Discussion Papers In Economics
And BusinessTransition from a Linear Economy toward a Circular Economy
in the Ramsey Model

Kiyoka Akimoto and Koichi Futagami

Discussion Paper 18-09

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANTransition from a Linear Economy toward a Circular Economy
in the Ramsey Model

Kiyoka Akimoto and Koichi Futagami

Discussion Paper 18-09

April 2018

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANTransition from a Linear Economy toward a Circular Economy
in the Ramsey Model∗
Kiyoka Akimoto†

Koichi Futagami‡

Abstract
We construct a Ramsey-type model where households recycle waste generated by consumption and the non-recycled waste has negative externality. The aim of this paper isthe following two points. Firstly, we examine a structural change process from “a linear
economy” based on consumption-disposal toward “a sound material-cycle economy” or “acircular economy” based on consumption-recycling. Secondly, we examine dynamics of optimal consumption tax and recycling subsidy. Additionally, we discuss the Environmental
Kuznets Curve (EKC). Previous theoretical literature explains the mechanism of the EKC
through changes in production sectors such as an introduction of abatement technology
or technological change. In contrast, this paper tries to explain the EKC through theaforementioned structural change process.

Keywords: Economic growth; Environmental Kuznets Curve; Environmental policies; Household waste recycling; Ramsey Model.
JEL Classification: O44; Q53; Q58.

∗

We would like to thank Martin Neve for his helpful discussions and suggestions, as well as the participants ofthe 2017 Asian Meeting of the Econometric Society, the 16th Journées Louis-André Gérard-Varet - International
Conference in Public Economics, the 23rd Annual Conference of the European Association of Environmental
and Resource Economists, the 18th Meeting of the Association for Public Economic Theory, and the 2017
Autumn Meeting of the Japanese Economic Association. Akimoto gratefully acknowledges the financial support
from the Grant-in-Aid for JSPS Fellows (Grant No. JP17J05419). Any errors are our responsibility.
†
Corresponding author; Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka,
Osaka 560-0043, JAPAN; e-mail: akimoto1015@gmail.com.
‡
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, JAPAN;
e-mail: futagami@econ.osaka-u.ac.jp

1

Introduction

The concept of a circular economy has been developed since the 1960s; for example, see Boulding (1966), Georgescu-Roegen (1971), and Pearce and Turner (1990). The circular structure
is comprised of production, consumption, and recycling. In this structure, recycling plays animportant role. Through recycling activities, used materials and resources are converted into
input factors for production. Consequently, the value of products, materials and resources
are maintained for as long as possible. In contrast, a linear economy is based on production,
consumption, and disposal. In the linear economy, only a small part of used materials andwaste is recycled, and a majority is thrown away. This causes great negative impacts on
the environment in the economy. Therefore, many countries have tried to transit toward a
circular economy despite the linear economy having been widespread around the world for along time.
According to Heshmati (2016), Germany, Japan, and China have especially developed
policy strategies related to the circular economy. Germany became the frontrunner of implementing the circular economy when the Closed Substance Cycle and Waste Management
Act was enacted in September of 1996. This law promoted closed cycle waste management toensure environmentally compatible disposal of waste. In Japan, the Basic Act on Establishinga Sound Material-Cycle Society was enacted in 2000 for forming a sound material-cycle economy to reduce environmental burdens. Additionally, Japan set forth the Fundamental Planfor Establishing a Sound Material-Cycle Society in 2003, which is reviewed every five years.
As a result, the recycling rate of municipal waste has risen; 5.3% in 1990, 14.3% in 2000,
and 20.6% in 2013. China introduced a number of laws and policies; the Cleaner Production
Promotion Law in 2003, the Law on Prevention and Control of Environmental Pollution by
Solid Waste in 2005, and the Circular Economy Promotion Law in 2009. Additionally, the European Union presented the “Closing the loop―An EU action plan for the circular economy”(the Circular Economy Package) in 2015. The Circular Economy Package includes proposals
such as recycling 65% of municipal waste and recycling 75% of packaging waste by 2030.
The aim of this paper is the following two points: (i) to examine the structural change
process from a linear economy based on consumption-disposal toward a circular economy

1based on consumption-recycling; and (ii) to examine dynamics of optimal consumption tax
and recycling subsidy. To analyze the transition path toward a circular economy and the
dynamics of the optimal tax-subsidy policy, we construct a simple Ramsey type model. In
our model, households recycle waste generated by consumption. The waste that is not recycled
has negative externality on the households’ utility. In the production sector, there are two
types of goods. One is virgin goods produced by capital and labor. The other is recycled goods
produced by recycled waste and virgin goods. We note that virgin goods and recycled goods
are homogeneous. In this paper, a circular economy is an economy where households recycle asmuch as possible. Firms produce goods, the households consume the goods, and they recycle
the waste. Then, the firms produce goods by using the recycled waste; that is, an economy
based on consumption-recycling. A linear economy is an economy where the households only
recycle a fraction of their waste. Firms produce goods, the households consume the goods,
and then they dispose of the waste; that is, an economy based on consumption-disposal.
Firstly, we examine the market equilibrium path and the social optimal path. The market
equilibrium path is as follows. An economy where the initial level of capital stock is low
starts as a circular economy. As capital accumulation progresses, the economic structurechanges from a circular economy to a linear economy. The linear economy approaches thesteady state. Hence, the recycling level is low and the amount of non-recycled waste is high.
On the other hand, in the social optimal path, through capital accumulation, the structure
can change from a circular economy to a linear economy, and at last to a circular economy.
Then, the circular economy is established in the steady state where recycling level is high andthen the amount of non-recycled waste is low. The diﬀerence between the market equilibrium
and the social optimum is due to externality in the market economy; that is, the households
do not take into account the negative externality of non-recycled waste. Because of thisexternality, recycling level remains low in the market economy even if capital accumulates
and consumption increases. Consequently, the market equilibrium path approaches a linear
economy. Secondly, we investigate the optimal tax-subsidy policy along the transition path.
We derive dynamics of the policy analytically. The dynamic policies require a government toincrease consumption tax and recycling subsidy along the transition path.
Furthermore, we discuss the Environmental Kuznets Curve (EKC). The curve postulates2

an inverted-U shaped relationship between pollution and per capita capital (see Grossman andKrueger (1991)). The previous theoretical literature shows various mechanisms to explain the
EKC. See, for example, Lopez (1994) and Andreoni and Levinson (2001) for static models,and John and Pecchenino (1994), Stokey (1998), Selden and Song (1995), Chimeli and Braden(2009), and Brock and Taylor (2010) for dynamic models. Mechanisms that most papers
provide are derived from production sectors such as technological progress from dirty to clean
technology, or increasing returns of abatement technology. In contrast with these studies, we
try to provide an explanation for the EKC through the structural change process of households’
recycling. In the social optimal path, that is, when the households take into account of thenegative externality by non-recycled waste, the relationship between the amount of nonrecycled waste and per capita capital becomes an inverted-U shape, like the EKC. At earlystages of economic growth, a circular economy is realized where consumption is small andrecycling is large, and thus the non-recycled waste is small. As capital accumulates, thestructure changes to a linear economy where consumption is large and recycling is low, andthus the non-recycled waste becomes large. Along the social optimal path, a circular economy
is realized again at a later stage. In this stage, consumption is large and the recycling level ishigh, thus decreasing non-recycled waste.
Economic literature has constructed dynamic models with respect to recycling (e.g., Smith(1972), Lusky (1976), Highfill and McAsey (1997, 2001), Huhtala (1999), Di Vita (2004, 2007),
André and Cerdá (2005, 2006), Pittel et al. (2010), and George et al. (2015))1 . However,most studies exclude economic development such as capital accumulation. The exceptions are
Di Vita (2004, 2007) and Pittel et al. (2010). While the stock of capital accumulates endogenously in their model, they focus on recycling by firms rather than households. Therefore, tothe best of our knowledge, this paper is the first study to show the transition of an economy
where households recycle waste. In addition, most previous models solve only social planner’s
problem because of diﬃculties that are caused by two or more stock variables such as pollution(or waste) and natural resources. In contrast, we treat pollution as a flow variable and then
we introduce the stock of capital into the model. This enables us to obtain not only the social
optimum but also the market equilibrium. By comparing among two solutions, our paper can
1

A few of related static models are Fullerton and Kinnaman (1995) and Fullerton and Wu (1998).

3

provide optimal dynamic policies.
The remainder of the paper is organized as follows. Section 2 considers the market economy and structural change path. Section 3 considers the social optimum and derives thetransition process toward the circular economy. Section 4 derives the dynamic schedule oftax-subsidy policy. Section 5 discusses the EKC through the transition path. Our conclusions
are summarized in Section 6.

2

Market economy

We consider a simple Ramsey-type model with recycling by households and negative externality by non-recycled waste. The population is constant over time and its size is normalized
to be unity. The economy consists of firms, households, and the government. In this section,
we consider the market economy and derive its equilibrium path.

2.1

Production sectors

There are two types of goods. One is virgin goods produced by capital and labor according
to the neoclassical production function per capita, f (kt ), where kt stands for the capitalstock per capita. The other is recycled goods produced by virgin goods and recycled waste
according to a production function. The production function of the recycled goods per capita
is represented by m(nt , et ), where nt and et stand for the virgin goods per capita used in the
production sector and recycled waste supplied by households, respectively. We assume thatm(nt , et ) is homogeneous with degree one. The characteristics of m(·) are as follows: mn > 0,
me > 0, mnn < 0, mee < 0, mne = men > 0, and mnn mee − (mne )2 > 0. We note that thesegoods are homogeneous and its price is normalized to 1.
Profit maximization of firms that produce the virgin goods yields:f ′ (kt ) = rt ,(1)

f (kt ) − f ′ (kt )kt = wt .

(2)

rt and wt represent the interest rate and wage rate, respectively. The firms that produce
4recycled goods maximize the profit, m(nt , et ) − nt − pt et , where the price of virgin goods is 1
and the price of recycled waste is pt . It yields:
mn (nt , et ) = 1,

(3)

me (nt , et ) = pt .

(4)

Since m(nt , et ) is a homogeneous function with degree one, mn (nt , et ) is homogeneous with
degree zero. Then, nt /et is determined from (3), and furthermore, nt is linear in et suchthat nt = bet where b > 0 is a constant parameter. Additionally, from (4), pt is constant.
We denote pt = p. The level of p depends on the function m(nt , et ). We assume thatmn (nt , et ) > me (nt , et ). The assumption ensures 1 > p, which means that the price of virgin
goods is higher than that of recycled waste.

2.2

Household

The representative household consumes goods2 , generates waste, and recycles the waste. Recycling activities impose psychological costs on the household because it is troublesome forthe household rather than disposing of the waste. The recycled waste can be sold to firms producing recycled goods at the price p. On the other hand, non-recycled waste is just disposedof and negatively aﬀects the utility of the household. Then, the preference of the household
is expressed as follows:∫ ∞U [0] =e−ρt [u(ct ) − v(et ) − z(ot )]dt,(5)0where u(·), v(·), and z(·) represent the instantaneous utility functions of consumption ct ,recycling et , and the non-recycled waste ot , respectively, and ρ is the rate of time preference.
We note that ot = ct − et > 0. The instantaneous utility functions, u(·), v(·), and z(·)
are twice diﬀerentiable and satisfy the following conditions, respectively: u′ > 0, u′′ < 0,limct →0 u′ = ∞, limct →∞ u′ = 0, v ′ > 0, v ′′ > 0, limet →0 v ′ = 0, z ′ > 0, z ′′ > 0, and
2
The goods the household consumes are the virgin goods and the recycled goods. The assumption thatthese goods are homogeneous makes our analysis simple.

5

limot →0 z ′ = 0.
The budget constraint isk̇t = rt kt + wt + (1 + τte )pet − (1 + τtc )ct + Gt .

(6)

The first term in the right hand side (RHS) of (6) is interest income from capital, the second iswage income, the third is income from selling the recycled waste, and the forth is consumption
expenditure, where τtc and τte represent consumption tax and recycling subsidy, respectively.
Additionally, Gt , the fifth term in the RHS, represents income transfer or lump-sum tax by
the government.
Gt = τtc ct − τte pet ,(7)

We assume that there is an upper limit on the recycling waste:et ≤ γct .

(8)The constant parameter γ ∈ (0, 1] can be regarded as the technology level of recycling in an
economy3 . If γ is small, the household recycles only a small amount of waste because thetechnology in the economy cannot convert the waste into materials for production.
In the market economy, the representative household disregards the externality of nonrecycled waste, z(ot ). Then, the household maximizes her lifelong utility:∫ ∞Û [0] =e−ρt [u(ct ) − v(et )]dt,(9)0

subject to the budget constraint (6) and the technological limitation on recycling (8). To
solve the maximization problem, we define the current value Hamiltonian as follows:H ≡ u(ct ) − v(et ) + λt [rt kt + wt + (1 + τte )pet − (1 + τtc )ct + Gt ] + µt (γct − et ).

(10)3
The purpose of this paper is to examine recycling activities by households. Thus, we omit the case ofγ = 0.

6

The first order conditions, the Kuhn-Tucker condition, and the transversality condition are:u′ (ct ) − (1 + τtc )λt + γµt = 0,

(11a)−v ′ (et ) + (1 + τte )pλt − µt = 0,

(11b)

rt λt = −λ̇t + ρλt ,(11c)

rt kt + wt + (1 + τte )pet − (1 + τtc )ct = k̇t ,µt (γct − et ) = 0,
lim e−ρt λt kt = 0.

t→∞

(11d)
γct − et ≥ 0,µt ≥ 0,

(11e)
(11f)

We consider the following two cases. One is that the level of the household’s recycling is lessthan the maximum level; that is, et < γct . The other is that the household recycles as muchas possible; that is, et = γct . We call the former a linear economy and the latter a circular
economy. A linear economy consists of a production-consumption-disposal structure. Mostof the waste generated by consumption is not repossessed for production since the household
recycles only a small amount of waste. On the other hand, a circular economy is based on a
production-consumption-recycling structure. In a circular economy, a high amount of recycled
waste is reworked as the input factor for production. Therefore, the materials circulate throughthe recycling activity. In the following two subsections, we consider the dynamics of these
economies, respectively.
Additionally, we assume that the government does not take any tax-subsidy policy in thissection. That is, τtc = τte = 0. Section 4 derives the dynamic schedule of the tax-subsidy
policy.

2.3

A linear market economy

We derive the dynamic equations of the linear economy in the market competitive equilibrium.
In the linear economy, et < γct . Substituting µt = 0 into FOCs, we rewrite (11a) and (11b)7

as follows:u′ (ct ) − λt = 0,

(12a)−v ′ (et ) + pλt = 0.

(12b)Firstly, from (11c) and (12a), we obtain the Euler equation:ċt = σlm (ct )(rt − ρ),u′ (ct )where σlm (ct ) ≡ − ′′> 0.
u (ct )

(13)

Next, (12a) and (12b) yield the ratio of marginal utility from consumption and recycling:v ′ (et )= p.
u′ (ct )(14)

By solving this equation for recycling, we can show that:et = eml (ct ),

(15)

where the subscript, l, and superscript, m, represent the linear structure in the market economy. The derivative is given by:detpu′′ (ct )
= ′′
< 0.
dct
v (et )

(16)

Thus, the level of recycling is decreasing in ct . Furthermore, eml (ct ) has the following characters:
lim et = ∞,lim et = 0.

ct →∞

ct →0

Figure 1 shows the relationship between recycling and consumption in the linear economy.
v ′ (et ) = pu′ (ct ) obtained from equation (14) provides an intuition of the shape of the graph′of eml (ct ). If the household increases her recycling level by one unit, she derives v (et ) unitsof disutility and gets income p. Thus, she can increase p units of consumption, that is, she

8

et
em
l (ct )

ct

O

Figure 1: Recycling and consumption in the linear-market economy.
derives pu′ (ct ) units of utility. When the level of consumption is still high, the marginalutility of consumption by increasing the recycling level is low. Thus, eml (ct ) is the decreasing
function of ct . In other words, recycling to yield income and increase consumption becomesless attractive for the household as consumption level increases.
Substituting (1), (2), and (15) into (6), we obtain the dynamic equation of the capitalstock per capita as follows:k̇t = f (kt ) + pem
l (ct ) − ct .

(17)

Equations (13) and (17) constitute the dynamic system of the linear market economy.

2.4

A circular market economy

We derive the dynamic equations of the circular economy where the household recycles asmuch as possible:emc (ct ) = γct .

9From (11a), (11b), and τtc = τte = 0, we obtain λt and µt as follows4 :λt =u′ (ct ) − γv ′ (γct ),1 − γpµt =pu′ (ct ) − v ′ (γct ).
1 − γp(18)µt > 0 requires that pu′ (ct ) − v ′ (et ) > 0 holds true. Thus, we assume that our analysis isconducted in the interval ct ∈ [0, c̄] and pu′ (ct ) − v ′ (et ) > 0 holds true for all ct ∈ [0, c̄]5 . Using(11a), (11b), (11c), and (18) yields the dynamic equation of consumption:ċt = σcm (ct )(rt − ρ) whereu′ (ct ) − γv ′ (γct )σcm (ct ) ≡ − ′′> 0.
u (ct ) − γ 2 v ′′ (γct )(19)

See Appendix A for derivation of equation (19).
From the equations, et = emc (ct ) = γct , (1), (2), and (6), we obtain the dynamic equationof capital:k̇t = f (kt ) − ct (1 − γp).

(20)Equations (19) and (20) constitute the dynamic system of the circular market economy.

2.5

Transition of the market economy and steady stateThis subsection examines the transition of the market economy and the steady state. Firstly,
we derive the ċt = 0 locus and the k̇t = 0 locus. The dynamics of consumption (13) in thelinear economy and (19) in the circular economy yield the identical ċt = 0 locus:f ′ (kt ) = ρ.

(21)

Next, to derive the k̇t = 0 locus in the market economy, we consider the cutoﬀ at which
the economic structure changes. Figure 2 shows et in the linear economy and in the circular
Since 0 < γ < 1 and p < 1, 1 − γp > 0.
For example, we can define c̄ as follows. If we introduce the depreciation rate of capital δ, the shape ofk̇t = 0 locus becomes an inverted-U shape. It yields the maximum stock level of per capita capital k̄. Then,
we obtain the maximum level of consumption which is ct = f (k̄) + m(nt , et ) − nt + (1 − δ)k̄. This implies thatthe household consumes all resources. We can represent this maximum level of consumption as c̄. Then, theanalysis is conducted in the interval 0 ≤ ct ≤ c̄.
4

5

10

et
em
l (ct )emc (ct )em (ct )

O

ct

c̃m

Figure 2: The cutoﬀ of economic structural change.
economy. Since et must satisfy et ≤ γct , the feasible level of et is represented by the solid line:em (ct ) =




γc

if ct ≤ c̃m ,

t



 em
l (ct )

if

ct

(22)

> c̃m .

Then, we denote the cutoﬀ c̃m that satisfies:
m
m
m m
em
l (c̃ ) = γc̃ = ec (c̃ ).

If the level of consumption in period t is less than c̃m , the structure of the economy is a circular
economy where the household recycles waste as much as possible, et = γct . If the consumption
level exceeds the cutoﬀ, the economic structure changes from a circular economy to a linear
economy. Thus, the level of recycling is decreasing in a linear economy. Additionally, at the
cutoﬀ c̃m , the k̇t = 0 locus of each structure crosses. We can write the k̇t = 0 locus in the11ctċt = 0k̇t = 0
c∗
c̃m

Em

ĉm
A
O

kt

k̃ m k ∗k0

Figure 3: The transition and steady state in the market economy.
economy from (17) and (20):f (kt )
,
ct = 1 − γpf (kt ) + pem (ct ),lif

ct ≤ c̃m ,if

ct > c̃m .

(23)Note that ct = f (kt ) + pem
l (ct ) does not go through the origin, while ct = f (kt )/(1 − γp) does.
m
When kt = 0, we denote ĉm that satisfies ĉm = peml (ĉ ).

Secondly, by drawing the ċt = 0 locus (21) and the k̇t = 0 locus (23) as in Figure 3, we
examine the characteristics of the structural change and the steady state. Thus, we focus on
the path on which structural change occurs6 . The steady state E m can exist above the borderline ct = c̃m . Then, we know that the structure of the economy is the linear economy where
the household disposes of a large share of waste. According to (21), (22) and (23), we obtainthe steady state levels of the variables as follows:f ′ (k ∗ ) = ρ,∗c∗ = f (k ∗ ) + peml (c ),∗e∗ = e ml (c ).

Additionally, the economy satisfies the saddle-path stability. In order to get on the stable
6
If the ċt = 0 locus and the k̇t = 0 locus cross at a point under the cutoﬀ c̃m , structural change does notoccur and then the circular economy is realized in steady state.

12ct em (ct )ctotem (ct )

O

c̃m

ct

Figure 4: The non-recycled waste in the market economy.
saddle path toward the steady state E m , the economy where the initial stock of per capita
capital is k0 starts at point A. Along the saddle path, the structural change occurs as follows.
When the level of capital stock is low, the structure of the economy is the circular economy
based on consumption and recycling in the interval k0 ≤ kt ≤ k̃ m . As capital accumulation
proceeds, the structure changes from a circular economy to a linear economy. Then, the linear
economy is realized in the interval k̃ m < kt ≤ k ∗ . Thus, the structure in the steady state E m
is a linear system. Along the path, the market economy cannot realize the circular economy
in the steady state E m .
The mechanism of this structural change process is as follows. At an early stage of economic growth, output is small and the level of consumption is low. Since marginal utility of
consumption is high, the household recycles a large amount of waste to earn income from recycling and to increase consumption. However, at a later stage, the household is able to engage
in high levels of consumption because there exists enough capital stock. Thus, the household’s
incentive to increase consumption levels by recycling decreases. Hence, the structure changes
from circular to linear.
Lastly, we examine the amount of non-recycled waste along the saddle path. Figure 4 shows
the relationship between consumption, recycling, and non-recycled waste. If the structure of
the economy is a circular economy where consumption is low and recycling is high, the amount13

of non-recycled waste is small. On the other hand, if the level of consumption exceeds the
cutoﬀ, a linear economy is realized where the amount of non-recycled waste is large. Therefore,
the non-recycled waste is increasing along with consumption at later stages.

3

Social optimum

In this section, we consider the socially optimal allocation where the social planner decides
the resource allocation. Taking account of negative externality from non-recycled waste, thesocial planner maximizes (5). The constraints for the social planner include the limitation on
recycling (8) and the resource constraint:k̇t = f (kt ) + m(nt , et ) − nt − ct .

(24)

To solve the maximization problem, we define the current value Hamiltonian as follows:H ≡ u(ct ) − v(et ) − z(ot ) + λt [f (kt ) + m(nt , et ) − nt − ct ] + µt (γct − et ).

(25)

Diﬀerentiating (25) with respect ct , et , nt , kt , and λt , we obtain FOCs:u′ (ct ) − z ′ (ot ) − λt + γµt = 0,(26a)−v ′ (et ) + z ′ (ot ) + λt me (nt , et ) − µt = 0,(26b)λt [mn (nt , et ) − 1] = 0,(26c)f ′ (kt )λt = −λ̇t + ρλt ,f (kt ) + m(nt , et ) − nt − ct = k̇t .

(26d)
(26e)

In the following two subsections, we consider the dynamics of variables in respective economic structures.

14

3.1

A linear social optimum

We derive the dynamic equations of the linear economy in the social optimum. Substitutingµt = 0 into FOCs derived above, we rewrite (26a) and (26b) as follows:u′ (ct ) − z ′ (ot ) − λt = 0,(27a)−v ′ (et ) + z ′ (ot ) + λt me (nt , et ) = 0.

(27b)Here, we assume that u′ (ct ) − z ′ (ot ) > 0 holds true for all ct ∈ [0, c̄] to obtain λt > 0. λt > 0
yields mn (nt , et ) = 1 from equation (26c). Then, me (nt , et ) becomes constant. We denoteme (nt , et ) = m̄e . Additionally, we note that m̄e = p.
Before deriving the dynamic equations of variables, we consider the relationship between
recycling and consumption in the linear social optimum. From (27a) and (27b), the ratio of
marginal utility from consumption and recycling is given by:v ′ (et ) − z ′ (ot )= m̄e .
u′ (ct ) − z ′ (ot )(28)Compared with equation (14) in the market economy, z ′ (ot ) appears in equation (28) because
the social planner takes into account the external eﬀects of non-recycled waste. Solving theequation for recycling, we obtain:
et = esl (ct ),where the derivative is:detm̄e u′′ (ct ) + z ′′ (ot )(1 − m̄e )=.
dctv ′′ (et ) + z ′′ (ot )(1 − m̄e )While the sign of det /dct is ambiguous, esl (ct ) has the following characters:
lim et = ∞,ct →0lim et = ∞.

ct →∞

15

(29)

et

esl (ct )ctOFigure 5: Recycling and consumption in the linear-social optimum.
Here, we assume that a unique and constant c exists which satisfies:det= 0.
dct ct =c
Additionally, we impose the following assumption that as ct goes to infinity, the slope of thefunction esl (ct ) is larger than γ:
lim

det

ct →∞ dct> γ.

Under these assumptions, the relationship between et and ct becomes U-shaped, as depicted
in Figure 5.
Next, we derive the dynamic equations of consumption, recycling, and capital. Firstly, we
can show the dynamic equation of consumption:ċt = σls (ct )[f ′ (kt ) − ρ],where

σls (ct ) ≡ −(30)[u′ (ct ) − v ′ (esl (ct ))]z ′′ (ot ) + [u′ (ct ) − z ′ (ot )]v ′′ (esl (ct ))> 0.
[u′′ (ct ) − v ′′ (esl (ct ))]z ′′ (ot ) + u′′ (ct )v ′′ (esl (ct ))16

Secondly, the dynamic equation of recycling is given by:ėt = σ̂ls (ct )[f ′ (kt ) − ρ],
whereσ̂ls (ct ) ≡ −(31)[u′ (ct ) − v ′ (esl (ct ))]z ′′ (ot ) + [v ′ (esl (ct )) − z ′ (ot )]u′′ (ct ).
[u′′ (ct ) − v ′′ (esl (ct ))]z ′′ (ot ) + u′′ (ct )v ′′ (esl (ct ))See Appendix B for derivations of equations (30) and (31).
Lastly, the resource constraint (24) yields the dynamic equation of capital:k̇t = f (kt ) + m̄e esl (ct ) − ct .

(32)

Equations (30) and (32) constitute the dynamic system of the linear economy in the social
optimum.

3.2

A circular social optimum

We derive the dynamic equations of the circular economy in the social optimum. From (26a),(26b) and et = γct , we obtain µt > 0. It yields me (nt , et ) = m̄e . Furthermore, we can calculate
λt and µt :u′ (ct ) − z ′ (ot ) − γ[v ′ (γct ) − z ′ (ot )],1 − γ m̄em̄e [u′ (ct ) − z ′ (ot )] − [v ′ (γct ) − z ′ (ot )].
µt =1 − γ m̄e
λt =

(33a)
(33b)

Firstly, we can show the dynamic equation of consumption as follows:ċt = σcs (ct )[f ′ (kt ) − ρ],
whereu′ (ct ) − γv ′ (γct ) − (1 − γ)z ′ (ot )> 0.
σcs (ct ) ≡ − ′′u (ct ) − γ 2 v ′′ (γct ) − (1 − γ)2 z ′′ (ot )See Appendix C for derivation of equation (34).

17

(34)

et
esl (ct )esc (ct )es (ct )

O

c̃s1

c̃s2

ct

Figure 6: The cutoﬀs in the social optimum.
Secondly, we obtain the dynamic equation of capital:k̇t = f (kt ) − ct (1 − γ m̄e ).

(35)

Thus, dynamic equations (34) and (35) constitute the dynamic system of the circular economy
under the social optimum.

3.3

Transition toward a circular economy and steady state

We derive cutoﬀs of economic structure in the social optimum. Figure 6 shows the relationshipbetween et and ct in each structures. Thus, there exist two cutoﬀs; c̃s1 and c̃s2 . If the levelof consumption is less than c̃s1 , the economic structure is a circular system. If consumption
is between c̃s1 and c̃s2 , a linear economy is realized. Finally, if the economy exceeds the cutoﬀ
of c̃s2 , the structure changes from a linear system to a circular system. Thus, the level of18

recycling in social optimum is given by:




γct



s
e (ct ) = es (ct )
l






γct

if ct ≤ c̃s1 ,if c̃s1 < ct < c̃s2 ,(36)

if c̃s2 ≤ ct .

From (30) and (34), the dynamics of consumption in each structures yields the identicalċt = 0 locus under the social optimum:f ′ (kt ) = ρ(37)Lastly, from (32) and (35), we can draw the k̇t = 0 locus:f (kt )1 − γ m̄e
ct = f (kt ) + m̄e es (ct )
l






f (kt )
1 − γ m̄e

if ct ≤ c̃s1 ,ifc̃s1 < ct < c̃s2 ,(38)

if c̃s2 ≤ ct .

When kt = 0, the equation ct = f (kt ) + m̄e esl (ct ) does not cross the origin. We denote ĉs thatsatisfies ĉs = m̄e esl (ĉs ).
Writing the ċt = 0 locus (37) and the k̇t = 0 locus (38) as in Figure 7, we examine thetransition of social optimum. Here, we also focus on the path toward a circular economy. Then,above the cutoﬀ c̃s2 , there can exist the steady state E s where the circular economy is realized7 .
Thus, according to (36), (37) and (38), the steady state levels of capital, consumption andrecycling are:f ′ (k ∗ ) = ρ,c∗ =f (k ∗ ),1 − γ m̄ee∗ = γc∗ .

There can be other two cases. The one is that a steady state exists within the range between c̃s1 and c̃s2 . In
this case, the structure of the economy changes from circular to linear and then the linear economy is realized
in the steady state. The other is that a steady state exists under the cutoﬀ c̃s1 , in which case the structuralchange does not occur and the economy maintains the circular structure in the steady state.
7

19

ct
ċt = 0
c∗
c̃s2

Es

k̇t = 0

c̃s1
ĉs

O

B

k0

k̃2s k ∗k̃1skt

Figure 7: The transition and steady state in the social optimum.
This economy satisfies the saddle-path stability. Thus, the economy with an initial capitalof k0 starts at point B and gets on the saddle path toward the steady state E s . The structuralchanges along the path are as follows. Firstly, if the level of capital stock is small, the structure
is a circular economy in the interval k0 ≤ kt ≤ k̃1s . Secondly, as capital accumulation proceeds
and the economy crosses the cutoﬀ, the structure changes from a circular economy to a linear
economy in the interval k̃1s < kt < k̃2s . Lastly, if capital accumulation is suﬃciently large, acircular economy is realized in the interval k̃2s ≤ kt ≤ k ∗ and reaches the steady state E s .
Along the structural change process, the level of recycling changes. Firstly, the recycling
increases until the first cutoﬀ c̃s1 . After crossing the first cutoﬀ, recycling decreases and thenincreases. However, the level of increase is still less than the maximum level of recycling.
Lastly, if the economy becomes a circular economy with large capital accumulation, the levelof recycling attains the maximum level.
Diﬀerences of structural changes between the market economy and the social optimum
are due to the externality of non-recycled waste. The incentive to increase consumption byrecycling activity is high at the early stage of economic growth and it becomes low afterwards. This low incentive results in the low level of recycling. Thus, the structure changes
from circular to linear. This mechanism is the same as in the market economy. In contrast
to the transition in the market economy, during the mid-to-later stages, the recycling level20

increases again. This is due to negative externality of non-recycled waste coming into play.
At these stages, there exists another incentive for recycling activities, which is to avoid thelarge negative eﬀects of non-recycled waste on utility. As the level of consumption increases,
negative externality grows larger if the recycling level remains low. Hence, the recycling level
increases again and the circular structure is realized in the social optimum.

4

Optimal tax-subsidy

In this section, the government determines the schedule for consumption tax and recycling
subsidy in order to make the market economy optimal. To obtain the dynamic equation of
tax and subsidy, we return to the model of the market economy and derive the dynamics ofvariables in each structure. Firstly, we consider the linear economy. The dynamic equationsof consumption and recycling are given by:
[

]
τ̇tc
(rt − ρ) −,1 + τtc[]τ̇teėt =σ̂lm (ct ) (rt − ρ) −,1 + τteċt =σlm (ct )

(39)
where

v ′ (em (ct ))σ̂lm (ct ) ≡ − ′′ lm.
v (el (ct ))(40)

See Appendix D for derivations of equations (39) and (40).
Next, we consider the circular economy. In the circular economy, the dynamics of recyclingis ėt = γ ċt and the dynamic equation of consumption is as follows:ċt = σcm (ct )[(rt − ρ) −]τ̇tc − γpτ̇te.
(1 + τtc ) − γp(1 + τte )(41)

See Appendix E for derivation of equation (41).
The government determines the dynamic schedule of τtc and τte by making the dynamics ofconsumption and recycling equal to the dynamics in the social optimum. Thus, in the linear
economy, from (30) and (39), the dynamics of τtc must satisfy:[
σlm (ct )]τ̇tc(f (kt ) − ρ) −= σls (ct )[f ′ (kt ) − ρ].
1 + τtc′

21

Then, we obtain the dynamic schedule of consumption tax as follows:τ̇tc =σlm (ct ) − σls (ct ) ′[f (kt ) − ρ](1 + τtc ).
σlm (ct )(42)

On the other hand, from (31) and (40), the recycling subsidy is determined to satisfy thefollowing equation:[σ̂lm (ct )]τ̇te(f (kt ) − ρ) −= σ̂ls (ct )[f ′ (kt ) − ρ],
1 + τte′and it yields the dynamic equation of recycling subsidy:τ̇te =

σ̂lm (ct ) − σ̂ls (ct ) ′[f (kt ) − ρ](1 + τte ).
σ̂lm (ct )(43)

Along the social optimum path, if the circular economy is realized, from (34) and (41),the government decides the dynamic schedule of tax and subsidy as follows:]τ̇tc − γpτ̇te= σcs (ct )[f ′ (kt ) − ρ](f (kt ) − ρ) −(1 + τtc ) − γp(1 + τte )σ m (ct ) − σcs (ct ) ′τ̇tc − γpτ̇te = c[f (kt ) − ρ][(1 + τtc ) − γp(1 + τte )].
σcm (ct )[σcm (ct )
⇔

′

(44)

We can show that (σlm − σls )/σlm > 0, (σ̂lm − σ̂ls )/σ̂lm > 0, and (σcm − σcs )/σcm > 0. See Appendix F. From equations (42) and (43), the government needs to increase both consumption
tax and recycling subsidy in an economy starting with a capital stock level of kt < k ∗ . The
sign of RHS in equation (44) depends on [(1 + τtc ) − γp(1 + τte )]. For simplicity, we assume that
only consumption tax is imposed, which is τte = 0 and τ̇te = 0. Then, we obtain τ̇tc > 0. Lastly,along the social optimal transition path, the government decides upon the tax-subsidy policy
as follows. Firstly, if the structure is the circular economy with small capital accumulation,
the optimal policy is given by (44). Secondly, if the circular economy changes to the linear
economy with the middle level of capital, (42) and (43) constitute the optimal policy. Finally,
if the circular economy is realized with large capital accumulation, it is given by (44).

22ct es (ct )ctes (ct )

ot

O

c̃s1

c̃s2

ct

Figure 8: The non-recycled waste in the optimal economy.

5

Discussion the Environmental Kuznets Curve

Here, we will discuss the Environmental Kuznets Curve (EKC). The EKC is the inverted-U
shaped relationship between environmental quality and economic development. One notable
empirical study is that of Grossman and Krueger (1991). Theoretical studies succeed in
explaining the EKC. See, for example, Lopez (1994), Andreoni and Levinson (2001), John
and Pecchenino (1994), Stokey (1998), Selden and Song (1995), Chimeli and Braden (2009),
and Brock and Taylor (2010). This paper also tries to explain the mechanism of this curvethrough the transitional path of optimal economy.
In this paper, the stock of capital per capita, kt , represents development. Along the transitional path of optimal allocation, the level of consumption increases as capital accumulation
proceeds. Thus, we examine the relationship between the non-recycled waste and the level ofconsumption. Firstly, we draw the level of recycling (36) and consumption in Figure 8. Then,the non-recycled waste in the optimum economy is given by:
ot = ct − es (ct ).
Figure 9 shows the relationship between the level of consumption and non-recycled waste.
The shape of this graph becomes an inverted-U shape, like the EKC. The inverted-U shaped

23

ot

O

c̃s1

c̃s2

ct

Figure 9: The Environmental Kuznets Curve.
relationship between economic development and non-recycled waste appears through the economic structural change process: the structure changes from circular to linear, and finallyback to circular. When the economy is less developed, the household recycles waste as muchas possible because the stock of capital is small. This results in a low amount of non-recycled
waste. As the economy develops, the level of recycling decreases and the non-recycled waste
increases. After the linear economy burdens the environment with a high amount of waste, thelevel of recycling rises because the negative externality becomes large. Finally, if the economy
realizes the circular structure, the household starts to recycle a large amount of waste anddispose less. Therefore, we can obtain the EKC along the optimum path.
While empirical and theoretical papers provide the inverted-U shape, some empirical studies exhibit the N-shaped relationship between pollution and development. For example, Grossman and Krueger (1995), De Bruyn and Opschoor (1997), and Sengupta (1996) provide empirical evidence of the N-shape. In contrast to large amounts of theoretical papers explaining
the inverted-U shape, only a few theoretical studies show the N-shaped relationship: Steger
and Egli (2007) and Alonso-Carrera et al. (2013). They analytically solve dynamic models andnumerically obtain the N-shape. On the other hand, we can show the N-shaped relationshipbetween the non-recycled waste and consumption analytically. If we extend the horizontal
line of Figure 9, the graph becomes N-shaped, as depicted in Figure 10. Thus, the inverted-U

24

ot

O

c̃s1

c̃s2

ct

Figure 10: The N-shaped relationship.
shape appears if the steady state level of per consumption c∗ is close to c̃s2 and the N-shapeappears if c∗ is far from c̃s2 . This implies that the shape can diﬀer among countries and thetime period that empirical analyses choose. This discussion is similar to Selden and Song(1994), Grossman and Krueger (1995), and Hill and Magnani (2002)8 .

6Conclusion

We constructed a Ramsey-type model with recycling activities by households. In the model,
the household recycles waste generated by consumption. The recycled waste is then repossessed for production. On the other hand, the non-recycled waste is disposed of and aﬀects
the household’s utility negatively. Using the model, we considered the transition path toward
a circular economy and the dynamic schedule of tax-subsidy policy.
Firstly, along the transition path of the market equilibrium, the structure of the economy
changes from a circular economy to a linear economy. Thus, the market economy cannot
realize the circular economy. Since the household ignores the negative externality from thenon-recycled waste, the household only recycles a small amount of waste. Secondly, in thesocial optimal path, the structure can change from a circular economy to a linear economy,and finally back to a circular economy. We showed the optimal tax-subsidy policy for the8

We note that Dinda (2004) and Kijima et al. (2010) give an overview on the EKC.

25

market economy to realize a circular economy. The policy raises the market equilibrium levelof recycling up to the social optimal level.

Appendix
A. The dynamics of the circular-competitive economy
Diﬀerentiating (11a) and (11b) with time yields:u′′ (ct )ċt − λ̇t + γ µ̇t = 0,

(A.1)−v ′′ (et )ėt + pλ̇t − µ̇t = 0.

(A.2)

By substituting (11c), (18), and (A.2) into (A.1) and using the equation ėt = γ ċt , we obtain
the dynamic equation of consumption (19):u′ (ct ) − γv ′ (γct )ċt = − ′′(rt − ρ).
u (ct ) − γ 2 v ′′ (γct )B. The dynamics of the linear-social optimal economy
Firstly, by taking the log of both sides of (27a) and (27b) and diﬀerentiating the equations
with time, we obtain:[]z ′′ (ot )u′ (ct ) − z ′ (ot ) λ̇t
ċt = ′′−ėt ,u (ct ) − z ′′ (ot ) λt u′ (ct ) − z ′ (ot )[]v ′ (et ) − z ′ (ot ) λ̇tz ′′ (ot )ėt = ′′+ċt .
v (et ) + z ′′ (ot ) λt v ′ (ct ) − z ′ (ot )

(A.3)
(A.4)

Substituting (26d) and (A.4) into (A.3) yields the dynamic equation of consumption (30):ċt = −[u′ (ct ) − v ′ (esl (ct ))]z ′′ (ot ) + [u′ (ct ) − z ′ (ot )]v ′′ (esl (ct )) ′[f (kt ) − ρ].
[u′′ (ct ) − v ′′ (esl (ct ))]z ′′ (ot ) + u′′ (ct )v ′′ (esl (ct ))On the other hand, by substituting (26d) and (A.3) into (A.4), the dynamic equation of26

recycling (31) is given by:ėt = −[u′ (ct ) − v ′ (esl (ct ))]z ′′ (ot ) + [v ′ (esl (ct )) − z ′ (ot )]u′′ (ct ) ′[f (kt ) − ρ].
[u′′ (ct ) − v ′′ (esl (ct ))]z ′′ (ot ) + u′′ (ct )v ′′ (esl (ct ))C. The dynamics of the circular-social optimal economy
Diﬀerentiating (26a) and (26b) with time yields:u′′ (ct )ċt − z ′′ (ċt − ėt ) − λ̇t + γ µ̇t = 0,

(A.5)−v ′′ (et )ėt + z ′′ (ċt − ėt ) + λ̇t me (nt , et ) − µ̇t = 0.

(A.6)

By substituting (26d), (33a), and (A.6) and using the equation ėt = γ ċt , we obtain the
dynamic equation of consumption (34):u′ (ct ) − γv ′ (γct ) − (1 − γ)z ′ (ot )[f ′ (kt ) − ρ].
ċt = − ′′u (ct ) − γ 2 v ′′ (γct ) − (1 − γ)2 z ′′ (ot )D. The dynamics of the linear economy with the tax-subsidy policy
We rewrite the FOCs, (11a) and (11b) since µt = 0 in the linear economy:u′ (ct ) − (1 + τtc )λt = 0,

(A.7)−v ′ (et ) + (1 + τte )pλt = 0.

(A.8)

By taking the log of both sides of (A.7) and (A.8) and diﬀerentiating with the equations withtime, we obtain:u′ (ct )ċt = ′′u (ct )v ′ (et )ėt = ′′v (et )

(
(

λ̇t
τ̇tc
+
λt 1 + τtc
λ̇t
τ̇te
+
λt 1 + τte

27

)
,
)
.

Substituting (11c) into these equations yields the dynamic equation of consumption and recycling with the tax-subsidy policy, (39) and (40), respectively:[]τ̇tcu′ (ct )′(f (kt ) − ρ) −ċt = − ′′,u (ct )
1 + τtc[]v ′ (em (ct ))τ̇te(f ′ (kt ) − ρ) −ėt = − ′′ lm.
v (el (ct ))1 + τteE. The dynamics of the circular economy with the tax-subsidy policy
By diﬀerentiating (11a) and (11b) with time, we obtain:u′′ (ct )ċt − λ̇t (1 + τtc ) − λt τ̇tc + γ µ̇t = 0,

(A.9)−v ′′ (et )ėt + λ̇t p(1 + τte ) − λt pτ̇te − µ̇t = 0.

(A.10)

Substituting (11c), (33a), and (A.10) into (A.9) and using the equation ėt = γ ċt yields the
dynamic equation of consumption with tax-subsidy policy (41):[]τ̇tc − γpτ̇teu′ (ct ) − γv ′ (γct )′(f (kt ) − ρ) −.
ċt = − ′′u (ct ) − γv ′′ (γct )(1 + τtc ) − γp(1 + τte )F. Motion of optimal policies
Firstly, from (13) and (30),σlm − σls =−u′ v ′′ z ′′ + u′′ v ′ z ′′ + u′′ z ′ v ′′.
−u′′ [(u′′ − v ′′ )z ′′ + u′′ v ′′ ]The denominator and numerator take positive values. Thus,
σlm (ct ) − σls (ct )> 0.
σlm (ct )Secondly, from (31) and (40),σ̂lm − σ̂ls =u′′ v ′ z ′′ − u′ v ′′ z ′′ + u′′ v ′′ z ′.
−v ′′ [(u′′ − v ′′ )z ′′ + u′′ v ′′ ]28

The denominator takes a positive value and the numerator takes a negative value. Thus,σ̂lm − σ̂ls < 0 andσ̂lm (ct ) − σ̂ls (ct )> 0.
σ̂lm (ct )Lastly, from (19) and (34),σcm − σcs =(1 − γ)2 z ′′ (−u′ + γv ′ ) + (1 − γ)u′′ z ′ − γ 2 (1 − γ)v ′′ z ′.
−(u′′ − γ 2 v ′′ )[u′′ − γ 2 v ′′ − (1 − γ)2 z ′′ ]The denominator and numerator take negative values. Thus,
σcm (ct ) − σcs (ct )> 0.
σcm (ct )29

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