ISSN (Print) 0473-453X

Discussion Paper No. 1110

ISSN (Online) 2435-0982

ENVIRONMENTAL POLICIESINA STAGNANT ECONOMY

Masako Ikefuji
Yoshiyasu Ono

December 2020

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Environmental policies in a stagnant economy∗
Masako Ikefuji

Yoshiyasu Ono

University of Tsukuba; ISER, Osaka University

ISER, Osaka University

December 7, 2020
Abstract: Using a dynamic optimization model of a monetary economy where persistent
unemployment can prevail, we examine the eﬀects of environmental policies on consumption and pollution emissions in a full-employment and a stagnant economy. If full employment prevails, environmental policies such as environmental tax and public pollution
abatement naturally crowd out commodity production and hence decrease consumption.
However, if unemployment appears, they stimulate consumption and production by increasing total employment and mitigating deﬂation, thus yielding additional pollution
emissions. Net pollution emissions may eventually increase. The optimal environmental
policy mix is also examined.
JEL Classiﬁcation: E24, O42, Q58.
Keywords: Secular stagnation, Unemployment, Public abatement, Environmental tax∗

The authors are grateful to the attendees at the conferences of SURED 2014 in Ascona, EEA-ESEM
2014 in Toulouse, and Public Economic Theory 2014 in Seattle; seminars at the University of Paris
Ouest-Nanterre La Défense 2014 and National Institute for Environmental Studies 2018; the Macroeconomic Workshop 2019 at Toyo University; and International Institute of Public Finance 2020 (virtually). This work was ﬁnancially supported by the Asahi Glass Foundation, JSPS KAKENHI Grant(JP15H05728, JP20H01477, JP20H05631) and the Program of the Joint Usage/Research Center for Behavioral Economics at the Institute of Social and Economic Research (ISER), Osaka University. E-mailaddresses: ikefuji.masako.gn@u.tsukuba.ac.jp (Ikefuji), ono@iser.osaka-u.ac.jp (Ono). Corresponding author: Masako Ikefuji. Faculty of Humanities and Social Sciences, University of Tsukuba, 1-1-1 Tennodai,
Tsukuba, Ibaraki, 305-8571 Japan; ISER, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047,
Japan

1

Introduction

It has long been recognized that the threat of global warming is increasing. In 2018,nearly 50% of human deaths and 90% of economic damage (estimated at US$ 123 billion)
were due to climatological, hydrological, or meteorological disasters (EM-DAT database,
OFDA/CRED). These are considered to have been caused by global warming. To stabilize greenhouse gas concentrations in the atmosphere and mitigate their threat, urgent
countermeasures are needed.
However, since the international ﬁnancial crisis of 2008, most countries have been
suﬀering from low (or even negative) economic growth with a lasting output gap (OECD,
2019; IMF, 2019), and environmental costs appear to be a signiﬁcant burden on stagnant
economies. IPCC (2014) expects an annual reduction of consumption growth by 0.04%
to 0.14% with mitigation policies over the current century relative to annual consumption
growth without mitigation policies. Currently, the CO2 reduction commitments of mostcountries are insuﬃcient and global emissions continue to increase.
Given this situation, we investigate the eﬀects of environmental policies, such as emission tax and public pollution abatement, on consumption, pollution emissions and welfare.
We do so in a full-employment and a stagnant economy context and compare the eﬀects
in the two cases.
In the literature, much attention has been paid to the eﬀects of environmental policies
on economic performance. Among others, Nielsen et al. (1995) assume that unemployment stems from the monopoly power of unions in an endogenous growth model, andshow that increased environmental concern and increased labor market distortions raise
the optimal environmental tax and lower economic growth. In the literature on the double
dividend hypothesis, which means that environmental tax reforms not only improve environmental quality but also stimulate economic performance, Koskela et al. (1998) modela wage employment bargaining process between trade unions and ﬁrms and explore the1case where increasing an energy tax drives ﬁrms to replace dirty energy by labor for
production and improves both employment and environment. Koskela and Schöb (1999)
show that the eﬀect of a revenue-neutral green tax reform on employment depends on
the structure of income taxes and unemployment beneﬁts. Bovenberg and van der Ploeg(1998) apply the job matching mechanism to the double dividend issue. They ﬁnd that anenergy tax promotes job matching by lowering the income of the unemployed who work in
the informal sector and improves production. Bye (2002) adopts a dynamic model withtrade union wage negotiations and shows that the eﬀect of an environmental tax reformon the double dividend may diﬀer in the short and long runs.
While those researchers treat unemployment as caused by job matching and tradeunion negotiations, many countries, including EU member states, Japan and the UnitedStates, have been facing involuntary unemployment with aggregate demand shortages
since the global ﬁnancial crisis of 2008. We consider this case using the model of Ono(1994, 2001), where people’s liquidity (or wealth) preference causes aggregate demand
shortages and involuntary unemployment to appear in the steady state.1
In this setup we analyze the eﬀects of environmental policies on consumption andpollution emissions in a full-employment and a stagnant economy.2 In a full-employment
economy, the policies lower pollution and consumption, or in other words, pollution abatement comes at the expense of lower consumption. In a stagnant economy, by contrast,
they not only abate pollution but also create employment and thereby stimulate consumption. The expansion in consumption will yield extra pollution, which may dominate
the ﬁrst-stage abatement. If the abatement technology is high, however, the ﬁrst-stage
abatement dominates the extra pollution and people can enjoy both business recovery
1This type of model has recently expanded in the literature, including Ono and Ishida (2014), Michaillat and Saez (2014), Michau (2018), and Illing et al. (2018).
2
Many studies have examined the eﬀects of public abatement activities in a static (Ligthart and vander Ploeg, 1999; Mayeres and Proost, 2001) and a dynamic framework (Bovenberg and Heijdra, 2002;
Pérez and Ruiz, 2007; Itaya, 2008; Fullerton and Kim, 2008; Angelopoulos et al., 2013; Beladi et al.,2013). However, none of them consider unemployment.

2and a better environment.
The remainder of the paper is organized as follows. Section 2 sets up the modeland characterizes the dynamics of the economy. Section 3 investigates the eﬀects ofenvironmental policies in a full-employment economy. Section 4 considers the case inwhich aggregate demand shortages and involuntary unemployment appear in a steadystate and shows the eﬀects of the policies. Section 5 examines the ﬁrst-best environmental
policy mix in a stagnant and a full-employment economy by taking into account thepossibility of a transition from stagnation to full employment thanks to the environmental
policies. Finally, Section 6 concludes.

2

The model

The household
We consider an economy populated by identical households whose population size is normalized to unity. Each household’s lifetime utility is∫ ∞U (ct , mt , zt ) =[u(ct ) + v(mt ) − δ(zt )] exp(−ρt)dt,(1)0where u(ct ) is the utility of consumption ct , v(mt ) is the utility of real money holdings
mt , and δ(zt ) is the disutility of pollution zt . The functions u(c), v(m), and δ(z) satisfy
the following properties:u′ > 0,u′′ < 0;v ′ > 0,v ′′ < 0;

δ ′ > 0,δ ′′ > 0.

The stock budget equation of the household is

at = bt + mt ,3(2)

where at and bt respectively denote total assets and equities.3 The ﬂow budget equationisȧt = rt at − Rt mt + wt ℓt − ct − θt ,(3)

where rt and Rt (= rt + πt when the inﬂation rate is πt ) are the real and nominal interest
rates, wt is the real wage, ℓt is the realized amount of labor supply, and θt is a lump-sum
tax. Each household inelastically supplies all labor endowment, which is normalized tounity. Since we consider the possibility of unemployment, ℓt is given by the short side oflabor demand ℓdt and labor supply 1:ℓt = min(ℓdt , 1).

(4)

The household maximizes (1) subject to (2) and (3). The ﬁrst-order optimal condition
is
′

ċt
v (mt )
ρ + η + πt = Rt = ′,ct
u (ct )

(5)

where η ≡ −u′′ (ct )ct /u′ (ct ) is the elasticity of the marginal utility of consumption. The
transversality condition islim λt at exp(−ρt) = 0,

t→∞

(6)

where λt denotes the shadow price of asset at .
The ﬁrm
In period t, the representative ﬁrm produces yt with the following production technology:

yt = Aℓp,t ,3(7)

It is shown in the market section of the model that bt is zero because the technology of the ﬁrm
sector is linear-homogeneous.

4where ℓp,t is the labor employed to produce commodities and A is constant labor productivity. The production process is assumed to yield a proportional amount of pollution
emissions:et = ϵyt .

(8)

The government imposes a real tax τ per unit of pollution emissions. The ﬁrm can reduce
the amount of pollution ea with homogeneous abatement technology:
ea = Φ(ℓa , e) = eΦ(h, 1) ≡ eφ(h),φ′ > 0,φ′′ < 0,0 ≤ φ ≤ 1,

h≡

ℓa
;
e

φ(0) = 0,(9)φ(∞) = 1,

where h and φ(h) respectively represent the ratio of labor employed in the abatement
sector to pollution emissions and the reduction rate of emissions.
Given emission tax τ and real wage wt , the ﬁrm chooses labor for commodity production ℓp and labor for pollution abatement ℓa to maximize proﬁts:
(
( ))
ℓa,t
Aℓp,t − wt (ℓp,t + ℓa,t ) − τ 1 − φϵAℓp,t .
et
Because et satisﬁes (8), the ﬁrst-order optimal conditions are
A = wt + τ ϵA (1 − φ(ht ) + φ′ (ht )ht ) ,wt = τ φ′ (ht ),(10)from which we immediately obtain

wt =




A (1 − τ ϵ (1 − φ(ht ) + φ′ (ht )ht )) = τ φ′ (ht ) ⇒ h = h(τ ) if τ > 0,
A and ℓa = 0

if τ = 0.

5(11)

If τ φ′ (ht ) that satisﬁes the ﬁrst equation in (11) is higher (lower) than the wt determinedin the labor market, the ﬁrm will hire inﬁnite (zero) labor. Thus, wt is determined sothat it satisﬁes (10) and (11). From (9) and (11), we ﬁndh′ (τ ) > 0.

(12)

The government
From (8) and (9), the amount of pollution emitted by the ﬁrm iset − ea,t = (1 − φ(h)) et .

The government reduces the amount of pollution eg by utilizing labor ℓg with the following
abatement technology:4

eg = ψ(ℓg );ψ ′ > 0,ψ ′′ < 0,ψ(0) = 0.

(13)

Then the amount of net pollution emissions zt is
zt = (1 − φ(h(τ ))) et − ψ(ℓg,t ).

(14)

For example, the public sector plants trees to absorb the CO2 emitted from the private
sector or disposes of industrial contaminated waste. The employment for public abatement
is ﬁnanced by the emission tax τ and lump-sum tax θ,

wℓg,t = τ (1 − φ(ht )) et + θt ,4

In Appendix B, we consider an alternative abatement technology of the government. We show thatthe following arguments hold under this technology as well.

6and hence can arbitrarily choose ℓg,t by changing θt .
The market
The commodity market equilibrium implies

ct = yt = Aℓp,t .

(15)

From (8) and (15),et = ϵct .
Thus, the net pollution emissions zt given by (14) and employment for private abatementℓa that satisﬁes (9) are respectively
zt = (1 − φ (h(τ ))) ϵct − ψ(ℓg,t ),(16)ℓa,t = h(τ )ϵct .

(17)

The money and equity markets always satisfy
Mt= mt ,
Pt

bt = 0,(18)

in which nominal money supply Mt is, for simplicity, assumed to be constant and thetotal value of bt is zero because the ﬁrm earns zero proﬁts with the linear-homogeneoustechnology.
It is assumed that, in the labor market, the nominal wage is ﬂexible under full em-

7ployment and sluggish in the presence of unemployment. Therefore,5W is perfectly ﬂexible if ℓd,t = 1,t


 Ẇt = γ(ℓd,t − 1) if ℓd,t < 1,

(19)

Wt

where from (15) and (17), total labor demand ℓd,t equals(ℓd,t (= ℓp,t + ℓa,t + ℓg,t ) = ct)1
+ ϵh(τ ) + ℓg,t .
A(20)

From (10) and (11), real wage wt (= Wt /Pt ) is constant over time once τ is given, andhence the movement of Wt represented by (19) yields
πt = γ(ℓd,t − 1) if ℓd < 1.

(21)

From (5), (18), (19) and (21), the dynamic equations of ct and mt under full employment arev ′ (mt )ċt− ρ − πt ,= ′ct
u (ct )
ṁt
= −πt ,
mt

η

(22)

while those in the presence of unemployment areċtv ′ (mt )= ′− ρ − γ(ℓd − 1),ct
u (ct )ṁt
= −γ(ℓd − 1).
mt

η

(23)

Hereinafter, we suppress the time index.
5

A microeconomic foundation for this wage adjustment is given by Ono and Ishida (2014). It is also
assumed by Schmitt-Grohé and Uribe (2016, 2017).

8

3

The case of full employment

If full employment prevails, the total labor demand is ℓd = 1 in (20), which immediately
gives c as a function of ℓg and τ . In this case, the price adjustment is perfect andthe dynamics are given by (22), which are the same as in the standard monetary growthmodel. Thus, the new steady state is immediately reached after policy parameter changes,
implying that ċ and π in (22) are always zero and that c and m satisfy

c = cf (τ, ℓg ) ≡A(1 − ℓg ),1 + Aϵh(τ )ρ=v ′ (m).
u′ (c)(24)

From (24), the eﬀects of the environmental policy changes on consumption and real money
balances in this state are−Acf ϵh′∂cf (τ, ℓg )
=
< 0,
∂τ
1 + Aϵh(τ )−A∂cf (τ, ℓg )
=
< 0,
∂ℓg
1 + Aϵh(τ ))(∂mf (τ, ℓg )ρu′′−Acf ϵh′
= ′′
< 0,
∂τ
v
1 + Aϵh(τ )()∂mf (τ, ℓg )
−A
ρu′′
= ′′
< 0.
∂ℓg
v
1 + Aϵh(τ )(25)Substituting cf (ℓg , τ ) in (24) into the net pollution emissions given by (16) yields
z f (τ, ℓg ) =(1 − φ(h(τ ))) ϵA(1 − ℓg )− ψ(ℓg ),1 + Aϵh(τ )which satisﬁes(())∂z f (τ, ℓg )Af′′
= −c ϵh (1 − φ)ϵ+ φ < 0,∂τ
1 + Aϵh(τ )()∂z f (τ, ℓg )−A= (1 − φ)ϵ− ψ ′ (ℓg ) < 0.
∂ℓg
1 + Aϵh(τ )

9

(26)

Because (24) shows cf to be a function of τ and ℓg , employment for private abatement ℓain (17) is represented as ℓa = ℓfa (τ, ℓg ). From (25), this function satisﬁes
ϵcf h′∂ℓfa (τ, ℓg )
=
> 0,
∂τ
1 + Aϵh∂ℓfa (τ, ℓg )Aϵh∂(ℓfa (τ, ℓg ) + ℓg )
1
=−
< 0,
=
> 0.
∂ℓg
1 + Aϵh∂ℓg
1 + Aϵh(27)

Intuitively, an increase in τ drives ﬁrms to reduce pollution emissions by hiring more labor for abatement ℓa , which crowds out commodity production and hence ∂cf (τ, ℓg )/∂τ <0 (see (25)). An increase in public abatement employment ℓg directly lowers pollution but
decreases private abatement. This is because a decrease in consumption due to an increasein ℓg reduces commodity production and private pollution emissions, which makes private
abatement less necessary. However, total abatement labor ℓa + ℓg increases, as shownin (27), and net pollution emissions decrease (see (26)), while commodity production is
crowded out (see (25)). Therefore, stricter environmental policies have negative welfare
eﬀects through reductions in consumption c and real money holdings m, and a positiveeﬀect through a reduction in pollution z.
We summarize the above discussion in the following proposition:
Proposition 1 If full employment prevails, an increase in the emission tax alleviates
pollution emissions and lowers commodity production. An increase in employment for
public abatement decreases private abatement but the net pollution emissions and commodity production decrease. Thus, the environmental policies create a trade-oﬀ between
environmental conservation and commodity production.

10

4

The case of stagnation

Let us now turn to the case in which aggregate demand shortages arise in the steadystate. The key assumption is that the desire for money holding is insatiable:6
lim v ′ (m) = β > 0.

m→∞

In this case, a full employment steady state exists only ifρ>βu′ (cf )⇔cf < c where ρ =βu′ (c).

(28)c̄ is the maximum level of consumption that a household chooses under no inﬂation (ordeﬂation) whereas cf , given in (24), is the household’s feasible consumption under full
employment, which equals its income under full employment.
However, if cf is larger than c, or equivalently if cf is so large as to satisfyβρ< ′ fu (c ))(v ′ (m)< ′ f for any m ,u (c )(29)

there is no solution for the second equation in (24). This property implies that the time
preference (i.e., the left-hand side) is less than the liquidity preference (i.e., the righthand side), urging the household to reduce consumption below cf and yielding aggregate
demand deﬁciency and unemployment (ℓd < 1). Therefore, deﬂation continues, following

6

Using aggregate quarterly data in Japan and Japanese survey data called NIKKEI RADAR, Ono etal. (2004) empirically ﬁnd this property to be well supported using both parametric and non-parametricmethods. Murota and Ono (2011) show that status preference with respect to money plays the samerole as a positive lower bound of the marginal utility of money in creating persistent stagnation. Ono
and Yamada (2018) propose a model of status preference that works in the same way as the insatiableliquidity preference and apply an experimental method to ﬁnd the validity of this property. If households
regard net wealth (= total wealth minus the present value of future tax payments) as the source of thewealth preference, as assumed by Michau (2018), the marginal utility of wealth is ﬁxed and hence itworks as if there is a lower bound of the marginal utility of wealth.

11

=

′(

′( )

)

=

+ (

− 1)

−

0
̅

Figure 1: Steady state in a stagnant economy(21), v ′ (m) approaches β, and eventually equation (5) reduces toβu′ (c)= ρ + γ(ℓd − 1).

(30)

Figure 1 illustrates the level of c that satisﬁes (30) in the case where the government
employs no labor to abate pollution (ℓg = 0). The dynamics around the steady state are
given by (23) and the stagnation steady state is saddle stable as proven in Appendix A.
Under the condition (29), we ﬁndβu′ (c)(=βu′ (cf ))> ρ + γ(ℓd − 1)(= ρ)when c = cf (ℓd = 1).

For the solution of c in (30) to exist in the range of (0, cf ), it must be valid thatβ(= 0) < ρ + γ(ℓd − 1) (= ρ − γ) when c = 0 (ℓd = 0),u′ (c)

12

or equivalently
ρ − γ > 0.
Furthermore, as is clear from Figure 1, around the steady state the left-hand side of (30)
is more inclined than the right-hand side:Ω≡−γ(1 + ϵAh)βu′′ (cs )
> 0.
2 −
′
s
A
(u (c ))

(31)

In this steady state, the transversality condition (6) is satisﬁed even though m persistently expands. In fact, from (2), when m → ∞,ṁȧ=
= −γ (ℓd − 1) .
am
Since λ̇ = 0 in the steady state, using (30), one obtains
λ̇ ȧβ+ − ρ = −γ (ℓd − 1) − ρ = − ′ s < 0,t→∞ λau (c )
lim

which implies that the transversality condition (6) is valid.
We next examine the eﬀects of environmental policies ℓg and τ on consumption in thestagnation steady state. As shown in Appendix A, the economy is always in the steadystate and hence, we have the following property:
Lemma 1 A new steady state is reached immediately after the environmental policies
change.
Taking into account this property, we obtain the consumption level under stagnation,
cs , for the given τ and ℓg by substituting (20) into (30). It satisﬁes
β=ρ+γ′u (cs )

((

)
)
1
s
+ ϵh(τ ) c + ℓg − 1 =⇒ cs = cs (τ, ℓg ).
A
13

(32)

Therefore, the eﬀects of environmental policies τ and ℓg on consumption cs given above
and employment for private abatement ℓa given by (17) are∂cs (τ, ℓg )γϵcs h′∂cs (τ, ℓg )γ=
> 0,
= > 0;
∂τ
Ω
∂ℓg
Ω
(
)
s
∂ℓa (τ, ℓg )γhϵ∂ℓsa (τ, ℓg )γhϵ= ϵcs h′ 1 +
> 0,
=
> 0,
∂τ
Ω
∂ℓg
Ω

(33)

where Ω > 0 as shown by (31). Real money balances m keep expanding because deﬂation
persistently continues in the presence of aggregate demand shortages. An increase in
private abatement ℓa caused by the emission tax τ and an increase in public abatementℓg create employment and mitigate deﬂation, which in turn makes holding money lessattractive than consumption. Thus, they stimulate consumption, as shown in (33).
This result is opposite to that under full employment mentioned in Proposition 1. Under full employment, the environmental policies crowd out private consumption, whereas
under stagnation, they create additional employment and stimulate consumption.
Figure 2 illustrates the eﬀects of the environmental policies on consumption. The
upper panel shows the eﬀect of an increase in τ on cs when ℓg is constant, whereas thelower panel exhibits the eﬀect of an increase in ℓg on cs when τ is ﬁxed. If τ and ℓg increase
suﬃciently to make consumption cs equal A(1 − ℓg )/(1 + ϵAh) in (32), full employment
is reached. That is, cs → c̄ when ℓd = 1, as shown in Figure 2. Thereafter, the eﬀects ofthe environmental policies are as described in Proposition 1.
The environmental policies directly reduce pollution emissions. However, because theycreate employment and hence consumption and production increase, pollution emissions
also increase, and hence ﬁrms increase private abatement. Net pollution emissions are
the sum of these eﬀects and responses; therefore net pollution emissions may increaseor decrease. To show this, substituting cs (τ, ℓg ) given by (32) into c in (16) yields z =

14

=
′(

′( )

)

↑
=

+ (

− 1)

−
̅
′

0

=
′(

′( )

)

↑
=

+ (

− 1)

−
̅
0

′

Figure 2: The eﬀects of the environmental policies on consumption in a stagnant economy.
z s (τ, ℓg ). From (33), they satisfy∂z s (τ, ℓg )= h′ ϵcs∂τ()(1 − φ)ϵγ′−φ ,Ω∂z s (τ, ℓg )(1 − φ)ϵγ
=
− ψ′.
∂ℓg
Ω

(34)

If the marginal eﬃciency of private abatement technology φ′ is large (small) enough, arise in the emission tax τ decreases (increases) net emissions. Similarly, if the marginaleﬃciency of public abatement technology ψ ′ is large (small) enough, more abatement
employment by the government decreases (increases) net emissions.
These results are diﬀerent from those in the case of full employment (see Proposition151). In the case of full employment, the environmental policies always reduce pollution
emissions by abating pollution and reducing private consumption. In the case of stagnation, they initially reduce pollution but expands abatement employment, which stimulates
consumption and production, worsens pollution, and requires additional employment forabatement. If private or public abatement technology is high (low) enough, net pollution
emissions decrease (increase).
In sum,
Proposition 2 If (29) holds, secular demand stagnation appears. Then, increasing theemission tax and employing more labor for public abatement expand total employment
and mitigate deﬂation, thereby stimulating consumption. However, the resultant pollution
emissions may increase or decrease because increase in consumption creates additional
pollution emissions.
By comparing Propositions 1 and 2, we ﬁnd that the directions of the eﬀects of theenvironmental policies on consumption under secular stagnation are opposite to thoseunder full employment. Under full employment, the environmental policies crowd out
consumption, whereas under stagnation, they increase consumption and commodity production. It is widely believed that there is a trade-oﬀ between environmental conservation and commodity production and that this trade-oﬀ seems particularly serious under
stagnation. However, Proposition 2 shows that under stagnation, stricter environmental
regulations stimulate consumption and expand commodity production. Moreover, despite
stricter regulations, pollution emissions may expand because of the increase in commodity
production.

16

5

Optimal policy mix

Let us next ﬁnd the optimal mix of the two policies ℓg and τ . We start the analysis byconsidering the case in which full employment prevails.
From (14) and (24), c, m, and z in the case of full employment are represented asfunctions of τ and ℓg : cf (τ, ℓg ), mf (τ, ℓg ), and z f (τ, ℓg ). Thus, welfare (1) isU (τ, ℓg ) ≡
f

∫ ∞

[ ( f
)
(
)
(
)]
u c (τ, ℓg ) + v mf (τ, ℓg ) − δ z f (τ, ℓg ) e−ρt dt

0

)
(
)
(
)]
1[ ( f=u c (τ, ℓg ) + v mf (τ, ℓg ) − δ z f (τ, ℓg ) .
ρUsing (25) and (26), when ℓg is given, the optimal condition of τ is
∂U f (τ, ℓg )
ρ
∂τ

=

[
(
)
]
′′
cf h′ ϵ′′ ρu′′
−A u + v ′′ + [(1 + Aϵh)φ + Aϵ(1 − φ)] δ
1 + Aϵhv= 0⇒ τ = τ (ℓg ).

(35)

From (25), (26) and (35), the optimal ℓg with τ = τ (ℓg ) satisﬁes
(
ρ

dU f
dℓg

)

(

)
∂U f (τ, ℓg ) ∂U f (τ, ℓg ) ′=ρ+τ (ℓg )∂ℓg∂τ
τ =τ (ℓg )
[
(
)
]
′′
1
′
′ ρu
′
′
=
−A u + v ′′ + [(1 + Aϵh)ψ + Aϵ(1 − φ)] δ
1 + Aϵhv= [ψ ′ (ℓg ) − φ′ (h(τ (ℓg ))] δ ′ = 0.

Thus, the optimal policy mix (τ f ∗ , ℓfg ∗ ) = (τ (ℓfg ∗ ), ℓfg ∗ )) is given byφ′ (h(τ (ℓfg ∗ )) = ψ ′ (ℓfg ∗ ).

(36)

Equation (36) implies that the optimal policy mix makes the marginal productivity
17

of public abatement equal that of private abatement. To show this, from (15), (16), (17)and (20) in which ℓd = 1, pollution emissions z under full employment satisfy))((
1 − Ac − ℓgz = 1−φϵc − ψ(ℓg ).
ϵc

(37)

Thus, once c is given, z is minimized whenφ′ (h (τ )) = ψ ′ (ℓg ),(38)which implies (36).
From (24) and (28), if full empolyment prevails with the optimal policy mix given by(36), consumption c must satisfyfc = c (τf∗

, ℓfg ∗ ) ≡A(1 − ℓfg ∗ )≤c
1 + Aϵh(τ f ∗ )

(
≡u

′−1

( ))
β
.
ρ

(39)

However, if cf (τ f ∗ , ℓfg ∗ ) > c, the household does not consume up to cf (τ f ∗ , ℓfg ∗ ). Therefore,
the optimal policy mix is such that labor used for commodity production is

ℓp =

c
,
A

and that the rest is allocated to the most eﬃcient mix of public and private abatement,
as long as full employment is maintained. From (20), in which ℓd = 1, and (38), the labor
allocation is achieved when τ and ℓg satisfyφ′ (h (τ )) = ψ ′ (ℓg ), h(τ )ϵc + ℓg = 1 −

c
.
A

(40)

We now consider the optimal combination of τ and ℓg in the case of secular stagnation,
where v ′ (m) ≈ β. Suppose that (τ s∗ , ℓs∗g ) is the optimal combination of (τ, ℓg ) in this case.
18

Then, we havec = cs (τ s∗ , ℓs∗g ) < c.
where cs (τ, ℓg ) is represented by (32). Because unemployment appears in this state, thegovernment can additionally allocate all the unemployed for private or public abatement,
and thereby realize better environment with the same consumption level, cs (τ, ℓg ), yielding higher utility. Thus, the ﬁrst-best optimal policy mix is never obtained under stagnation. From (39), once full employment is realized, the optimal policy mix is (τ f ∗ , ℓfg ∗ ) if
cf (τ f ∗ , ℓfg ∗ ) ≤ c. If not, the optimal policy mix satisﬁes (40).
We summarize the above discussion as follows:
Proposition 3 The optimal policy mix must accompany full employment. This mix is
given by (36) if cf (τ f ∗ , ℓfg ∗ ) ≤ c. If not, it is given by (40).

6Conclusion

We examine the eﬀects of an environmental tax and employment for public abatementon consumption, pollution, and welfare. If full employment prevails, these policies crowdout consumption, and hence, there is a trade-oﬀ between consumption and environmental
conservation. In contrast, under secular stagnation of aggregate demand, the policies create employment for private and public abatement and hence stimulate consumption. The
stimulated consumption requires larger production, therefore increasing pollution emissions, which may dominate the direct decrease in emissions due to the initial abatement
and eventually worsen pollution.
We then examine the ﬁrst-best optimal mix of the two policies. Under the optimal
policy mix, the sum of private and public employment for abatement must be large enoughto achieve full employment and the marginal productivity of private abatement must equalthat of public abatement.
19

Appendix A: Stability
We ﬁrst deﬁne x ≡ 1/m. Then, the dynamic system can be written as[]
c v ′ (1/x)
ċ =− ρ − γ(ℓd − 1) ,σ u′ (c)ẋ = xγ(ℓd − 1),

where ℓd = (1 + ϵAh(τ )) c/A + ℓg . Applying a ﬁrst-order Taylor expansion of these equations around the steady state yields ċ  
 =
ẋ

c
η

(

βu′′ (cs )γ(1+ϵAh(τ ))− (u′ (cs ))2 −A)′′ (1/x)− ηc vu′ (c)x2xγ(1+ϵAh(τ ))
A

π




∗

 c − c .
x − x∗The eigenvalues ν1 and ν2 satisfycν1 ν2 =
η()
βu′′ (c)γ(1 + ϵAh(τ ))c v ′′ (1/x) xγ(1 + ϵAh(τ ))
·
.
−
−
π+
A
η u′ (c)x2A(u′ (c))2(A)

In the full-employment steady state, π = 0; thus, ν1 ν2 < 0, which implies that one of theeigenvalues is positive and the other is negative because v ′′ < 0. Since c is jumpable andx is non-jumpable, the steady state is saddle stable.
When stagnation persists, ℓd < 1 and π < 0 in the steady state and x approaches zero.
Hence, from (31), the ﬁrst term in (A) is negative and the second term is also negative,
which implies that the dynamic path is saddle-point stable.
Furthermore, once v ′ (m) approaches β, the dynamics of c reduce to[]βc− ρ − γ (ℓd (c, τ ) − 1) .
ċ =σ u′ (c)Since c is a jumpable variable, the economy reaches the new steady state immediately

20

after an exogenous shock

Appendix B: Alternative abatement technology
In this appendix we assume that the public abatement technology is given by the following
homogenous form:(eg = Ψ(ℓg , ep ) = ep Ψψ ′ > 0,ψ ′′ < 0,

)
( )
ℓg
ℓg
, 1 = ep ψ,ep
ep

0 ≤ ψ ≤ 1,

ψ(0) = 0,

ep = (1 − φ(h))ϵc,
ψ(∞) = 1,(B1)

instead of (13), and show that all three propositions in the text are valid under thistechnology as well.
Note that neither cf (τ, ℓg ) given by (24) nor cs (τ, ℓg ) given by (32) depends on the
public abatement technology. Thus, the results regarding the eﬀects of the environmental
policies on consumption given by (25) and (33) are valid under this technology.
Under (B1) the amount of net pollution emissions z given by (14) turns to
(
( ))
ℓg
z = ep 1 − ψ
.
ep

(B2)

Substituting ep in (B1) to (B2) and partially diﬀerentiating the result with respect to τand ℓg yields()()(1 − φ) ∂cj∂z j (τ, ℓg )′ ℓg′
= (1 − ψ) + ψ− φ ϵcj h′ ,∂τep
h′ cj ∂τ()∂z j (τ, ℓg )∂cj
∂cj /∂ℓg= (1 − φ)(1 − ψ)ϵ− 1−
ψ ′ , j = f, s.
∂ℓg∂ℓg
ep /ℓg(B3)

Noting that (25) still holds under full employment, we ﬁnd ∂cf /∂τ < 0 and ∂cf /∂ℓg < 0,and hence, from (B3) ∂z f (τ, ℓg )/∂τ < 0 and ∂z f (τ, ℓg )/∂ℓg < 0. Therefore, Proposition 1
21

is valid.
Under stagnation, (33) holds. Substituting ∂cs /∂τ in (33) to the ﬁrst equation in (B3)gives∂z s (τ, ℓg )
=
∂τ

(
)(
)
(1 − φ)ϵγ′ ℓg′(1 − ψ) + ψ− φ ϵcj h′ .
ep
Ω

(B4)

Comparing (B4) with the ﬁrst equation in (34), we ﬁnd that the sign is the same between
the two. Thus, it can be either positive or negative. Substituting ∂cs /∂ℓg in (33) to thesecond equation in (B3) yields()∂z s (τ, ℓg )(1 − φ)(1 − ψ)ϵγ
γ ℓg
=
− 1−
ψ′.
∂ℓg
Ω
Ω ep

(B5)

If ℓg ≈ 0, we have ψ(ℓg /ep ) ≈ 0, and the value of (B5) is the same as ∂z s /∂ℓg in (34),which can be either positive or negative, as mentioned below (34). Thus, Proposition 2
is valid.
The same logic for the proof of Proposition 3, given above the proposition in the text,
still holds. Therefore, this proposition is also valid under the technology represented by(B1).
Thus, all three propositions are valid under the alternative technology of public abatement.

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