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.
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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:
�
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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.
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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.