained economy. The
increase in incumbent value, in turn, boosts entrants’
incentives to innovate (and to
become the next incumbent). In equilibrium, financing frictions
result in fewer entrants,
which can be more valuable though. That is, we show that
financing frictions mostly affect
the extensive (rather than intensive) innovation margin, in line
with Nanda and Rhodes-
Kropf (2013). Overall, we illustrates that financing frictions
importantly affect industry
composition via incentives to innovate, providing theoretical
grounds to empirical studies
relating access to financing and composition of aggregate
R&D investment.3
2Relatedly, Stangler (2009) finds that cold markets lead to the
funding of more successful companies.Anecdotal evidence also
suggests that hot periods in innovation financing are associated
with less valuablefirms being financed (see Gupta, 2000).
3See, e.g., Brown et al. (2009); Brown et al. (2013); Gao, Hsu,
and Li (2014); Kerr and Nanda (2009).
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We next analyze how these firm dynamics affect economic growth.
We focus on a
“balanced growth path” equilibrium featuring a constant,
endogenous growth rate. In-
cumbents’ and entrants’ contributions to growth are obtained by
aggregating their inno-
vation rates via the distribution of cash, which assigns a
probability weight to innovation
rates associated with different levels of cash reserves. Thus,
the shape of entrants’ and
incumbents’ distribution of cash determines the equilibrium
impact of firms’ policies.
Via this “composition” of firms’ policies in equilibrium,
financing frictions have two
offsetting effects on growth. First, financing frictions
decrease the entrants’ contribution
to economic growth. In fact, although entrants with large cash
reserves invest more in
innovation in the constrained economy than in the unconstrained
economy, their aggregate
impact is modest because there is a large mass of entrants
holding small levels of cash
reserves. This is what we call the “hurdle” effect of financing
frictions on growth, which
is growth-decreasing. Second, incumbents increase their
investment in innovation as they
expect their profits to last longer before losing their market
position due to creative
destruction. This is what we call the “haven” effect of
financing frictions on growth,
which is growth-enhancing. Which of the two effects dominates
depends on the relative
share of entrants’ and incumbents’ contributions to growth,
which is itself endogenous.
We find that the haven effect overtakes the hurdle effect when
the entrants’ contribution
to growth is relatively small, which is the case when setup
costs are sufficiently large.
Two key predictions follow. First, financing frictions
importantly affect the compo-
sition of growth and place entrants at a disadvantage. This
finding is consistent with
Brown et al. (2009) and Nanda and Nicholas (2014), who show that
tighter constraints
are mostly detrimental for R&D of young firms. Second,
financing frictions do not need
to be detrimental to growth. They can be growth-enhancing if the
haven effect overtakes
the hurdle effect.
Related literature Our paper contributes to the literature
studying cash management
models, which includes Riddick and Whited (2009); Décamps,
Mariotti, Rochet, and Vil-
leneuve (2011); Bolton, Chen, and Wang (2011, 2013); Hugonnier,
Malamud, and Morellec
(2015); and Décamps, Gryglewicz, Morellec, and Villeneuve
(2016). In this strand, the
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papers analyzing the relation between cash reserves and
investment consider neoclassi-
cal models of investment and capital accumulation, either
incremental (as in Bolton et
al., 2011, 2013) or lumpy (as in Hugonnier et al., 2015). To the
best of our knowledge,
our paper is the first to study cash management in a
Schumpeterian framework.4 This
problem is economically important given the documented relations
between cash reserves
and R&D and between R&D and economic growth (e.g.,
Caballero and Jaffe, 1993; Ak-
cigit and Kerr, 2016; Kogan et al., 2016), and is not trivial
because it involves solving a
problem with an infinite number of (sequentially arriving)
stochastic growth options.
The paper also contributes to the literature on innovation
financing. Lyandres and
Palazzo (2016) and Ma, Mello, and Wu (2014) investigate and test
the relation among
competition, R&D, and cash reserves. Falato, Kadyrzhanova,
and Sim (2013) focus on
the relation between investment in intangible capital and
corporate cash in a neoclassical
model with no growth. We contribute to this strand by studying
cash, production, and
innovation decisions altogether, as well as their ensuing impact
on economic growth. Our
model studies the relation between R&D and target cash
reserves and between R&D
and deviations from this target. While acknowledging that
R&D prompts firms to hoard
precautionary cash, we characterize how firms will adjust their
investment in innovation
when operating shortfalls erode cash reserves below their target
level.
The link between innovation and corporate cash is supported by
vast empirical evi-
dence. Hall (2005) and Hall and Lerner (2010) document that
innovation is best financed
through internal funds because it is subject to asymmetric
information, it is not pledge-
able, and it has uncertain returns. Brown, Fazzari, and Petersen
(2009) document that
innovation decisions are related to the supply of internal and
external equity. They re-
port that young publicly traded firms in high-tech industries
finance R&D almost entirely
with internal and external equity, as debt financing is
difficult due to non-pledgeability.5
Consistently, Hall (2005); Hall and Lerner (2010); Rajan (2012);
Brown, Martisson, and
4In neoclassical capital accumulation models, investment implies
an immediate increase in output. InSchumpeterian models, the payoff
from R&D investment arrives at uncertain (Poisson) times.
5Citing from Brown, Fazzari, and Petersen (2009), “Our results
suggest that more attention shouldbe given to equity finance [...]
for models that emphasize innovation. [...] External equity is the
morerelevant substitute for internal cash flow for young high-tech
firms.”
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Petersen (2013); and Acharya and Xu (2016) emphasize the
relative importance of equity
rather than debt for the financing of innovation, as we do in
our model.
Schumpeter (1942) underlined the link between innovation and
economic growth by
introducing the concept of “creative destruction,” which spurred
the development of a
subfield of macroeconomics known as Schumpeterian models of
endogenous growth (for
a survey, see Aghion, Akcigit, and Howitt, 2014). We nest in
this literature by assuming
that growth is spurred by endogenous technological change (see
Romer, 1990; Grossman
and Helpman, 1991; or Klette and Kortum, 2004) and that both
incumbents and en-
trants invest in innovation (as in Acemoglu and Cao, 2015;
Akcigit and Kerr, 2016; and
Acemoglu, Akcigit, Bloom, and Kerr, 2013). Our paper also
relates to the literature
studying the effects of financial constraints on economic
growth; see Levine (2005) and
Beck (2012). Thus far, however, little attention has been paid
to the role of corporate
cash in providing flexibility to innovative firms. Our model
seeks to fill this gap.
The paper is organized as follows. Section 2 describes the
model. Section 3 solves
the model in the unconstrained economy, which serves as a
benchmark to single out the
effects of financing frictions on innovation and growth. Section
4 solves the model in
the constrained economy. Section 5 provides a numerical
implementation of the model.
Section 6 concludes. Technical developments are in the
Appendix.
2 The model
Time is continuous and uncertainty is modeled by a probability
space, (Ω,F,P), equipped
with a filtration, (Ft)t≥0, that represents the information
available at time t. We study
an economy in which the representative household maximizes
discounted inter-temporal
utility with constant relative risk aversion (CRRA)
preferences:
E
[∫ ∞0
e−ρtC1−θt − 1
1− θ
]dt . (1)
In this expression, Ct is consumption at time t, ρ is the
discount rate, and θ is the inverse
of elasticity of intertemporal substitution. The household
supplies one unit of labor at
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any time and receives a competitive wage. Population size is
constant, normalized to one.
2.1 Consumption good sector
There is a unique consumption good, whose output is denoted by
Yt. The consumption
good serves as the numeraire of the economy. The consumption
good is produced com-
petitively using labor and a multiplicity (a continuum) of
inputs, which are indexed by j.
As in Grossman and Helpman (1991), the continuum of inputs has
measure one, j ∈ [0, 1].
The production technology of the consumption good is
Yt =1
1− β
∫ 10
X̄1−βjt qβjt dj , β ∈ (0, 1), (2)
where X̄jt denotes the quantity and qjt the quality of input j
at time t. Improvements
in the quality of inputs stem from innovation. In the tradition
of Schumpeterian growth
models, innovation advances the technological frontier and spurs
economic growth via
creative destruction—i.e., new inputs drive the old ones out of
the market. Thus, only
the frontier (highest-quality) version of each input is used by
the consumption good sector.
2.2 Input sector
There are two types of firms in the input sector: (1) incumbent
firms, which actively
produce the frontier inputs and invest in innovation to further
enhance their quality; and
(2) entrant firms, which only invest in innovation.
Incumbents The latest innovator of industry j enforces a patent
on the frontier version
of input j and becomes the incumbent monopolist of that
industry. While producing input
j, the incumbent keeps investing in innovation. Patents last
forever but do not prevent
firms from innovating and further improving the quality of
inputs.
We adopt the modeling of innovation that is standard in the
growth literature, as
surveyed by Aghion, Akcigit, and Howitt (2014). We denote by zjt
the innovation intensity
(or innovation rate) of incumbent j at time t. Innovation is
costly, and its outcome is
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uncertain. Specifically, an incumbent bearing the flow cost
Φ(zjt, qjt) = ζz2jt2qjt, ζ > 0 (3)
increases the quality of its input at Poisson rate φzjt. These
Poisson events represent
technological breakthroughs, which are more frequent if zjt is
larger. When the incumbent
of industry j attains a breakthrough, the quality of the input
jumps from qjt− to
qjt = λqjt− ,
where λ > 1 represents the size of the quality
improvement.6
Taking the demand schedule of the consumption good sector, each
incumbent opti-
mally sets the production rate and the associated selling price,
which we denote respec-
tively by X̄jt and pjt. The dynamics of incumbents’ cash flows
(associated with these
endogenous choices) satisfy
dΠjt =[X̄jt(pjt − 1)− Φ(zjt, qjt)
]dt + σX̄jtdZjt , (4)
where Zjt is a standard Brownian motion representing operating
shocks. Operating shocks
are firm-specific and independent across firms. To ease the
notation, the marginal cost of
production is normalized to one.
The cash flow process (4) implies that incumbents can make
operating profits and
losses. If external financing was costless, losses could be
covered with fresh funds whenever
needed. We depart from this assumption and assume that firms
face financing costs. As in
previous cash management models, financing costs are modeled in
a reduced-form fashion
to capture limited enforcement, asymmetric information, or
limited pledgeability.
Specifically, incumbents can raise external funds to cover
operating shortfalls and
the flow cost of R&D. This “routine” financing is incurred
by the firm during the “dis-
covery” phase of R&D and entails a proportional cost � ∈ [0,
1] and a fixed cost ωqjt.6As in Aghion, Howitt, and Mayer-Foulkes
(2005), Acemoglu, Aghion, and Zilibotti (2006), and
Acemoglu and Cao (2015), we abstract from skilled labor in the
inputs sector.
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The fixed cost scales with input quality, which directly affects
firm size in our model.
The assumption of scalability of the fixed cost with firm size
is shared with extant cash
management models (e.g., Bolton, Chen, and Wang, 2011, 2013;
Décamps, Gryglewicz,
Morellec, and Villeneuve, 2016).7 Incumbents can also raise
funds when they attain tech-
nological breakthroughs—i.e., at the “marketing” phase of
R&D—by ceding a surplus
share α ∈ [0, 1] to financiers.8
To maintain financial flexibility, incumbents retain earnings in
cash reserves. We
denote cash reserves at time t by C̄jt. Incumbents’ cash
reserves earn a rate of return δ,
which is lower than the market interest rate r. As previous cash
management models,
we interpret the difference r − δ (the cost of holding cash) as
an agency cost of free cash
flows. Alternatively, this cost can be interpreted as a
liquidity premium related to the
scarce supply of liquid assets in the economy (see Krishnamurthy
and Vissing-Jorgensen,
2012). The dynamics of cash reserves satisfy
dC̄jt = δC̄jt + dΠjt − dD̄jt + dF̄jt + dF̄ Ijt − dΩ̄jt. (5)
In this equation, D̄jt, F̄jt, F̄Ijt, and Ω̄jt are non-decreasing
processes that respectively
represent cumulative payouts, cumulative financing obtained
during the discovery and
the marketing phases, and cumulative issuance costs. Equation
(5) is an accounting
identity illustrating that cash reserves increase with the
interests on cash and cash flows
(the first and second terms) and financing (the fourth and fifth
terms) and decrease with
payouts (the third term) and issuance costs (the last term).
Each incumbent sets production, innovation, financing, and
payout policies to maxi-
mize future net dividends subject to the budget constraint (5)
and C̄jt ≥ 0. We denote
incumbents’ value by V (t, C̄j, qj), which is a function of cash
reserves and quality at time
t. When creative destruction hits industry j, the associated
incumbent exits. The ex-
7While keeping the analysis tractable, this assumption is
motivated by the observation that “negativeincentive effects of a
more diluted ownership may have costs that are proportional to firm
size,” assuggested by Bolton et al. (2011). See also Décamps et
al. (2016) and the discussion therein.
8Nash bargaining over surplus between the firm and financiers
provides a micro-foundation for ourassumption. Denoting the
bargaining power of financiers by α ∈ [0, 1] and the surplus
created by S, therents extracted by financiers are Γ∗ = arg maxΓ≥0
Γ
α [S − Γ]1−α = αS, whereas the firm retains (1−α)S.
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iting incumbent entirely recovers its cash reserves but only
some fractions ψ and ψE of
the values of mature productive assets (denoted by KTjt) and
R&D assets (denoted by
KEjt). We assume that these values increase in quality and are
given by KTjt = κT qjt and
KEjt = κEqjt for some κT , κE > 0. Because R&D assets are
less tangible than productive
assets, they are more costly to liquidate and, thus, ψ > ψE.
Because liquidation is costly,
incumbents never choose to liquidate the firm when cash reserves
are positive.
Entrants A mass of penniless entrepreneurs seeks to obtain
financing to buy the R&D
technology and start investing in innovation. Entrepreneurs who
obtain financing become
active entrants. The mass of active entrants is endogenous and
is denoted by mE.
Entrants differ from incumbents in three respects. First,
entrants do not produce any
inputs and only invest in innovation (recall that frontier
inputs are patent protected).
Second, entrants are more financially constrained than
incumbents, as documented by
Nanda and Rhodes-Kropf (2016), among others. Third, entrants’
innovations have wider
breadth than incumbents’ innovations, which results in ex-ante
uncertainty regarding the
input that an entrant may improve. This uncertainty dissolves
when the entrant targets
an industry after achieving a breakthrough. As a result, the
entrants’ cost of innovation
is a function of the average quality of frontier inputs, as we
will further explain. Because
entrants’ characteristics are not input-specific, we describe a
“representative” entrant.
Active entrants choose their optimal innovation rate, zEt.
Similar to incumbents,
zEt regulates the Poisson rate of technological breakthroughs,
which is given by φEzEt,
φE > 0. Technological breakthroughs by entrants lead to jumps
in quality by a factor
Λ > 1. Sustaining the innovation rate zEt entails a flow
cost
ΦE(zEt, q̄t) = ζEz2Et2q̄t.
While this expression has a structure similar to (3), it differs
along two dimensions. First,
this cost is proportional to the average quality of frontier
inputs
q̄t =
∫ 10
qjtdj,
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rather than to the quality of a specific input j. Second, ζE may
differ from ζ.9
We model entrants’ financing frictions following entrepreneurial
finance schemes, ac-
cording to which financiers tend to stage their investments,
learn about the firm’s po-
tential, and deny further financing if no milestone or
breakthroughs are attained (see
Gompers, 1995; Lerner et al., 2012; and Nanda and Rhodes-Kropf,
2016). To keep the
analysis tractable, we model this financing scheme as follows.
Consider an active en-
trant that has obtained initial financing to set up the R&D
technology, whose cost is
proportional to average quality and is given by K̄Et = κE q̄t.
At the outset, the entrant
faces proportional and fixed financing costs denoted
respectively by �E and ωE q̄t, as in-
cumbents. These costs increase if the entrant seeks to raise
more funds without having
attained a breakthrough. In these cases, successive financing
rounds can be so expensive
that the unsuccessful entrant may prefer to liquidate rather
than continue. For simplicity,
we assume that this happens at the second financing round (in
Appendix A.3, we micro-
found this assumption via learning about an entrant’s type).10
Conversely, technological
breakthroughs are milestones that wane adverse selection. In
these cases, entrants can
raise funds by promising a fraction αE ≥ α of surplus to
financiers, as incumbents do.
Because external financing is costly, entrants raise some slack
at the outset. This
slack serves to finance the flow cost of R&D and is stored
as cash reserves. We denote
by C̄Et an entrant’s cash reserves at time t. We assume that the
rate of return on cash
is lower for entrants than for incumbents; for simplicity, we
set this rate to zero. This
assumption can be motivated by a harsher free cash flow problem
caused by entrants’
less tangible assets, which is consistent with Gompers (1995).
If we interpret the cost of
cash as a liquidity premium, the lower return on entrants’ cash
stems from their need to
keep cash in vaults (rather than investing in money-like
securities with higher returns)
because entrants have no cash flows to smooth the flow cost of
R&D. Thus, the dynamics
9We do not impose a priori restrictions on the relations between
λ and Λ, φ and φE , and ζ and ζE .10In Appendix A.3, we assume that
there are two entrant types: good and bad. Good entrants
eventually attain breakthroughs, while bad entrants never do so.
Financiers start with a prior probabilityabout an entrant’s type,
which they update using Bayes rule. Conditional on observing no
breakthroughs,the posterior estimate of the probability of the
entrant being of good type is monotonically decreasing overtime. It
is then straightforward to show that there is a deterministic time
t∗ after which an unsuccessfulentrant exits. When the prior is
sufficiently small, only one round of financing is optimal, in
agreementwith our assumption.
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of an entrant’s cash reserves satisfy
dC̄Et = −ΦE(zEt, q̄t)dt (6)
until the entrant attains a technological breakthrough or it
liquidates after depleting cash
reserves and being denied further financing. As in practice, not
all active entrants succeed
(see, e.g., Ewens and Rhodes-Kropf, 2015).
We denote entrant value by VE(t, C̄E, q̄), which is a function
of cash reserves and of the
average quality of frontier inputs at time t. Each entrant
maximizes its value by setting
innovation, financing, and cash management decisions until it
attains a breakthrough
or exits. If it attains a breakthrough, the entrant improves the
quality of the input in
industry j from qjt− to qjt = Λqjt− and needs to pay a cost KTjt
= κT qjt to set up the
productive assets. If it exits, the entrant recovers a fraction
ψE of the investment in R&D
assets, K̄Et. Just like incumbents, entrants never choose to
stop operations when holding
positive cash balances because liquidation is costly.
2.3 Balanced growth path
We focus on a balanced growth path equilibrium in which
aggregate quantities grow at
the constant rate g, which is endogenously determined.
Competition among entrants de-
termines the measure of active entrants, which is pinned down by
the free entry condition
VE(t, C̃Et, q̄t;mE) = (1 + �E)(C̃Et + K̄Et) + ωEt .
In this equation, K̄Et + C̃Et is the amount raised at entry
(setup cost and slack) and
�E(C̃Et + K̄Et) + ωEt is the associated financing fees.11 The
measure of entrants and the
entrants’ innovation rates together define the equilibrium rate
of creative destruction.
To solve the model, we first derive incumbents’ and entrants’
optimal policies by taking
the market interest rate and the growth rate as given. We then
aggregate firms’ decisions
11Note that in equilibrium, C̃Et depends on mE , the mass of
entrants. We consciously suppress thedependence of endogenous
quantities on mE to keep the notation simple.
13
-
and derive these equilibrium quantities. An equilibrium is an
allocation such that: (i)
incumbents set production, innovation, cash retention and
payout, and financing decisions
to maximize their value; (ii) entrants choose innovation,
financing, and cash policies to
maximize their value; (iii) the measure of entrants makes the
free entry condition binding;
(iv) the consumption good sector maximizes profits; (v) the
representative household
maximizes utility from consumption; and (vi) all markets
clear.
2.4 Discussion: Corporate finance meets growth
We nest a dynamic corporate finance model with financing
frictions into a model of
endogenous growth. Our growth model has a Schumpeterian nature,
as innovation is
the engine of growth and new inputs replace the old ones—i.e.,
growth involves creative
destruction. As we clarify in this section, our key assumptions
are borrowed from the
extant literature (see Aghion et al., 2014, for an excellent
survey).
Specifically, the innovation technology is drawn from the
theoretical industrial orga-
nization and patent race literature (Tirole, 1998). Firms choose
their innovation rates,
which affect the Poisson rate of occurrence of technological
breakthrough—i.e., larger
innovation rates increase the likelihood of attaining a
breakthrough. While early endoge-
nous growth models focus on innovations by new firms only, we
contribute to a growing
literature assuming that both incumbents and entrants innovate
(e.g., Acemoglu and Cao,
2015; Akcigit and Kerr, 2016), which is motivated by the
observation that a large fraction
of R&D in the United States is done by incumbents (see
Acemoglu et al., 2013).
Breakthroughs come as improvements in input quality. Following
the literature, we
assume that only the frontier version of any input is used in
the production of the final
good, so that the latest innovator is the monopolist of the
industry. We rule out the case
of limit pricing to keep the analysis tractable and motivate it
as previous contributions.
For instance, Aghion and Howitt (1992) assume that innovations
are always drastic, so
that a monopolist is unconstrained by potential competition from
the previous patents.
Akcigit and Kerr (2016) assume that current and former
incumbents of the same product
line enter a two-stage price-bidding game whereby each firm pays
a fee to announce its
14
-
price. Under this assumption, only the new incumbent pays the
fee and announces its
price. We adopt similar assumptions in our setup.
Finally, the consumption good is the numeraire of the economy
and is produced using
labor and inputs j.12 The interpretation of j as inputs implies
that g is the rate of growth
of final good output. If we interpreted j as consumption goods,
g would represent the
growth rate of a quality-adjusted consumption index (see
Grossman and Helpman, 1991).
3 A benchmark: the unconstrained economy
We first solve the model in an identical, but unconstrained
economy (featuring no financ-
ing frictions). In this setting, operating shortfalls can be
covered by raising external funds
at no cost. Thus, firms have no incentive to keep cash.
We start by solving the optimization problem of the consumption
good sector,
maxX̄j
1
1− β
∫ 10
X̄1−βjt qβjtdj −
∫ 10
pjtX̄jtdj,
which delivers the demand curve for each frontier input: X̄jt
=(qβjt/pjt
) 1β. Taking the
demand schedule of the consumption good sector as given,
incumbents in the input sector
maximize profits and set the following monopoly price:
pjt =1
1− β≡ p∗. (7)
This price implies a constant markup above the marginal cost of
production and is asso-
ciated with the production rate X̄∗jt = qjtX∗, where
X∗ = (1− β)1β (8)
denotes the optimal production rate scaled by quality. Working
with scaled quantities
eases the model solution because of their time-invariant
nature.
12Because labor is normalized to one and is supplied
inelastically, it does not show up in (2).
15
-
We denote by V (t, q) the value of incumbents in the
unconstrained economy (hence-
forth, the subscript j will be suppressed when it causes no
confusion). Following standard
arguments, V (t, q) satisfies the Hamilton-Jacobi-Bellman (HJB)
equation:
r∗V (t, q)− Vt(t, q) = maxz∗
{βq (1− β)
1β−1 − (z
∗)2
2ζq + φz∗ [V (t, λq)− V (t, q)]
+ x∗d [ψκT q + ψEκEq − V (t, q)]}.
(9)
The terms on the left-hand side represent the return required by
investors and the change
in firm value as time elapses, respectively. The first two terms
on the right-hand side
represent operating cash flows net of production and innovation
costs. The third term
is the probability-weighted change in value when the incumbent
markets a technological
breakthrough. The fourth term captures the effect of creative
destruction on incumbent’s
value. An incumbent is replaced by a new firm at a rate x∗d,
otherwise it preserves its
monopoly power. When creative destruction hits, the incumbent
exits and recovers just
a fraction of its investment in productive and R&D assets.
The quantities r∗ and x∗d are
endogenously determined later in the analysis.
To solve incumbents’ problem, we conjecture incumbent value to
be linear in qt,
V (t, q) = V (qt) = qtv∗ , (10)
for some v∗ > 0 representing scaled (time-invariant)
incumbent value. Substituting (10)
into (9) gives an equation that is independent of quality (see
equation (37) in Appendix
A.1), which we differentiate to obtain the optimal innovation
rate:
z∗ =φ
ζ(λ− 1)v∗ . (11)
The optimal innovation rate increases with firm value v∗, with
the Poisson parameter φ,
and with the size improvement λ. Moreover, it decreases with the
cost coefficient ζ.
The rate of creative destruction in (9) is determined by
aggregating the innovation
rate of active entrants, which we now derive. The value of an
active entrant is denoted by
V E(t, q̄), where q̄t represents the average quality of frontier
inputs. Following standard
16
-
arguments, entrant value satisfies the following HJB
equation:
r∗V E(t, q̄)− V Et (t, q̄) = maxz∗E
{− (z
∗E)
2
2ζE q̄ + φEz
∗E
[V (t,Λq̄)− κTΛq̄ − V E(t, q̄)
] }. (12)
The terms on the left-hand side represent the return required by
the investors and the
change in value as time passes. The first term on the right-hand
side represents the flow
cost of R&D. The second term represents the
probability-weighted change in value when
the entrant attains a technological breakthrough.13 To solve
this maximization problem,
we conjecture that the value of an active entrant is linear in
q̄t:
V E(t, q̄) = q̄tv∗E, (13)
where v∗E represents entrant value scaled by average quality.
Substituting (13) into equa-
tion (12) yields a time-independent version of the HJB equation
(see equation (39)).
Differentiating this HJB equation yields the entrants’ optimal
innovation rate:
z∗E =φEζE
max(Λv∗ − v∗E − ΛκT , 0). (14)
The optimal z∗E is strictly positive if the inequality Λv∗ >
v∗E + ΛκT holds.
14 Substituting
(14) into the HJB equation, we can solve for v∗E (see Appendix
A.1). Combining (13)
with the free entry condition (given by V E(t, q̄;m∗E) = K̄Et in
the unconstrained economy)
gives
v∗E(m∗E) = κE.
The free entry condition pins down the equilibrium measure of
active entrants, which,
together with z∗E, gives the equilibrium rate of creative
destruction:
x∗d = m∗E φE z
∗E = m
∗E
φ2EζE
(Λv∗ − v∗E − ΛκT ) . (15)
13Recall that entrants do not know ex-ante to which industry
they will contribute until they target aspecific industry after a
breakthrough. When a breakthrough occurs, firm value jumps to V
(t,Λq̄t) −κT q̄t =
∫ 10
[V (t,Λqjt)− κTΛqjt] dj. The first term in the integral is the
value of the successful entrantafter uncertainty dissolves, whereas
the second term represents the setup cost of productive assets.
14If it did not hold, entrants would not have incentives to
enter and invest in innovation, and theeconomy would feature
incumbents only.
17
-
In the model, economic growth stems from advancements in the
quality of inputs,
which are triggered by the successful innovations of incumbents
and entrants. Improve-
ments occur at independent Poisson times in different industries
j ∈ [0, 1]. By the law of
large numbers (see Appendix A.1), the rate of economic growth is
given by
g∗ = (λ− 1)φz∗ + (Λ− 1)x∗d .
The first (respectively, second) term is the jump in quality due
to incumbents’ (entrants’)
breakthroughs times the associated rate of occurrence. Because
aggregate consumption
grows at rate g∗ along the balanced growth path, the
maximization problem of the rep-
resentative household delivers the market interest rate via the
standard Euler equation:
r∗ = ρ+ θg∗.
While the interest rate and the growth rate are the outcome of
incumbents’ and entrants’
optimal policies, these rates feed back into corporate decision.
Then, equilibrium rates
and corporate policies are determined by a fixed point problem
(see (42) in Appendix A.1),
which we solve numerically. In section 5, we analyze the
equilibrium of the unconstrained
economy and compare it to the constrained economy, which we
derive in the next section.
4 The constrained economy
In the constrained economy, firms face financing frictions and
have incentives to hoard
cash reserves. We first study the value-maximizing decisions of
incumbents and entrants.
We then derive their stationary distributions of cash, which
serve to aggregate firm deci-
sions and derive the equilibrium of the constrained economy.
4.1 Incumbents’ optimal policies
Each incumbent selects retention, payout, financing, production,
and innovation policies
to maximize its value. Consider first retention and payout
policies. Because precautionary
18
-
concerns are relaxed when cash reserves are large, the benefit
of cash is decreasing in cash
reserves. The cost of cash is constant. We then conjecture the
existence of a target cash
level, C̄∗(qt), which trades off costs and benefits (as for the
unconstrained economy, we
drop the index j when it causes no confusion). It is optimal to
pay out cash exceeding
this target level to shareholders. Below this target level, it
is optimal to retain earnings
in cash reserves.
Consider now financing decisions. Because routine financing is
costly, incumbents
tap it when cash reserves are depleted. To economize on the
fixed cost, it is optimal
to raise a lumpy amount. We denote this amount by C̄∗(qt), which
is endogenously
determined. As we show, proportional financing costs imply that
C̄∗(qt) is smaller than
C̄∗(qt). Besides this routine financing, incumbents can raise
funds when marketing a
higher-quality product after a technological breakthrough, at
the cost of ceding a surplus
share to financiers. Incumbents find it optimal to raise an
amount that replenishes cash
reserves to C̄∗(λqt). The surplus from this transaction is S(t,
C̄, q) = V (t, C̄∗(λq), λq) −
(C̄∗(λq)− C̄)− V (t, C̄, q), but incumbent value only increases
by (1− α)S(t, C̄, q).15
Next, consider production and innovation decisions. As in the
unconstrained econ-
omy, incumbents choose their optimal production rate by taking
the demand schedule of
the consumption good sector as given. Differently, optimal
production (and innovation)
decisions depend on the level of cash reserves. Using standard
arguments, it possible to
show that firm value satisfies the following HJB equation for c
∈ [0, C̄∗]
rV (t, C̄, q)−Vt(t, C̄, q) = maxz,X̄
{σ22X̄2Vcc(t, C̄, q) +
[δC̄ + (p− 1)X̄ − z
2
2qζ
]Vc(t, C̄, q)
+ φz(1− α)[V (t, C̄∗(λq), λq)− (C̄∗(λq)− C̄)− V (t, C̄, q)
]+ xd[ψκT q + ψEκEq + C̄ − V (t, C̄, q)]
}. (16)
The left-hand side admits an interpretation analogous to (9).
The first and second terms
15If an incumbent does not raise funds when marketing a
breakthrough, its value jumps from V (t, C̄, q)
to V (t, C̄, λq). If it raises funds, the surplus created (S̃(t,
C̄, q) = V (t, C̄∗(λq), λq) − (C̄∗(λq) − C̄) −V (t, C̄, λq)) needs
to be shared with financiers. While we assume that financiers
receive a fraction α
of S = S̃ + (V (t, C̄, λq) − V (t, C̄, q)) for tractability,
this is without loss of generality and it is alwaysoptimal for the
firm to raise funds upon a breakthrough, see Appendix A.2.1.
19
-
on the right-hand side capture the effect of cash flow
volatility and cash accumulation.
The third term captures the change in value when the incumbent
attains a technological
breakthrough. The fourth term captures the effect of exit due to
creative destruction.
To solve the incumbents’ problem, we conjecture that incumbent
value scales with qt:
V (t, C̄, q) = qt v
(C̄tqt
)≡ qt v(c) (17)
for some function v(c), which represents scaled
(time-independent) firm value. Moreover,
c ≡ C̄t(qt)/qt, C∗ ≡ C̄∗(qt)/qt C∗ ≡ C̄∗(qt)/qt
denote scaled cash reserves, the scaled target cash level, and
the scaled amount raised at
(routine) refinancing events, respectively. Optimal policies are
determined by solving the
HJB equation for v(c):
maxz,X
{σ22X2v′′(c) + v′(c)
[δc + X1−β −X − z
2
2
ζ
L
](18)
+ (1− α)φz [λ(v(C∗)− C∗)− (v(c)− c)] + xd [ψκT + ψEκE + c−
v(c)]− rv(c)}
= 0,
where X(c) ≡ X̄t/Qt represents scaled production. As in extant
cash management mod-
els, the marginal value of cash is monotone decreasing in c
(that is, v′′(c) ≤ 0). The
maximization problem (18) has an interior solution for any
c.
Differentiating (18) with respect to X yields the optimal
production rate:
X(c) = F (A(c)), (19)
where F is a decreasing function of A:
A(c) = −σ2v′′(c)
v′(c)(20)
which represents the incumbent’s effective “risk aversion” and
depends on the level of cash
reserves (see Appendix A.2.1). Moreover, differentiating (18)
with respect to z yields the
20
-
optimal innovation rate:
z(c) =φ (1− α)
ζ
(λw∗ − v(c) + c)v′(c)
, (21)
where w∗ represents the scaled value of an incumbent holding its
target cash level, net
of cash: w∗ ≡ w∗(C∗) = v(C∗) − C∗. The numerator in (21) is a
function of c and
represents the change in firm value when the incumbent achieves
a breakthrough, which
represents the marginal gain of innovation. The denominator is
also a function of c and
represents the marginal loss of using cash to finance
innovation, which is the marginal
cost of innovation.
To derive incumbent value and the two thresholds C∗ and C∗, we
substitute (19) and
(21) into (18) and solve the resulting differential equation
(equation (45)) subject to the
following boundary conditions:
v(0) =v(C∗)− C∗(1 + �)− ω, (22)
v′(C∗) =1 + �, (23)
v′(C∗) =1, (24)
v′′(C∗) =0 . (25)
Condition (22) means that an incumbent raises the amount C∗
whenever cash reserves
are depleted. We only consider parameter values for which the
following inequality
v(C∗)− C∗(1 + �)− ω > ψκT + ψEκE
holds, which guarantees that firm value after refinancing (the
left-hand side) is preferred
to liquidation (the right-hand side). The optimal issue amount
C∗ equalizes the marginal
benefit and cost of financing, such that (23) holds. Condition
(24) means that the value
of one dollar at C∗ equals its value if paid out. When cash
reserves are close to C∗, the
firm’s effective risk aversion, A(c), goes to zero and
precautionary concerns are gradually
relaxed. Above C∗, excess cash is paid out and incumbent value
is linear in cash reserves;
21
-
i.e., v(c) = v(C∗) + c −C∗ for c > C∗. Finally, condition
(25) guarantees that the target
cash level maximizes firm value. We summarize our findings in
the next proposition.
Proposition 1 The value of an incumbent is given by (17), where
v(c) is the unique
concave solution to (18) that satisfies the boundary conditions
(22)-(25) (see (45) in
Appendix A.2.1). The value of an incumbent holding its target
level of cash reserves is
given by w(C∗) +C∗, where w(C∗) obtains by substituting
conditions (24)-(25) into (45):
w(C∗) =v(C∗)− C∗ (26)
=2 [µ∗ + xd(ψκT + ψEκE)− (r − δ)C∗]
xd + r + [(xd + r)2 − 2ϕ2(λ− 1)2[µ∗ + xd(ψκT + ψEκE)− (r −
δ)C∗]]1/2
with µ∗ ≡ β (1− β)1β−1 and ϕ2 ≡ φ
2(1−α)2ζ
.
So far, we have taken the rate of creative destruction as given.
We now micro-found
it by studying entrants’ maximization problem.
4.2 Entrants’ optimal policies
We now derive entrants’ optimal policies by starting from cash
retention. As it is for
incumbents, the marginal benefit of entrants’ cash decreases
with cash reserves. Thus,
there should be a target cash level at which the marginal value
of cash equals one, mak-
ing it optimal to pay out excess cash. Yet, entrants never reach
this target and never
pay out cash. Because entrants do not produce any goods, their
cash reserves decrease
monotonically as cumulative R&D expenditures grow (see
equation (6)).
Using standard arguments, entrant value satisfies the following
HJB equation:16
rV E(t, C̄E, q̄)− V Et (t, C̄E, q̄) = maxzE
{− z
2E
2ζE q̄t V
Ec (t, C̄E, q̄)
+ φEzE(1− αE)[V (t, C̄∗,Λq̄)− κTΛq̄ − (C̄∗(Λq̄)− C̄E(q̄))− V
E(t, C̄E, q̄)
] }.
(27)
16Similar to the unconstrained economy, V (t, C∗,Λq̄t)− κTΛq̄ =∫
1
0[V (t, C∗,Λqjt)− κTΛqjt] dj is the
jump in firm value after a breakthrough. The first term in the
integral is firm value when targeting anindustry after a
breakthrough, whereas the second one is the setup cost of
productive assets.
22
-
To solve this optimization problem, we scale entrant value and
cash reserves by the
average quality of frontier inputs, which we denote respectively
by vE(cE) = VE/q̄t and
cE = C̄Et/q̄t. Moreover, we denote by ĈE ≡ C̃Et/q̄t the scaled
amount raised at entry.
Substituting these expressions into (27) delivers a
time-independent HJB equation (see
equation (47) in Appendix A.2.1), which we differentiate to
obtain the optimal innovation
rate:
zE(cE) =φE (1− αE)
ζE
max(Λw∗ − vE(cE) + cE − ΛκT , 0)v′E(cE)
. (28)
Similar to the unconstrained economy, the inequality Λw∗ −
vE(cE) + cE − ΛκT > 0
guarantees that technological breakthroughs create value for
entrants. To derive entrant
value as well as the threshold ĈE, we substitute (28) into (47)
and solve the resulting
differential equation (equation (48)) subject to the following
boundary conditions:
v′E(ĈE) = 1 + �E, (29)
vE(ĈE) = (1 + �E)(ĈE + κE) + ωE, (30)
vE(0) = ψEκE (31)
At the outset, entrants raise financing to cover the setup cost
(κE) and to have some slack
(ĈE) to fund R&D costs. Condition (29) means that ĈE is
optimally set to equate the
marginal benefit (the left-hand side of (29)) and the marginal
cost (the right-hand side)
of external financing. Condition (30) represents the free entry
condition, which we use to
pin down the equilibrium mass of entrants.17 The left-hand side
of (30) is the value of an
incumbent right after entry, whereas the right-hand side is the
total cost of entry (setup
cost, cash, and financing fees). Finally, condition (31) implies
that when cash reserves
are depleted, entrants recover a fraction ψE of their investment
in R&D assets.
Notably, the analysis illustrates that incumbents’ and entrants’
decisions are inter-
twined. Entrants’ innovation rates affect the rate of creative
destruction, which deter-
mines incumbents’ exit threats and their incentives to invest in
innovation (by affecting
17Because entrant value is concave in cE , the condition Λw∗−
vE(cE) + cE −ΛκT ≥ 0 that guarantees
zE(c) > 0 is satisfied for any cE > 0 if it holds at ĈE .
By (30), this is the case if Λw∗ ≥ κE + ΛκT +
�E(κE + ĈE) + ωE .
23
-
w∗ in equation (21)). Any change in incumbents’ value, in turn,
affects entrants’ incen-
tives to innovate and to become the next incumbent (note that w∗
also enters equation
(28)). We now delve deeper into the properties of production and
innovation decisions.
4.3 Analyzing optimal policies
Production and markups Equation (19) illustrates that the
dynamics of X(c) depend
on the firm’s effective risk aversion A(c) via the function F .
The function F is monotone
decreasing in A and, hence, so is X(c) (see Appendix A.2.1).
Because A(c) decreases with
c, X(c) increases with cash reserves. The intuition behind the
math is the following. When
an incumbent’s cash reserves decrease, the associated increase
in effective risk aversion
makes it optimal to limit operating risk and scale down
production. Because effective
risk aversion is negligible at the target cash level (condition
(25) implies A(C∗) = 0), the
production rate in the constrained economy equals the
unconstrained production rate (8)
only when c = C∗. As a result, X(c) < X∗ for all c <
C∗.
Given the demand schedule of the consumption good sector,
selecting quantities is
equivalent to setting prices:
p(c) = X(c)−β ≥ (X∗)−β = 11− β
.
Notably, financing frictions lead incumbents to deviate from the
constant price associated
with the unconstrained economy. In the constrained economy, the
markup (p(c) − 1)
varies with the level of cash reserves and exceeds the markup
that the same firm would
set in the unconstrained economy. In the wake of negative shocks
depleting cash reserves,
constrained incumbents decrease production and increase markups.
As a result, financial
constraints cause markups to be countercyclical to firm’s
operating shocks. Corollary 2
summarizes these results.
Corollary 2 Financing frictions lead incumbents to decrease
their production rate and
charge larger markups in comparison to the unconstrained
economy. The decrease in the
production rate (and the increase in markups) is greater as cash
reserves decrease.
24
-
Extant models of endogenous growth have stressed that firms’
incentives to invest
in innovation arise from the perspective of earning monopoly
rents upon attaining a
breakthrough (e.g., Aghion and Howitt, 1992; Aghion, Akcigit,
and Howitt, 2014; Romer,
1990). Corollary 2 warns that financing frictions and liquidity
constraints may generate a
distinct, reverse relation, which is novel to the literature.
Innovative firms drain their cash
reserves faster, which makes it optimal to decrease their
production rate, limit operating
volatility, and charge higher markups.
In our model, production decisions affect the volatility of cash
flows, which is a func-
tion of c and given by σX(c). This result has interesting
implications for the relation
between cash flow volatility and cash reserves (e.g., Bates,
Khale, and Stulz, 2009). As
previous contributions, our model suggests that cash flow
volatility affects firm’s cash
management. On top of this relation, our model sheds light on a
potential feedback effect
from cash reserves to cash flow volatility via optimal
production decisions. After positive
(respectively, negative) operating shocks, cash reserves
increase (decrease), the curvature
of the value function decreases (increases), and the firm is
willing to take on more (less)
risk. The optimal production rate rises (decreases), and so does
cash flow volatility.
Innovation By (21), the optimal innovation rate is a multiple of
the ratio (λw∗−v(c)+c)v′(c)
.
As explained, the numerator is the gain from investing in
innovation, whereas the de-
nominator represents the marginal “cost” of using cash to
finance innovation. Both the
gain and the cost decrease with cash reserves. The gain
decreases in c because v′(c) ≥ 1,
which implies that any breakthrough-driven jump in firm value is
more valuable when
cash reserves are small. The cost decreases in c because the
precautionary benefit of cash
is greater when cash reserves are small. As a result, z(c)
decreases with cash reserves
if the gain increases at a higher rate than the cost as cash
reserves shrink. A direct
calculation delivers the following result.
Corollary 3 z(c) is decreasing in cash reserves at a given cash
level c if the following
inequality holds:v′(c)− 1
λw∗ − v(c) + c≥ σ−2A(c). (32)
25
-
If two firms have the same fundamental characteristics but
different cash reserves, the
firm with smaller cash reserves may invest more in innovation if
(32) holds. That is, the
firm may substitute production for innovation to increase the
probability of attaining a
technological breakthrough. When a breakthrough occurs, the firm
earns monopoly rents
related to the brand-new technology and can raise outside funds
in light of a “success”
rather than a “failure” (i.e., running out of funds due to
operating losses).
While this result may be surprising in light of the documented
positive relation be-
tween R&D and corporate cash (see the references in footnote
1), one should not confuse
the relation between R&D and target cash reserves with the
relation between R&D and
deviations from this target. The first relation refers to
ex-ante heterogeneity—firms differ
in their R&D technologies (e.g., in the parameters φ, λ ζ),
choose different innovation
rates, and set different C∗—whereas the second focuses on
ex-post heterogeneity—ex-ante
identical firms (setting the same C∗) have different cash levels
because of idiosyncratic
operating shocks. Extant works have focused on the first
relation, whereas we shed light
on the second by characterizing how firms adjust their
innovation rates when operating
shocks erode their cash reserves below the target level.
Although the richness of our
model prevents us from deriving analytical comparative statics,
section 5.1 provides a
thorough investigation of which firm characteristics prompt the
increase in z(c) as cash
reserves decrease. We find that constrained firms with more
volatile profits and lower
margins should increase their innovation rate in the face of
decreasing cash reserves.
As it is for incumbents, equation (28) reveals that entrants’
optimal innovation rate
is also a multiple of the gain-cost ratio, (Λw∗−vE(cE)+cE−ΛκT
)
v′E(cE). Both the gain and the cost
decrease with cash reserves, as for incumbents. Yet, the level
and the rate of decrease
of the cost (i.e., the marginal value of cash) is larger for
entrants than for incumbents.
The reason is that cash is more valuable for entrants as it is
the only resource available
to fund R&D expenditures. Indeed, entrants do not produce
any good and, thus, lack
regular cash flows that can serve to finance R&D and
replenish cash reserves. Because
the marginal value of cash is relatively large for entrants, the
increase in the marginal
cost of innovation as cash reserves decrease shadows the
increase in the gain. Entrants
decrease their innovation rate when cash reserves shrink, in
order to deplete cash reserves
26
-
at a slower pace and have relatively more time to achieve a
breakthrough before running
out of funds. While analytical comparative statics are not
viable, we study the entrants’
innovation rate across different parameterizations in section
5.1.
4.4 Aggregation
We now aggregate incumbents’ and entrants’ optimal choices. To
do so, we derive the
stationary cross-sectional distributions of cash of incumbents
and entrants, which we
denote respectively by η(c) ∈ [0, C∗] and ηE(c) ∈ [0, ĈE].
The distributions of cash Our analysis so far shows that
constrained firms adapt
their production and innovation choices to their cash positions,
which provide firms with
financial flexibility. The distribution of cash allows us to
weigh these decisions across the
population of active firms with different levels of cash
reserves.
Incumbents’ level of cash reserves varies because of operating
profits, losses, and in-
novation expenditures. Moreover, cash reserves are reflected at
C∗ because of payouts,
and jump upward at refinancing events. We show that the
distribution of this controlled
process (see equation (54)) satisfies the following Kolmogorov
forward equation:
1
2
(σ2(c)η(c))
)′′ − (µ(c)η(c))′ − xdη(c) − φz(c)η(c) = 0 (33)with µ(c) =
δc+X1−β(c)−X(c)− z
2(c)
2ζ and σ(c) = σX(c).
We interpret this equation heuristically and report the
technical details in Appendix
A.2.2. In any time interval, the probability mass can move to
adjacent points because
of operating cash flows and volatility (the first two terms).
Else, it moves because of
creative destruction (which hits at rate xd) or because of
technological breakthroughs
(which occur at rate φz(c)). Equation (33) is solved for a
continuous η(c) subject to the
27
-
following boundary conditions:
η(0) = 0
σ2(0)η′(0) + σ2(C∗)[η′(C∗+)− η′(C∗−)] = 0∫ C∗
0
η(c)dc = 1.
(34)
The first condition implies that there is no probability mass at
zero, as incumbents move
to C∗ as soon as they deplete cash reserves. This financing
policy implies an inflow of
mass at C∗ and a kink in the distribution, as captured by the
second condition (in which
η′(C∗+) 6= η′(C∗−)). The last condition guarantees that the area
under the curve is one.
We next turn to the distribution of entrants’ cash. Each
entrant’s cash reserves
deterministically decrease over time, until the entrant attains
a breakthrough or depletes
cash reserves and liquidates. The exiting firm is immediately
replaced by a new entrant,
so that the free entry condition is always binding. The
distribution of entrants’ controlled
cash reserves (see equation (58)) satisfies:
−(−z
2E(c)
2ζEηE(cE)
)′− zE(cE)φEηE(cE) = 0 .
In any time interval, the probability mass is driven down by
R&D expenditures (the first
term). Moreover, the probability mass moves because of
technological breakthroughs,
which happen at rate φEzE(c). To uniquely pin down the
distribution, we impose the
unit mass condition:∫ ĈE
0ηE(cE)dcE = 1.
General equilibrium quantities We next use the distributions of
cash to derive the
equilibrium quantities of the constrained economy. Along the
“balanced growth path,”
all aggregate quantities grow at the endogenous, constant rate
g. As it is for the uncon-
strained economy, the law of large numbers implies that the
growth path is smooth and
is driven by improvements in quality of inputs (i.e.,∫ 1
0qjtdj = e
gt). In equilibrium, the
28
-
growth rate is the sum of incumbents’ and entrants’
contributions to growth:
g = (λ− 1)φ∫ C∗
0
z(c)η(c)dc︸ ︷︷ ︸gI
+ (Λ− 1)φEmE∫ ĈE
0
zE(cE)ηE(cE)dcE︸ ︷︷ ︸gE
. (35)
Because innovation decisions are independent and identically
distributed across firms, the
law of large numbers implies that the contribution of incumbents
to economic growth (the
first term in (35)) is given by the size of their quality
improvements multiplied by their
average innovation rate. Similarly, the contribution of entrants
(the second term in (35))
is given by the size of their quality improvements times the
average innovation rate across
the measure of active entrants, which represents the rate of
creative destruction:
xd = φEmE
∫ ĈE0
zE(cE)ηE(cE)dcE.
The measure of active entrants is pinned down by the free entry
condition (30).
Equilibrium consumption also grows at the endogenous rate g. The
household’s in-
tertemporal choice delivers the standard Euler equation
g =r − ρθ
.
Together with (35), the Euler equation pins down the equilibrium
interest rate:18
r = ρ+ θ
[(λ− 1)φ
∫ C∗0
z(c; r)η(c; r)dc+ (Λ− 1)xd(r)]. (36)
Equations (35) and (36) illustrate that while incumbents and
entrants set optimal
policies by taking r as given, their policies actually feed back
into r. This feedback implies
that any direct increase (or decrease) in g has a “dampened”
effect in equilibrium via its
impact on r. As an illustration, consider an exogenous
improvement in the incumbents’
R&D technology; e.g., an increase in φ. Incumbents’ value
and their innovation rates
increase, which makes entry more attractive and spurs creative
destruction. These effects
18We numerically solve this fixed point equation under the
technical constraints r > δ and r > g.
29
-
lead to a direct increase in g. The rise in g leads to a rise in
r via the representative
household’s intertemporal problem. The larger r, however,
implies a greater cost of
capital for firms, which depresses firm values and innovation
rates. This countervailing
effect dampens the direct increase in g. Similarly, any decrease
in aggregate innovation
rates leads to a decrease in g, which, in turn, reduces r. Yet,
the lower r implies a lower
cost of capital and, thus, larger innovation rates that partly
offset the initial decline in g.
In section 5.2, we further analyze the properties of the
equilibrium and compare it to
the unconstrained economy.
5 Model analysis
This section provides additional results by comparing the
constrained and unconstrained
economies via numerical examples. Table 1 reports the baseline
parameterization. We set
λ = 1.04 and Λ = 1.10 to capture the observation that
innovations by entrants tend to be
path-breaking, whereas innovations by incumbents tend to be
incremental (e.g., Gao et
al., 2014; Acemoglu et al., 2013). Because they are more
radical, entrants’ breakthroughs
have longer gestation periods and are more costly than those of
incumbents. Hence, we
set φE = 1.5 to be lower than φ = 3 and ζE = 2 to be larger than
ζ = 1.
As external funds are more expensive for entrants than for
incumbents, we set � = 0.06
and ω = 0.01 for incumbents and �E = 0.10 and ωE = 0.03 for
entrants, which are in the
range of the estimates of Hennessy and Whited (2007) and
Altinkilic and Hansen (2000).
We set the surplus to financiers when funding a breakthrough
equal to α = 0.02 and
αE = 0.07. We set ψ = 0.9, which implies that incumbents lose
10% of their investment
in productive assets at exit (see Hennessy and Whited, 2007). As
R&D assets are less
tangible, we set ψE = 0.6. Finally, we set β = 0.16 and σ =
0.40, implying that the cash
flow volatility σX(c) varies between 5% (when c = 0) and 13.8%
(when c = C∗).
30
-
5.1 Production, innovation, and firm values
Figure 1 shows optimal production and innovation decisions as
well as firm values. The
top panel shows that the incumbents’ production rate associated
with the constrained
economy increases with cash reserves. It is equal to the
production rate associated with
the unconstrained economy only when the firm holds its target
cash level (at which effec-
tive risk aversion is zero as financial constraints are
relaxed), while it is below otherwise.
Cash flow volatility also increases with cash reserves.
In the middle-left panel of figure 1, the incumbents’ innovation
rate associated with
the constrained economy is larger than that associated with the
unconstrained one, and
more so for small cash reserves (z(c) is 38% larger than z∗ for
c close to zero). The
innovation rate decreases with cash reserves when these are
sufficiently small, and it is
quite flat otherwise. Recall that this pattern arises if the
gain from innovation increases
more than the cost as cash reserves decrease. Comparing the
middle-left panel of figure
1 with figure 2 reveals that this is the case for firms with
volatile cash flows (high σ) and
low margins (small β). Firms with smaller β choose larger
production rates and charge
lower markups, which makes it less costly (in terms of foregone
profits) to substitute
production for innovation. Firms with larger σ quickly replenish
or deplete their cash
reserves because of their volatile profits. For these firms, the
marginal cost of innovation
increases at a lower rate than the gain as cash reserves wane.
Notably, if β is sufficiently
large or σ is sufficiently small, z(c) increases with cash
reserves.
In figure 3, we vary the magnitude of financing costs. Fixed
financing costs make
it optimal to raise funds in lumps, which makes refinancing
events more expensive (and
cash more valuable) compared with a setup with only proportional
costs. For larger fixed
costs, the increase in the marginal cost of innovation
associated with decreasing cash
reserves can locally offset the rising gain, which can cause
z(c) to be non-monotonic in
cash when cash reserves are almost depleted (and refinancing is
more likely).19 Yet, the
change in z(c) is not very sizable—Brown et al. (2013)
consistently find that shifts in
the supply of equity (and the change in financing frictions
thereof) have little impact
19As we explain in section 5.2, the aggregate impact of this
drop is negligible because the mass ofincumbents with low cash
reserves is small (η(0) = 0).
31
-
on innovation decisions of mature firms. Absent fixed costs,
larger financing costs � are
associated with an upward shift in z(c) because the gain from
innovation increases at
a larger rate than the cost.20 Differently, a larger α is
associated with a lower z(c) as
it trims incumbents’ gain from marketing a breakthrough.21
Interestingly, the financing
friction that discourages incumbents’ innovation is the one
incurred at the “marketing”
phase (α) rather than those incurred during the “discovery”
phase (�, ω).
Via the threat of creative destruction, incumbents’ innovation
choices also depend on
the financing frictions borne by entrants. The middle-right
panel of figure 3 shows that
a greater ωE is associated with higher z(c). The reason is that
an increase in entrants’
financing frictions reduces the threat of creative destruction
for incumbents (as we confirm
in section 5.2). As a result, w∗ in (21) increases and, thus,
incumbents have greater
incentives to invest in innovation. Similar arguments imply that
z(c) shifts upward as a
result of larger entry costs (see figure 3, bottom-left
panel).
Turning to entrants, the bottom-left panel of figure 1 shows
that zE(cE) exceeds z∗E
if cash reserves are sufficiently large. The entrants’
innovation rate increases with cash
reserves. As previously explained, cash is more valuable for
entrants than for incumbents
because it is the only resource available to finance the flow
cost of R&D. When cash
reserves decrease, the increase in the marginal cost of
innovation more than offsets the
rising gain and, thus, zE(cE) increases with cE.
Section 5.2 illustrates that greater setup costs or financing
frictions reduce the rate
of creative destruction. Yet, comparing the bottom-left panel of
figure 1 with figure 4
highlights that a larger (smaller) κE is associated with larger
(smaller) zE(cE), especially
for high cash levels. Moreover, more severe financing costs have
a modest effect on zE(cE).
The reason is that larger entry costs or financing frictions
reduce entrants’ innovation via
the extensive margin—i.e., the measure of active entrants, mE
(see figure 6)—rather than
20We do not display this environment in the figure. In a
previous version of the paper, we solved forthe case with
proportional costs only (see Appendix A.2.1). Absent fixed costs,
firms find it optimal toraise an infinitesimal amount whenever cash
reserves are depleted. At c = 0, the marginal gain and costof
raising funds are equated: v′(0) = 1 + �. Absent fixed costs, the
marginal value of cash is relativelylower for any c. The plots
associated with this environment are available upon requests.
21For very large values of α, z(c) can be smaller than z∗. Note,
however, that in our model werealistically impose that α ≤ αE .
32
-
the intensive one—i.e., the innovation rate of active entrants,
zE(cE). In equilibrium,
fewer entrants increase the value of incumbents as well as the
value of becoming the next
incumbent for an active entrant (indeed, w∗ enters both (21) and
(28)). As a result,
innovation rates of active entrants can even increase with
larger entry hurdles.
The bottom-right panel of figure 4 shows that higher recovery
rates of R&D assets are
associated with larger innovation rates for small levels of cash
reserves. Larger (lower)
recovery rates mean that liquidation is less (more) costly,
which makes entrants more (less)
willing to invest in innovation, consistent with Acharya and
Subramaniam (2009). From
the incumbents’ perspective, this positive effect is offset by
an increase in the threat of
creative destruction, which in turn reduces incumbents’
incentives to invest in innovation
(see figure 3, bottom-right panel).
Importantly, our analysis shows that financing frictions can
lead to an increase in firm
values (figure 1, middle- and bottom-right panels). In the
unconstrained economy, entrant
value is equal to v∗E = κE by the free entry condition. In the
constrained economy, the
free entry condition accounts for issuance costs and the cash
reserves with which entrants
optimally start. The value of constrained entrants ranges from
vE(0) = ψEκE < v∗E to
vE(ĈE) > v∗E (see equation (30)), implying that constrained
entrants exceed their uncon-
strained value if cash reserves are sufficiently large. When
cash reserves are large, entrants
invest more in innovation and have a larger probability of
attaining a breakthrough, which
increases their value. When cash reserves are small, conversely,
financing and liquidation
costs depress the value of entrants below their unconstrained
value. Because the rate of
creative destruction is smaller in the constrained economy (as
we discuss in section 5.2),
incumbents invest more in innovation and are more valuable than
in the unconstrained
economy. These findings are consistent with Stangler (2009), who
finds that cold markets
(i.e., times characterized by tighter financial constraints)
lead to funding of more success-
ful firms, as well as with anecdotal evidence suggesting that
hot markets are associated
with less valuable firms being financed (see Gupta, 2000).
33
-
5.2 Financing frictions and growth
Figure 5 shows the distribution of cash of incumbents and
entrants. Incumbents’ cash
reserves can grow because of retained earnings and external
financing, whereas entrants
build cash reserves only by raising external funds. An entrant’s
cash reserves decrease
over time, until it attains a breakthrough or runs out of funds.
Entrants decrease their
innovation rate when cash reserves are small, which helps
deplete reserves at a slower
pace. The ensuing entrants’ distribution of cash is
non-monotonic, and a large mass is
concentrated around cE = 0, in line with the evidence (e.g.,
Lins, Servaes, and Tufano,
2010).22 Conversely, the incumbents’ cash reserves are
relatively large most of the time
as most of the mass is concentrated around large levels of cash
reserves.
The shape of the distributions plays a key role in determining
how firms’ policies
aggregate in equilibrium. Incumbents’ and entrants’
contributions to growth (respectively,
gI and gE in (35)) are obtained by weighing firms’ innovation
rates using the distributions
of cash. This implies that financing frictions affect growth not
only through firms’ optimal
innovation rates (which are functions of cash reserves), but
also through the probability
with which firms will find themselves with a given level of cash
reserves (and, thus,
different degrees of financial flexibility).
In our baseline parameterization, the rate of economic growth is
about 2.00% (re-
spectively, 2.55%) in the constrained (unconstrained) economy.
Entrants generate 69.8%
of growth in the unconstrained economy, compared with only 54.9%
in the constrained
economy.23 The reason is that the entrants’ distribution has a
large mass around low
cash levels, for which the constrained entrants’ innovation rate
falls below the uncon-
strained rate (see figure 1). Conversely, even if the
contribution of incumbents with very
low levels of cash reserves is modest, the incumbents’
contribution to growth is larger
as their innovation rates in the constrained economy exceed
those in the unconstrained
economy. Notably, financing frictions importantly change the
composition of growth and
place entrants at a disadvantage, consistent with Brown et al.
(2009) and Nanda and
22Differently, existing cash management models (e.g., Bolton,
Chen, and Wang, 2011) generate adistribution that is monotonically
increasing and has most mass concentrated at the target cash
level.
23In Acemoglu, Akcigit, Bloom, and Kerr (2013), entrants
generate 58% of economic growth.
34
-
Nicholas (2014).
Figure 6 compares the constrained and unconstrained economies
and investigates how
the composition of growth changes as entry costs vary. First,
the figure shows that the
entrants’ contribution to growth decreases with the magnitude of
entry costs. Larger
entry costs do not decrease the optimal innovation rate of
active entrants (as discussed in
section 5.1) but importantly decrease the equilibrium measure of
active entrants (third
panel of figure 6). By acting as additional entry barriers,
financing frictions lead to a
further reduction in the measure of active entrants and, thus,
to a further decrease in the
entrants’ contribution to growth. Nanda and Rhodes-Kropf (2013)
consistently find that
shifts in capital supply largely impact the extensive margin of
innovation by new firms.
We label this growth-decreasing effect as the hurdle effect of
financing frictions on growth
because financing frictions act as entry barriers that prevent
innovations by new firms.
Second, figure 6 shows that the incumbents’ contribution to
growth increases with
entry costs and is larger in the constrained economy. In fact,
the decrease in the entrants’
contribution to growth reduces the incumbents’ probability of
being hit by creative de-
struction. The lower threat of creative destruction shifts the
incumbents’ optimal inno-
vation rate upward (as shown in figure 3 and discussed in
section 5.1). When aggregated
across the population of incumbents via the stationary
distribution, this shift translates
into a larger contribution of incumbents to growth. We label
this growth-enhancing effect
as the haven effect of financing frictions on growth because
financing frictions effectively
shield incumbents from exit threats.
Figure 7 shows the compound effect of these strengths and
illustrates that financing
frictions have a non-monotonic impact on growth.24 If κE is low,
the hurdle effect domi-
nates and an increase in financing frictions leads to a decrease
in growth. In this case, the
decrease in the rate of creative destruction more than offsets
the increase in incumbents’
innovation rate. Conversely, if κE is sufficiently large, the
haven effect can dominate and
an increase in financing frictions may lead to a rise in g. In
this case, the decrease in
the rate of creative destruction is more than offset by the
increase in incumbents’ inno-
24In this figure, we plot g up to the critical κE for which xd
is zero. Above this level (which variesacross parameterizations),
innovation is pursued by incumbents only.
35
-
vation rates. A key implication of our model is that financing
frictions do not need to
be detrimental to growth. Indeed, financing frictions can spur
growth if the haven effect
dominates—i.e., when entry barriers are sufficiently large.
6 Conclusion
In the aftermath of the recent financial crisis, financial
constraints have raised much
attention among academics and policymakers. While their effects
have been mostly in-
vestigated from a financial stability perspective, their impact
on economic growth is still
poorly understood. This paper seeks to fill this gap by building
a model of endogenous
growth featuring financially constrained firms.
Comparing our constrained economy with an identical but
unconstrained economy, we
single out the effects of financing frictions on growth. We show
that financing frictions do
not need to be detrimental to innovation. In fact, constrained
incumbents characterized
by low and volatile profits substitute production for innovation
in the face of shrinking
cash reserves (and decreasing financial flexibility). Also,
financing frictions increase the
entrants’ investment in innovation above the unconstrained level
if their cash reserves are
sufficiently large (i.e., right after a financing round). In
equilibrium, financing frictions
have a strong impact on industry composition and result in fewer
but more valuable en-
trants. On aggregate, financing frictions importantly change the
composition of growth by
deterring the entrants’ contribution to economic growth (the
hurdle effect) but prompting
the incumbents’ contribution (the haven effect). If the haven
effect dominates, financing
frictions can spur growth, which occurs if setup costs are
sufficiently large.
36
-
A Appendix
A.1 The unconstrained economy
In this appendix, we report additional calculations and results
referred to the uncon-strained economy described in section 3. The
maximization problem of the consumptiongood sector (as reported in
the main text) gives the demand curve for each frontier in-put, the
price of each input (equation (7)), and the incumbents’ optimal
production rate(equation (8)). Substituting (7) and (8) into (2),
we obtain the aggregate output of theconsumption good sector:
Y∗t = (1− β)1β−2∫ 1
0
qjtdj ,
and the competitive labor wage W ∗t = β(1− β)1β−2 ∫ 1
0qjtdj .
V (t, q) denotes the value of an incumbent operating in the
unconstrained economy (thesubscript j will be suppressed when it
causes no confusion). The incumbent producesat the value-maximizing
rate X̄∗t and pays out any operating profit to shareholders.
Asimple calculation gives an expression for operating profits on
each time interval:
β qt (p∗)1−
1β = βqt (1− β)
1β−1 .
Following standard arguments, V (t, q) satisfies the
Hamilton-Jacobi-Bellman (HJB) equa-tion (9). To solve incumbents’
maximization problem, we conjecture incumbent value tobe linear in
qt, V (t, q) = V (qt) = qtv
∗ for some v∗ > 0 representing scaled
(time-invariant)incumbent value. Substituting into (9), we
obtain
maxz∗
{µ∗ − (z
∗)2
2ζ + φz∗(λv∗ − v∗) + x∗d (ψκT + ψEκE − v∗)
}= r∗v∗, (37)
where the auxiliary quantity
µ∗ ≡ β (1− β)1β−1 (38)
represents scaled profits. The maximization of (37) delivers the
optimal innovation ratein (11). Substituting the optimal innovation
rate into (37) gives
µ∗ +φ2
2ζ(λ− 1)2(v∗)2 + x∗dψκT + x∗dψEκE = (x∗d + r∗)v∗,
which we solve with respect to v∗ (see the system of equation
(42)).
Let us now consider entrant firms. Their value in the
unconstrained economy isdenoted by V E(t, q̄). As explained in the
main text, q̄t represents average quality offrontier inputs at time
t. Following standard arguments, entrant value satisfies the
HJBequation (12). To solve this maximization problem, we conjecture
that the value of anactive entrant scales with q̄t, V
E(t, q̄) = q̄tv∗E, where v
∗E represents the scaled entrant
37
-
value. Substituting into (12) gives:
maxz∗E≥0
{− (z
∗E)
2
2ζE + φEz
∗E(Λv
∗ − v∗E − ΛκT )}
= (r∗ − g∗)v∗E, (39)
which we differentiate to obtain the entrants’ optimal
innovation rate, which is given byequation (14). Substituting (14)
into (39) gives
(r∗ − g∗)v∗E =φ2E2ζE
(Λv∗ − v∗E − ΛκT )2,
which we solve for v∗E :
v∗E = Λ(v∗ − κT ) +
r∗ − g∗ −√
(r∗ − g∗)2 + 2φ2EΛ(v∗ − κT )(r∗ − g∗)/ζEφ2E/ζE
. (40)
We plug the above expression into the free entry condition
v∗E(m∗E) = κE. The free entry
condition, together with equation (14), implies that the
inequality Λv∗ > κE +ΛκT needsto hold, otherwise the innovation
rate would be negative. Also, the free entry conditionpins down the
equilibrium measure of active entrants and the rate of creative
destruction,see (1