Determinants of the Optimal Waste Disposal Charge Rate: A Numerical Simulation with Comparative Statics

1. The Background: Waste Disposal Charge System in Korea

In Korea, the Framework Act on Resource Circulation came into effect in 2018. This Act includes the clauses for Waste Disposal Charge System. Korea’s Waste Disposal Charge System is designed as follows. Municipalities or business waste emitters are levied 10~30 KRW/kg for landfill and 10 KRW/kg for incineration of wastes. When the one who should pay the charge recycles the wastes in three years or he/she recovers the heat energy more than 50%, it is exempt. Small businesses are lessened or exempt for the charges. Designated wastes which are very difficult to recycle, wastes in islands, wastes from disaster area are eligible for exempt or reduced charges. Revenues from the charges are used to promote the social recognition for resource circulation, to support for small scaled resource circulation facility, to help classified collection of wastes, and to support the production and distribution of circular resources and recycled products.

2. Objective of the Study

Although the Waste Disposal Charge has been in practice since 2018 in Korea, there is a room for controversy on whether the charge rate is designed close enough to the optimum. Despite that the fundamental principle of economics teaches waste disposal charge should reflect externality, in reality, the decision makers in the governments do not seem to consider unit external cost as the single most important factor that determines the environmental tax rate or disposal charge rate. Elasticity is sometimes used as an important parameter, occasionally considered even more important than the unit external cost. The market is usually assumed competitive, implicitly or explicitly, even though in reality the market is often in an imperfect competition. In practice, neither of these factors are seriously considered because there is no sufficient information on the unit external cost, market elasticity, or market structure. In many cases the expected or desired revenue seems to be the most important consideration to the decision making government authorities. Maybe this is the reason why policy makers are sometimes concerned with elasticity while market structure or unit external cost are not seriously considered.

This paper attempts to provide explanations on this question using comparative statics and numerical simulations with a model built on the waste treatment services market. The motivation of this study is to answer the following questions: 1) How should we determine the optimal disposal charge rate?, 2) How much do the external cost of final disposal, elasticity, and market structure matter?, and 3) What are the policy implications?

3. Related Previous Studies and Their Implications for This Study

In standard environmental economics textbooks, optimal environmental tax rate is unit external cost in a competitive product or environmental service market. However, when the market is in imperfect competition, optimal environmental tax rate differs from unit external cost. Studies on the optimal environmental taxes in imperfectly competitive markets are focused either on a product market or environmental services market.

There are many studies focusing on product markets. Misolek (1980), Barnett (1980) and many other studies posit that when polluting industry is monopoly, the optimal environmental tax rate will be lower than the one for competitive case. Requate (2005) reaffirms that with monopoly the optimal tax rate is lower than Pigiouvian tax rate. Requate (2005) also showed that in oligopoly optimal environmental tax rate decreases with the increase in the number of symmetric firms. This is equivalent with the statement that lower market concentration reduces the optimal environmental tax rate. Bovenberg and Gould (1996) state that if there are taxes other than environmental tax, the optimal environmental tax rate is lower than the external cost. Sandmo (1975) showed that the optimality of the Pigouvian tax can be achieved by adjusting the indirect tax system when an environmental tax is implemented as an element in a more comprehensive indirect tax system. However, this does not refute the fact that the optimal tax diverges from the Pigouvian tax if other taxes still exist. Kim and Kim (2002) showed that when a tax on capital exists, the optimal environmental tax rate is lower than the environmental externality cost. Lho (2002) introduced a generalized utility function and constructed a model in which citizens maximize utility as a function of consumption, leisure, and environmental pollution, and environmental pollution occurs from consumption. By using this social utility function, the optimal environmental tax rate is numerically derived by given specific values of parameters. In addition, a sensitivity analysis is conducted on the optimal environmental tax rate over the various values of parameters. From the numerical simulation and sensitivity analysis, Lho (2002) showed that the optimal environmental tax rate is a decreasing function of the concavity parameter of the social welfare function and an increasing function of the elasticity of substitution of leisure and final consumer goods.

There are a few studies focusing on environmental services market. Baumol (1995) and Feess and Muehlheusser (1999, 2002) acknowledged the existence of the eco-industrial sector, but they did not explicitly address the consequences of imperfect competition on the optimal environmental tax rate. David and Sinclair-Desgagné (2005) analyzed the optimal pollution tax when pollution abatement technologies and services are provided by an imperfectly competitive environment industry. They found that under endogenous market elasticity, the optimal pollution tax will be higher than the marginal social cost of pollution. David, Nimubona and Sinclair-Desgagné (2011) examined the effect of emission taxes on pollution abatement and social welfare, where abatement goods and services are provided by a Cournot oligopoly with free entry. They showed that under endogenous market structure, it is highly probable that the welfare maximizing emission tax is lower than the simple Pigouvian tax. ¹⁾

There are fewer studies regarding environmental tax in waste treatment service market. Kaneko (2009), using a partial equilibrium comparative statics model based on a vertically linked competitive waste treatment services markets, demonstrated that the waste generation reduction effect differs by the point of levying industrial waste tax. However, Kaneko (2009) did not provide the welfare implication of these tax options.

Kim and Chang (2015) calculated potential environmental benefits (with a given unit external cost) and tax revenue from Korea’s final disposal charge by conducting simulations, apparently with the assumption of a competitive market. Elasticity is considered as the major factor in determining the charge rate for final disposal in Kim and Chang (2015), without explicitly considering social welfare.

4. Characteristics of This Study Demarcating from Previous Studies

Adding to the unit external cost, which is the starting point and basic element in determining the optimal disposal charge rate, several studies have examined additional factors. Barnett(1980), Requate(2005) and others suggested the role of market structure in optimal disposal charge rate. Kim and Chang (2015) demonstrated the importance of elasticity. Kaneko(2019) showed the importance of considering levying method in designing the charge schemes. David et al.(2011) analyzed the optimal environmental tax rate in the oligopolistic environmental service firms, and Lho (2002) applied a numerical simulation method to a social utility function to derive the optimal environmental tax rate.

This study aims to identify the relative importance of the determinants of the optimal waste disposal charge rate. This can be achieved by synthesizing various elements of previous studies and adding new ones. Those characteristics differentiating this study from previous ones are as follows. First, assuming a generalized imperfectly competitive market, the market structure is expressed in a continuum from perfect competition to pure monopoly/monopsony, so we can parameterize the market structure for buyer’s and seller’s side. Hence, now we have three parameters of market structure, market elasticity, and unit external cost. Second, compared to previous studies that analyzed products or input markets, this study is focused on the waste treatment service market so that the market structure and elasticity in the waste service sector are explicitly considered so the policy implications are more appropriate to the environmental policy practitioners. Third, by conducting comparative statics and numerical simulations, it is possible to compare the relative size of the impact of the different parameters in determining the optimal disposal charge rate. This is an improvement compared with the previous studies.

1. Definition of Variables, Structure of the Waste Treatment Service Market, and Objective Functions

Waste generators generate constant amount of waste . Waste generators have two choices in dealing with the generated waste. The first choice is to transfer the waste to the waste treatments service providers ²⁾ by the amount of w. In this case, the waste treatment service providers receive fee of p per quantity of waste delivered. The value of p is determined in the market. The second choice is waste reduction. ³⁾ The quantity of waste reduction, which is the quantity of waste not delivered to the waste treatment service providers, is e-w. The cost incurred in the process of reduction is a function of e-w, denoted C_E(e-w). ⁴⁾

The cost of waste treatment is a function of w, denoted C_T(w), which includes the costs of collection, transportation, pre-treatment, treatment, and final disposal. Waste disposal charge (t per quantity of waste disposed) is to be levied on the final disposal and paid by the waste generators. ⁵⁾

Waste generators’ problem is maximizing profit by choosing w, given constant e and market-determined variable p. Waste generators’ cost is the sum of waste reduction cost C_E(e-w) plus the payment to waste treatment service providers pw. ⁶⁾ So waste generators’ profit maximization problem becomes in fact a cost minimization problem.

Waste treatment service providers, after receiving w from waste generators, provide waste treatment services and finally dispose wastes. Waste treatment service providers receive pw as revenue for their services and incur treatment cost of C_T(w). ⁷⁾ Waste treatment service providers maximize profits by choosing w, given p that is determined in the market.

These relationships are depicted in the following <Figure 1>.

<Figure 1>

Flow of Service and Money in the Waste Treatment Service Market

2. Parameters of Market Structure, Market Elasticity, and Externality

In the model, we have variables w and p and a constant e, as well as several parameters for elasticity, market structure, and environmental externality. The objective of this study is to assess and compare the relative importance of three parameter groups of unit external costs, market structure, and market elasticity in determining the optimal charge rate for the final disposal of wastes.⁸⁾

Regarding market elasticity, we have parameters g of h and . The slope of marginal cost function of waste reduction , C_E^"(e-w), which is the minus of the slope of market demand curve of the waste treatment service, ⁹⁾ assumed to be a non-negative constant, is denoted g as a parameter for the inverse of demand elasticity. The slope of market supply curve of waste treatment service, which is C_T^"(w)¹⁰⁾, assumed to be non-negative constant, is denoted h as a parameter for the inverse supply elasticity. The condition for the inverses of slopes of the demand and supply curves to be elasticity is that the price and quantity are assumed to normalized to unity at the equilibrium. In this study, price (p) and quantity (w) of waste treatment services are assumed to be unities at the equilibrium.

Regarding externality, the unit external cost of final disposal is denoted θ. Regarding market structure, the uniform market share is β_g for waste service provider (seller of the service) and β_h for waste generator (buyer of the service) , where 0≦β_g≦1 and 0≦β_h≦1. ¹¹⁾

This study assumes the market is in an imperfect competition (Cournot oligopoly and oligopsony). Following Appelbaum(1982). Ji and Chung(2016), and Chung(2017), in an imperfectly competitive oligopoly/oligopsony, when the market is composed of uniformly sized firms, the market power or Lerner index is expressed as $$ \frac{\beta }{\eta }$$, where β is the market share for uniform firms and η is the elasticity for demand or supply. ¹²⁾ In other words, the market power of individual firms is the market share of each uniform firm (market structure parameter) multiplied by the reciprocal of market elasticity.

In this study, market elasticity is assumed to be the reciprocal of the slope of the demand or supply curve at the equilibrium. ¹³⁾ Therefore, we have the equations for the market power of individual sellers and buyers, $$ \frac{{\beta }_g}{{\eta }_g}={\beta }_{g}g \mathrm{a}\mathrm{n}\mathrm{d} \frac{{\beta }_h}{{\eta }_h}={\beta }_{h}h$$, where g is the slope of the demand curve and h is the slope of supply curve, and β_g is seller’s market share and β_h is buyer’s market share, η_g is market elasticity of demand and η_h is market elasticity of supply. Please remind that demand curve slope represents the seller’s market power and supply curve slope represents the buyer’s market power, not the other way around. Also remind that market demand or market supply curve is different from the demand curve faced by individual sellers or the supply curve faced by individual buyers.

Market power of individual sellers or buyers is equal to the inverse of elasticity of demand or supply faced by individual sellers or buyers, which is seller’s or buyer’s market share divided by demand or supply elasticity. Market power of individual sellers or buyers, which is inverse of elasticity of demand or supply faced by individual sellers or buyers, is proportional to the market share. In this study, the inverse of elasticity is the slope of demand or supply curve. Hence, the slope of demand or supply curve faced by individual sellers or buyers is proportional to the market share. The slope of the demand curve faced by individual sellers is seller’s market share times the slope of market demand curve (β_gg). The slope of the supply curve faced by individual buyers is the buyer’s market share times the slope of the market supply curve (β_hh).

3. Behaviors of market participants and social planner

The objective of waste generators is profit maximization regarding waste related activity. Non-waste related variables assumed to remain constant. In addition, the quantity of waste generated is also assumed to be constant. Therefore, the profit maximization for waste generator becomes equivalent to the minimization of waste related costs. Waste generators’ objective is minimizing the sum of cost of waste reduction, waste treatment fee paid to the waste treatment service providers, and disposal charge paid to the government.

Waste reduction cost is a function of (e-w), that is C_E(e-w). Treatment fee is the product of fee rate and the quantity of waste transferred, p^D(w). Here p^D(w) is buyers’ (waste generators’) ex-ant offer price for the waste treatment services, i.e., inverse demand function of waste treatment services. The value of disposal charge paid to the government is tw where t is the disposal charge rate.

Waste treatment service providers are maximizing profit, which is the revenue from waste treatment service minus the cost of waste treatment. Revenue from waste treatment service is p^S(w), where p^S(w) is sellers’ (waste treatment service providers’) ex-ant offer price of waste treatment services, i.e., inverse supply function of waste treatment service. We have p^D(w) = p^S(w) in equilibrium.

Given profit maximizing behaviors of the market participant and the market equilibrium condition, the social planner maximizes the social welfare. The social planner maximizes the social welfare after observing the profit maximizing behaviors of waste generators and waste treatment service providers.

Behaviors of waste generators and waste treatment providers are given as follows.

Since the social planner determines the optimal disposal charge rate after observing the behaviors of market participants given a disposal charge rate, the sequence of obtaining the optimal disposal charge rate is as follows.

By getting the first order conditions for the waste generator and the waste treatment service provider regarding w, and combined with the equilibrium condition, we get the followings.

Since $$ \frac{\partial C_E}{\partial w}=-C_E^{\text{'}}=-gw,\frac{\partial C_T}{\partial w}=C_T^{\text{'}}=hw\text{, and}\frac{\partial p^D}{\partial w}=-{\beta }_{g}g$$, and $$ \frac{\partial p^S}{\partial w}={\beta }_{h}h$$, we get the followings, ¹⁴⁾.

The Stackelberg condition is satisfied by considering w as a function of t. Hence the social planner’s problem is as follows.

By getting first order condition regarding t from (Equation 4), we get $$ \frac{\partial w}{\partial t}(-\theta +\frac{\partial C_E}{\partial w}-\frac{\partial C_T}{\partial w})=0 \mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e} \frac{\partial C_E}{\partial w}=-gw \mathrm{a}\mathrm{n}\mathrm{d} \frac{\partial C_T}{\partial w}=hw$$, we get –gw–hw–θ=0

So we have the following three equations with three variables w, p, and t. ¹⁵⁾

4. Comparative statics results and discussions

By applying the implicit function rule for the previous equations in (Equation 5), we get the following results for comparative static derivatives regarding the impact of parameters on the optimal disposal charge rate. ¹⁶⁾C_E^'

First, θ is the unit external cost and $$ \frac{\partial t^{*}}{\partial \theta }$$ is the impact of unit external cost on the optimal disposal charge rate. Second, g is minus of the slope of demand curve, which measures the inverse market elasticity of demand, where $$ \frac{\partial t^{*}}{\partial g}$$ is the impact of inverse market elasticity of demand on the optimal disposal charge rate. Third, h is the slope of supply curve, which measures the inverse market elasticity of supply, where $$ \frac{\partial t^{*}}{\partial h}$$ is the impact of inverse market elasticity of supply on the optimal disposal charge rate. Fourth, β_g is seller’s market share., where $$ \frac{\partial t^{*}}{\partial {\beta }_g}$$ is the impact of seller’s market share on the optimal disposal charge rate. Fifth, β_h is buyer’s market share., where $$ \frac{\partial t^{*}}{\partial {\beta }_h}$$ is the impact of buyer’s market share on the optimal disposal charge rate.

1. Need for Numerical Simulations

Although we obtained several inequalities from comparative static analysis, we still do not have information regarding the relative importance of parameters. It is important to have information regarding the relative sizes of the values of the comparative static derivatives. Since it takes costs to estimate the values of the parameters and the resource for the estimation is limited, it is important to know the relative benefit from the estimation of each parameter. The welfare gain from the preciseness of parameters increases with the sensitivity of optimal disposal charge rate to the parameters. Higher sensitivity means higher welfare gains from correct estimation of the parameter. Having information about which comparative static derivative has the larger value helps policymakers to focus on that specific parameter, considering the cost of information.

Information about the relative magnitudes of the comparative static derivatives can be obtained through numerical simulations. Numerical simulations are conducted by computing values of comparative static derivatives by plugging ranges of values to the parameters chosen for the simulation, while keeping other parameters and variables constant. We call these chosen parameters as reference varying parameters. Reasonable ranges of values are assigned to the reference varying parameters. In each simulation scenario, average values are assigned to the non-reference fixed parameters, while normalized fixed values are assigned for variables.

By examining the computed values of comparative static derivatives, we can draw implications on which parameter we should allocate more effort and resources in determining the optimal disposal charge rate. ¹⁹⁾

2. Assumptions for the Parameters and Variables

Market shares(β_g and β_h) are distributed from 0 to 1 and elasticities(η_g and η_g) are also distributed from 0 to 1, which are normalized values. g is reciprocal of η_g and h is reciprocal of η_h, which are slopes of straight lines, so β_g and β_h, η_g and η_h and h are all normalized values, free from measurement units. For variables, which are not free from measuring units, variable w (waste treatment quantity) is normalized to be a fixed value of 1, and variable p (price of waste treatment service) is also normalized to be a fixed value of 1. ²⁰⁾

For each simulation, two parameters are designated as reference varying parameters and other parameters and variables are fixed. For the reference varying parameters, reasonable and realistic boundaries are used instead of the whole possible ranges of parameters. For the reference varying parameters of β_g and β_h, simulations are conducted for the range between 0 and 0.4. ²¹⁾ For the varying reference parameter of β_GAP, which is β_g - β_h , simulations are conducted for the range between –0.4 and 0.4. For the reference varying parameters of g and h, the ranges are set between 2.5 and 20.²²⁾ As average values for β_g and β_h, we assigned 0.2 for each. As average values for g and h, we assigned 5 for each.

3. Numerical Simulation Scenarios and Results

1) Simple Average Values

By simply plugging average values of parameters and normalized fixed values of variables, we can obtain the average values for the comparative static derivatives. The result is shown in the last column of <Table 1>. In terms of absolute values, buyer’s market share (β_h) and seller market share(β_g) are the most influential parameters, where the average value of comparative static derivative with regard to β_h, which is $$ \frac{\partial t^{*}}{\partial {\beta }_h}$$, and the average value of comparative static derivative with regard to seller’s concentration , $$ \frac{\partial t^{*}}{\partial {\beta }_g}$$, are -11.25. Among the five derivatives, the ones for elasticities have the smallest values. The simple averages of the comparative static derivative with regard to demand elasticity, $$ \frac{\partial t^{*}}{\partial {\beta }_g}$$, and the comparative static derivative with regard to supply elasticity, $$ \frac{\partial t^{*}}{\partial {\beta }_h}$$, are all zeroes. The simple average of the comparative static derivative with regard to unit external cost $$ \frac{\partial t^{*}}{\partial \theta }$$ is 1.2, which is larger than the ones with regard to elasticities but smaller than the ones with regard to market shares.

<Table 1>

Numerical Simulation Scenarios and Results

2) Scenarios and Results of Numerical Simulations

Numerical simulations are conducted to examine the distribution’s of the derivatives according to the changes of the reference varying parameters.

Regarding the comparative static derivative of optimal disposal charge rate with respect to the unit external cost $$ (\frac{\partial t^{*}}{\partial \theta }=1+\frac{{\beta }_{g}g+{\beta }_{h}h}{g+h})$$, numerical simulation is conducted over the range of reference varying parameters for elasticity and market structure. For $$ \frac{\partial t^{*}}{\partial \theta }$$, we have two simulations with respect to elasticity related parameters (g and h) and market shares (β_g and β_h). By simulating the value of $$ \frac{\partial t^{*}}{\partial \theta }$$ over the values of g (inverse of demand elasticity), and h (inverse of supply elasticity), which are varying from 2.5 to 20, while keeping β_g=0.2 and β_h=0.2, we get the result 1.2. By simulating the value of $$ \frac{\partial t^{*}}{\partial \theta }$$ over the values of β_g(seller’s average market share) and β_h(buyer’s average market share), which are varying from 0 to 0.4, we get the result distributed between 1 and 1.4, with average of 1.2. The overall range of the simulated values for $$ \frac{\partial t^{*}}{\partial \theta }$$ is between 1 and 1.4, and the average is 1.2, which is the same as the simple average. The simulation result is demonstrated with a three dimensional diagram in <Figure 2>.

<Figure 2>

Numerical Simulations Result for ∂t*∂θ=1+βgg+βhhg+h and over βg and βh with g=h=5

The simulations of optimal disposal charge rate regarding the inverse of elasticities are conducted keeping w=1, a normalized value and keeping β_GAP= 0.2 or β_GAP= - 0.2. Although the average value of β_GAP is equal to zero, if we choose zero as the fixed value for β_GAP, the simulation results for $$ \frac{\partial t^{*}}{\partial g}=\frac{-wh{\beta }_{GAP}}{g+h} \mathrm{a}\mathrm{n}\mathrm{d} \frac{\partial t^{*}}{\partial h}=\frac{wg{\beta }_{GAP}}{g+h}$$ will be zero, which is meaningless, so we instead chose +0.2 and –0.2 for the fixed values of β_GAP.

The comparative static derivative of optimal disposal charge rate regarding the inverse of demand elasticity $$ (\frac{\partial t^{*}}{\partial g}=\frac{-wh{\beta }_{GAP}}{g+h})$$ is simulated over the values of g (inverse of demand elasticity) and h (inverse of supply elasticity), which are varying from 2.5 to 20, while keeping w=1 and β_GAP=0.2, we get the result between -0.1628 and –0.0997 with the average of –0.1. The simulation result is demonstrated with a three dimensional diagram in <Figure 3>. The same one $$ (\frac{\partial t^{*}}{\partial g}=\frac{-wh{\beta }_{GAP}}{g+h})$$ is simulated over the values of g (inverse of demand elasticity), and h (inverse of supply elasticity), which are varying from 2.5 to 20, while keeping w=1 and β_GAP= - 0.2, we get the result between 0.0997 and 0.1628 with the average of +0.1. The simulation result is demonstrated with a three dimensional diagram in <Figure 4>.

<Figure 3>

Numerical Simulations Result for ∂t*∂g=-whβGAPg+h over g and h, with βGAP=+0.2

<Figure 4>

Numerical Simulations Result for ∂t*∂g=-whβGAPg+h over g and h, with βGAP=-0.2

The same one $$ (\frac{\partial t^{*}}{\partial g}=\frac{-wh{\beta }_{GAP}}{g+h})$$ is simulated over the values of β_GAP, which are varying from –0.4 to +0.4, while keeping w=1 and g=5 and h=5, we get the result between -0.2 and +0.2 with the average of zero. We get the result for $$ \frac{\partial t^{*}}{\partial g}$$ between -0.2 and +0.2 with the average of zero.

The comparative static derivative of optimal charge rate regarding the inverse of supply elasticity $$ (\frac{\partial t^{*}}{\partial g}=\frac{wh{\beta }_{GAP}}{g+h})$$ is simulated over the values of g (inverse of demand elasticity), and h (inverse of supply elasticity), which are varying from 2.5 to 20, while keeping w=1 and β_GAP=0.2, we get the result between 0.0997 and 0.1628 with the average of +0.1. The simulation result is demonstrated with a three dimensional diagram in <Figure 5>. The same one $$ (\frac{\partial t^{*}}{\partial g}=\frac{wh{\beta }_{GAP}}{g+h})$$ is simulated over the values of g (inverse of demand elasticity), and h (inverse of supply elasticity), which are varying from 2.5 to 20, while keeping w=1 and β_GAP=-0.2, we get the result between –0.1628 and -0.0997 with the average of -0.1. The simulation result is demonstrated with a three dimensional diagram in <Figure 6>.

<Figure 5>

Numerical Simulations Result for ∂t*∂h=wgβGAPg+h over g and h, with βGAP=+0.2

<Figure 6>

Numerical Simulations Result for ∂t*∂h=wgβGAPg+h over g and h, with βGAP=-0.2

The same one $$ (\frac{\partial t^{*}}{\partial g}=\frac{wh{\beta }_{GAP}}{g+h})$$ is simulated over the values of β_GAP, which are varying from –0.4 to +0.4, while keeping w=1 and g=5 and h=5. We get the result for $$ \frac{\partial t^{*}}{\partial {\beta }_h}$$ between -0.2 and +0.2 with the average of zero.

The comparative static derivative of optimal charge rate regarding the seller’s market share $$ (\frac{\partial t^{*}}{\partial {\beta }_g}=-gw)$$ is simulated over the values of g, which is varying from 2.5 to 20, while keeping w=1, we get the result distributed between –2.5 and -20, with average of –11.25. By simulating the comparative static derivative of optimal charge rate regarding the seller’s market share $$ (\frac{\partial t^{*}}{\partial {\beta }_h}=-hw)$$ over the values of h, which is distributed between 2.5 and 20, we get the result distributed between -2.5 and -20, with average of -11.25.

Simulation results are summarized in the following <Table 1>.

4. Summary of the Numerical Simulations

Unit external cost is the only important factor in determining the disposal charge rate in a competitive market. In a competitive market, it is always true that t^*=θ and $$ \frac{\partial t^{*}}{\partial \theta }$$=1. In an imperfectly competitive market, not only unit external cost determines the optimal charge rate, but elasticity and market structure also play key roles. But the relative importance of parameters is not clearly visible if we only look at the comparative statics results mathematically. It can be seen by conducting numerical simulations.

The followings are derived from the numerical simulations. In an imperfectly competitive market, the value of optimal disposal charge rate is smaller than in competitive markets. This relationship is manifested by the negative value of the comparative static derivative of optimal charge rate regarding market share, which dictates that the optimal charge rate will be smaller as the market structure becomes less competitive either in supply or demand side. As shown in the following <Table 2>, market structures has the largest impacts on t^*(optimal disposal charge rate). Considering that the policy makers and practitioners usually do not take market structure into consideration in determining the disposal charge rate, this has a fairly important implication.

<Table 2>

Summary of the Numerical Simulation Results

Although unit external cost is not the sole factor for the optimal charge rate in an imperfectly competitive market, it is still fairly important. The average value of comparative static derivative of optimal charge rate is 1.2, which is larger than the one in a competitive market (=1).

In an imperfectly competitive market, market elasticity has some but minor role, much smaller than the other parameters. The average values are zeroes. In addition, when the buyer and seller have symmetric market structures, including the case of perfect competition, elasticity has zero effect in optimal charge rate decisions.