Skill Formation and the Production of Environmental Goods:
The Role of Public Education

Fatt Seng CHONG* and Yasuyuki SUGIYAMA†

Abstract
We construct a model to examine the impacts of an increase in the skilled labor input topublic education on the output of final goods, environmental goods as well as the amountof pollution emissions, in which skill formation is incorporated into the traditional RV
model. Due to the skill formation, one of the specific factors and mobile factor which are
represented by skilled labor and unskilled labor are endogenously determined compared tothe traditional RV model. We then reveal that an increase in public education service may
bring negative effects to the output above due to the change in wages of skilled labor andunskilled labor as well as the rental rate of real capital which are caused by the skill
formation process. However, we also find some conditions for the role of public education
to ensure the policy to achieve the goals, in particular environmental protection.

Corresponding Author
Name: Yasuyuki Sugiyama
Affiliation: Faculty of Economics, Fukui Prefectural University
Address: 4-1-1 Matsuoka-Kenjojima, Eiheiji Town, Fukui Prefecture,910-1195, JAPAN.
Tel: +81-776-61-6000 (EX.2402)
Fax: +81-776-61-6014
E-mail: sugiyama@fpu.ac.jp*

Faculty of Policy management, Shobi University, JAPAN†

Faculty of Economics, Fukui Prefectural University, JAPAN.

1. Introduction
The role of education in the issue of sustainable development has long been discussed actively ona global scale. 1 In the context of sustainable development, environment is one of the mostimportant issues for the sustainability. On the other hand, according to the United States
Environmental Protection Agency (EPA), the environmental education can play a great role toprotect and improve the environment.
One of the main contents of the environmental education is to learn the skills for identifying
and resolving environmental challenges. 2 Then, environmental goods (EG) such as soil/water
separation systems, catalytic converters, and chemical remediation and bioremediation can beproduced with not only unskilled labor but also skilled. For example, bioremediation means thatusing microorganisms, fungi, plants, or their enzymes restore contaminated land or groundwaterto restore the natural environment contaminated with harmful substances. Professional expertiseon the environment are essential for its supply. Hence, in this paper, we consider the role of1

For example, UNESCO World Conference on Education for Sustainable Development (ESD) held in

Aichi-Nagoya, Japan, from 10 to 12 November 2014, discussed the role of education that fosters theleadership needed to build a sustainable society. For further information, see the official website of Japanese
National Commission for UNESCO (http://www.mext.go.jp/en/unesco/index.htm).
2

According to the information provided by United States Environmental Protection Agency(https://www.epa.gov/), environmental education includes various activities and programs, besides theprofessional development.

1education for environmental preservation which affects the level of supply for skilled labor
employed in EG sector, that is, the environmental industry. As related papers, Merrifield (1988),
Copeland (1991), Chua (2003), Abe and Sugiyama (2010), and Sugiyama and Saito (2016)
examine the vertical relationship between upstream EG sector and downstream final goods (FG)sector which discharges emissions or dumps wastes. However, these papers above do not include
skill formation. 3
Although we follow the basic concept of Findlay and Kierzkowski (1983) to describe theskill formation, our model incorporates the publicly provided education into the traditional
Ricardo=Viner (RV) Model. 4 In particular, we focus on the skill formation through the public
education provided by the central and local governments, instead of privately provided one,because such education courses are sometimes offered by national and public universities. 5
Wong and Chong (1999), Janeba (2003) and Kreickemeier (2009) also modeled skill
formation in HO or RV model. However, environmental-related issues are not included in these3

Wan, et.al (2018) considered the impact of trade liberalization in final goods as EGs with or without thepollution taxes on dirty upstream sectors.
4

Findlay and Kierzkowski (1983) assume that education is provided by the private sector instead of the

public sector, and incorporates the skill formation into the traditional Hechsher=Ohlin (HO) Model, insteadof RV Model.
5

In this meaning, the public education can be considered as public intermediate goods in our model. Refer

to Tawada and Abe (1984), Abe (1990), and Yanase and Tawada (2012), in which public intermediate goodis incorporated into HO Model and RV Model. See also Ishikawa (2000), where RV model is applied witha sector-specific (private) intermediate good.

2

studies. As explained below, our model assume that skilled labor can either be employed in EG
sector or in the public education sector as an “educator”.6 Then, unlike the traditional RV model,
the supply of skilled labor used in EG sector that is one of the specific factors is endogenously
determined due to skill formation under the public education.
We then reveal the impacts of an increase in skilled labor toward public education, in otherwords, an increase in the educator service on the output of EG, FG, the wage differential between
skilled and unskilled labor, and the national income in addition to the amount of pollution
emissions. One of the important results we obtain in our analysis is, the policy that increases theeducator service may not help to increase the output of EG, even if the supply of skilled labor
employed in EG sector can be increased by the policy. On the other hand, this policy also
decreases the output of FG necessarily, and the amount of emissions decreases under someconditions. The relative wage of unskilled labor to skilled labor can always be improved by thispolicy. However, the national income may decline with a decrease in the output of FG.7
The remainder of the paper is organized as follows. Section 2 presents our model. Section 3
gives preliminary analysis for comparative statics. Section 4 confirms the impact of an increase6

In this paper, the supply of education service provided the government depends only on the number ofeducators employed in the education sector. Hence, an increase in the number of educators implies anincrease in the education service as well. We refer this as “educator service”.
7

EG may increase or decrease in our analysis.

3

in the educator service on the supply side of the economy, that is, skilled and unskilled wages, therental rate of capital, the supply of skilled labor, the output of EG and FG, and the amount ofemissions. Moreover, we discuss the effect of this policy on national income. Section 5 provides
some concluding remarks.

2. The Model

We consider a small open economy consisting of two private sectors and one public sector which
provides public education. Private sectors correspond to a final goods (FG) sector and an
environmental goods (EG) sector, that is, an environmental industry. Let the output of FG and
EG be denoted by 𝑋 and 𝐴, respectively. The production process for FG generates pollution
emissions as a by-product, whereas EG produced in the upstream environmental industry are used
for abating emissions in FG sector.8 In this sense, two private sectors are vertically related. We
assume that the government imposes an emission tax, 𝑡, on polluting downstream producers to
reduce pollution emissions.
There are three factors of production in our model. Let unskilled labor be denoted by 𝐿,
which can move between FG and EG sectors, while 𝐾 denotes the real capital, is a specific factor

8

In general, environmental industries can supply either the environmental goods or the environmental

services. Contrary to our model, abatement programs or services such as consulting, monitoring, andauditing in the polluting FG sector can also be provided by EG sector, with skilled workers who possess
advanced environmental knowledge. However, for simplicity and tractability, we stick only our attention
on the skilled labor who contributes the production of EG.

4

of production for FG. 𝐻𝐴 denotes the skilled labor employed in EG sector who serves as humancapital with professional environmental knowledge, which is a specific factor of production for
EG.9 The assumptions above are exactly same as those in the traditional RV model.
In addition, the government provides public education to produce the skilled labor. Since thegovernment also employs the skilled labor as educators for the public education, the skilled labor
is employed not only in EG sector as input for the production but also in the public education
sector as educators.10 Let 𝐻𝐺 denote the educator service, that is, the skilled labor employed in
the public education sector. These two parts of skilled labor, then, can be expressed as
𝐻 = 𝐻𝐺 + 𝐻𝐴 ,where 𝐻 denotes the whole supply of the skilled labor in the economy, which can also be referredas the whole size of “human capital” through the skill formation.
Hence, unlike the conventional specific factor model, the input of skilled labor as a specificfactor used in the production of EG, 𝐻𝐴 , is immobile between two private sectors, i.e. EG and FG
sectors, but is mobile between EG and public education sectors. Moreover, as explained in the9

Notice that skilled labor is not a number of workers in this paper.

10

Unlike the traditional RV model, the skilled labor supply is not inelastic, and in fact, it can move between

EG and public education sectors. Moreover, the unskilled labor supply is also not inelastic in our model.
Since the skilled labor acquires special environmental knowledges via the skill formation process, that is,
the public education, we stick on the terminology of “skilled labor” consists of both the “quality” and the“amount” of individuals, instead of “the number of workers”. We will see this more later.

5

next subsection, the whole supply of skilled labor, 𝐻, is also a production function of skilled laborproduced in the public education sector. As a result, 𝐻𝐴 is endogenously determined in our model.
We represent the production functions for FG and for EG as
𝑋 = 𝐹 𝑋 (𝐾𝑋 , 𝐿𝑋 , 𝐴𝑋 , 𝑍),
𝐴 = 𝐹 𝐴 (𝐻𝐴 , 𝐿𝐴 ),respectively, where 𝐿𝑖 , (𝑖 = 𝑋, 𝐴) is the input of unskilled labor for both goods, 𝐾𝑋 is the inputof real capital, 𝐴𝑋 denotes the input demand for EG, and Z is the amount of pollution emissions.
We assume that 𝐹 𝑋 (∙) exhibits constant returns to scale in four variables including emissions Z,
and 𝐹 𝐴 (∙) is also constant returns to scale in two variables, and both are assumed to be quasiconcave. In the following analysis, we assume the substitutability between unskilled labor andskilled labor, (i.e., 𝐿𝐴 and 𝐻𝐴 ) in EG sector, as well as unskilled labor and real capital, (i.e., 𝐿𝑋
and 𝐾𝑋 ) in FG sector. We will explain in detail whether 𝑍 and 𝐿𝑋 or 𝐾𝑋 is substitutive orcomplementary are critical when we derive the results of comparative statics with respect to anincrease in the educator service, 𝐻𝐺 .
2.1 Skill formation
In this economy, 𝑁 individuals are born at each period and are uneducated in the beginning.
Assume that each of them lives for two periods, then the population of the economy at each period6

is 2𝑁.11 At each period, 𝑁 individuals are split into two parts right after being born, in which they
can choose to either become skilled labor or unskilled labor at the first period. Those who choose
to become skilled labor, need to be educated through the public education for free, which isprovided by the government, but they need to forgo the opportunity to participate in the market
for unskilled labor during the education period (i.e., the first period). In other words, educatedindividuals can only earn their income at the second period as skilled labor.
Let 𝑈𝐺 denote those who decide to earn their income under wage rate 𝑤𝐻 as skilled labor at
the second period after having enjoyed the public education at the first period, which can also bereferred as “student”. On the other hand, those who decide not to enjoy the education, can choose
to participate in the unskilled labor market, to earn their income under wage rate 𝑤𝐿 , at the first
period and the second period (i.e., their lifetime). Let the number of individuals choose to become
unskilled labor be denoted by 𝑈𝐿 . Hence, the population in this economy can be expressed as
2𝑁 = 2(𝑈𝐺 + 𝑈𝐿 ).12 This equation is equivalent to
𝑁 = 𝑈𝐺 + 𝑈𝐿 .

11

(1)

The assumption for two periods is for tractability. If we assume any 𝑇 periods, there is no essential

difference. Similarly, our main results do not change if we assume constant population growth rate.
12

At each period, the unskilled labor consists of 2 types of 𝑈𝐿 , since 𝑈𝐿 born at the present period and 𝑈𝐿

born at the previous period will work together at the present period, which results in 2𝑈𝐿 . In contrast, ateach period, only 𝑈𝐺 born at the previous period participate in the skilled labor market at the present period,
while 𝑈𝐺 born at the present period who are being educated do not participate.

7

As mentioned above, 𝐻 is the whole supply of skilled labor in the economy, the skill
formation in the public education sector can be expressed with the production function as follows:
𝐻 = 𝐹 𝐺 (𝐻𝐺 , 𝑈𝐺 ),(2)

𝐹 𝐺 (∙) is also assumed to exhibit constant returns to scale in two variables, where 𝐻𝐺 and 𝑈𝐺 areused as the input factors for the skill formation in the public education. Noting this assumption,
𝐹 𝐺 (∙) is rewritten asℎ = 𝑓(ℎ𝐺 ),where ℎ and ℎ𝐺 are defined as ℎ ≡ 𝐻/𝑈𝐺 and ℎ𝐺 ≡ 𝐻𝐺 /𝑈𝐺 , respectively. The former represents
the skill acquired by each individual of 𝑈𝐺 , and the latter denotes the educator-student ratio. In
addition, we assume 𝑓 (0) > 0, 𝑓′(ℎ𝐺 ) > 0, and 𝑓′′(ℎ𝐺 ) < 0.
Next, we present the lifetime income of educated and uneducated individuals, and then thearbitrary condition. For simplicity, we assume that the domestic capital stock is constant at each̅ and owned by all the individuals. Moreover, there is perfectly
period which is denoted by 𝐾
equality in distribution of the capital stock. 13 Then, in each period, each individual receives 𝑟𝜅̅/2𝑁 and 𝑟 is the factor price (i.e., rental rate) of the real capital.
equally, where 𝜅 ≡ 𝐾
On the other hand, the government imposes an emission tax for FG sector where pollution
emissions are generated. For simplicity, we also assume that the tax revenue is distributed to all

13

Many studies assume this, for example, see Gupta (1994).

8

individuals equally at each period, which can be expressed as 𝑡𝑧, where 𝑡 denotes the emission
tax rate, and 𝑧 ≡ 𝑍/2𝑁.
Let 𝑤𝐿 denotes the wage for each uneducated individual (i.e., 𝑈𝐿 ) earn in period 1 and period
2 as unskilled labor (i.e., 𝐿), while 𝑤𝐻 𝑓 (ℎ𝐺 ) is earned by each individual in period 2 for EG
production if he or she decides to acquire the skill in period 1 through the public education, where𝑤𝐻 denotes the wage for a unit of skill. 14
Assume that the public education provided by the government is financed by equal income
tax among all individuals, which can be shown as
𝑤𝐻 𝐻𝐺 = 𝜏{𝑤𝐻 (𝐻𝐴 + 𝐻𝐺 ) + 𝑤𝐿 𝐿 + 𝑟𝐾},
where 𝜏 denotes the income tax rate.15 Therefore, the lifetime income of 𝑈𝐿 and 𝑈𝐺 at present
value are, respectively, can be expressed as follows:
𝑉𝐿 = (1 − 𝜏){𝑟𝜅 + 𝑤𝐿 + (𝑟𝜅 + 𝑤𝐿 )/(1 + ρ)} + 𝑡𝑧 + 𝑡𝑧/(1 + 𝜌),
𝑉𝐺 = (1 − 𝜏){𝑟𝜅 + (𝑟𝜅 + 𝑤𝐻 𝑓 (ℎ𝐺 ))/(1 + 𝜌)} + 𝑡𝑧 + 𝑡𝑧/(1 + 𝜌),
where 𝜌 is the fixed interest rate.16

14

Notice that 𝑤𝐻 is not the wage for an individual, 𝑈𝐺 , but for a unit of 𝑓(ℎ𝐺 ) which is owned by eachindividual, 𝑈𝐺 , as a result, each of them earns 𝑤𝐻 𝑓(ℎ𝐺 ) in period 2.
15

See Abe (1990). Notice also that 𝜏 is determined with this equation alone if other endogenous variables

are all determined in the model. In addition, the emission tax revenue distributed to the individuals is
assumed not subject to the income taxation. However, even without this assumption, we will obtain thesame result under the arbitrary condition.
16

Note that the interest rate, 𝜌, does not necessarily coincide the price of real capital since we assume a

9When the arbitrary condition holds for an individual's choice regarding whether he or she
enjoys the education or not, that is, 𝑉𝐿 = 𝑉𝐺 , the condition is expressed as follows:17
𝑓 (ℎ𝐺 ) = (𝑤𝐿 /𝑤𝐻 )𝑅,(3)

where 𝑅(≡ 2 + 𝜌) is constant. As explained above, given 𝑅 and the educator service, that is, theskilled labor employed in the public education sector, 𝐻𝐺 , the number of individuals who decide
to become skilled labor, 𝑈𝐺 , is determined from equation (3) as the function of the relative wageof unskilled relative to skilled, 𝑤𝐿 /𝑤𝐻 . Moreover, after 𝑈𝐺 is determined, the number of
uneducated individuals, 𝑈𝐿 , who work as unskilled labor, 𝐿, is determined from equation (1),given individuals 𝑁 born in each period.
Moreover, considering ℎ𝐺 = 𝐻𝐺 /𝑈𝐺 = 𝐻𝐺 /(𝑁 − 𝑈𝐿 ) from equation (1), equation (3) isrewritten as
𝑓(

𝐻𝐺
𝑁−𝑈𝐿

) = (𝑤𝐿 /𝑤𝐻 )𝑅.

(3)’

From equation (3)’, we obtain the relation between the number of uneducated individuals, 𝑈𝐿 ,and the relative wage of unskilled relative to skilled, 𝑤𝐿 /𝑤𝐻 , given the educator service, 𝐻𝐺 , and𝑅(≡ 2 + 𝜌).

short run with special factors of production.
17

Since 𝑑𝑉𝐺 ⁄𝑑𝑈𝐺 < 0 holds, 𝑈𝐺 must be determined under the arbitrary condition, as well as 𝑈𝐿 .

10

2.2 Equilibrium Conditions
In the following analysis, we designate the price of FG as the numeraire and the relative price of
EG as 𝑝. Considering a small open economy and the assumption of constant returns-to-scaletechnology about production functions, 𝐹 𝑖 (∙), (𝑖 = 𝑋, 𝐴) , the zero-profit conditions for the
production of FG and EG are represented as
1 = 𝑐 𝑋 (𝑤𝐿 , 𝑟, 𝑝∗ , 𝑡),(4)𝑝∗ = 𝑐 𝐴 (𝑤𝐿 , 𝑤𝐻 ),(5)

where 𝑐 𝑖 (∙), (𝑖 = 𝑋, 𝐴) is the unit cost function for private sectors, and 𝑝∗ denotes the world
relative price of EG. 18 Note that each unit cost function is homogeneous of degree one andconcave with respect to the vector of corresponding variables.
Next, we provide full employment conditions of production factors. Note that we assume
two periods and unskilled labor is the input for the production of EG and FG. Then, the supply of
unskilled workers, 𝐿, at each period is expressed as follows:19
𝐿 = 2𝑈𝐿 .

(6)

18

An asterisk denotes the corresponding foreign variables in this paper.

19

Since unskilled labor does not acquire skill at all, the supply of unskilled labor is just the number of theindividuals. In contrast, the supply of the skilled labor is not the number of educated individuals but the
total skill they acquire. Notice also that the skilled input in the public sector, namely the educator service is
not represented as the number.

11

On the other hand, as mentioned above, noting the skilled labor are used for the production of EG
and the public education (i.e., 𝐻 = 𝐻𝐴 + 𝐻𝐺 ), and considering ℎ ≡ 𝐻/𝑈𝐺 and ℎ = 𝑓(ℎ𝐺 ), we
obtain the following equation:
𝐻𝐴 = 𝑓 (ℎ𝐺 )𝑈𝐺 − 𝐻𝐺 .

(7)

Since the educator service, that is, the input of skilled labor in the public education, 𝐻𝐺 , is given,and then the number of uneducated individuals, 𝑈𝐺 , is determined from equation (3), the input ofskilled labor, 𝐻𝐴 , used in the production of EG is also determined from equation (7).
In the following analysis, 𝑐𝑗𝑖 , (𝑖 = 𝑋, 𝐴; 𝑗 = 𝐻, 𝐿, 𝑟, 𝑝∗ , 𝑡) denotes the partial derivatives of
the unit cost functions with respect to the factor prices, which are the skilled and unskilled wages,
the rental rate of real capital, the world price of EG, and the emission tax rate. Considering
equation (7), the full employment condition for skilled labor is then expressed as follows:20
𝐻𝐴 = 𝑐𝐻𝐴 (𝑤𝐿 , 𝑤𝐻 )𝐴.

(8)

Moreover, considering ℎ𝐺 = 𝐻𝐺 /𝑈𝐺 = 𝐻𝐺 /(𝑁 − 𝑈𝐿 ) in equation (7), then equation (8) isrearranged as
𝐻𝐺) (𝑁 − 𝑈𝐿 ) − 𝐻𝐺 = 𝑐𝐻𝐴 (𝑤𝐿 , 𝑤𝐻 )𝐴.
𝑓 (𝑁−𝑈
𝐿

20(8)’

From Shephard’s lemma and the envelope theorem, 𝑐𝑗𝑖 , (𝑖 = 𝑋, 𝐴; 𝑗 = 𝐻, 𝐿, 𝑟, 𝑝∗ , 𝑡) is equal to skilled,

unskilled, real capital and EG requirements for producing one unit of the i-th good, and the emissions per
FG output for the partial derivative with respect to the emission tax. For example, 𝑐𝐿𝑋 (∙) ≡ 𝜕𝑐 𝑋 ⁄𝜕𝑤𝐿 is theunskilled labor requirements for producing one unit of FG.

12

Similarly, the full employment conditions for unskilled labor is expressed as
𝐿 = 𝑐𝐿𝑋 (𝑤𝐿 , 𝑟, 𝑝∗ , 𝑡)𝑋 + 𝑐𝐿𝐴 (𝑤𝐿 , 𝑤𝐻 )𝐴,(9)

which yields the following equation from equation (6):2𝑈𝐿 = 𝑐𝐿𝑋 (𝑤𝐿 , 𝑟, 𝑝∗ , 𝑡)𝑋 + 𝑐𝐿𝐴 (𝑤𝐿 , 𝑤𝐻 )𝐴.

(9)’

The full employment conditions for real capital is represented as
̅ = 𝑐𝑟𝑋 (𝑤𝐿 , 𝑟, 𝑝∗ , 𝑡)𝑋.
𝐾

(10)

On the other hand, since we assume a small open economy which allows FG to be tradable
in the world market, the equilibrium condition for EG is expressed as 𝐴 + 𝐴∗ = 𝐴𝑋 , which isrepresented as
𝐴 + 𝐴∗ = 𝑐𝑝𝑋∗ (𝑤𝐿 , 𝑟, 𝑝∗ , 𝑡)𝑋,

(11)

where 𝐴∗ denotes the traded volume of EG in the world market.21
Finally, noting that 𝑐𝑡𝑋 (𝑤𝐿 , 𝑟, 𝑝∗ , 𝑡) denotes the emissions per output of FG, the amount ofpollution emissions is expressed as follows:
𝑍 = 𝑐𝑡𝑋 (𝑤𝐿 , 𝑟, 𝑝∗ , 𝑡)𝑋.

(12)

There are 12 endogenous variables 𝐿, 𝑈𝐿 , 𝑈𝐺 , 𝐻, 𝐻𝐴 , 𝑤𝐿 , 𝑤𝐻 , 𝑟, 𝑋, 𝐴, 𝐴∗ , and 𝑍 fromequations (1) to (12) except for (3)’, (8)’, and (9)’. We then confirm the economic andenvironmental impacts of the educator service, that is, an increase in the input of skilled labor

21

Notice that if we assume EG is to be imported, 𝐴∗ > 0, and vice versa.

13toward the public education, 𝐻𝐺 . Moreover, these equilibrium conditions above are summarized
into the equations (3)’, (4), (5), (8)’, (9)’, and (10) to solve the variables of, 𝑈𝐿 , 𝑤𝐿 , 𝑤𝐻 , 𝑟, 𝑋 and𝐴.

3. Preliminaries

Totally differentiating the equations (3)’, (4), (5), (8)’, (9)’, and (10), we obtain the following
simultaneous equation system:
ℎ𝐺
𝑈𝐺

𝑓′

0
0
(1 − 𝜀𝐻 )𝑓
−2
[
0

𝑅

𝑤𝐿 𝑅

𝑤𝐻
𝑐𝐿𝑋
𝑐𝐿𝐴
𝐴
𝑐𝐻𝐿
𝐴

2
𝑤𝐻

−

−𝛿𝐿
𝑋
𝑐𝑟𝐿
𝑋

0

0

𝑐𝑟𝑋
0
𝐴
𝑐𝐻𝐻 𝐴
0
𝐴
𝑋
𝑐𝐿𝐻 𝐴 𝑐𝐿𝑟 𝑋
𝑋
0
𝑐𝑟𝑟
𝑋

0
0
0
𝑐𝐿𝑋
𝑐𝑟𝑋

0
𝑐𝐻𝐴

𝐻

ℎ

𝑓′

0

−𝑈
𝑑𝑈𝐿
𝑔
𝑑𝑤𝐿
0
0
𝑑𝑤𝐻 =
0
0
𝑑𝐻𝐺 ,
𝜀
𝐴
𝑑𝑟
1 − 𝜃𝐻
𝑐𝐻
𝐺
𝑑𝑋
𝑐𝐿𝐴 [
0
]
𝑑𝐴
[ 0 ]
0]

(13)

ℎ

𝐺
𝑋
𝐴
where 𝛿𝐿 ≡ −𝑐𝐿𝐿
𝑋 − 𝑐𝐿𝐿
𝐴, 𝜃𝐺 ≡ 𝐻 +𝐻
= 𝑓𝐺 , and 𝜀𝐻 ≡ 𝑓 ′ ･ 𝑓𝐺 , which also imply that 𝛿𝐿 > 0,𝐺𝐴

0 < 𝜃𝐺 < 1, and 0 < 𝜀𝛨 < 1 since 𝐹 𝐺 (𝐻𝐺 , 𝑈𝐺 ) is a linear homogeneous function.
𝑖
𝑐𝑗𝑘, (𝑖 = 𝑋, 𝐴; 𝑗, 𝑘 = 𝐻, 𝐿, 𝑟, 𝑝∗ , 𝑡) is the second-order partial derivatives for 𝑐𝑗𝑖 . As the unit

cost function 𝑐 𝑖 , (𝑖 = 𝑋, 𝐴), is concave with respect to the vector of corresponding variables, the𝑖𝑋𝐴𝑋signs of the 𝑐𝑗𝑘
are negative when 𝑗 = 𝑘. In particular, we have, 𝑐𝐿𝐿< 0, 𝑐𝑟𝑟< 0, and 𝑐𝐻𝐻< 0,𝑖which also implies that 𝛿𝐿 > 0. On the other hand, when 𝑗 ≠ 𝑘, the signs of 𝑐𝑗𝑘
depend on the𝐴𝑋
substitutability between factor 𝑗 and 𝑘. In our model, 𝑐𝐻𝐿> 0 and 𝑐𝑟𝐿> 0 are satisfied because

unskilled labor is substitutive to real capital in FG sector and to skilled labor in EG sector. In
addition, 𝑐𝑗𝑡𝑋 > (<)0 , (𝑗 = 𝐿, 𝑟) means that the unskilled labor or real capital is substitutive

14

(complementary) to the emissions in FG sector.22
Notice also that 𝜀𝐻 denotes the output (i.e., 𝐻) elasticity of the educator service (i.e., 𝐻𝐺 ) in
the public education sector, whereas 𝜃𝐺 is the share of the educator service from the whole supply
ℎ

𝐻𝐺

𝑓

𝐻𝐺 +𝐻𝐴

of human capital. Since 𝜀𝐻 ≡ 𝑓 ′ ･ 𝐺 and 𝜃𝐺 ≡
𝜀

=

𝐻𝐺
𝐻

=

𝐻𝐺 /𝑈𝐺
𝐻/𝑈𝐺

which results in 𝜃𝐺 =

ℎ𝐺
𝑓

,

𝜀

we also have 𝜃𝐻 = 𝑓′(ℎ𝐺 ). Hence, 1 − 𝜃𝐻 = 1 − 𝑓′(ℎ𝐺 ) tends to be positive if ℎ𝐺 is sufficiently𝐺𝐺

large which brings a significant small size of 𝑓 ′ or 𝜀𝐻 , and vice versa. We conclude as follows:

LEMMA 1: If educator-student ratio (ℎ𝐺 ) is sufficiently large (small), 𝜀𝐻 < 𝜃𝐺 (𝜃𝐺 < 𝜀𝐻 ) holds,
𝜀

𝜀

𝜃𝐺

𝜃𝐺

thus we have 1 − 𝐻 > 0 (1 − 𝐻 < 0).

Let 𝐽 represent the coefficient matrix on the left-hand side of equation (13). Then, noting𝑝 ∗ = 𝑐 𝐴 (𝑤𝐿 , 𝑤𝐻 ) = 𝑤𝐻 𝑐𝐻𝐴 + 𝑤𝐿 𝑐𝐿𝐴 , the determinant of 𝐽 , that is, |𝐽 | results in the following
equation:|𝐽 | =
22ℎ𝐺

𝑝 ∗𝑐𝑟𝑋 𝑅𝐺

2𝑤𝐻𝑓′ 𝜓 +
𝑈{(1 − 𝜀𝐻 )𝑓𝑐𝐿𝐴 + 2𝑐𝐻𝐴 } > 0,2where 𝜓 ≡ 𝑐𝐻𝐴 𝑋∆ 𝑋 + 𝑐𝑟𝑋 𝐴∆ 𝐴 > 0, ∆𝑋 ≡ 𝑐𝐿𝑋 ∆1𝑋 + 𝑐𝑟𝑋 ∆2𝑋 > 0, and ∆𝐴 ≡ 𝑐𝐿𝐴 ∆1𝐴 + 𝑐𝐻𝐴 ∆2𝑋 > 0, since
𝑋
𝑋
𝑋
𝐴
𝐴
𝑋
∆1𝑋 ≡ 𝑐𝑟𝑋 𝑐𝑟𝐿− 𝑐𝐿𝑋 𝑐𝑟𝑟
> 0 , ∆2𝑋 ≡ 𝑐𝐿𝑋 𝑐𝑟𝐿− 𝑐𝑟𝑋 𝑐𝐿𝐿
> 0 , ∆1𝐴 ≡ 𝑐𝐻𝐴 𝑐𝐻𝐿− 𝑐𝐿𝐴 𝑐𝐻𝐻
> 0 and ∆2𝐴 ≡𝐴𝐴
𝑐𝐿𝐴 𝑐𝐻𝐿− 𝑐𝐻𝐴 𝑐𝐿𝐿
> 0.

22

𝑋
Without the analysis for the change in 𝑝∗ , the discussion for 𝑐𝐿𝑝∗ is omitted.

15

4. Comparative Statics

In this section, we first reveal the impacts of the change in the employment of skilled labor as
educators in the public education sector (i.e., the role of public education), on the output of FG
and EG, as well as the amount of emissions. Before this, we first confirm the effects on the wagesof skilled and unskilled labor as well as the rental rate of the real capital, which will change theskill formation as well as the supply of skilled labor employed in EG sector.
4.1 Wages, rental rate and supply of skilled labor
In order to see the change in the supply of skilled labor employed in EG sector due to the change
in the educators employed in the public education, we need to see also the change in 𝑈𝐺 , where𝑑𝑈𝐺 = −𝑑𝑈𝐿 , as well as the change in 𝑤𝐿 , 𝑤𝐻 , and 𝑟, simultaneously.
From equation (13), we obtain
𝑑𝑤𝐿

𝑓′

𝐺

𝑑𝑤𝐻
𝑑𝐻𝐺
𝑑𝑟
𝑑𝐻𝐺

2

= |𝐽 |−1 𝑈 𝑐𝑟𝑋 𝑐𝐻𝐴 [2𝑐𝐻𝐴 + 𝑐𝐿𝐴 𝑓 (1 − 𝜃𝐺 )] > 0

𝑑𝐻𝐺

𝑓′

2

= −|𝐽 |−1 𝑈 𝑐𝑟𝑋 𝑐𝐿𝐴 [2𝑐𝐻𝐴 + 𝑐𝐿𝐴 𝑓(1 − 𝜃𝐺 )] < 0𝐺𝑓′= −|𝐽 |−1 𝑈 𝑐𝑟𝑋 𝑐𝐿𝑋 𝑐𝐻𝐴 [2𝑐𝐻𝐴 + 𝑐𝐻𝐴 𝑓 (1 − 𝜃𝐺 )] < 0
𝐺

(14)
(15)
(16)

These results with respect to the impacts on factor prices are summarized in the following
proposition.

16

PROPOSITION 1: An increase in the educator service, always brings unambiguous effects on
the factor prices, in which the wage of unskilled labor rises, whereas the wage of skilled laborand the rental rate of real capital decline. As a result, the relative wage of unskilled labor toskilled labor always rises.

These results are so familiar compared to the results in the traditional RV model, in which
an increase in one of the specific factors, always increases the factor price of mobile factor while
decreases the factor prices of immobile factors. In this paper, unskilled labor is a mobile factor
whereas skilled labor and real capital are immobile factors between EG and FG sector,
respectively. However, in fact, skilled labor is actually mobile between EG and public education
sector. Moreover, the supplies of skilled labor and unskilled labor which are equal to the amountemployed in FG and EG sector are not inelastic compared to the traditional RV model.
Since 𝐻𝐺 is only an increase in educator service instead of the supply of skilled labor
employed in EG sector, the change in the 𝐻𝐴 needs to be revealed as well.
From equation (13), we obtain
𝑑𝑈𝐿
𝑑𝐻𝐺

𝑓′

𝜀

2

= −|𝐽 |−1 [𝑈 𝜓 + (1 − 𝜃𝐻 ) 𝑐𝐿𝐴 𝑐𝑟𝑋 𝑝∗ ] < 0.
𝐺

𝐺

(17)

As we have discussed in Lemma 1, if the educator-student ratio is sufficiently large, 𝜀𝐻 < 𝜃𝐺𝜀𝑑𝑈holds, which implies 1 − 𝜃𝐻 > 0, then we have 𝑑𝐻𝐿 < 0.
𝐺

𝐺

17

Moreover, considering equation (17) and 𝑑𝑈𝐺 = −𝑑𝑈𝐿 from equation (1), we can show theeffect on the unskilled labor, which implies
𝑑𝑈𝐺
𝑑𝐻𝐺
𝑑𝑈𝐺

Similarly,

𝑑𝐻𝐺= |𝐽 |−1 [

𝑓′
𝑈𝐺

𝜀

2

𝜓 + (1 − 𝐻 ) 𝑐𝐿𝐴 𝑐𝑟𝑋 𝑝∗ ] > 0.

(18)

𝜃𝐺

> 0 must hold if the educator-student ratio is sufficiently large.

LEMMA 2: If educator-student ratio is sufficiently large, an increase in educator service in the
public education sector decreases the unskilled labor supply, while increases students in the
public education sector.

Totally differentiating equation (7) and rearrange it, we have23
𝑑𝐻𝐴
𝑑𝐻𝐺

𝑑𝑈

𝜀

= (1 − 𝜀𝐻 )𝑓 𝑑𝐻𝐺 − (1 − 𝜃𝐻 ).
𝐺

(19)

𝐺

The first term and the second term in the RHS of equation (16) are the indirect effect and the
direct effect of 𝐻𝐺 on 𝐻𝐴 , which are positive and negative, respectively.24 Paying attention to 1 −𝜀𝑑𝑈𝜀𝐻 > 1 − 𝜃𝐻 in the case of the educator-student ratio is sufficiently large, if 𝑓 𝑑𝐻𝐺 > 1 holds, we𝐺𝐺

can obtain
𝑑𝐻𝐴
𝑑𝐻𝐺

> 0,

2

23

Concretely, 𝑑𝐻𝐴 /𝑑𝐻𝐺 = (1 − 𝜀𝐻 )𝑓|𝐽|−1 {𝑓 ′ 𝜓/𝑈𝐺 + (1 − 𝜀𝐻 /𝜃𝐺 )𝑐𝐿𝐴 𝑐𝑟𝑋 𝑝∗ 𝑅/𝑤𝐻2 } − (1 − 𝜀𝐻 /𝜃𝐺 ).

24

Recalling 𝑑𝑈𝐺 /𝑑𝐻𝐺 > 0 may not be satisfied if 𝜀𝐻 < 𝜃𝐺 does not hold.

18

unambiguously.25 In particular, equation (19) can also be expressed as
𝑑𝐻𝐴
𝑑𝐻𝐺

𝜎𝑈

𝜀

= (1 − 𝜀𝐻 ) 𝜃𝐺 − (1 − 𝜃𝐻 ),
𝐺

(19)’

𝐺

𝐻 𝑑𝑈

where 𝜎𝐺𝑈 ≡ 𝑈𝐺 𝑑𝐻𝐺 denotes the output elasticity of educator service input, which is smaller than𝐺𝐺

1. This can be proved by totally differentiating equation (3)’ and using the results obtained from equations(14) and (15), in which the impacts of the educator service on the wages of skilled and unskilled have been𝑤

𝐻

discussed. From the results we know that an increase in 𝐻𝐺 raises the ratio of 𝑤 𝐿 , then 𝑓′(𝑈𝐺) increases as
𝐻

𝐺

𝐻well, which also implies that the ratio of 𝑈𝐺 rises, hence we know that 𝑈𝐺 is diminishing returns to 𝐻𝐺 . As𝐺a result, 𝜎𝐺𝑈 must be smaller than 1. Notice also that if 𝐻𝐺 is too large, 𝑈𝐺 turns to decrease, but 𝜎𝐺𝑈 < 1still holds.

Hence, we can conclude that if 𝜎𝐺𝑈 > 𝜃𝐺 , then

𝑑𝐻𝐴
𝑑𝐻𝐺

> 0 is satisfied, which can be concluded

as the following proposition: 26

PROPOSITION 2: If the output elasticity of educator service input is larger than the educatorstudent ratio, an increase in educator service in the public education sector increases the skilledlabor supply for EG sector.

25

One may argue that if 1 − 𝜀𝐻 /𝜃𝐺 < 0, then 𝑑𝐻𝐴 /𝑑𝐻𝐺 becomes positive unambiguously, which is nottrue, since 𝑑𝑈𝐺 /𝑑𝐻𝐺 > 0 requires the condition in the opposite way (i.e., 1 − 𝜀𝐻 /𝜃𝐺 > 0) to be determinedunambiguously.
26

Considering 0 < 𝜃𝐺 < 1 and 𝜎𝐺𝑈 < 1, we have also 0 < 𝜃𝐺 < 𝜎𝐺𝑈 < 1.

19

Table 1. The impacts on wages, rental rate, students, and supply of skilled labor for EG

𝑯𝑮

𝒘𝑳

𝒘𝑯

𝒓

𝑼𝑮

𝑯𝑨

+

−

−

+∗

+∗∗

The results from equations (14) to (19) are summarized in Table 1. Signs with (*) and (**)
in Table 1 are determined under conditions, which require a significantly large ℎ𝐺 and 𝜎𝐺𝑈 > 𝜃𝐺 ,respectively.
4.2 The Output of FG and EG
In this section, we reveal the impacts on the output of FG and EG, which are shown andsummarized in Table 2. Here, (*) in Table 2 is determined under conditions. We will confirm thecondition in detail.

Table 2. The impacts on environmental goods and final goods
𝑯𝑮

𝑿

𝑨

+

+∗

From equation (13), we obtain the effects on FG and EG. The effects of the educator service on
FG is as follows:𝑑𝑋
𝑑𝐻𝐺= −|𝐽 |−1 𝑐𝐻𝐴 𝑋∆1𝑋 ∙

𝑓′
𝑈𝐺

𝑐𝐴

𝜀

𝑈𝐺

𝜃𝐺

{2𝑐𝐻𝐴 + 𝐿 [(1 − 𝜀𝐻 )𝐻 − (1 − 𝐻 ) 𝐻𝐺 ]} < 0,

𝜀

(20)

since 𝐻 > 𝐻𝐺 and (1 − 𝜀𝐻 ) > (1 − 𝐻 ). That is, the educator service definitely decreases the𝜃𝐺20

output of FG.
PROPOSITION 3: An increase in educator service in the public education sector always
decreases the output of FG.

On the other hand, the effects of the educator service on EG is as follows:
𝑑𝐴
𝑑𝐻𝐺

2 𝑓′
𝑓′
𝜀
𝑅
= |𝐽 |−1 [𝑈 2 ̃
Δ𝜉 + 2𝑐𝑟𝑋 {𝑈 𝐴Δ𝐴1 − (1 − 𝜃𝐻 )𝑝 ∗ 𝑤 2 }],
𝐺

𝐺

𝐺

(21)

𝐻

𝜀

2

where ̃Δ ≡ 𝑐𝐻𝑋 𝑋Δ𝑋 + 𝑐𝑟𝑋 𝐴Δ𝐴2 > 0 and 𝜉 ≡ {(1 − 𝜀𝐻 )𝐻 − (1 − 𝐻 ) 𝐻𝐺 } > 0 as we have argued
𝜃𝐺

previously.
Reminding the production function of EG which is represented as 𝐴 = 𝐹 𝐴 (𝐻𝐴 , 𝐿𝐴 ), we knowthat the change in output of EG can be decomposed into two parts (i.e., 𝑑𝐻𝐴 and 𝑑𝐿𝐴 ). As wehave revealed, an increase in 𝐻𝐺 always decreases the output of FG which brings a positive effecton 𝐿𝐴 which is similar to the result in the traditional RV model.
However, on the other hand, as we have seen, an increase in 𝐻𝐺 increases 𝐻𝐴 but also
decreases 𝐿 which also brings a negative effect on 𝐿𝐴 . As a result, the input demand of 𝐿 in EG
sector which is represented by 𝐿𝐴 may increase or decrease due to many reasons such as thechange in wages, wage elasticity of unskilled labor demand, substitutability between factors,
outputs, etc. If this disturbance brings a negative total effect which is significantly large, the21

output of EG may decrease. 27
PROPOSITION 4: Even though an increase in educator service in the public education sector
can raise up the level of the supply of skilled labor employed in EG sector, it may decrease theoutput of EG due to the decrease in the whole supply of unskilled labor and hence the unskilledlabor demand in EG sector.

4.3 The output of emissions and national income
In this section, we reveal the impacts on the output of emission and the national income. This canbe done by using the results what we have examined above. The effects are ambiguous as shownin Table 3. However, some arguments for the conditions required are discussed in this section.

Table 3. The output of emissions and national income

𝑯𝑮

𝒁

𝒀

?

?

Totally differentiating equation (12), we have as follows:28
𝑑𝑍
𝑑𝐻𝐺

27

𝑑𝑟

𝑑𝑤

𝑑𝑋

𝐺

𝐺

𝐺

𝐿
𝑋
𝑋
) 𝑋+𝑐𝑡𝑋= (𝑐𝑡𝑟
+ 𝑐𝑡𝐿
,
𝑑𝐻
𝑑𝐻
𝑑𝐻

(22)

One may argue that 𝑑𝐴/𝑑𝐻𝐺 > 0 can also be obtained if we assume 1 − 𝜀𝐻 /𝜃𝐺 < 0 , however, thisviolates the condition, 1 − 𝜀𝐻 /𝜃𝐺 > 0, which is required for the result simultaneously.
28

Substituting equations (14), (16) and (18) into equation (20), we will see the indeterminacy in more

concrete form.

22where 𝑑𝑝∗ = 𝑑𝑡 = 0.
The second term in the RHS is negative whereas the first term is not determined. In general,𝑋𝑋
if 𝑐𝑡𝑟< 0 and 𝑐𝑡𝐿> 0 due to the complementary between real capital and emissions, and thesubstitutability between unskilled labor and emissions, respectively, the impacts on the emissions
𝑋
𝑋
is ambiguous. On the contrary, if 𝑐𝑡𝑟> 0 and 𝑐𝑡𝐿< 0, which is the opposite case, then

𝑑𝑍
𝑑𝐻𝐺

<0

is satisfied.

PROPOSITION 5: An increase in the educator service decreases the emissions if the real capitalis substitutable while the unskilled labor is complementary for the emissions. Otherwise, theemissions may increase.

Next, the national income can be expressed as follows:29
𝑌 = 𝑋 − 𝑝∗ (𝐴𝑋 − 𝐴).

(23)

Totally differentiating equation (23), we have
𝑑𝑌
𝑑𝐻𝐺

𝑑𝑋

𝑑𝑟

𝑑𝑤

𝑑𝐴

𝐺

𝐺

𝐺

𝐺

= (1 − 𝑝∗ 𝑐𝑝𝑋∗ ) 𝑑𝐻 − 𝑝∗ 𝑋 (𝑐𝑝𝑋∗ 𝑟 𝑑𝐻 + 𝑐𝑝𝑋∗ 𝐿 𝑑𝐻 𝐿 ) + 𝑝∗ 𝑑𝐻 ,(24)

̅ = 𝑑𝑡 = 0.
where 𝑑𝐾As we have revealed previously, the first term in the RHS of equation (24) is negative since29

The national income can also be defined as 𝑌 = 𝑟𝐾 + 𝑤𝐻 𝐻𝐴 + 𝑤𝐿 𝐿, which provides the same result andargument.

23

1 − 𝑝∗ 𝑐𝑝𝑋∗ > 0 where 𝑝∗ 𝑐𝑝𝑋∗ is the income share of the input demand of EG. 𝑐𝑝𝑋∗ 𝑟 and 𝑐𝑝𝑋∗𝐿 arepositive if the real capital and the unskilled labor are both substitutable for EG, hence the sign of
𝑑𝑟

𝑑𝑤

𝐺

𝐺

the second term in the RHS is not determined since 𝑑𝐻 < 0 and 𝑑𝐻 𝐿 > 0. As we have argued
previously, an increase in the educator service may decrease the output of EG, the third term in
the RHS can also be negative. As a result, an increase in the educator service may decrease thenational income if the second term in the RHS turns to be negative.

5. Concluding Remarks

In this paper, we construct a skill formation model to investigate the role of public education
which aims to promote the skilled labor in EG sector. Under this skill formation model, we study
that even though the educator service in the public education sector can increase the whole supplyof skilled labor, the amount of skilled labor that can be employed in EG sector does not increase
necessarily if the share of skilled labor employed in the public education sector as educator is toolarge.
In addition, unlike the traditional RV model, even if the amount of skilled labor employed in
EG sector increases, the output of EG does not increase necessarily as well due to the decrease in
the whole supply of unskilled labor in the economy. Whether the increase in skilled labor as
specific factor can increase the output of EG or not depends on some conditions which are notrequired in the traditional RV model. For example, if the wage elasticity of unskilled labor demand24

in the environmental goods sector is sufficiently large, the educator service may decrease theoutput of EG.
On the other hand, the output of polluting FG always decreases. However, this may not help
to decrease the emissions necessarily as well due to the disturbance from the substitutability
between emissions and real capital or unskilled labor when the factor prices change. Although westudy that the role of public education can decrease the emissions if the real capital is substitutable
while the unskilled labor is complementary for the emissions, this assumption is quite perverse
considering in most cases the pollution emissions are generated by real capital and unskilled laboris substitutable for the polluting real capital. Moreover, from this argument, we also study that thenational income may decrease due to the policy.
The impacts on the wages of skilled and unskilled and the rental rate of real capital are very
similar to those in the traditional RV model. The results remain robust even though we incorporate
the skill formation into the RV model in which the supply of skilled labor is endogenously
determined in this paper. In the sense of improving the relative wage of unskilled labor to skilledlabor, the role of public education always brings a positive impact on this. However, this does nottake the skill acquired by the individuals into account. In other words, even though the wage ofskilled labor decline, the skilled labor may still be better off under the policy.
Further research is required to address the following points. First, as long as the lifetime

25

income of skilled and unskilled labor at present value are always equal in our model, it seems thatto investigate the impacts on the lifetime income is more important compared to the wage
differential. This analysis can solve for not only the optimum level of public education service
but also the tax rate for emissions which takes more examination.
Second, in the context of international trade, the analysis for the impacts of change in theworld price of EG can also be done. However, this has been omitted in this paper since we arefocusing on the role of public education. By showing the impacts, we will be able to supplement
the actual discussion on trade liberalization in EG.
Finally, the environmental industry is depicted to produce EG instead of environmental
service, which does not make difference in our model. However, in the context of environmental
service, if we consider that abatement activities in polluting FG are carried out through the service
by the environmental experts solely, the model we construct in this paper can be modified.
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