Discussion Papers In Economics
And BusinessExamination and Approval of New Products in an
Endogenous Growth Model
Kiyoka Akimoto and Takaaki Morimoto

Discussion Paper 17-33

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANExamination and Approval of New Products in an
Endogenous Growth Model
Kiyoka Akimoto and Takaaki Morimoto

Discussion Paper 17-33

November 2017

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANExamination and Approval of New Products in an
Endogenous Growth Model∗
Kiyoka Akimoto†Takaaki Morimoto‡

Abstract
This paper introduces examination of newly developed varieties into a standard “variety expansion” and “lab-equipment” type R&D-based growth model. Producing newly
developed varieties requires approval, and their examination incurs both cost and time.
Reducing the examination duration increases the unit cost of examination. This paper
investigates the eﬀects of reducing the examination duration on examination backlogs(the number of varieties under examination), economic growth, and welfare. Examination backlogs and examination duration have an inverted-U shaped relationship, because
reducing the examination duration on the one hand decreases the examination backlogs
by accelerating the examination process, but on the other hand increases backlogs bypromoting R&D and increasing the number of applications. Reducing the examination
duration promotes economic growth; while it tightens the resource constraint and thus
seems to hurt initial consumption, but the numerical analysis shows that this is not always
the case. Nevertheless, a drastic reduction in the examination duration is detrimental toinitial consumption, so there is an optimal examination duration.
Keywords: Endogenous growth; R&D; Examination duration.
JEL Classification: O31; O34; O38.

∗

We would like to thank Koichi Futagami, Akihisa Shibata, Masaru Inaba, and Yoshiyasu Ono for theirhelpful comments and suggestions. We also thank participants of the seminars at Otaru University of Commerce, Osaka University, Hosei university, and participants of the 18th meeting of the Association for Public
Economic Theory at Université Panthéon-Assas Paris II. Any errors are our responsibility. Takaaki Morimoto
acknowledges financial support from the Grant-in-Aid for JSPS Fellows of the Japan Society for the Promotionof Science No. JP15J04580.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,
JAPAN; e-mail: akimoto1015@gmail.com.
‡
Corresponding author; Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, JAPAN; Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka,
Ibaraki, Osaka 567-0047, Japan; e-mail: t.morimoto.econ@gmail.com.

1 Introduction
As innovation has been growing all over the world, a lot of new technology and products havebeen generated. Most of these newly invented technology and products must be examined tocheck their safety, durability, environment friendliness, and other criteria. Agrichemicals,
medicines, and automobiles are a few examples. On the other hand, the examination candeprive firms of incentives to innovation. During examinations, firms obtain no profit since
they cannot produce the products without approval. Thus, if examinations take much time,
innovation yields less expected benefits, which can stall innovative activities. Moreover,delays in examinations have been pointed out in recent years. Due to growth in innovation,
the number of applications increases, causing an increase in backlogs of applications waitingfor examination or approval and lengthening the duration from application to approval. In
other words, the larger the backlogs, the longer the examination duration.
Therefore, to cope with many backlogs, some countries try to reduce examination durations. We briefly show a recent trend in examination duration of drugs and patents. Regarding drugs, the Pharmaceuticals and Medical Devices Agency (PMDA) in Japan, the Food andDrug Administration (FDA) in the United States, and the European Medicine Agency (EMA)
in the European Union are responsible for examinations. Kagayama et al. (2016) define duration from application to approval as review time and provide the median time of standard
reviews in each oﬃce. Table 1 shows the median time for every five years from 2000 to2015. The FDA keeps relatively short review time since the Prescription Drug User Fee Act
was created in 1992. From its foundation in 2004, the PMDA has aimed to decrease the lag
between the review time in Japan and that in the United States. They did so by increasing
the number of workers: 256 in FY2004, 521 in FY2009, and 753 in FY20141 . In addition,
the number is planned to increase to 1065 by March 2018. Next, patent oﬃces set first action(FA) targets and try to reduce actual FA pendency. The FA pendency is the average time
lapse between the request for research and the preliminary decision by a patent oﬃce. The
annual FA target of the United States Patent and Trademark Oﬃce (USPTO) is 10 months
1

The PMDA annual reports: https://www.pmda.go.jp/english/about-pmda/annual-reports/0001.html. We
note that the fiscal year refers to the duration from October 1 through September 30 in the United States andfrom April 1 through March 31 in Japan.

1

PMDA (Japan)
FDA (US)EMA(EU)

2000
25.4
12.0
16.6

2005
20.9
13.0
15.8

2010
15.7
12.9
13.7

2015
10.6
11.7
12.2

Table 1: The median review time of standard review of drugs (months)

by FY2019. The USPTO achieved 16.4 months of FA pendency in 20152 . Between the endof FY2014 and FY2015, FA pendency decreased by 1.7 months. The Japan Patent Oﬃce
(JPO) achieved a long-term FA goal proposed in 2004 that would reduce FA pendency to 11
months by FY2013 (the goal is called FA11). After achieving FA11, the JPO set a new target
to reduce FA pendency to less than 10 months by FY2023. Both oﬃces have increased thenumber of examiners to reach the goals: in the USPTO, 3681 in 2004 and 7928 in 20133 , andin the JPO, 1243 in 2004 and 1701 in 20134 . Therefore, drug and patent examination oﬃces
can reduce their examination durations by employing a high number of examiners
However, it remains unclear whether the policy for reducing the examination duration tocope with the backlogs is better for economic growth and social welfare because the researchand development (R&D) based growth literature pays less attention to the examination duration. Thus, this study investigates the eﬀects of reducing the examination duration and showsthat an optimal duration exists. We use the Romer (1987)’s standard “variety expansion” and“lab-equipment” type R&D-based growth model, where final goods are used for R&D and to
produce intermediate goods. Furthermore, the variety of intermediate goods expands throughR&D. We introduce the examination duration into the standard model. Our model makes
two assumptions. First, an authority provides approval after examinations. The authority
examines each application from firms that develop new varieties of intermediate goods. The
amount of final goods used for the examination depends on the duration of the examination.
That is, reducing the examination duration requires more resources. Second, new entrant
firms cannot start producing newly invented goods until they pass the examinations and ob2

The IP5 statistics reports: http://www.fiveipoﬃces.org/statistics/statisticsreports.html.
The USPTO performance and accountability reports: https://www.uspto.gov/about-us/performance-andplanning/uspto-annual-reports.
4
The JPO annual reports: http://www.jpo.go.jp/shiryou e/toushin e/kenkyukai e/annual report2013.htm.
32

tain the approval. Thus, there is a lag between invention and production. In the dynamic
general equilibrium models in the previous literature on innovation, firms start production assoon as they succeed in innovation. That is, there is no lag because previous studies do not
focus on examination. Our model provides the following results. First, the steady state ratioof the number of varieties under examination to the number of approved varieties, which canbe considered as a measure of the examination backlogs, is an inverted-U shaped functionof the examination duration. Therefore, reducing the examination duration does not always
reduce the examination backlogs. Second, the shorter the examination duration, the fasterthe economic growth. Reducing the examination duration increases the expected profits for
new entrants. The high expected profits accelerate R&D, and furthermore, economic growth
proceeds. However, a rather short examination duration needs considerable resources as theexamination has a higher cost. Thus, extremely short examination durations hurt households’
welfare. Thus, we third show that an optimal examination duration exists that maximizes
welfare by a numerical analysis. In the numerical analysis, we assume that the economy isinitially in a steady state corresponding to a given initial examination duration and then reduce the examination duration. Reducing the examination duration seems to be detrimental
to the initial consumption because it requires more resources for the examination cost, but
the numerical analysis shows that this is not always the case.
To our knowledge, there is no macroeconomic literature analyzing examination duration
based on R&D-based growth models. Thus, this paper is the first study on the relationshipbetween examination duration and examination backlogs, economic growth, and welfare.
The literature on patent policies is closely related with this study. We can divide the literature
into three groups based on policies: patent length, patent breadth, and patentability. On
patent length, for example, see Judd (1985), Iwaisako and Futagami (2003), and Futagami
and Iwaisako (2007). On patent breadth, see Li (2001), Goh and Olivier (2002), and Chu andFurukawa (2011). Many seminal studies on patent length and patent breadth exist. On theother hand, there are only a few studies on patentability, such as Hunt (1999), Koléda (2004),
O’donoghue and Zweimüller (2004), Chu and Furukawa (2013), and Kishi (Forthcomming).
In addition, our study relates to works on intellectual property rights protection in a NorthSouth model (Helpman (1992), Lai (1998), Glass and Saggi (2002), Kwan and Lai (2003),
3

Tanaka et al. (2007), and Tanaka and Iwaisako (2014)). However, as far as we know, there are
no studies on patent examination duration. Since our model can apply to patent examination
duration, we provide a simple comparison between the results from previous studies on patent
policies and our results. Comparing our results with prior studies on patent length is more
important because both focus on duration. Reducing the examination duration implies anincrease in patent length. Thus, some papers on patent length obtain the same results as we doregarding the eﬀects on economic growth and social welfare. An increase in patent length (adecrease in examination duration) increases economic growth. Moreover, there are invertedU shaped relationships between social welfare and patent length (examination duration). On
the other hand, only our model yields the inverted-U shaped relationship between backlogs
and examination duration. Since we incorporate the process of obtaining a patent or approvalin Romer (1987)’s standard model, our model can analyze the eﬀects of the process, yielding
similar results to those of previous studies.
The remainder of this paper is organized as follows. Section 2 presents the basic model,
mainly based on Romer (1987). Section 3 describes the dynamics of the model and steadystate. Section 4 investigates the eﬀect of reducing the examination duration on the steady statevariables, the steady state growth rate, and welfare. The results are numerically confirmed in
section 5. Section 6 provides concluding remarks.

2 ModelConsider a variety-expansion and lab-equipment type R&D-based growth model as in Romer(1987). The model includes a representative household, a representative final good firm, acontinuum of intermediate good firms, potential entrants to the intermediate goods sector, andan examining authority. A representative final good firm produces final goods by using intermediate goods and labor in a competitive market, whereas intermediate goods are produced
by using final goods in a monopolistically competitive market. New varieties are developed
by R&D using final goods. Most of the settings of the model follow Romer (1987). In contrast to the standard variety-expansion and lab-equipment type endogenous growth model,
production requires approval for some institutional or legal reasons. That is, a newly devel-4oped variety cannot be produced until the variety obtains an approval to the production fromthe examining authority. It takes time and cost for the examining authority to examine andapprove a newly developed variety, and the cost is met by labor income tax.

2.1 Households
The households live infinitely, supply L units of labor inelastically, and have a constant relative risk aversion utility function:
U=

∫ ∞
0

c1−σ − 1e−ρt t
dt,
1−σ

(1)

where c is consumption, ρ > 0 is the discount factor, and σ > 0 is the rate of relative risk
aversion or inverse of the intertemporal elasticity of substitution. Assume that the discountrate is high enough to satisfy ρ > (1 − σ)r at the equilibrium so that the utility will be welldefined. The budget constraint isẆt = rt Wt + (1 − τt )wt L − ct L,(2)

where W is the asset, r is the interest rate, and w is the wage rate. Maximizing (1) with respect
to (2) yields the following Euler equation:ċt1= (rt − ρ).
ct σ

(3)

2.2 Final GoodA representative firm produces final goods Y by using labor L and intermediate goods x. The
production technology is specified as
Yt = L∫ At
1−αxt ( j)α d j,

(4)0

where A is the number of varieties produced, that is, the number of approved varieties. Profit
maximization yields the following first order conditions:Yt
wt = (1 − α) ,L[
]α−1xt ( j)pt ( j) = α,
Lwhere p is the price of intermediate goods.
5

(5)
(6)

2.3 Intermediate Good
Each approved intermediate good firm translates γ units of final goods into one unit of adiﬀerentiated intermediate good. The intermediate good firm maximizes
πt ( j) = max {pt ( j)xt ( j) − γxt ( j)} ,(7)

subject to (6). The optimal prices, quantities, and profits areγ,α1( 2 ) 1−ααL,xt ( j) = x ≡γα( 2 ) 1−ααπt ( j) = π ≡ (1 − α)αL.
γpt ( j) = p ≡

(8)
(9)
(10)

The optimal prices (8), quantities (9), and profits (10) are the same for all j ∈ [0, At ] and timeinvariant.

2.4 R&D and Request for Approval
Assume that production of a newly developed variety requires approval of an examining
authority for some institutional or legal reasons. New entrants to the intermediate goods
sector should first develop new varieties and, then, request for approval to the production of
the new variety from the examining authority.
In order to develop one variety, firms need to invest η units of final goods in R&D activities. An entrant that has developed a new variety requests to approval to the examining
authority. The entrant obtains a value Vt in exchange for the R&D costs5 . Therefore, theprofits of new entrant firms are Vt − η. As long as the profits are positive, new entrants enter
the intermediate goods sector. In contrast, when the profits are negative, there are no entrants.
Since new entrants develop new varieties, the number of developed varieties Dt , or sum of
the number of approved and unapproved varieties, increases when there are new entrants and5

For simplicity, we assume that an examination fee is zero. Even though the examintion fee is positive andconstant ϕ, the qualitaive results does not change because this modification only change the cost of new entrants
η to η + ϕ.

6

vice versa. Therefore, we obtain




= ∞,
if Vt > η,




Ḋt 
∈ [0, ∞), if Vt = η,




= 0,
if Vt < η.

(11)

In equilibrium, R&D is finite, that is, Vt ≤ η, Ḋt ≥ 0, and (Vt − η) Ḋt = 0.

2.5 Examination
After developing a new variety and requesting for approval to the examining authority, an examination starts. We assume that the examination obays Poisson process with instantaneous
probability, µ. Roughly speaking, in each point of time, an examination is finished and avariety is approved with probability µ. Then, an examination duration is stochastically determined and the expected examination duration is 1/µ. The law of large numbers implies thatthe examining authority approves a fraction µ of the unapproved varieties, Dt − At . Therefore
an increase of the number of approved varieties isȦt = µ(Dt − At ).

(12)

In each point of time, it takes δ(µ) units of final goods for the examining authority to
examine one variety. Since the shorter examination duration takes more costs, δ(µ) is increasing in µ, that is, δ′ (µ) ≥ 0. The cost of the examining authority is paid by labor income
tax. Therefore the government budget constraint is,δ(µ)(Dt − At ) = τt wt L.

(13)

2.6 No Arbitrage Condition
Since the expected examination duration is independent of when a variety was developed,
the discounted sum of expected profits is the same for all unapproved varieties, Dt − At .
Therefore, the value of unapproved varieties Vt is also independent of when a variety wasdeveloped. The no arbitrage condition about unapproved varieties requires that their value
are just the discounted sum of expected profits, that is,
∫ ∞
∫ ∞ ∫
u
−µ(s−t)Vt =µee− t rz dz πu duds.
t

s

7

(14)

∫ ∞ ∫u
The term s e− t rz dz πu du in (14) is the discounted sum of profits if the variety is approved at
s > t, and the term µe−µ(s−t) is the probability that the variety is approved at s > t, given thatit has not been approved at t.
The approved varieties, At , do not face any uncertainty, so the no arbitrage condition
about the approved varieties requires that the value of approved varieties, VAt , are just thediscounted sum of profits.
VAt =

∫ ∞

∫u

e− t rz dz πu du.

(15)

t

2.7 Market Clearing
In this model, there is a final good market, a continuum of intermediate goods markets of[0, At ), a labor market, and an asset market. The intermediate goods market clears sinceeach intermediate good firm behaves with considering the demand (6). The labor market also
clears. The final goods are used as consumtion, input for production of intermediate goods,
input for R&D, and input for examination. So, the final goods market clearing condition is∫ At
Y t = ct L +γxd j + ηḊt + δ(µ)(Dt − At ).
(16)
0

There are two kinds of asset, stocks of approved and unapproved varieties. Thus, the asset
market clearing condition is
Wt = VAt At + Vt (Dt − At ),(17)

which is cleared using Walras’ law focusing on the final goods market (16).

2.8 Equilibrium
In equilibrium, the output of intermediate goods, x, is symmetric for all j ∈ [0, At ) from (9),and thus the final good production (4) can be written asα( 2 ) 1−ααYt = At
L.
γ

8

(18)

From (5), (9), (10), and (18), the final good output is distributed to the input for the intermediate good production, labor income, and profit of intermediate good firms as follows:∫ At
2
α Yt =
γxd j,(19)
0(1 − α)Yt = wt L,(1 − α)αYt = At π.

(20)

Since πt and Vt are constant in an equilibrium with positive R&D from (10) and (11), only
rt is variable in (14). Therefore, the no arbitrage condition about unapproved varieties (14)yields the interest rate when R&D is conducted as follows (see Appendix):
(

)1/2

µ 
π
r = r(µ) ≡  1 + 4− 1 .
2
µη

(21)

2

r
= µ(µ+2r)> 0 and limµ→∞ r(µ) = π/η. Thus, the interest rate r is
Moreover, note that dr(µ)
dµincreasing in µ and converges to π/η as µ goes to infinity.

3 Dynamics and Steady State
This section investigates the dynamics of the number of developed varieties Dt , the number ofpatented varieties At , and the consumption ct when R&D is conducted6 . These three variables,
Dt , At , and ct , and their diﬀerential equations are translated into two new variables and theirdiﬀerential equations. Then, a steady state of the new variables is investigated. This steadystate corresponds to a balanced growth path on which the original three variables, Dt , At ,and ct , grow at the same constant growth rate. Then, the growth rate at the steady state is
calculated.

3.1 Dynamics
Substituting (18) and (19) into (16) and rearranging it yield the dynamics of the developed
varieties as follows:

[
]
1 1+α
Ḋt =
πAt − ct L − δ(µ)(Dt − At ) .
η
α

6

(22)

For some levels of initial state variables, it is optimal not to invest in R&D. In the case of no R&D, theequilibrium conditions diﬀer from that in the case of positive R&D. See Appendix.

9αNote that the term (1 − α2 )(α2 /γ) 1−α L is rewritten as 1+απ in (25) by using (10). The first termαin the brackets of the left hand side of (22) is the net output of the final goods, that is, theoutput of final goods minus the input for the intermediate good production. The second andthird terms in the brackets are the consumption and the input for examination, respectively.
That is, the net output of the final goods is used as consumption and the input for examinationand then the rest of the final goods are injected to R&D. Therefore, the more unapproved
varaieties (or examination backlogs) Dt − At , the less R&D. Together with (3) and (12),equation (22) describes the dynamics in the case of positive R&D.
Because R&D is non-negative, Ḋt should be non-negative. From (22) and Ḋt ≥ 0, thefollowing feasibility condition must hold:ct L ≤

1+α
πAt − δ(µ)(Dt − At ),α(23)where equality holds when Ḋt = 0. Feasibility condition (23) implies that a high consumption
level is not feasible when there are huge examination backlogs because it takes enormous
resouces to examine the examination backlogs.
To reduce the number of variables, define
ωt =

Dt
,
At

χt =

ct L
.
At

(24)

Because ωt −1 = (Dt −At )/At is the ratio of unapproved varieties, or the examination backlogs,
to approved varieties, ωt can be considered as a measure of the examination backlogs whereas
χt represents a measure of consumption level. Substituting (12), (18), and (20) into (22) yields
the dynamics of ω as

{
}
1
1+α
ω̇t =
−[µηωt + δ(µ)](ωt − 1) +π − χt .
ηαThe dynamics of χ can be obtained from (3) and (12) as{}]1[
χ̇t = χtr(µ) − ρ + µ − µωt .
σ

(25)

(26)

Feasibility condition (23) is rewritten as
χt ≤

1+α
π − δ(µ)(ωt − 1).
α
10

(27)

χt

χ̇t = 0Eω̇t = 0
O

ωt

ω∗

1

Figure 1: Phase diagram
In addition, ωt ≥ 1 must hold for all t since developed varieties are always more than approved
varieties, or Dt ≥ At for all t. Gathering (25), (26), (27), and ωt ≥ 1, a phase diagram of thiseconomy can be depicted as in Figure 1. The shaded area in Figure 1 is not feasible.

3.2 Steady StateIn this model, the steady states of ωt and χt correspond to the balanced growth paths on which
ct , At , Dt , and Yt grow at the same constant rate g. The point E in Figure 1 represents thesteady state which is saddle path stable (see Appendix). The economy converges to the steadystate E from upper right or from lower left depending on the initial state, ω0 .
From (25) and (26), the steady state values of ω and χ areω∗ = 1 +1[r(µ) − ρ],σµχ∗ = −[µηω∗ + δ(µ)](ω∗ − 1) +

(28)
1+α
π.
α

(29)

From (3), the growth rate at the steady state isȦt Ḋt ċt==
At Dt ct]1[=r(µ) − ρ .
σ

g≡

11

(30)

Then, examining a necessary condition for existence of the balanced growth path, we canobtain the following proposition.
Proposition 1 (Necessary condition for existence of BGP). 7 Assume that
π
> ρ.
η

(A1)

Then, a saddle point stable balanced growth path is exists only if
ρ
> 0.
−1
ρη

µ > µ0 ≡ π(31)Proof. For the existence of the balanced growth path, ω∗ should be greater than one, or gshould be positive. Solving r(µ) > ρ for µ and applying (A1) yields (31).
Since limµ→∞ r(µ) = π/η and g = [r(µ) − ρ]/σ, Assumption (A1) ensures positive steadystate growth when µ goes to infinity; that is, it ensures positive steady state growth in thestandard variety expansion type endogenous growth model. Then, there is a lower boundof µ, above which the positive steady state growth is ensured since the steady state growthrate is increasing in µ. Proposition 1 specifies the lower bound of Poisson arrival rate, µ0 . In
other words, for the existence of the balanced growth path, examination should be suﬃciently
short. Hereafter, we assume (A1) in this paper.

4 Comparative Statics
This section investigates the eﬀects of reducing the examination duration, or increasing µ, on
the steady state values, economic growth, and welfare.

4.1 Eﬀects on Steady State Values and GrowthFirst, we examine the eﬀects of an increase in µ on the steady state values, ω∗ and χ∗ . Then,
we discuss the initial jumps in the control variable χ0 . Next, the eﬀect on the steady state
growth rate is examined.
Proposition 1 provides only the necessary condition that assures ω∗ > 1. However, for existance of BGP,
χ > 0 as well as ω∗ > 1. For a large value of σ, χ∗ is likely to be positive because the absolute value of firstnegative term in (29) is small when σ is large.
7

∗

12

Substituting (21) into (28) and diﬀerentiating it with respect to µ yields

 

 
dω∗
1 
π/η≥ 0, if µ ≤ µ1 ,
 

ρ
−
=


(
)

1/2 
< 0, if µ > µ1 , dµσµ2 
1+4π

(32)

µη

4π

where µ1 ≡ ( π )η2
be stated.

ρη

3 π −ρ

−1> µ0 because µ1 − µ0 = ( π η)2ρη−1> 0. Then, the following proposition can

Proposition 2 (The steady state ratio of developed varieties to approved varieties). The steadystate ratio of developed varieties to approved varieties, ω∗ , is an inverted-U shaped functionof the Poisson arrival rate of the examination, µ, and maximized at µ1 , which is greater thanthe lower bound of the Poisson arrival rate, µ0 .
Since ωt can be considered as a measure of the examination backlogs, Proposition 2 implies that when the examination duration is suﬃciently long (µ < µ1 ), a reduction in the
duration increases the examination backlogs. In contrast, when the examination duration is
suﬃciently short (µ > µ1 ), a reduction in the duration reduces the examination backlogs.
Therefore, reducing the examination duration does not always reduce examination backlogs.
This is because reducing the duration accelerates both R&D activities, Ḋt , and the examination, Ȧt . When the examination duration is suﬃciently long, R&D becomes more active thanthe examination (Ḋt > Ȧt ), and thus the examination backlogs (ω∗ ) increase and vice versa.
Diﬀerentiating (29) with respect to µ yields
[
] dω∗
dχ∗
= −[ηω∗ + δ′ (µ)](ω∗ − 1) − µη(2ω∗ − 1) + δ(µ)
.
dµ
dµ

(33)

Since δ′ (µ) > 0 and ω∗ ≥ 1, the first term of the right hand side of (33) takes a negative value.
The second term is also negative for µ ≤ µ1 since dω∗ /dµ ≥ 0 for µ ≤ µ1 . Therefore, the signof dχ∗ /dµ is negative at least for µ ∈ [µ0 , µ1 ] and ambiguous for µ > µ1 .
Summarizing the preceding arguments, when µ increases, in Figure 1, the steady state
shifts to the lower right for µ ∈ [µ0 , µ1 ] (Point E ′ in Figure 2), whereas it shifts to the upper
left or lower left for µ > µ1 (Point E ′ in Figure 3). The new steady state is also saddle
path stable, and the economy converges to the steady state from the lower left or upper right.
Since ωt is a state variable and χt is a jump variable, when the steady state shifts to the lower
13

χt

E

E′
X
O

ωt

Figure 2: Comparative statics when µ ∈ [µ0 , µ1 ] (Points E and E ′ represent the initial andnew steady states, respectively.)
right, χt jumps to the lower level (Point X in Figure 2). In contrast, when the steady state
shifts to the upper left or lower left, χt jumps to the upper or the lower (Points X1 and X0 in
Figure 3, respectively). Since At is also a state variable, a lower (upper) jump in χt implies alower (upper) jump in consumption, ct . That is, when µ ∈ [µ0 , µ1 ), a marginal decrease in theexamination duration definitely hurts the initial consumption, whereas when µ > µ1 , it may
not do so. Thus, an increase in µ leads to lower initial consumption, or a static loss, at leastfor µ ∈ [µ0 , µ1 ).
Next, we examine the eﬀect on the steady state growth rate. Since r(µ) is increasing in µ,
the steady state growth rate is also increasing in µ, that is,
dg 1 dr(µ)=> 0.
dµ σ dµ(34)

Therefore, the shorter the examination duration is, the faster the growth due to the higher
expected return from R&D. Moreover, since the interest rate is constant over time, the Euler
equation (3) is also constant. Therefore, consumption grows at a constant rate, not only in thesteady state, but also in the transition paths. So, an increase in µ leads to higher consumption
growth, that is, a dynamic gain.

14

χt

X1
E
E′

X0

ωt

O

Figure 3: Comparative statics when µ > µ1 (Points E and E ′ represent the initial and newsteady states, respectively.)
4.1.1 Eﬀect on Welfare
We define welfare as the households’ indirect utility. Substituting (3) and (21) into (1) yields
the equilibrium (not only at the steady state) welfare level with positive R&D, as follows:
σc1−σ
1
0
U = W(c0 , r(µ)) ≡−.
2(1 − σ)ρ − (1 − σ) r(µ) ρ(1 − σ)(35)

Assume first that the economy initially is in the steady state E, and second, that theexamination duration is reduced at time 0. The eﬀect of reducing the duration on welfare isdW(c0 , r(µ)) dW(c0 , r(µ)) dr(µ) dW(c0 , r(µ)) dc0
=
+
.
dµ
dr(µ)
dµ
dc0
dµ

(36)

Since an inequality ρ > (1 − σ)r(µ) is assumed so that the utility is well-defined, dW/dr anddW/dc0 are both positive. A marginal decrease in the examination duration has two eﬀects
on welfare. The first eﬀect through an increase in the interest rate, r, which is represented
by the first term of the right hand side of (36), is positive. This eﬀect is the dynamic gain of
the marginal decrease in the examination duration. The second eﬀect through a change in theinitial consumption, c0 , which is represented by the second term, is negative for µ ∈ [µ0 , µ1 )and ambiguous for µ > µ1 as discussed above. Therefore, the second eﬀect is a static loss forµ ∈ [µ0 , µ1 ) and can be static gain for µ > µ1 .
15

To investigate the optimal examination duration, we need to calculate the initial jumps inconsumption, which converge to the new steady states for each change in µ. However, it is notpossible to solve those jumps analytically. Therefore, the next section provides a numericalcalculation and shows that there can exist an optimal examination duration.

5 Numerical Analysis
To investigate the optimal examination duration, this section conducts a numerical analysis.
Because the model is highly stylized, the purpose of this section is not to calibrate the model,
but to provide a numerical example that helps to clarify the results of the model.
In order to calculate the initial jump in χ0 , we use the Relaxation Algorithm of Trimborn
et al. (2008). At time 0, the economy is in the steady state and µ changes. For each change inµ, we calculate transition paths of the economy, which converges to new steady states given
the initial state variable ω0 . From these paths, we obtain the initial jump in χ0 . The initial
number of approved varieties, A0 , is normalized to one. Therefore, the initial jump in χ0 is
the same as the initial jump in consumption, c0 . We can thus calculate welfare (35) for eachchange in µ. Then, we can obtain the optimal µ that maximizes the welfare given the initialstate ωo . For calculation, δ(µ) is specified as δ̃µε . Some parameters and results have beenrounded to no more than three significant figures.
Table 2 provides the baseline parameters. Because there is a rich store of data for patentexamination, some parameters are suited to such data. The initial inverse of examination
duration or the Poisson arrival rate, µinit , is suited to the inverse of the average US final action
pendency from 2013 (2.42 years) to 2014 (2.28 years)89 . The population size is normalized
to 1. σ is chosen to ensure 2% growth in the initial steady state. ρ is 0.02, and α is 1/3.
The R&D technology parameter, η, is normalized to 1. The parameter of examination cost,δ̃, is chosen to ensure that the initial steady state ratio of the examination cost to the R&D
expenditures equals the ratio of the average expenditure of the USPTO10 to the average US
8

Final action pendency is the time lapse between the application for patents and the final decision by a patentoﬃce.
9
The USPTO performance and accountability report for FY2015: https://www.uspto.gov/aboutus/performance-and-planning/uspto-annual-reports.
10
The USPTO congressional budget justifications: https://www.uspto.gov/about-us/performance-and-16µinit
0.426

L
1

σ
ρ
α η
2.5 0.02 1/3 1δ̃γ
ε0.0639 0.826 4

Table 2: Baseline parameter values

0.307

χ0

0.306

χ*
initial µ

0.305
0.304

χ

0.303
0.302
0.301
0.3
0.299
0.298
0.297
0.4

0.6

0.8

1
µ

1.2

1.4

1.6

Figure 4: χ0 and χ∗ (The solid curve represents χ0 (= c0 ) and the dashed curve represents χ∗for each change in µ. The vertical dotted line is µinit .)of the USPTO
R&D expenditure11 from 2010 to 2013; that is, δ(µηḊ )Ȧ = expenditure= 0.0047. The
the US R&D expenditure
init

parameter of intermediate goods production, γ, is set to ensure r = 0.07. The parameter ρ andα, and the target level of the growth rate and the interest rate are set following Grossmann etal. (2013, 2016) that calibrate the R&D based growth model using the Relaxation Algorithmof Trimborn et al. (2008). Finally, to illustrate a clear and interesting result, ε is set to 4.
Figure 4 numerically shows the initial jumps in consumption represented by the solid
curve. For each change in µ (horizontal axis), χt first jumps to the solid curve, and thenconverges to the new steady state represented by the dashed curve. Since the initial A isnormalized to one, χ0 (= c0 /A0 ) is equal to c0 . Therefore, in this case, the initial consumption
planning/budget-and-financial-information.
11
OECD (2017), Gross domestic spending on R&D (indicator). doi: 10.1787/d8b068b4en (Accessed on 20
October 2017).

17

-39

-40

welfaresteady state welfare
initial µwelfare

-41

-42

-43

-44

-45

-46
0.4

0.6

0.8

1
µ

1.2

1.4

1.6

Figure 5: Welfare (The solid curve represents welfare and the dashed curve represents steadystate welfare. The vertical dotted line is µinit .)jumps higher for a moderate increase in µ; that is, there exists a static gain.
We now have the initial jumps in consumption for each change in µ, so we can calculate
welfare which is illustrated in Figure 5. The solid curve represents welfare, which includes
consumption along the transition path. In contrast, the dashed curve represents the steadystate welfare calculated as if the economy is in the new steady state corresponding to each µfrom the beginning. Welfare (solid curve) is maximized at µ = 1.32, which can be stated as
the optimal µ, whereas the steady state welfare is maximized at a smaller value, µ = 1.22.
This is because there is the static gain as well as the dynamic gain.
Since an increase in µ raises the unit cost of examination, it seems to be detrimental to theinitial consumption by putting pressure on the resource constraint. However, this is not thecase under this parameter set. The reason the initial consumption jumps higher is as follows.
The ratio of the number of developed varieties to the number of the approved varieties, ω, canbe considered as the measure of the developed but unused technology, as well as the measureof examination backlogs. An increase in µ encourages the activation of such technology.
By activating unused technology, the economy can maintain higher economic growth, eventhough the economy reduces R&D activities for a while. If the increase in the examination
18

0.07

initial g
gA
gc

0.06

gD

gA, gc , gD

0.05

0.04

0.03

0.02

0.01
10

20

30

40

50

60

70

time

Figure 6: Time paths of the growth rates of A, c, and D for the change in µ to the optimum(The asterisk represents the initial steady state growth rate (2%). The dotted, solid, anddashed curves represent the time paths of the growth rates of A, c, and D, respectively.)

cost is lower than the cost savings from such reduction in R&D activities, there are extra
resources that can be consumed. So, in this case, the initial consumption can jump higher fora moderate increase in µ.
An increase in µ reduces R&D, or Ḋt for a while, so a rather large increase in µ makes
R&D negative. However, R&D is non-negative. In such case, the dynamics should change(see Appendix for the dynamics in case of no R&D). Therefore, we check whether R&D ispositive or not in the results illustrated in Figures 4 and 5 using Figure 612 . Figure 6 shows
the time paths of gA ≡ Ȧ/A, gc ≡ ċ/c, gD ≡ Ḋ/D when µ changes to the optimal value

Figure 6 is related to why we choose ε = 4. For ε above or below 4, the inverted-U shaped relation between
initial consumption c0 and µ appears. However, for a lower ε, for example 2, a decrease in static gain or an
increase in static loss of an increase in µ is small. Therefore, there exists a large incentive for an increase in µ.
As mentioned above, a rather large increase in µ makes R&D negative. So, when ε = 2, the change in µ to the
optimum changes the dynamics. In contrast, for a higher ε, for example 5, the resource constraint is more likely
to be tight by increasing µ. Also in this case, the change in µ to the optimum makes R&D zero and changes the
dynamics. However, as mentioned above, USPTO expenditure is about 0.47% of the US R&D expenditures, so
it seems strange that the increase of examination cost leads to no R&D. Therefore, we choose the parameter εto ensure that R&D is not zero.
12

19

that maximizes welfare13 . From Figure 6, it can be confirmed that R&D is positive for allt because gD is positive. It can be also confirmed that the growth rates of each variables
converge to the new steady state growth rate as time goes on.

6 Conclusion
This paper constructs a variety expansion and lab-equipment type R&D-based growth modelin which producing intermediate goods requires the approval of an examining authority. The
examination for approval takes time and cost. Therefore, the intermediate goods firms cannot immediately start production after developing new varieties, and there are two kinds ofintermediate goods: approved varieties that are producible and unapproved varieties that areunder examination and not producible.
This study investigates the eﬀects of reducing the examination duration on the examination backlogs, economic growth on a balanced growth path, and welfare. Because theexamination backlogs are the number of unapproved varieties, the ratio of the number ofunapproved varieties to the number of approved varieties is considered as a measure of examination backlogs. Then, the relation between this measure of examination backlogs andthe examination duration is found to be inverted-U shaped; that is, reducing the examination
duration increases examination backlogs when the examination is suﬃciently long. This isbecause reducing the examination duration on the one hand reduces the examination backlogs by speeding up examinations, and on the other hand, increases the examination backlogs
by increasing the return of R&D and encouraging R&D activities. So, the shorter examination, the faster the economic growth on the balanced growth path. Thus, there is a dynamic
gain from reducing the examination duration. However, the shorter examination raises theunit cost of the examination since it takes more cost to examine sooner. Therefore, reducing the examination duration seems to be detrimental to an initial consumption by raising
the examination costs and putting pressure on the resource constraint. Therefore, a numerical analysis is conducted to investigate the eﬀect on welfare and shows this is not always
By confirming that R&D is non-negative for the change in µ to the optimum, it can be shown that the
optimal µ exists at the range of positive R&D. Of course, in this case, for all changes in µ ∈ (µinit , 1.6] in Figures
4 and 5, R&D is positive.
13

20

the case. Although there can be a static loss from reducing the examination duration undersome parameter set, the numerical analysis shows that there is a static gain rather than a static
loss. That is, a moderate reduction in the examination duration increases initial consumption as well as the growth rate. Initial consumption increases for the following reason. The
examination backlogs can be considered as the developed but unused varieties. Reducing
the examination duration accelerates the activation of unused varieties and high economic
growth can be maintained even though the R&D drops for a while. When the increase in
the examination cost is less than the cost savings by reducing R&D, there are extra resources
consumed. Nevertheless, a significant reduction hurts initial consumption, so there is an optimal examination duration shown in the numerical analysis. Since there is static gain, theoptimal examination duration is shorter than that which maximizes the steady state welfare.

AppendixDynamics of the Interest Rate with Positive R&D
The value of one variety that is developed at t and approved at s is
∫ ∞ ∫uVt,s =
e− t rz dz πu dus

The examination duration is stochastically determined according to the Poisson process with
the instantaneous probability µ. Therefore, the probability that an unapproved variety has notbeen approved during the time period (s − t) is e−µ(s−t) , and that it is approved at the end of
this period is µe−µ(s−t) . Therefore, the expected value of one variety that is developed at t is
∫ ∞
∫ ∞
∫ ∞ ∫
u
−µ(s−t)−µ(s−t)Vt =µe
Vt,s ds =µee− t rz dz πu duds.
tts

Diﬀerentiating this with respect to time, we obtain
∫ ∞ ∫
∫ ∞
∫ ∞ ∫
u
u
− t rz dz
2 −µ(s−t)V̇t = − µπu du +eµe
e− t rz dz πu duds
t
s
∫ ∞t
∫ ∞
∫u
+
µe−µ(s−t)
rt e− t rz dz πu duds
t
s
∫ ∞ ∫
u
=−µ
e− t rz dz πu du + (µ + rt )Vt .
t

21

(37)

The profit, π, is constant and the firm value, V is constant and equal to η, as long as R&D
occurs. Therefore, (37) can be rewritten as(µ + rt )η = µ

∫ ∞

∫u

e− t rz dz πdu.

tDiﬀerentiating both sides with respect to t results in∫ ∞ ∫uṙt η = −µπ + rt µe− t rz dz πdut= −µπ + rt (µ + rt )η,Thus, we obtainṙt = rt (µ + rt ) −µπ.
ηSince ṙt is increasing in rt , rt decreases with time when rt is small and vice versa; that is, thedynamics of the interest rate is unstable. Since the interest rate is jumpable, it initially jumps
to the steady state defined as follows:r(µ + r) π= .
µη
By solving this, we obtain (21).

Case of No R&D
Since the economy converges to the steady state E from the lower left or upper right in Figure
1, the economy cannot take a saddle path converging to the steady state E when the initialω is large enough. Figure 6 illustrates such a case. A dotted arrow represents the saddle
path converging to the steady state E from the upper right. In Figure 6, the initial ω is toolarge to initially take the saddle path. In order to reach the steady state, the economy should
start without R&D, and initial χ satisfies the feasible condition (27) with equality, that is, thefollowing no R&D condition:
χt =

1+α
π − δ(µ)(ωt − 1).
α(38)The economy goes to the upper left along the no R&D condition (38) (a dashed arrow) until
it reaches point E ′ . After reaching E ′ , R&D becomes positive, and the economy starts for
22

χt

E′

χ0
E

O

ω∗

1

ω0

ωt

Figure 7: Phase diagram with high initial ω
the steady state taking on the saddle path (a dotted arrow). That is, when the initial ω is
suﬃciently large, there is initially no R&D.
When there is no R&D, Ḋt = 0 and Vt < η. Therefore, rt , Vt , and VAt are diﬀerent fromthose in case of positive R&D. Since πt is constant regardless of whether R&D is positive
or not, Vt and VAt are determined by rt from (14) and (15). Therefore, it is necessary toinvestigate how rt is determined in case of no R&D.
In order to reach point E ′ , the economy should move along the no R&D condition (27).
Therefore, the ratio of the change of χt to the change of ωt should be the same as the slope ofthe no R&D condition (38):
χ̇t
dχt
=
= −δ(µ).
ω̇t Ḋt =0 dωt Ḋt =0(39)

Substituting (38) into (25) and (26) yields
ω̇t |Ḋt =0 = −µωt (ωt − 1),
[
][
]
1+α
1
χ̇t |Ḋt =0 =π − δ(µ)(ωt − 1)
(rt − ρ) − µ(ωt − 1) .
ασSubstituting (40) and (41) into (39) and solving for rt yields
[
]
σ 1+α
π
+
δ(µ)
µ(ωt − 1)
α
rt =
+ ρ.
1+α
π − δ(µ)(ωt − 1)
α
23

(40)
(41)

(42)

Equations (40), (41), and (42) describe the dynamics in case of no R&D.
Since rt is not constant in case of no R&D, welfare is also diﬀerent from (35). Let t˜ denotethe period when the economy reaches point E ′ in Figure 2. Substituting (3), (21), and (42)
into (1) yields the welfare level when there is initially no R&D, as follows:∫ ˜
1˜  c1−σt∫σ [(1−σ)r(µ)−ρ]t 
σe
1
1 t

 0
W≡
e σ 0 [(1−σ)rt −ρ]ds dt +
−
.



 0
ρ − (1 − σ)r(µ)  1 − σ ρ(1 − σ)Stability
The linearized dynamic system in the neighborhood of steady state E isżt = Hzt ,
where

(

)
ωt − ω∗zt =,χt − χ∗()−2µω∗ + µ − ηδ − 1η
H=
.
−µχ∗
0

The matrix H has one positive characteristic root and one negative characteristic root sinceµdet H = − χ∗ < 0η
Therefore, the steady state E is a saddle point.

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