Journal of Applied Input-Output Analysis, Vol. 15, 2009

A Dynamic Leontief Pollution Model with Environmental Standards

byImre Dobos1 and Adel Floriska2

Abstract

Despite known simplifications the input-output analysis is an effective tool to investigate theenvironmental impacts of the producing economy on the natural environment. The emissionof pollutants is introduced direct in the model studied as joint products of the process of
production. The interdependence of the economic activities and the associated pollution
emission is analyzed in this paper in a generalized input-output model. The pollution con
trol of the government is defined as an upper bound for the production process, so called
environmental standards. The mathematical form of the proposed model consists of a dy
namic linear difference equation system of production and a system of inequalities for pol
lution. We examine the balanced path compared to the environmental standards.

Keywords: Environmental economics, Input-output model, Renewable resources, Pollution emission, Environmental standards

1

Introduction

In this study we will investigate the effects of environmental standards on the trajecto
ries of the activities in an open dynamic Leontief model. Environmental standards are aregulation policy of the government to control the environmental pollution. Thismethod of regulation sets an upper bound on the emission. (Pearce and Turner (1990))
This type of regulation is a good tool to decrease of pollution of renewable resources,
like air or water.

The environmental standards in a Leontief model was first examined by Luptacik andBohm (1994), but they have investigated a static Leontief model. In the model they
have introduced abatement activities, and the net pollution is controlled by environ
mental standards for the whole economy. Ebiefung and Udo (1999) have shown another environmental application of static Leontief model on industrial pollution emis1 Institute of Business Economics, Corvinus University of Budapest, H-1093 Budapest, Fovam ter 8,
Hungary, imre.dobos@uni-corvinus.hu - corresponding author
2 Institute of Mathematics and Informatics, Szent Istvan University, H-2100 Godollo, Pater Karoly u.l,Hungary, E-mail:floriska.adel@gek.szie.hu

A Dynamic Leontief Pollution Model with Environmental Standards

41sion. Wirl (1991) has investigated the effects of environmental standards in a dynamic
Arrow-Karlin model. In this model the standards are an upper bound on the production
for an one-product firm. In this paper we extend the dynamic Leontief model (Leontief(1986)) with pollution constraints. We analyze the environmental standards in a dy
namic context. We assume that the Leontief economy increases in a constant percent
age in every time period. It is assumed as well, that the government allows a constantly
increasing emission. We will ask, how do these standards influence the balanced
growth path of the economy.

The paper organizes, as follows. In section 2 the model is posed and we make someassumption about the model. The next section characterizes the properties of the bal
anced growth path in case of effective environmental regulation of the government. In
section 4 we illustrate our results by the help of a simple numerical example, and thelast section summarizes the results of the paper and proposes some possibilities for fur
ther research.

2 Description of the model
Our model is based on the equations of the dynamic multi-sector input-output model
well known in the literature. (Leontief (1986)) We extend this model with a system ofinequalities for pollution emission in order to analyze the balanced path compared tothe environmental standards imposed by the government.

Suppose that there are n economic industries each industry producing a single eco
nomic commodity and m types of pollutants released by industries. The input-output
balance of the economy is described by the equation of economic goods and the inequality of pollutants. The equation of goods describes the balance between the total
output of goods of production and the sum of total input of goods of all activities ofthe economy and the consumed goods.
xt = Axt + B ( xt+i - xt) + ct

(1)

The inequality of pollution emission describes the control of emission by imposing
separate environmental standards on each source of emission. That is, the amount of
the emitted pollutants is not allowed to (should not) exceed the prescribed legal limit.
Ex, < et

(2)

where

-

Xt is the n-dimensional vector of gross industrial outputs,
ct is the n-dimensional vector of final consumption demands for economic-

A is the nxn matrix of conventional input coefficients, showing the input ofgoods that are required to produce a unit of product,
B is the nxn matrix of capital coefficients, showing the invested products tocommodities,-

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Journal of Applied Input-Output Analysis, Vol. 15, 2009--

increase the output of the producing sectors by a unit,
E is the mxn matrix of pollution output coefficients of producing sectors,
showing the quantity of pollutants generated during producing a unit of indus
trial product,e, is the m-dimensional vector of environmental standards imposed, to limit thepollution emission,
T is the length of the planning horizon.

Assumption 1.
Throughout the paper it is assumed that the matrices A, B and E are nonnegative, B is
nonsingular and c, is a nonnegative vector. In a previous work Dobos and Floriska(2005) have already studied the balanced growth solution of the system (1) for non
renewable resources corresponding to a given growth rate a (a > 0) supposing thatboth production and consumption increase at the same rate a. A similar investigation
was made by Schoonbeek (1990). Under these assumptions the balanced growth solu
tion of the model (1) has the form

xt = (1 + a)'-xo and ct = (1 + ay-c0(3)where a > 0. After substituting the former expressions for x, and c in the equation (1)we have got the following relation(/- A - CC'B) • x0 = Co .

(4)

After that we have established conditions for the existence of nonnegative output con

figuration x0. The output configuration x0 corresponding to equation (4) exists and it isnonnegative if a ^ [0, a0), where ceo is the marginal growth rate such that Xi(A + a0
B) = 1, i.e. it is the balanced growth rate of the closed dynamic Leontief model.
Where Xi(M) denotes the Frobenius root of an arbitrary nonnegative square matrix M,
it is the nonnegative real dominant eigenvalue of M. If the former condition for the ex
istence of nonnegative x0 is fulfilled then the output configuration x0 has the followingform:

xo (a, Co)-(I - A - aB)~['c0 .

(5)

3 Properties of the balanced growth path
We assume a balanced growth for the level of the environmental standards as well. Let
P denote the rate of growth permitted by government prescribed by the control author

ity. Then the vector of environmental standards has the form of e, = (1 + f5){e0 where e0
denote the initial limit level of emissions. Let us substitute this expression and the relations (3) and (5) in the inequality (2) then we obtain the following inequality(y^ )'E(I -A- aPVco <eo .

(6)Lemma 1. The growth rate of production and consumption a is limited by an upper

A Dynamic Leontief Pollution Model with Environmental Standards

43bound a* due to environmental regulations. That is the following limitation must hold
0 < a < a* where

w

Max
'

/ E(I-A - a*B)-lc0)i \

-*

V

—

(e0)i

t

}—^ =1i(7)and (•), denotes the /'''component of the respective vector.
Proof. We assume that we are at the beginning of the examined time period i.e. t = 0.

Using the relation (6), we determine the maximal growth rate a* for which the quantity of the pollutants generated is not more than the allowed legal limit. Then for thisa* must hold the equality (7).
Remark 1.

By applying environmental standards as the control of pollution, we should impose
more strict restriction on the chosen growth rate a than we have made previously (according to Dobos and Floriska (2005) the upper bound for a is the marginal growthrate a0, i.e. the balanced growth rate for the closed dynamic Leontief model ). Consid

ering ceo for the value of a * in the equality (7), the left-hand side of it will be an unbounded function for a0 . This implies that a* should be less than a0. That is the fol
lowing inequalities must hold: 0 < a ^ a* < a0.
Assumption 2.
In the next we suppose that at the beginning of the studied time period, the level ofpollution emission is less than the legal limit and the growth rate of production a isgreater than the growth rate of emission standards (3, a >/3 . So the environmental
standards are defined as

ft = (1 + py-e* for* = 1,2, ..., T,
and e0 is a given pollution level at the beginning of the planning horizon.

Under these assumptions, there will be exist a time t* such that the amount of one particular pollutant generated by industries will be equal to the allowed environmental
limit.

Lemma 2. The time t* can be calculated by the following formula:' =7TT^ir ln I (EW-A-aBVc )
lnTVp

(8)

where (•)/ denotes the /"' component of the respective vector and the ith row of the respective matrix.

Proof. By a simple mathematical calculation we express t* from the inequality (6).
This lemma gives estimation for the time interval without an adjustment process on en
vironmental regulation, i.e. environmental standard. After this point of time the econ-

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Journal of Applied Input-Output Analysis, Vol. 15, 2009

omy must change either production level or consumption rate, or both. In our model
we assume that first the production rate is adjusted to the environmental regulation andthen the consumption level. It can be proven that this kind of adjustment process leads
to a higher consumption level then another choice, i.e. first adjusted consumption andthan production.

Lemma 3. After the time t* (i.e. for t > t*), the maximum growth rate of production is
pProof. Denote y the growth rate of production after the time t*. Then the balanced
growth path of production has the form xt=(\ + y)'~'*xt. for t > t\ The balanced growth
path of the environmental standards for the prescribed growth rate p has the forme,=(l+ py~'*e,. for t > t\ Substituting these expressions for x,and e, the inequality (2) we obtain/ 1 + p \ <-<*

-—V 1
1 +
+ 7
y /

et* > Ex,* for t > t .

If y > P, for t —> oc the former inequality will be 0 > Ext*. This obviously is not ful
filled for every x,* ^ 0. If y = p, the former inequality will be et* ^ Ext*. This is obvi
ously fulfilled for the time t\ This concludes that the maximal value for y, i.e. themaximum growth rate of production is p.
This lemma allows us to construct the path of the production level. The productionlevel is grown with a growth rate a until point of time t* and after this point thegrowth rate is p. The growth rate can be determined as follows:(1 + a)''Xo
xt =t < t*(1 +/?r'-(l + a)tm-xo t*< t< T 'The next proposition makes it possible to calculate the consumption levels along theplanning horizon. Let us now define the growth rate of the production level in theplanning horizon as function of the time:_ Ia

t < t*

7'~\p t*< t< T'
The results of lemmas 1, 2, and 3 can be summarized in the next

Proposition 1. In case of a balanced growth solution of the model (1) and (2), for agiven rates of growth, the following must hold
(I-A - y - B)xt = ch fort = 1,2, ..., T.

Proof . This relation can be proved in similar way as we have got the relation (4).
The consumption rate can be constructed as(9)

A Dynamic Leontief Pollution Model with Environmental Standards(1 + a)''Co
Ct =

45

t < t*(1 +/?)'• (t^~|)V-A - P•fl)-(/-A - a •Byl-co t*< t<T 'Remark 2. An overview of the model.

The growth rate a of the balanced growth path of the system (1) could be at most a*according to the Lemma 1. In so far as this rate of growth is greater than the growthrate of environmental standards P, then this balanced path with rate of growth a, canbe continued at most to the time t according to Lemma 2. After the time t the maxi
mal growth rate of the balanced path is P, according to Lemma 3. The production cor

responding to such a path is growing with a rate of growth a until the time t*, andwith a rate of growth /? after this time. In case of different growth rates, to a given
level of production correspond different levels of consumption in a given time.
In the next lemma we analyze this change of the consumption level.

Lemma 4. The consumption level at the time t* is not less than it was at the time r*-l.
That is ct* ^ o-i.

Proof. For a ^ 0 the next inequality is obviously fulfilled:
ct* - o-i > ct* - (1 + a)c,*-i(10)Using the formula (9), Lemma 3 and the Remark 2 we get thatc* - (1 + a)c,*-, > (/ - A - PB)x,* - (1 + a)(I - A - aB)x,*-i .

(11)

For the rate of growth a we have

(1 + a)x,*-x -xt*

(12).

By substituting the equation (12) into the inequality (11) finally we obtain the inequality (10) in the following form:c,*-c,*-i > (a - P)Bx,*

The right-hand side of the previous inequality is nonnegative for a > P, B nonnega
tive matrix and x,* nonnegative vector. This concludes that c, - c,*-\ > 0.
Remark 3.

The decrease of the growth rate of production from the value a to the value P results
an excess supply of economic products in the year t\ Because the less growth rate ofproduction induces less investments in capital goods. This surplus of goods results a

sudden growth of the consumption level in the year t*.

4 A numerical example
In this section we will demonstrate the functioning of the proposed model. Let us as-

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Journal of Applied Input-Output Analysis, Vol. 15, 2009

sume that the investigated economy produces three goods and emits two pollutants.
The matrices of input coefficients, capital coefficients and emission coefficients are thefollowing:

A =

B =

E =

0.2

0.3

0.2

0.5

0.3

0.1

0.2

0.3

0.3

0.07

0.03

0.02

0.06

0.07

0.04

0.07

0.06

0.03

0.6

0.2

0.1

0.4

0.5

0.3

and

Using the results of assumptions 1 the marginal growth rate of the model is 0.376 (a0= 0.376). It means that a rational growth rate must be lower than this growth rate.
Let us assume next that the balanced growth rate a is equal 0.10 i.e. 10% and the vec
tors of initial consumption level cO and the initial environmental standards eO arec0=

and e0=90
130

We will assume that the growth rate of the environmental standards /3 is equal to 0.03,
i.e. 3 %. The planning horizon of the economy is T =35 years. Applying the equation(5) the initial output of the economy is
28.215*o(0.10) =35.59732.616The balanced growth path for the economy is as follows:(l+a)'-coCi=t < 18(l+y3r,9-c, 18 < t < 35 ^where the new initial consumption rate ce = (1 + a)"-c0 is
27.8
c=38.91933.616

The pollution for the first type of emission is depicted in Figure 1. with the environ
mental standards as an upper bound. (Pollution is depicted with a dotted line.) For thesecond type of pollution the upper bound is not achieved.
The second figure presents production level for the first activity. The dotted line showsbalanced growth path in case of no environmental regulation. It can be seen that theA Dynamic Leontief Pollution Model with Environmental Standards

47

Figure 1: The emission and the environmental standard for pollution 1
ouu

1

1

1

1

l

1

600

yy

yv

y^

400

-

(l+ay\Ex0], ^^^
el(00

200

_

Zj(j)j
i

i

i

i

1

1

10

15

20

25

30

35

Figure 2: The balaced growth path for the first sector without and with environ
mental standards
1500

1000

500

growth path will be lower after the environmental standard is attained.

Figure 3 shows the development of the consumption level in time. The dotted line rep
resents the consumption level in case of environmental standards. This numerical ex
ample supports the result of remark 3. After the environmental standards is achieved,
the consumption level is higher then without it. But after three time period the con
sumption level is lower then with environmental standard. The consumption increases
because less goods will be invested to increase the production level. It is a positive ef
fect of environmental regulation on the consumption.

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Journal of Applied Input-Output Analysis, Vol. 15, 2009

Figure 3: The path of the consumption level for the first good without and withenvironmental standards
30

5 Conclusion and further research

In this paper we have investigated the effect of environmental standards on production
and consumption in a dynamic Leontief model in case of balanced growth path. If theenvironmental standards are effective, i.e. it is a constraint on the production, then thegrowth rate of the production and consumption will be lower after the allowed levels

are attained. This means that a rigorous governmental regulation on environmental pol
lution restricts the emission of harmful materials, if the allowed growth rate of emis
sion is smaller than that of without restriction. In this case the emission standard is a

useful tool to avoid such kind of industrial activity of mankind, as global warming. Of
course, we can not offer the restriction of consumption in this process. The consump
tion level is higher after the new growth rate for some period compared to the path
without standards.

The investigated model assumes that in the economy there is no technological develop
ment. In a further research we could analyze the economy with technological develop
ment, i.e. the matrices of the model could be changed in time. In a modern economy
the research and development will develop new technologies to save the environment.
References

1. Dobos, I. and Floriska, A. (2005), "A Dynamic Leontief Model with Non-renewable
Resources", Economic Systems Research, Vol.17, pp.319-328.
2. Ebiefung, A.A., Udo, G. (1999), An Industrial Pollution Emission Control Model,
Computers & Industrial Engineering, Vol.37, pp.371-374.

A Dynamic Leontief Pollution Model with Environmental Standards

49

3. Leontief, W. (1986), Input-Output Economics, Oxford University Press, Oxford.
4. Leontief, W.W. (1970), Environmental Repercussions and Economic Structure: An
Input-Output Approach, The Review of Economics and Statistics, Vol.52, pp.262-271.

5. Luptacik, M. and Bohm, B. (1994), Reconsideration of Non-negative Solutions for
the Augmented Leontief Model, Economic Systems Research, Vol.6, No.2, pp. 167170.

6. Pearce, D.W. and Turner, R.K. (1990), Economics of Natural Resources and the Evironment, The Johns Hopkins University Press.
7. Schoonbeek, L. (1990), The Size of the Balanced Growth Rate in the Dynamic Leontief Model, Economic Systems Research, Vol. 2, No.4, pp.345-349.
8. Wirl, F. (1991), Evaluation of Management Strategies under Environmental Con
straints, European Journal of Operational Research, Vol.55, pp. 191-200.