Economic Growth, Patent Race, andthe Distribution of R&D Firms
Tetsugen Haruyama
January 2015
Discussion Paper No.1504

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANEconomic Growth, Patent Race, andthe Distribution of R&D Firms
Tetsugen HARUYAMA∗
Graduate School of Economics
Kobe University
January 2015

Abstract
The paper constructs a general equilibrium model model where the rate of technical progress and the distribution of R&D expenditure by heterogeneous researchfirms are simultaneously determined. Using the model, we explore the effects ofthe following policy measures on those two endogenous variables: (i) subsidies toflow variable R&D costs, (ii) subsidies to flow fixed R&D costs, (iii) an increase inentrant firms into a series of patent races, and (iv) an increase in the supply of human capital as inputs to R&D. Contrasting results are demonstrated. For example,
subsidies to flow variable R&D costs promote technical progress and induce the exitof R&D firms with low R&D productivity. That is, the policy accelerates technological progress through R&D by “elite” firms. On the other hand, the opposite
result holds if subsidies are applied to flow fixed R&D costs.

Keyword: R&D, Patent race, technical progress, heterogeneous firms, firm distribution∗

Email: haruyama@econ.kobe-u.a.cjp, Web: http://www.econ.kobe-u.ac.jp/~haruyama/11

Introduction

It is widely recognized that the primary engine of long-run growth in modern economies
is technological progress mainly driven by research and development (R&D). Roughly ahalf of GDP per capita growth is attributed to technological progress, and this fact isfelt throughout our daily life (e.g. smartphones and medical instruments). On the other
hand, market failures are an inherent feature of R&D activities. Therefore, the role ofgovernments in affecting the incentives of firms to conduct R&D is of primary importanceto promote long-run growth.
The purpose of this paper is to re-examine the impact of government policy on R&D
in the private sector, using a model of endogenous technological progress. In particular,
we emphasize the explicit role of heterogeneity of R&D firms, which is neglected in theliterature on dynamic general equilibrium models. There are many precursory studies onpolicy effects.1 A departure of the present paper from them is to examine the determination of the distribution of R&D firms as well as R&D expenditure in the analysis of policy
effects. That is, we explore how policy affects the firm and R&D expenditure distributionin addition to the issue of whether or not government policy promotes innovation.2 In
doing so, we identify types of firms that conduct R&D (and types of firms that do not),the amount of R&D investment each of them does, the number of such firms.
Policy measures considered in the present paper are (i) subsidies to flow variable
expenditure on R&D, (ii) subsidies to flow fixed expenditure on R&D, (iii) an increase in
the number of entrant firms into R&D (increasing competition), and (iv) an increase in
the supply of human capital used in R&D.
The importance of explicitly considering the distribution of R&D firms and theirexpenditure is three-fold. First, the pace of technical progress in an economy is determined
by R&D activities of all firms rather than a small number of “big” firms. Naturally,
the analysis of the R&D firm distribution is imperative to understand the economywide effects of public policy. Second, industrial policy like R&D tax/subsidy is verymuch likely to affect incentives of all firms in different ways. Such policy effects may berealized directly for some and indirectly for others, e.g. in response to changes in R&D
expenditure of the directly affected rival firms. Capturing those differences is importantto understand the aggregate effects of public policy. Third, industrial policy is also likely
to cause exit/entry of R&D firms. Though analysis based on homogeneous firms can deal
with the issue, it does not allow us to identify types of firms which start or stop investingin R&D when policy changes. Such information is certainly important for policy makers.
In the model, patent races continuously take place across industries, driving growthin the long run.3 This allows us to examine the afore-mentioned issues in a dynamic
1

For example, see Hall and Rosenberg (2010) and Reinganum (1989).
One of the key issues regarding R&D policy is whether it complements or substitutes private R&D
expenditure. For example, Zúñiga-Vicente, Alonso-Borrego, Forcadell, and Galán (2014)see for survey.
3
Pioneering studies on a patent race include Lee and Wilde (1980), Loury (1979), Dasgupta and22general equilibrium model. Like many theoretical studies, our model shows that subsidies
to flow variable R&D costs increase the rate of technical progress. A contribution of
the present study is that a new channel is identified where this pro-innovation effect isrealized. It is what we call the cleansing effect, which results from two factors; firms withrelatively low R&D productivity exit patent races and the remaining firms expand R&D
investment. That is, innovation accelerates due to increased R&D investment by “elite”
firms with high productivity. On the other hand, the reverse of the cleansing effect arisesin the case of subsidies to flow fixed R&D costs. Innovation decelerates with the entryof low productivity firms. Quantitatively those two policy measures are diametric in thesense that nothing changes if subsidies to flow variable and fixed R&D costs are applied
simultaneously.4 The result indicates that the assessment of the impact of R&D subsidies,
which are often taken as a useful tool to correct market failures in R&D, may require amore careful examination.
Given the above results, we consider the following questions. Should different R&D
firms be subsidized (or taxed) at the same rate? If not, how differently? One approach to
tackle the questions is to consider them in terms of achieving social optimum. Instead, we
assess the issue, taking a balanced government budget as a constraint. More specifically,
we explore the case where subsidies to flow variable R&D costs are financed out of taxes
on flow fixed R&D costs. Note that the both policy are pro-growth. We argue thatthe approach considered here is important because the source of financing subsidies isoften neglected in the literature for the reason of simplification. Our model offers aframework where the question is answered, explicitly taking a balanced government budget
into account. Our analysis demonstrates that firms with high productivity should besubsidized and low productivity firms be taxed in net. This implies that resources are
required to shift from low to high productivity firms. In this sense, our result is in linewith the idea of minimizing overall R&D costs in an economy as a whole.
We also analyze the effect of a measure to increase the number of firms entering apatent race. An important message of the result is that it generates a cleansing effectlike subsidies to flow variable R&D costs: low productivity firms are forced to exit andinnovation accelerates. In contrast, boosting human capital used as inputs for R&D
accelerates innovation by inducing low productivity firms to enter a patent race.
The structure of the paper is as follows. Section 2 develops the model. Equilibrium
conditions are derived in Section 3, which also explores the effects of industrial policy.
Section 4 concludes.
Stiglitz (1980) and Reinganum (1982). Partial equilibrium models in those studies contributed to thedevelopment of general equilibrium models of endogenous technical progress. In particular, Lee andWilde (1980) laid the foundation of what is now called the quality-ladder models of Aghion and Howitt(1992) and Grossman and Helpman (1991a) where technical progress is endogenized by introducing profitseeking monopoly firms. Our model belongs to this literature. See Aghion, Akcigit, and Howitt (2014)
for a recent survey.
4
It assumes that the initial rates of subsidies to flow variable and fixed R&D costs are the same.

3

2

The Model

2.1

Consumers, Final and Intermediate Goods

Our model is built on variety expansion models of endogenous technical progress developedby Romer (1990) and Grossman and Helpman (1991b, Ch 3). However, ours departs fromtheirs in R&D technology in order to introduce a series of patent races.
We assume that there are two types of workers, skilled and unskilled. Although they
are different in skill levels, they share the same preferences. The utility function isZ ∞U=e−ρt ln Ct dt(1)0

where ρ the subjective rate of time preference and Ct is final consumption. Utility maximization requires the Euler condition Ėt /Et = rt − ρ where rt is the interest rate and Et isconsumption expenditure. Following Grossman and Helpman (1991b, Ch 3), consumption
expenditure is taken to be the numeraire (Et = 1), so that rt = ρ.
Final output is produced under perfect competition, using intermediate products. To
be more specific, we assume a continuum of intermediate goods industries j ∈ [0, 1], ineach of which ktj varieties are produced at time t. ytji denotes the amount of variety
product i in industry j. Given these assumptions, the production function of final output
Yt is assumed to take the following form:
Yt = 

Z 1X
ktj
0

1/α
α
ytji
dj 

,

1 > α > 0.

(2)

i=1

Technical progress is captured by an increase in ktj , which occurs whenever innovation
occurs in a series of patent races. Given that the production function (2) is CES in nature,derived demand for intermediate goods can be shown to take the form of−ytji = Z01

ptji1−α
ktj
1X

p

α
− 1−α
tji0

.

(3)

dj

i0 =1

where ptji is the price of intermediate goods ytji .
Turning to the intermediate goods industry, products can be produced only after their
blueprints are successfully created through R&D. Since those new ideas are protectedby patents, monopoly firms produce the goods, facing the demand (3) with the priceelasticity being −1/(1 − α). Assume that one unit of the intermediate goods is produced

4with one worker. Then, firms set the following monopoly price
ptj =

ωt
≡ pt
α

(4)

where ωt is unskilled wages. Using (4), demand for a variety good (3) is reduced to1ytji =≡ yt ,
pt K tZ 1
where Kt =ktj dj.

(5)0

Kt is the total number of variety intermediate goods. The results (4) and (5) allow us toderive monopoly profits per variety:
πtj =

1−α
≡ πt .
Kt(6)

Note that profits in (6) are the same for all products across industries. Let Vt denote thesum of discounted future profits arising from selling a variety intermediate good. It isdetermined by
ρVt = πt + V̇t .
(7)
Vt is interpreted as the value of a single patent. Since Vt is independent of the industry
index j, R&D is conducted in all intermediate goods industries j ∈ [0, 1] in equilibrium if
it is active at least in any one of them.

2.2

Patent Races of Heterogeneous Firms

Firms have to invest and succeed in R&D in order to create a new intermediate productand earn monopoly profits protected by patents. Firms compete in a race to obtain apatent for new variety products in a given intermediate goods industry. The first firm
which generates innovation in a patent race becomes monopoly in the industry, and others
which fail cannot produce goods.5
2.2.1

Assumptions

Heterogeneity of R&D firms is due to differences in research productivity, denoted by a. In
particular, we assume that such heterogeneity arises probabilistically. Firms do not know
their own true productivity before entry into a patent race, and it is revealed only after
entry. R&D productivity levels are distributed according to the following distributionfunction:Z(a),a ∈ [0, aH ],0 < aH < ∞,(8)which is assumed to be known. On the other hand, we assume that rival firms’ R&D
productivity levels are not observable even during a patent race. We assume that N firms
5

The basic structure of a patent race is based on Lee and Wilde (1980).

5

enter a patent race whenever it starts afresh. For simplicity, N is taken as exogenously
given.6
After entry, firms incur flow R&D costs which consist of variable and fixed components.
Consider a firm with productivity a. Let Rtj (a) be the number of R&D workers the firmchoose to employ in industry j. Use f to denote a fixed number of R&D workers required
as long as the firm stays in a patent race irrespective of whether or not the variable numberof R&D workers Rtj (a) is positive. More specifically, if the firm employs Rtj (a)+f workers,
then innovation occurs at a Poisson arrival rate ofah(Rtj (a)),

h(Rtj (a)) ≡ Rtj (a)µ ,0 < µ < 1.

(9)

In the R&D technology (9), there is no term representing externalities which are an
important characteristic of R&D. Instead, we make the following assumption. When afirm succeeds in R&D, it is able to creates blueprints for λKt number of variety products.
A parameter λ > 0 measures the degree of positive externalities in R&D, and it plays animportant role in sustaining the rate of technical progress in the long run. Therefore, awinner in a patent race achieves the firm value λKt Vt .
Note that because of the presence of flow fixed costs f , relatively low productivity firms
exit the race after their productivity levels are observed, and relatively high productivity
firms stay in the race. In other words, there exists the threshold level of productivity,
denoted by Atj (0 < Atj ≤ aH ), at which firms are indifferent between continuing and exiting the race. This threshold productivity level is an important factor in determining the
distribution of R&D firms in equilibrium. The next section shows how Atj is determined.
A patent race ends once innovation occurs, and another race immediately starts withthe entry of another set of N firms.7 Note that although the intermediate goods industries
are symmetric in structure, the number of varieties is different across those industries sinceinnovation occurs stochastically.
2.2.2

Optimal Decisions

R&D firms with productivity a are called firms a. The value of R&D firm a, denoted by
vtj (a), is determined by the following recursive equation:
ρvtj (a) = [λKt Vt − vtj (a)]ah(Rtj (a)) − wt [(1 − sR )Rtj (a) + (1 − sf )f ] − vtj (a)Itj−1 . (10)The first term on the right-hand side is the expected flow benefit of R&D (an expected
increment of the firm’s value), and the second term gives flow R&D costs with wt as
skilled wages. There are two policy variables in the flow costs. sR is the rate of subsidies
6

In principle, N can be endogenized by introducing fixed entry costs. But, we focus on the case of a
fixed N because it is sufficient to establish our key results.
7
Following the literature, it is assumed that incumbent firms which earn profits in the intermediate
goods industries do not engage in R&D.

6

to variable costs, and sf is the rate of subsidies to fixed costs. Itj−1 in the third term is the
expected Poisson rate of rival firms succeeding in R&D. That is, −vtj (a)Itj−1 captures therisk of the firm a losing in the patent race with its value being reduced to nil. Note that
the Poisson rate Itj−1 is the same for all firms competing in the race in industry j becauseof the assumption that rival firms’ R&D productivity levels are unobservable (incomplete
information).
Taking Itj−1 as given, a firm a chooses Rtj (a) to maximize the right-hand side of (10).
The first-order condition is[λKt Vt − vtj (a)] ah0 (Rtj (a)) = (1 − sR )wt .

(11)

Its left-hand side gives the expected marginal benefit of employing an additional R&D
worker, and the associated marginal cost is on the right-hand side.
To explore some implications of this result, we use (10) and (11) to derive
λKt Vt ah(Rtj (a)) − wt [(1 − sR )Rtj (a) + (1 − sf )f ]λKt Vt −ρ + ah(Rtj (a)) + Itj−1!
ah0 (Rtj (a)) = (1 − sR )wt .

(12)
Let Xtj (a) denote the value of Rtj (a) which satisfies (12) (see (28)). Xtj (a) reflects thefirm’s optimal decision on R&D employment, taking rival firms’ decisions as given. It iseasy to show that a firm with a higher a employs more workers.8
Next, substitute Xtj (a) into (10) to obtain
vtj (a) =λKt Vt ah(Xtj (a)) − wt [(1 − sR )Xtj (a) + (1 − sf )f ].
ρ + ah(Xtj (a)) + Itj−1(13)This equation allows us to examine how the R&D firm value vtj (a) changes as productivity
a changes. An increase in a impacts on the firm value directly via a on the right-handside of (13) and indirectly through Xtj (a). Invoking the envelope theorem, however, theindirect effect via the Xtj (a) can be ignored. Therefore, taking Itj−1 as given, the following
result holds:∂vtj (a)> 0.
(14)∂a constant I −1
tj

This demonstrates that firms with higher R&D productivity have a greater value of competing in a patent race.
Based on the above results, we next characterize the types of firms which continue
investing in flow R&D expenditure after entry. A patent race starts with N firms. At the
beginning of the race, those firms observe their productivity a and form the expectationof their firm value vtj (a). It is obvious that firms stay in the race as long as its value
vtj (a) is positive. On the other hand, firms exit the race for vtj (a) < 0. The threshold
8

This result can be established by totally differentiating (12) and using the envelope theorem.

7

productivity Atj at which firms are indifferent between staying and exiting a patent raceis defined by
vtj (Atj ) = 0,(15)which means
λKt Vt Atj h(Xtj (Atj )) = wt [(1 − sR )X(Atj ) + (1 − sf )f ].

(16)

This condition determines the threshold productivity Atj .9
In addition, making use of (11) and (16), we can show the following result:µ 1 − sff ≡ X̄A
1 − µ 1 − sR
λKt Vt(1 − sR )µ (1 − sf )1−µ F=≡ Λ̃(Atj )wtAtj

Xtj (Atj ) =

(17)
(18)

1−µ

f
where F ≡ µµ (1−µ)
1−µ . There are two implications regarding (18). First, the thresholdproductivity is the same for all industries, i.e. Atj ≡ At . Second, the threshold At
depends on the value of a patent Vt . Intuitively, an increase in the value of a patent
boosts the value vtj (a) of all firms competing in the patent race, so that even relatively
low productivity firms can afford to stay in the race, incurring flow R&D costs. Becauseof this property, the effective value of a “prize” λKwttVt in the patent race is negatively
related to the threshold productivity At . That is, a greater prize of winning in a patent
race induces even lower productivity firms to invest in flow R&D. This mechanism accords
with our intuition. But, we will show that there is a counter-acting effect in the following
analysis.

2.3

Labor Market

There are two types of workers, skilled and unskilled. The latter type of workers are
employed for the production of intermediate goods only. Using (4), (5) and L to denotethe number of unskilled workers, the following full-employment condition holds:

L=

Z 1X
ktj
0

ytji dj = Kt yt =

i=1

α
.
ωt

(19)

Skilled workers, on the other hand, are used for R&D only. Using H to denote their
total number, the following condition holds for their full employment.
Z 1 Z aH
H=N

[Rtj (a) + f ]dZ(a)dj.
0

9

Atj

The “interior” value of Atj exists if aH is sufficiently large.

8

(20)

On the right-hand side is the sum of variable and fixed R&D workers employed for flow
R&D activities in the continuum of the intermediate goods industries j ∈ [0, 1]

2.4

Technical Progress

The number of firms making flow R&D investment is N [1 − Z (At )], because firms withproductivity greater than the threshold At stay in a patent race and others exit. Therefore,
the average Poisson rate of innovation in industry j, denoted by ιtj , is given by
Z aH
ιtj =

ah(Rtj (a))
At

dZ(a)
.
1 − Z(At )(21)

Because the industry-wide Poisson rate Itj is equivalent to the product of the number ofactive firms and the average Poisson rate, we have
Itj = N [1 − Z(At )] ιtj .

(22)This equation is equivalent to the sum of the Poisson rates of all firms conducting flow
R&D, given the assumption that R&D productivity a is randomly distributed.
Similarly, the average Poisson rate of rival firms is
Itj−1 = [N (1 − Z(At )) − 1] ιtj .

(23)

Note that this rate is the same for all firms, since R&D productivity levels of rival firms areunobservable. Making use of the above equations, the following relations can be derived
Itj−1 = m(At ; N )Itj
m(At ; N ) = 1 −

(24a)

1
,
N [1 − Z(At )]

∂m
< 0,
∂At

∂m
> 0.
∂N

(24b)

This equation relates the Poisson rate of rival firms to the industry-wide Poisson rate.
Next, let us consider the growth of final output. Making use of (19), the production1−αK̇tfunction (2) can be reduced to Yt = Kt α L. Hence, its growth rate is 1−α. To calculate
α Kt
it, note that the number of variety goods increases by λKt in each industry wheneverinnovation occurs at a rate of Itj . Given a continuum of intermediate goods industries,
R1
we can write K̇t = 0 λKt Itj dj or

K̇t
=λ
Kt

Z 1
Itj dj.
0

9

(25)

3

Steady State Analysis

3.1

Equilibrium Conditions

This section focuses upon steady state, dropping the subscripts t from relevant variables.
Because innovation follows a stochastic process, the timing and the number of variety
products differ across the intermediate goods industries. Other endogenous variables, on
the other hand, are the same for all industries due to the symmetric structure of themodel. This feature allows us to drop the industry subscript j from all variables otherthan the number of varieties goods ktj .
We derive equilibrium conditions which determine two variables (I, A). For this, substitute (18) and (24a) into (12) and rewrite the resulting equation as

1 − sf 
f
1 − sR  0 ah (R(a)) = 1
ρ + ah(R(a)) + m(A; N )I

Λ(A; sR , sf )ah(R(a)) − R(a) −Λ(A; sR , sf ) −

(26)

where
Λ(A; sR , sf ) ≡

∂Λ
< 0,
∂A

Λ̃(A)
,
1 − sR

∂

∂Λ

 >0
1−sf
1−sR

(27)

and Λ̃(A) is defined in (18). (26) implicitly defines the optimal number of workers employed by a firm a. It is re-expressed as
R(a) = X (a, I, A; sR , sf , N ) ,
∂X(a)
> 0,
∂k1

k1 = a, I, N, sR ,

∂X(a)
< 0,
∂k2

(28)
k2 = A, s2 .

The signs of the derivatives can be verified by invoking the envelope theorem. Note thatR&D subsidies sR and sf have opposite effects on R(a). The implications of this result
will be explored later.
Next, substitute (28) into (21) and rewrite the resulting equation, using (22) and(24a), to obtain
Z
aH

I=N

ah(X(a, I, A; sR , sf ))dZ(a).

(29)

A

This equation determines the industry-wide Poisson rate I, taking the threshold productivity A as given. It is called the R&D equilibrium condition because it is interpreted tosummarize optimal behaviors of all firms conducting flow R&D in all intermediate goods
industries.
To succinctly express the relation between the two variables (I, A) in (29), rewrite it
as
∂Φ(A)
∂Φ(A)
∂Φ(A)

 > 0.
< 0,
> 0,
(30)
I = Φ(A; N, sR , sf ),
1−s
∂A
∂N
∂ 1−sRf
10

The signs of the derivatives are due to the assumption regarding “stability” of the model:10
Z aH
1−N

ah0 (X(a))

A

∂X(a)
dZ(a) > 0.
∂I

(31)

The R&D equilibrium condition (30) is downward-sloping in Figure 1. To develop an
intuition, recall that it determines I for a given A and suppose that the threshold productivity A falls. Then, relatively lower productivity firms stay in a patent race and conduct
flow R&D, boosting the industry-wide Poisson rate of innovation. This gives rise to anegative relationship between A and I.11
To derive another equilibrium condition, substitute (28) into (20) to derive the skilledlabor market condition in terms of two variables (I, A).
Z aH
H=N

[X(a, I, A; sR , sf ) + f ]dZ(a).

(32)

A

It is depicted as an upward-sloping curve in Figure 1. To understand intuition behind theslope, suppose that the Poisson rate I rises. It requires a greater number of skilled workers
in flow R&D (see (28)), which results in an increase in skilled wages, i.e. higher flow R&D
costs. Consequently, relatively lower productivity firms can no longer afford flow R&D
and exit a patent race, raising the threshold A. Such adjustment instantaneously occurs
so that the labor market condition holds.
The two endogenous variables (I, A) are determined in equilibrium in the system of
two equations (30) and (32). In Figure 1, the intersection point E0 between the two curves
define steady state equilibrium.12

3.2

Policy Effects

3.2.1

R&D Subsidies

This section considers the effects of two different types of R&D subsidies. sR is the rateof subsidies to variable flow R&D expenditure, and sf is the equivalent to flow fixed R&D
expenditure. The following proposition summarizes the results concerning the former
industrial policy.
Proposition 1. Following an increase in sR ,10
Although there may be cases where (31) does not hold, we focus upon a “normal” in which (31) ismet. (31) is essentially equivalent to the assumption that Lee and Wilde (1980) uses to prove Theorem
1 of their paper.
11
Furthermore, there is an additional channel where the following chained effects work: I ↑ ⇒ v(a) ↓⇒ [V − v(a)] ↑ ⇒ R(a) ↑ for a ∈ [A, aH ] (the third “⇒” is due to (11)). This again leads to a higher I.
12
To derive a condition which determines skilled wages, note that wt , Rt (a), At and v(a) are constant insteady state. Therefore, inspecting (11) shows that V̇t /Vt = −K̇t /Kt . Using this, (7) is rewritten as Vt =(1−sR )µ (1−sf )1−µ F
λ(1−α)
πt /(ρ + K̇t /Kt ). This equation, (6) and (25) allows us to reexpress (18) as (ρ+λI)w=
A
Thus, equilibrium skilled wage w is defined in this equation once (I, A) are determined.

11

(i) the threshold R&D productivity A rises (the number of active firms N [1 − Z(A)]decreases),(ii) the industry-wide Poisson rate I increases, and(iii) the total and average number of variable R&D workers R(a) employed by active firms
for a ∈ [A, aH ] increases.
Proof. See Appendix A.
In Figure 1, the initial equilibrium is given at a point E0 , which moves to a point E1
following a rise in the rate of subsidy sR . To understand this result, consider (12) which
determines the optimal number of variable flow R&D workers for a firm a. A higher
subsidy rate sR reduces the marginal cost of R&D directly. Its indirect effect comes fromthe fact that the expected marginal benefit of R&D also falls because the effective “prize”
of a patent race, captured by Λ̃, decreases (see (18) and (26)). Such indirect effect isrealized through the entry/exit process of R&D firms. A higher sR induces relatively
low productivity firms to stay in the race, intensifying competition, and consequently theeffective “prize” of a patent race falls. Because the direct effect dominates the indirect
effect, firms increase variable R&D workers.
But, this fact generates two opposite effects. First, a greater employment of variable
R&D workers raises the industry-wide Poisson rate I, causing a rightward shift in the R&D
equilibrium condition in Figure 1. This is the partial equilibrium effect. On the other
hand, a greater demand for variable R&D workers creates an upward pressure for skilledwages, given a fixed supply of skilled workers. This induces relatively low productivity
firms to exit a patent race, leading to a higher threshold productivity A. The second
effect is reflected in an upward shift of the (32) curve. It is the general equilibrium effectof the subsidy policy which works via the labor market.
The subsidy policy unambiguously raises the threshold productivity A because both ofthe partial and general equilibrium effects of the policy reinforce each other (Proposition
1-(i)). On the other hand, those two effects work on the industry-wide Poisson rate I
in opposite directions. The partial equilibrium effect of the policy tends to boost the
Poisson rate, whereas it tends to fall due to the general equilibrium effect. Since theformer effect outweighs the latter effect, the industry-wide Poisson rate I unambiguously
rises (Proposition 1-(ii)).
Regarding changes in the employment of variable R&D workers R(a), the result isambiguous in general. A rise in sR and a resulting higher I both encourage firms toemploy more skilled workers (see (28)). On the other hand, a higher threshold A which
results from the policy change tends to reduce skilled workers employed, counteracting
the afore-mentioned effect. However, it can be easily established that the total numberof flow variable workers and its average are both unambiguously increase (Proposition
1-(iii)).
12

Next, let us consider a higher rate of subsidy to flow fixed costs. The results are
summarized in the next proposition.
Proposition 2. Following a rise in sf ,(i) the threshold productivity A falls (the number of active firms N [1−Z(A)] increases),(ii) the industry-wide Poisson rate I drops, and(iii) the total and average numbers of variable R&D workers R(a) employed by active
firms for a ∈ [A, aH ] decrease.
Proof. See Appendix B.
This is diametric to Proposition 1. The marginal cost of R&D is independent of sf . On
the other hand, as in the case of sR , the expected marginal benefit of R&D is a decreasing
function of sf through the entry/exit process of firms. Thus, firms tend to reduce R&D
employment with a drop in the Poisson rate I for a given threshold A. This is reflected
in a leftward shift of the (30) curve (the R&D equilibrium condition) in Figure 1. On theother hand, the (32) curve (the labor market condition) shifts downward. Intuitively, thereduction of demand for flow variable R&D workers tends to reduce skilled wages. This
makes it possible for relatively lower productivity firms, which did not invest in flow R&D
before, start doing so. As a result, the threshold productivity A falls.
It is clear from Figure 1 that the threshold A unambiguously drops, given the directionsof shifts of the equilibrium curves, as explained above (Proposition 2-(i)). The figure also
shows that the industry-wide Poisson rate falls with the entry of less productive firms(Proposition 2-(ii)).
Having established the above two propositions, we next explore the case where subsidies are applied to both of flow variable and fixed costs simultaneously without distinguishing them. The result is summarized below:
Proposition 3. Suppose s ≡ sR = sf . A change in s affects none of the threshold A, thePoisson rate I and variable R&D employment R(a).
This proposition indicates that the effects of two types of subsidies sR and sf exactly
cancel each other because they are qualitatively and quantitatively opposite in their effects. This intuition can also be confirmed in the fact that (26) becomes independentof those policy variables for sR = sf . Proposition 3 contrasts with results reported in
the precursory studies including Aghion and Howitt (1992) and Grossman and Helpman(1991b). In the literature on endogenous technical progress, R&D subsidies are shown topromote innovation and instrumental in restoring the socially optimal level of R&D investment. The difference in policy implications is due to the assumption of heterogeneity
in R&D firms, which is largely neglected in the literature.

13

3.2.2

Entry of Firms

Due to the assumption of the Poisson process of innovation, the average length of timefor each patent race is 1/I. Once a race ends in a given industry, a new patent race starts
afresh with an entry of N firms. This section considers how the degree of competition
measured by N affects innovation and the firm distribution.
Proposition 4. As the number of entrant firms N in each patent race increases,(i) the threshold productivity A increases,13(ii) the industry-wide Poisson rate I increases, and(iii) the total and average numbers of variable R&D workers R(a) employed by active
firms for a ∈ [A, aH ] increase.
Proof. See Appendix C.
The results are explained using Figure 2. The two curves shift in the same directions
as in the case of a higher subsidy rate to flow variable costs sR , but for different reasons.
Consider the R&D equilibrium condition (30). As entrant firms rise in number, there
are more firms which stay in the race and invest in flow R&D, intensifying competition.
This means a higher risk of losing a race, thereby reducing the value v(a) of each andevery active firm. As a result, the expected marginal benefit of R&D increases (see (11)),
causing a rightward shift of the (30) curve. With other things being equal, both of thethreshold A and the Poisson rate I increase. This is the partial equilibrium effect ofthe policy change. The general equilibrium effect is realized via the skilled labor market
condition (32). More entrant firms boost demand for skilled workers, which is amplified
due to intensified R&D competition, mentioned above. This raises skilled wages, causing
relatively low productivity firms to exit. This causes the threshold productivity A to risewith an upward shift of the the (32) curve.
It is not clear whether the direct or indirect effects dominates. If the former is dominant, equilibrium is established at E3 . In the opposite case, equilibrium settles at E30 .
In other words, a higher A due to the partial equilibrium effect is reinforced by the general equilibrium effect. On the other hand, those two effects work in opposite directions
regarding their impact on the Poisson rate I, hence its overall change is ambiguous.
3.2.3

Skilled Workers

Skilled workers only can serve as inputs for R&D. In this sense, they may be interpreted
as human capital in a narrow sense. Then, how does its increase, e.g. through education,affect equilibrium?
Proposition 5. Following an increase in H,
13

A change in the number of active firms N [1 − Z(A)] is ambiguous.

14

I
A

sR
+
+

sf
−
−

sR = sf
0
0

N
±
+

H
+
−

+

−

0

+

+

+

−

0

+

+

Z 1
R(a)dZ(a)
N
0
Z 1
dZ(a)
R(a)
1 − Z(A)0

R1
Table 1: Summary of comparative statics results. N 0 R(a)dZ(a) is the total flow variable
R1dZ(a)R&D employment, and 0 R(a) 1−Z(A)is the average employment.

(1) the threshold productivity A falls (the number of active firms N [1 − Z(A)] decreases),(2) the industry-wide productivity I increases, and(3) all of active firms for a ∈ [A, aH ] expands flow variable R&D workers R(a) (the total
and average numbers of variable R&D workers R(a) increase).
Proof. See Appendix D.
To understand this proposition, note that the supply of skilled workers H affects thelabor market condition (32) only. In Figure 2, the (32) curve only shifts downward, giving
rise to the results Proposition 5-(1) and (2). It is also easy to confirm an unambiguous
increase in R(a) for all active firms, using (28).
Intuitively, a greater supply of skilled workers cause skilled wages to fall, reducing themarginal cost of R&D. This induces relatively low productivity firms to start investingin flow R&D, lowering the threshold productivity A. In addition, this leads to a greater
effective “prize” for a winner of a patent race, as (18) shows. This fact incentivizes active
firms to boost R&D employment, accelerating technical progress.
The growth literature shows that an increase in the supply of skilled workers is conducive to innovation, and it is called the scale effect. Though the prediction is not supported (e.g. Jones (1995)), education for that purpose draws much attentions in most of
economies. Our result demonstrates that accelerated innovation due to a greater humancapital stock is accompanied by the entry of relatively low productivity R&D firms.

3.3

Self-financed R&D Policy

Table 1 summarizes the effects of changes in policy variables. It shows that the qualitative
effects (i.e. signs) of subsidies to flow variable and fixed costs, sR and sf , are opposite.
Their quantitative effects are also opposite in the sense that the net effects are nil if theboth policy variables marginally change by the same amount for s = sR = sf initially. A
higher sR is pro-innovation, whereas a higher sf is anti-innovation.
15

Figure 3 illustrates such differences in terms of the firm distribution. The verticalaxis measures the number of firms, and R&D productivity is on the horizontal axis. The
area under the distribution is equivalent to the number of firms which stay in a patent
race and invests in flow R&D. An increase in sR induces lower productivity firms to exita patent race and makes it possible for “elite” firms only to contribute to accelerated
innovation. In contrast, a greater sf retards innovation, giving a greater incentives forlower productivity firms to join a patent race.
The above results have two important implications for policy makers. First, the literature shows that R&D subsidies are best used to correct market failures which are prevalent
in R&D. Our result shows that this prediction fails if the policy is applied to flow fixed
costs only, and it even worsens the situation. Second, the literature usually ignores thesource of financing R&D subsidies in developing normative implications for the reason of
simplification. Our paper gives a possible justification for such omission. To understandthis claim, note that in our model, the innovation-promoting effect of subsidizing flow
variable R&D costs can be replicated by taxing flow fixed R&D expenditure. That is,the combined subsidy-tax policy measures can be self-financed with neutral fiscal stance,
and its total effect is unambiguously pro-innovation. In this sense, policy analysis in theliterature may be justified if flow variable and fixed R&D costs are clearly distinguished.
To explore this second implication further, note that the following budget constraint
should hold for self-financed R&D subsidy/tax policy:14
Z aHw(−sf )N [1 − Z(A)]f = wsR N

R(a)dZ(a)(33)

Awhere −sf > 0 is now interpreted as a tax rate on flow fixed R&D costs. The left-handside of (33) is the tax revenue, and the right-hand side gives government spending on
subsidies to flow variable R&D costs. Suppose that the tax rate −sf is endogenously
adjusted for the balanced budget constraint (33) to hold in equilibrium.15 Rearranging(33) givesw(−sf )f = wsR R̃(A)
(34)
R aH
dZ(a)
is the average flow variable R&D workers. This equation
where R̃(A) ≡ A R(a) 1−Z(A)shows that gross tax payments that each R&D firm makes, w(−sf )f , are equal to subsidies
to a firm with the average flow variable R&D employment, wsR R̃(A). Therefore, net tax
14

Lahiri and Ono (1999) considers a partial equilibrium model of R&D competition between oligopoly
firms with asymmetric costs. In this sense, their study is similar to our heterogenous firms structure.
However, no discussion is given on the source of financing subsidies in their study.
15
In this extended case, three endogenous variables are (I, A, sf ). To solve it, substitute (32) into (33)
to deriveH−sf = sR−1 .
N [1 − Z(A)]f(I, A, sf ) are determined by the system of three equations (30), (32) and the above equation.

16

payments are given by w(−sf )f − wsR R(a) = wsR (R̃(A) − R(a)).16
This demonstrates that firms with flow variable employment lower than the average
are taxed in net, whereas those with the above average employment are subsidized innet.17 Empirical studies show that the social rate of return is much higher than theprivate counterpart, implying underinvestment in R&D. To correct the market failure,
the literature shows that R&D subsidies should be used. Following this widely recognizedargument, suppose that self-financed subsidies are used to address the issue of R&D
underinvestment. According to our analysis, it would be necessary to tax relatively lower
productivity firms and subsidize subsidize relatively higher productivity firms to improvewelfare.

4

Conclusion

The paper developed a model of endogenous technical progress where heterogeneous firms
compete in a series of patent races. In equilibrium, the rate of technical change andthe distribution of R&D firms and their research expenditure are determined. Using themodel, we identified several interesting results concerning the effects of government policy.
In particular, subsidies to flow variable costs and policy to intensify competition with firms
entry generate the cleansing effect in the sense that relatively low productivity firms exit
patent races and innovation accelerates. On the other hand, subsidies to flow fixed R&D
expenditure gives rise to the opposite result. We also explored the issue of whether firms
with high (or low) productivity should be subsidized (or taxed) in net when the industrial
policy is self-financed. Our analysis demonstrated that relatively high productivity firms
should be more favorably treated than lower productivity firms.
Finally, we comment on possible extensions of the model. First, the distribution of
R&D firms and their expenditure can be further endogenized. The firm distribution ischaracterized by the distribution function Z(a), the threshold productivity A and thenumber of entrant firms N . The second variable A only is enogenized in the paper, but
in principle N can also be optimally chosen by firms. One possible way is to assume afree entry process at the beginning of each patent race with fixed sunk costs, say fN . Free
entry continues until the expected benefit of entry into a patent race is equalized to the
fixed cost. By so doing, the “hight” of the firm distribution in Figure 3 would respondto policy changes, and new insight could be gained. Second, the current paper focussed
16

Using the equation in footnote 15, one can verify the following:


dsf
dsR
sR
z(A)
−
=
+
+1
dA
sf
sR
sf
1 − Z(A)for sf 6= 0 and sR 6= 0. It says that the proportionate rate of changes in sf must be greater than that ofsR when the rate of subsidy sR is changed. It means that a change in the tax rate must be greater thanthat of the subsidy rate in order for the budget constraint (33) to hold.
17
Note that firms with R̃(A) are not necessarily those with the average R&D productivity for a ∈[A, aH ], given a general form of the distribution function Z(a).

17upon steady state. But, off steady state analysis is desirable to examine the stability
properties of steady state equilibrium. In addition, dynamic changes in the distribution
in response to policy changes may generate interesting insights.

AppendixA.

Proof of Proposition 1

Totally differentiating (32) and (29) gives
 
 H
 N

 dI
 S
M1
M1
P11 P12  
M1
dH=
N dN +S dS +
M2H
M2
M2
P21 P22 dA
1 − sfS=,
1 − sR

(A1)

Z aH

∂X(a)
dZ(a) > 0,
∂I
A
Z aH
∂X(a)
dZ(a) − N z(A)f < 0,
P12 = −N X(A)z(A) + N
∂A
A
Z aH
∂X(a)
P21 = 1 − N
ah0 (X(a))dZ(a) > 0,
∂I
A
Z aH
∂X(a)
P22 = N Ah(R(A))z(A) − N
ah0 (X(a))dZ(a) > 0,
∂A
A
Z aH
Z aH
∂X(a)
∂X(a)
S
S
ah0 (X(a))
M1 = −NdZ(a) < 0,
M2 = NdZ(a) > 0,
∂S
∂S
A
A
Z aH
Z aH
∂X(a)
N
M1 = −
X(a)dZ(a) −− [1 − Z(A)]f < 0,
∂N
A
A
Z aH
Z aH
∂X(a)
N
M2 =
ah(X(a))dZ(a) + N
ah0 (X(a))dZ(a) > 0,∂N
AA
M2H = 0,M1H = 1,
P11 = N

(A2)
(A3)
(A4)
(A5)
(A6)
(A7)
(A8)
(A9)

where z(a) is the density function of productivity a. The following can be confirmed:
D = P11 P22 − P21 P12 > 0.

(A10)

S
Result (i): Using (A1), it is easy to show DA= P11 M2S − P21 M1S > 0. By Cramer’s rule,DS
dA= A >0
dS
D

dA
dA dS
=
> 0.
dsR
dS dsR⇒(A11)

Result (ii): Using (A1),
DIS = M1S P22 − M2S P12 = N 2



Z aH
∂X(a)
z(A)
Θ(a, A)dZ(a) + ∆(A)
∂S
A

18

(A12)

whereΘ(a, A) = ah0 (X(a))[X̄A + f ] − Ah(X̄A ),Θ(A, A) = 0,∂Θ(a, A)∂Θ(a, A) ∂ah0 (X(a))∂ah0 (X(a))== [X̄A + f ]> 0.
∂a∂ah0 (X(a))
∂a
∂a

(A13)
(A14)
(A15)

The sign of (A15) can be confirmed, using (11). Hence, Θ(a, A) > 0. In addition,
Z aH
Z aH
∂X(a)
∂X(a)
∆(A) =dZ(a)ah0 (X(a))
dZ(a)
∂S
∂A
A
A
Z aH
Z aH
(A16)
∂X(a)
∂X(a)
0
−
ah (X(a))
dZ(a)
dZ(a)
∂S
∂A
A
A
Z aH
Z aH
∂X(a)
∂X(a)
=
dZ(a)
ah0 (X(a))
dZ(a)
∂S
∂A
A
A
Z aH
Z aH
(A17)
∂X(a) ∂A
∂X(a) ∂S
0−ah (X(a))
dZ(a)
dZ(a)
∂A ∂S
∂S ∂A
A
A
=0
(A18)
is given in (A11). Using this result, we have DIS > 0, andwhere ∂A∂S
DS
dI
= I >0
dS
D

dA
dA dS
=
> 0.
dsR
dS dsR⇒

Result (iii): Rearranging (32) gives
Z aH
dZ(a)
H
R(a)
=
− f.
1 − Z(A)N [1 − Z(A)]
A

(A19)

(A20)

This shows that as the average flow variable employment on the left-hand side increases
as A increases. Hence, (A20) along with Result (i) proves the result.

B.

Proof of Proposition 2

dA dS
dA
=
< 0.
ds2
dS ds2dA
dA dS
Result (ii): From (A19),
=
< 0.
dsf
dS dsfResult (iii): It is obvious from Result (i) and (A20).
Result (i): From (A11),C.

Proof of Proposition 3

Result (i): From the second equation of (A2), (A4), (A7) and (A8),dA
P11 M2N − P21 M1N
=
> 0.
dN
D

19

Result (ii): From (A3), (A5), (A7) and (A8),dI
M N P22 − M2N P12
= 1
T 0.
dN
D
Result (iii): It is obvious from Result (i) and (A20).

D.

Proof of Proposition 4

Result (i): From the second equation of (A2), (A4) and (A9),dA
P11 M2H − P21 M1H
P21
=
=−
< 0.
dH
D
D
Result (ii): From (A3), (A5) and (A9),dI
M H P22 − M2H P12
P22
= 1
=
> 0.
dH
D
D
Result (iii): It is obvious from Result (i) and (A20).

References
Aghion, P., U. Akcigit, and P. Howitt (2014): “What Do We Learn From Schumpeterian Growth Theory?,” in Handbook of Economic Growth, ed. by P. Aghion, andS. N. Durlauf, vol. 2B, chap. 1, pp. 515–563. Elsevier, Amsterdam.
Aghion, P., and P. Howitt (1992): “A Model of Growth Through Creative Desctruction,” Econometrica, 60(2), 323–351.
Dasgupta, P., and J. Stiglitz (1980): “Industrial Structure, and the Speed of R&D,”
Bell Journal of Economics, 11(1), 1–28.
Grossman, G., and E. Helpman (1991a): “Quality Ladders in the Theory of Growth,”
Review of Economic Studies, 58(1), 43–61.
Grossman, G. M., and E. Helpman (1991b): Innovation and Growth in the Global
Economy. MIT Press, Cambridge MA.
Hall, B. H., and N. Rosenberg (2010): Handbook of the Economics of Innovation
Vol.1-2. Elsevier, Amsterdam.
Jones, C. I. (1995): “Times Series Tests of Endogenous Growth Models,” Quarterly
Journal of Economics, 110, 495–525.
20

Lahiri, S., and Y. Ono (1999): “R&D Subsidies Under Asymmetric Duopoly: A Note,”
Japanese Economic Review, 50(1), 104–111.
Lee, T., and L. L. Wilde (1980): “Market Structure and Innovation: A Reformulation,” Quarterly Journal of Economics, 94(2), 429–436.
Loury, G. C. (1979): “Market Structure and Innovation,” Quarterly Journal of Economics, 93(3), 395–410.
Reinganum, J. F. (1982): “A dynamic game of R and D: Patent protection and competitive behavior,” Econometrica, 50(3), 671–688.
(1989): “The Timing of Innovation: Research, Development, and Diffusion,”in Handbook of Industrial Organization, ed. by R. Schmalensee, and R. Willig, vol. 1,
chap. 14, pp. 849–908. Elsevier, Amsterdam.
Romer, P. (1990): “Endogenous Technological Change,” Journal of Political Economy,
98, S71–S102.
Zúñiga-Vicente, J. A., C. Alonso-Borrego, F. J. Forcadell, and J. I. Galán(2014): “Assessing the Effect of Public Subsidies on Firm R&D Investment: A Survey,”
Journal of Economic Surveys, 28(1), 36–67.

A
aH

The Skilled Labor Market
Condition (32)
E1
E0
E2

The R&D Equilibrium
Condition (30)I0Figure 1: Following an increase in sR , the rate of subsidies to flow variable R&D costs,
equilibrium moves from E0 to E1 . On the other hand, it moves to E2 after an increase in
sf , the rate of subsidies to flow fixed R&D costs.

21

A
aH

The Skilled Labor Market
Condition (32)
E30

E3

E0

E4
The R&D Equilibrium
Condition (30)I0Figure 2: An increase in the number of entrant firms N moves equilibrium to E3 or E30 .
Equilibrium moves to E4 after the number of skilled workers H increases.

N z(a)The distribution after a rise in sf

The distribution after a rise in sR

The initial
distribution

aH

0

A

Figure 3: The initial distribution of R&D firms is given by thick curves, and the area
inside the curves is equivalent to the number of firms which invest in flow R&D. As therate of subsidies to flow variable R&D costs sR increases, the distribution shrinks. On
the other hand, a higher rate of subsidies to flow fixed costs sf expands the distribution.

22