The Effects of a Tax on Pollution
Course:
Date:
Teacher:
5114, Advanced Topics in Micro and General Equilibrium Theory
2003-10-14
Lars Bergman
Jonna Olsson 18875
Jeanette Reinbrand 18404
1
Table of Contents
TABLE OF CONTENTS ...................................................................................................................................... 2
INTRODUCTION ................................................................................................................................................. 3
QUESTIONS AT ISSUE ...................................................................................................................................... 3
THE MODEL ........................................................................................................................................................ 5
Production of goods ...................................................................................................................................... 5
The Households ............................................................................................................................................. 6
Equilibrium conditions .................................................................................................................................. 7
REVIEW OF THE RESULTS ............................................................................................................................. 8
CONCLUSION.................................................................................................................................................... 12
2
Introduction
The purpose of this report is to investigate the welfare effects of a tax on pollution. This will be
done in a general equilibrium model programmed in GAMS. The background is that pollution is
something that has negative effects on society. If there is no cost associated with the production
of pollution it can be assumed that initially, more pollution than what is socially optimal will be
produced. Taxing the pollution harmful to the environment could be a way to correct this and
therefore increase the overall welfare.
In theory, the market functions well when the price of a good is the same as the social cost for
producing that good. However, sometimes the social cost to produce a good exceeds the cost the
company face, and hence also the price the consumer pays. One common example of this is
pollution. If pollution is not associated with any cost for the company, there will be a social cost
which will not show up in the final price of the good. The pollution can be viewed as a negative
externality, and initially, before correction, more pollution than what is socially desirable is
produced.
According to theory, this can be corrected for if the company has to incorporate the cost for the
negative externality. This can be done by taxing pollution. However, such a tax on pollution can
also be negative for the household. Taxes may have distorting consequences and can also prevent
households to consume goods in optimal proportions. The question is then how these different
effects could be balanced.
A question like this is very suitable to study in a general equilibrium model. The relations between
taxes, changes in goods prices, changes in factor prices, and changes in consumption patterns are
such a complex issue that a partial equilibrium model hardly would give a fair picture. To capture
all the aspects of the question, a general equilibrium model has to be used.
Questions at issue
The questions we want to investigate in our model are:
1. Does a tax on environmentally harmful pollution reduce the production of it?
For a good with neither perfectly inelastic supply nor perfectly inelastic demand, an increase in
price will lead to a decrease in the number of products sold. Therefore, in equilibrium, the
production of the good must decrease as well. By taxing the production of the good harmful to
the environment, which is done by taxing pollution, the production cost increases and hence also
the price of the good (in equilibrium price equals marginal cost). Assuming that the good harmful
to the environment has neither perfectly inelastic supply, nor perfectly inelastic demand, a price
increase will thus lead to a decrease in production.
A decrease in the production of the good harmful to the environment will also free resources that
can be used to produce the environmentally friendly good. Hence, an increase in the production
of the environmentally friendly good can be expected when pollution is taxed.
2. Can a tax of this kind increase total welfare?
Assuming that the households value pollution negatively, like a “bad good”, a decrease in
pollution should lead to an increase in welfare. If a tax can lower the amount of pollution, total
welfare should consequently increase. However, it is not evident that all tax rates will lead to an
increase in welfare. See next question for a discussion of this.
3
3. If it can, which tax rate is optimal?
Even though a tax on pollution will lead to a decrease in pollution, it is not evident that all tax
rates will lead to an increase in welfare. Not only a reduction of pollution affects the welfare of
the households. A large consumption of the good harmful to the environment, as well as a large
consumption of the environmentally friendly good, increases welfare. Therefore, it is possible
that there is a trade-off between the decrease in consumption of the good harmful to the
environment and the reduction of pollution. This means that if the tax rate is too high, the
decrease in consumption of the environmentally harmful good cannot be compensated by the
benefits from a decrease in pollution and the increase of consumption of the environmentally
friendly good. At the same time, a too low tax rate would mean that the benefit from a large
consumption couldn’t compensate the welfare losses from the pollution. The challenge is to find
the tax rate that maximizes the household’s welfare, by reducing the pollution and on the same
time not having too distortion effects.
4. In what ways does a tax on pollution affect factor prices and income distribution?
As mentioned earlier, the introduction of the tax will probably affect the relative prices of the
goods. Since the goods use production factors in different proportions it seems plausible that the
return to the different production factors also will be affected.
Since different households own capital and labor in different proportions a change in factor
prices might affect income distribution. If we assume that the return to labor will decrease, such a
change would have a more negative effect on the households that own relatively more labor than
on the households which own relatively more capital. In our model, we assume that the poor
households own relatively more labor than the rich households. A decrease in the return to labor
would hence mean that the income of the poor households would decrease, while the income of
the rich households would increase.
Therefore, from a redistributional perspective, it is also interesting to investigate the effects on
factor prices that a tax on pollution might have, and the effects on income distribution this
consequently will have.
5. Can the optimal tax rate be different for different types of households, depending on their resources and
preferences?
One of the criticisms against a tax on pollution is that it would disadvantage low income
households and people on the countryside. With such a tax, these groups would have to spend a
larger part of their income on the environmentally harmful good, which would have a negative
effect on welfare. This way of reasoning presupposes that the good harmful to the environment
is some sort of basic commodity that low-income households consume relatively more of.
The background of this argument is that this sort of tax often is introduced on energy. Energy is
something that everyone has to consume a minimum quantity of. The demand for such a good
harmful to the environment is very close to perfectly inelastic, until a certain amount is consumed.
Assuming that the good harmful to the environment is such a basic commodity, what will happen
to the consumption patterns?
It is plausible that households in different income groups will value the reduction of pollution
differently. With a high tax rate the households that consume relatively more of the good harmful
to the environment will have a lower utility than the households that consume the same good
relatively less. The question is then if and how the optimal tax rate differs between different
households.
4
The Model
We use two variations of one basic model. Initially, we have only one household, which owns all
production factors. In the second version we use two households, of which one represents “rich”
households, and the other represents “poor” households. Otherwise, the two models are identical.
Our model is a closed economy, no foreign trade is included. A description of the most
important characteristics of the model follows, where i = 1, 2.
Production of goods
Production function
In the economy two goods, good 1 and good 2, are produced with capital and labor. The
production function for both goods is assumed to be of Cobb-Douglas type and looks as follows:
xi  An i k 1 i
where xi is the production volume, n the amount of labor, k the amount of capital used and A
and α both are constants.
Good 1 is assumed to be labor intensive and good 2 capital intensive. For good 1 we set α, the
output elasticity, to 0.75 and for good 2 0.25.
Pollution
The pollution from the production is defined in the following way:
POLLi  Pi  xi
where Pi is the amount of pollution from the production of one unit of good i. P1 is assumed to
be 0, and P2 is assumed to be 1. The production of the capital intensive good, good 2, is hence
assumed to create pollution, while the labor intensive good is assumed to be produced with a
environmentally friendly technology.
The assumption that the capital-intensive good is harmful to the environment can definitely be
discussed. We have chosen to assume this because of the current energy production situation in
Sweden. Very capital-intensive coal burning and nuclear power plants represent a large share of
total production. At the same time, it can be argued that services, which are labor intensive, often
are environmentally friendly.
Tax
In the model a tax on pollution is included. We incorporated this tax by increasing the cost of
production by the tax rate times the amount of pollution caused by production. In model 1, the
tax is transferred back to the single household as a lump sum transfer. In model 2, with two
households, the transfer is initially divided equally between the two households.
Demand for production factors
The cost function hence looks like the following:
i w, r , xi   wni  rk i  TAX POLL  Pi  xi
5
where w represents the return to labor, r represents the return to capital and ni and ki the amount
of labor and capital respectively. TAXPOLL is the tax rate per unit of pollution.
From this cost function we can derive the demand for n and k:
 ir 

ni  A 
 1   i w 
1 i
1
i
 ir 

k i  A 


1


w
i


 xi
 i
1
i
 xi
Supply of goods
With perfect competition the price of a good equals the marginal cost of production. The
following relation therefore determines the price of the different goods:
pi 
i ( w, r , xi )
 1
 Ai1 ii 1   i  i wi r 1i   TAX POLL  Pi
xi
The Households
The household’s budget restriction
In model 1 there is only one household, which owns all labor and all capital. The household gets
all its income from these factors. The supply of both labor and capital is assumed to be constant.
We assume the total amount of labor to be 100, and the total amount of capital to be 100 as well.
The following relation therefore gives the income of the households:
m  wn  rk  tax transfer
and the budget restriction is therefore:
m  p1c1  p2 c2
Utility function
The household/households are assumed to have a utility function that builds on the classical
Cobb-Douglas function, but where pollution is added as a separate term. The utility function
then looks like the following:
U H   c1  c12   e  POLL
where c1 and c2 represent the consumption of good 1 and good 2 respectively, POLL represents
the total amount of pollution in society, and β states how the households value their
consumption of good 1 and good 2. γ, finally, is the parameter that states how the households
value pollution. If γ is positive the households are assumed to appreciate pollution, and if γ is
negative the households dislike pollution. The smaller the γ, the more the households dislike
pollution.
Initially, we assume β to be 0.5, i.e. that the households value the two goods equally.
6
Demand
When we combine the budget restriction and the utility function of the household we can derive
the demand function for the different goods:
c1  p1 , p2 , m  
m
and
p1
c2  p1 , p2 , m  
(1   )m
p2
Equilibrium conditions
We then have a number of equilibrium conditions that together define our economy. They can be
divided into two main groups: one that concerns the goods market and one that concerns the
factor market.
Goods market
The supply (i.e. the production) should equal the demand (i.e. the consumption of the
households):
x1 
x2 
m
p1
1   m
p2
The price should equal marginal cost of production:
p1  A111i 1  1 
1 1 1 11 
w r
 TAX POLL  P1
2 1 2 12 
 TAX POLL  P2
p2  A21 2i 1   2 
w r
Factor market
The supply (i.e. the households’ total resources) should equal the demand for the different factors:
11
 1r 

n  A 
 1   1 w 
1
1
1 2
  2r 

x1  A 
 1   2 w 
1
2
1
x2
2
 1r 
  2r 
 x1  A21 
 x2
k  A 
 1   1 w 
 1   2 w 
1
1
We then have six equations and six endogenously determined variables: x1, x2, p1, p2, w and r.
7
Review of the results
In this section we investigate what results our model gives with respect to the five questions
posed before.
1. Does a tax on environmentally harmful pollution reduce the production of it?
We use our initial model with one household in order to investigate if a tax on environmentally
harmful pollution reduces the production of it. In the base case, where the tax rate is set to 0 %,
100 units of each type of good, good 1 (environmentally friendly good) and good 2
(environmentally harmful good), are produced (see figure 1).
Figure 1. Social accounting matrix for one household, tax rate equal to 0 %.
x1
x2
n
k
h
Total
x1
x2
75
25
25
75
100
100
n
k
100
100
100
100
h
100
100
Total
100
100
100
100
200
200
With a tax rate greater than 0 %, the production of the environmentally harmful good is reduced.
When we arbitrarily set the tax rate to 10 %, the production and consumption of the
environmentally harmful good are reduced from 100 units to 96.45 units. The production of
pollution is thereby also reduced. At the same time, the production and consumption of the
environmentally friendly good increase, from 100 to 103.51 units (see figure 2). These changes in
production can easily be understood: A reduced production of the environmentally harmful good
frees capital and labor, which now can be used in the production of the environmentally friendly
good. It should be noted that no matter what (positive) tax rate we use, we always get the same
results, namely that the production of the environmentally harmful good is reduced.
Figure 2. Social accounting matrix for one household, tax rate equal to 10 %..
x1
x2
n
k
h
Total
x1
x2
76.702
26.810
23.244
73.203
103.512
96.447
n
k
100
100
100
100
H
103.512
96.447
Total
103.512
96.447
100
100
200
200
The price of good 2 has become higher relative to the price of good 1 (see GAMS print-out,
appendix 1). According to the household’s utility function of Cobb-Douglas type, the household
wants to consume equal amounts of both goods ( p1c1  p2 c2 ). The household therefore spends
the same amount of income on the consumption of each of the two goods.
2. Can a tax of this kind also increase total welfare?
and
3. If it can, which tax rate is optimal?
In order to examine these two problems we start from our initial model with one household. In
the base case, when the tax rate is equal to 0 %, the household has a total utility (welfare) of 95.12.
When setting the tax rate to 10 %, utility increases (see GAMS print-out, appendix 1). The reason
for this is that the household values pollution negatively. Utility thus increases when pollution is
reduced.
8
However, if the tax rate is arbitrarily set to 20 %, utility decreases again. This outcome is due to
the fact that there is an optimal tax rate that maximizes the welfare of the households. The reason
for this is that there is a trade-off between the reduced consumption of the environmentally
harmful good and the reduced amount of pollution (for a more thorough discussion of this
relationship, see the above section “Questions at issue”). From the diagram we see that there
indeed is a tax rate that maximizes total welfare. In our case, i.e. when
γ = -0.0005, the optimal tax rate amounts to about 11 % (see figure 3).
Figure 3. The relationship between tax rate and welfare when γ = -0.0005.
95,3
95,2
95,2
Välfärd
W
e
l
f
a
r
e
95,1
95,1
95,0
95,0
94,9
94,9
0%
10%
20%
30%
TaxSkattesats
rate
The optimal tax rate depends on how negatively the households value pollution. With a smaller
value on γ, the optimal tax rate would have been higher. If we, on the other hand, had chosen a
higher γ, the optimal tax rate would have been lower. Figure 4 depicts the relationship between
welfare and tax rate when γ = 0. In this case, the optimal tax rate is 0 %.
Figure 4. The relationship between tax rate and welfare when γ = 0.
100,0
99,8
Välfärd
W
e
l
f
a
r
e
99,6
99,4
99,2
99,0
0%
10%
20%
30%
Skattesats
Tax
rate
4. In what ways does a tax on pollution affect factor prices and income distribution?
In order to examine this problem we need to introduce one more household in our model. We
assume that one of the households, the ‘rich’ household, owns more resources than the other.
Consequently we assume that the other household, the ‘poor’ household, owns less resources.
Total supply of labor and capital is the same as before. Furthermore, we assume that the rich
9
household owns relatively more capital, while the poor household owns relatively more labor.
Figure 5 shows the base case of this economy.
Figure 5. Social accounting matrix for two households, tax rate equal to 0 %.
x1
x2
n
k
h_rich
h_poor
Total
x1
X2
75
25
25
75
100
100
n
k
60
40
100
80
20
100
h_rich
70
70
h_poor
30
30
140
60
Total
100
100
100
100
140
60
In the base case, when the tax rate is set to 0 %, the price of capital is 1 and the price of labor is 1.
The factor income is divided in the following manner: the poor household receives 30 % and the
rich household 70 % of total factor income. However, when introducing a tax, factor prices and
thereby also the distribution of factor income are changed.
Figure 6. The distribution of factor income at different tax rates.
Tax rate
w
r
0%
5%
10 %
15 %
20 %
30 %
50 %
1
1.025
1.048
1.071
1.093
1.135
1.213
1
1
1
1
1
1
1
Rich household’s share of
total factor income
70 %
69.9 %
69.8 %
69.7 %
69.6 %
69.4 %
69.0 %
Poor household’s share of
total factor income
30.0 %
30.1 %
30.2 %
30.3 %
30.4 %
30.6 %
31.0 %
Total factor
income
100 %
100 %
100 %
100 %
100 %
100 %
100 %
It is worth noting that we at this stage only investigate factor income. The total amount of
resources possessed by the households is not examined. The reason for this is that we do not
want to include effects of tax transfers in our analysis at this stage. Instead we want to isolate the
effects of changes in factor prices. Since the rich and the poor household share the same utility
function, both households will have the same set of preferences and both will consume the two
goods in equal proportions. The distribution of tax transfers will therefore not affect factor prices.
From figure 6 it is seen that a tax on pollution will increase the poor household’s share of factor
income. Since we have assumed that it is the capital intensive good that causes pollution, the
price of this good will rise. Consequently, the demand for and thereby also the production of this
good will decrease, while the demand for the environmentally friendly, labor intensive, good will
increase. As a result, the demand for labor increases. If instead the labor intensive good would
have been environmentally harmful, these conclusions would have been reversed, i.e. that the
poor household’s share of factor income would have decreased.
These results have interesting implications for potential policy decisions. When evaluating how
different groups in society are affected by a tax on an environmentally harmful good,
consideration also has to be taken to the distribution of production factors between different
groups as well as what production factor is used intensively in the production of the taxed good.
5. Can the optimal tax rate be different for different types of households, depending on their resources and
preferences?
In order to examine this problem we want the different households to consume good 1 and good
2 in different proportions. Ideally, we want the households to have the same utility function: a
utility function where the goods are consumed in different proportions depending on the
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household’s level of income. However, a utility function of this kind is very difficult to model. As
an approximation we therefore let the two households in our model have different values of β.
Rich:
Poor:
β = 0.5
β = 0.48
This means that the poor household value good 2, the environmentally harmful good, higher.
Since the two households now have differing preferences and therefore do not consume the two
goods in equal proportions, we can no longer ignore tax transfers. We begin this analysis by
investigating what the relationship between welfare and tax rate looks like when 50 % of the tax
revenue is transferred to the poor household and the other 50 % is transferred to the rich
household. The result of this is displayed in figure 7.
Figure 7. The relationship between tax rate and welfare for two households when γ = -0.0005.
50 % is transferred to the rich household, 50 % to the poor household.
(Note that welfare is measured as a share of welfare when tax rate equals 0%.)
1,15
1,10
Välfärd
W
e
l
f
a
r
e
1,05
Poor household
1,00
Fattigt hushåll
Rich household
0,95
Rikt hushåll
0,90
0,85
0,80
0%
10%
20%
30%
Skattesats
Tax
rate
Figure 7 clearly shows that higher taxes are beneficial for the poor household, but are
disadvantageous to the rich household. The explanation for this is that a 50 % transfer of the tax
revenue to each of the two households in practice is nothing but a transfer from the rich to the
poor household. This is thus the relationship which is seen in the results.
In order to deepen our analysis we now choose to distribute the transfers from the tax revenue in
a manner which will make the distribution of the total disposable income equal to what it was
before the introduction of the tax. A very close approximate of this distribution is obtained when
75 % of the tax revenues is transferred to the rich household and 25 % to the poor household. At
a tax rate of 10%, the rich household has a total disposable income of 140.17 and the poor
household has a disposable income of 59.81 (see figure 8 and GAMS print-out, appendix 1). This
distribution of transfers gives the same distribution of disposable income also for other tax rates.
11
Figur 8. Social accounting matrix for two households, tax rate equal to 10 %.
75 % transfer to the rich household, 25 % to the poor household.
X1
X2
N
k
H_rich
H_poor
Total
x1
x2
76.26
26.38
23.65
73.69
102.63
97.34
n
k
60
40
100
80
20
100
h_rich
72.78
67.39
h_poor
29.85
29.95
140.17
59.81
Total
102.63
97.34
100
100
140
60
We can now analyze how different tax rates affect the welfare of the two different types of
households. Figure 9 shows that the two households do not have the same optimal tax rate. Since
the poor household values the environmentally harmful good higher than the rich household
does, the poor household will have a lower optimal tax rate than the rich household.
Figure 9. The relationship between tax rate and welfare for two households when γ = -0.0005.
75 % transfer to the rich household, 25 % to the poor household.
(Note that welfare is measured as a share of welfare when the tax rate equals 0%.)
1,002
1,001
Välfärd
W
e
l
f
a
r
e
1,000
Poor
Rikt household
hushåll
0,999
Rich
household
Fattigt
hushåll
0,998
0,997
0%
10%
20%
Skattesats
Tax rate
We can thus conclude that the effects of a tax on pollution for different groups in the society
depend on several factors. In order to obtain a correct understanding of the effects of such a tax
consideration has to be taken to the fact that different households own different proportions of
production factors, that different households can have different preferences and how the transfer
scheme is set up.
Conclusion
By using a general equilibrium model programmed in GAMS, we have analyzed the effects of
taxing an environmentally harmful good. We have seen that a tax on an environmentally harmful
good reduces the production of this good, while the production of the good without the tax, the
environmentally friendly good, increases.
Furthermore, we have seen that introducing a tax can increase total welfare. The extent to which
welfare is increased is, however, dependent on the size of the tax rate. We have shown that there
is an optimal tax rate that maximizes utility. The size of the optimal tax rate depends on how
negatively the households value pollution: the more a household dislike pollution, the higher the
optimal tax rate will be.
12
When introducing two households in our model, we have seen that there is a number of different
factors that have be taken into account when analyzing how a tax on an environmentally harmful
good affects different households. One effect of such a tax is that factor prices change, which in
turn affects the distribution of factor income between households. Moreover, different
households can have different preferences, which give different values of the optimal tax rate.
The welfare resulting from imposing a tax is also dependant on how the tax transfer scheme is set
up.
We can thus conclude that imposing a tax has a number of different effects. In order to capture
all these effects it is necessary to study the introduction of this kind of tax in a general
equilibrium model. A partial equilibrium model would not have been able to provide the same
kind of comprehensive understanding.
Today’s discussion concerning the green tax shift inspired us to analyze the effects of imposing a
tax on environmentally harmful goods. A green tax shift means that the revenues generated by
taxing an environmentally harmful good are used for reducing labor costs, for example in the
form of reduced payroll tax. It would have been interesting to combine our study with the effects
of such a shift.
In order to improve our model and to a larger extent make it reflect reality, reasonable values on
the parameters and input data of the model are needed. Although difficult to estimate, they are
necessary for making the model more useable for policy decisions.
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