GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 250

Dynamic analysis of an endogenous growth model withinvestment-specific technological change

Kizuku Takao

March 2012GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, JapanDynamic analysis of an endogenous growth model withinvestment-specific technological change∗
Kizuku Takao†
Osaka University
March 16, 2012

Abstract
This paper examines some dynamic properties of the endogenous growth model developedby Krusell[Krusell, P., 1998. Investment-speciﬁc R&D and the decline in the relative price ofcapital. Journal of Economic Growth 3, 131-141], in which investment-speciﬁc technologicalprogress occurs endogenously due to ﬁrm-speciﬁc R&D performed by monopolistic ﬁrms. I
show that when the function of R&D technology is assumed to be linear to labor inputs,
the steady state is saddle-point stable under a continuous time setting. Furthermore, I also
show that a combination of the time-invariant subsidy for investment and the time-variant
subsidy for R&D can replicate the market equilibrium to the socially optimal allocation inthis setting.
Keywords: Investment, R&D, Transitional dynamics
JEL classiﬁcation: D92, O32, O41∗

I am very grateful to Koichi Futagami for his invaluable advice and encouragement. I also thank David Flath,Shinsuke Ikeda, Tatsuro Iwaisako, Katsuya Takii, and the participants at the 2011 Autumn Annual Meeting of theJapanese Economic Association in the University of Tsukuba. Of course, all remaining errors are mine.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan. Email address: kge006tk@mail2.econ.osaka-u.ac.jp1

1

Introduction

Investment-speciﬁc technological change means technological progress in the production ofcapital goods. This technological progress lowers the relative price of capital goods, thereby
encouraging capital accumulation and promoting economic growth. In fact, in the postwar
period, the relative price of capital goods has fallen and the ratio of capital investment to
GDP has risen substantially in the US. For instance, information and computer technologies
have advanced dramatically in this period. Greenwood, Hercowitz, and Krusell(1997) quantitatively show that extending Gordon(1990)’s price index, the presence of this technologicalprogress accounts for about 60% of the US economic growth after World War II.
Although recently this technological progress has been recognized as a major source ofeconomic growth, only a few papers have attempted to develop an endogenous growth modelwith investment-speciﬁc technological change 1 . As the seminal paper, Krusell(1998) presents
a model in which investment-speciﬁc technological change occurs endogenously due to ﬁrmspeciﬁc R&D performed by monopolistic ﬁrms. However, Krusell(1998) analyzes only thesteady state and consequently does not explore the dynamic properties of his model. Further,he does not analyze the policy implication, which is an important goal of this paper.
In this paper, I examine the dynamic properties of the model developed by Krusell(1998).
To get clear results, I use a continuous time model for simplifying the analysis, although
Krusell(1998) uses a discrete time model. I assume that R&D technologies are speciﬁed bya linear function of labor inputs. In this setting, I re-construct the model and analyze thetransitional dynamics, further enhancing the analysis of Krusell(1998) in two respects.
First, I show that the steady state is saddle-point stable. Hence, indeterminacy andcyclical movements can be ruled out. If the steady state is unstable, a policy analysis of
the equilibrium is not meaningful because no equilibrium path converges to the steady state.
Second, I also analyze an optimal subsidy policy that replicates the market equilibrium tothe socially optimal allocation.
In the same manner as Krusell(1998), the economy has two distortions: monopoly powerand social knowledge spillovers in R&D. Monopoly power makes ﬁrms produce and innovate
at less than an optimal level. Technological advances in the production of capital goods
decrease the rental price of capital goods, which lowers revenues generated from capital stocks
possessed by producers of capital goods. This means that capital goods produced in the past
become obsolete. Each ﬁrm internalizes this obsolescence eﬀect of the technological advance,
which discourages the ﬁrm’s inventive to innovate. This is a salient feature, as compared to
the familiar endogenous growth models based on R&D. Another distortion is the presence ofsocial knowledge spillovers in R&D, which too distort a ﬁrm’s incentive to innovate.
As the main contribution, I show that a combination of the time-invariant subsidy forinvestment and the time-variant subsidy for R&D can replicate the market equilibrium to thesocially optimal allocation. Because the speed of convergence may be slow2 , it is important
to note that a simple subsidy policy cannot replicate the market economy to socially optimal
1

Boucekkine(2003) develops an endogenous growth model where learning-by-doing is the engine of investmentspeciﬁc technological change. Boucekkine(2005) and Huﬀman(2007, 2008) develop an endogenous growth model
where investment-speciﬁc technological change is determined through R&D. However, their models examine only
the steady state.
2
For example, Steger(2003) shows that the speed of convergence is slow in a calibrated version of the model
constructed using a quality-ladder model developed by Segerstrom(1998).

2

allocation during the transition to the steady state3 .
The rest of the paper is structured as follows: Section 2 describes the model, following
Krusell(1998). Section 3 analyzes the dynamic system of the model. Section 4 analyzes
the socially optimum allocation and compares the market economy with socially optimal
allocation. Finally, section 5 analyzes an optimal subsidy policy.

2

Model

In this section, I set up a model, following Krusell(1998). In contrast to Krusell(1998),
I use a continuous time model to simplify the analysis of transition paths. Except for the
speciﬁcation of R&D technology, the environment of the model is the same as in Krusell(1998).
An economy is composed of a ﬁnal goods sector, a capital goods sector, and households. The
production technology of ﬁnal goods and capital goods are diﬀerent. First, I consider theﬁnal goods sector.

2.1

Final goods sector

The ﬁnal good, Yt , is produced by the following production function:
∫ 1
α
Kjt
dj L1−αYt =Y t , α ∈ (0, 1),(1)0where LY t and Kjt , respectively, represent labor inputs and inputs of the jth capital goodat time t. Perfect competition prevails in the ﬁnal goods market. The price of ﬁnal goods isnormalized to one. Therefore, I obtain the following proﬁt-maximization conditions:∫ 1αwt = (1 − α)Kjt
dj L−α(2)Yt ,0

α−1 1−αpjt = αKjt
LY t ,(3)

where wt and pjt , respectively, represent the wage rate and the rental price of jth capital
good at time t.

2.2

Capital goods sector

Monopolistic competition prevails in the capital goods market and there is a continuum ofgoods, indexed by type j ∈ [0, 1]. Each ﬁrm accumulates its speciﬁc capital goods andrents them to producers of ﬁnal goods exclusively. Initially, this monopoly is assumed to beprotected by perfect patent protection, and the entry of new ﬁrms is not considered. Investing
one unit of ﬁnal goods produces Tjt units of capital goods. That is, Tjt represents the levelof the jth ﬁrm’s production technology of producing capital goods at time t. Further, each3

As related studies, Arnold(2000a,b) examine the stability of an equilibrium in Romer(1990)’s model and showthat a combination of the time-invariant subsidy for production and the time-variant subsidy for R&D can replicate
the market equilibrium to the socially optimal allocation.

3

ﬁrm can conduct cost-reducing innovations with labor inputs. The proﬁt of the jth ﬁrm attime t is
πjt = pjt Kjt − Ijt − wt LAjt ,where Kjt , Ijt , and LAjt , respectively, represent capital stocks, inputs of ﬁnal goods (hereafter,
referred to as investment), and labor inputs to R&D of the jth ﬁrm at time t. The law of
motion of the capital stocks of the jth ﬁrm isK̇jt = Tjt Ijt − δKjt ,(4)

where δ represents the physical depreciation rate of capital stocks. As the production technology improves, more capital goods can be produced with one unit of ﬁnal goods. This
form of technological progress is characterized as “investment-speciﬁc technological change,”
which is the only engine of economic growth in this model. Moreover, the law of motion of
the production technology of the jth ﬁrm isṪjt = ψTjtγ T̄t1−γ LAjt , γ ∈ [0, 1],(5)∫1
where ψ > 0 represents the productivity in the R&D technology and T̄t ≡ 0 Tjt dj represents
the average level of the production technology across ﬁrms. In contrast to Krusell(1998)4 ,I assume that the R&D technology is speciﬁed to be linear in labor inputs, following thespeciﬁcation of standard endogenous growth models based on R&D; for instance, Grossman
and Helpman(1991, ch3) and Romer(1990). Tjt captures in-house knowledge spillovers and T̄tcaptures social knowledge spillovers of R&D. γ measures the relative importance of dynamic
returns to R&D. If γ > 0, each ﬁrm internalizes such in-house knowledge spillovers. If
γ = 1, each ﬁrm internalizes these spillovers completely. Higher values of γ make each ﬁrm
internalize the eﬀects more aggressively.
The present value of the sum of the jth ﬁrm’s operating proﬁt at time 0 is
( ∫ t
)
∫ ∞
Vj0 =
(πjt ) exp −rs ds dt,
(6)00where rt represents the return on safe assets at time t. Symmetry across ﬁrms is assumed,
and as such I can drop the subscript j. Hence, in equilibrium,
Tt = T̄t .
Each ﬁrm maximizes (6), subject to (3), (4), and (5), given T̄t 5 . To solve the inter-temporalmaximization problem, I deﬁne the following Current-Value Hamiltonian as
γ 1−γ
H = αKtα L1−α
LAt ],
Y t − It − wt LAt + µt [ Tt It − δKt ] + qt [ ψTt T̄twhere the co-state variables, µt and qt , respectively, represent the shadow value of investment
and R&D at time t. In this paper, I restrict the analysis to an interior solution. I obtain theγ 1−γ
Krusell(1998) assumes that the R&D technology is Tt+1 = Tjt
T̄jt H(LAjt ), where H ′ () > 0, H ′′ () < 0.
In this model, incumbent ﬁrms innovate ﬁrm-speciﬁc technology repeatedly. This formulation is similar to
Peretto(1998) and Smulderts and van de Klundert(1995).
4

54

following ﬁrst-order conditions:
1,Tt
rt µt − µ̇t = α2 Ktα−1 L1−α
Y t − δµt ,
wt
qt =
,
ψTtγ T̄t1−γµt =rt qt − q̇t = µt It + qt ψγTtγ−1 T̄t1−γ LAt .

(7)
(8)
(9)
(10)

(7) means that the shadow value of investment must be equal to installed costs. (9) meansthat the shadow value of R&D must be equal to innovation costs. Further, (8) and (10),respectively, represent the no-arbitrage condition of investment and R&D.
the
( ∫Furthermore,)tfollowing transversality conditions must be satisﬁed: limt→∞ µt Kt exp − 0 rs ds = 0 and( ∫)tlimt→∞ qt Tt exp − 0 rs ds = 0.
From (7) and (8), the amount of capital stocks at time t is as follows:1

2

1

Kt = α 1−α LY t Tt1−α utα−1 ,(11)where ut is deﬁned as:
Ṫt.
Tt
From (3) and (11), the rental price of capital goods at time t is given byut ≡ rt + δ +

pt =

(12)

1 1
ut ,
α Tt

where ut T1t represents the user-cost of capital at time t, which is the marginal cost to produce one unit of capital goods. Each ﬁrm charges a monopoly markup price. Technologicalprogress decreases the rental price of capital goods, which lowers the revenue generated from
capital invested as a preliminary. On the other hand, technological progress enhances the
productivity of R&D through in-house knowledge spillovers. Therefore, each ﬁrm creates aschedule of R&D internalizing the above eﬀects and costs.

2.3

Households

Here, I consider a representative household’s problem. Population size, L, is constant overtime. Each individual supplies one unit of labor inelastically. Each household maximizes thefollowing lifetime utility function6 :∫ ∞U0 =(log ct ) exp (−ρt) dt, ρ > 0,0where ct and ρ, respectively, represent consumption of ﬁnal goods per-capita at time t andthe individual discount rate. Solving the inter-temporal optimization problem, I obtain thefollowing Euler equation:c˙t= rt − ρ.
(13)
ct
6

For simplicity, the utility function is speciﬁed to be of log-utility type although it is speciﬁed to be of CRRA
type in Krusell(1998).

5

Furthermore, the following transversality condition must be satisﬁed: limt→∞ c1t at exp (−ρt) =0.

3Market equilibrium and dynamics

In this section, I derive the market equilibrium and construct a dynamic system based on themodel in the preceding section. The market equilibrium condition of ﬁnal goods is
Yt = ct L + It ,(14)

where ct L and It , respectively, represent aggregate consumption and investment at time t.
Further, the market equilibrium condition of labor is
L = LY t + LAt .

(15)

To derive the dynamic system of the economy, I deﬁne
1

Zt ≡ Tt α−1 Kt ,αSt ≡ Tt α−1 ct ,α−1

Qt ≡ qt α Tt

2α−1
α

.

In Appendix A, I show that the following three equations constitute the dynamic system of
the economy:
1Żt
ψL= κQt + κQt1−α Zt − St Zt−1 L −
− δ,
Zt
1−α
1
Ṡt
ψL
= α2 κQt + κQt1−α Zt −− (δ + ρ),
St
1−α
1
α
Q̇t
(1 − α)2
1 − α 1−α=
(1 − γ)κQt1−α Zt − α(1 − α)κQt −Qt St L
Qt
α
α
]
[
1−α
α + (1 − α)γ
+
ψL +
δ,
α
α
1−α

α−1

(16)
(17)

(18)

where κ ≡ (1 − α) α ψ α . In Appendix B, I show that the steady state {Z ∗ , S ∗ , Q∗ } isdetermined by][1α(δ + ρ)ψL∗∗+Q∗ α−1 − α2 Q∗ α−1 ,Z = F (Q ) ≡(1 − α)κ
κ
]
[
α
{(1
−
α)(1 − γ) + 1} δ (1 − α)(1 − γ)ρ
ψ
∗
∗
+
+
Q∗ α−1S = G(Q ) ≡
1−α
L
L
[
]
(1 − α)(1 − γ) + 1 2 ∗ 2α−1−
α κQ α−1 ,L√−N + N 2 − 4M P
∗
Q =
,
2κM

6

where M , N , and P , respectively, are deﬁned as
M ≡ α4 + α2 (1 − α)(1 − γ),ψL(−α2 ) + δ[−α2 − (1 − α)(1 − γ)]N≡1−α+ ρ[1 − 2α2 − (1 − α)(1 − γ)],
[
]
ψL
P ≡
+ (δ + ρ) ρ.
1−αIn the steady state, {LY t , LAt , rt , ut } are constant.
Proposition 1. The steady state is saddle-point stable.
Proof. See Appendix C.

4Socially optimal growth path

In this section, I consider the ﬁrst-best resource allocation in which a social planner maximizes
the lifetime utility function of a representative household, subject to the market constraints ofﬁnal goods and labor, and the law of motion of capital stocks and the production technology
of capital goods. Therefore, the inter-temporal optimization problem of the social planner
can be described as follows:∫ ∞(log ct ) exp (−ρt)dt,
max
It ,Kt ,LAt ,Tts.t0

Yt = ct L + It ,L = LY t + LAt ,K̇t = Tt It − δKt ,Ṫt = ψTt LAt .

In Appendix D, I show that the following three equations constitute the dynamic system of
the socially optimal allocation:
1Żt
ψL= κQt + κQt 1−α Zt − St Zt −1 L −
− δ,
Zt
1−α
1
Ṡt
ψL
= ακQt + κQt 1−α Zt −− (δ + ρ),
St
1−α
α
Q̇t
1 − α 1−α1
1−α
= −(1 − α)κQt −Qt
St L + ψL +
δ.
Qt
α
α
α

(19)
(20)
(21)

Because the dynamic systems are too complicated, I resort to numerical simulations to
see the diﬀerences between the market equilibrium and the socially optimal allocation. Using
linearized approximation around the steady state, I numerically calculate the transition pathto the steady state.
As a benchmark, I choose the parameter values as follows. The time preference rate, ρ,and the physical depreciation rate of capitals, δ, respectively, are set as 0.05 and 0.2, as is

7

conventional in macroeconomic literature. α is set as 0.8, which implies that the mark-up
rate of capital goods is 25%. The scale parameter, L, is normalized to one. ψ is set as 0.025
so that the economic growth rate of the economy is around 2%. To examine the transition
path to the steady state, I choose Z0 = Z ∗ × 0.5 as the initial value of the state variable.
Figure 1 shows how the linearly approximated transition path of {LAt } evolves over time
under the benchmark parameter case. The solid line stands for the transition path in themarket equilibrium, and the dotted line stands for that in the socially optimal allocation.
Figure 1 shows that labor inputs to R&D in the market equilibrium are less than in thesocially optimal allocation. Each ﬁrm internalizes the obsolescence eﬀect of a decline in therental price. Furthermore, social knowledge spillovers in R&D also discourage the ﬁrm’s
inventive to innovate.
LAt
0.8
0.7
0.6
0.5
0.4
0.3
0.2

0

10

20

30

40

50

t

Figure 1: The transition path of labor inputs to R&D

5

Optimal subsidy policy

In this section, I analyze the subsidy policy that enables the market equilibrium to replicate
the socially optimal allocation. The following two aspects must be considered. First, themarket economy must attain the steady state of the socially optimal allocation. Second, thetransition path to the steady state too must correspond with that of the socially optimal
allocation.
To see this, I introduce two policy tools in the market economy: the proportional subsidy
for investment and R&D. Let τtI and τtR be the rate of subsidy for investment and for R&D attime t, respectively. I assume that the government can impose a lump-sum tax on households
to cover these subsidies. The net proﬁt of the jth ﬁrm at time t is rewritten as
πjt = pjt Kjt − Ijt (1 − τtI ) − wt LAjt (1 − τtR ).
Each ﬁrm maximizes the present value of the sum of proﬁt, subject to (4) and (5), given
the rate of these subsidies. The dynamic system of the economy is given by the following

8three equations:
1Żt
ψL= κQt + κQt1−α Zt − St Zt−1 L −
− δ,
Zt
1−α
[
]
1
Ṡt
1
ψL
τ̇tI
1−α
2
= α κQt
+
κQ
Z
−
−
(δ
+
ρ)
−
,
t
t
St
1−α
1 − τtI
1 − τtI
]
[
1
Q̇t
1−α
1 − τtI1−α=
κQt Zt −2α + 1 − (1 + γ)(1 − α) +Qt
α1 − τtR[][]α
11 − α 1−α
1 − τtI− α(1 − α)κQt−
Q t St Lα
1 − τtI
1 − τtR[][]
α + (1 − α)γ
1−α
1−α
τ̇tI
+
ψL +
δ+
α
α
α
1 − τtI
[
]
τ̇tR
1−α
−
.
α
1 − τtR

(22)
(23)

(24)

Now, I examine the optimal rate of two subsidies at which (22), (23), and (24) correspondwith (19), (20), and (21). With or without the subsidy policy, (22) corresponds with (19).
When γ = 1, τtI = 1 − α replicates (23) to (20), and τtR = 1 − α replicates (24) to (21).
However, when γ ̸= 1, the R&D subsidy must be time-variant to replicate (24) to (21). For(24) to correspond with (21), the following equation must be satisﬁed:
[
[
]
]
1
α
α
α
op
op
op 1−αop 1−αZt −α − γ(1 − α) +St L 1 −κQt+ Qt
1 − τtR
1 − τtRτ̇tR,+ (γ − 1) ψL =1 − τtR
where {Ztop , Stop , Qopt } represent the equilibrium values during the transition to the steadystate in the socially optimal allocation. Rewriting the above equation, I obtain the following
diﬀerential equation:
τ̇tR + A(t)τtR = B(t),(25)
where A(t) and B(t) are deﬁned as follows:
1

α

1

α

1−α Z op [−α − γ(1 − α)] + Qop 1−α S op L + (γ − 1)ψL,
A(t) ≡ κQoptttt
1−α Z op [−α − γ(1 − α)] + Qop 1−α S op L
B(t) ≡ κQop
t
t
t
t
[
]
1
α
op 1−αop
op 1−αop+ (γ − 1)ψL + α κQt
Zt − Qt
St L .

First, I examine the optimal rate of the R&D subsidy in the steady state, τ ∗R . By settingτ̇tR = 0 in (25), I obtainτ ∗R = 1 −αρ[ακQ∗∗ − δ] − [α + γ(1 − α)][ακQ∗∗ − (δ + ρ)],

where Q∗∗ is the steady-state equilibrium value in the socially optimal allocation. τ∗R is
a decreasing function of γ. This is because each ﬁrm does not further internalize social
knowledge spillovers as γ moves toward 0.

9I can analytically obtain the optimal transition path of τtR by solving the above diﬀerential
equation, (25). However, because (25) is too complicated, I resort to numerical simulations.
Calculating under the benchmark parameter sets as previously discussed, I obtain the numerical example of the linearly-approximated optimal transition path of τtR as given in Figure
27 . It is shown that when the social knowledge spillover in R&D exists, a higher subsidy ratefor R&D is required during the early stage of the transition. To sum up, I state the following
proposition.
Proposition 2. When the social knowledge spillover in R&D does not exist (that is, γ = 1),
a combination of the time-invariant subsidy for investment and for R&D can replicate themarket equilibrium to the socially optimal allocation. When social knowledge spillovers in
R&D exist (that is, γ ̸= 1), a combination of the time-invariant subsidy for investment andthe time-variant subsidy for R&D can replicate the market equilibrium to the socially optimal
allocation.
ΤtR
0.300
0.295
0.290
0.285
0.280
0.275
0.270
0.265
0

10

20

30

40

50

t

Figure 2: The optimal transition path of subsidy rate for R&D

6

Conclusion

This paper analyzes some dynamic properties of a modiﬁed version of Krusell(1998)’s model,
in which investment-speciﬁc technological change occurs endogenously. I make and prove
the two propositions. First, I show the steady state to be saddle-point stable. Second, Ialso show that a simple subsidy policy cannot replicate the market economy to the socially
optimal allocation, and consequently, a complex subsidy policy is required in this model.

7

An alternative choice of Z0 does not change the basic result.

10

A

Appendix A

In this appendix, I derive the dynamic system of the economy.
From (2) and (9), I can eliminate wt and derive(1 − α)Ktα L−α
Y t = ψqt Tt .
From the above equation and (15), I can eliminate LY t and derive(1 − α)Ktα (L − LAt )−α = ψqt Tt .
Then, LAt is derived as−1−1

LAt = L − (1 − α) α ψ − α qt α Tt α Kt
11

1

= L − (1 − α)ψ −1 κQt 1−α Zt .

(A-1)

From (11), (15), and (A-1), ut is derived as
ut = α2 κQt ,(A-2)

From (12), (A-1), and (A-2), rt is derived as
1

rt = α2 κQt + (1 − α)κQt1−α Zt − ψL − δ.
Substituting (A-3) into (13), I obtain1c˙t= α2 κQt + (1 − α)κQt1−α Zt − ψL − (δ + ρ).
ct

From (1), (4), (14), (15), and (A-1), I obtain
K̇t= κQt − St Zt−1 L − δ.
Kt
Substituting (A-1) into (5), I obtain
1Ṫt= ψL − (1 − α)κQt1−α Zt .
Tt

From (1), (7), (9), (10), (14), (15), (A-1), and (A-3), I obtain1
1
q˙t= α2 κQt + (1 − α)(1 + γ)κQt1−α Zt − κQt1−α Ztqtα+ Qt1−α St L − (1 + γ)ψL − δ.
From the deﬁnitions of {Zt , St , Qt }, I can derive (16), (17), and (18).

11

(A-3)

B

Appendix B

In this appendix, I provide a detailed derivation of the steady state in the market economy.
From (16), (17), and (18), the steady state, {Z ∗ , S ∗ , Q∗ }, must satisfy the following equations:κQ∗ + κQ∗ 1−α Z ∗ − S ∗ Z ∗ −1 L −11

α2 κQ∗ + κQ∗ 1−α Z ∗ −ψL− δ = 0,
1−α

(B-1)

ψL
− (δ + ρ) = 0,
1−α

(B-2)

1
α
(1 − α)2
1 − α ∗ 1−α(1 − γ)κQ∗ 1−α Z ∗ − α(1 − α)κQ∗ −
Q
S∗L
α
α
[
]
α + (1 − α)γ
1−α
+
ψL +
δ = 0.
αα(B-3)

From (B-2) and (B-3), I obtain Z ∗ = F (Q∗ ) and S ∗ = G(Q∗ ), respectively. Therefore, from(B-1), Q∗ is derived from the following equation:
H(Q∗ ) ≡ κQ∗ + κQ∗ 1−α F (Q∗ ) − G(Q∗ )F (Q∗ )−1 L −1ψL− δ = 0.
1−αCalculating the above equation, I can rewrite this as the following quadratic equation:
κ2 M Q∗ 2 + κN Q∗ + P = 0.
Solving the above quadratic equation, I obtain√−N + N 2 − 4M P
∗
Q =
or
√2κM
−N − N 2 − 4M P
.
Q∗ =
2κM

(B-4)
(B-5)

Here, I assume that ψL is high enough, and δ and ρ are low enough to obtain an interior
solution of the positive growth rate. Further, N < 0 and κ2 N 2 − 4κ2 M P > 0 is assumed tobe satisﬁed. Under the above assumptions, (B-5) cannot be an interior solution of positive
growth rate because labor inputs to R&D activity are negative, that is, L∗A < 0. Therefore,
I focus only on (B-4) as the solution of the steady state.

C

Appendix C

In this appendix, I consider the stability of the steady state. Linearizing (16), (17), and (18),around the steady state, I obtain
 


Żt
Zt − Z ∗ Ṡt  = J  St − S ∗ ,
Qt − Q∗
Q̇t

12



J11

where J ≡ J21
J31
given by
J13
J23 , and each element in this coeﬃcient matrix, respectively, is
J33

J12
J22
J32

J11 = κQ∗ 1−α Z ∗ + S ∗ Z ∗ −1 L,
1

J12 = −L,
J13 = κZ ∗ +ακQ∗ 1−α Z ∗ 2 ,1−α1

J21 = κQ∗ 1−α S ∗ ,J22 = 0,ακQ∗ 1−α Z ∗ S ∗ ,
1−α
2
2−α
(1 − α) (1 − γ)κ ∗ 1−α
J31 =
Q
,
α
1
α − 1 ∗ 1−α
J32 =
Q
L,
α
1
(1 − α)(1 − γ)κ ∗ 1−α
J33 =
Q
Z∗
α
α
− α(1 − α)κQ∗ − Q∗ 1−α S ∗ L.

J23 = α2 κS ∗ +As the dynamic system has one state variable, Zt , and two jump variables, St , and Qt ,to ensure that the steady state is saddle-point stable, the characteristic equation with thiscoeﬃcient matrix must have one positive eigenvalue and two negative eigenvalues.
If TrJ > 0 and det J < 0, the following two cases arise. In the ﬁrst case, three eigenvalues
are real numbers, of which two are positive and the third is negative. In the second case, twoeigenvalues are complex with positive real parts, and the third is a negative real number. In
each case, the steady state is saddle-point stable. In what follows, I show that TrJ > 0 and
det J < 0.
The determinant of J is given bydet J = A ×1

κQ∗ 1−α Z ∗ + S ∗ Z ∗ −1 L
1Q∗ 1−α(1 − α)(1 − γ)κQ∗ακ
κZ ∗ + 1−αQ∗ 1−α Z ∗ 2

−1
0
−1

α

1
α2 + 1−αQ∗ 1−α Z ∗αα(1 − γ)κZ ∗ − α2 κQ∗ α−1 − 1−αQ∗ −1 S ∗ L

1∗ 1−α S ∗ L. Calculating this, I obtain
where A ≡ 1−αα κQ

det J = A × [−2κM Q∗ − N ].
Therefore, substituting (B-4) into this, I can derive√
det J = A × [− N 2 − 4M P ] < 0.

13,

The trace of J is given by
TrJ = κQ∗ (1 − α)[1 − α(1 − α)(1 − γ)][]
ψL(1 − α)2 (1 − γ)+
(1 − α)(1 − γ) +
1−α
α
]
[
]
[
2
(1 − α)2 (1 − γ)
(1 − α) (1 − γ)
+ρ 2+
> 0.
+δ
α
α
Therefore, it is proved that the steady-state equilibrium is saddle-point stable.

D

Appendix D

In this appendix, I derive the dynamic system of the socially optimal allocation. To solve theinter-temporal optimization problem of the social planner, the current-value Hamiltonian is
set up as follows:[]
H p = log Ktα (L − LAt )1−α − It + µpt [Tt It − δKt ] + qtp [ψTt LAt ],where µpt and qtp represent the co-state variables. Restricting the analysis to an interior
solution, I obtain the ﬁrst-order conditions as follows:µpt Tt =1,
Ktα (L − LAt )1−α − It

(D-1)

αKtα−1 (L − LAt )1−α
− δµpt ,Ktα (L − LAt )1−α − It
(1 − α)Ktα (L − LAt )−α
qtp ψTt =,Ktα (L − LAt )1−α − It
ρqtp − q̇tp = µpt It + qtp ψLAt .

ρµpt − µ̇pt =

(D-2)
(D-3)
(D-4)

Furthermore, the (following
transversality
conditions must)( ∫be∞satisﬁed:)∫∞
limt→∞ µpt Kt exp − 0 rs ds = 0 and limt→∞ qtp Tt exp − 0 rs ds = 0.
From (D-1) and the resource constraint of ﬁnal goods, I obtain
µpt =

1
.
ct LTt(D-5)

From (D-1), (D-2), and (D-5), I obtain

Kt = α

1
1−α

1
1−α

Tt

(
(L − LAt )

ċt
Ṫt
+ρ+δ+
ct
Tt

) 1

α−1

.

(D-6)

From (D-3) and the resource constraint of ﬁnal goods, I obtainqtp =(1 − α)Ktα (L − LAt )−α
1
×
.
ψTt
ct L

14

(D-7)

In the market economy, qt =given by(1−α)Ktα (L−LAt )−α. Therefore, the relation between qt and qtp is
ψTt

qt = qtp ct L.

(D-8)

From (D-7), (D-8), and the resource constraint of labor, I obtain
1

LAt = L − (1 − α)ψ −1 κQt 1−α Zt(D-9)

From the law of motion of the production technology and (D-9), I obtain
1Ṫt= ψL − (1 − α)κQt1−α Zt .
Tt

(D-10)

From (D-6), (D-9), and (D-10), I obtain1c˙t= ακQt + (1 − α)κQt1−α Zt − ψL − (δ + ρ).
ct(D-11)

From the law of motion of capital stocks, the resource constraints of ﬁnal goods and labor,and (D-9), I obtain
K̇t= κQt − St Zt−1 L − δ.
Kt
From the resource constraints of ﬁnal goods and labor, (D-4), (D-5), (D-7), (D-9), (D-10),
and (D-11), I obtain1
1q˙t= ακQt + 2(1 − α)κQt1−α Zt − κQt1−α Ztqtα+ Qt1−α St L − 2ψL − δ.
Thus, from deﬁnitions of {Zt , St , Qt }, I obtain (19), (20), and (21).

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