Once the equilibria for all firms and the government are derived as discussed in
23
previous. We have derived mixed and private oligopoly, and we assume that the number of firms can be observed, if n = 10.
Proposition 3.3 When the government’s preference and pollution rate are higher, pollution tax under unionized mixed oligopoly is higher than the one under unionized privatize oligopoly.
Thus, the differences in the optimal tax, Define that ∆t ≡ 𝑡𝑚− 𝑡𝑝, ∆t courve describes the contour of the ∆t = 0 along which the productivity and firm's objective function are identical between mixed and private oligopoly.
Figure 3.1 Illustration of Proposition 3.3
By the diagram above, the less pollution ratio that pollution tax paid by the firms are also relatively small. Privatization firms production to maximize profits so that the output will increase. Therefore, the higher the government preference will make the privatization of the manufacturers of higher pollution tax.
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When the degree of pollution is high, the government preference is higher will make pollution tax under mixed oligopoly are higher than privatized oligopoly.
Because in the mixed oligopoly, the production higher cause of social welfare the higher, even if the government levied a higher pollution tax does not reduce social welfare. Therefore, pollution tax under mixed oligopoly are higher than privatized oligopoly.
Corollary 1 When the government preference and pollution ratio the higher, social welfare under mixed oligopoly are higher than privatized oligopoly.
Thus, the differences in the optimal tax, Define that ∆SW ≡ 𝑆𝑆𝑚− 𝑆𝑆𝑝, ∆t courve describes the contour of the ∆SW = 0 along the productivity and the output are identical between mixed and private oligopoly.
Figure 3.2 Illustration of Corollary1
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Corollary 2 When the government preference and pollution ratio the higher, consumer surplus under mixed oligopoly are higher than privatized oligopoly.
Thus, the differences in the optimal tax, Define that ∆CS ≡ 𝐶𝑆𝑚− 𝐶𝑆𝑝, ∆t courve describes the contour of the ∆CS = 0 along which the productivity and firm's objective function are identical between mixed and private oligopoly.
Figure 3.3 Illustration of Corollary 2
Corollary 3 When the government preference and pollution ratio the higher, environmental damage under mixed oligopoly are higher than privatized oligopoly.
Thus, the differences in the optimal tax, Define that ∆ED ≡ 𝐸𝐸𝑚 − 𝐸𝐸𝑝, ∆t courve describes the contour of the ∆ED = 0 along which the productivity and firm's objective function are identical between mixed and private oligopoly.
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Figure 3.4 Illustration of Corollary 3
3.5 Concluding Remark
We consider a mix-oligopoly situation for a homogeneous good that is supplied by a public firm and n(≥ 1), private firms and the good produced by each firms will lead to pollution. This section analyses that the impact of government preference for tax on social welfare. The results obtained are the following.
Firstly, in the mixed oligopoly, firm production will cause pollution, social welfare increase with an increase government preference for tax revenues. Secondly, a unionized private oligopoly that there are production will cause pollution, social welfare increase with an decrease government preference for tax revenues. Lastly, when the government preference and pollution ratio are higher, social welfare, consumer surplus and environmental damage under mixed oligopoly are higher than privatized oligopoly.
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CHAPTER FOUR: The Effects of Pollution Taxes on Urban Areas with an Endogenous Plant Location and Pollution Emitted.
This result is different from Hwang and Mai (2004) who indicate that the tax revenue is independent of a change in the distance between plant and CBD case. In this paper, we show that the pollution tax revenue has positive correlation with the distance between plant and CBD.
We find that government makes upgrade the resident utility if a higher pollution tax may lead a lower level of pollution measured at CBD. Further to say, a stricter pollution policy such as a higher pollution tax lead to lower pollution damage to the CBD irrespective of the type of Return to scale.
Although the paper assumes the markets to be of monopoly, the intuition derived in this paper is robust in other market structures, such as oligopoly or perfect competition.
4.1 Basic Model
Figure 4.1 Locational triangle.
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The analysis is based on the following assumptions:
(a) There is a monopoly industry in which firm produce homogeneous output.
(b) The firm uses two transportable inputs 𝑀1 and 𝑀2 , which are available at A and B, respectively ,in the production of output q , which it supplies to the consumption center C, as illustrated in Figure4.1. The firm is interested in finding the optimum production location E. In figure4.1, the distance a and b and the angle 𝛽 are known; h is the distance between the plant location E and the CBD(C);𝑧1and 𝑧2 are the distance of plant E from A and B, respectly;θ
(c)The production function of the firm can be specified as:
q = f(𝑀1, 𝑀2) with 𝑖𝑀1 ≡ ∂M1∂q > 0, 𝑖𝑀2 ≡ ∂𝑀∂q
2 > 0, 𝑖𝑀1𝑀1 ≡∂M1∂2q2< 0, 𝑖𝑀2𝑀2 ≡∂M2∂2q2 < 0 (4.1) (d)To simplify our analysis, we first derive the total cost function by minimizing total cost subject to a given output level. That is
Min
(𝑤1+ 𝑟1𝑧1)𝑀1+ (𝑤2+ 𝑟2𝑧2)𝑀2s.t.
q = f(𝑀1, 𝑀2) (4.2)where 𝑤1 and 𝑤2 are the base price of 𝑀1 and 𝑀2 at A and B ,respectively, which are assumed to be constant; 𝑟1and 𝑟2 are constant transport rates of 𝑀1 and 𝑀2 respectively; and 𝑧1 and 𝑧2 may be defined by the low of cosines as follows:
𝑧1 = √𝑎2+ ℎ2− 2𝑎ℎ cos 𝑑
(4.3)
𝑧2 = �𝑏2+ ℎ2− 2𝑏ℎ cos( 𝛽 − 𝑑)
(4.4)
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(e)For homothetic production function, the cost function can be written as the product of two functions of factor prices only. We have:
C(q) = c(𝑤1+ 𝑟1𝑧1, 𝑤2+ 𝑟2𝑧2)H(q) (4.5)
Hence, the average cost and marginal cost can be written as:
AC =𝐶(𝑞)𝑞 =𝑐𝐻𝑞 (4.6)
MC=𝐶𝑞 = 𝐶𝐻𝑞 (4.7)
Following Hanoch (1975), form(4.6)and(4.7), we obtain the following relation:
𝐻
𝑞 > (=, <)𝐻𝑞 (4.8)
if the production function is increasing (constant, decreasing) returns to scale,i.e.,IRS(CRS,DRS).
(f) The industry inverse demand function for output is given by
P = P(q), 𝑃𝑞 < 0 (4.9)
(g)The prices of inputs and output are evaluated at the plant location E. The cost of purchasing inputs in the price of input at the source plus the freight cost, and the price of output is the market price minus the freight cost.
(h) Transportation rates are constant.
(i)The pollution tax revenue function G(q) is specified as follows:
G(q) = em(h)y(q) (4.10)
where e is the pollution tax, m(h) is the relationship between the pollution level at CBD and y(q) is the a mount of pollution generated by the production process which depends on the amount of output produced.
30
where G(q) = em(h)y(q) and t is the constant transport rate of shipping one unit of the output to the CBD.
We assume throughout the paper that emission rises linearly with output(i.e.,𝑒𝑞 >
0 and 𝑒𝑞𝑞 = 0). Given this assumption, we can immediately derive that 𝐺𝑞 = 𝑒𝑚𝑒𝑞 > 0, 𝐺𝑞𝑞 = 𝑒𝑞 > 0and 𝐺𝑞 > 0. Moreover, an increase in e indicates that the government adopts a stricter pollution policy.
With these assumptions, the profit maximizing location problem of the representative firm is given by
Max π=[P(q)-th]q-c(θ,h)H(q)-G(q) (4.11)
q, θ,h
The first-order conditions for profit maximization are:
𝜋𝑞= (𝑃 − 𝑡ℎ) + 𝑃𝑞𝑞 − 𝑐𝐻𝑞− 𝐺𝑞 = 0 𝜋𝜃 = −𝐶𝜃H = 0
𝜋ℎ = −𝑡𝑞 − 𝐶ℎ𝐻 − 𝑒𝑒𝑚ℎ= 0
Moreover, taking the total different and using Cramer,s rule, we can examine the effects of a change in the demand function on the optimum location:
�
31
We can derive the effects of an increase in the pollution tax on and q、h and θ respectively, as
Since the effect of a change in the pollution standard on production is important in understanding the economic forces controlling the optimal plant location and the measurement of pollution emission, we shall address this issue first. It follows immediately from Equation (4.13) that:
When the production is DRS then exhibits three outcomes, a higher pollution taxes may leads to a lower, invariant or higher output level.
We now turn to the effect on locational choice. It follows from Equation (4.14) that: When the production is DRS then exhibits three outcomes, a higher pollution tax may leads closer to, invariant or farther away from the CBD.
32
According to Equations (4.8) and (4.17), we can derive:
Proposition 4.1 The output level of the firm is higher as a result of a higher pollution tax, if the production function exhibits decreasing returns to scale.
The effect of pollution tax on the optimum output level is, perhaps, surprising.
According to HM (2004), tax revenue is independent of a change in the distance between plant and CBD, an increase in the pollution tax rate will decrease the output level. But the above result shows at HM，s result can not apply to the this case. The economic interpretation behind Proposition 1 is given as follows.
In this paper, the first-order condition for profit maximization of location is
𝜋ℎ = −𝑡𝑞 − 𝐶ℎ𝐻 − 𝑒𝑒𝑚ℎ. The relationship between the firm determining the location and the pollution level at CBD is negative correlation (𝑚ℎ < 0). In other words, the pollution tax revenue not only about output level but also have relationship the distance which between the plant location E and the CBD. As a result, it will make the output level of the firm will increase as the pollution tax rate increase if the production function is decreasing returns to scale.
In Hwang and Mai (2004), is not consider space factor of the pollution tax. In other words, the pollution tax revenue is unrelated the distance h that between the plant location E and the CBD.
Lemma 4.1 (Hwang-Mai 2004): The plant location of the firms is invariant with
respect to a change in the pollution functions is CRS. Nevertheless, the plant location moves closer to (farther away from) the CBD as a result of a higher pollution tax if production is DRS (IRS).33
Proposition 4.2 The plant location of the firm is further away from the CBD as a result of a higher pollution tax, which is irrespective of the type of return to scale.
The logic behind this proposition is straightforward. In Hwang and Mai (2004) , the location determine is independent with pollution tax(𝜋ℎ𝑞 = 0). In this paper, the firm choice the location that in order to maximum profit and pollution tax are positive correlation (𝜋ℎ𝑞 > 0). We know that the plant location moves closer to the CBD can decrease transport cost, but the distance of plant E from CBD more closer that the amount of pollution is more higher.
As a result, the effect on location choice that not only determine production function but also to depend on the relation between distance and pollution rate.
4.2 The Impact of the Resident Utility on Pollution Taxes
Next, we examine the impact of pollution taxes on the pollution level at the CBD. The pollution level measured at the CBD is lower than at the plant location and is affected by the distance between the plant location the CBD. Following Hwang and Mai (2004), the pollution level at CBD is specifies as:
𝑋∗ = 𝑚(ℎ)𝑋 (4.18)
where X= total pollution measured at E, 𝑋∗=total pollution measure at C, m(h)=relationship between the pollution at CBD and the plant location. With 𝑚ℎ < 0 and 𝑚ℎℎ > 0,implying that as the distance between CBD and the plant goes up, the pollution measured at C not only declines but also declines at a decreasing rate.
Then, the impact of a higher pollution tax (i.e., an increase of e) on 𝑋∗ is derivable as follows:
∂𝑋∗
∂e = 𝑚𝑒𝑞𝑞𝑞+ 𝑚ℎℎ𝑞𝑒 = output effect + location effect (4.19)
34
when 𝜕𝑋∗
𝜕𝑞 < 0, there are three situations can be increase in e may lead to a lower level of pollution measured at CBD
i.
ℎ𝑞 > 0, 𝑞𝑞 < 0 ⇨ the production function is (ALL, ALL)ii.
ℎ𝑞 > 0, 𝑞𝑞 < 0 ⇨ the production function is (ALL, DRS)iii.
ℎ𝑞 < 0, 𝑞𝑞 < 0 ⇨ the production function is (DRS, ALL)which ℎ𝑞 > 0, 𝑞𝑞 < 0 is necessary condition can be sure be increase in e lead to a lower level of pollution measured at CBD.
Now, we can set a function to measure the resident utility at C in Figure4. 1, is given by:
G(𝑋∗) = U(𝑋∗) + se𝑋∗, 𝑈𝑋∗ < 0
Where U is resident utility at CBD, and s > 0 is percent rate of the government subsidies to residents. We take partial derivative of G with respect of e to obtain
∂G
∂e= [𝑈𝑋∗+ 𝑠𝑒]𝜕𝑋𝜕𝑞∗+ 𝑠𝑋∗ (4.20)
we can show 𝑈𝑋∗+ 𝑠𝑒 < 0,and 𝑠𝑋∗ > 0, Thus, we can conclude that
Lemma 4.2 (Huang-Mai 2004): If the production function is DRS, then a higher
pollution tax may increase the pollution damage to CBD residents.Proposition 4.3 When the government protects the resident and implements pollution taxes, it will increase the utility of residents. Moreover, if the production function of plant is irrespective of the type of return to scale , than a higher pollution tax will definitely decrease the pollution to the CBD.
If the Government takes pollution taxes, would increase the utility of
35
residents �
∂G∂e> 0�, than
the pollution taxes and total pollution measure at CBD mustbe negatively correlated �
∂𝑋∂e∗< 0�. Form
∂𝑋∂e∗< 0, we can know the
necessary condition isℎ
𝑞> 0 and 𝑞
𝑞< 0, regardless of the plant is IRS, CRS or DRS.
4.3 Concluding Remarks
This result is quite different from Hwang and Mai (2004) who indicate that the tax revenue is independent of a change in the distance between plant and CBD case.
In this paper, we show that the pollution tax revenue has positive correlation with the distance between plant and CBD.
In Hwang and Mai (2004) conclusion ”Our paper has show that a stricter pollution policy such as a higher pollution tax may lead to higher pollution damage to CBD residents when the plant location is endogenous and the production technology of the firm exhibits DRS .” This statement may not hold true in our spatial model.
Instead of, the government makes upgrade the resident utility if a higher pollution tax may lead a lower level of pollution measured at CBD (I.e., ∂𝑋∗
∂e < 0) . From (4.19), we know 𝑑𝑞
𝑑𝑞 < 0 and 𝑑ℎ𝑑𝑞 > 0 are necessary conditions if a tougher pollution control usually results in less pollution damage. Further to say, a stricter pollution policy such as a higher pollution tax lead to lower pollution damage to the CBD irrespective of the type of Return to scale.
Although the paper assumes the markets to be of monopoly, the intuition derived in this paper is robust in other market structures, such as oligopoly or perfect competition.
36
CHAPTER FIVE:CONCLUSION
In chapter 2, we explored the design of environmental policy under the consideration of union bargaining and homogeneous firm in duopoly. We find that emission and profit tax are imposed that the firms of two identical, the social welfare under centralized wage will be lower than the one under decentralized wages, while the environmental damage and firm’s profit under centralized wage will be higher than the one under decentralized wages.
In chapter 3, we consider a mix-oligopoly situation for a homogeneous good that is supplied by a public firm and private firms and the good produced by each firms will lead to pollution. We found that even if the Government favored tax, the output of the entire market will not decreased instead also increase of total output and social welfare will also increase.
In chapter 4, we consider space into the effect of a direct pollution control on the pollution damage and the pollution tax will change with location. We find that the government make upgrade the resident utility if an higher pollution tax may lead a lower level of pollution measured at CBD. Further to say, a stricter pollution policy such as a higher pollution tax lead to lower pollution damage to the CBD irrespective of the type of return to scale.
37
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Mai, C.C., and Hwang, H.,(1992). “Production-Location Decision and Free Entry Oligopoly.” Journal of Urban Economics, 31, 252-271.
Mai, C.C., and Hwang, H.,(2004). “The Effects of Pollution Taxes on Urban Areas with an Endogenous Plant Location.’’ Environment & Resource Economics, 29, 57-65.
Mathur, V.K., (1976). “Spatial Economics Theory of Pollution Control.” Journal of
Environmental Economics and Management, 3, 16-28.
Mujumdar, S., Pal, D., (1998). “Effects of indirect taxation in a mixed oligopoly. ”
Economics Letters, 58, 199–204.
Mukherjee, A., (2010). “Product Market Cooperation, Profits and Welfare in the Presence of Labor Union. ” Journal of Competition Industry and Trade, 10, 151-160.
Wang, L.F.S., Mukherjee, A. and Hsu, C.C., (2012). “To Be Unionized or Not to Be?
A Case for Environment Concern and Firm Heterogeneity. ” Unpublished paper.
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Environment and Resource Economics, 6, 359-369
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APPENDIX
𝑡𝑚− 𝑡𝑝
=−4 + 52𝛼 + 3𝑛(−3 + 19𝛼 + 𝑛(−1 + 5𝛼)) + 8𝜏 − 2(−3𝑛(3 + 𝑛) + 8(2 + 𝑛)(4 + 3𝑛)𝛼)𝜏 (1 + 𝑛 + 4(2 + 𝑛)𝛼)(−2 + 8𝛼 + 𝑛(−3 + 6𝛼))𝜏
𝑆𝑆𝑚− 𝑆𝑆𝑝 = (−(2𝛼22 (−10 + 3𝑛(3 + 𝑛)(−2 + 𝛼) + 4𝛼 + 24𝜏 + 2𝑛(23 + 9𝑛)𝜏)(4 − 52𝛼 + 3𝑛(3 + 𝑛 − 19𝛼 − 5𝑛𝛼) − 8𝜏 + 2(−3𝑛(3 + 𝑛) + 8(2 + 𝑛)(4 + 3𝑛)𝛼)𝜏)))/(1 + 𝑛 + 4(2 + 𝑛)𝛼)2 (−2 + 8𝛼 + 𝑛(−3 + 6𝛼))2
𝐶𝑆𝑚− 𝐶𝑆𝑝 = 2(−2𝑛𝜏 − 3𝑛22 𝜏 − 20𝛼𝜏 − 30𝑛𝛼𝜏 − 9𝑛22 𝛼𝜏 + 8𝛼22 𝜏 +26𝑛𝛼22 𝜏 + 12𝑛22 𝛼22 𝜏 + 4𝑛𝜏22 + 6𝑛22 𝜏22 + 48𝛼𝜏22 + 76𝑛𝛼𝜏22 +24𝑛22 𝛼𝜏22) ⁄ (1 + 𝑛 + 8𝛼 + 4𝑛𝛼)(−2 − 3𝑛 + 8𝛼 + 6𝑛𝛼)
𝐸𝐸𝑚− 𝐸𝐸𝑝 = (((−4 − 8𝛼 + 8𝜏 + 3𝑛2 (1 + 4𝛼)(−1 + 𝛼 + 2𝜏) + 8𝛼(3𝛼 + 2𝜏) +2𝑛(−4 + 8𝜏 + 𝛼(−12 + 19𝛼 + 26𝜏)))(3𝑛2 (1 + 4𝛼)(−1 + 𝛼 + 2𝜏) + 4𝛼(−5 +2𝛼 + 12𝜏) + 𝑛(−2 + 4𝜏 + 2𝛼(−15 + 13𝛼 + 38𝜏))))) ⁄
((2(1 + 𝑛 + 4(2 + 𝑛)𝛼)2(−2 + 8𝛼 + 𝑛(−3 + 6𝛼))2))