Pareto-Improving Supply Subsidy in a Simple General Oligopoly Equilibrium Model with Pollution Permits


We introduce a competitive pollution permit market in a two-sector oligopoly equilibrium model. In this model, one commodity is inelastically supplied by one competitive trader and another one is produced by a finite set of oligopolists, using the first commodity as an input. The production of the second commodity is a polluting activity. We study both the competitive and oligopoly equilibria. We provide some conditions under which a supply subsidy given to the oligopolists that is financed by a tax on the competitive agent is Pareto-improving.

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  1. 1.

    Our approach belongs to the Cournot–Walras equilibrium tradition opened by [11] and pursued by [1, 21]. In this approach, there are both strategic and price-taking traders. The market demand is endogenous, and the determination of the relative prices depends on agents’ preferences. By taking into consideration the Walras price correspondence, which depends on their supplies, the strategic traders interact in a Cournot quantity setting game and determine their equilibrium strategies. Thus, the strategic traders manipulate the price system by controlling the amount of commodity they send to the market.

  2. 2.

    For a survey of imperfectly competitive emission permit market, see [25]. The pollution permit market may also be modeled as a two-stage strategic market game, where firms are allowed to behave strategically [5].

  3. 3.

    Therefore, the source of market distortions stems from consumption decisions by the strategic traders.

  4. 4.

    Some contributions rather study strategic behavior in the permit market, either by assuming from the outset market power market (see [16, 19], among others), or by considering that this market power is endogenous (see for instance [23, 24]).

  5. 5.

    In general equilibrium models with imperfect competition, taxation can achieve two objectives. On the one hand, it aims at correcting the distortions generated by the market mechanism in the presence of imperfectly competitive behavior [9, 27]. On the other hand, it serves as a tool for redistributive purposes [9, 14, 15]. [14] compares two taxation schemes in a general oligopoly equilibrium setting where the firms maximize the utility of their shareholders. A lump-sum tax may increase the price of the commodity produced by the oligopolists and decrease welfare, while a tax on profits may decrease the price and increase welfare. Otherwise, an ad valorem tax may decrease the price and increase welfare. Our contribution differs in two ways from those quoted above. First, we do not determine whether taxation can decentralize Pareto-efficient allocations. We rather investigate the indirect effects of a tax policy on the permit markets. Second, we do not compare different kinds of taxes, but concentrate on the effect of a supply subsidy.

  6. 6.

    This is meant to capture the fact that in the real world a share of the production is stocked up, or exported and so on, but is not directly consumed.

  7. 7.

    Notice that here we do not consider taxing market power as in [9, 15]. Nor do we pay attention to ad valorem or per unit taxes (as in [14]). See also [4] for a recent contribution to the study of taxation in oligopolistic equilibrium.

  8. 8.

    We thank a referee for having suggested this interpretation.

  9. 9.

    For simplicity, pollution is not explicitly taken into account in the utility function. Indeed, we assume that pollution enters the utility function in a separable way, and that that part of the utility function can be neglected since for most of the analysis, the emission ceiling is considered as being constant (see footnote 11).

  10. 10.

    As in [2931], we suppose that the permit market is competitive. Indeed, according to [25]: “An empirical revision of the functioning of past and existing permit markets shows no indication of market power that can be of concern” (p. 26).

  11. 11.

    The aggregate level of emissions E is held constant, so the pollution will be the same in all the equilibria studied in this paper. Indeed, our objective is not to study the optimal amount of pollution emissions per se, but rather to focus on the conditions under which a supply subsidy is welfare improving, an emission ceiling being given. That is, we want to see how to increase the oligopolists’ supply without polluting more (the authorized quantity E of pollution emissions being allocated among firms along with a pollution permit market). We shall see, however, that the existence of the different equilibria considered in the paper, as well as the effects of a supply subsidy, depends on the authorized level of emissions.

  12. 12.

    The detailed computation of the competitive equilibrium for a given allocation of emission rights \(\left( \lambda ^{i}\right) _{i\in \mathbf {\mathcal {O}}}\) is presented in Appendix 1.

  13. 13.

    The quantity of permits actually used by each agent i is undetermined and is immaterial for him since it yields zero-profit. The only source of income of agent i is the proceeds of the sale of his share \(\lambda _{i}E\) of the authorized aggregate pollution.

  14. 14.

    The equilibrium concept belongs to the class of Cournot–Walras equilibria ([1, 11, 21]).

  15. 15.

    The term \(\frac{\gamma }{\beta }\alpha \frac{n-1}{n^{2}s}\) is the marginal value of an additional unit of pollution (the increase in production due to a unit increase in pollution, i.e., \(\frac{\gamma }{\beta }\), times the marginal value of a unit increase in production, i.e., \(\alpha \frac{n-1}{n^{2}s}\)). The term \(\gamma +q\) represents the marginal cost of a unit increase in pollution. To increase pollution by one unit requires increasing the input used by \(\gamma\) units, which results in an increase by \(\gamma\) (recall that \(p_{2}=1\)). Moreover, to pollute one more unit, a trader must buy an additional permit (or decrease its permit sales by one unit) and therefore has to pay (or to bear a loss equal to) q.

  16. 16.

    That is because of the relation \(\tilde{p}\frac{\gamma }{\beta }\left( 1-\frac{1}{n}\right) =\gamma +\tilde{q}\). The higher \(s^{i}\), the lower \(\tilde{p}\), and the lower \(\tilde{q}\) must be in order to satisfy this relation.

  17. 17.

    This comparative statics exercise is performed by assuming a small feasible deviation \(dE>0\) from E, with \(0<E+dE<\frac{\alpha }{\gamma }\dfrac{n-(1-\alpha )}{n}\).

  18. 18.

    By using examples of pure exchange economies with two types of oligopolists, [9] shows that a Pareto-optimal allocation can be reached by taxing the endowment of each type of oligopolists and transferring the product of the tax to the other type.

  19. 19.

    By using a partial equilibrium model, [12] notably shows that first-best allocations can be achieved by tax/tax-refunding schemes with and without pre-investment.


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Competitive Equilibrium

Taking the price vector \(\mathbf {p}=\left( p,1\right)\) as given, the competitive agent chooses his demands for both goods 1 and 2 by solving the problem:

$$\begin{aligned} \underset{\left( x_{1},x_{2}\right) }{\max }\alpha \ln x_{1}+(1-\alpha )\ln x_{2}\text { s.t.\ }px_{1}+x_{2}\leqslant 1. \end{aligned}$$

It is straightforward to show that his demand functions are given by \(\left( x_{1},x_{2}\right) =\left( \frac{\alpha }{p},1-\alpha \right)\).

The income of trader i is equal to his profit \(\Pi ^{i}\left( s^{i},e^{i}\right)\), where

$$\begin{aligned} \Pi ^{i}\left( s^{i},e^{i}\right) =p\frac{\gamma }{\beta }e^{i}-\gamma e^{i}+q\left( \lambda ^{i}E-e^{i}\right) \text {, }i\in \mathbf {\mathcal {O}}\text {.} \end{aligned}$$

Trader i takes the price vector \(\left( \mathbf {p},q\right)\) as given and determines his demands for both goods 1 and 2 by solving the problem:

$$\begin{aligned} \max _{(x_{1}^{i},\,x_{2}^{i})}\text { }\alpha \ln x_{1}^{i}+(1-\alpha )\ln x_{2}^{i}\text { s.t. }px_{1}^{i}+x_{2}^{i}\leqslant \Pi ^{i}\left( s^{i},e^{i}\right) \text {, with}\,x_{1}^{i}\geqslant 0\text {, }x_{2}^{i}\geqslant 0\text {.} \end{aligned}$$

Assuming \(\Pi ^{i} > 0,\) the demand functions of agent \(i\in \mathbf {\mathcal {O}}\) are then given by \(\left( x_{1}^{i},x_{2}^{i}\right) =\left( \frac{\alpha \Pi ^{i}}{p},(1-\alpha )\Pi ^{i}\right)\). In addition, as a producer, agent i’s demand for the input satisfies \(z^{i}=\gamma e^{i}\).

The determination of the competitive equilibrium, i.e., a price vector \((\mathbf {p}^{*},q^{*})\), and an allocation \(\mathcal {A}^{*}=\{\left( x_{1}^{*},x_{2}^{*}\right) ;\left( (x_{1}^{i})^{*},(x_{2}^{i})^{*}\right) _{i\in \mathbf {\mathcal {O}}}\}\) uses the market-clearing conditions. These conditions may be written as

$$\begin{aligned} \sum _{i\in \mathbf {\mathcal {O}}}\frac{\alpha \Pi ^{i}}{p}+\frac{\alpha }{p}&=\sum _{i\in \mathbf {\mathcal {O}}}\frac{\gamma }{\beta }e^{i}, \end{aligned}$$
$$\begin{aligned} \sum _{i\in \mathbf {\mathcal {O}}}(1-\alpha )\Pi ^{i}+\sum _{i\in \mathbf {\mathcal {O}}}\gamma e^{i}+1-\alpha =1, \end{aligned}$$
$$\begin{aligned} \sum _{i\in \mathbf {\mathcal {O}}}e^{i} =E\text {.} \end{aligned}$$

But as agent i’s profit (\(i\in \mathbf {\mathcal {O}}\)) is linear in \(e^{i}\). Thus, there exists an equilibrium only if the following condition holds

$$\begin{aligned} p=\beta \left( 1+\frac{q}{\gamma }\right) \text {.} \end{aligned}$$

With such a price, the demands of agent i may be written \(\left( x_{1}^{i},x_{2}^{i}\right) =\left( \alpha \frac{q}{p}\lambda ^{i}E,(1-\alpha )q\lambda ^{i}E\right)\), \(i\in \mathbf {\mathcal {O}}\). Using the equilibrium condition on the permit market, the market-clearing condition for good 2 may be obtained as

$$\begin{aligned} \ \sum _{i\in \mathbf {\mathcal {O}}}(1-\alpha )q\lambda ^{i}E+\sum _{i\in \mathbf {\mathcal {O}}}\gamma E+1-\alpha =1, \end{aligned}$$

or, since \(\sum _{i\in \mathbf {\mathcal {O}}}\lambda ^{i}=1\), as

$$\begin{aligned} \ (1-\alpha )qE+\gamma E+1-\alpha =1. \end{aligned}$$

The term on the left-hand side of this equation represents the aggregate demand for commodity 2, where \((1-\alpha )qE\) and \(\gamma E\) are, respectively, the demands for consumption and for the input by the n agents who belong to the set \(\mathbf {\mathcal {O}}\), and \(1-\alpha\) is the demand by the representative consumer. The term on the right-hand side is the total supply of commodity 2. By solving (60), we obtain (11), which by using \(p=\beta (1+\frac{q}{\gamma })\) yields (12). Then, equations (13)-(16) follow.

Effects on an Increase in the Emission Ceiling E

The first part of the proposition stems immediately from the expressions of (23) and (24). In addition, from (22), one easily sees that \(\frac{\partial \tilde{s}^{i}}{\partial E}>0\) since \(\tilde{s}^{i}=\frac{\alpha }{n}\frac{1}{\tilde{p}}\). By using the expression of (25), one can write \(\tilde{e}^{i}\) as

$$\begin{aligned} \tilde{e}^{i}&= f(E) \equiv \frac{(a - b E)E}{c -d E} \end{aligned}$$


$$\begin{aligned} a&= \left\{ (1-\alpha ) \dfrac{n-(1-\alpha )}{n^{2}}+\lambda ^{i} \left[ \alpha \dfrac{n-(1-\alpha )}{n}\right] \right\} \\ b&= \lambda ^{i} \gamma \\ c&= \dfrac{n-(1-\alpha )}{n} \\ d&= \gamma . \end{aligned}$$

We have

$$\begin{aligned} f^{\prime }(E)&= \frac{(c-dE) (a - 2b E)+d(a-bE)E}{(c-dE)^2} \end{aligned}$$

The sign of \(f^{\prime }(E)\) is the same as that of \((c-dE) (a - 2b E)+d(a-bE)E\). The latter expression can be rewritten as

$$\begin{aligned} P(E)&= db E^2 - 2 bc E + ac. \end{aligned}$$

Clearly, this polynomial takes strictly negative values only if it has 2 positive real roots (because \(P(0)=ac\), \(P^{\prime }(E) = 2dbE - 2 bc\), and \(P^{\prime }(0) = - 2 bc < 0\)). This is the case whenever the discriminant \(\triangle\) is strictly positive:

$$\begin{aligned} \triangle&= 4 (bc)^2 - 4 abcd = 4bc (bc - ad)> 0. \end{aligned}$$

Now, we have

$$\begin{aligned} bc&> ad \end{aligned}$$

that is if

$$\begin{aligned} \lambda _{i} \gamma \dfrac{n-(1-\alpha )}{n} >&\gamma \left\{ (1-\alpha ) \dfrac{n-(1-\alpha )}{n^{2}}+\lambda ^{i} \left[ \alpha \dfrac{n-(1-\alpha )}{n}\right] \right\} , \end{aligned}$$


$$\begin{aligned} \lambda _{i} >&\frac{1}{n}. \end{aligned}$$

This condition cannot be simultaneously satisfied for all the traders. Moreover, E must be located between the two roots, which is possible, but not always possible.

Inefficiency of the OE with Pollution Permits

Let \(\theta ^{i}\), where \(\theta ^{i}>0\), for \(i=1,\dots ,n\), with \(\sum _{i}\theta ^{i}=1\), be the weight of oligopolist i in total welfare. Observe that all solutions to the problem.

$$\begin{aligned} \underset{(x_{1}^{i},x_{2}^{i},x_{1},x_{2};z^{i})}{\max } \nonumber \\ \sum _{i=1}^{n}\theta ^{i}\left( \alpha \ln x_{1}^{i}+(1-\alpha )\ln x_{2}^{i}\right) +\alpha \ln x_{1}+(1-\alpha )\ln x_{2} \nonumber \\ \text {s.t.} \sum \nolimits _{i=1}^{n}x_{1}^{i}+x_{1}=Y \nonumber \\ \sum \nolimits _{i=1}^{n}x_{2}^{i}+x_{2}+\sum \nolimits _{i=1}^{n}z_{1}^{i}=1 \end{aligned}$$

with \(\frac{\sum _{i=1}^{n}z^{i}}{\beta }=Y\) and \(\frac{\sum _{i=1}^{n}z^{i}}{\gamma }=E\) are Pareto-optimal.

The first-order optimality conditions for this problem are

$$\begin{aligned} \frac{\theta ^{i}}{x_{1}^{i}}=\frac{1}{\frac{\gamma }{\beta }E-\sum _{j=1}^{n}x_{1}^{j}}, \end{aligned}$$
$$\begin{aligned} \frac{\theta ^{i}}{x_{2}^{i}}=\frac{1}{1-\gamma E-\sum _{j=1}^{n}x_{2}^{j}}. \end{aligned}$$

Summing the first condition across j yields after a little algebra

$$\begin{aligned} x_{1}^{i}=\frac{\theta _{i}\frac{\gamma E}{\beta }}{1+\sum _{j=1}^{n}\theta ^{j}}. \end{aligned}$$

By the same reasoning, we also obtain

$$\begin{aligned} x_{2}^{i}=\frac{\theta ^{i}(1-\gamma E)}{1+\sum _{j=1}^{n}\theta ^{j}}. \end{aligned}$$

Using the two equations above, one gets

$$\begin{aligned} \frac{x_{2}^{i}}{x_{1}^{i}}=\frac{1-\gamma E}{\frac{\gamma }{\beta }E}. \end{aligned}$$

Using (26) to compute the ratio of the consumptions of an oligopolist in an oligopoly equilibrium, we obtain after some algebra

$$\begin{aligned} \frac{\tilde{x}_{2}^{i}}{\tilde{x}_{1}^{i}}=\frac{\frac{n-(1-\alpha )}{n}-\gamma E}{\frac{\gamma }{\beta }E}. \end{aligned}$$

Comparing (C6) with (C7), we see that

$$\begin{aligned} \frac{\tilde{x}_{2}^{i}}{\tilde{x}_{1}^{i}}<\frac{x_{2}^{i}}{x_{1}^{i}}\text {.} \end{aligned}$$

Welfare Comparison

We now compare \(\tilde{U}^{i}\) with \((U^{i})^{*}\). From (15) and (28), one has

$$\begin{aligned} \tilde{U}^{i}-(U^{i})^{*}= & {} \alpha \ln (1-\gamma E)+\ln \left( \frac{\alpha (1-\alpha )}{n^{2}}+\lambda ^{i}\left[ \alpha \frac{n-(1-\alpha )}{n}-\gamma E\right] \right) \nonumber \\&-\alpha \ln \left( \frac{n-(1-\alpha )}{n}-\gamma E\right) -\ln \lambda ^{i}(\alpha -\gamma E)\text {.} \end{aligned}$$

This leads to

$$\begin{aligned} \tilde{U}^{i}-(U^{i})^{*}=\alpha \ln \left( \frac{1-\gamma E}{\frac{n-(1-\alpha )}{n}-\gamma E}\right) +\ln \left( \frac{\frac{\alpha (1-\alpha )}{n^{2}}+\lambda ^{i}\left[ \alpha \frac{n-(1-\alpha )}{n}-\gamma E\right] }{\lambda ^{i}(\alpha -\gamma E)}\right) . \end{aligned}$$

The first term in brackets is positive. The second term is positive if \(\lambda ^{i}\leqslant \frac{1}{n}\). Then, for all \(i\in {\mathcal {O}}\) for which \(\lambda ^{i}\leqslant \frac{1}{n}\), it is certainly true that \(\tilde{U}^{i}-(U^{i})^{*}>0\). Finally, one can have \(\tilde{U}^{i}<(U^{i})^{*}\) for the oligopolists for which \(\lambda ^{i}>\frac{1}{n}\).

We now compare \(\tilde{U}\) with \(U^{*}\). From (16) and (), one gets

$$\begin{aligned} \tilde{U}-U^{*}=\alpha \ln \left( \left( \frac{n-1}{n}\right) \frac{1-\gamma E}{\frac{n-(1-\alpha )}{n}-\gamma E}\right) . \end{aligned}$$

We see that this expression is negative since all the terms in the logarithm are lower than 1.

OE with a Supply Subsidy

Using equation (39), we obtain after some tedious algebra a polynomial of degree two in the strategy s

$$\begin{aligned} P(s)=as^{2}+bs-c=0\text {,} \end{aligned}$$

where \(a\equiv \tau (1-\alpha )(\alpha +n)\), \(b\equiv \alpha (\frac{n-(1-\alpha )}{n}-\gamma E)-\tau \frac{\gamma }{\beta }\frac{E}{n}(1-\alpha )(\left( 1-\alpha \right) n+\alpha )\), and \(c\equiv \alpha (1-\alpha )\frac{\gamma }{\beta }\frac{E}{n}(1-\frac{1}{n})\).

To prove the existence of an OE with a subsidy, we must show that there is a positive solution to the above equation which satisfies: \(s<\min \{\frac{\gamma }{\beta n}E,\frac{1}{\tau n}\}\). This last condition ensures that the supply of the oligopolists is feasible and that the after-tax income of the competitive agent is positive.

We first prove that there is a positive solution lower than \(\gamma E/\beta n\). After some computations, we find that

$$\begin{aligned} P(0)=-\alpha (1-\alpha )\frac{\gamma }{\beta }\frac{E}{n}(1-\frac{1}{n})<0, \end{aligned}$$
$$\begin{aligned} P\left( \frac{\gamma E}{\beta n}\right) =\alpha \frac{\gamma E}{\beta n}(\alpha -\gamma E)>0. \end{aligned}$$

It follows that there is a unique positive root of the polynomial function located in \(]0,\frac{\gamma E}{\beta n}[\). The value of this root is

$$\begin{aligned} s(\tau )=\frac{-\alpha (\frac{n-(1-\alpha )}{n}-\gamma E)+\tau \frac{\gamma E}{\beta n}(1-\alpha )(\left( 1-\alpha \right) n+\alpha )+\sqrt{\Omega (\tau )}}{2\tau (1-\alpha )(\alpha +n)}\text {,} \end{aligned}$$


$$\begin{aligned} \Omega (\tau )= & {} \left( \alpha (\frac{n-(1-\alpha )}{n}-\gamma E)-\tau \frac{\gamma }{\beta }\frac{E}{n}(1-\alpha )(\left( 1-\alpha \right) +\alpha )\right) ^{2} \nonumber \\&+4\tau \alpha (1-\alpha )^{2}(\alpha +n)\frac{\gamma E}{\beta n}(1-\frac{1}{n}). \end{aligned}$$

It remains to check that the positive root of the polynomial is lower than \(1/\tau n\). We have

$$\begin{aligned} P(\frac{1}{\tau n})=\frac{1}{\tau n}\left( \frac{(1-\alpha )(1+n)}{n}+\frac{1}{\tau }(\alpha \frac{n-(1-\alpha )}{n}-\gamma E)\right) -(1-\alpha )\frac{\gamma E}{\beta n}. \end{aligned}$$

Since we will only study the case where \(\tau\) is close to 0, it follows that \(P(\frac{1}{\tau n})\) will always be positive.

Supply Subsidy and Supply of Good 1

Applying the implicit function theorem to equation (39), we obtain after some tedious algebra

$$\begin{aligned} \frac{\partial s}{\partial \tau }= \frac{ \frac{\gamma E}{\beta n}(\alpha +n(1-\alpha )) - s (\alpha +n)}{\tau (\alpha +n) + \frac{\gamma E}{\beta n s^2}(n-1)} \text {.} \end{aligned}$$

Since \(s(\tau )\) is continuous at 0, we may define \(\frac{\partial s}{\partial \tau }_{\mid _{\tau =0}}=\lim _{\tau \rightarrow 0+}\frac{\partial s}{\partial \tau }\), where

$$\begin{aligned} \lim _{\tau \rightarrow 0+}\frac{\partial s}{\partial \tau }=\frac{\frac{\gamma E}{\beta n}(\alpha +n(1-\alpha ))-s(0)(\alpha +n)}{\tau (\alpha +n)+\frac{\gamma E}{\beta ns^{2}(0)}(n-1)}\text {.} \end{aligned}$$

Using the expression of s(0) in the above definition, we get the result.

Notice that

$$\begin{aligned} \frac{1}{s^{2}(0)}\frac{\partial s}{\partial \tau }_{\mid _{\tau =0}}=\frac{1}{n-1}\left( n(1-\alpha )+\alpha -\frac{(\alpha +n)(1-\alpha )(1-\frac{1}{n})}{\frac{n-(1-\alpha )}{n}-\gamma E}\right) . \end{aligned}$$

Proposition 6 follows.

Effect of a Supply Subsidy on Oligopolist i’s Welfare

Differentiating \(V^{i}(\tau )\) with respect to \(\tau\) yields

$$\begin{aligned} \frac{\partial V^{i}(\tau )}{\partial \tau }=\frac{-\alpha }{\frac{\gamma }{\beta }\frac{E}{n}-s}\frac{\partial s}{\partial \tau }+(1-\alpha )^{2}\frac{s+\tau \frac{\partial s}{\partial \tau }}{\frac{\alpha -\gamma E}{n}+\tau s(1-\alpha )}\text {;} \end{aligned}$$


$$\begin{aligned} \lim _{\tau \rightarrow 0+}\frac{\partial V^{i}(\tau )}{\partial \tau }=\frac{-\alpha }{\frac{\gamma }{\beta }\frac{E}{n}-s}\frac{\partial s}{\partial \tau }_{\mid _{\tau =0}}+(1-\alpha )^{2}\frac{s}{\frac{\alpha -\gamma E}{n}}. \end{aligned}$$

From the first-order conditions for oligopolists i’s problem, we know that

$$\begin{aligned} \frac{\alpha }{\frac{\gamma }{\beta }e^{i}-s^{i}}=\frac{(1-\alpha )\left( p+\tau -s^{i}\frac{\alpha (1-T)}{\left( \sum _{k=1}^{n}s^{k}\right) ^{2}} \right) }{(p+\tau )s^{i}-(\gamma +q)e^{i}+q\frac{E}{n}}. \end{aligned}$$

In equilibrium, we get

$$\begin{aligned} \frac{\alpha }{\frac{\gamma }{\beta }\frac{E}{n}-s}=\frac{(1-\alpha )\frac{\alpha }{ns}(1-\frac{1}{n})}{\frac{\alpha -\gamma E}{n}}. \end{aligned}$$

Using this last equality, and (G2) when \(\tau\) goes to zero, we obtain

$$\begin{aligned} \lim _{\tau \rightarrow 0+}\frac{\partial V^{i}(\tau )}{\partial \tau }=-\frac{(1-\alpha )\frac{\alpha }{ns}(1-\frac{1}{n})}{\frac{\alpha -\gamma E}{n}}\frac{\partial s}{\partial \tau }_{\mid _{\tau =0}}+(1-\alpha )^{2}\frac{s}{\frac{\alpha -\gamma E}{n}}. \end{aligned}$$

Therefore, the sign of \(\lim _{\tau \rightarrow 0+}\frac{\partial V^{i}(\tau )}{\partial \tau }\) is also that of

$$\begin{aligned} \Sigma =-\frac{\alpha }{ns(0)}(1-\frac{1}{n})\frac{\partial s}{\partial \tau }_{\mid _{\tau =0}}+(1-\alpha )s(0). \end{aligned}$$

Pareto-improving supply subsidy

Observe that the bounds \(\frac{1-\alpha }{\alpha }n\) and \(\frac{1-\alpha }{\alpha }\frac{n^{2}}{n-1}\) in condition (51) are both decreasing in \(\alpha\), and go to \(+\infty\) when \(\alpha\) goes to 0, and both of them vanish when \(\alpha = 1\). Now, the expression \(\frac{1}{s^{2}(0)}\frac{\partial s}{\partial \tau }_{\mid _{\tau =0}}\) is an increasing function of \(\alpha\) which always takes finite values. Indeed, from equation (82) we have

$$\begin{aligned} \frac{\partial }{\partial \alpha }\left( \frac{1}{s^{2}(0)}\frac{\partial s}{\partial \tau }_{\mid _{\tau =0}}\right)&= -1 + \frac{2 \alpha + n-1}{\frac{n+\alpha -1}{n}-\gamma E} + \frac{(\alpha +n)(1-\alpha )}{n\left( \frac{n+\alpha -1}{n}-\gamma E \right) ^2}>0. \end{aligned}$$

Moreover, we have

$$\begin{aligned} \lim _{\alpha \rightarrow 0} \frac{1}{s^{2}(0)}\frac{\partial s}{\partial \tau }_{\mid _{\tau =0}} = \frac{n}{n-1} \frac{- \gamma E}{\frac{n-1}{n}-\gamma E}<0, \end{aligned}$$


$$\begin{aligned} \lim _{\alpha \rightarrow 1} \frac{1}{s^{2}(0)}\frac{\partial s}{\partial \tau }_{\mid _{\tau =0}} = 1/(n-1). \end{aligned}$$

Therefore, the graph of this function crosses that of the two bounds (viewed as functions of \(\alpha\)). The value of \(\underline{\alpha }\) is given by the value of \(\alpha\) at which the graph of \(\frac{1}{s^{2}(0)}\frac{\partial s}{\partial \tau }_{\mid _{\tau =0}}\) crosses the lower bound \(\frac{1-\alpha }{\alpha }n\). Similarly, the value of \(\overline{\alpha }\) is the value of \(\alpha\) at which the graph \(\frac{1}{s^{2}(0)}\frac{\partial s}{\partial \tau }_{\mid _{\tau =0}}\) crosses that of the upper bound \(\frac{1-\alpha }{\alpha }\frac{n^{2}}{n-1}\).

We still have to check that the values of \(\alpha\) in \([\underline{\alpha }, \overline{\alpha }]\) are consistent with those ensuring the existence of an oligopoly equilibrium. First, notice that by construction of the set \([\underline{\alpha }, \overline{\alpha }]\), we have \(\frac{\partial s}{\partial \tau }>0\). But Proposition 6 implies that we must then have \(\alpha >\frac{\gamma nE}{1+(n-1)\gamma E}\). To ensure the existence of the oligopoly equilibrium, it is sufficient to check that \(\alpha \frac{n-(1-\alpha )}{n}>\gamma E\). In turn, this last condition holds whenever

$$\begin{aligned} \frac{\alpha }{n-\alpha (n-1)}&< \alpha \frac{n-(1-\alpha )}{n}. \end{aligned}$$

But the condition above is equivalent to

$$\begin{aligned} 0&< (1-\alpha ) \left( n^2 - n (2-\alpha ) - \alpha \right) . \end{aligned}$$

This condition is always satisfied for \(n\ge 2\) and \(\alpha >0\).

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Crettez, B., Jouvet, PA. & Julien, L.A. Pareto-Improving Supply Subsidy in a Simple General Oligopoly Equilibrium Model with Pollution Permits. Environ Model Assess (2021).

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  • Oligopoly equilibrium
  • Pollution permit markets
  • Pollution
  • Supply subsidy

Mathematics Subject Classification

  • D43
  • D51
  • H2