The Cleansing Effect of R&D Subsidies
Tetsugen Haruyama

October 2014
Discussion Paper No.1425

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANThe Cleansing Effect of R&D Subsidies
Tetsugen Haruyama∗
Graduate School of Economics
Kobe University
October 2014

Abstract
The paper develops a patent race model of firms which differ in R&D productivity. It is demonstrated that R&D subsidies generate the cleansing effect where relatively lower productivity firms
drop out of the race and innovation accelerates due to expanded R&D investment by the remaining
firms and new entrants with higher productivity than those that exit.

Keywords: Patent race, R&D, industrial policy, cleansing effectJEL Code: L10, L20, L52, O32∗ Email: haruyama@econ.kobe-u.ac.jp
This research is funded by Grant-in-Aid for Scientific Research C (23530277).

1

Introduction

Governments affect private R&D directly (e.g. grants) and indirectly (e.g. tax credits) in order toalleviate inefficiency caused by, e.g. the “public good” characteristics of its output. A key question in
the theoretical literature on R&D is to understand how firms respond to policy changes in making R&D
decisions and to inform policy makers of policy options to tackle inefficiency.
The present paper complements the literature by showing that R&D subsidies generate the cleansing
effect where relatively lower productivity firms drop out of R&D competition and innovation accelerates
due to expanded R&D investment by the remaining firms and new entrants with higher productivity thanthose that exit. This new result is established in an extended model of a patent race where firms differ
in R&D productivity. Intuitively, greater R&D subsidies increase the value of all R&D firms, inducing
even relatively lower productivity firms to start R&D. This is called the value-boosting effect. On theother hand, greater R&D subsidies create incentives to expand R&D investment and induce more firms
to enter a patent race. As a result, competition intensifies and the value of all firms falls, which in turncauses relatively lower productivity firms to exit an R&D race. This is called the competition effect. The
latter effect dominates the former, giving rise to the cleansing effect of R&D subsidies.
The present paper is structured as follows. In Section 2, the basic model is developed, and equilibrium
is characterized. Section 3 explores the effects of R&D subsidies on variables related to the distributionof research firms and their flow R&D expenditure.

2

A Patent Race Model

2.1

Assumptions

We consider a one-shot patent race in continuous time. Firms have to incur two types of sunk costs before
investing in flow R&D. First, firms do not know their own R&D productivity before entry, and their true
value, labelled a, is revealed once the first type of sunk costs fN are incurred. R&D productivity is
randomly drawn according to the known probability density function z(a), a ∈ [0, aH ], aH < ∞.1 The
probability distribution function is denoted by Z(a). Second, after R&D productivity a is revealed, firms
have to incur another type of fixed costs fA in order to stay in the race. Those which cannot afford topay fA simply exit the race at this stage.
Next consider flow R&D expenditure. For a firm with productivity a, innovation occurs with the
Poisson rate ah(R(a)), h0 > 0 > h00 where R(a) is the number of workers employed. The race ends when
one of firms succeeds in R&D and gains a prize V . The model is solved “backward”.

2.2

R&D Investment

Let us consider a firm investing in flow R&D. The value of the firm is given by2
v(a) =

V ah(R(a)) − (1 − sR )R(a)r + ah(R(a)) + I−1(1)

where r is an interest rate and sR is the rate of R&D subsidy to flow expenditure. The first term in
the numerator is the expected gross benefit of R&D, and the second term is flow R&D costs. In the
denominator, the second term is the instantaneous probability that the firm loses its value v(a) due
1 In this paper, information on R&D productivity is updated after sunk costs are paid. In Malueg and Tsutsui (1997)
and Choi (1991), updating occurs as flow costs are incurred.
2 (1) can be derived from a recursive equation rv(a) = [V − v(a)] ah(R(a)) − (1 − s )R(a) − v(a)I−1 . The first-orderR
condition for maximizing its right-hand side is equivalent to (2).

1

to its own innovation. The last term I−1 is the expected instantaneous probability that one of rival
firms succeeds first. It is independent of a because rival firms’ productivity levels are assumed to beunobservable, although the number of firms investing in flow R&D is known.
A firm chooses R(a) to maximize the right-hand side of (1), taking I−1 as given. The FOC is[V − v(a)] ah0 (R(a)) = 1 − sR(2)after rearrangement. The left-hand side is the expected marginal net benefit of employing a research
worker, which is equated to the marginal cost on the right-hand side. Substituting (1) into (2) gives

V ah(R(a)) − (1 − sR )R(a)V −r + ah(R(a)) + I−1ah0 (R(a)) = 1 − sR .

(3)

It implicitly defines the optimal number of R&D workers employed, which is rewritten in an implicit
form:
∂X(a)
> 0,
j = a, I−1 , sR
for a > A
(4)
R(a) = X(a, I−1 ; sR ),∂jwhere A is a certain threshold value, which will be defined below. It shows that employment is higher infirms with higher productivity.

2.3

Stay or Exit Decisions

Before the start of making flow R&D investment, firms have to pay sunk costs fA . To identify types offirms which compete in flow R&D, substitute (4) into (1) to derivev(a) =

V ah(X(a, I−1 ; sR )) − (1 − sR )X(a, I−1 ; sR ).
r + ah(X(a, I−1 ; sR )) + I−1(5)

It is the gross value of staying in the patent race. Let us consider how it changes as productivity a changes.
Using the fact that the indirect effect through X(a, I−1 ; sR ) can be ignored due to the envelope theorem,> 0. Therefore, firms with a non-negative net value v(a) − fA stay
we can easily confirm ∂v(a)∂agiven I−1

in the race, and others exit the race at this stage. Hence, for a sufficiently high aH , there is the thresholdproductivity aH > A > 0 such thatv(A) = (1 − sA )fA(6)
where sA is the rate of subsidy to sunk costs fA .
Making use of (6), the first-order condition (2) for firms with the threshold productivity becomes[V − (1 − sA )fA ] Ah0 (R(A)) = 1 − sR or
R(A) = X(A, I−1 ; sR , sA ),

2.4

∂X(A)
> 0,
∂j

j = A, I−1 , sR , sA .

(7)

Poisson Rates

Before we proceed further, we explore various Poisson rates which are useful in analysis. First, the average
Poisson rate of innovation across firms is given by
Z aHι=ah(X(a, I−1 ; sR ))AdZ(a)≡ ι(A, I−1 ; sR ),1 − Z(A)

2

∂ι
> 0,
∂j

j = A, I−1 , sR .

(8)

Therefore, from the viewpoint of a given firm which cannot observe rival firms’ productivity, the expected
Poisson rate of rival firms is Z aH1
I−1 = [N (1 − Z(A)) − 1]ι = 1 −ah(X(a, I−1 ; sR ))dZ(a)(9)
N [1 − Z(A)]A−1< 0, ∂I∂j−1 > 0, j = N, sR and N (1 − Z(A)) − 1 is the number of rival firms. The signs of thosewhere ∂I∂A
derivatives are based on an assumption regarding the “stability” of equilibrium.3 Using (8) and (9), theexpected industry-wide Poisson rate of innovation, denoted by I, can be written as

I = I−1 + ι(A, I−1 ; sR ).

2.5

(10)

Entry into a Patent Race

Firms have to pay sunk costs fN in order to enter the race. Given that firms do not know their own
productivity, the expected benefit of entry is
Z aH[v(a) − (1 − sA )fA ] dZ(a).

Π=

(11)

A

Note that v(a), hence Π, falls as the number of entrant firms N rises. To show it, note that in (9) the
Poisson rate of rival firms I−1 increases with N , i.e. a greater number of entrant firms means a higher
possibility that one of rival firms innovates first. In addition, the value of an R&D firm v(a) in (5)decreases in I−1 because intensified competition increases the risk of losing the race. Thus, using sN todenote the rate of subsidy to sunk costs fN , the following holds because of free entry:4
Π = (1 − sN )fN

3

Equilibrium and Policy Effects

3.1

Equilibrium Conditions(12)

We characterize equilibrium in two variables (I−1 , A). First, let us consider the determination of thethreshold productivity A, given I−1 . Using (1), (6) and (7), we obtain(1 − sA ) fA =V Ah(X(A, I−1 ; sR , sA )) − (1 − sR )X(A, I−1 ; sR , sA ).
r + Ah(X(A, I−1 ; sR , sA )) + I−1(13)This is called the value cut-off condition and shown as an increasing curve in Figure 1. An intuition is
simple. An increase in the Poisson rate I−1 is tantamount to an increased risk of losing in the race, andthe value of each of all firms falls, ceteris paribus. Hence, the net value of staying in the race v(a) − fA
becomes negative for relatively lower productivity firms, and they exit the race before reaching the stageof investing in flow R&D. This results in a higher threshold productivity A.
Next, making use of (4), (5), (11) and (12), we can derive the following condition:
Z aH 
(1 − sN )fN =A
3 Formally, 1 −V ah(X(a, I−1 ; sR )) − (1 − sR )X(a, I−1 ; sR )− (1 − sA )fA dZ(a)r + ah(X(a, I−1 ; sR )) + I−1

(14)


R
∂X(a,I−1 )aH
10
1 − N [1−Z(A)]dZ(a) > 0 is assumed. This is essentially equivalent to an
A ah (X(a, I−1 ))∂I−1assumption in Theorem 1 of Lee and Wilde (1980). The assumption is made in order to avoid taxonomic analysis, so that
we focus on “normal” equilibrium.
4 We assume parameter values such that Π > (1 − s )f
N N for N = 1.

3

It is the free entry condition which determines the Poisson rate I−1 , given A. It is depicted as a verticalline in Figure 1 because the condition is independent of the threshold A at the margin.

3.2

The Cleansing Effect of R&D Subsidies

This section conducts a comparative static analysis with respect to R&D subsidy rates without examining how the industry moves from one equilibrium to the other. The analysis should be interpreted ascomparing two one-shot patent races with different subsidy rates rather than considering the effect ofpolicy shifts on a given patent race.
First, consider subsidy to flow R&D expenditure. The result is summarized below:
Proposition 1. A patent race with a higher sR has a greater value of each of A, I−1 , I, N , R(a) fora ≥ 0.
Proof. See Appendix.
To develop an intuition behind a higher value of the threshold productivity A, consider the value cutoff condition (13). A greater sR reduces R&D costs, and hence, boosts the net value of firms investingin flow R&D, v(a) − fA . It induces low productivity firms (which are willing to pay fN , but not fA ) to
incur fixed costs fA to compete in flow R&D. This shifts downward the value cut-off condition (13) inFigure 1, tending to lower the threshold productivity A. This is called the value boosting effect.
An opposing effect comes from the other condition. A higher sR reduces the marginal cost of flow
R&D, encouraging firms to expand flow R&D employment. In addition, more firms enter a patent race
paying fixed costs fN (a higher N ). This increases the number of firms conducting flow R&D, N z(a),at productivity levels a ≥ A.5 These two facts shift rightward the free entry condition (14) in Figure
1, tending to increase the the threshold productivity A. An intuition for an upward pressure for A is
that, as competition intensifies (a higher I−1 ), the net value of staying in R&D v(a) − fA falls, causing
relatively lower productivity firms to exit the race. This is called the competition effect.
The proposition shows that the competition effect dominates the the value boosting effect with theresult of a higher threshold productivity A. That is, a greater rate of subsidy to flow R&D generates
the cleansing effect where relatively lower productivity firms are driven out of a patent race, and an
industry-wide Poisson rate I rises due to expanded flow R&D investment by the remaining firms and newentrant firms which have higher productivity than the exiting firms.
The cleansing effect also exists in the case of subsidies to two types of fixed costs:
Proposition 2. A patent race with a higher sA or sN has a greater value of each of A, I−1 , I, N , R(a)for a ≥ 0.
Proof. The proof is omitted since it is similar to that of Proposition 1.

4

Conclusion

This paper developed a patent race model of firms which differ in R&D productivity. R&D subsidies areshown to generate the cleansing effect where lower productivity firms exit a patent race and the total
R&D investment in the industry expands.

Appendix: Proof of Proposition 1
5 The total number of firms investing in flow R&D is N (1 − Z(A)). The direction of its change after the policy shift isambiguous.

4

Totally differentiating (13) and (14) gives

P11
P21



 dI−1 
0 
 = S1 dsR
P22
S2
dA

(A1)

where
Z aH

v(a)
dZ(a) < 0,
r + ah(X(a)) + I−1
A[V − (1 − sA )fA ]h(X(A))(1 − sA )fA< 0, P22 => 0,
P21 = −
r + Ah(X(A)) + I−1
r + Ah(X(A)) + I−1
Z aH
X(a)
X(A)
S1 = −
dZ(a) < 0, S2 = −< 0.
r+ah(X(a))+Ir+
Ah(X(A))
+ I−1
−1
A

P11 = −

(A2)
(A3)
(A4)

after rearrangement and invoking the envelope theorem which allows us to ignore the impact of changes
in I−1 and sR through X(a, I−1 ; aR ) for a ∈ [A, aH ]. Note that
D = P11 P22 < 0,

DI−1 = S1 P22 < 0,

DA = P11 S2 − P21 S1 =
where

Z aH
D(A) =
A

(A5)

D(A)
<0
r + Ah(X(A)) + I−1(A6)v(a)X(A) − (1 − sA )fA X(a)dZ(a) < 0
r + ah(X(a)) + I−1(A7)

whose sign can be confirmed by noting that dD(A)dA > 0 and D(aH ) = 0. Using Cramer’s rule,
DI−1
dI−1
> 0,
=
dsR
D

dA
DA
> 0.
=
dsR
D

(A8)

dR(a)
dI
dN
dsR > 0, dsR > 0 and dsR > 0 can be confirmed from (4), (7), (9), (10) and (A8).

References
Choi, J. P. (1991): “Dynamic R&D Competition under “Hazard Rate” Uncertainty,” RAND Journalof Economics, 22(4), 596–610.
Lee, T., and L. L. Wilde (1980): “Market Structure and Innovation: A Reformulation,” Quarterly
Journal of Economics, 94(2), 429–436.
Malueg, D. A., and S. O. Tsutsui (1997): “Dynamic R&D Competition with Learning,” RAND
Journal of Economics, 28(4), 751–772.
A

The free entry
condition (14)The value cut-off
condition (13)Figure 1: A patent race at E1
has a higher rate of subsidy toflow R&D, sR , than the one at
E0 .

E0

E1

I−1

0

5