ISSN (Print) 0473-453X

Discussion Paper No. 1139

ISSN (Online) 2435-0982

CLIMATE POLICYAND
WEALTH
DISTRIBUTION
Thang Dao

August 2021

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanClimate policy and wealth distribution*
Thang Dao

August 5, 2021

Abstract
We set up a model with intergenerational bequest transfers and climate damage on the wealth ofheterogeneous households. We show that, under credit market imperfections and depending on wealthdistribution across households, a balanced budget climate policy may widen the wealth inequality gap
between the rich and poor. Climate policy may create positive eects on the wealth of households, butthese eects are asymmetric across households in terms of both magnitude and the transmission of gains
from a climate policy within households. The gains of the poor from a climate policy are mainly transmitted into improving living standards and the investment in human capital due to the higher marginal
return to education investment. By contrast, the gains of the rich from a climate policy are transmitted biasedly into physical capital accumulation and thereby enhance their monopolistic position in the
production of intermediate inputs. We show that, for any climate policy, there exists a corresponding
threshold of aggregate physical capital. When the aggregate physical capital of the economy exceeds thisthreshold, the corresponding climate policy may widen the intergenerational bequest transfers among
heterogeneous households, thereby contributing to widening the wealth inequality gap between the richand poor in the long run.

JEL: D62, D63, O15, Q52, Q54.
Keywords: Climate policy, balanced budget policy, credit market imperfections, intergenerationalbequest transfer, wealth inequality.

1 Introduction
Two issues emerging in the twenty-rst century are climate change and income/wealth inequality.
The Paris Agreement under the United Nations Framework Convention on Climate Change, whichaims to limit global warming to well below 2 °C above pre-industrialization levels, indeed requires
strict transitions to clean economies.

These transitions can be implemented through a carbon

I thank the editor and two anonymous referees for their very helpful and constructive comments. I am grateful to Kerstin Burghaus,
Ottmar Edenhofer, Achim Hagen, Matthias Kalkuhl, Sang Pham, and Martin Quaas for insightful discussions and encouragements.
I also thank the participants of the Workshop of Macroeconomics and Climate Policy at the Mercator Research Institute on Global
Commons and Climate Change, the European Association of Environmental and Resource Economists Conference (EAERE 2021), andthe Osaka International School of Public Policy Lunch Seminar for their comments and discussions. I am indebted to Kimiyo Moriuchi
and Yoshie Tachikawa for their excellent administrative support. Finally, I gratefully acknowledge nancial support from the KAKENHI
Startup Grant-in-Aid for Research Activity, Japan Society for the Promotion of Science (20K22121). The scientic responsibility is
assumedby myself as the author.
 Institute of Social and Economic Research, Osaka University, 6 - 1 Mihogaoka, Ibaraki, Osaka 567-0047 Japan, email:daonguyen@iser.osaka-u.ac.jp.
*

1

1

INTRODUCTION

tax policy in which the government imposes a Pigouvian tax on the dirty production sector, andthen uses the tax revenue to subsidize the clean production sector (Acemoglu et al. 2012, 2016).
Such a climate policy may improve environmental quality and the living conditions of households.
However, whether the eects of climate policy may be asymmetric to heterogeneous households
needs to be investigated. Wealth inequality has been a topic of debate among economists and social
scientists for a long time, especially since the publication of 

Capital in the twenty-rst century 

by Piketty (2014). However, the link between climate policy and wealth/income inequality has notbeen suciently investigated in the literature. This paper aims to contribute a theoretical model
linking climate policy and wealth inequality to identify the conditions under which climate policy
may widen the wealth inequality gap between the rich and poor.
Numerous studies have considered the optimal paths for global emissions and optimal taxation
strategies for the decentralization (e.g.

Nordhaus 1992, 1993; Nordhaus and Boyer 2000; Pizer

1999; Acemoglu et al. 2012; Golosov et al. 2014; among others). These studies, however, haveignored the asymmetric eects of tax policy on the wealth of households in an economy.

Inaddition, while a large body of literature focuses on the relationship between climate policy andinequality between countries, theoretical considerations of the relationship between climate policy
and inequality between households remain limited. The present paper aims to ll this gap in theliterature.
A sizable body of literature has also attempted to explain the relationship between wealth/income
inequality and development, as well as persistent wealth/income inequality (e.g. Galor and Zeira
1993; Galor and Moav 2004, 2006; Piketty 1997; Piketty and Zucman 2014; Piketty et al. 2019;
Lakner and Milanovic 2013; Liberati 2015, among others).

These previous studies stress that

wealth inequality can persist because of credit market imperfections and intergenerational bequest
transfers within households. The presence of credit market imperfections under wealth inequality creates a long-lasting eect on investment in human capital and entrepreneurial activities,
contributing to the persistence of wealth inequality.

How is the link between climate change policy and wealth/income inequality ? Surprisingly, few
studies, particularly from the theoretical aspect, have been conducted on the interactions between
and dynamics of this link.

In an alternative approach to examine climate policy under wealthinequality, Vasconcelos et al. (2014) focused on on the conicting policies and challenges faced inachieving cooperation between rich and poor countries.

Blonz et al.

(2011) and William III etal. (2014) studied empirically near-term eects of climate policy on the welfare of heterogeneous
households by considering the eects on populations with dierent age groups. Dennig et al. (2015)
advanced the Dynamic Integrated Climate Economy model to stress the importance of accounting
inequality within regions and point out the asymmetric eects of climate policy on the welfare
of households.

Ravallion et al.

(2000) empirically showed that higher income inequality, bothwithin and between countries, is associated with lower carbon emissions at given average incomes.
Grunewald et al. (2011) reported a U-shaped relationship between carbon emissions and income
inequality. Chancel and Piketty (2015) argued that the rich should be responsible for the cost of2

2

THE BENCHMARK MODEL

climate policy because, in terms of consumption-based emissions, they contribute more to global
warming.
In contrast to the related literature, in this paper, we consider theoretically not only the eects ofclimate policy on wealth distribution, but also the conditions under which climate policy can widen
the wealth gap between the rich and the poor, who cannot escape the poverty trap. We set up an
overlapping generations model with heterogeneous households under credit market imperfections
and a climate externality on the wealth of households.

In particular, we identify an aggregate

capital threshold corresponding to each climate policy. When the aggregate physical capital exceeds
this threshold, such climate policy tends to widen the wealth gap between the rich and poor.
This occurs because the gain from the climate policy can be transmitted intergenerationally toinvestment in human capital for the poor or may improve the living conditions of the poor without
intergenerational transfer, whereas it is transmitted biasedly into the accumulation of physicalcapital for the rich. The improvement in human capital for the whole economy benets the prots of
the intermediate producing rms owned by the rich biasedly, thereby amplifying wealth inequality
and enhancing the accumulation of physical capital. This increased accumulation of physical capitalleads to more physical capital being allocated to the production of dirty intermediate inputs, which
generates an increased burden on future climate policy.
The remainder of the paper is organized as follows. Section 2 introduces the building blocksof the model.

Section 3 characterizes the equilibria and dynamics of the model.

The eects

of climate policy on macroeconomic variables and household wealth are presented in section 4.
Section 5 provides a discussion and concludes.

2 The benchmark model
We consider a discrete time overlapping generations economy with a constant population. Following Acemoglu et al. (2012) and Dao and Edenhofer (2018), we assume that there is one homogeneous nal output produced by human capital and intermediate inputs. In each period t ∈ N,
we dene It as a nonempty set of adult (or working) individuals/households, i.e. each individual i
who becomes an adult in period t belongs to the set It . Each individual i ∈ It lives for two periods
in childhood in t − 1 and adulthood in t. Along with choosing the optimal education investment,
individuals allocate their wealth (when they are adults), which comes from labour income, capitalincome and monopoly prot, between consumption and bequest transfer for their children so as tomaximize their lifetime utility.
In the production sectors of this economy, we follow Dao and Edenhofer (2018) with a slightmodication of aggregate human capital instead of xed aggregate labour. In the household sector,
we follow Galor and Moav (2004, 2006), in particular, the intergenerational bequest motives andeducation investment decisions. Diering from Galor and Moav (2004, 2006), however, we consider
the eects of monopolistic prot sharing and climate damage on the dynamics of intergenerationalbequest transfers.

3

2.1

Final goods sector

2

THE BENCHMARK MODEL

2.1 Final goods sector
There is one homogeneous nal output produced by human capital and intermediate inputs under
the following (aggregate) production function1αYt = Ht1−α Ac1−α Xctα + Ad1−α Xdt;

α ∈ (0, 1)(1)where Yt is the aggregate nal output and Ht is the aggregate human capital employed in the nalgoods production in period t, i.e.

ZHt =hit diIt
iin which ht is the human capital of individual i ∈ It .
The subscripts ”c” and ”d” denote clean  and  dirty , respectively. Thus, Xct and Xdt are

the amounts of clean and dirty inputs employed in the nal goods production, respectively.

In

addition, Ac > 0 and Ad > 0 are quality (or aggregate total factor productivity) indexes of thecorresponding intermediate inputs.
Suppose that the nal goods sector operates under a perfectly competitive environment. The
prot maximization problem of producing rms in this sector ismax Ht1−α

Ht ,Xct ,Xdt

X

α
A1−α
Xvt
− wt Ht −vv∈{c,d}X

pvt Xvtv∈{c,d}given the return on human capital, wt , and the price of intermediate input v ∈ {c, d}, pvt , in the
period t.
The rst-order conditions with respect to Ht and Xvt givewt =1 − α X 1−α α
Av XvtHtα(2)v∈{c,d}α−1
pvt = αHt1−α Av1−α Xvt;(3)

as the inverse demand functions of human capital and intermediate inputs.

1 Such a nal good production function is introduced extensively in Aghion and Howitt (2009). Our paper focuses on the eect ofclimate policy on wealth distribution rather than on the transition to a clean economy, so to keep the model as simple as possible, we
adopt a production function (1) in which the intermediate inputs are completely substitutable rather than complementary. As apparent
in the subsequent sections, with this production function, we can derive explicitly the rule of capital allocation for intermediate
input production, which depends on the relative quality (or aggregate total factor productivity) of the intermediate inputs. The
characterization of the features of the capital allocation rule is crucial for analysing the eects of climate policy on capital allocationin producing intermediate inputs, and subsequently on other macroeconomic variables such as labour income, the capital rental rate,
and aggregate monopolistic prots. In their online appendix, Dao and Edenhofer (2018) provide a robustness check for a more generalproduction function, in particular

Y = H 1−α


 
−1
α −1
α −1
A1−α
Xc  + A1−α
Xd 
c
d

where  ∈ (0, +∞) is a constant elasticity of substitution between intermediate inputs. With this production function, Dao and
Edenhofer (2018) point out implicitly that the capital allocation rule has similar properties to the production function in this paper. In
particular, physical capital will be allocated biasedly to the intermediate sector with higher aggregate total factor productivity.
42.2

Intermediate sectors and climate policy

2

THE BENCHMARK MODEL

2.2 Intermediate sectors and climate policy
For the sake of simplicity, we assume that each intermediate input indexed by v ∈ {c, d} is producedin period t according to the following production function

Xvt = Kvt

(4)

where Kvt is the amount of physical capital used as input in the intermediate sector v ∈ {c, d}. We
assume that the physical capital fully depreciates in each period t of use, given that each period
lasts around 30 years.2

Thus, the cost of producing Xvt units of intermediate input v ∈ {c, d}in period t is rt Kvt , where rt is the rental rate of physical capital in period t. The producer ofthe intermediate good vj in period t decides the quantity Xvt to be produced to maximize its
monopolistic prot. Therefore, the monopolist prot of the entrepreneur vj in period t is

πvt = max (1 − τvt )pvt Kvt − rt Kvt

(5)

kvjt

given rt and τvt , where τvt < 1 is the Pigouvian tax rate (or subsidy if negative) imposed by thegovernment on the production of each intermediate good v ∈ {c, d} in period t. These tax rates
represent the climate policy of the government.

In this paper, we consider a climate policy inwhich τdt ∈ [0, 1) and τct ≤ 0, i.e. in any period t, the government imposes a Pigouvian tax rateτdt ∈ [0, 1) on the production of dirty intermediate sectors and uses this tax revenue to subsidize
production of the clean sector at a rate −τct ≥ 0. The extreme values (τct , τdt ) = (0, 0) imply thecase that no climate policy is carried out.

2.3 Returns on factors of production and monopolistic prots
As apparent in Appendix A0, under the set-up above, the returns on physical capital and humancapital, as well as the monopolistic prots, are1−α
rt = α2 

Ht
Kt

1

X

(1 − τvt ) 1−α Av 
1−α

wt = (1 − α) 

(6)v∈{c,d}X(1 − τvt )

1
1−α



Av 

v∈{c,d}

X

α
(7)

1−α


Πt =

Kt
Ht

πvt = α(1 − α) Htv∈{c,d}X

1(1 − τvt ) 1−α Av 

Ktα

(8)

v∈{c,d}given the aggregate physical capital Kt , the aggregate human capital Ht , and the balanced budget
climate policy (τct , τdt ) ∈ <− × [0, 1).

In subsection 2.4, we assume the rate of decay of greenhouse gases to belong to interval (0, 1), capturing the fact that physicalcapital and greenhouse gases evolve at dierent timescales. I thank an anonymous referee for pointing out this.
2

5

2.4

Dynamics of greenhouse gas stock

2.4

2

THE BENCHMARK MODEL

Dynamics of greenhouse gas stock

We assume that only the use of dirty intermediate inputs degrades the environment by emitting
greenhouse gas. So the dynamics of greenhouse gas stock is characterized by

Pt = P̄ + (1 − δ)(Pt−1 − P̄ ) + ξXdt
where Pt is the greenhouse gas stock in period t, which can be viewed as the carbon concentrationin the atmosphere. The term ξXdt is the ow of greenhouse gases in period t emitted from dirty
intermediate inputs, and ξ > 0 is the dirtiness coecient of the dirty intermediate inputs. P̄ is thenatural state of greenhouse gas concentration in the atmosphere, i.e. the state of the ecologicalsystem without any human activity, and δ ∈ (0, 1) is the decay rate of greenhouse gases whichmeasures the convergent speed of greenhouse gas stock to the natural state P̄ .

For the sake ofsimplicity, without losing any generality, we normalize P̄ = 0. Note that Xdt = Kdt , and with theallocation rule of physical capital in (21), the dynamics of greenhouse gas stock can be written as
follows

Pt = (1 − δ)Pt−1 + ξKdt

(9)

1

(1−τdt ) 1−α Ad
Kt is aggregate physical capital which is allocated in the production
1P(1−τvt ) 1−α Av

where Kdt =v∈{c,d}of the dirty intermediate inputs d (see the rule of capital allocation in Appendix A0).

2.5 Individuals/households
Throughout this paper, we use the terms individual(s) and household(s) interchangeably.
assume that, in each periodWet ∈ N, the economy is populated by a constant and continuous

population of working individuals who are identical in innate talent and preference.

Without

losing any generality, we normalize the size of the population by 1. Each working individual in
any period t has a single parent and a single ospring, and lives for two periods, say t − 1 andt. In the rst period of life, period t − 1 (e.g. childhood), individuals spend all of their time forhuman capital formation. In the second period of life, period t (e.g. adulthood), they supplyecient units of labour to the market inelastically and allocate their wealth (from labour income,capital income, and probably monopolistic prot) between consumption and bequest transfers totheir ospring.

Human capital increases if the time for education is supplemented with capitali
investment in education. The human capital of adult individual i ∈ It , ht , ishit = h(eit )iwhere et ≥ 0 is capital investment in education for individual i ∈ It , who becomes an adult inperiod t. The human capital formation function satises the following assumption 1.

Assumption 1. h(0) = 1, h0 (e) > 0, h00 (e) < 0, lim+ h0 (e) = +∞ and lim h0 (e) = 0.
e→0

6

e→+∞

2.5

Individuals/households

2

THE BENCHMARK MODEL

Assumption 1 implies that, without capital investment in education, each individual is naturally
endowed one unit of labour. Human capital formation is a strictly increasing and concave function
in education investment and satises Inada conditions.

i
The utility of individual i ∈ It comes from the consumption ct and the bequest to its ospring

bit+1 , obeying the following utility function
U (cit , bit+1 ) = (1 − γ) ln cit + γ ln(θt + bit+1 );θt > 0(10)

where γ ∈ (0, 1) is the preference weight towards the individual's ospring.3
The budget constraint of individual i ∈ It is

cit + bit+1 ≤ Wti φ(Pt )(11)iwhere Wt is the wealth of individual i ∈ It , and φ(Pt ) ∈ (0, 1) is the damage eect of greenhouse0
gas stock Pt in period t and satises φ (Pt ) < 0. We assume that the greenhouse gas stock in eachperiod damages the wealth of individuals and acts as a negative externality. We deneψ(Wti , Pt ) = Wti φ(Pt ) and ψ(wt , Pt ) = wt φ(Pt )respectively as

disposable wealth and minimal disposable wealth of household i ∈ It . The ψ(wt , Pt )is constituted only by a unit of physical labour endowment of the household.

i
The ln θt , which appears in the utility function when bt+1 = 0, is the minimal expected utility
derived from the ospring of household i∈ It .

The θt can be viewed as the expectation of aminimal disposable wealth of its ospring in period t + 1. This expectation isbased on the individual's own minimal disposable wealth, i.e. ψ(wt , Pt ) = wt φ(Pt ). Indeed, thehousehold of theminimal disposable wealth ψ(wt , Pt ) is what an adult household i ∈ It observes in period t. Basedon this observation, it assigns an expectation to the minimal disposable wealth that its ospring
may own in the next period. Thus, we impose the following assumption:

Assumption 2. θt = θψ(wt , Pt )andγ
θ > 1−γ.

The parameter θ may change over time depending (endogenously) on other factors that are
not captured in the model. However, θ is not a focus of the model, and hence, for simplication
without lessening the power of the model, we treat θ as a constant.4
The household i ∈ It maximizes its utility (10) subject to the budget constraint (11), given θt ,i
Wt , and Pt . The optimal bequest transfer is

3 We follow the utility function from Galor and Moav (2004, 2006), which allows a corner solution for bequest transfer, i.e. b= 0,when the wealth of individual i ∈ I is too small. For simplication and consistency with Galor and Moav (2004, 2006), we abstractfrom consumption when old without changing the qualitative results of the households. Indeed, with the log-linear utility, consumption
when old can be incorporated into the utility function, and then the dynamics of the model is altered only by a multiplicative constant.
One of the main features in analysing the heterogeneity in wealth among households is convex saving behaviour, which implies thatthe marginal propensity to save higher for the rich than for the poorsee Stiglitz (1969) and Bourguignon (1981). The qualitative
results of our model would be valid under the convex saving function. The specic log-linear function (10) allows us to derive such theclassical
saving function while guaranteeing the dynamics of savings in terms of intergenerational bequest transfers to be traceable.
4θ >holds for plausible values of θ and γ, θ ≥ 1 (i.e. households place an expectation on the minimal disposable wealththat their ospring can earn at least as much as they could) and γ ' 1/4 that is used in the literature (see Lagerlöf 2006 and Dao 2016).
θ can be probably interpreted as the minimal income that the society can expect to be set by the government.
i
t+1

t

γ
1−γ

e.g.

t

7

2.5

Individuals/households

2

bit+1 = max

THE BENCHMARK MODELθt (1 − γ)i0, γ ψ(Wt , Pt ) −γ(12)Figure 1: Bequest transferi
The intergenerational bequest transfer bt≥ 0 that household i ∈ It receives from its parenti ∈ It−1 is allocated between education investment, eit , and capital saving, sit , which is lent to thecapital market in period t to earn capital income. Hence,
bit = eit + sit
and the wealth of household i ∈ It is5

Wti = wt h(eit ) + (bit − eit )rt + ρit Πtiiiiin which wt h(et ) is the labour income, (bt − et )rt is the capital income, and ρt Πt is the income fromithe monopolistic prots. The ρt ∈ (0, 1) is the share of monopolistic prots assigned for householdi ∈ It ; hence, {ρit }i∈It holdsZρit di = 1

It
iIn the absence of borrowing constraints, the rational household i ∈ It will choose et such that
eit ∈ arg max wt h(eit ) + (bit − eit )rt + ρit Πt

(13)

eit

i
i
Note that the term ρt Πt in the optimization problem (13) above does not depend on et . Underassumption 1 and in the absence of the borrowing constraint, the optimal education investment is
5 There can be a joint decision between the household i ∈ I and its parent i ∈ I
for allocating the intergenerational bequest
transfer b between investing in human capital and savings so as to maximize the wealth of household i ∈ I .
t

t−1

i
t

t

8

2.5

Individuals/households

2

eit = e∗t∀i ∈ It

where h0 (e∗t ) =THE BENCHMARK MODELrtwt

We assume that a household cannot borrow for education investment because of the imperfectioniiof credit markets, i.e. education investment et is limited by the bequest transfer bt .

Proposition 1.
(i) Under the imperfections of credit markets, the optimal education investment eit of individuali ∈ It depends on the transfer bequest bit that the individual receives from the parent, in particular
eit = min bit , e∗t(ii) For any strictly positive aggregate stock of bequest transfer bt =et ∈ (0, bt ) is a unique solution toh0 (et ) =

(14)

R

, e∗t > et where

bi di > 0It t

α2 h(et )1 − α bt − etMoreover, et = e(bt ) is an increasing function of bt .
Proof. See Appendix A1.

Figure 2: Education investmenti
Proposition 1 implies that the education investment et of household i ∈ It is a bounded function
iof the intergenerational bequest transfer bt that she/he receives from its parent i ∈ It−1 . In any∗
period t, the corresponding upper bound of education investment is et , which depends on the wage
rate wt and the rental rate of capital rt , in particulare∗t = h0−1 (

rt
)
wt

Under perfect credit markets, the last equation equalizes the marginal benets of education
investment and capital savings

i.e. wt h0 (e∗t ) = rt guaranteeing the maximal the income/wealth9

3of household i ∈ It .

EQUILIBRIA AND DYNAMICS

Under imperfect credit markets, for a given wage rate wt> 0 and rental

rate of capital rt > 0, households prioritize investing in education over savings. When the inter-∗i
generational bequest transfer is suciently low, particularly bt ≤ et , the household i allocates theeit = bit . When the bequest transfer is sucientlyi∗∗high, particularly bt > et , the household i allocates et for education investment and the remaining
bit − e∗t for savings in terms of physical capital.
bequest transfer for education investment, i.e.

The statement (ii) of Proposition 1 implies that whenever the capital stock of the economy

t is strictly positive, the demand for human capital investment of household i withiiintergenerational bequest transfer bt > 0 always exists, i.e. et > 0. In particular, the optimalin period

education investment under perfect credit markets is bounded from below by some strictly positive
level, depending positively on the aggregate intergenerational bequest transfer. This is because,with a strictly positive stock of capital, the rental rate of capital is nite. Therefore, a householdwith a strictly positive bequest transfer received from its parent always can improve its potential
income by allocating some amount from the bequest transfer in education investment.
We need to impose essential assumptions about the ownership of monopoly rms. We assume
that in each period t, monopolists of intermediate goods represent a fraction λ ∈ (0, 1) of theλλpopulation. We denote It as the set of monopolists in period t, then It ⊂ It . The owners of themonopolistic rms own the monopolistic prots.

Assumption 3. The share of monopolistic prots is as follows
ρit > 0iff

bit > e∗t + k̂,

with k̂ > 0

Assumption 3 implies that to own monopoly rms and earn monopolistic prots, the bequestitransfer bi that individual i receives from his or her parent must exceed a threshold at which∗
after investing in education at level et , the individual i must have sucient physical capital, i.e.

bit − e∗t > k̂ holds, to run a monopoly rm.

This assumption admits the widely observed factthat when a potential entrepreneur accesses the capital market to borrow capital for operating amonopoly producing rm, the banks or nancial intermediate institutions

always require an initial

capital capacity in addition to the entrepreneurial skills of the applicant.6k̂ can be linked to theminimum equity of the rm owned by the individual in order for that individual to have rights inmaking decisions on production and entrepreneurial activities to obtain monopolistic prots.

3 Equilibria and dynamics
In this section, we dene equilibria. Then, we characterize the dynamics of the model and obtain
the reduced dynamics of intergenerational bequest transfers for any household i ∈ It .

Denition 1. ( Equilibria ) Given Pt−1 and {bit , ρit }i∈It , under the balanced budget climate policy(τct , τdt ) ∈ <− × [0, 1), the competitive equilibria in period t is characterized by: (i) the determi6 In this model, we assume that individuals have identical innate talent. Hence, the dierence in education investment generates adierence in entrepreneurial skills.

10

3

EQUILIBRIA AND DYNAMICS

nations of returns on production factors in (6) and (7); (ii) the optimal choices of each household i ∈ It in (12) and (14); (iii) the allocation rule of physical capital in intermediate sectors
1

Kvt =

(1−τvt ) 1−α Av
Kt
1P(1−τv0 t ) 1−α Av0

with v ∈ {c, d}; and (iv) the dynamics of greenhouse gas stock in (9) .

v 0 ∈{c,d}The evolution of intergenerational bequest transfer within each family dynasty i is

bit+1 = maxθwt (1 − γ)iiii0, γφ(Pt ) wt h(et ) + (bt − et )rt + ρt Πt −γ

(15)

We study the dynamics of intergenerational bequest transfer under the following assumption

Assumption 4. h(e(λk̂)) > θ 1−γ.
γAssumption 4 implies that wt h(e(λk̂))φ(Pt ) > θwt φ(Pt )(1 − γ)/γ , i.e. the disposable wealth of∗a non-monopolistic household at the education level e(λk̂) ∈ (0, et ) is high enough at which thehousehold leaves a part of wealth, as bequest transfer, for its ospring. Note that by construction,Z

bit diλk̂ < inft∈Ni.e. λk̂ is a lower bound of bequest stock bt =RIt

bi di of the economy.
It t

Proposition 2. Under assumptions 1, 2, 3 and 4, there exists a threshold b ∈ (0, e∗t ) such that
 = 0 if bi ∈ [0, b]tibt+1
 > 0 ifbit > b

where

−1

b=h



θ 1−γ
γ



Proof. See Appendix A2.
Proposition 2 implies that a household i ∈ It will transfer to its ospring i ∈ It+1 a strictlyiipositive bequest bt+1 > 0 if and only if the bequest transfer bt , that it received from its parent−1ihousehold i ∈ It−1 , is suciently high, in particular bt > b = h



θ 1−γ
γ



∈ (0, e∗t ). This is because

in the model set-up that focuses on the persistent eect of intergenerational bequest transfers on
wealth distribution, the bequest transfer that a household receives from its parent household isthe source/determinant of its wealth. Indeed, this bequest transfer is allocated for human capitalformationwhich generates the labour incomeand capital savingswhich generate the capitalincome and probably monopolistic prot.

i
When the bequest transfer bt that household i ∈ Iti
receives from its parent i ∈ It−1 is too smallin particular, bt ≤ bthen household i ∈ It allocatesiiall of this bequest transfer for education investment to generate wealth Wt = wt h(bt ) ≤ wt h(b).
This wealth, however, is not high enough for household i ∈ It to be able to transfer optimally aiistrictly positive bequest bt+1 > 0 to its ospring household i ∈ It+1 . As a consequence, bt+1 = 0.
Propositions 1 and 2 allow us to represent the dynamics of bequest transfers as follows

11

3

bit+1 =










0

EQUILIBRIA AND DYNAMICS

ifhbit ∈ [0, b)i

γφ(Pt ) wt h(bit ) − θwt (1−γ)
ifbit ∈ [b, e∗t )γihθwt (1−γ)∗i∗if bit ∈ [e∗t , e∗t + k̂)
γφ(P
)
w
h(e
)
+
(b
−
e
)r
−

t
t
t
t
t
t

γ

h
i


 γφ(Pt ) wt h(e∗ ) + (bi − e∗ )rt + ρi Πt − θwt (1−γ) if
bit ≥ e∗t + k̂
t
t
t
t
γ

(16)

The dynamics of intergenerational bequest transfer between generations t and t + 1 in family
dynasty i can be classied into four parts, as presented in equation (16). How the bequest transfer

bit+1 is determined depends on which segment that bit belongs to. If bit ∈ [0, b) then, as stated inii∗i
Proposition 2, bt+1 = 0. If bt ∈ [b, et ), then household i ∈ It allocates all the bequest bt received
ifrom its parent householdin education investment to generate wealth wt h(bt ) by its labour inhiθwt (1−γ)iicome. Thereby, it transfers a bequest bt+1 = γφ(Pt ) wt h(bt ) −to its ospring householdγi∗ ∗under the eect of climate damage φ(Pt ). If bt ∈ [et , et + k̂), then household i ∈ It chooses the∗education investment level et at which the marginal eect of education investment on improvingthat household's wealth equals the marginal return of capital savings, i.e.

wt h0 (e∗t ) = rt .

The

∗
i
household i ∈ It then allocates the remaining bt − et ∈ [0, k̂) as capital savings to seek capital∗i∗
income. In this case, the wealth of household i ∈ It in period t is wt h(et ) + (bt − et )rt , andhiθwt (1−γ)i∗i∗
to its ospring household.
it transfers an amount bt+1 = γφ(Pt ) wt h(et ) + (bt − et )rt −γi∗
Finally, if bt ≥ et + k̂ , then, diering from the last case, this bequest transfer is suciently highto allow household i ∈ It to own monopolistic producing rms. Therefore, the current wealth ofhousehold i ∈ It comes from not only the labour and capital incomes, but also the monopolistic∗i∗iprots. Hence, its wealth is wt h(et ) + (bt − et )rt + ρt Πt and the transfer to its ospring householdiis bt+1 = γφ(Pt )hiwt h(e∗t ) + (bit − e∗t )rt + ρit Πt − θwt (1−γ)
.
γ

i i
2
It would be interesting to study the evolution of bequest transfers in the space (bt , bt+1 ) ⊂ <+ .
We refer to thisevolution in this space as conditional evolution in the sense that we have to x othervariables that have their own evolution and interactions with the evolution of the bequest transfers.
This simplication would be helpful for focusing on the mechanisms leading to wealth inequality.
We would note, however, that this approach is valid when we focus on any two successive periodsof time, t and t + 1. Therefore, the whole evolution of bequest transfers, indeed, is the sequence

conditional evolution under thefollowing conditions, which guarantee the existence of multiple conditional steady states .
of replication of what we try to do in this section. We study this∗∗(i) φ(Pt )wt [γh(et ) − θ(1 − γ)] ≥ et ;(ii) γφ(Pt )rt < 1;h

wt h(e∗t ) + k̂rt − θwt (1−γ)γ

i≤ e∗t + k̂ ; andihθwt (1−γ)∗i> e∗t + k̂(iv) γφ(Pt ) wt h(et ) + k̂rt + ρt Πt −γ(iii) γφ(Pt )∀i ∈ Itλ ⊂ It .

Under these conditions, the evolution of intergenerational bequest transfers are depicted ingures 3 and 4. These gures suggest that the model can generate multiple clusters of convergencein intergenerational bequest transfers across households. The direction of conditional convergence

12

4

EFFECTS OF CLIMATE POLICYi
of bequest bt+1 , in which the household i ∈ It transfers to its ospring household i ∈ It+1 , depends
ion the bequest bt that the household receives from its parent household i ∈ It−1 .7

Figure 3: Conditional evolution of bequest transfers (monopoly prot shares are even across monopolists)

4 Eects of climate policy
In this section, we study the eects of balanced budget climate policy on the structure of production
factors (or more precisely, the allocation of resources constituting aggregate production factors),
on bequest transfers and wealth inequality across households.

Denition 2. We dene a set of balanced budget climate policies in any period t as follows:

7 In this model, we assume that households/individuals are identical in innate talent and preferences, and that there are no stochastic
factors aecting their wealth or income. Hence, the inequality in wealth across households originates from the heterogeneity in intergenerational bequest transferswhich leads to heterogeneity in education investment and capital savings. Therefore, the dynamics ofbequest transfers indicate the tendency of wealth and income inequality.

13

4

EFFECTS OF CLIMATE POLICY

Figure 4: Conditional evolution of bequest transfers (monopoly prot shares are uneven across monopolists)






X
α
1−α
Ct ≡ (τct , τdt ) ∈ <− × (0, 1) such thatτvt (1 − τvt ) Av = 0v∈{c,d}Proposition 3. (On resource allocation) In any period t, the allocation of (bequest) resources
constituting aggregate physical capital Kt and human capital {eit , hit }i∈It is independent from climate
policy (τct , τdt ) ∈ Ct .
Proof. See Appendix A3.
Proposition 3 suggests that the eects of climate policy on individual i's disposable wealthoccurs through its eects on the return on human capital, the rental rate of physical capital, andthe stock of greenhouse gases.

Proposition 4. (On macroeconomic variables) In any period t, climate policy (τct , τdt ) ∈ Ct
reduces the rental rate of physical capital, the return on human capital, and the aggregate monopoly
14

4

EFFECTS OF CLIMATE POLICY

prot compared with the case of no climate policy (τct , τdt ) = (0, 0). In addition, it holds
 P
1−α
1
(1 − τvt ) 1−α Avrt (τct , τdt )wt (τct , τdt )Πt (τct , τdt )Yt (τct , τdt ) v∈{c,d}

P
=
=
=
=


rt (0, 0)
wt (0, 0)
Πt (0, 0)
Yt (0, 0)Avv∈{c,d}Proof. See Appendix A4.
The economic intuition supporting proposition 4 is as follows: the climate policy (τct , τdt ) ∈ Ct
distorts the optimal allocation rule of capital in producing intermediate inputs, and therefore
reduces the aggregate nal output. The aggregate nal output, indeed, is distributed for aggregate
labour income, aggregate physical capital income, and aggregate monopoly prot.

The rule of

distributing the aggregate nal output for labour income, physical capital income and monopoly
prot is indeed independent from climate policy (τct , τdt )∈ Ct .

This is because, as proven inproposition 3, Kt and Ht are independent from (τct , τdt ) ∈ Ct . Hence, the aggregate labour income(and return on human capital), aggregate physical capital income (and rental rate of physicalcapital), and aggregate monopolistic prot are distorted in exactly the same way by the distortionin the optimal allocation rule of capital in producing intermediate inputs.

iiFrom the dynamics equation (16) of bequest transfer, we nd that when bt ∈ [0, b], then bt+1 = 0
regardless of the climate policy (τct , τdt ) ∈ Ct . We now consider the eects of climate policy on theiibequest transfer bt+1 of household i ∈ It characterized by bt > b. Along with the equality statedin proposition 4, we can determine

 P
1−α
1
(1 − τvt ) 1−α Av
bit+1 (τct , τdt ) φ(Pt (τct , τdt ))v∈{c,d}

P
=
i


b (0, 0)Av
φ(Pt (0, 0))

if

bit > bt+1v∈{c,d}For exposition purposes without crucially lessening the power of the analyses, we follow Golosov
et al. (2014) to specify the functional form of the climate damage as follows8φ(P ) = exp(−P ) for P ≥ 0
For this functional form of climate damage, and from the dynamics of greenhouse gas stockicharacterized in (9), we have that when bt > b, the following holds

 
 P
1−α
1

(1 − τvt ) 1−α Av

  A

bit+1 (τct , τdt ) v∈{c,d}PPd=
exp ξ −ibt+1 (0, 0)
Av
Av


v∈{c,d}v∈{c,d}8

Such a damage function can also be found in Dao et al. (2017).
15


1






(1 − τdt ) 1−α Ad 
 Kt
P
1
 
1−α
(1 − τvt ) Av
v∈{c,d}(17)

4

EFFECTS OF CLIMATE POLICY

From the last equation, we nd that the overall eect of climate policy (τct , τdt ) ∈ Ct on theiiidirection of change in bequest transfer bt+1 (τct , τdt )/bt+1 (0, 0) when bt > b depends on the climate
policy itself in the relation to the size of aggregate physical capital in the economy.

Proposition 5. (On bequest transfers and inequality) In the economy set up above,(i) for any household i ∈ It with bit > b, it holds that bit+1 (τct , τdt ) > (=)(<) bit+1 (0, 0) withclimate policy (τct , τdt ) ∈ Ct if, and only if,



α
 1−α



 α 
1−τdt 1−α
τct − τdt
(τdt − τct )1−α τct − τdt 1−τctKt > (=)(<)
lnα  = K̂t (τct , τdt )
1


1−α
ξ
1−α
(τct − τdt )(1 − τdt )dt(1 − τct )τct τdt
1 − 1−τ
1−τct


1−τdt
1−τct

Moreover, K̂t (τct , τdt ) is bounded for all (τct , τdt ) ∈ Ct .
(ii) Climate policy (τct , τdt ) ∈ Ct alters the disposable wealth ψ(Wti , Pt ) of all households i ∈ It
by the same multiplier, andψ(Wti (τct , τdt ), Pt (τct , τdt )) > (=)(<) ψ(Wti (0, 0), Pt (0, 0))⇐⇒

Kt > (=)(<)K̂t (τct , τdt )Proof. See Appendix A5.

Proposition 5 tells us that, for climate policy (τct , τdt ) ∈ Ct , which is carried out in period t,
if the stock of physical capital is rather high, in particular Kt> K̂t (τct , τdt ), then that climate

policy (τct , τdt ) ∈ Ct may increase the inequality in bequest transfers across households. This isbecause climate policy improves environmental quality, and thereby enhances disposable wealthamong households. Under the climate policy (τct , τdt ) ∈ Ct where Kt > K̂t (τct , τdt ), while the choiceof bequest transfers to the ospring of the poor households is unchanged and set at 0 when thebequest the households receive from their parents does not exceed b, the bequest transfers to theospring of the rich and middle-class households increase.
This proposition implies that for a very unequal economy in bequest transfers, the climate
policy (τct , τdt ) ∈ Ct may enhance inequality. For a given aggregate bequest transfer, the aggregate
physical capital tends to be higher in an unequal economy than in a more equal one because when
the bequest is owned too biasedly by the rich, then a higher fraction of bequest will be transformed
into physical capital, making Kt > K̂t (τct , τdt ), since the education investment is bounded. Hence,in this case, the climate policy (τct , τdt )∈ Ct enhances inequality.

However, when the bequest

spreads more equally to the poor and middle classes, then they are transformed more into education
investment, thereby reducing the aggregate physical capital in the economy. Therefore, it is more
likely for climate policy (τct , τdt ) ∈ Ct to satisfy Kt < K̂t (τct , τdt ), which may reduce the inequalityiin bequest transfers {bt+1 }i∈It for the next generation.

16

To assess more precisely the eects of5

CONCLUDING REMARKS

Figure 5: Eects of climate policy on bequest transfer dynamics.
climate policy (τct , τdt ) ∈ Ct on the inequality in bequest transfers for generation t + 1, we requireiinformation about the distribution of {bt }i∈It .

5 Concluding remarks
We set up an overlapping generations economy with credit market imperfections and a climate
damage on the wealth of heterogeneous households to investigate theoretically the link between
two emerging issues in the twenty-rst century: climate change and wealth inequality. For eachbalanced budget climate policy, we identify a corresponding threshold of aggregate physical capital
above which such a climate policy may enlarge the wealth gap between the rich and poor. This
theoretical result derived from the model may be helpful for investigating the eects of climate
policy on wealth inequality, as well as for designing a composite policy to improve equity andprotect the environment.
The model suggests that, for a given stock of aggregate intergenerational bequest transfers, phys-

17

5

CONCLUDING REMARKS

ical capital stock tends to be higher in a more unequal economy. This is because of the asymmetry
in allocating bequest transfers to human capital investment and physical capital accumulation
among heterogeneous households in the presence of credit market imperfections.

Following thestatement in Proposition 5, this makes climate policy more likely to benet the rich in a biased
manner. In addition, while climate policy may not help the poor escape the poverty trap, in thelong run, the rich may become richer because of the positive eects of improving aggregate humancapital on monopolistic prots. This would contribute to widening the wealth gap between therich and poor. Hence, from the perspectives of equity, we should think about taxing monopolisticprots to subsidize the clean production sector and/or education for the poor to reduce inequality
in wealth distribution.
For simplication, this model ignores the role of the research and development sector, in which
the tax revenue from the dirty production sector is used to subsidy clean technology innovations (seeAcemoglu et al. 2012, 2016). Introducing innovation sectors would be challenging, interesting, andpromising using the present framework. The theoretical results of our model hinge specically on aplausible assumption of credit market imperfections. Relaxing this assumption and/or introducing
policies that eliminate the imperfections of credit markets would be important and interesting
research. These ideas, among others, are left for further research in the future.

Appendix
A0. Monopolistic prots and returns on physical and human capital
Substituting (3) and (4) into (5), we haveαπvt = (1 − τvt )αHt1−α A1−α
Kvt
− rt Kvt
v

αwith Kvt ∈ arg max (1 − τvt )αHt1−α A1−α
Kvt
− rt Kvt
v
Kvt

Hence,α2 (1 − τvt )
Kvt =
rt


1
 1−α

Ht Av

(18)

From (18), we can compute the aggregate physical capital in period t,

Kt =

X


Kvt =

v∈{c,d}

α2
rt

1
 1−α

Ht

X

1

(1 − τvt ) 1−α Av(19)v∈{c,d}From (19), the rental rate of capital is determined by

1−α
rt = α2 

1Ht X(1 − τvt ) 1−α Av Ktv∈{c,d}

18

(20)

5

CONCLUDING REMARKS

Substituting (20) into (18) we determine the allocation rule of physical capital across intermediate sectors vj in period t
1(1 − τvt ) 1−α Av
KtP1(1 − τv0 t ) 1−α Av0

Kvt =

(21)

v 0 ∈{c,d}The monopolist prot πvt is determined by substituting (21) and (20) into (5), that is

α


Kt


1
πvt = α(1 − α)Ht1−α (1 − τvt ) 1−α Av  P

(1 − τv0 t )

1
1−α

Av




0

v 0 ∈{c,d}In addition, the aggregate monopolistic prot in period t is

1−α


X

Πt =

πvt = α(1 − α) Ht

1X(1 − τvt ) 1−α Av 

Ktα

(22)

v∈{c,d}v∈{c,d}By substituting (21) into (2), given that Xvt = Kvt for v ∈ {c, d}, we can determine the returnon human capital,αP
wt = (1 − α) "(1 − τvt ) 1−α Avv∈{c,d}#αP(1 − τvt )

1
1−α



Kt
Ht

α
(23)

Av

v∈{c,d}The government balanced budget constraint requires

X

α

X

⇐⇒

τvt pvt Xvt = 0v∈{c,d}τvt (1 − τvt ) 1−α Av = 0(24)v∈{c,d}Lemma 1. Under the government balanced budget constraint of the climate policy (τct , τdt ) ∈ <− ×[0, 1), the following relation holds
X1(1 − τvt ) 1−α Av =v∈{c,d}Xα(1 − τvt ) 1−α Avv∈{c,d}Proof.
Indeed, under government balanced budget condition (24) we haveX1(1 − τvt ) 1−α Av =v∈{c,d}X

1X(1 − τvt ) 1−α Av +v∈{c,d}v∈{c,d}19ατvt (1 − τvt ) 1−α Av =X

α(1 − τvt ) 1−α Avv∈{c,d}5

CONCLUDING REMARKS

Under the government balanced budget constraint (24), and from lemma 1, the return on humancapital (23) becomes

1−α
wt = (1 − α) X

(1 − τvt )

1
1−α



Av 

v∈{c,d}

Kt
Ht

α
(25)

A1. Proof of Proposition 1i(i) Indeed, the existence of optimal solution et is guaranteed by the compactness of the domainiiset, 0 ≤ et ≤ bt

∀i ∈ It , and continuity in eit of the objective function. When bit = 0, it is trivial
i
i
∗
i
i
that et = 0. When bt ∈ (0, et ], we suppose a contradiction that et < bt and prove that there existsε ∈ (0, bit − eit ) such thatwt h(eit + ε) + (bit − eit − ε)rt > wt h(eit ) + (bit − eit )rt
which is equivalent toh(eit + ε) − h(eit )
rt
>
ε
wt

⇐⇒

h(eit + ε) − h(eit ) > εh0 (e∗t )0 ∗since h (et ) = rt /wt . The last inequality trivially holds because: h(e) is an increasing and concavei∗iifunction of e, and et + ε < et . Hence, in this case et < bt is not the optimal education investment.
∗iii
Therefore, et = bt when bt ∈ [0, et ].

> e∗t , then the optimal education investment is set at the level that equalizes the
0 imarginal return of education investment and marginal capital income, i.e. wt h (et ) = rt , which∗igives us et = et .
i
When bt(ii) We haveh0 (et ) =α2 h(et )1 − α bt − et⇐⇒G(bt , et ) ≡ h0 (et ) −α2 h(et )=0
1 − α bt − e t(26)Under assumption 1,

Ge (bt , et ) = h00 (et ) −i.e.

α2 h0 (et )(bt − et ) + h(et )<01−α(bt − et )2G(bt , et ) is monotonically decreasing in et ∈ (0, bt ).

In addition,

lim G(bt , et ) = +∞ andet →0+

lim G(bt , et ) = −∞. Thus, there exists a unique et ∈ (0, bt ) satisfying (26).

et →+∞

By applying the implicit function theorem for function G(bt , et ) = 0, we have et is a function of

bt , i.e. et = e(bt ), in whiche0 (bt ) =α2 h(et )>0α2 [h0 (et )(bt − et ) + h(et )] − (1 − α)(bt − et )2 h00 (et )∗Now we prove that et ≥ et . In eect, From (20) and (25) which determine rt and wt respectively,

20

5

CONCLUDING REMARKS

we have0h (e∗t ) =Rh(eit )di
rt
α2 Ht
α2
It
R
=
=
wt
1 − α Kt
1 − α It (bit − eit )di∗We suppose a contradiction that et < et , henceZh(et ) >h(eit )diZ
and et >It

eit di since eit ≤ e∗t ∀i ∈ It .

It

Note that, since we normalize the size of the population by 1, et and h(et ) are respectively
aggregate education investment and human capital when each adult individuali ∈ It has an

education investment et . Thus, it would hold that0h (e∗t ) =Rh(ei )diα2 h(et )α2
R It i t i<= h0 (et )1 − α It (bt − et )di1 − α bt − et⇐⇒e∗t > et∗∗which contradicts the assumption et < et . Therefore, et ≥ et .

A2. Proof of Proposition 2
i∗In eect, when bt ∈ (0, et ), under assumptions 2 and 3, the disposable wealth of individual i ∈ Itiis φ(Pt )wt h(bt ). From (15), we havebit+1 > (=) 0⇐⇒φ(Pt )wt h(bit ) > (≤)θwt φ(Pt )(1 − γ)γ⇐⇒h(bit ) > (≤)θ(1 − γ)γ

Under assumption 1, the last inequalities are equivalent to

bit > (≤) h−1θ(1 − γ)γ≡bUnder assumption 4, we haveh(b) < h(e(λk̂))=⇒h(b) < h(e∗t ) since e∗t ≥ e(λk̂)∗
Therefore, b ∈ (0, et ).

A3. Proof of Proposition 3
We have in any period t, the aggregate bequest bt =R

bi di is given. From equations (20) andIt t(25) determining, respectively, the rental rate of physical capital and the return on human capital
under climate policy (τct , τdt ) ∈ Ct , we havert (τct , τdt )α2 Ht (τct , τdt )=wt (τct , τdt )1 − α Kt (τct , τdt )21

5CONCLUDING REMARKS

where rt (τct , τdt ) is the rental rate of physical capital in period t under climate policy (τct , τdt ) ∈ Ct ;and the analogous logic applied for other variables in period t. We suppose a negation that

Ht (τct , τdt )Ht (0, 0)<
Kt (τct , τdt )Kt (0, 0)for some (τct , τdt ) ∈ Ctthene∗t (τct , τdt ) > e∗t (0, 0)
Note that we prove in proposition 1 thateit (τct , τdt ) = min {bit , e∗t (τct , τdt )}, hence, the lastinequality implies that



∗
i
i
i


 et (τct , τdt ) = et (0, 0) if bt ∈ [0, et (0, 0)] ei (τct , τdt ) > ei (0, 0) if
t
t

=⇒







Ht (τct , τdt ) =RIt

bit > e∗t (0, 0)h(eit (τct , τdt ))di >RIth(eit (0, 0))di = Ht (0, 0)



R
R

 Kt (τct , τdt ) = bt − ei (τct , τdt )di < bt − ei (0, 0)di = Kt (0, 0)
t
It
It t
which leads to a contradiction that Ht (τct , τdt )/Kt (τct , τdt ) > Ht (0, 0)/Kt (0, 0).

Analogous logic

can be applied to the case of supposing a negation that Ht (τct , τdt )/Kt (τct , τdt ) > Ht (0, 0)/Kt (0, 0).
Therefore, it must hold that

Ht (τct , τdt )Ht (0, 0)=
Kt (τct , τdt )Kt (0, 0)∀(τct , τdt ) ∈ Ct

which, along with proposition 1, trivially gives us for any climate policy (τct , τdt ) ∈ Ct that

Kt (τct , τdt ) = Kt (0, 0),eit (τct , τdt ) = eit (0, 0) and hit (τct , τdt ) = hit (0, 0) ∀i ∈ It .

A4. Proof of Proposition 4
Indeed, under climate policy (τct , τdt ) ∈ Ct the allocation of physical capital Kvt (τct , τdt ), as shownin (21), is1

Kvt (τct , τdt ) =(1 − τvt ) 1−α Av
Kt 6=
P
1
(1 − τv0 t ) 1−α Av0v 0 ∈{c,d}

A
Pv

Av0

Kt = Kvt (0, 0);v ∈ {c, d}v 0 ∈{c,d}for given aggregate physical capital Kt .
By solving the following optimization problem, which maximizes the nal output in any period

t,

22

5

max

{Kvt }vj∈{c,d}×Jt

Ht1−α

X

α
Av1−α Kvt

X

subject tov∈{c,d}CONCLUDING REMARKS

Kvt = Ktv∈{c,d}∗given Kt and Ht ,9 we nd that the allocation rule Kvt = Kvt (0, 0) is the unique optimal allocationof capital.
By substituting the physical capital allocation rules Kvt (τct , τdt ) under (τct , τdt ) ∈ Ct and Kvt (0, 0)
into the nal good production function, along with the government balanced budget constraint,
as mentioned in lemma 1, we have1−α

Yt (τct , τdt ) = Ht

X

1

1−α


X

Ktα < Ht

(1 − τvt ) 1−α Av 

Av 

Ktα = Yt (0, 0)v∈{c,d}v∈{c,d}Hence, with (τct , τdt ) ∈ Ct , it holds that

X

1

X

(1 − τvt ) 1−α Av <v∈{c,d}Av

v∈{c,d}Now, it is quite straightforward from equations (20), (22) and (25), which determine, respectively, the rental rate of physical capital, the aggregate monopolistic prot and the return on humancapital, and the nal good production, that, with balanced budget climate policy (τct , τdt ) ∈ Ct ,they hold

 P
1−α
1
(1 − τvt ) 1−α Avwt (τct , τdt )Πt (τct , τdt )Yt (τct , τdt ) rt (τct , τdt )v∈{c,d}

P
=
=
=
=
< 1.


rt (0, 0)
wt (0, 0)
Πt (0, 0)
Yt (0, 0)Avv∈{c,d}A5. Proof of Proposition 5(i) The proof for this statement is fairly straightforward from equation (17) when we evaluate theiibequest ratio bt+1 (τct , τdt )/bt+1 (0, 0) in comparison with 1. Indeed,

bit+1 (τct , τdt )> (=)(<) 1
bit+1 (0, 0)
if, and only if

 P


1
(1 − τvt ) 1−α Av v∈{c,d} Ad+ξ PP(1 − α) ln 
−



Av
Av
v∈{c,d}v∈{c,d}1(1 − τdt ) 1−α Ad 
 Kt > (=)(<) 0
P
1

1−α
(1 − τvt ) Avv∈{c,d}9 Note that in the nal good production function, we replace X by K because of the production function X= K in theintermediate sector v ∈ {c, d}.
vt

23

vt

vt

vt

5

CONCLUDING REMARKS

which is equivalent to

−1



P


ln 
 P

v∈{c,d}1(1 − τdt ) 1−α Ad P1(1 − τvt ) 1−α Av

Ad
1−α
 P
−
Kt > (=)(<)
ξAv
v∈{c,d}v∈{c,d}Av(1 − τvt )

1
1−α

Av


>0


v∈{c,d}αP

Using the balanced government budget conditionτvt (1 − τvt ) 1−α Av = 0 as mentioned inv∈{c,d}(24) and substituting it into the right-hand side of the last inequality, we obtain K̂t (τct , τdt ).
We have the balanced budget conditionαατct (1 − τct ) 1−α Ac + τdt (1 − τdt ) 1−α Ad = 0
By applying the implicit function theorem for the last equation with respect to τct and τdt , wehave τct as function of τdt , in whichlim τct = lim − τct = 0, andτdt →0+τdt →1∂τct
Ad=−τdt →0 ∂τdt
Ac
lim +

We have

−1





P


ln 
 P

v∈{c,d}

1

1−α
Ad
 P
K̂t (τct , τdt ) =
−

ξ
Av

(1 − τdt ) 1−α Ad 

P
1

1−α
(1 − τvt ) Avv∈{c,d}v∈{c,d}Av(1 − τvt )

1
1−α

Av





v∈{c,d}K̂t (τct , τdt ) > 0 for all (τct , τdt ) ∈ Ct , then we only need to prove that K̂t (τct , τdt ) isbounded from above. Suppose a negation that there existed (τct , τdt ) ∈ Ct such that K̂t (τct , τdt )!
PP
1
could approach +∞. We have the term ln
Av /(1 − τvt ) 1−α Av > 0, which is always
Sincev∈{c,d}bounded from above because 0 <P(1 − τvt )11−αv∈{c,d}PAv <v∈{c,d}Av < +∞. Hence, for K̂t (τct , τdt ) tov∈{c,d}1

AdP

approach +∞, the only possibility is that(1−τdt ) 1−α Adapproaches 0 from the right.
1P(1−τvt ) 1−α Av−
Avv∈{c,d}v∈{c,d}This possibility occurs only if τdt approaches 0.10 However,τdt → 0

=⇒



P


ln 
 P

v∈{c,d}Av(1 − τvt )

1
1−α

Av


→0


v∈{c,d}It is straightforward to show that∀(τ , τ ) ∈ C , which trivially holds.
10

ct

dt

1

Adt
P
v∈{c,d}Avt−(1−τdt ) 1−α Adt
1P(1−τvt ) 1−α Avtv∈{c,d}t24> 0 ∀(τct , τdt ) ∈ Ct

. Because it is equivalent to

1−τct
1−τdt

>1

6

Therefore, by applying l'Hopital rule, and noting thath1−αlim + K̂t (τct , τdt ) =lim
τdt →0ξ τdt →0+ Addlim ∂τct = − A, we haveAcτdt →0+ ∂τdt

∂τct
Ac
Ad + ∂τdt

P

REFERENCES

i P

Av

v∈{c,d}

i =0
h
∂τct
A
Av − Ad Ad + ∂τcdtv∈{c,d}That is to say, K̂t (τct , τdt ) is bounded.

Declarations
Funding: This research had been conducted under the nancial support from the KAKENHI Startup Grantin-Aid for Research Activity, Japan Society for the Promotion of Science (20K22121).

Conicts of interest: There are no conicts of interest.
Competing interests: There are no competing interests.
Availability of data and material: There is no data used for this research.
Author's contributions: The author set up the theoretical model, conducted analyses, and wrote the wholemanuscript for this research.

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