Full text of Working Papers (Federal Reserve Bank of Chicago) : On the Timing of Innovation in Stochastic Schumpeterian Growth Models, Working Paper 2004-11

The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.

Federal Reserve Bank of Chicago

On the Timing of Innovation in
Stochastic Schumpeterian
Growth Models
Gadi Barlevy

Revised August 9, 2005
WP 2004-11

On the Timing of Innovation in
Stochastic Schumpeterian Growth Models∗
Gadi Barlevy
Economic Research
Federal Reserve Bank of Chicago
230 South La Salle
Chicago, IL 60604
e-mail: gbarlevy@frbchi.org
August 9, 2005

Abstract
Economists have recently revived the notion that recessions play a useful role in fostering
innovation and growth. But in practice, a major source of innovation, R&D, is procyclical.
In fact, R&D is procyclical even for firms that do not appear to be financially constrained.
This paper argues the reason R&D is procyclical is because of a dynamic externality inherent
to R&D that makes entrepreneurs short-sighted and concentrate their innovation in booms
even though it is optimal to concentrate it in recessions. Thus, what previous authors have
argued is a desirable feature of fluctuations in the previous literature — creating opportunities
for intertemporal substitution — turns out to be a social liability in equilibrium.

∗

I am grateful to Jeff Campbell, Marty Eichenbaum, Huw Lloyd-Ellis, Kiminori Matsuyama, Alex Monge,
Joanne Roberts, and Fabrizio Zilibotti for their comments, as well as seminar participants at Northwestern, the
Stockholm School of Economics, Queens’ University, the Chicago Fed, the NBER, the Canadian Macro Study
Group, and the Society of Economic Dynamics.

Introduction
In recent years, economists have revived the Schumpeterian idea that recessions promote various
activities that contribute to long-run productivity.1 Modern reincarnations of this hypothesis
emphasize the role of intertemporal substitution: since the opportunity cost of investing in
growth — the forgone output or sales that could have been obtained instead — is lower in
recessions, there is more incentive to undertake such activities in downturns. Recessions should
therefore stimulate long-run productivity growth, and cyclical fluctuations may raise welfare by
allowing the economy to grow at a lower overall resource cost.
Although some growth-enhancing activities do appear to be countercyclical, one of the major
sources of long-run productivity growth — research and development (henceforth R&D) — is
clearly procyclical. That is, spending on R&D falls rather than rises in recessions, despite
the fact that R&D is relatively less costly in downturns. Griliches (1990) summarizes the
evidence for procyclical R&D activity. More recent work by Fatas (2000) reaffirms Griliches’
conclusions. Still more recently, Comin and Gertler (2004) find that R&D is strongly procyclical
at frequencies of between 8 and 50 years, suggesting sustained low output growth is associated
with low R&D activity.
How can we reconcile this apparent conflict between theory and data? Is procyclical R&D
inefficient, and if so is there a role for policy intervention? One hypothesis is that recessions
do not really create opportunities for intertemporal substitution, e.g. because R&D does not
use resources that could otherwise be used in production. Indeed, Aghion and Saint Paul
(1998) show that if productivity-improving activities use final goods rather than factor inputs,
innovation will be procyclical.2 But Aghion and Saint Paul are also quick to dismiss this
explanation. As Griliches (1984) observes, the main input into R&D is labor, not produced
goods. Moreover, productivity in the goods sector is procyclical, even after correcting for
variable utilization as in Burnside, Eichenbaum, and Rebelo (1993) and Basu (1996), whereas
Griliches (1990) concludes productivity in the R&D sector (with patents as a proxy for output)
is acyclical. This suggests that the forgone output from employing resources in R&D is indeed
lower in recessions, and procyclical R&D may be socially inefficient.
1
See, for example, Hall (1991), Mortensen and Pissarides (1994), and Gomes, Greenwood, Rebelo (2001) on
the effects of recessions on search; Cooper and Haltiwanger (1993), Aghion and Saint Paul (1998), and Canton
and Uhlig (1999) on the effects of recessions on technical change; and DeJong and Ingram (2001), Dellas and
Sakellaris (2003), and Barlevy and Tsiddon (2004) on the effects of recessions on human capital accumulation.
2

Following Rivera-Batiz and Romer (1991), this is known as the lab-equipment model. Comin and Gertler
(2004) also assume R&D uses final goods, and note that this assumption helps to generate procyclical R&D.

1

So why would firms fail to concentrate R&D in recessions when it is relatively less costly? One
natural explanation is that firms are somehow constrained from doing so. For example, Aghion,
Angeletos, Banerjee, and Manova (2005) argue that credit constraints may discourage firms from
undertaking growth-enhancing activities in recessions. In their model, the opportunity cost of
R&D is lower in recessions, but downturns also reduce the amount of internal funds that firms
can use to finance ongoing R&D projects. Aghion et al show that if firms had unlimited access
to credit, they would choose to concentrate growth-enhancing activities in recessions. But if
firms anticipate they will be constrained from borrowing, they may focus their R&D efforts
in booms. This logic would suggest that facilitating borrowing in recessions should serve to
restore efficiently countercyclical R&D.
While credit constraints are undoubtedly responsible for part of the cyclical pattern in R&D,
there are reasons to suspect they cannot account for all of it. First, as I document below, R&D
remains procyclical even for firms that are relatively unconstrained. Second, other investments
in productivity that are just as vulnerable to credit constraints do not appear to be procyclical.
For example, Francois and Lloyd-Ellis (2003) cite various studies that show firms concentrate
reorganization, retraining, and machine upgrading in periods of weak demand. Since these
activities can be just as costly as R&D, binding credit constraints would presumably lead these
activities to also turn procyclical, a point Aghion et al (2005) argue explicitly. Thus, there
appears to be something about R&D that makes it particularly prone to being procyclical,
independently of credit market conditions.
This paper argues that the distinguishing feature of R&D that helps to explain its procyclicality is a dynamic externality inherent to the research process. In particular, when an innovator
comes up with a new idea, she allows others to build on and further profit from her insights,
often at her own expense. This is distinct from a firm upgrading machines to newer existing
models, which improves its productivity but confers no spillovers to other firms.
It is already well-known from the work of Grossman and Helpman (1991) and Aghion and
Howitt (1992) that if innovators cannot appropriate the spillovers from their research, there
may be too little steady-state R&D activity. But there is also an important temporal aspect to
this externality that previous work has ignored, and which holds the key to the procyclical bias
in R&D. Since rival innovators are more likely to succeed in improving on a new idea the more
time they have to refine and work on it, the benefits from a new idea that pay off in the future
are increasingly more likely to accrue to someone other than the original innovator. Hence,
the incentives to engage in R&D depend on the short-term benefits of successful innovation.
Since profits are procyclical, innovators chasing short-term profits will undertake more R&D in
2

booms than is socially optimal. If profits are sufficiently procyclical, R&D will turn procyclical,
even though booms are precisely when the cost of R&D is highest.3
This paper demonstrates the above intuition in a general equilibrium model. The main
difficulty lies not with establishing the procyclical distortion in R&D, but in getting the model
to deliver sufficiently volatile profits. The standard Schumpeterian model predicts equilibrium
R&D will exhibit a procyclical bias, but it also implies that profits are no more volatile than
other macroeconomic series. As a result, the incentive to shift R&D towards booms will not be
enough to turn R&D procyclical. However, when I modify the model to accord with the fact
that profits are highly procyclical, namely by introducing fixed costs and insuring that markups
are not too countercyclical, the absolute level of R&D will be higher in booms, even though the
optimal path still dictates it should be countercyclical.
I then use the model to explore the welfare implications of such inefficiently procyclical R&D.
I show that the distorted timing of R&D increases the cost of achieving productivity growth.
Thus, cyclical fluctuations impose a social cost. This cost is three times as large as what Lucas
(1987) estimated for the cost of business cycles based only on risk aversion.
The paper is organized as follows. Section 1 reviews the evidence on the procyclicality of
R&D and argues it cannot be entirely explained by credit constraints. Section 2 formalizes
the intuition for why equilibrium innovation suffers from a procyclical bias. Section 3 extends
these results to a more realistic environment that can be calibrated to assess the welfare implications of procyclical R&D. Section 4 conjectures whether the results would survive additional
modifications. Section 5 concludes.

1. Empirical Evidence
I begin by briefly reviewing the evidence on the cyclicality of R&D. As noted in the Introduction,
the classic reference on the cyclical properties of R&D is Griliches (1990). He argues that
3
At first, this result appears to reiterate the insight in Shleifer (1986) that enterpreneurs introduce new technologies in booms to capture high profits. However, Shleifer examines when firms implement new technologies,
not when they undertake R&D. When Francois and Lloyd-Ellis (2004) endogenize innovation in Shleifer’s model,
they find it is countercyclical: entrepreneurs undertake innovation in recessions, but wait to implement their ideas
in booms. Here, firms implement new ideas immediately. However, it will still be socially optimal to concentrate
R&D in recessions, which creates strong incentives to concentrate R&D during recessions. The contribution of
this paper is to explain why private entrepreneurs may concentrate R&D in booms despite these incentives.

3

both R&D spending and its output, patents, are procyclical.4 Figure 1 reproduces some of
these findings. In particular, it plots the growth rate of real GDP against the growth rate
of two distinct measures of R&D. The first measure is the growth rate of inflation-adjusted
expenditures on R&D funded and performed by private industry, as reported by the National
Science Foundation (NSF). The second is the growth rate of full-time equivalent R&D scientists
and engineers employed in companies performing R&D. The latter only captures part of the
inputs into R&D, but it has the advantage that it doesn’t depend on the price of R&D inputs.
This is important, since inflation adjustments to nominal R&D expenditures may not accurately
reflect changes in the prices of R&D inputs.
As evident from the figure, the two series track each other relatively closely, and both tend
to track real GDP growth. In particular, the growth rate of R&D declines in nearly all NBER
recession years before 1980. However, R&D appears to be less synchronized with NBER recession years in the second half of the sample. For example, growth in R&D was essentially
flat through the 1980 recession. While growth in R&D employment fell in the 1991 recession,
growth in R&D expenditures only started to decline after the recession ended. Conversely,
although R&D expenditures fell dramatically during the 2001 recession, employment growth in
R&D began to decline several years earlier. But even if the exact timing does not correspond to
NBER dating conventions between 1980 and 2002, R&D growth and GDP growth are positively
correlated over this period.
Figure 1 uses aggregate data. To gain some insight on the role of credit constraints in accounting for the procyclical pattern, I turn to firm level data on R&D expenditures. In particular,
I examine whether firms that are relatively less credit constrained have less procyclical R&D
expenditures. In this regard, the Standard & Poor’s Compustat database provides data on
R&D expenditures for publicly traded companies. Although this sample does not capture all of
the R&D activity in the NSF data, it turns out that large firms account for the vast majority
of R&D activity, and thus tracks the aggregate time series quite well. Figure 2 reports the
average growth rate of real R&D over the previous year among all Compustat firms reporting
positive R&D spending, together with inflation adjusted private R&D as reported by the NSF
and depicted in Figure 1. The two series again track each other quite closely; if anything, R&D
growth is more systematically correlated with NBER recession dates among Compustat firms.
4

The fact that patents are synchronized with the business cycle might seem surprising given the time it
presumably takes to undertake research. However, Griliches argues that micro data shows R&D leads to patenting
without significant delay: “[T]he evidence is quite strong that when a firm changes its R&D expenditures, parallel
changes occur also in its patent numbers. The relationship is close to contemporaneous with some lag effects
which are small and not well estimated (Hall, Griliches, and Hausman, 1986). This is consistent with the
observation that patents tend to be taken out relatively early in the life of a research project.” (p1674).

4

I consider two ways of identifying firms that are relatively unconstrained financially. One
is based on the cash flow available to the firm, which would mitigate the need to borrow
externally in order to finance expenditures. That is, I look at whether the growth rate of R&D
expenditures for firms that report at least $50 million of cash (in 1996 dollars) in the year in
which they undertake R&D appears less procyclical. This corresponds to the top quintile of the
firms in my sample as ranked by their cash flow within each year. As an alternative indicator
of constrainedness, I use the net worth of a firm, given that net worth can be used as collateral
against which the firm can borrow. That is, I look at the cyclicality of the growth rate of R&D
expenditures for firms reporting at least $150 million in net worth (in 1996 dollars) in the year
they undertake R&D. Not surprisingly, there is a fair degree of overlap in the two samples.
Figure 3 illustrates the growth rate of real R&D expenditures for these two groups. The results
are striking: the growth rate of R&D among these relatively unconstrained firms is actually
more synchronized with the business cycle than for the sample as a whole: the correlation with
real GDP growth is higher, and the average growth rate of R&D falls in each NBER recession.
Changing the cutoff levels does not change these qualitative results.
It would therefore appear that even firms that are relatively free to concentrate their R&D
activity in downturns choose not to do so. But since the opportunity cost of innovation appears
to be countercyclical, we need to explain why firms would deliberately pursue such a policy.

2. A Model of Schumpeterian Growth
For my model, I use a variation of the Grossman and Helpman (1991) quality-ladder model that
allows for fluctuations in the relative productivity of the goods sector. To maintain tractability,
I initially impose certain restrictions on preferences and technology that are unrealistic but
convey the intuition more transparently. I then relax these in the subsequent section.
The economy consists of a representative agent whose instantaneous utility is given by
U (Ct ) = Ct

(2.1)

I relax the assumption of risk neutrality in the next section. As will become clear, this assumption plays a role in the analysis, but is not essential. Utility is discounted at rate ρ.
The agent is endowed with a constant labor endowment L pet unit time and an initial capital
stock normalized to one. For now, I assume capital is not accumulable and does not depreciate,
i.e. it is a fixed factor (e.g. land). This assumption will also be relaxed in the next section.
5

Labor and capital can be converted into consumption goods according to a two-stage process.
First, labor is converted into a series of intermediate goods indexed by j ∈ [0, 1]. Second,
intermediate goods are combined with capital to produce a non-storable consumption good.
At the second stage, I assume the intermediate goods xjt must first be assembled into a
composite good, whose quantity I denote by Xt , using a Cobb-Douglas technology
·Z 1
¸
Xt = exp
ln xjt dj
(2.2)
0

Given Xt units of this composite good and Kt units of capital, it is possible to produce Yt units
of the consumption good, where
Yt = zt Ktα Xt1−α
(2.3)
Here zt reflects productivity in the final goods sector. To capture the fact that productivity in
the goods sector varies over the business cycle, I assume zt follows a Markov switching process
between two states, Z1 ≥ Z0 , with a constant hazard rate µ. I treat these fluctuations as
exogenous, although one could potentially derive them endogenously.5
Turning next to the production for intermediate goods, I assume each good j can be produced
from labor according to a linear technology
xjt = λjt Ljt

(2.4)

where Ljt denotes the amount of labor employed in the production of good j at date t. The
coefficient λjt is given by
λjt = λmjt
(2.5)
where λ > 1 is a constant and mjt is an integer that denotes the generation of technology used
for producing good j at date t. Each good j starts out at generation mj0 , respectively, but
agents can advance to higher-generation technologies by engaging in research. That is, starting
with generation mj , devoting Rj units of labor to research on good j gives rise to a hazard
φRj of discovering generation mj + 1 in the next instant, which will be more productive given
λ > 1. Once a new generation is discovered, research can begin on the next generation. This
last assumption captures the spillovers inherent to research: when one researcher succeeds in
discovering a new generation, she allows others to build on her work and develop the next
successive technology. In line with the evidence on the acyclicality of productivity in the
research sector, I assume φ is fixed over time.
5

For example, Benhabib and Farmer (1994) describe an economy with spillovers in which there are equilibria
where the scale of production, and thus the productivity of individual producers, fluctuates over time.

6

To recap, labor in this economy has two uses: production and innovation. Agents must therefore choose between producing more consumption goods now and employing labor in research
activities that allow for more consumption goods to be produced in the future.
R1
To see this tradeoff formally, let Mt = 0 mjt dj denote the average generation across intermeR1
diate goods, and let Rt = 0 Rjt dj denote aggregate employment in R&D. We can now express
the output of consumption goods Yt directly in terms of labor resources. In particular, suppose
each sector uses the same amount of labor, i.e. Ljt = L − Rt , which is both optimal and holds
in any equilibrium. Since the supply of capital is normalized to 1, it follows that
¤1−α
£
Yt = zt K α Xt1−α = zt λMt (L − Rt )

(2.6)

The indirect productivity of labor in terms of final goods thus depends on both an exogenous
term zt and an endogenous term λ(1−α)Mt . We assume the law of large numbers holds across
intermediate goods producers, which implies Ṁt = φRt . The growth rate of the endogenous
component of labor productivity is therefore given by
d (1−α)Mt
λ
= (1 − α) Ṁt ln λ = (1 − α) φRt ln λ
dt

(2.7)

Equations (2.6) and (2.7) show the essential tradeoff: faster growth requires a higher Rt , which
leaves fewer resources to produce goods in the current instant.6 Note that with risk-neutrality,
the utility of the agent is finite only if the growth rate does not exceed the discount rate ρ, so
an optimal policy exists only if utility is bounded for any feasible innovation, i.e.
ρ > (1 − α) φL ln λ

(2.8)

It will not be necessary to restrict ρ this way when I allow for curvature in the utility function
in the next section.
In the next two subsections, I solve for how the optimal and equilibrium paths of Rt vary with
zt . In particular, I show that equilibrium R&D suffers from a procyclical bias. However, due to
the model’s counterfactually low profit volatility, equilibrium R&D remains countercyclical. In
the last subsection, I modify the model to correct its counterfactual implication and show that
when profits are sufficiently volatile, the optimal path remains countercyclical but equilibrium
innovation will turn procyclical.
6
One might ask whether specialized labor employed in R&D is really substitutable for production workers at
high frequencies. While some R&D expenditures involve scientists and engineers who may not be easily shifted
to production, NSF data suggests that on average 40% of wage payments in R&D is allocated to support staff.

7

2.1. The Social Planner’s Problem
The neo-Schumpeterian view argues it is desirable to concentrate innovation in periods of low
productivity in the goods sector, since it reduces the overall cost of achieving a given average rate
of productivity growth. Solving the planning problem that maximizes the utility of the agent
confirms this intuition. Formally, let Zi for i ∈ {0, 1} denote the initial level of productivity,
and recall that M0 denotes the initial value of the average generation across all goods. The
expected utility of the agent under the optimal path starting from z0 = Zi is given by
¯
¸
·Z ∞
¤1−α −ρt ¯
£ Mt
¯
(2.9)
zt λ (L − Rt )
e dt ¯ z0 = Zi
Vi (M0 ) = max E
Rt
0

subject to the constraint

Ṁt = φRt

We can rewrite (2.9) recursively as
¾
½
¤1−α
£ M
∂Vi
φR
ρVi (M ) = max Zi λ (L − R)
+ µ (V1−i (M ) − Vi (M )) +
∂M
R∈[0,L]

(2.10)

Given the stationarity of the environment, the planner will choose a constant level of employment R for a given Zi . Thus, finding the optimal policy reduces to finding a pair of numbers
(R0 , R1 ). Note that this solution does not specify how innovation varies across intermediate
goods j, so wlog we can assume Rj = R for all j ∈ [0, 1]. I now demonstrate the existence of
an optimal path and argue that it undertakes more innovation when productivity in the final
goods sector is low. It is a special case of the more general Proposition 3 below. The proof of
that proposition, along with those of all remaining propositions, is contained in an Appendix.
Proposition 1: If (2.8) is satisfied, there exists a unique solution to the social planner’s
problem, and innovation is (weakly) countercyclical along the optimal path, i.e. R0 ≥ R1 .
Note that while a countercyclical policy allows the economy to achieve growth at a lower cost,
it also makes output more volatile: fewer inputs will be allocated to production precisely when
productivity is already low. This is irrelevant given the agent is assumed to be risk neutral.
But it may make countercyclical R&D undesirable when the agent is risk averse. I will return
to this issue in the next section.

2.2. Decentralized Equilibrium
I now turn to the decentralized equilibrium of this economy. All goods — both intermediate and
final — are produced by profit-maximizing firms. The technology for producing final goods is
8

freely available, so profits in this sector will equal zero in equilibrium. By contrast, intermediate
goods producers enjoy some market power: the entrepreneur who discovers the m-th generation
for producing good j earns a patent that grants him exclusive rights to use this technology.
Since no firm would undertake innovation without patent protection, some monopoly power is
necessary in this environment in order to sustain growth.
In what follows, I focus on equilibria where Rj is the same across all goods j, and where their
common value only depends on the value of aggregate productivity Zi . Formally, I restrict
attention to symmetric Markov perfect equilibria. This is natural given these features are true
for the optimal path. Solving for an equilibrium proceeds in several steps. Briefly, I first
express the equilibrium profits of intermediate goods producers π in terms of aggregate R&D
employment R. As an important aside, I observe that equilibrium profits are essentially as
volatile as wages, an important but counterfactual implication of the model. I then use the
expression for profits π to express the value of a successful innovation, v, strictly in terms of
(R0 , R1 ), aggregate employment in R&D for the two respective levels of aggregate productivity.
The uninterested reader can skip ahead to Proposition 2.
I begin by solving for the price pjt the producer of intermediate good j would charge. Given
the Cobb-Douglas aggregator X, the demand of final goods producers for each intermediate
good j will be unit elastic. Thus, each intermediate-goods producer would want to charge as
high a price as possible: his revenue will be constant regardless of the price he charges, but at
higher prices he can produce fewer goods and lower his costs. However, if he were to charge
more than the marginal cost of his next most efficient competitor, the latter could steal away
his business. Thus, in equilibrium, only the monopolist with the most productive technology
will supply goods, at a price pjt equal to the marginal cost of his most efficient competitor.
As Grossman and Helpman observe, incumbent producers benefit less from extending their
lead than new entrants do from overtaking the lead, so only entrants engage in innovation
in equilibrium. Hence, the next most efficient producer will use the (mjt − 1)-th generation
technology.7 Normalizing the wage to 1, the marginal cost of the next most efficient producer
is λ−(mjt −1) , the number of labor units he requires to produce a unit of good j.
Let ejt = pjt xjt denote total expenditures by final goods producers on intermediate good j.
Given the Cobb-Douglas specification for X, final goods producers will equalize expenditures
7

As in Grossman and Helpman (1991), this requires that a firm’s R&D expenditures on a particular intermediate good are unobservable, so an incumbent has no incentive to undertake R&D to discourage entry.

9

across intermediate good j, i.e.
ejt = (1 − α) Pt Yt ≡ et
where Pt denotes the price of the final good. With the wage normalized to 1, the cost of
production is just the number of employed workers λ−mjt xjt . Since xjt = et /pjt , this cost is
equal to λ−1 et . Hence, the profits of the incumbent firm that supplies good j are given by
¢
¡
π jt = et − λ−1 et = 1 − λ−1 (1 − α) Pt Yt

Profits are thus the same for all goods j, i.e. π jt = π t for all j. To express these profits in
terms of Rt , we use the fact that total spending on consumption goods must equal the income
of the representative agent in equilibrium. Thus, Pt Yt equals the sum of aggregate profits Πt
and payments to factors,
Pt Yt = Πt + rt K + L
Z 1
π t dj − Rt + rt K + L
=

(2.11)

0

where rt denotes the rental rate of capital at date t. Substituting in for π t from above and
using the fact that cost minimization by final goods producers implies rt K = αPt Yt allows us
to express equilibrium profits π t in terms of research employment Rt :
π t = (λ − 1) (L − Rt )

(2.12)

Note that nominal profits do not depend on zt . Thus, for a fixed level of innovation R, profits
are just as cyclically volatile as the numeraire good, labor. As noted above, this counterfactually
implies that the volatility of profits is commensurate with the volatility of the cost of R&D.
This implication will figure prominently below.
Entrepreneurs who succeed in innovation earn profits (2.12) as long as their technology is the
most advanced. To calculate the value of a successful innovation, let Ijt denote an indicator
which equals 1 if the entrepreneur is the leading-edge producer of good j and zero otherwise,
and let vj denote the value to a claim on the profits of a successful innovation at date 0. Since
the representative agent owns all claims in equilibrium, the price vj must leave him indifferent
between buying and selling an additional claim. This indifference condition implies
¯ ¸
·Z ∞
¯
U 0 (Ct ) /Pt
−ρt
vj = E
Ijt · 0
π t e dt ¯¯ z0
U (C0 )/P0
¯ ¸
· Z0 ∞
¯
P0
−ρt
= E
(2.13)
Ijt ·
π t e dt ¯¯ z0
Pt
0
where the expectation above is taken over all possible paths for zt and Ijt . Non-incumbent
firms choose Rj to maximize the expected value from a successful innovation net of R&D costs
10

φRj vj − Rj . It follows that φvj ≤ 1 in equilibrium, with strict equality if Rj > 0. Recall that I
restrict attention to equilibria in which Rjt = Rt for all j. This implies the value of a successful
innovation is the same for each intermediate good j, i.e. vj = v for all j ∈ [0, 1].
To express v in terms of R0 and R1 , the values that Rt assumes when zt = Z0 and Z1 ,
respectively, I need to express the price of final goods Pt in terms of Rt . Since the production
of final goods is competitive, the equilibrium price Pt equals the minimum cost to produce a
single unit of the good in equilibrium, i.e.
½Z 1
¾
Pt = min
pjt xjt dj + rt Kt
xjt ,Kt
0 ³
i´1−α
hR
1
=1
s.t. zt Ktα exp 0 ln xjt dj
Using the fact that pjt = λ−(mjt −1) and rt = αPt Yt , one can show that
Pt =

λ (L − Rt )α

(1 − α) zt λ(1−α)Mt

(2.14)

Let vi denote the value of a successful innovation if initial productivity z0 = Zi . Substituting
in the above expression Pt into (2.13) yields
¯
"Z
#
µ
¶
¯
∞
zt λ(1−α)Mt L − R0 α
¯
−ρt
vi = E
It ·
(λ − 1) (L − Rt ) e dt ¯ z0 = Zi
(2.15)
¯
z0 λ(1−α)M0 L − Rt
0
Next, for any zt -measurable function X (·), the value of the integral
¯
·Z ∞
¸
¯
(1−α)Mt
−ρt
It · λ
X (zt ) e dt ¯¯ z0 = Zi
Wi (M0 ) = E
0

subject to Ṁt = φRt can be characterized by the recursive equation
·
¸
∂Wi
(1−α)M
− Wi (M ) φRi
(ρ + µ) Wi (M ) = λ
X (Zi ) + µW1−i (M ) +
∂M

The method of undetermined coefficients confirms that Wi (M ) = wi λ(1−α)M where
wi =

ω (R1−i ) X (Zi ) + µX (Z1−i )
ω (Ri ) ω (R1−i ) − µ2

and
ω (R) = ρ + µ + (1 − (1 − α) ln λ) φR
The value of a successful innovation vi can thus be expressed in terms of R0 and R1 :

vi (Ri , R1−i ) = (λ − 1)

Z1−i
(L − R1−i )1−α (L − Ri )α
Zi
ω (Ri ) ω (R1−i ) − µ2

ω (R1−i ) (L − Ri ) + µ

11

(2.16)

A symmetric Markov-perfect equilibrium is any pair (R0 , R1 ) in [0, L]2 where φvi (Ri , R1−i ) ≤ 1,
with strict equality if Ri > 0. The next proposition characterizes these equilibria, and provides
a sufficient condition for a unique equilibrium to exist.8
Proposition 2: Innovation along the equilibrium path is weakly countercyclical in any
symmetric Markov-perfect equilibrium, i.e. R0 ≥ R1 along any equilibrium path. Moreover, if
1
λ < e 1−α , there exists a unique symmetric Markov-perfect equilibrium.
At first glance, these results seem to deny the intuition presented in the Introduction: despite
the presence of spillover effects, equilibrium R&D is countercyclical. However, as I now argue,
the problem lies not with the intuition but with the inability of the model to generate sufficiently
procyclical profits. In particular, there is a precise sense in which the decentralized economy
exhibits a procyclical bias in R&D. It is only because profits are not sufficiently procyclical in
the model that this bias is not enough to turn R&D procyclical.
To appreciate why R&D in the market economy suffers from a procyclical bias, let us consider
the ratio of the real value of a successful innovation in a boom to its real value in a recession.
This ratio is of interest because the higher it is, the more incentive entrepreneurs have to
undertake their R&D in booms than in recessions. To compute this ratio in equilibrium, we
divide the value of a successful innovation vi in (2.15) by the equilibrium price P of final goods
in (2.14), evaluated at zt = Zi . This ratio reduces to
¯
¸
·Z ∞
¤1−α −ρt ¯
£ Mt
It · zt λ (L − Rt )
e dt ¯¯ z0 = Z1
E
v1 /P1
0
¯
¸
(2.17)
= ·Z ∞
¯
¤1−α
£ Mt
v0 /P0
−ρt
¯
E
It · zt λ (L − Rt )
e dt ¯ z0 = Z0
0

where It is an indicator that is equal to 1 if the entrepreneur at date 0 remains the leading
edge producer by date t. For the social planner, the analogous value for a successful innovation
when zt = Zi is given by ∂Vi /∂M , where Vi (M ) is given by (2.10). Using the solution to the
planner’s problem in the Appendix, the ratio of the two values is given by
¯
·Z ∞
¸
£ Mt
¤1−α −ρt ¯
¯
E
zt λ (L − Rt )
e dt ¯ z0 = Z1
∂V1 /∂M
¯
¸
= · Z 0∞
(2.18)
¯
¤1−α
£ Mt
∂V0 /∂M
−ρt
¯
E
zt λ (L − Rt )
e dt ¯ z0 = Z0
0

8
Canton and Uhlig (1999) show that equilibrium innovation is countercyclical in a similar model. Aghion and
Saint Paul (1998) also show equilibrium innovation is countercyclical, but their model assumes only one firm
undertakes all innovation, so there is no externality that could lead to procyclical innovation.

12

Now, suppose we tried to implement the optimal program and set R0 > R1 in the decentralized
market. Under this program, more output will be produced in booms than in recessions, since
¤1−α
¤1−α
£
£
Z1 λMt (L − R1 )
> Z0 λMt (L − R0 )

(2.19)

Since the probability that It = 1 decreases with t, the integrals in (2.17) assign more weight to
output at dates close to t = 0 than the integrals in (2.18). Combined with the fact that zt is
mean-reverting, it follows that the ratio in (2.17) is higher than the ratio in (2.18). Thus, if we
tried to implement the optimal path in the decentralized economy, entrepreneurs would assign
too much value to innovations in booms relative to recessions, and they will have incentive to
deviate from the optimal path and instead concentrate more of their R&D in booms. This is
the inherent procyclical bias of R&D in the decentralized economy.9
However, this bias is not enough to turn R&D procyclical. As noted above, this is because
equilibrium profits in the model are only as volatile as the cost of R&D: in a recession, both the
cost of R&D and profits fall in proportion to zt . But the expected discounted value of future
profits falls by less than today’s profits, since zt is stationary and future profits are expected to
eventually revert to their unconditional mean. Hence, the value of a successful innovation falls
by less than the cost of R&D, giving incentive to undertake more innovation in recessions. For
R&D to turn procyclical, the value of a successful innovation must fall by more than the cost
of R&D in recessions, which in turn requires profits to fall by more than the cost of R&D.
Empirically, of course, profits are far more volatile than the cost of R&D. Mansfield (1987)
constructs R&D price indices from 1969 to 1983 and finds the R&D deflator is closely synchronized with the GDP deflator at high frequencies, implying real R&D costs are not very
cyclical. This is not surprising given R&D is labor intensive and real wages are only mildly
procyclical. The remainder of this section modifies the model so it can accord with the larger
relative volatility of profits.

2.3. Fixed Costs and the Volatility of Profits
In analyzing the behavior of profits over the business cycle, Ramey (1991) argues an important
piece of evidence is the strongly procyclical pattern in the ratio of aggregate profits to aggregate
9
The inefficiency concerns relative rather than absolute values of R&D; that is, the tendency towards procyclical R&D in independent of whether the overall level of R&D is too high or too low relative to the socially
optimal level. Hence, this inefficiency is distinct from the one emphasized by Grossman and Helpman (1991) and
Aghion and Howitt (1992) concerning the potential inefficiency of steady-state growth.

13

sales. By contrast, in the model, this ratio is given by
πt
1
=1−
et
λ
which does not vary with zt . In principle, we could make profits more volatile than sales in
the model by making the markup λt procyclical. However, empirical evidence summarized
in Rotemberg and Woodford (1999) suggests markups are moderately countercyclical. This
observation leads Ramey to conclude that the reason observed profits are so volatile over the
business cycle is the presence of fixed costs of production. To see why, suppose producers of
intermediate goods had to pay some fixed amount F to initiate production. In this case, the
profit-to-sales ratio equals
¶
µ
πt
F
1
−
= 1−
(2.20)
et
λt
et
As long as F is constant and the markup λt is not too countercyclical, the ratio of profits to
sales will increase with sales et . Thus, to properly reconcile the model with evidence of highly
volatile profits, we need to introduce fixed costs of production, and at the same time ensure
that markups are not too countercyclical. The remainder of this section modifies the model to
incorporate these two features, and confirms that equilibrium R&D will indeed turn procyclical
when profits are sufficiently more volatile than the cost of R&D.
The first step is to introduce fixed costs of production. Suppose that in order to produce any
intermediate good j ∈ [0, 1], a producer must first purchase F units of the final good. To insure
that the economy doesn’t outgrow this cost, we need to further scale this fixed cost so that it
grows at the same rate as the economy. Let us therefore assume that at date t, the amount of
the final good required to initiate production is equal to λ(1−α)Mt F . The notion that fixed costs
grow with the rest of the economy seems plausible; for example, overhead labor will naturally
become more expensive as overall labor productivity increases.10
The next step is to ensure markups are not too countercyclical. It turns out that introducing
fixed costs of production leads to strongly countercyclical markups. To see why, note that since
demand for each intermediate good is unit elastic, the markup an intermediate goods producer
charges is the gap between his own cost and the price at which his next most efficient competitor
10
This begs the question why I did not model the fixed cost directly in terms of labor. The reason is that in
a frictionless model, the price of labor changes with zt . As a result, the cost of overhead labor rises in booms,
cutting into profits and preventing them from rising too much. Empirically, wages do not appear to respond
much to short term variations over the business cycle, although they certainly grow over longer time horizons.
Assuming the fixed cost is denominated in final goods is a way of capturing rigidity in the salaries of overhead
labor without unnecessarily complicating the model. Since I assume the same fixed cost when I solve the planner’s
problem, the planner will take this rigidity into account when choosing R&D.

14

breaks even. In a boom, any competitor would earn proportionately higher gross profits at any
given price. At the same time, the fixed cost the competitor faces would remain unchanged. To
keep the rival from entering in a boom, then, an intermediate goods producer would have to
lower his price. In the model, this fall in the markup is large enough so that profits would only
be as volatile as sales, despite the presence of a fixed cost. We therefore need to modify the
model to prevent markups from being so countercyclical. In other words, we need to separate
between the price an incumbent charges and the price at which the producer using the previous
generation technology breaks even.
One reason the two prices might differ is that new ideas can be partly imitated, and producers
have to set prices to also deter entry from imitators. That is, suppose entrepreneurs can engineer
knock-off versions of the latest generation of any technology that, while inferior to the leadingedge technology, are more profitable than the best version of the previous generation. As long
as the price at which an imitator breaks even is not too countercyclical, profits will be more
volatile than sales. The latter will be true if knock-off versions involve lower fixed costs but
higher variable costs than the leading edge technology. This assumption is plausible; after all,
imitators would not incur the costs of patent protection that a leading-edge producer incurs,
and inferior knock-offs would presumably involve higher variable costs of production than the
technology they imitate. In what follows, I assume the inferior version of each technology is
such that imitators break even at a price of λ times the marginal cost of the leading technology,
implying a constant markup.11 This is a convenient simplification; we would obtain similar
results with moderately countercyclical markups.
Before examining the effects of more volatile profits on equilibrium R&D, let us first consider
how these modifications affect the planner’s problem. Since the planner would always employ
the leading technology, the presence of inferior knockoffs is irrelevant. However, the planner
will react to the presence of fixed costs. The analog to equation (2.10) now corresponds to
h
i



 λ(1−α)M Zi (L − R)1−α − F +
(2.21)
ρVi (M ) = max
∂Vi
R∈[0,L] 
µ (V1−i (M ) − Vi (M )) +
φR 
∂M
That is, the planner takes into account that a fixed amount of the output produced must be
spent on overhead. While this changes the optimal level of innovation, there is no reason to
expect it will affect the timing of innovation with respect to zt . The next proposition confirms
that the optimal path is indeed countercyclical, at least at an interior optimum.
11
An example of such a knock-off technology is one where the variable cost is λ times the variable cost under
the new technology and which requires no fixed cost of production. This technology is clearly more profitable
than the leading-edge version of the previous generation.

15

Proposition 3: If (2.8) is satisfied, there exists a unique solution to the social planner’s
problem. If the optimal level of R&D is always positive, then innovation is countercyclical
along the optimal path, i.e. R0 > R1 . For small F , the optimal path is weakly countercyclical
even if the optimal path involves periods of zero innovation, i.e. R0 ≥ R1 .
I now turn to the decentralized economy. Again, the uninterested reader can skip ahead to
Proposition 4. Since my assumptions imply pjt = λ−(mjt −1) , profits will equal
¢
¡
π jt = 1 − λ−1 et − λ(1−α)Mt Pt F

(2.22)

Once again, we can express profits directly in terms of Rt using the aggregate resource constraint. However, the original constraint in (2.11) must be revised to reflect the fact that the
household only purchases those goods that are not used by intermediate goods producers, i.e.
³
´
Pt Yt − λ(1−α)Mt F = Πt + rt K + L
(2.23)
Substituting this aggregate resource constraint into the expression for profits yields
π t = (λ − 1) (L − Rt ) − λ(1−α)Mt Pt F

(2.24)

instead of (2.12). Since Pt varies over the cycle, profits π t will be more volatile than the price of
the numeraire good. Solving for Pt , which turns out to be identical to the expression in (2.14),
we obtain the following as the value of a successful innovation:

vi (Ri , R1−i ) = (λ − 1)

Z−i
(L − R1−i )1−α (L − Ri )α
Zi
−
ω (Ri ) ω (R1−i ) − µ2
λ (L − Ri )α F
ω (R1−i ) + µ
ω (Ri ) ω (R1−i ) − µ2 (1 − α) Zi

ω (R1−i ) (L − Ri ) + µ

(2.25)

Since profits are now more volatile than the cost of R&D, we would expect equilibrium R&D
can turn procyclical. To confirm this conjecture, I begin with the following lemma:
1

Lemma: Suppose λ < e 1−α . Then for any F > 0, there exists a unique R∗ < L such that
φv0 (R∗ , R∗ ) = φv1 (R∗ , R∗ ). Moreover, there exists a F ∗ > 0 such that φvi (R∗ , R∗ ) < 1 for
F < F ∗ and φvi (R∗ , R∗ ) > 1 for F > F ∗ .
The lemma above establishes that for any fixed cost F , there is a unique level of innovation
that leaves the nominal value of a successful innovation v constant over the cycle. From the
proof of the lemma, one can show that this value v (R∗ , R∗ ) increases with F and ranges from
zero to infinity, so there must be some F ∗ for which it equals 1/φ. As the next proposition

R∗

16

establishes, if F is greater than F ∗ , we are assured of finding a pair (R0 , R1 ) where R1 > R0
and which satisfies the condition that φvi (Ri , R1−i ) = 1 for both i ∈ {0, 1}.
1

Proposition 4: Suppose λ < e 1−α . If F > F ∗ , where F ∗ is defined in Lemma 2 , there
exists a pair R0 < R1 such that φvi (Ri , R1−i ) = 1 as required of an interior equilibrium.
Proposition 4 suggests that for sufficiently large fixed costs, equilibrium innovation will covary positively with productivity.12 More precisely, it states that for large enough fixed costs
we will always be able to find a solution for the system of equations that characterize an interior equilibrium such that R1 > R0 . However, this does not guarantee that the solution
(R0 , R1 ) lies in [0, L]2 as required of an interior equilibrium. Numerically, though, the solution does appear to lie in the interior of [0, L]2 for large L, mirroring a result in Grossman
and Helpman (1991) for the case of no fixed cost. As to whether this equilibrium is unique,
the set {(R0 , R1 ) | φvi (Ri , R1−i ) = 1} can have multiple solutions. However, these additional
solutions do not appear to correspond to equilibria; rather, they appear to involve very high
levels of innovation (close to L) for which the revenue of intermediate goods producers does not
cover their fixed costs. Experimenting with several parameter values always led to a unique
symmetric Markov equilibrium that was countercyclical if F < F ∗ but procyclical if F > F ∗ .
In sum, when rivals are likely to steal away future profits, entrepreneurs act short-sightedly
when they undertake R&D. They will therefore tend to undertake too much R&D in periods
of high profits than is socially optimal. For larger fixed costs of production, which imply more
volatile equilibrium profits, relatively more R&D will be shifted towards booms, until eventually
R&D turns procyclical. However, even for large fixed costs, the optimal path continues to dictate
that R&D be countercyclical.
An important practical question is whether empirically plausible fixed costs are enough to
account for the procyclical pattern of R&D in the data, and if so how costly is this procyclicality
of R&D. To address these questions, I need to relax some of the assumptions above to make the
model more amenable to quantitative analysis. This is precisely what I do in the next section.
As an aside, it is worth noting here that although unrealistic, the assumption of risk neutrality
does highlight some of the model’s stark welfare implications. Consider the implied welfare cost
of volatility in the model, i.e. the cost of moving from an economy with constant productivity
12
One has to be careful about referring to this variation as procyclical, since it is possible that output will fall
when productivity Z rises. In all of my numerical simulations, though, output comoves with productivity.

17

Z to one in which zt fluctuates between Z0 and Z1 where E [zt ] = Z. In the stable environment,
the optimal path mandates a constant level of R. Under risk neutrality, the planner can always
achieve the same expected utility in the stochastic environment by adopting the same R, since
at any date t
h
³
³
´i
´
E λ(1−α)Mt zt (L − R)1−α − F
= λ(1−α)Mt E [zt ] (L − R)1−α − F
³
´
= λ(1−α)Mt Z (L − R)1−α − F

Since Proposition 3 tells us the optimal (interior) path will vary R with zt , it follows that the
planner can achieve an even higher utility. By contrast, in the decentralized economy, welfare
can be lower in the volatile environment than in the stable one. Hence, cyclical fluctuations
can reduce welfare even when they allow a benevolent planner to achieve a higher utility. This
cost is due to the suboptimal use of resources in response to time-varying productivity, and
is distinct from the cost of consumption volatility in Lucas (1987) due to risk aversion. It is
also distinct from previous work that argues cycles are costly because they affect growth, e.g.
Barlevy (2004), which involves the volatility of innovation rather than its timing. Once I allow
for risk-aversion, fluctuations may no longer necessarily allow the planner to achieve a higher
utility than in a stable environment. Nevertheless, fluctuations still allow the planner to achieve
growth at a lower overall cost, which the inefficient timing of innovation in the decentralized
equilibrium precludes.

3. Schumpeterian Growth with Concave Utility and Accumulable Capital
As noted in the previous section, concentrating innovation in recessions lowers the average
cost of growth but increases the volatility of output. Under the assumption of risk neutrality,
this volatility is inconsequential. However, when the utility function exhibits curvature, this
volatility may make it undesirable to concentrate innovation in recessions. This section modifies
the model to allow for concave utility, and examines whether procyclical innovation remains
inefficient for empirically plausible assumptions.
In introducing risk aversion, it will be important to also relax the assumption that capital
is not accumulable. Otherwise, the only way to smooth consumption over the cycle is to vary
R&D with productivity, implying procyclical R&D may in fact be optimal. In practice, though,
there are other options to smooth consumption such as inventories and capital accumulation,
and since these activities do not occur at the expense of current production, they presumably
dominate R&D for purposes of consumption smoothing.
18

Formally, I modify the model in two ways. First, I replace (2.1) with a more reasonable utility
U (Ct ) = ln Ct

(3.1)

Log utility also allows us to drop the restriction on ρ in (2.8). Second, I replace the assumption
that Kt ≡ 1 for all t with the assumption that capital satisfies the law of motion
K̇t = It − δKt

(3.2)

where It denotes investment, i.e. net output of final goods that is not consumed, and δ denotes
the rate at which capital depreciates. Finally, since capital accumulation contributes to growth,
we need to scale the fixed cost of production differently. Specifically, an intermediate goods
producer will now require λMt F units of consumption good to initiate production at date t.

While these modifications are simple to describe, they greatly complicate the analysis by
introducing additional state variables. In the next two subsections, I sketch out how to derive
the optimal and equilibrium paths of R in this environment. I then solve for these paths
numerically for parameter values meant to replicate certain features of U.S. data, and show
that equilibrium innovation is indeed inefficiently procyclical at empirically plausible parameter
values.

3.1. The Social Planner’s Problem
I begin with the planner’s problem. In lieu of (2.9), the planner’s problem is now given by
¯
¸
·Z ∞ ³
´
¯
¤1−α
£ Mt
Mt
α
−ρt
¯
ln zt Kt λ (L − Rt )
− λ F − It e dt ¯ z0 = Zi
Vi (K0 , M0 ) = max E
Rt ,It

0

s.t. 1. Ṁt = φRt
2. K̇t = It − δKt

To solve this problem, define k = λ−M K and ι = λ−M I. Using the law of motion for M , one
ln λ
, where vi (k) satisfies
can show that Vi (K0 , M0 ) = vi (k) + M0
ρ
 ³

´ φR ln λ


 ln Zi kα (L − R)1−α − F − ι +

+
ρ
ρvi (k) = max
(3.3)
∂vi

ι,R 

(ι − (δ + φR ln λ) k) + µ (v1−i (k) − vi (k)) 
∂k
19

The planner can now control two variables, investment and R&D. The first-order conditions for
the maximization problem with respect to these two variables are
1
Zi

k α (L − R)1−α

−F −ι

kα (L − R)−α

(1 − α) Zi
Zi k α (L − R)1−α − F − ι

∂vi
∂k
¶
µ
1
∂vi
−k
φ ln λ
=
ρ
∂k
=

Substituting the first equation into the second yields the following formula for Ri , the value of
R&D when productivity is equal to Zi :
¶
¸−1/α
·µ
φ ln λ
1
−k
Ri = L −
ρ (∂vi /∂k)
(1 − α) Ziα k α
Rather than two numbers R0 and R1 , an optimal plan now corresponds to two functions R0 (k)
and R1 (k). I will refer to a policy as procyclical if it assigns R1 (k) > R0 (k) for any k in the
limiting set of capital-to-productivity ratios for this economy, i.e. for any level of k that occurs
infinitely often along the optimal path with probability 1, and countercyclical if R1 (k) < R0 (k)
for all such k. Note that this definition may not correspond to the way R&D would appear to
covary with zt in data generated by this model, since changes in capital accumulation over the
cycle may offset the response of R&D to changes in productivity for a fixed k.
To solve for Ri (k), we need an expression for the value function vi (k). Since the system in
(3.3) does not yield a closed-form solution, I need to solve it numerically. My implementation
uses a collocation method whereby I approximate each vi (k) with a polynomial in k.13

3.2. Decentralized Equilibrium
Next, I characterize the equilibrium allocation. Certain features of the equilibrium remain
unchanged from the previous section. For example, intermediate good producers will continue
to charge the price at which their next most efficient rival breaks even. Once again, I assume
new ideas can be imperfectly imitated, and the price at which an imitator breaks even is
pjt = λ−(mjt −1) . As such, nominal profits for each producer once again correspond to (2.24).
The price of final goods P is analogous to but a little different from (2.14), namely
λ1−M (L − R)α
P =
z (1 − α) kα
13

(3.4)

More precisely, I obtain an n-th order polynomial approximation of each function by choosing n+1 coefficients
such that when I replace the true vi (k) with the approximate function, (3.3) is exactly satisfied at n+1 particular
values of k. These particular values correspond to the roots of the first n + 1 Chevyshev polynomials, adapted
to the limiting interval for k. The results reported in the paper are based on fourth-order polynomials.

20

As in the previous section, I only consider symmetric Markov equilibria. In equilibrium, the
path for R must satisfy the free entry condition φv = 1, where recall v denotes the expected
present discount value of profits to the leading producer,
¯ ¸
·Z ∞
¯
U 0 (Ct ) /Pt
−ρt
v=E
It · 0
π t e dt ¯¯ z0
U (C0 )/P0
0

With risk aversion, U 0 (Ct ) is no longer a constant. Hence, the value of a successful innovation
depends on the consumption of the household. In order to solve for an equilibrium, then, we
would need to solve the household problem.

The household problem can be characterized as follows. At any point in time, a household
must decide how to divide its wealth between physical capital and claims on entrepreneurial
profits. It does this taking as given the distribution of future prices, e.g. the price of capital, the
rate of return on capital, the profits it expects entrepreneurs to earn, and so on. In addition,
the household must choose how much of its wealth to use to finance consumption.
Since the household must own all of the claims to entrepreneurial profits in equilibrium, it will
be convenient to pretend as if there was a mutual fund company that pooled all entrepreneurs
into a single portfolio on behalf of the household. Arbitrage requires the value of this portfolio
R1
to be the same as the cost of buying up all firms, which is just 0 vdj = v. To insure the fund
continues to own all incumbents, it must pay the research expenses of any potential innovator
in exchange for the rights to the patent if the innovator is successful, i.e. the fund deducts an
operating expense R out of dividends. Thus, as far as the household is concerned, it can either
allocate its wealth to physical capital or to an asset whose price is v and which yields a dividend
of Π = π − R per unit time, where π is given by (2.22).
Let w denote the household’s nominal wealth and σ denote the fraction of this wealth the
household allocates to capital. The instantaneous change in the household’s nominal wealth
from its investments in physical capital derives from rental income and capital gains. As long
as aggregate productivity remains constant over the next instant, the return per unit of capital
is r + Ṗ , and the number of units of capital it holds is σw/P . Similarly, as long as aggregate
productivity zt remains constant, the return per share of the mutual fund it owns is Π + v̇,
and the number of shares it owns in the mutual fund is (1 − σ) w/v. Note that the free entry
condition implies v̇ = 0. In addition to the returns to its assets, the household also earns
labor income and spends some of its resources on consumption. As long as zt is constant, then,
nominal wealth w would evolve continuously according to the law of motion
"Ã
#
!
r
Ṗ
Π
ẇ =
+
(3.5)
σ + (1 − σ) w + L − λM P c
P
P
v
21

where c = λ−M C. If instead productivity zt changes in the next instant, the nominal value of
the physical capital the household owns will jump as the price of capital P itself jumps. The
nominal value of wealth held in the mutual fund does not change, however, since the value of
the mutual fund v = φ−1 independently of aggregate productivity. Hence, the wealth of the
household will jump from w to w∗ where
·
¸
P1−i
w∗ =
σ + (1 − σ) w
(3.6)
Pi
Let W denote the aggregate wealth of the economy. In equilibrium, of course, w = W .
However, since individual households act as price takers, they treat the path of W as given
and assume it determines the values of all relevant economic variables. Let Ri (W ) denote the
equilibrium employment in R&D when zt = Zi and aggregate wealth is Wi . If we knew Ri (W ),
we would be able to derive all remaining equilibrium quantities. For example, using the fact
that W = P K + v = λM P k + φ−1 and the expression for P in (3.4), we can solve for the
capital-to-productivity ratio ki (W ):
"¡
# 1
¢
W − φ−1 Zi (1 − α) 1−α
ki (W ) =
λ (L − Ri (W ))α
We can similarly express the nominal quantities r, P , and Π as functions of W . This implies
we can express the household problem recursively in terms of two state variables, w and W :




 ln c + φR ln λ + ∂Vi ẇ + ∂Vi Ẇ +

ρ
∂w
∂W
(3.7)
ρVi (w, W ) = max
σ,c 


µ (V1−i (w∗ , W ∗ ) − Vi (w, W )) 

subject to (3.5) and (3.6), the free entry condition φv = 1, and the laws of motion for W , i.e.
if zt remains constant over the next instant, then
³
´
Ẇ = r + Ṗ λM k + Π + L − λM P c (W, W )
(3.8)
while if zt changes over the next instant, W will jump to W ∗ , where
W ∗ = λM P1−i ki (W ) + φ−1
The first order conditions for the household problem with respect to σ and c are given by
Ã
!
·
¸
Ṗ
P1−i
r
∂V1−i (w∗ , W ∗ ) /∂w∗
+
1−
(3.9)
− φΠ = µ
P
P
∂Vi (w, W ) /∂w
Pi
1
P c (w, W )

=

∂Vi
∂w

(3.10)
22

An equilibrium therefore corresponds to a set of functions wi∗ (w), Vi (w, W ), and Ri (W ) which
satisfy equations (3.6), (3.7), and (3.9).
Again, I can only solve this equilibrium numerically. In particular, I approximate Ri (W )
and wi∗ (w) using n-th degree polynomials in W and w, respectively, and I approximate the
n n−k
P
P
function Vi (w, W ) with the polynomial
ak wk W . The coefficients of these polynomials
k=0 =0

are chosen so that the equations above hold exactly at particular values of (w, W ).14

3.3. Calibration and Results
I now proceed to solve the model numerically for particular parameter values. Since this version
of the model corresponds to a standard real business cycle model, only with an endogenously
determined growth rate, I can build on previous literature in assigning many of its parameters.
The particular values I use are summarized in Table 1.
Table 1
ρ 0.05
α 0.33

Z0 0.94
Z1 1.06

µ
δ

0.20
0.08

λ 1.20
φ 0.10

F 3.6
L 30.8

Normalizing a unit of time in the model to correspond to a year, the discount rate ρ is set to
5%. The share of capital in the production of final goods is set to one third. To accord with
an unconditional standard deviation of detrended productivity growth of 6%, I set Z0 to 0.94
and Z1 to 1.06. The transition rate µ is set so that the average length of a complete cycle is 10
years, slightly longer than the 8 year frequency often used to identify business cycle fluctuations.
The depreciation rate is set to 8% per year. For λ, I follow Rotemberg and Woodford (1999)
in calibrating the markup to 20%. The productivity term φ turns out to be a pure scaling
parameter, so I normalize it to 0.10.
The remaining two parameters, F and L, are chosen to yield a growth rate of 2% per year
and a share of R&D in output of 2%, in accordance with the average share of total R&D
expenditures (private and public, since both can contribute to long-run growth) in GDP. It is
not obvious whether the model counterpart to the latter statistic is the share of R&D in gross
14

In particular, the set of points w that I use correspond to the roots of the Chebyshev polynomials, adapted
to the limiting interval for w. The set of values for W are the same as for w, but I focus on the triangular
array {wi , Wj }1≤i≤j≤n+1 . Note that I need to approximate V (w, W ) on and off the equilibrium path (in which
w = W ) to approximate both ∂Vi /∂w and ∂Vi /∂W .

23

output or output net of the amount used by intermediate goods producers to cover fixed cost.
However, at F = 3.6, R&D accounts for 2.0% of gross output and 2.2% of net output, which are
virtually indistinguishable. Either way, R&D accounts for a tiny share of GDP, as it does in the
data. Simulating the model for these parameter values reveals that the model also generates
reasonable time variation in R&D: the standard deviation of the log R&D share over time is
0.139 and 0.136 for gross and net output respectively, compared to 0.137 for the log share of
total R&D between 1953 and 2002.
Measured against output, the value of F implies that fixed costs account for 8.1% of gross
output (and 8.8% of net output). By comparison, Basu (1996), following Ramey (1991), suggests
using non-production workers as a proxy for overhead labor. Non-production workers account
for about 20% of the labor force during the post-War period. Since labor accounts for two-thirds
of total output, this suggests an even larger overhead cost of 13% of output.
Figure 4 plots both optimal and equilibrium Ri (k) for the parameter values in Table 1. In each
case, only the limiting set for k is depicted. For the parameter values in Table 1, optimal R&D
policy is countercyclical, despite the fact that the representative household is risk-averse. This
is consistent with the notion that capital accumulation is a much more efficient way to smooth
consumption that varying R&D. Equilibrium R&D, by contrast, is unambiguously procyclical,
even after accounting for more rapid capital accumulation in booms.
Comparing the axes of the two panels in Figure 4 reveals that optimal R&D is an order of
magnitude larger than equilibrium R&D. This result mirrors the findings of Jones and Williams
(2000), although the wedge between the optimal and equilibrium levels of R&D is larger than in
their analysis. This is because Jones and Williams allow for diminishing returns to R&D; with
linear returns to R&D, the planner would want to devote almost all available labor resources
to innovation. One should therefore be skeptical about this model’s predictions for optimal
R&D. However, as long as we limit attention to small perturbations around the non-stochastic
equilibrium, we can use this model to analyze the efficiency of observed R&D. This is because
the equilibrium of the model is calibrated to match average long-run growth in the data, and
curvature in the production function for R&D is irrelevant when shocks are small enough since
the production function is locally linear. Thus, the calibrated model should be informative as
whether the procyclical R&D we observe in the data is socially inefficient.
Formally, suppose aggregate productivity fluctuates between Z0 = 1−ε and Z1 = 1+ε, where
ε is small. Let R denote non-stochastic steady state equilibrium R&D when Z0 = Z1 = 1. For
ε small, we can approximate the equilibrium of the stochastic economy by R0 = (1 − ε0 ) R
24

and R1 = (1 + ε0 ) R for some ε0 > 0. We can then check numerically the effect of changing ε0
on welfare. For the parameter values in Table 1, the welfare of the representative household
proves to be decreasing in ε0 . Thus, independently of what our calibration implies for the
global optimum, the representative household would be better off if the timing of R&D were
countercyclical as opposed to procyclical, to a degree that depends on the exact process for zt .
Note the cost of procyclical R&D in the model is fundamentally a cost of macroeconomic
volatility; in a stable environment where Z0 = Z1 = 1, there is no opportunity to distort
the timing of R&D in a way that would make growth more costly. Thus, the implied cost of
procyclical R&D is related to the implied cost of macroeconomic volatility in the model. Let
us therefore compare the utility of the household when Z0 = Z1 = 1 with the utility of the
same household when Z0 and Z1 are as in Table 1. For concreteness, I evaluate both utilities
starting at the deterministic steady-state level of k, and for the stochastic environment I assume
z0 is distributed according to the invariant distribution for zt . Following Lucas (1987), we can
express the change in utility in terms of the fraction of lifetime consumption agents would be
willing to give up to remain in the stable environment. By a standard Taylor approximation
argument, the cost of moving to a volatile environment is proportional to the variance of
aggregate productivity σ 2z . For the parameter values in Table 1, I calculate this constant of
proportionality to be approximately 1. We can then decompose this cost into a cost due to
volatility in zt holding R&D fixed at its deterministic equilibrium level (but allowing agents to
accumulate capital), and a cost due to changes in equilibrium R&D. The former cost reflects
the cost of volatility due to risk aversion, while the latter reflects the cost due to the effect of
fluctuations on growth. For the parameter values in Table 1, I calculate the cost of volatility
holding R&D fixed to be approximately 13 σ 2z . Taking into account the inefficient response of
R&D to aggregate fluctuations fully triples the cost of business cycles relative to models in
which the growth rate is treated as exogenous and fixed.
Of course, since σ 2z is small empirically, the implied welfare cost of procyclical R&D is small.
For example, when σ z = 6% as in Table 1, the cost of fluctuations is 0.36% of lifetime consumption. This is indeed three times as large as Lucas’ estimated cost of business cycles using direct
evidence on consumption (as opposed to calibrating to productivity shocks). That said, this
estimate probably understates the true cost of the inefficient timing of R&D over the cycle. This
is because the model substantially underpredicts the volatility of trend growth φR ln λ. The
standard deviation of trend growth in the model is about 0.2 percentage points. By contrast,
when Barlevy (2004) estimates the volatility of trend growth, he finds a standard deviation
of 1.8 percentage points. To be sure, not all of this variation is due to R&D. However, as is
well-known, the preferences I use fail to generate asset prices that are as volatile as in the data.
25

This is important, since asset prices affect the incentives of firms to engage in innovation, and
more volatile asset prices would result in more volatile trend growth. Recent work has argued
that allowing for time non-separable preferences can help to reconcile RBC models with asset
prices. Analyzing the model with this class of preferences would be an important next step for
getting a better sense of the cost of the inefficient timing of R&D.
Finally, since the fixed cost F plays an essential role, some discussion of the robustness of the
results to changes in this parameter are in order. In particular, how do the results change as we
vary F while adjusting L to maintain a steady-state growth rate of 2% per year? As F is driven
to zero, equilibrium innovation turns countercyclical. How large does F have to be for R&D
to be procyclical? For the parameters values in Table 1, the cutoff level above which R&D is
procyclical is approximately 3.3. This corresponds to a fixed cost of about 8.6% of net output,
implying that fixed costs cannot be much smaller than those I calibrated. For larger values of F
than in Table 1, equilibrium R&D is procyclical. Moreover, a similar perturbation analysis to
the one reported above reveals that this procyclical pattern remains inefficient for much larger
values of F . Interestingly, the optimal path for R&D — which recall may be suspect given the
absence of diminishing returns — actually turns procyclical for high values of F . Intuitively,
increasing F acts in a similar way to increasing the curvature of the utility function. Since along
the optimal path the planner uses almost all labor resources for R&D rather than production,
fluctuations in consumption are large in a proportional sense. Thus, a very risk-averse planner
would resort to using even R&D to smooth consumption, especially given that the amount of
resources devoted to R&D at the optimal path is so large. But since the equilibrium R&D
share of GDP is only 2%, there is very little to gain from using R&D to smooth consumption
in practice, and the procyclical pattern in equilibrium R&D remains suboptimal even when the
optimal path turns procyclical.

4. Extensions
While the previous section goes some lengths to render the model more plausible, it still abstracts from certain features that may be relevant for its conclusions. This section briefly raises
some of these considerations and speculates on whether the results above are likely to survive
the introduction of further modifications.
One feature the model abstracts from is that innovations take time to bring to fruition,
so firms should not expect to be able to bring their innovations on-line as soon as aggregate
conditions improve, which they conceivably can under a Poisson technology for innovation.
26

Introducing diffusion lags might make entrepreneurs reluctant to undertake more innovation in
booms if they are inherently temporary. Formally, suppose an idea discovered at date t can
only be used for production at date t + T . By continuity, inefficient timing would continue to
arise even for very small T . For larger T , we would have to explicitly solve the model. This is
quite difficult, since the whole continuum of productivity levels [zt−T , zt ] enter as state variables.
But intuitively, R&D should remain procyclical as long as shocks are persistent. That is, since
Pr (zt+T = zt | zt ) > Pr (zt+T 6= zt | zt ), a potential entrant who is constrained to implement
his invention only after T units of time should still expect profits at the time of implementation
to be higher starting in a boom than in a recession. Diffusion lags mitigate the benefits to
procyclical R&D, but R&D should still be procyclical for sufficiently large fixed costs.
Another feature the model abstracts from is the possibility that firms strategically delay the
time they implement their innovation, a point raised by Shleifer (1986). This is particularly
relevant when aggregate productivity varies over time, since firms can potentially undertake
innovation in recessions when the cost of R&D is low but wait until booms to implement them.15
While this intuition is suggestive, allowing for strategic delay need not eliminate the inefficient
timing of innovation. This is best illustrated using the version of the model in Section 2. The
next Proposition provides conditions under which the equilibrium in Proposition 4 survives
even when firms can strategically delay implementing their innovations:
Z0
µ
<
. Then there exists a F 0 > F ∗ as defined
ρ+µ
Z1
in Lemma 2 such that the (R0 , R1 ) identified in Proposition 4 remains an equilibrium for all
F ∈ (F ∗ , F 0 ) even if agents can delay implementation.
Proposition 5: Suppose λ < e1−α and

Intuitively, as long as the discount rate ρ is large and regime switches are infrequent so µ is
small, it will not pay to delay innovation until a boom given the long expected wait until it
arrives. The notion that firms do not strategically delay innovation does have some empirical
support. As already noted in an earlier footnote, Griliches (1990) notes that firms tend to take
out patents — and thus reveal their new ideas — early on in the research process, long before
they put their new ideas to use. Moreover, strategic delay would imply a mismatch between
R&D activity and patenting over the cycle, whereas Griliches reports that R&D and patents
remain highly synchronized over the business cycle.
If the assumptions of Proposition 5 are not satisfied, firms might very well prefer to delay
15
Indeed, this is what Francois and Lloyd-Ellis (2003) find when they endogenize R&D in Shleifer’s model.
However, in their model the recession emerges endogenously as a result of R&D activity rather than as the result
of an exogenous change in productivity.

27

implementation until aggregate productivity is high, and the equilibrium with procyclical innovation may not survive. In this case, something else would be required to account for the
procyclical pattern in R&D we observe in the data. However, if in equilibrium some of the
new ideas discovered in recessions are only implemented in booms, the inefficiency described
here would simply carry over from innovation to implementation: a benevolent planner would
choose to concentrate both innovation and implementation in recessions, but in the decentralized economy implementation would be procyclical even if innovation were not. The inefficient
implementation of new ideas — as opposed to the inefficient allocation of resources in coming
up with new ideas — is related to results in Caballero and Hammour (1996), who show in a
different setting that the process of adopting new technologies can be distorted in the presence
of frictions. Unfortunately, strategic delay in the presence of aggregate fluctuations is difficult
to analyze. Although Francois and Lloyd-Ellis (2003) can fully characterize strategic delay in
the same model as this one but where aggregate productivity zt is fixed over time, it turns out
that their approach cannot be easily extended to the case of time-varying productivity.

5. Conclusion
In recent years, a growing number of economists has argued that recessions encourage agents
to invest in making their technologies more productive. But one of the important channels for
productivity growth, R&D effort, appears to be procyclical. This paper provides an explanation
for why R&D is procyclical even though, as the neo-Schumpeterian view argues, it is efficient to
concentrate it in recessions. This captures the spirit of recent work by Caballero and Hammour
(2004), who also question whether recessions promote the reallocation of resources to more
productive uses. Focusing on job reallocation, they conclude that “on the contrary, cumulative
restructuring is lower following recessions... yet — contrary to the cost-of-liquidations view —
this stifling of reallocation is costly.” This paper considers R&D rather than job reallocation,
but it similarly argues that recessions lead to lower R&D activity, not higher, and that this
stifling of R&D is costly. Recessions thus fail to act as an incubator for productivity growth,
even when in principle they ought to play that role.16
At the same time, as already noted in the Introduction, some growth-enhancing activities are
countercyclical. One prominent example is schooling. Betts and McFarland (1995) and Dellas
and Sakellaris (2003) document countercyclical college enrollment for the U.S., especially for
16
On a related note, Barlevy (2002) argues that search frictions may prevent recessions from reallocating
resources to their most appropriate uses, even though recessions should ameliorate productive inefficiency by
cleansing out less productive uses of resources.

28

part-time students, while Sepulveda (2002) finds participation in training courses, both on and
off the job, are countercyclical. In the opposite direction, King and Sweetman (2002) argue that
the number of workers who quit their job to return to school is procyclical, although full-time
commitment to school is a special form of human capital accumulation that may be sensitive to
the difficulty of finding jobs in recessions. The fact that schooling is countercyclical but R&D
is not is consistent with the notion that schooling is not associated with dynamic externalities
that would cause agents to behave in a short-sighted manner.
The quantitative analysis in this paper suggests the inefficient timing of R&D leads to a
small welfare cost, although the implied cost of business cycles is larger than in Lucas (1987).
Interestingly, the optimal policy response does not involve stabilization. In fact, since cyclical
fluctuations allow the economy to grow at a lower overall cost, policymakers might very well
welcome fluctuations (provided agents are not too risk averse), at least if they can induce
agents to substitute intertemporally and undertake R&D in recessions. This could presumably
be achieved through subsidies to R&D in recessions. By contrast, given evidence that relatively
unconstrained firms continue to concentrate their R&D in booms, a policy of easing credit
conditions in recessions by itself may not suffice to induce countercyclical R&D.
While this paper focuses primarily on business cycles, it also contributes to the literature
on long-run growth. Whereas most of this literature considers steady-state growth, this paper
explores non-steady-state dynamics in traditional growth models. It reveals that spillovers
inherent to the R&D process can lead not only to too much or too little steady-state growth,
as previous work has already documented, but also to an inefficient response to shocks around
the steady state. Thus, whereas previous work suggests that there is a special case in which
various forces cancel out so that steady-state equilibrium growth is optimal, the results here
suggest that even in this special case, the response of R&D to shocks around its steady state is
necessarily suboptimal. The ability of the decentralized market to achieve an efficient level of
growth would therefore appear to be even more fragile than suggested in previous work.

29

Appendix
Proof of Propositions 1 and 3: For given values of {Ri }i=0,1 , the system given by (2.10) reduces to ordinary
linear differential equations in V (Zi , M). Standard theorems ensure this system has a unique solution. Hence,
starting with values for Ri , we can use the method of undetermined coefficients to find the unique value functions
V (Zi , M) associated with a given pair (R0 , R1 ). I conjecture that the value function V (·, ·) takes the form
V (Zi , M) = vi λM (1−α)
Differentiating this function with respect to M yields
∂V
= (1 − α) vi λM(1−α) ln λ
∂M
which simplifies the differential equations above to a system of independent linear equations in the coefficients
vi :
ρvi = Zi (L − Ri )1−α − F + µ (v1−i − vi ) + (1 − α) vi φRi ln λ

This yields a unique solution (v0 , v1 ) as functions of (R0 , R1 ).

Since the RHS of (2.10) is strictly concave in Ri , the first order condition is both necessary and sufficient to
characterize the optimal Ri . The first order condition is given by
− (1 − α) Zi λM (1−α) (L − Ri )−α +

∂V
φ≤0
∂M

with equality if Ri > 0. Substituting the expression for V (·, ·), we obtain

1
α

Zi
Zi
L−
if vi >
α ln λ
Ri =
v
φ
ln
λ
φL
i

0
else

(5.1)

(5.2)

If we substitute this expression into the asset equation (2.10), we obtain a pair of equations with v1−i as a
function of vi that hold at the optimal Ri :

1
α 1
F
Zi
(ρ + µ − (1 − α) φL ln λ)

vi − Ziα (vi φ ln λ)1− α +
if vi >

α ln λ
µ
µ
µ
φL
v1−i = g1−i (vi ) =
1−α
F

 ρ + µ vi − Zi L
+
else
µ
µ
µ

The optimal program corresponds to any pair (v0∗ , v1∗ ) which solves the equations
v1∗

=

g1 (v0∗ )

v0∗

=

g0 (v1∗ )

The function g1−i (·) is continuous and differentiable, since the left and right hand derivatives at vi =

Zi
φLα ln λ

ρ+µ
∂g1−i (vi )
. Since ρ > (1 − α) φL ln λ, it follows that
> 1 for all vi . The functions g1−i (·)
µ
∂vi
∗
are illustrated in Figure A1, suggesting that there is a unique solution (v0 , v1∗ ). To establish this formally, I use
dg1−i
the fact
> 1 > 0 for all vi implies g1−i (·) is invertible. An equilibrium therefore involves a value v0∗ such
dvi
that g1 (v0∗ ) − g0−1 (v0∗ ) = 0. Differentiating this condition with respect to v0∗ yields
are both equal to

dg1
d
g1 (x) − g0−1 (x) =
−
dx
dx

dg0
dx

−1

>0

This monotonicity insures there is at most one value of v0∗ . To establish existence, note that g1 (0) < 0 while
−1
dg0
dg1
> 1 > lim
g0−1 (0) > 0. Hence, g1 (0) − g0−1 (0) < 0, and is finite. The fact that lim
implies
x→∞ dx
x→∞
dx
∂
g1 (x) − g0−1 (x) is strictly bounded away from 0, and so g1 (x) − g0−1 (x) → ∞ as x → ∞. The existence of
∂x
∗
v0 follows from continuity. This implies there is a unique social solution to the social planner’s problem.
Next, suppose that the optimal path dictates Ri > 0 for both i. I need to show R0 > R1 . The proof proceeds
in two steps. First, I argue that v1∗ > v0∗ . Since Ri > 0, the asset equations imply
∗
v1−i

=
≡

(ρ + µ − (1 − α) φL ln λ) ∗ α α1 ∗
F
1− 1
vi − Zi (vi φ ln λ) α +
µ
µ
µ
1
1
F
1−
avi∗ − bZiα (vi∗ ) α +
µ

Consider the fixed point vi which solves
1

1

vi = avi − bZiα (vi )1− α +

F
µ

It is easy to show vi exists and is unique. Implicit differentiation implies

dvi
=
dZi

so that Z0 < Z1 ⇒ v0 < v1 . Since
Hence,

b
α

vi
Zi

1
1− α

1−α
b
(α − 1) +
α

1
α

Zi
vi

>0

dgi−1
< 1, we know that for any x < v1 , it follows that x − g0−1 (x) < 0 .
dx
g1 (v0 ) − g0−1 (v0 )

=
<

v0 − g0−1 (v0 )

0

where the inequality uses the fact that v0 > v1 . Since g1 (v0∗ ) − g0−1 (v0∗ ) = 0 and g1 (x) − g0−1 (x) is increasing
dgi−1
> 1, the fact that g1 (v0 ) = v0 implies g1 (x) > x for any x > v0 .
in x, it follows that v0∗ > v0 . But since
dx
Hence, g1 (v0∗ ) > v0∗ . But since v1∗ = g1 (v0∗ ), it follows that v1∗ > v0∗ .
v1
v0
<
, which is sufficient to establish R1 < R0 from the
Z1
Z0
∗
first-order condition above. Combining the equations v1−i = g1−i (vi∗ ) for both values yields the equation
Next, I use the fact that v1∗ > v0∗ to argue

1

1
1− α

av0∗ − bZ0α (v0∗ )

1

which can be rearranged to yield
v0∗
=
v1∗

1
1− α

− v1∗ = av1∗ − bZ1α (v1∗ )
(a + 1) − b

Z1
v1∗

(a + 1) − b

Z0
v0∗

so that
v1∗ > v0∗ ⇔

v1∗
Z1
<
v0∗
Z0

But given the expression for Ri in (5.2), this implies R0 > R1 .

1
α

1
α

− v0∗

Finally, suppose Ri = 0 for some i. The proposition follows if we can rule out the case where R0 = 0 and
v∗
Z1
. Let vi∗ (Z0 , Z1 ) denote the
R1 > 0 for sufficiently small F . Once again, it will be enough to show that 1∗ <
v0
Z0
values of vi∗ given Z0 and Z1 . It will be enough to prove that
∂
∂Z1

Z0 v1∗ (Z0 , Z1 )
<0
Z1 v0∗ (Z0 , Z1 )

(5.3)

This is because by integrating (5.3) with respect to Z1 , we obtain
Z0 v1∗ (Z0 , Z1 )
Z1 v0∗ (Z0 , Z1 )

Z1

=

Z0 v1∗ (Z0 , Z0 )
+
Z0 v0∗ (Z0 , Z0 )

<

Z0 v1∗ (Z0 , Z0 )
=1
Z0 v0∗ (Z0 , Z0 )

Z0

∂
∂Z1

Z0 v1∗ (Z0 , Z1 )
dZ1
Z1 v0∗ (Z0 , Z1 )

Note that (5.3) holds if and only if
∂
ln
∂Z1

Z0 v1∗ (Z0 , Z1 )
Z1 v0∗ (Z0 , Z1 )

<0

or alternatively if

∂v ∗ /∂Z1
∂v1∗ /∂Z1
< 1 + 0∗
∗
v1 /Z1
v0 /Z1
Differentiating the asset equations with respect to Z1 yields

v1
(L − R1 ) φ ln λ




φ (L − R1 ) ln λ + αφR1 ln λ Z1

∂v1
=

∂Z1

v1
(ρ + µ) Z0 L1−α



(ρ + µ) Z1 L1−α + µ (Z0 L1−α + A) − (ρ + 2µ) F Z1
∂v1
µ
∂v0
=
∂Z1
ρ + µ − (1 − α) φR1−i ln λ ∂Z1

if R1 > 0

if R1 = 0

where A is defined by
Z0 L1−α + A = max Z0 (L − R0 )1−α + (1 − α) v0 φR0 ln λ
R0

∂v1
v1
∂v0
∂v1∗ /∂Z1
µ
Z0 L1−α ,
<
, and
> 0. This insures
< 1 <
so that A ≥ 0. Provided F <
ρ + 2µ
∂Z1
Z1
∂Z1
v1 /Z1
∗
∂v /∂Z1
, completing the proof. ¥
1 + 0∗
v0 /Z1

µ
Z1 α Z0 1−α
ξ −
ξ
, where ω (L) = ρ + µ + (1 − (1 − α) ln λ) φL. There exists a
ω (L) Z0
Z1
∗
∗
∗
∗
unique ξ > 0 such that 1 − ξ T h (ξ ) if ξ S ξ . This unique solution ξ ∗ lies in the interval (0, 1).
Lemma 1: Let h (ξ) =

Proof of Lemma 1: First, I claim there exists a ξ ∗ ∈ (0, 1) for which 1 − ξ = h (ξ). This is straightforward:
if ξ = 0, we have
1 − ξ = 1 > 0 = h (ξ)

while if ξ = 1, we have

1−ξ =0<

Z1
µ
Z0
= h (ξ)
−
ω (L) Z0
Z1

where the inequality relies on the fact that ω (L) > 0 given that ρ > (1 − α) φL ln λ. The claim follows from
continuity.

To prove ξ ∗ is unique, I proceed in two steps. Differentiating h (·) yields
µ
Z0 −α
Z1
ξ
α ξ α−1 − (1 − α)
ω (L)
Z0
Z1

h0 (ξ) =
For ξ ≥ 1, we have
α

Z1 α−1
Z0 −α
ξ
− (1 − α)
ξ
Z0
Z1

>

− (1 − α)

>

−1

Z0 −α
ξ
Z1

Since at ξ = 1, h (ξ) > 1 − ξ, a necessary condition for there to exist a ξ ∗ > 1 such that 1 − ξ ∗ = h (ξ ∗ ) is that
there exists a ξ > 1 such that h0 (ξ) < −1. Thus, there exists no ξ > 1 for which 1 − ξ = h (ξ).
Next, I need to show there is a unique ξ ∗ ∈ (0, 1) for which 1 − ξ ∗ = h (ξ ∗ ). Consider first the case where
α > 12 . Differentiating h (·) establishes that h0 (ξ) ≥ 0 if and only if
α
1−α
For α >
α
1−α

1
,
2

Z1
Z0

2

Z1
Z0

> ξ 1−2α

0

1 − 2α < 0, and so h (ξ) is negative if 0 < ξ <
2

1
1−2α

α
1−α

Z1
Z0

. Since h (0) = 0, it follows that h (ξ) < 0 for 0 < ξ <

1 − ξ > 0 > h (ξ) for ξ <

α
1−α

Z1
Z0

2

1
1−2α

0

2

1
1−2α

and positive if ξ >
α
1−α

< 1. Since h (ξ) is strictly positive for

follows that ξ ∗ is unique and 1 − ξ ∗ T h (ξ ∗ ) if ξ S ξ ∗

Z1
Z0
α
1−α

2

1
1−2α

. Hence,
Z1
Z0

2

1
1−2α

, it

1
µ
Z1 − Z0
ξ 2 which is monotonically increasing in ξ while 1 − ξ is monotonω (L)
Z1 Z0
ically decreasing. This again insures ξ ∗ is unique and 1 − ξ ∗ T h (ξ ∗ ) if ξ S ξ ∗

If α = 12 , h (ξ) simplifies to

Finally, if α < 12 , it is enough to prove that h0 (ξ) > −1 for all ξ ∈ (0, 1]. Differentiating h (·) twice yields
h00 (ξ) =

α (1 − α) µ Z0 −α−1 Z1 α−2
ξ
−
ξ
ω (L)
Z1
Z0

so that

h00 (ξ) R 0 ⇔ ξ R

Z1
Z0

2
1−2α

2

Z1 1−2α
which for α < 12 is strictly greater than 1. But it was
Z0
previously argued that h0 (ξ) > −1 for all ξ ≥ 1. Hence, h0 (ξ) > −1 for all ξ ∈ (0, ∞) if α < 12 .
Thus, the derivative attains a minimum at ξ =

Lastly, since at ξ = 0, 1 − ξ = 1 > 0 = h (ξ), continuity implies 1 − ξ > h (ξ) for all ξ < ξ ∗ . Likewise, at ξ = 1,
1 − ξ = 0 < h (1), so by continuity it follows that 1 − ξ < h (ξ) for ξ > ξ ∗ . This establishes the lemma. ¥

Proof of Proposition 2: Since Ri ≥ 0, the case where R1 = 0 trivially satisfies the claim. If R0 = 0, we need
to verify that R1 = 0. To show this, suppose not, i.e. suppose R1 > 0 = R0 . Then it follows that φv1 = 1 ≥ φv0 .
Substituting in for vi , we get
 


  ω (R1 ) (L − R0 ) +

 ω (R0 ) (L − R1 ) +
Z0
Z1
≥
1−α
α
1−α
α
µ
µ
(L − R0 )
(L − R1 )  
(L − R1 )
(L − R0 ) 

Z1
Z0

Since v1 (R1 , R0 ) = 0 if R1 = L, then R1 < L in any such equilibrium. This allows us to define ξ such that
R0 = ξR1 + (1 − ξ) L

Note that since R1 < L, by construction, ξ ≥ 0, and R0 > R1 implies ξ ∈ [0, 1) while R0 < R1 implies ξ > 1.
After substituting in for ω (R) and rewriting R0 in terms of R1 , we can rewrite the inequality v1 ≥ v0 in terms
of ξ:
µ
Z1 α Z0 1−α
1−ξ ≥
ξ −
ξ
ω (L) Z0
Z1
Applying lemma 1, it follows that ξ < ξ ∗ < 1, which implies R0 > R1 , a contradiction. Thus, R0 = 0 implies
R1 = 0.
1
, which in turn implies that
φ
∗
1 − ξ = h (ξ). But from the lemma, the unique ξ which satisfies this equation is less than 1, which implies
R0 ≥ R1 .
Finally, if R1 and R0 are both positive, it must be true that v1 = v0 =

1

Next, I show that there exists a unique symmetric Markov-perfect equilibrium when λ < e 1−α . This condition
implies that (1 − α) ln λ < 1, which implies
ω0 (R) = (1 − (1 − α) ln λ) φ > 0
∂vi
∂vi
< 0 and
< 0. Recall from the
∂Ri
∂R1−i
proof of Proposition 2 that in any Markov-perfect equilibrium in which φvi (Ri , R1−i ) = 1 for both i, the levels
of innovation for the two levels of productivity are related by R0 = ξ ∗ R1 + (1 − ξ ∗ ) L, where ξ ∗ is a constant.
Hence, there can be at most one equilibrium in which φv0 = φv1 = 1. For suppose there were two such equilibria,
(R0 , R1 ) 6= (R00 , R10 ) where wlog R00 > R0 . Since ξ ∗ is constant, it follows that R10 > R1 , but since vi is decreasing
1
0
)= .
in both Ri and R1−i it is impossible that vi (Ri , R1−i ) = vi (Ri0 , R1−i
φ
Using (2.16), the fact that ω0 (R) > 0 can be shown to imply that

∂vi
< 0, it follows that for any (R0 , R1 ) 6= 0 there always
∂Ri
exists some i ∈ {0, 1} such that φvi < 1 but Ri > 0, which is inconsistent with equilibrium. In this case, (0, 0)
would be the unique equilibrium. Without loss of generality, then, I henceforth assume that if an equilibrium
exists, it is not equal to (0, 0).
If (R0 , R1 ) = (0, 0) is an equilibrium, given that

∗
I begin by arguing that if there exists an equilibrium (R0∗ , R1∗ ) where φvi (Ri∗ , R1−i
) = 1 for both i, there exists
no other equilibria in which Ri = 0 and φvi (0, R1−i ) < 1 for some i ∈ {0, 1}. For each i, define the contour sets

Ωi = {(Ri , R1−i ) | φvi (Ri , R1−i ) = 1}
for all values of (R0 , R1 ) ≥ (0, 0). These sets are illustrated in Figure A2. Using the implicit function theorem
∂vi
∂vi
and the fact that
and
are both strictly negative, we can establish that the graphs of Ωi form
∂Ri
∂R1−i
connected, downward sloping curves in (R0 , R1 ) space. If there exists an equilibrium (R0∗ , R1∗ ) 6= (0, 0) such that
∂vi
∗
∗
φvi (Ri∗ , R1−i
) = 1 for both i, the sets Ωi must both be nonempty. Since
< 0 and φvi (Ri∗ , R1−i
) = 1,
∂R1−i

then φvi (Ri∗ , 0) > 1. Since φvi (L, 0) = 0, there exists an Ri0 ≥ 0 such that (Ri0 , 0) ∈ Ωi by continuity. Hence,
The graph of Ωi intersects the Ri axis.
00
00
> 0 as the value of R1−i such that φvi (0, R1−i
) = 1. If no such value exists, I adopt the
Next, define R1−i
00
00
0
convention that R1−i = ∞. I now argue that R1 > R1 . The statement follows trivially if R100 = ∞. If R100 < ∞, I
argue that φv1 (R100 , 0) > 1. For suppose not, i.e. suppose φv1 (R100 , 0) ≤ 1. Since φv0 (0, R100 ) = 1, it follows that
either (0, R100 ) constitutes an equilibrium, or there exists some R1000 ∈ (R100 , L) such that φv1 (R1000 , 0) = 1, from
which it follows that (0, R1000 ) is an equilibrium. Since R0 ≥ R1 in any Markov-perfect equilibrium, it follows that
∂vi
R100 = R1000 = 0. Since φv0 (R0∗ , R1∗ ) = 1 and (R0∗ , R1∗ ) 6= (0, 0) by assumption,
< 0 implies φv0 (0, 0) > 1,
∂Ri
00
0
a contradiction. Hence, φv1 (R1 , 0) > 1 = φv1 (R1 , 0). Since vi is decreasing in Ri , it follows that R100 > R10 as
claimed.

The fact that R100 > R10 can be used to establish that R000 > R00 as well. First, though, I argue that at the
equilibrium (R0∗ , R1∗ ),
dR1
dR1
>
dR0 φv1 =1
dR0 φv0 =1
To see this, consider a neighborhood around (R0∗ , R1∗ ). Recall that for any admissible (R0 , R1 ) where R0 is defined
as ξR1 + (1 − ξ) L for some ξ ≥ 0, the proof of Proposition 2 above implies that v1 (R1 , R0 ) > v0 (R0 , R1 ) if and
only if 1−ξ > h (ξ), which from Lemma 1 holds if and only if ξ < ξ ∗ . Since for any ε > 0, R0∗ +ε = ξR1∗ +(1 − ξ) L
for some ξ < ξ ∗ , it follows that
v1 (R1∗ , R0∗ + ε) > v0 (R0∗ + ε, R1∗ )
1
Subtracting v0 (R0∗ , R1∗ ) = v1 (R1∗ , R0∗ ) = from both sides, dividing by ε, and taking the limit as ε → 0 implies
φ
∂v1 (R0∗ , R1∗ )
∂v0 (R1∗ , R0∗ )
≥
∂R0
∂R0
∂v0 (R1∗ , R0∗ )
∂v1 (R0∗ , R1∗ )
=
, the derivative
∂R0
∂R0
of 1 − ξ − h (ξ) with respect to ξ would be equal to 0 at ξ = ξ ∗ , which is contradicted by the proof of Lemma 1
above. Similarly, for any ε > 0, Lemma 1 implies
We can further establish this inequality is strict. This is because if

v0 (R0∗ , R1∗ + ε) > v1 (R1∗ + ε, R0∗ )
and by an analogous argument,

∂v1 (R1∗ , R0∗ )
∂v0 (R0∗ , R1∗ )
≥
∂R1
∂R1

Taking into account the fact that ∂vi /∂Ri and ∂vi /∂R1−i are both negative, it follows that
∂v0 /∂R0
∂v1 /∂R0
>
∂v0 /∂R1
∂v1 /∂R1
which implies
dR1
dR0

φv1 =1

=−

∂v1 /∂R0
∂v0 /∂R0
dR1
>−
=
∂v1 /∂R1
∂v0 /∂R1
dR0

φv0 =1

Since there can be only one point at which φvi (Ri , R1−i ) = 1, the two contours sets Ωi intersect only at (R0∗ , R1∗ ),
and by continuity it follows that R000 > R00 .
With these observations, I can finally establish that there exists no other equilibrium

Ri , R1−i

in which

Ri = 0 for some i ∈ {0, 1} and φvi 0, R1−i < 1. This is because if such an equilibrium existed, by definition it

00
00
must be true that φvi 0, R1−i ≤ 1. Since φvi (0, R1−i
) = 1 by definition, monotonicity implies R1−i ≥ R1−i
>

0
0
. But since R1−i > R1−i
, it follows that φv1−i (R1−i , 0) < 1. For this to be an equilibrium, R1−i = 0. But
R1−i
∗
∗
) = 1, it follows that φvi (0, 0) > 1, a contradiction.
since there exists an (R0 , R1∗ ) 6= (0, 0) such that φvi (Ri∗ , R1−i

Finally, suppose there exists no pair (R0∗ , R1∗ ) such that φv0 (R0∗ , R1∗ ) = φv1 (R0∗ , R1∗ ) = 1. We need to establish
that there still exists a unique Markov-perfect equilibrium. Suppose first that φv0 (0, 0) ≤ 1. Then for any
(R0 , R1 ) ≥ (0, 0) where (R0 , R1 ) 6= (0, 0), it must be the case that φv0 (R0 , R1 ) < 1. This implies R0 = 0 in any
equilibrium, and since R0 ≥ R1 in any equilibrium according to Proposition 2, R0 = R1 = 0 must be the unique
equilibrium. If we rewrite R0 as ξR1 + (1 − ξ) L, then ξ = 1 > ξ ∗ . But recall from Lemma 1 that this implies
v1 (0, 0) < v0 (0, 0). Since φv0 (0, 0) ≤ 1, it follows that (0, 0) is in fact an equilibrium.
This leaves the case where (i) there exists no pair (R0∗ , R1∗ ) such that φv0 (R0∗ , R1∗ ) = φv1 (R0∗ , R1∗ ) = 1 and
(ii) φv0 (0, 0) > 1. By continuity, there exists an R00 < L such that φv0 (R00 , 0) = 1. Then I claim (R0 , R1 ) =
(R00 , 0) is the unique Markov perfect equilibrium. Consider again two cases. First, suppose that φv1 (0, 0) ≤ 1.
In that case, monotonicity implies φv1 (0, R00 ) < 1 given that R00 > 0, so that φv1 (0, R00 ) ≤ 1 and (R00 , 0) is
indeed an equilibrium. Moreover, since φv1 (0, 0) ≤ 1, then R1 = 0 in any equilibrium, and it follows that (R00 , 0)
is the unique equilibrium. Lastly, suppose φv1 (0, 0) > 1. Once again, define R000 such that φv0 (0, R000 ) = 1,
with the convention of setting R000 = ∞ if no such value exists. We need to show that R00 > R000 , which insures
φv1 (0, R00 ) ≤ 1. Suppose not, i.e. suppose R000 ≥ R00 . But using the same argument as before, we know that
R100 > R10 . If R000 ≥ R00 , then by continuity Ω0 and Ω1 must intersect, which contradicts the supposition that
there exists no solution (R0∗ , R1∗ ). Hence, φv1 (0, R00 ) ≤ 1, so that (R00 , 0) is an equilibrium, and since R1 = 0 in
any equilibrium, which insures the equilibrium above is unique.¥

Proof of Lemma 2 (in text): Consider the equation v0 (R∗ , R∗ ) = v1 (R∗ , R∗ ). It implies
Z1
Z0
ω (R∗ ) − µ
Z1
µ Z0
Z0
Z1
−
−
=
λ Z1
Z0
(1 − α) Z1 (L − R∗ )1−α − F
(1 − α) Z0 (L − R∗ )1−α − F
ω (R∗ ) − µ

Differentiate left hand side yields
To establish the existence and uniqueness of R∗ , define yi = L − Ri . Using the assumption that Ri < L implies
yi 6= 0, we can rearrange the condition v0 (R, R) = v1 (R, R) by dividing both sides by y α and expanding out
ω (L − y) to obtain
µ (Z1 + Z0 )

λ (ω (L) + µ)
λ
y 1−α
+
(1 − (1 − α) ln λ) φy =
F
(λ − 1) (1 − α)
(λ − 1) (1 − α)

(5.4)

The LHS of this equation is monotonically increasing in y, and ranges from 0 to ∞ as y ranges from 0 to ∞.
Since the RHS above is strictly positive, there exists a unique value y ∗ for which the equation is satisfied. This
translates into a unique value R∗ = L − y ∗ .
ρ + 2µ
(1 − (1 − α) ln λ) φ
as F → ∞, at which point ω (L − y ∗ ) = −µ. If we evaluate vi (R∗ , R∗ ) and substitute in from (5.4), we obtain
Note that y ∗ is monotonically increasing in F . Taking limits, y ∗ → 0 as F → 0, while y ∗ → L+

vi

=

(λ − 1)

ω (L − y) + µ

Z1−i
(ω (L − y) + µ) λF (y ∗ )α
y∗ −
Zi
(λ − 1) (1 − α) Zi
ω2 (L − yF∗ ) − µ2

Z1−i
µ (Z1 + Z0 ) y ∗
y∗ −
Zi
Zi
ω2 (L − yF∗ ) − µ2

ω (L − y ∗ ) + µ

=

(λ − 1)

=

(λ − 1) y ∗
ω (L − y ∗ ) + µ

Hence, vi (R∗ , R∗ ) is monotonically increasing in y ∗ , which in turn is monotonically increasing in F . As noted
ρ + 2µ
, which implies vi ranges between 0 and ∞.
above, for different values of F , y ∗ ∈ 0, L +
(1 − (1 − α) ln λ) φ
∗
The existence of F follows from continuity. ¥
Proof of Proposition 4: Consider a fixed value F > F ∗ , where F ∗ is defined in Lemma 2. As in the proof
of Lemma 2, it will be useful to work with yi = L − Ri . Consider the set
S = {(y0 , y1 ) | v0 (L − y0 , L − y1 ) = v1 (L − y1 , L − y0 )}
The proposition follows if I can show that within this set there exists an element (y0 , y1 ) ∈ S such that y0 > y1
and φvi (L − yi , L − y1−i ) = 1 for i ∈ {0, 1}.
Consider first the case where y1 = 0. For this value, we have

v0

=

v1

=

ω (L) + µλF α
y
(1 − α) Z0 0
ω (L − y0 ) ω (L) − µ2

(λ − 1) ω (L) y0 −
0

Hence, there are exactly two values of y0 for which v0 (L − y0 , L) = v1 (L, L − y0 ), namely y0 = 0 and
(ω (L) + µ) λF
ω (L) (λ − 1) (1 − α) Z0

y0 = y0 ≡

1
1−α

By the implicit function theorem, there exist continuous functions y0 (·) defined in a neighborhood of y1 = 0 such
that y0 (y1 ) → 0 and y0 (y1 ) → y0 as y1 → 0 which satisfy v0 (L − y0 (y1 ) , L − y1 ) = v0 (L − y1 , L − y0 (y1 )).
For y1 6= 0, we can rewrite the equation v0 = v1 in terms of y1 and ξ =

y0
. The condition that v1 = v0 can
y1

be rewritten as

where

λF y1α−1
(A0 − A1 ξ − A2 ξ α ) − 1 + ξ + h (ξ) = 0
Z1 (λ − 1) (1 − α)
A0

=

A1

=

A2

=

ω (L) + µ
ω (L)
(1 − (1 − α) ln λ) φy1
ω (L)
ω (L − y1 ) + µ Z1
ω (L)
Z0

(5.5)

and as in Lemma 1,
µ
Z1 α Z0 1−α
ξ −
ξ
ω (L) Z0
Z1
For notational convenience, I will rewrite (5.5) more compactly as
h (ξ) =

Q (ξ; y1 ) = 0
The implicit functions y0 (y1 ) described above which limit to 0 and y0 establish the existence of functions ξ (y1 )
1
Z0 α
and ξ = ∞, respectively, as y1 → 0.
defined locally near y1 = 0 that limit to ξ =
Z1
Define y ∗ as in Lemma 2. Differentiate Q (ξ; y1 ) with respect to ξ twice to obtain
λF ξ α−2
∂2Q
00
A2 y1α−1
2 = h (ξ) + α (1 − α)
Z1 (λ − 1) (1 − α)
∂ξ
Substituting in for h00 (ξ), we obtain
∂2Q
= α (1 − α)
∂ξ 2

λF ξ α−2
Z0 −α−1 Z1 α−2
µ
+
A2 y1α−1
ξ
−
ξ
ω (L) Z1
Z0
Z1 (λ − 1) (1 − α)

Now, note that
A2 y1α−1

=
=

ω (L − y1 ) + µ Z1 α−1
y
ω (L)
Z0 1
ω (L) + µ α−1 (1 − (1 − α) ln λ) φy1α
y1 −
ω (L)
ω (L)

Z1
Z0

∂2Q
∂2Q
0
> 0 for all y1 ≤ y10 .
2 > 0 for some y1 , it follows that
∂ξ
∂ξ 2
For y1 = y ∗ , we know from the proof of Lemma 2 that y ∗ satisfies
is decreasing in y1 . Hence, if we can show that

µ
λF (y ∗ )α−1
=
Z1 (λ − 1) (1 − α)
ω (L − y ∗ ) + µ
Substituting this into the expression for

Z1 + Z0
Z1

∂2Q
yields
∂ξ 2

∂2Q
= α (1 − α)
∂ξ 2

µ Z0 −α−1
µ α−2
ξ
ξ
+
ω (L) Z1
ω (L)

>0

Hence, for all y1 ≤ y ∗ , Q (ξ; y1 ) is convex in ξ. This will prove important in what follows.
Before I proceed, I introduce the notation (y0 , y1 ) Ã (y00 , y10 ) to denote the case in which there exists a
continuous mapping y1 (τ ) > 0 and a continuous mapping y0 (τ ) defined for τ ∈ (0, 1) such that
1. lim y1 (τ ) = y1 and lim y0 (τ ) = y0
τ →0

2. lim y1 (τ ) =
τ →1

y10

τ →0

and lim y0 (τ ) = y00
τ →1

3. For all τ ∈ (0, 1), (y0 (τ ) , y1 (τ )) ∈ S, i.e. Q (ξ (τ ) ; y1 (τ )) = 0 for ξ (τ ) =

y0 (τ )
y1 (τ )

The notation lim y (τ ) = ∞ denotes, as usual, that for every N > 0, there exists a τ N such that y (τ ) > N
τ →1

for all τ > τ N . Thus, we can describe a path in which yi0 = ∞ for some i. Note that if we can establish that (y ∗ , y ∗ ) Ã (y0 , 0), the statement of the proposition follows from a simple continuity argument: since

φv0 (L − y ∗ , L − y ∗ ) > 1 for F > F ∗ but φv0 (y0 , 0) = 0, there exists some τ for which (y0 (τ ) , y1 (τ )) ∈ S and
where φvi (L − yi (τ ) , L − y1−i (τ )) = 1. Since v0 (L − y, L − y) = v1 (L − y, L − y) if and only if y = y ∗ , and
since y0 > 0, it follows that y0 (τ ) > y1 (τ ) by continuity.
∂Q
evaluated at ξ = 1 and
∂ξ
∗
y1 = y . For convenience, a graphical view of the set S corresponding to each of the three cases (for particular
parameter values) is provided in Figure A3.
I now break down my analysis into different cases, depending on the sign of

Case I:

∂Q (1; y ∗ )
>0
∂ξ

I claim this is enough to establish that (y ∗ , y ∗ ) Ã (y0 , 0), which from above is enough to establish the
y0
proposition. I first argue that there exists a y10 ∈ [0, y ∗ ) such that (y ∗ , y ∗ ) Ã (y00 , y10 ) where 00 = ∞. For
y1
∂Q (1; y1 )
∗
suppose not. Since
> 0 for all y1 , it follows that Q (1; y1 ) < 0 for all y1 < y . Furthermore, since
∂y1
∗
Q (ξ; y ) is convex in ξ, it also follows that Q (ξ; y ∗ ) > 0 for all ξ > 1. By continuity, then, the assumption that
(y ∗ , y ∗ ) 6Ã (∞, y1 ) for all y1 ∈ (0, y ∗ ) implies that for each y1 ∈ (0, y ∗ ) there must exist some ξ (y1 ) > 1 such
that Q (ξ (y1 ) ; y1 ) > 0. Applying the intermediate value theorem, we can deduce that for every y1 ∈ (0, y ∗ ) there
exists a ξ ∗ (y1 ) > 1 such that Q (ξ ∗ (y1 ) ; y1 ) = 0. Since Q is continuous and convex in ξ for all y1 ≤ y ∗ , the root
ξ ∗ (y1 ) is the unique value of ξ > 1 such that Q (ξ ∗ (y1 ) ; y1 ) = 1, is continuous in y1 , and limy1 ↑yF∗ ξ ∗ (y1 ) = 1.
Hence, (y ∗ , y ∗ ) Ã (ξ ∗ (y1 ) , y1 ). Taking the limit as y1 ↓ 0, it follows that (y ∗ , y ∗ ) Ã (y0 , 0), since the unique root
y0
greater than one for which lim Q (ξ; y1 ) = 0 limits to ∞. But then (y ∗ , y ∗ ) Ã (y00 , y10 ) such that 00 = ∞, which
y1 →0
y1
∂Q
is a contradiction. Since lim
< 0 for all y1 > 0, there can exist at most one y1 for which lim Q (ξ; y1 ) = 0.
ξ→∞ ∂y1
ξ→∞
Since lim Q (ξ; 0) = 0, it follows that (y ∗ , y ∗ ) Ã (y0 , 0).
ξ→∞

Case II:

∂Q (1; y ∗ )
= 0.
∂ξ

Since Q (1; y ∗ ) is strictly convex, it follows that Q (ξ; y ∗ ) > 0 for all ξ 6= 1. The fact that (y ∗ , y ∗ ) Ã (y0 , 0)
then follows from the same argument as in Case I.

Case III:

∂Q (1; y ∗ )
<0
∂ξ

In this case, the arguments above can no longer be used to establish that (y ∗ , y ∗ ) Ã (y0 , 0). However, I argue
y0 (τ )
> 1 along
that if (y ∗ , y ∗ ) 6Ã (y0 , 0), then there exists some y1 > 0 such that (y ∗ , y ∗ ) Ã (∞, y1 ) where lim
τ →1 y1 (τ )
any such connecting path. As I argue below, this condition is also sufficient to establish the proposition.
Again, the proof is by contradiction. Suppose the claim is false, i.e. suppose (y ∗ , y ∗ ) 6Ã (y0 , 0) and (y ∗ , y ∗ ) 6Ã
(∞, y1 ) for all y1 > 0, including y1 = ∞. If we differentiate ξ with respect to y1 along the curve Q (ξ; y1 ) = 0,
we obtain
∂Q/∂y1
dξ
>0
=−
dy1 (ξ;y1 )=(1;y∗ )
∂Q/∂ξ
∂Q (1; y1 )
> 0 for all y1 . Hence, if (y ∗ , y ∗ ) Ã (y0 , y1 ) for
∂y1
some y0 > y ∗ , it follows from continuity and the uniqueness of y ∗ that y0 > y1 . Next, since (y ∗ , y ∗ ) 6Ã (∞, y1 )

where the last inequality follows from the fact that

for all y1 (including y1 = ∞) by assumption, it follows that
y = sup {y0 | (y ∗ , y ∗ ) Ã (y0 , y1 ) for some y1 }
is finite. It follows that for any y0 > y ∗ , it must be the case that (y ∗ , y ∗ ) 6Ã (y0 , y1 ) if y1 ≥ y. Thus, any
continuous path that originates at (y ∗ , y ∗ ) for which y0 (τ ) > y ∗ is bounded in its y1 term from above by y.
But the fact that limy1 →0 Q (ξ; y1 ) < 0 for all finite ξ > 0, together with continuity and the uniqueness of y ∗ ,
y0 (τ )
= ∞. Since
implies that this occurs only if (y ∗ , y ∗ ) Ã (y0 , y1 ) for some y0 > y ∗ and some y1 such that lim
τ →1 y1 (τ )
∗
∗
y0 (τ ) ≤ y for all τ , this requires that lim y1 (τ ) = 0. But this contradicts the fact that (y , y ) 6Ã (y0 , 0). It
τ →1

follows that either (y , y ) Ã (y0 , 0) or (y ∗ , y ∗ ) Ã (∞, y1 ) where lim
∗

∗

τ →1

y0 (τ )
> 1 along this path.
y1 (τ )

y0 (τ )
> 1 implies there exists a
y1 (τ )
solution (y0 , y1 ) with y0 > y1 such that φv0 (L − y0 , L − y1 ) = φv1 (L − y1 , L − y0 ) = 1. Consider
The final step is to prove that the fact that (y ∗ , y ∗ ) Ã (∞, y1 ) where lim

τ →1

lim vi (L − yi , L − y1−i ) = lim

yi →∞

Z1−i 1−α (ω (L − y1−i ) + µ) λF
y
−
Zi 1−i
(1 − α) (λ − 1) Zi
ω (L − yi ) ω (L − y1−i ) − µ2

(λ − 1) ω (L − y1−i ) yi + µ

yi →∞

yiα

As yi → ∞, the numerator converges to ±∞, depending on the sign of ω (L − y1−i ), and since ω (L − yi ) → −∞,
the denominator converges to ±∞, again depending on the sign of ω (L − y1−i ). Applying L’Hopital’s rule, we
obtain
(λ − 1) ω (L − y1−i )
<0
lim vi (L − yi , L − y1−i ) = − 0
yi →∞
ω (·) ω (L − y1−i )

Hence, since v0 (L − y0 , L − y1 ) < 0 as y0 → ∞, it follows by continuity that there exists a pair (y0 , y1 ) such
that φvi (L − yi , L − y1−i ) = 1. Again, since v0 (L − y, L − y) = v1 (L − y, L − y) if and only if y = y ∗ and
y0 (τ )
> 1, it follows that y0 (τ ) > y1 (τ ) by continuity.
lim
τ →1 y1 (τ )
∂Q (1; y ∗ )
are possible, depending on parameter values. In cases II and III, there will be
∂ξ
multiple solutions to the problem φvi (Ri , R1−i ) = 1, i.e. in addition to the solution identified above, there also
exists a second solution with R0 > R1 . However, the existence of multiple solutions does not necessarily imply
multiple equilibria, since these solutions may involve negative values of Ri . ¥
Remark: all cases for

Proof of Proposition 5: Given that firms maximize expected profits, a firm that has successfully innovated
will choose the time to implement by solving
max Et e−ρs
s

1/Pt+s
vt+s
1/Pt

Suppose the solution in Proposition 4 is an equilibrium. The Proposition follows if we can show that s = 0.
Clearly, if zt+s = zt , there is no benefit from delay, since vt+s = vt and so in the best case scenario the firm
becomes the leader and earns vt discounted at a positive rate. Thus, given other agents are implementing
immediately and innovating in accordance with Proposition 4, a firm will only delay implementation until a
1
change in the level of productivity. If the current level of productivity is equal to Z1 , then given v0 = v1 = in
φ
equilibrium, waiting until a regime change yields at most
µ Z0
µ P1
v=
ρ + µ P0
ρ + µ Z1

L − R1
L − R0

α

v

assuming the firm is the leading producer when it implements. Since R1 > R0 , it follows that this is less than v.
Thus, there is no reason to delay an innovation uncovered when productivity is high. Conversely, there will be
no reason to delay an innovation that is discovered when productivity is low if
µ Z1
ρ + µ Z0

L − R0
L − R1

α

<1

µ Z1
< 1. Moreover, by continuity, the solution (R0 , R1 ) identified in Proposition 4 limits
ρ + µ Z0
∗
∗
∗
to (R , R ) as F → F . Thus, there will be no benefit from delay even though R1 > R0 for F close to F ∗ .

By assumption,

Figure 1

Growth in R&D Measures vs. Growth in Real GDP
(Shaded regions denote NBER recessions)

R&D growth

GDP growth

0.40

0.08
0.06

0.35

0.04
0.30
0.02
0.25
0.00
0.20

-0.02

0.15

-0.04
-0.06

0.10

-0.08
0.05
-0.10
0.00
1960

1970

1980

1990

-0.05
-0.10

2000

-0.12
-0.14

Growth rate, real GDP
growth
Growth rate, real R&D expenditures

-0.16

Growth rate, number of full-time-equivalent R&D scientists and engineers

Source: National Science Foundation

Figure 2

R&D Growth, Aggregate vs. Compustat Firms
0.30

0.25

0.20

0.15

0.10

0.05

0.00
1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002
-0.05

-0.10

Compustat universe

Aggregate (from NSF)

Source: National Science Foundation and S&P's Compustat database

Figure 3

Growth in R&D in Relatively Unconstrained Firms
(Shaded regions denote NBER recessions)
0.30

0.25

0.20

0.15

0.10

0.05

0.00
1954 1956 1958 1960 1962 1964 1966 1968 1970 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002

-0.05

-0.10

Lagged Cash Flow > $50 M

Lagged Net Worth > $150 M

Source: S&P Compustat database

Figure 4

21.9

R

21.7

21.5

21.3

21.1

R0
R1

20.9

k
20.7
4.55

4.65

4.75

4.85

4.95

5.05

5.15

5.25

Optimal R&D as a function of capital ratio k

R

1.30
1.25
1.20
1.15
1.10
1.05

R0
R1

1.00
0.95
0.90

k
0.85
88

90

92

94

96

98

100

102

104

Equilibrium R&D as a function of capital ratio k
Note: arrows indicate direction of capital accumulation for each respective level of productivity

106

108

110

v1
v1 = f1(v0)

v0 = f0(v1)

(v*0, v*1 )

45°
v0

v0
v1

Figure A1: Uniqueness of Social Optimum

R1

R1’’

Ω0 ≡ { (R0, R1) | φv0(R0, R1) = 1}

R1’
R1*
Ω1 ≡ { (R0, R1) | φv1(R1, R0) = 1}

R0*

R0’

R0’’

R0

Figure A2: Uniqueness of Markov-Perfect Equilibrium

(yκ*, yκ*)
(yκ*, yκ*)

~0, 0)
(y
(0, 0)

(yκ*, yκ* )
(yκ*, yκ*)

y1

~0, 0)
(y
(0, 0)

y1

(yκ*, yκ*)
(yκ*, yκ* )

~y0

0

y0

~y0

0

(A)

(B)
(yκ*, yκ*)
(yκ*, yκ*)

?

~0, 0)
(y
(0, 0)

y1

(yκ*, yκ*)

~y0

0

y0

(C)

Figure A3: The Evolution of the set S in Proposition 4

Panel (A) corresponds to Case I in proof of Proposition 4
Panel (B) corresponds to Case II in proof of Proposition 4
Panel (C) corresponds to Case III in proof of Proposition 4

y0

References
[1] Aghion Philippe, George-Marios Angeletos, Abhijit Banerjee, and Kalina Manova, 2005.
“Volatility and Growth: Credit Constraints and Prodcutivity-Enhancing Investment”
NBER Working Paper #11349.
[2] Aghion, Philippe and Peter Howitt, 1992. “A Model of Growth through Creative Destruction” Econometrica, 60(2), March, p323-51.
[3] Aghion, Philippe and Gilles Saint Paul, 1998. “Virtues of Bad Times: Interaction between
Productivity Growth and Economic Fluctuations” Macroeconomic Dynamics, September,
2(3), p322-44.
[4] Barlevy, Gadi, 2002. “The Sullying Effect of Recessions” Review of Economic Studies, 69
(1), January, p65-96.
[5] Barlevy, Gadi, 2004. “The Cost of Business Cycles under Endogenous Growth” American
Economic Review, September, 93(4).
[6] Barlevy, Gadi and Dani Tsiddon, 2004. “Earnings Inequality and the Business Cycle”
NBER Working Paper #10469 and forthcoming, European Economic Review.
[7] Basu, Susanto, 1996. “Procyclical Productivity: Increasing Returns or Cyclical Utilization?” Quarterly Journal of Economics, 111(3), August, p719-51.
[8] Benhabib, Jess and Roger Farmer, 1994. “Indeterminacy and Increasing Returns” Journal
of Economic Theory, 63(1), June, p19-41.
[9] Betts, Julian and Laurel McFarland, 1995. “Safe Port in a Storm: The Impact of Labor
Market Conditions on Community College Enrollments” Journal of Human Resources,
30(4), p741-65.
[10] Burnside, Craig, Martin Eichenbaum, and Sergio Rebelo, 1993. “Labor Hoarding and the
Business Cycle” Journal of Political Economy, 101(2), April, p245-73.
[11] Caballero, Ricardo and Mohamad Hammour, 1996. “On the Timing and Efficiency of
Creative Destruction” Quarterly Journal of Economics, August, 111(3), p805-52.
[12] Caballero, Ricardo and Mohamad Hammour, 1999. “The Costs of Recessions Revisited: a
Reverse-Liquidationist View” NBER Working Paper #W7355, September.
[13] Canton, Eric and Harald Uhlig, 1999. “Growth and the Cycle: Creative Destruction versus
Entrenchment” Journal of Economics, 69(3), p239-66.

[14] Cooper, Russell and John Haltiwanger, 1993. “The Aggregate Implications of Machine
Replacement: Theory and Evidence” American Economic Review, June, 83(3), p181-186.
[15] Cooper, Russell, John Haltiwanger, and Laura Power, 1999. “Machine Replacement and
the Business Cycle: Lumps and Bumps” American Economic Review, 89(4), September,
p921-46.
[16] Dellas, Harris and Plutarchos Sakellaris, 2003. “On the Cyclicality of Schooling: Theory
and Evidence” Oxford Economic Papers, January, 55(1), p148-72.
[17] DeJong, David and Beth Ingram, 2001. “The Cyclical Behavior of Skill Acquisition” Review
of Economic Dynamics, July, 4(3), p536-61.
[18] Fatas, Antonio, 2000. “Do Business Cycles Cast Long Shadows? Short-Run Persistence
and Economic Growth” Journal of Economic Growth, 5(2), June, p147-62.
[19] Francois, Patrick and Huw Lloyd-Ellis, 2003. “Animal Spirits Through Creative Destruction” American Economic Review, June, 93(3), p530-50.
[20] Gomes, Joao, Jeremy Greenwood, and Sergio Rebelo, 2001. “Equilibrium Unemployment”
Journal of Monetary Economics, 48(1), p109-52.
[21] Griliches, Zvi, 1984. “Comment on R&D and Innovation: Some Empirical Findings” in
R&D, Patents, and Productivity, edited by Zvi Griliches, Chicago: University of Chicago
Press, p148-149
[22] Griliches, Zvi, 1990. “Patent Statistics as Economic Indicators: a Survey” Journal of
Economic Literature, December, 28(4), p1661-1707.
[23] Grossman, Gene and Elhanan Helpman, 1991. “Quality Ladders in the Theory of Growth”
Review of Economic Studies, 58(1), January, p43-61.
[24] Hall, Robert 1991. “Labor Demand, Labor Supply, and Employment Volatility” NBER
Macroeconomics Annual. Cambridge, Mass: MIT Press.
[25] Hall, Robert, 2000. “Reorganization” Carnegie Rochester Conference Series on Public Policy, 52(0), June, p1-22.
[26] Jones, Charles and John Williams, 2000. “Too Much of a Good Thing? The Economics of
Investment in R&D” Journal of Economic Growth, 5(1), March, p65-85.
[27] King, Ian and Arthur Sweetman, 2002. “Procyclical Skill Retooling and Equilibrium
Search” Review of Economic Dynamics, 5(3), July, p704-17.

[28] Lucas, Robert, 1987. Models of Business Cycles. Oxford: Basil Blackwell.
[29] Mansfield, Edwin, 1987. “Price Indexes for R and D Inputs, 1969-1983” Management
Science, 33(1), January, p124-9.
[30] Ramey, Valerie, 1991. “Comment on Markups and the Business Cycle” NBER Macroeconomics Annual, vol 6, p134-9.
[31] Rivera-Batiz, Luis and Paul Romer, 1991. “Economic Integration and Endogenous
Growth,” Quarterly Journal of Economics, 106(2), May, p531-55.
[32] Rotemberg, Julio and Michael Woodford, 1999. “The Cyclical Behavior of Prices and
Costs” in Handbook of Macroeconomics (1B), vol. 15, edited by John Taylor and Michael
Woodford. Amsterdam: Elsevier Science, p1051-1135.
[33] Sepulveda, Facundo, 2002. “Training and Business Cycles” Mimeo, Michigan State University.
[34] Shleifer, Andrei, 1986. “Implementation Cycles” Journal of Political Economy, 94(6), December, p1163-90.

Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and economic topics.
Does Bank Concentration Lead to Concentration in Industrial Sectors?
Nicola Cetorelli

WP-01-01

On the Fiscal Implications of Twin Crises
Craig Burnside, Martin Eichenbaum and Sergio Rebelo

WP-01-02

Sub-Debt Yield Spreads as Bank Risk Measures
Douglas D. Evanoff and Larry D. Wall

WP-01-03

Productivity Growth in the 1990s: Technology, Utilization, or Adjustment?
Susanto Basu, John G. Fernald and Matthew D. Shapiro

WP-01-04

Do Regulators Search for the Quiet Life? The Relationship Between Regulators and
The Regulated in Banking
Richard J. Rosen
Learning-by-Doing, Scale Efficiencies, and Financial Performance at Internet-Only Banks
Robert DeYoung
The Role of Real Wages, Productivity, and Fiscal Policy in Germany’s
Great Depression 1928-37
Jonas D. M. Fisher and Andreas Hornstein

WP-01-05

WP-01-06

WP-01-07

Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy
Lawrence J. Christiano, Martin Eichenbaum and Charles L. Evans

WP-01-08

Outsourcing Business Service and the Scope of Local Markets
Yukako Ono

WP-01-09

The Effect of Market Size Structure on Competition: The Case of Small Business Lending
Allen N. Berger, Richard J. Rosen and Gregory F. Udell

WP-01-10

Deregulation, the Internet, and the Competitive Viability of Large Banks
and Community Banks
Robert DeYoung and William C. Hunter

WP-01-11

Price Ceilings as Focal Points for Tacit Collusion: Evidence from Credit Cards
Christopher R. Knittel and Victor Stango

WP-01-12

Gaps and Triangles
Bernardino Adão, Isabel Correia and Pedro Teles

WP-01-13

A Real Explanation for Heterogeneous Investment Dynamics
Jonas D.M. Fisher

WP-01-14

Recovering Risk Aversion from Options
Robert R. Bliss and Nikolaos Panigirtzoglou

WP-01-15

Economic Determinants of the Nominal Treasury Yield Curve
Charles L. Evans and David Marshall

WP-01-16

1

Working Paper Series (continued)
Price Level Uniformity in a Random Matching Model with Perfectly Patient Traders
Edward J. Green and Ruilin Zhou

WP-01-17

Earnings Mobility in the US: A New Look at Intergenerational Inequality
Bhashkar Mazumder
The Effects of Health Insurance and Self-Insurance on Retirement Behavior
Eric French and John Bailey Jones

WP-01-18

The Effect of Part-Time Work on Wages: Evidence from the Social Security Rules
Daniel Aaronson and Eric French

WP-01-20

Antidumping Policy Under Imperfect Competition
Meredith A. Crowley

WP-01-21

WP-01-19

Is the United States an Optimum Currency Area?
An Empirical Analysis of Regional Business Cycles
Michael A. Kouparitsas

WP-01-22

A Note on the Estimation of Linear Regression Models with Heteroskedastic
Measurement Errors
Daniel G. Sullivan

WP-01-23

The Mis-Measurement of Permanent Earnings: New Evidence from Social
Security Earnings Data
Bhashkar Mazumder

WP-01-24

Pricing IPOs of Mutual Thrift Conversions: The Joint Effect of Regulation
and Market Discipline
Elijah Brewer III, Douglas D. Evanoff and Jacky So

WP-01-25

Opportunity Cost and Prudentiality: An Analysis of Collateral Decisions in
Bilateral and Multilateral Settings
Herbert L. Baer, Virginia G. France and James T. Moser

WP-01-26

Outsourcing Business Services and the Role of Central Administrative Offices
Yukako Ono

WP-02-01

Strategic Responses to Regulatory Threat in the Credit Card Market*
Victor Stango

WP-02-02

The Optimal Mix of Taxes on Money, Consumption and Income
Fiorella De Fiore and Pedro Teles

WP-02-03

Expectation Traps and Monetary Policy
Stefania Albanesi, V. V. Chari and Lawrence J. Christiano

WP-02-04

Monetary Policy in a Financial Crisis
Lawrence J. Christiano, Christopher Gust and Jorge Roldos

WP-02-05

Regulatory Incentives and Consolidation: The Case of Commercial Bank Mergers
and the Community Reinvestment Act
Raphael Bostic, Hamid Mehran, Anna Paulson and Marc Saidenberg

WP-02-06

2

Working Paper Series (continued)
Technological Progress and the Geographic Expansion of the Banking Industry
Allen N. Berger and Robert DeYoung

WP-02-07

Choosing the Right Parents: Changes in the Intergenerational Transmission
of Inequality  Between 1980 and the Early 1990s
David I. Levine and Bhashkar Mazumder

WP-02-08

The Immediacy Implications of Exchange Organization
James T. Moser

WP-02-09

Maternal Employment and Overweight Children
Patricia M. Anderson, Kristin F. Butcher and Phillip B. Levine

WP-02-10

The Costs and Benefits of Moral Suasion: Evidence from the Rescue of
Long-Term Capital Management
Craig Furfine

WP-02-11

On the Cyclical Behavior of Employment, Unemployment and Labor Force Participation
Marcelo Veracierto

WP-02-12

Do Safeguard Tariffs and Antidumping Duties Open or Close Technology Gaps?
Meredith A. Crowley

WP-02-13

Technology Shocks Matter
Jonas D. M. Fisher

WP-02-14

Money as a Mechanism in a Bewley Economy
Edward J. Green and Ruilin Zhou

WP-02-15

Optimal Fiscal and Monetary Policy: Equivalence Results
Isabel Correia, Juan Pablo Nicolini and Pedro Teles

WP-02-16

Real Exchange Rate Fluctuations and the Dynamics of Retail Trade Industries
on the U.S.-Canada Border
Jeffrey R. Campbell and Beverly Lapham

WP-02-17

Bank Procyclicality, Credit Crunches, and Asymmetric Monetary Policy Effects:
A Unifying Model
Robert R. Bliss and George G. Kaufman

WP-02-18

Location of Headquarter Growth During the 90s
Thomas H. Klier

WP-02-19

The Value of Banking Relationships During a Financial Crisis:
Evidence from Failures of Japanese Banks
Elijah Brewer III, Hesna Genay, William Curt Hunter and George G. Kaufman

WP-02-20

On the Distribution and Dynamics of Health Costs
Eric French and John Bailey Jones

WP-02-21

The Effects of Progressive Taxation on Labor Supply when Hours and Wages are
Jointly Determined
Daniel Aaronson and Eric French

WP-02-22

3

Working Paper Series (continued)
Inter-industry Contagion and the Competitive Effects of Financial Distress Announcements:
Evidence from Commercial Banks and Life Insurance Companies
Elijah Brewer III and William E. Jackson III

WP-02-23

State-Contingent Bank Regulation With Unobserved Action and
Unobserved Characteristics
David A. Marshall and Edward Simpson Prescott

WP-02-24

Local Market Consolidation and Bank Productive Efficiency
Douglas D. Evanoff and Evren Örs

WP-02-25

Life-Cycle Dynamics in Industrial Sectors. The Role of Banking Market Structure
Nicola Cetorelli

WP-02-26

Private School Location and Neighborhood Characteristics
Lisa Barrow

WP-02-27

Teachers and Student Achievement in the Chicago Public High Schools
Daniel Aaronson, Lisa Barrow and William Sander

WP-02-28

The Crime of 1873: Back to the Scene
François R. Velde

WP-02-29

Trade Structure, Industrial Structure, and International Business Cycles
Marianne Baxter and Michael A. Kouparitsas

WP-02-30

Estimating the Returns to Community College Schooling for Displaced Workers
Louis Jacobson, Robert LaLonde and Daniel G. Sullivan

WP-02-31

A Proposal for Efficiently Resolving Out-of-the-Money Swap Positions
at Large Insolvent Banks
George G. Kaufman

WP-03-01

Depositor Liquidity and Loss-Sharing in Bank Failure Resolutions
George G. Kaufman

WP-03-02

Subordinated Debt and Prompt Corrective Regulatory Action
Douglas D. Evanoff and Larry D. Wall

WP-03-03

When is Inter-Transaction Time Informative?
Craig Furfine

WP-03-04

Tenure Choice with Location Selection: The Case of Hispanic Neighborhoods
in Chicago
Maude Toussaint-Comeau and Sherrie L.W. Rhine

WP-03-05

Distinguishing Limited Commitment from Moral Hazard in Models of
Growth with Inequality*
Anna L. Paulson and Robert Townsend

WP-03-06

Resolving Large Complex Financial Organizations
Robert R. Bliss

WP-03-07

4

Working Paper Series (continued)
The Case of the Missing Productivity Growth:
Or, Does information technology explain why productivity accelerated in the United States
but not the United Kingdom?
Susanto Basu, John G. Fernald, Nicholas Oulton and Sylaja Srinivasan

WP-03-08

Inside-Outside Money Competition
Ramon Marimon, Juan Pablo Nicolini and Pedro Teles

WP-03-09

The Importance of Check-Cashing Businesses to the Unbanked: Racial/Ethnic Differences
William H. Greene, Sherrie L.W. Rhine and Maude Toussaint-Comeau

WP-03-10

A Structural Empirical Model of Firm Growth, Learning, and Survival
Jaap H. Abbring and Jeffrey R. Campbell

WP-03-11

Market Size Matters
Jeffrey R. Campbell and Hugo A. Hopenhayn

WP-03-12

The Cost of Business Cycles under Endogenous Growth
Gadi Barlevy

WP-03-13

The Past, Present, and Probable Future for Community Banks
Robert DeYoung, William C. Hunter and Gregory F. Udell

WP-03-14

Measuring Productivity Growth in Asia: Do Market Imperfections Matter?
John Fernald and Brent Neiman

WP-03-15

Revised Estimates of Intergenerational Income Mobility in the United States
Bhashkar Mazumder

WP-03-16

Product Market Evidence on the Employment Effects of the Minimum Wage
Daniel Aaronson and Eric French

WP-03-17

Estimating Models of On-the-Job Search using Record Statistics
Gadi Barlevy

WP-03-18

Banking Market Conditions and Deposit Interest Rates
Richard J. Rosen

WP-03-19

Creating a National State Rainy Day Fund: A Modest Proposal to Improve Future
State Fiscal Performance
Richard Mattoon

WP-03-20

Managerial Incentive and Financial Contagion
Sujit Chakravorti, Anna Llyina and Subir Lall

WP-03-21

Women and the Phillips Curve: Do Women’s and Men’s Labor Market Outcomes
Differentially Affect Real Wage Growth and Inflation?
Katharine Anderson, Lisa Barrow and Kristin F. Butcher

WP-03-22

Evaluating the Calvo Model of Sticky Prices
Martin Eichenbaum and Jonas D.M. Fisher

WP-03-23

5

Working Paper Series (continued)
The Growing Importance of Family and Community: An Analysis of Changes in the
Sibling Correlation in Earnings
Bhashkar Mazumder and David I. Levine

WP-03-24

Should We Teach Old Dogs New Tricks? The Impact of Community College Retraining
on Older Displaced Workers
Louis Jacobson, Robert J. LaLonde and Daniel Sullivan

WP-03-25

Trade Deflection and Trade Depression
Chad P. Brown and Meredith A. Crowley

WP-03-26

China and Emerging Asia: Comrades or Competitors?
Alan G. Ahearne, John G. Fernald, Prakash Loungani and John W. Schindler

WP-03-27

International Business Cycles Under Fixed and Flexible Exchange Rate Regimes
Michael A. Kouparitsas

WP-03-28

Firing Costs and Business Cycle Fluctuations
Marcelo Veracierto

WP-03-29

Spatial Organization of Firms
Yukako Ono

WP-03-30

Government Equity and Money: John Law’s System in 1720 France
François R. Velde

WP-03-31

Deregulation and the Relationship Between Bank CEO
Compensation and Risk-Taking
Elijah Brewer III, William Curt Hunter and William E. Jackson III

WP-03-32

Compatibility and Pricing with Indirect Network Effects: Evidence from ATMs
Christopher R. Knittel and Victor Stango

WP-03-33

Self-Employment as an Alternative to Unemployment
Ellen R. Rissman

WP-03-34

Where the Headquarters are – Evidence from Large Public Companies 1990-2000
Tyler Diacon and Thomas H. Klier

WP-03-35

Standing Facilities and Interbank Borrowing: Evidence from the Federal Reserve’s
New Discount Window
Craig Furfine

WP-04-01

Netting, Financial Contracts, and Banks: The Economic Implications
William J. Bergman, Robert R. Bliss, Christian A. Johnson and George G. Kaufman

WP-04-02

Real Effects of Bank Competition
Nicola Cetorelli

WP-04-03

Finance as a Barrier To Entry: Bank Competition and Industry Structure in
Local U.S. Markets?
Nicola Cetorelli and Philip E. Strahan

WP-04-04

6

Working Paper Series (continued)
The Dynamics of Work and Debt
Jeffrey R. Campbell and Zvi Hercowitz

WP-04-05

Fiscal Policy in the Aftermath of 9/11
Jonas Fisher and Martin Eichenbaum

WP-04-06

Merger Momentum and Investor Sentiment: The Stock Market Reaction
To Merger Announcements
Richard J. Rosen

WP-04-07

Earnings Inequality and the Business Cycle
Gadi Barlevy and Daniel Tsiddon

WP-04-08

Platform Competition in Two-Sided Markets: The Case of Payment Networks
Sujit Chakravorti and Roberto Roson

WP-04-09

Nominal Debt as a Burden on Monetary Policy
Javier Díaz-Giménez, Giorgia Giovannetti, Ramon Marimon, and Pedro Teles

WP-04-10

On the Timing of Innovation in Stochastic Schumpeterian Growth Models
Gadi Barlevy

WP-04-11

7