Discussion Papers In Economics
And Business

Environmental Policy Effects: An R&D-based Economic
Growth Model with Endogenous Labour Supply

Yoshihiro Hamaguchi
Discussion Paper 16-06

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEnvironmental Policy Effects: An R&D-based Economic
Growth Model with Endogenous Labour Supply

Yoshihiro Hamaguchi
Discussion Paper 16-06

April 2016

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEnvironmental Policy Eﬀects: An R&D-based Economic
Growth Model with Endogenous Labour Supply∗
Yoshihiro Hamaguchi†
April 2, 2016

Abstract
In this article, we investigate the eﬀect of environmental policy on economic growthusing an R&D-based growth model with endogenous labour supply. A government
implements a pollution permit as environmental policy. As a result, we conduct a
numerical analysis and find that a decrease in pollution permit levels positively eﬀects
economic growth.

Keywords: Labour supply, Pollution permit, Innovation, Economic growth.
JEL Classications: J22, O31, Q58.

∗

The author is grateful to Koichi Futagami for his valuable advice. Additionally, the author sincerely
thanks Yutaro Hatta and Yuki Uchida for their assignments. The author is responsible for all of the errors
in this article.
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka, 560-0034,
Japan, E-mail address: kge012my@mail2.econ.osaka-u.ac.jp

1

Introduction

Recently, environmental pollution has significantly aﬀected economic activities worldwide.
The economic cost of air pollution such as the cost of particulate matter (PM) has been
extensively reported by the OECD (2014). Air pollution in China is particularly serious.
Serious air pollution such as PM-10 forces China to abstain from outdoor economic activities
and diminishes labour productivity because of sickness caused by air pollution.1
When discussing a relation between environment policy and economic growth in an endogenous growth model, researchers often discuss the growth eﬀect from environmental policies.2 For example, Bovenberg and Smulders (1995) find that an environmental tax has apositive growth eﬀect by considering the positive externality of environmental quality. By
using the variety expansion model of Romer (1990), Grimaud (1999) shows that a decreasein pollution permit levels has a negative growth eﬀect. In contrast, Ono (2002) shows thata decrease in pollution permit levels has a negative growth eﬀect. Considering the creative
destruction model of Aghion and Howitt (1992), Nakada (2004) finds that an environmental
tax has a positive growth eﬀect. Although we do not reach any consistent conclusion, theprevious works show that a decrease in pollution permit levels has a growth-degeneratingeﬀect.
Researchers argue that an endogenous labour supply clarifies the growth-enhancing eﬀect
of an environmental policy. Hettich (1998) shows that an environmental tax has a growthenhancing eﬀect in a Uzawa-Lucas model with endogenous labour supply. This is because
leisure time that is decreased by an environmental tax increases study time and boosts
economic growth. Using a learning by doing model, Itaya (2008) shows that the growthenhancing eﬀect from an environmental tax exists when an indeterminacy of equilibrium
occurs. Although the author shows that a labour supply increase from an environmental

1

For example, Hanna and Oliva (2011) and Yang et al. (2013) argue that pollution reduces labour supply,and Graﬀ Zivin and Neidell (2011) argue that pollution decreases worker productivity.
2
Ricci (2007) is a recent survey that introduces theoretical papers on growth-enhancing eﬀects fromenvironmental policies.

1policy stimulates growth rates, no study analyzes the growth eﬀect from a reduction in
pollution permit levels in an R&D model. This paper attempts such an analysis.
The model in this study is based on the variety expansion model by Romer (1990). We
consider a pollution permit an environmental policy. We conduct a numerical analysis andfind that a decrease in the pollution permit levels can have a positive eﬀect on economic
growth. Additionally, we present a numerical example where the environmental policy has apositive eﬀect on welfare.
The present paper is composed of the following sections. Section 2 shows the dynamic
general equilibrium model. We investigate the stability of the dynamic system in Section 3.
Section 4 depicts an eﬀect of environmental policy on economic growth and welfare. Finally,
Section 5 concludes.

2

The model

We consider an economy that consists of a representative household, a final good sector,
an intermediate good sector, and an R&D sector. Perfect competition exists in the final
good sector. The final good is produced by employing labour and intermediate goods. The
pollution flow is produced using capital stock. The level of pollution flow can be reduced
using an abatement good produced by the final good. The intermediate good firms produce
the intermediate goods using capital from household rents. The firms in the R&D sector
employ labour to produce new designs. The number of households is normalized to one andlives infinitely in the economy. The population is normalized to one. While the household
who supplies labour acquires a positive utility from consumption and leisure, the household
suﬀers from the negative externalities of pollution.

2

2.1

Final good sector

Following Gradus and Smulders (1993), we assume that net pollution flow is produced by
the following mechanism:
∫ At
Pt =
where

∫ At
0

0

xj,t dj,
Zt(1)

xj,t dj ≡ Kt represents aggregate stock of physical capital, Zt is an abatement good,

xj,t is a quantity of intermediate good j, and At is a number of intermediate goods. Whilethe net pollution flow increases by using the aggregate stock of physical capital, the flow
decreases by employing the abatement good produced from the final good.3
To internalize the negative environmental externalities, the government implements anenvironmental policy of pollution permits. We explain the market for the pollution permit.
The government distributes quotas for permits to the firms (P̄ ) in each period. The firms
freely trade the distributed quotas in the competitive pollution permit market. The unit
price of the pollution permit is denoted by pet . The firms that emit pollution in excess of thepollution permit (Pt > P̄ ) using the intermediate good must purchase the pollution permit of(Pt − P̄ > 0) in the market at the price pet . On the other hand, the firms that emit pollution
under the pollution permit (Pt < P̄ ) by employing the abatement good can sell the pollution
permit of (P̄ − Pt > 0) in the market at the price pet . The pollution permit market must be
cleared.
The final good is produced by the following production function:∫ At
Yt = L1−αY,txαj,t dj, 0 < α < 1,(2)0

where Yt is the output of final goods. We employ the final good as the numeraire good. LY,tis labour input. The final good firms choose their inputs, taking the factor prices as given.
Thus, the final good firms maximize the following:
3

In the present model, reducing the level of pollution permit decreases pollution by employing the abatement good as an end-of-pipe technology.

3

max

LY,t ,xj,t ,ZtΠt = Yt − wt LY,t −∫ At

pj,t xj,t dj − Zt − pet (Pt − P̄ ),0

where wt is a wage rate in the final good sector, pj,t is the price of the intermediate good j,
pet is the price of the pollution permit, and P̄ is the permit quotas given to a firm in eachperiod. The first order conditions of profit maximization are given by:

(∫ A

t

wt = (1 − α)

)α
,

)1−α
LY,t
,
xj,t
∫ At
pet 0 xj,t dj
,
1=
Zt
Zt

pe
pj,t + t = αZt(xj,t dj
0
LY,t

(3)
(4)
(5)

where (3), (4) and (5) state that the firms hire labour, the intermediate goods i and theabatement good until their marginal products are equal to their factor prices.

2.2

Intermediate good sector

Each intermediate good firm is a monopoly firm. The firms buy designs from the R&D sector
by paying the fixed-cost investment and maximize their profits by taking the inverse demand
function for their intermediate good as given. The variable costs are the interest costs. Thus,
the firms maximize the following:max πj,t = pj,t xj,t − rt xj,t ,xj,t()1−α ( )1
LY,t−s.t pj,t = α
.
xj,t
Pt
The first order conditions of profit maximization are given by

4

)
(
1
1−α
pt =
,
rt +
α
Pt
(
)
1−α
1
πt =
rt +
xt .
α
Pt

(6)
(7)

The quantity of the intermediate good x is determined by substituting the price into theinverse demand function for the intermediate good j. Therefore, the prices and the output
level of all intermediate goods firms become the same.

2.3

R&D sector

A new variety of intermediate good is developed by the following technology:Ȧt = δAt LA,t ,(8)

where At is the stock of the variety’s intermediate good, LA,t is labour input and δ > 0 is aparameter of productivity. PAt is the price of a new design. A perfect competition prevails
in the R&D sector. Thus, free entry into the R&D sector results in the following:δPA,t At = wt .

2.4

(9)

Household

The representative household maximizes the following:

∫ ∞
Ut =

[β log Ct + (1 − β) log lt − ηP log Pt ] e−ρt dt,

(10)

0

where Ct is consumption, lt is leisure time, 0 < β < 1 represents the weight on the utility
attached to consumption and leisure , ηp > 0 shows the weight on the utility attached to5pollution, and ρ > 0 is a subjective rate of time preference.
The budget constraint isẆt = rt Wt + wt (LY,t + LA,t ) − Ct ,(11)

where Wt is a financial asset held by the household. The time constraint is

1 = LY,t + LA,t + lt .

(12)

The household maximizes (10) by choosing a consumption stream and an allocation oftime between leisure and labor supply. The first order conditions becomeβ= λt ,
Ct
1−β= wt λt ,lt−λ̇t + ρλt = rt λt ,lim λT WT e−ρT = 0,

T →∞

(13)
(14)
(15)
(16)

where λt is the shadow price of assets, and (16) is the transversality condition. Substituting(13) into (14), we obtain the following:wt =1 − β Ct
,
β lt

(17)

where (17) states that a marginal rate of substitution between consumption and leisure isequal to the wage rate. From (13) and (15), we obtain the following Euler equation:
6

Ċt= rt − ρ.
Ct

2.5

(18)

Market

The economy is composed of the pollution permit market, the labour market, the capital
market, the stock market and the good markets. In equilibrium, the pollution emitted by the
final good firms coincides with the pollution permits distributed by the government(Pt = P̄ ).
The labour market is cleared(1 = LY,t +LA,t +lt ). Because each intermediate good firm holding
the patent rents capital from households, we obtain the following equilibrium condition forthe capital market:∫ At

xj,t dj = Kt ⇔ At xt = Kt .

(19)

0

The no-arbitrage equation is the following:

πt + P˙At= rt .
PA,t(20)

Finally, in the good market, the following holds:

Yt = Ct + K̇t + Zt .

7

(21)

3

Equilibrium

3.1

Dynamic system

By defining two jump variables (yt ≡ Yt /Kt , zt ≡ Ct /Kt ) and one state variable (ωt ≡ Kt /At ),we obtain the following dynamic system:

{( )
}
1
1−α
y˙t
1
2
=
+ αδyt 1−α ωt − α yt ,
yt
α
P̄
( )
{
}
1
α
1
−
β
ω̇t
1
= y t − zt −− δ 1 − yt 1−α ωt −yt 1−α ωt zt ,ωtβ(1 − α)P̄z˙t= zt − (1 − α2 )yt − ρ.
zt

(22)
(23)
(24)

Appendix A shows their derivations.

3.2

Steady state

The steady state is determined by y˙t = ω̇t = z˙t = 0. Then, we obtain the following steadystate:ω ∗ (P̄ ) =α2 y ∗ (P̄ ) − 1/P̄
1αδy ∗ (P̄ ) 1−α,z ∗ (P̄ ) = (1 − α2 )y ∗ (P̄ ) + ρ,
)
(
β
β−α
1+α
∗
y (P̄ ) =+ D(P̄ ) ,δ+ρ+2α(1 + α)β(1 − α)
αβ P̄
[(
] 12
)2
β−α
1+α
4ρ(1 + α)(1 − β)D(P̄ ) ≡δ+ρ++.
β(1 − α)αβ P̄β 2 (1 − α)P̄
We show the derivation of steady state in Appendix B. In the steady state, the leisure time
and the labour times spent in the final good sector and in the R&D sector become8l∗ (P̄ ) =α1−β ∗y (P̄ ) 1−α ω ∗ (P̄ )z ∗ (P̄ ),β(1 − α)1L∗Y (P̄ ) = y ∗ (P̄ ) 1−α ω ∗ (P̄ ),1

L∗A (P̄ ) = 1 − y ∗ (P̄ ) 1−α ω ∗ (P̄ ) −α1−β ∗y (P̄ ) 1−α ω ∗ (P̄ )z ∗ (P̄ ).
β(1 − α)We show their derivations in Appendix B. In a steady state, the growth rate becomesg(P̄ ) = α2 y ∗ (P̄ ) −

1
− ρ.
P̄

(25)

Appendix B shows the derivation of growth rate.

3.3

Stability

By linearizing the dynamic system around the steady state (y ∗ , ω ∗ , z ∗ ), we obtain the following
linearized system:

  


2−α
1
∗ 1−α
2 ∗
∗
0
 y˙t   α y − αP̄ δ(1 − α)y yt − y 
(
) 
  


1−β
1
ω̇  = 
∗
∗ 
∗ ,
α
−
ω
J
J
−ω
+
ω
−
ω
21
22
 t  t
β(1−α)αP̄ y ∗
  


∗
2 ∗
∗
z˙t
z
−
z
−(1 − α )z0ztwhere J21 and J22 areω∗J21 = ω +
1−α
∗

(
α−

1
αP̄ y ∗)(α(1 − β)z ∗1+β(1 − α)y ∗We apply the Routh-Hurwitz theorem:

9

)

1
1−β
, J22 = αy −+β(1 − α)
αP̄
∗

(
α−

1
αP̄ y ∗)
z∗.

Theorem 1 (Routh-Hurwitz Theorem)
The number of roots with positive real parts involved in the characteristic equation is equal
to the number of variations in the sign of the scheme:−1, trJ, −BJ +detJ, detJ.
trJ

To check the stability, we obtain the following signs of trJ and detJ:

trJ > 0 and detJ < 0.

We show their derivations in Appendix C. Because trJ > 0 and detJ < 0 hold, the eigenvaluesof the Jacobi matrix have one stable root and two unstable roots. Hence, the steady state islocally saddle-point stable.

4

Comparative statics

In this section, we numerically investigate how changes in the pollution permit levels aﬀect
the growth rate and welfare.

4.1

The eﬀect of environmental policy on the growth rate

By following De Hek (1999) and Oueslati (2002), we set the following parameters: α =0.25, β = 0.8, ρ = 0.01 and δ = 0.5. Using their parameters, we present a decreases in P̄ asa growth eﬀect. Figure 1 to 7 show the results. Figure 1, Figure 2, Figure 4 and Figure 6
reveal that decreases in P̄ have a positive eﬀect on y ∗ , z ∗ , LY ∗ , and LA ∗ . Figure 3 and Figure
5 reveal that decreases in P̄ have a negative eﬀect on ω ∗ and l∗ . Thus, Figure 7 shows that
decreases in P̄ have a positive eﬀect on the growth rate.
[Inserted Figure.1-8]
10

These results are explained by the following. If P̄ declines, the final good firms must employ
the abatement good to conserve pollution. The increases in the abatement good raise theprice of the intermediate good and the monopoly profit rises(See (7)). This increases thedividend per stock and the demand for a new design. The demand for labour from the R&D
sector also increases along with the sector wage rates. Hence, the households supply labourto the firms in the R&D sector. Finally, the decreases in P̄ stimulate R&D activity and boost
the growth rate.

4.2

The eﬀect of environmental policy on welfare

We investigate numerically the eﬀects of a decrease in pollution permit levels on the welfare
level of the steady state. Our welfare measure is (10). By substituting Ct = z ∗ (P̄ )K0 egt and(25) in (10), we rewrite (10) as follows:

Ut (P̄ ) =()β log z ∗ (P̄ )1−βηP+ β g(P̄ ) − ρ t log K0 +log l∗ (P̄ ) −log P̄ .
ρt
ρt
ρt

(26)

The terms from first to third show the indirect eﬀect of a decrease in P̄ on the welfare level
through consumption, the growth rate, and leisure. The fourth term shows the direct eﬀectof a decrease in P̄ on the welfare level. By following De Hek (1999) and Ouealati (2002), weshow the welfare eﬀect using the following parameters: α = 0.25, β = 0.8, ρ = 0.01, δ =0.5, ηP = 0.140974. We assume K0 = A0 = t = 10. The result is presented in Figure 8.4
Figure 8 presents the following numerical example: a reduction in pollution permit levels has
a positive eﬀect on welfare.

4

U (P̄ ) is negative when P̄ > 14.38 holds.

11

5Conclusion

We considered the eﬀect of an environmental policy on economic growth in an R&D-based
growth model with endogenous labour supply. Then, we analyzed how a reduction pollution
permit levels aﬀects the growth rate and welfare. As the result of analysis, we conduct a
numerical analysis and find that a decrease pollution permit levels has a positive eﬀect on theeconomic growth rate. Moreover, we presented a numerical example where environmental
policy has a positive eﬀect on welfare.

AppendixA

The derivation of the dynamic system

Using (1) and Pt = P̄ , we rewrite (5) as the following:

pet
1
Zt
= .
= ∫ AtZt
P̄xj,t dj(A. 1)

0

Using (A. 1), we rewrite (4) as the following equation:(
pt = α

LY,t
xj,t

)1−α
−

1
.
P̄

(A. 2)

Substituting (A. 2) into (6), we obtain the following:

(
α

LY,t
xj,t

)1−α

1
1
− =
α
P̄

)
( )1−α
(
1
rt
At
1
1−α
1−α
−
⇔
= αLY,t⇔ rt = α2 yt − .
rt +
α
Kt
P̄
αP̄
P̄
(A. 3)

12

Using (A. 3) and (7), we obtain the following:

πt = α(1 − α)yt ωt .

(A. 4)Yt.
LY,t(A. 5)We rewrite (3) as the following:

wt = (1 − α)Substituting (A. 5) into (9), we obtain the following:δPA,t At = (1 − α)yt

Kt
.
LY,t

(A. 6)We rewrite Yt = Kt α (At LY,t )1−α as the following:(yt =At LY,t
Kt

)1−α

1

⇔ yt 1−α

1
1
LY,t
Kt
=
⇔
= yt − 1−α At .
At
Kt
LY,t

(A. 7)

Substituting (A. 7) into (A. 6), we obtain the following:1

δPA,t At = (1 − α)yt 1− 1−α At ⇔ yt = [(1 − α)(δPA,t )−1 ] α .
1−α(A. 8)

Using ωt = (Kt )/(At ), we rewrite yt = [(At LY,t )/(Kt )]1−α as the following:

(
yt =

LY,t
ωt

)1−α

1

⇔ LY,t = yt 1−α ωt .

Substituting (A. 3) and (A. 4) to (20), we obtain the following:13(A. 9)

˙
˙
PA,t
πt
PA,t
= rt −⇔= α 2 yt −
PA,t
PA,t
PA,t

( )
1
α(1 − α)yt ωt
−
.
PA,t
P̄

(A. 10)Substituting (A. 8) into (A. 10), we obtain the following:˙PA,t= α 2 yt −
PA,t

( )
1
1
− αδyt 1−α ωt .
P̄(A. 11)

Using the time derivative of (A. 8), the following holds:

˙
α y˙t
PA,t
=−
.
1 − α yt
PA,t

(A. 12)

We obtain (22) by substituting (A. 11) into (A. 12).
Using (1) and Pt = P̄ , we rewrite (21) as the following:K̇t= y t − zt −
Kt

( )
1
.
P̄

(A. 13)

Substituting (9) into (17), we obtain the following:δPA,t At =α
1 − β C t Kt 1
1−β
1 − β Ct
⇔ lt =⇔ lt =
yt 1−α ωt zt .
β lt
βδ Kt At PA,tβ(1 − α)(A. 14)Using (12), (A. 7) and (A. 14), we rewrite (8) as the following:

(
)
1
α
Ȧt
1
−
β
= δ 1 − yt 1−α ωt −yt 1−α ωt zt .
Atβ(1 − α)14

(A. 15)

We obtain (23) using (A. 13) and (A. 15).
Substituting (A. 3) into (18), we obtain the following:Ċt= α 2 yt −
Ct

( )
1
− ρ.
P̄

(A. 16)

We obtain (24) using (A. 13) and (A. 16).

B

The derivation of steady state and growth rate

The steady-state is determined by y˙t = ω̇t = z˙t = 0. Using (22), we obtain ω ∗ (P̄ ). Using (24),we also obtain z ∗ (P̄ ). By substituting ω ∗ (P̄ ) and z ∗ (P̄ ) into (23), we obtain the following
equation:{}α(1 + α) ∗ 2(β − α)ρ 1 + αρ(1 − β)f (y ) ≡y − δ++y∗ −=0ββ(1 − α)αβ P̄αβ(1 − α)P̄∗

(A. 17)(A. 17) is a quadratic equation of y ∗ (P̄ ). By solving (A. 17), we obtain a positive
solution(y ∗ (P̄ ) > 0) and a negative solution(y ∗ (P̄ ) < 0). We choose the positive solution(y ∗ (P̄ ) >0) as the solution of (A. 17). Then, we obtain the following solution:

(
)
β
β−α
1+α
y (P̄ ) =+ D(P̄ ) ,δ+ρ+2α(1 + α)β(1 − α)
αβ P̄
[(
] 21
)2
β−α
1+α
4ρ(1 + α)(1 − β)D(P̄ ) ≡δ+ρ++.
β(1 − α)αβ P̄β 2 (1 − α)P̄∗Using (18) and (A. 3), we obtain (25).

15(A. 18)(A. 19)C

Proof on the signs of the trace and the determinantof the Jacobian matrix
1

Using (A. 3) and ω ∗ , we obtain the following equations: δy ∗ 1−α ω ∗ − α(1 − α)y ∗ = α2 y ∗ −α11/(αP̄ ), δy ∗ 1−α ω ∗ = αy ∗ −1/(αP̄ ) and δy ∗ 1−α ω ∗ = α−1/(αP̄ y ∗ ). We use these equations to
calculate the Jacobian matrix. By calculating trJ = J11 + J22 + J33 , we obtain the following:(
2 ∗trJ = α y −

1
αP̄

(

)
∗

+ αy −

1
αP̄

)

1−β
+
β(1 − α)(α−1αP̄ y ∗)z∗ + z∗.

(A. 20)Using (A. 3) and z ∗ , we rewrite (A. 20) as the following:

)
)
(
(
1
(1 + α)r∗(1 − β)r∗ (1 − α2 )y ∗ + ρ
1
1
2 ∗
∗
−
+ ρ. (A. 21)
trJ =+− α y −αy −ααβ(1 − α)
y∗
α
P̄
P̄
We rewrite (A. 3) as the following equation:r∗ = α2 y ∗ − 1/P̄ ⇔ r∗ + α(1 − α)y ∗ = αy ∗ − 1/P̄ .
By substituting this equation into (A. 21), we obtain the following:

trJ =

2r∗
(1 − β)r∗ (1 − α2 )y ∗ + ρ++ ρ + (1 − α)y ∗ > 0.
∗ααβ(1 − α)yThen, we obtain trJ > 0.
By calculating detJ = J11 J22 J33 + J12 J23 J31 − J12 J21 J33 , we obtain the following:

16

(
) {(
)
) }
(
1
1
1−β
1
2 ∗
∗
detJ =z α y −
αy −
+
z∗
α−
β(1 − α)αP̄αP̄αP̄ y ∗
)}
{
(
1
−
β
1
2
∗ ∗ ∗ 2−α+ (1 − α )δ(1 − α)z ω y 1−α 1 −α−β(1 − α)αP̄ y ∗
{
(
)(
)}
∗
1
1
α(1
−
β)z
∗ ∗ ∗ 2−α− δ(1 − α)z ω y 1−α 1 +
α−
1+
.
1−α
β(1 − α)y ∗αP̄ y ∗∗

(A. 22)

1

Using αδ y ∗ 1−α ω ∗ = α2 y ∗ − 1/P̄ = r∗ /α, we rewrite (A. 22) as the following:

(
){ ∗
}
1 − β z ∗ r∗
1r
2 ∗detJ =z α y −+αP
α2 αβ(1 − α) y ∗{}(1 − α)y ∗ z ∗ r∗
(1 + α)(1 − β)r∗r∗
(1 − β)r∗ z ∗
2
−
α +
+
+
. (A. 23)ααβy ∗α(1 − α)y ∗ β(1 − α)2 y ∗ 2∗Rewriting (A. 23), we obtain the following:

1+αdetJ = −
αP̄

( ∗
)
r
(1 − β)z ∗ r∗
(1 − α)z ∗ r∗ 2
(1 − β)(1 − α2 )r∗ 2 z ∗
∗ ∗ ∗
+
−
−
α(1
−
α)y
z
r
−
< 0.
α
αβ(1 − α)y ∗α2α2 βThen, we obtain detJ < 0.
We obtain BJ by calculating BJ = J11 J22 + J12 J21 + J12 J23 + J22 J33 . By proving detJ <0 < trJ, we conclude that the eigenvalues of the Jacobi matrix have one stable root and two
unstable roots. Thus, we do not show BJ.

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Figure 1: y ∗ (P̄ )Figure 2: z ∗ (P̄ )Figure 3: ω ∗ (P̄ )Figure 4: LY ∗ (P̄ )Figure 5: l∗ (P̄ )Figure 6: LA ∗ (P̄ )Figure 7: g(P̄ )Figure 8: U (P̄ )Note: α = 0.25, β = 0.8, ρ = 0.01, δ = 0.5,
ηP = 0.140974, K0 = A0 = t = 10.

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