The Model of Homogeneous Recycling Firms

In this section, we study how the EPA determines the optimal number of recycling licenses in the recycling market to maximize the social welfare. From Chapter 3, we know that recycling firms usually collect e-scrap products in distinct market segments;

that is, when there are m areas in the recycling market, there are m recycling firms in the recycling market. In other words, the EPA has the political power to determine geographically exclusive areas in the recycling market; that is, the EPA has to divide the recycling market into different areas, where only one recycling firm

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exists in one area and it is responsible for recycling.

In Chapter 3, we assume that recycling firms are heterogeneous. For simplicity, in this chapter, we assume that recycling firms are homogeneous. For the recycling firms, the net cost for recycling one unit of e-scrap products, r, is the same, r 0. A decrease in the recycling quantity in a recycling firm caused by a unit of increase in the reward money paid by another recycling firm, k is the same between any two recycling firms. As mentioned in Chapter 3, we know that a legal recycling firm exists in one area and proceeds with recycling, and under the assumption that recycling firms are homogeneous, we assume that the situations in different areas are the same. We let c and d denote the intercept and slope parameters of the recycling quantity function of each recycling firm in the recycling market.

Furthermore, we use the parameters and assumptions mentioned above to solve for the policies of each participant under the social welfare model and fund balance model.

5.1.1 The Social Welfare Model

In the social welfare model, the EPA aims to maximize the total social welfare when it makes policies. Then the MIS firms and recycling firms aim to maximize their profits according to the level of ARFs and subsidy fees announced by the EPA.

In this chapter, we study the impact of the number of recycling licenses in the recycling market, so the production quantity of the MIS firms and the level of ARFs, t, are the same as the analytical solutions in Section 3.2.3. Let P_w and q_c denote the reward money and recycling quantity of each recycling firm respectively. Given

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that the EPA issues ^m recycling licenses in the recycling market; that is, the EPA divides the recycling market into m areas. According to (3.9) and the assumption that recycling firms are homogeneous, the recycling quantity function is listed as follows:

  .

c w

q  c d mk k P (5.1)

According to (3.32), (3.33), and the assumption that recycling firms are homogeneous, the reward money and recycling quantity of each recycling firm are written as follows:

Adding up recycling firms’ recycling quantity together, we obtain the total recycling quantity, Q_c, as follows:

The EPA maximizes the total social welfare while determining the level of the ARFs and subsidy fees. Under the assumption that recycling firms are homogeneous, the social welfare defined in Section 3.2.3 can be written as follows:

 

According to (3.27) and the assumption that recycling firms are homogeneous, the level of the subsidy fees, s, is written as follows:

   

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5.1.2 The Fund Balance Model

In the fund balance model, which is the current practice model, the EPA determines the level of the ARFs and subsidy fees on the basis of balance between the total tax revenue and the total subsidy expenditure. Then the MIS firms and recycling firms aim to maximize their profits according to the level of fees announced by the EPA.

As mentioned in Section 5.1.1, the production quantity of the MIS firms and the level of ARFs, t, are the same as the analytical solutions in Section 3.3.3. The parameters, ^r, k, c, d, in this section are the same as the parameters in Section 5.1.1. Let P_w' and q_c' denote the reward money and recycling quantity of each recycling firm respectively. Given that the EPA issues m' recycling licenses in the recycling market; that is, the EPA divides the recycling market into m' areas.

According to (3.9) and the assumption that recycling firms are homogeneous, the recycling quantity function is listed as follows:

 

' ' '.

c w

q  c d m k k P (5.7)

According to (3.57), (3.58), and the assumption that recycling firms are homogeneous, the reward money and recycling quantity of each recycling firm are written as follows:

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Adding up recycling firms’ recycling quantity together, we obtain the total recycling quantity, Q_c', as follows:

The EPA determines the level of the ARFs and subsidy fees on the basis of balance between the total tax revenue and the total subsidy expenditure. In order to have a fair basis for comparison, we study the impact of the number of recycling licenses on the value of social welfare, total recycling quantity, reward money, and subsidy fees on the basis of fund balance in this section. Under the assumption that recycling firms are homogeneous, the social welfare defined in Section 3.2.3 is as follows:

According to (3.56) and the assumption that recycling firms are homogeneous, the level of the subsidy fees, ^s', is written as follows:

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5.1.3 The Optimal Number of Recycling Licenses

In this section, we study how the EPA determines the optimal number of recycling licenses in the recycling market to maximize the social welfare. In the social welfare model, the EPA aims to maximize the total social welfare. All other parameters remaining the same, the value of (5.5) may increase or decrease as the value of m increases. Therefore, a value, m^*, which satisfies the maximum of (5.5), is the optimal number of recycling licenses in the recycling market in the social welfare model. However, in the fund balance model, the EPA aims to establish the level of the ARFs and subsidy fees on the basis of balance between the total tax revenue and the total subsidy expenditure. For a fair basis for comparison, we let the value of social welfare be the performance measure in the fund balance model. All other parameters remaining the same, the value of (5.11) may increase or decrease as the value of m' increases. Therefore, a value, ^m'*, which satisfies the maximum of (5.11), is the optimal number of recycling licenses in the recycling market in the fund balance model.

Furthermore, we utilize a set of numerical experiments to study how the EPA determines the optimal number of recycling licenses in the recycling market to maximize the social welfare and illustrate the impact of the number of recycling