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In this paper, we first show that if the firm’s production leads to environmental damage and the government does not implement any environmental policy by using a two-stage game model, the “excess-entry” theorem holds. We then show that entry can be socially insufficient in the presence of production externality and policy mix is needed for pollution control in oligopoly industry with endogenous market structure. Hence, the anti-competitive entry regulation policy suggested by the “excess-entry” theorem does not always hold.

The wave of economic liberalization is being guided by the World Trade Organization around the world. Competition policies in many imperfect market sectors have facilitated the entry and removal of new entries. Competition policy plus many other intervention policies, such as subsidies, tariffs, environmental regulations, etc., should be reviewed from the perspective of long-term welfare improvement. For example, the welfare effects of environmental policies and liberalization policies and the implementation of these two policies should be the main focus of policy decisions.

Is free entry desirable for social efficiency? In an influential work, Mankiw and Whinston [^{1}, Ghosh and Morita [

^{1}Recent works show that: 1) Entry can be insufficient in an oligopolistic market with scale economies if there are vertical relationships [

Is free entry desirable for social efficiency in a pollution industry? We see that in those papers which addressed the effects of environmental policy mostly focused on how it affects profits, environmental damage and social welfare in an oligopoly model with a fixed number of firms, but even allowing free entry it did not asking whether free entry is desirable or not for social efficiency. On early studies of environmental policies in oligopolistic framework, Simpson [

^{2}Lehmann [

In this paper, we show that if the firm’s production leads to environmental damage and the government does not implement any environmental policy, the “excess-entry” theorem holds in a pollution industry. However, the anti-competitive entry regulation policy suggested by the “excess-entry” theorem does not always hold if the environmental policy is implemented. Furthermore, policy mix is needed for pollution control in an oligopoly industry with endogenous market structure.^{2}

This paper is organized as follows. Section 2 provides the basic model and examines the case of no environmental policy in an oligopoly with restricted entry and at free entry. Section 3 does a similar analysis when the environmental policy is implemented. Section 4 further analyzes the policy implication of excess entry in the model when the firms also decide the abatement level in addition to the output decision. Section 5 presents concluding remarks.

Assume the market demand is P = a − Q , and there are n identical firms producing homogeneous good in the market. Each firm produces q i , and the total output is Q = ∑ q i . The production of this good leads to pollution e i = θ q i .^{3} Environmental damage is measured by the quadratic form E D = ( ∑ e i ) 2 / 2 .

The cost function is measured by a quadratic form, q i 2 / 2 for the firms, and the profit functions are,

π i = P q i − q i 2 2 − f 2 (1)

where f 2 is the fixed cost of market entry.

In the absence of environmental policy, the social welfare is defined as

W = C S + ∑ π i − E D (2)

where C S = Q 2 / 2 .

Using a two-stage game model, in the first stage, the firms decide whether to enter or not to enter. The second stage is the monopolistic firm’s production decisions. We solve the maximization problems by backward induction under the premise of Subgame Perfect Nash equilibrium (SPNE).

^{3}See Ulph [

All the firms choose output only; we differentiate Equation (1) with respect to q i and obtain that

∂ π i ∂ q i = a − ( n + 2 ) q i = 0 (3)

From the above first-order conditions, we obtain the short-run equilibrium results with regulated entry

q i * = a 2 + n , π i * = 3 a 2 2 ( 2 + n ) 2 , C S i * = a 2 n 2 2 ( 2 + n ) 2 , W * = a 2 n ( 3 + n − n θ 2 ) 2 ( 2 + n ) 2 .

Consider the case where entry occurs in the market, firm i’s net profit is given by

π i = 3 a 2 2 ( 2 + n ) 2 − f 2 .

The free entry equilibrium number of firms is given by

π i = 3 a 2 2 ( 2 + n ) 2 − f 2 = 0 , or 3 a 2 2 ( 2 + n ) 2 = f 2 (4)

The social welfare is given as

W = a 2 n ( 3 + n − n θ 2 ) 2 ( 2 + n ) 2 − n f 2 (5)

The welfare-maximizing number of firms is given by

∂ W ∂ n = 0 , ⇒ a 2 ( 6 + n − 4 n θ 2 ) 2 ( 2 + n ) 3 = f 2 (6)

Defining Δ = [LHS of (6) – LHS of (4)], we have

Δ = − a 2 n ( 1 + 2 θ 2 ) ( 2 + n ) 3 (7)

Entry is socially excessive if and only if Δ < 0 .^{4} We find that Δ < 0 , suggesting excessive entry when the firm’s production leads to environmental damage. The reasoning is that if the firm’s production leads to environmental damage and the government does not implement any environmental policy, then the total output will be too much for social optimum which echo with the excessive entry argument of Mankiw and Whinston [

In this section, we assume each firm has to pay an environmental tax t per unit of pollutant emitted and the profit of firm i is given by

π i = P q i − q i 2 2 − t e i − f 2 (1')

The objective of the government is to maximize social welfare, which is expressed as

W = C S + ∑ π i − E D + T (2')

where T = t ∑ e i denotes tax revenues collected by the government.

^{4}In Mukherjee [

All the firms choose output and differentiate Equation (1') with respect to q i , and obtain that

∂ π i ∂ q i = a − ( n + 2 ) q i − t θ = 0 (8)

From the above first-order conditions, we need the condition a − t θ > 0 for having positive output and obtain the short-run equilibrium results with regulated entry

q i * = a − t θ 2 + n , π i * = 3 ( a − t θ ) 2 2 ( 2 + n ) 2 , C S i * = ( a − t θ ) 2 n 2 2 ( 2 + n ) 2 ,

W = n ( a − t θ ) { a [ 3 + n ( 1 − θ 2 ) ] + t θ ( 1 + n − n θ 2 ) } 2 ( 2 + n ) 2 .

Consider the case where entry occurs in the market, firm i’s net profit is given by

π i = 3 ( a − t θ ) 2 2 ( 2 + n ) 2 − f 2 .

The free entry equilibrium number of firms is given by

π i = 3 ( a − t θ ) 2 2 ( 2 + n ) 2 − f 2 = 0 , or 3 ( a − t θ ) 2 2 ( 2 + n ) 2 = f 2 (9)

The social welfare is given as

W = n ( a − t θ ) { a [ 3 + n ( 1 − θ 2 ) ] + t θ ( 1 + n − n θ 2 ) } 2 ( 2 + n ) 2 − n f 2 (10)

The welfare-maximizing number of firms is given by

∂ W ∂ n = 0 , ⇒ ( a − t θ ) { t θ [ 2 + n ( 3 + 4 θ 2 ) ] + a [ 6 + n ( 1 − 4 θ 2 ) ] } 2 ( 2 + n ) 3 = f 2 (11)

Defining Δ = [LHS of (11) − LHS of (9)], we have

Δ = − ( a − t θ ) { n ( 1 + 2 θ 2 ) ( a − t θ ) − 2 t θ ( 2 + n ) } ( 2 + n ) 3 (12)

We find that Δ < ( > ) 0 if t < ( > ) t ^ ≡ a n ( 1 + 2 θ 2 ) θ [ 2 ( 2 + n ) + n ( 1 + 2 θ 2 ) ] , suggesting entry is socially excessive (insufficient) depending on the level of environmental taxes.

Proposition 1. If t < ( > ) t ^ , entry of the firms is socially excessive (insufficient).

Proposition 1 shows that entry can be socially insufficient in the presence of production externality. Hence, the anti-competitive entry regulation policy suggested by the “excess-entry” theorem does not always hold. It points out that environmental taxes should be designed properly and combined with (anti-)competition policy to correct multiple distortions, market distortion and production externality. Hence, a higher environmental tax should be combined with entry-promotion policy and vice versa from the standpoint of social welfare maximization.

In this section, we suppose that firm i chooses pollution abatement level a i , the emission level of each firm is e i = θ q i − a i and each firm has to pay an environmental tax t per unit of pollutant emitted. The cost of pollution abatement of firm i is a i 2 / 2 . The profit of firm i is given by

π i = P q i − q i 2 2 − a i 2 2 − t e i − f 2 (1")

We differentiate Equation (1") with respect to q i and a i , and obtain that

∂ π i ∂ q i = a − ( n + 2 ) q i − t θ = 0 (13)

∂ π i ∂ a i = t − a i = 0 (14)

Equation (14) shows that all the firms abate pollution to the point where marginal abatement cost equals the tax. From the above first-order conditions, we obtain the short-run equilibrium results with regulated entry

q i * = a − t θ 2 + n , π i * = 3 ( a − t θ ) 2 + ( 2 + n ) 2 t 2 2 ( 2 + n ) 2 , C S i * = ( a − t θ ) 2 n 2 2 ( 2 + n ) 2 ,

W * = n { a 2 ( 3 + n − n θ 2 ) + 2 a t θ [ − 1 + n ( 2 + n + θ 2 ) ] − t 2 { 4 + θ 2 + n [ 8 + 5 n + n 2 + ( 5 + 2 n ) θ 2 + θ 4 ] } } 2 ( 2 + n ) 2 .

Consider the case where entry occurs in the market, firm i’s net profit is given by

π i = 3 ( a − t θ ) 2 + ( 2 + n ) 2 t 2 2 ( 2 + n ) 2 − f 2 .

The free entry equilibrium number of firms in the pollution industry is given by

π i = 3 ( a − t θ ) 2 + ( 2 + n ) 2 t 2 2 ( 2 + n ) 2 − f 2 = 0 , or 3 ( a − t θ ) 2 + ( 2 + n ) 2 t 2 2 ( 2 + n ) 2 = f 2 (15)

The social welfare is given as

W = n { a 2 ( 3 + n − n θ 2 ) + 2 a t θ [ − 1 + n ( 2 + n + θ 2 ) ] − t 2 { 4 + θ 2 + n [ 8 + 5 n + n 2 + ( 5 + 2 n ) θ 2 + θ 4 ] } } 2 ( 2 + n ) 2 − n f 2 (16)

The welfare-maximizing number of firms is given by

∂ W ∂ n = 0 , ⇒ W ′ = f 2 (17)

Defining Δ = [LHS of (17) − LHS of (15)], we have

Δ = − a 2 ( n + 2 n θ 2 ) + t 2 { ( 1 + n ) ( 2 + n ) 3 + [ 4 + n ( 11 + 6 n + n 2 ) ] θ 2 + 2 n θ 4 } − a t θ { 4 + n [ n ( 6 + n ) + 4 ( 3 + θ 2 ) ] } ( 2 + n ) 3 (18)

We find that Δ > ( < ) 0 if

t > ( < ) t ¯ = a θ { 4 + n [ n ( 6 + n ) + 4 ( 3 + θ 2 ) ] } − a ( 2 + n ) − 4 n ( 1 + n ) ( 2 + n ) + ( − 2 + n 2 ) 2 θ 2 2 n ( 2 + n + θ 2 ) [ ( 2 + n ) 2 + 2 θ 2 ] − 4 θ 2

suggesting entry is socially excessive (insufficient) depending on the level of environmental taxes. Similar to the reasoning provided for Proposition 1, the possibility for socially insufficient is that if the environmental tax is set too high, the firms need to increase their abatement level which makes the industry profit down and the number of firm at free entry is smaller, the entry is socially insufficient. Under such circumstance, the policy implication is not entry regulation but rather entry promotion coupled with a stringent environmental policy from the standpoint of social welfare maximization.

Regardless of whether firms have conducted pollution prevention, it points out that environmental taxes should be designed properly and combined with (anti-)competition policy to correct multiple distortions, market distortion and production externality.

In this paper, we showed that if the firm’s production leads to environmental damage and the government does not implement any environmental policy, the “excess-entry” theorem holds in a pollution industry. However, the anti-competitive entry regulation policy suggested by the “excess-entry” theorem does not always hold if the environmental policy is implemented. In particular, if the environmental tax is set too high, the firms need to increase their abatement level which makes the industry profit down and the number of firm at free entry is smaller, the entry is socially insufficient. The policy implication is not entry regulation but rather entry promotion coupled with a stringent environmental policy. Hence, policy mix is needed for pollution control in oligopoly industry with endogenous market structure.

The authors declare no conflicts of interest regarding the publication of this paper.

Tsai, T.-C., Chen, S.-S., Lu, Y.-S., Hsu, C.-C and Lee, J.-Y. (2019) Environmental Policy and Social Efficiency under Free Entry. Modern Economy, 10, 2110-2119. https://doi.org/10.4236/me.2019.109132