Timing of adoption of clean technologies, transboundary pollution and international trade

We consider a symmetric model composed of two countries and a (cid:133)rm in each country. Firms produce the same good by means of a polluting technology that uses fossil energy. However, these (cid:133)rms can adopt a clean technology that uses a renewable energy and that has a lower unit cost. Surprisingly, opening markets to international competition increases the per-unit emission-tax and decreases the per-unit production subsidy. Interestingly, the socially-optimal adoption date under a common market better internalizes transboundary pollution than that under autarky, and than the optimal adoption date of regulated (cid:133)rms. However, the optimal adoption date of non-regulated (cid:133)rms completely don(cid:146)t internalize transboundary pollution. In autarky (resp. a common market), regulated (cid:133)rms adopt earlier (resp. later) than what is socially-optimal, whereas non-regulated (cid:133)rms adopt later than the socially-optimal adoption date and than the optimal adoption date of regulated (cid:133)rms. Therefore, in autarky (resp. a common market) regulators can induce (cid:133)rms to adopt at the socially-optimal adoption date by giving them postpone ( resp. speed up) adoption subsidies. Opening markets to international trade, speeds up the socially-optimal adoption date and delays optimal adoption dates of regulated and non-regulated (cid:133)rms.


Introduction
This paper tries to study the relation that may exist between the timing of adoption of clean technologies, transboundary pollution and opening markets 1 to international competition. Typical examples of clean production technologies are those using renewable energy such as solar energy, whereas polluting production technologies usually use fossil energy. Our research is related to at least four literature …elds.
The …rst …eld deals with renewable energies and clean technologies. Dosi and Moretto (1997) studied the regulation of a …rm which can switch to a clean technology by incurring an irreversible investment cost. To bridge the gap between the private and the policy-maker's desired timing of innovation, they recommended that the regulator stimulate the innovation by subsidies and by reducing the uncertainty concerning the pro…tability of the clean technology by appropriate announcements. Dosi and Moretto (2010) extended the previous study to oligopolistic …rms and studied the incentives of not being the …rst …rm adopting the clean technology. Soest (2005) analyzed the impact of environmental taxes and quotas on the timing of adoption and found that neither policy instrument is always preferred to the other. Nasiri and Zaccour (2009) proposed a game-theoretic model and analyzed the process of utilizing biomass for power generation. They considered three players: distributor, facility developer, and participating farmer, characterized the subgame-perfect Nash equilibrium and discussed its features. Wirl and Withagen (2000) showed that pollution-control policy is not necessarily optimal in the sense of giving the social optimum. Fischer, Withagen and Toman (2004) developed a model of a uniform good that can be produced by either a polluting or a clean technology, and showed that the optimal transition path is quite di¤erent with a clean or polluting initial environment. Ben Youssef (2010) showed that the instantaneous regulated monopoly adopts the clean technology earlier than what is socially-optimal, while the non-regulated monopoly adopts later than what is socially-optimal. The regulator can induce the monopoly to adopt at the socially-optimal date by a postpone adoption subsidy. Fujiwara (2011) developed a dynamic game model of an asymmetric oligopoly with a renewable resource and showed that increasing the number of e¢cient …rms reduces welfare. Reichenbach and Requate (2012) considered a model with two types of electricity producers and showed that a …rst-best policy requires a tax in the fossil-fuel sector and an output subsidy for the renewable energy sources sector.
Many empirical studies have been interested in clean technologies, among which Whitehead  The second …eld deals with transboundary pollution. Chander and Tulkens (1992) showed that non-cooperating behavior of countries is not Pareto-optimal. Mansouri and Ben Youssef (2000) showed the necessity of cooperation between countries to e¤ectively internalize all the transboundary pollution, while reaching the …rst best. Nevertheless, some studies showed that non-cooperating countries can reach the …rst best under some conditions (Hoel (1997), Zagonari (1998)). Ben Youssef (2009) showed that free R&D spillovers and the competition of …rms on the common market help non-cooperating countries to better internalize transfrontier pollution. Ben Youssef (2011) established that the investment in absorptive R&D enables non-cooperating regulators to better internalize transfrontier pollution.
The third …eld deals with international trade. Because pollution crosses the borders, Copeland and Taylor (1995) showed that uncoordinated regulation of pollution at the national level and free trade don't necessarily raise welfare. Under incomplete information, Péchoux and Pouyet (2003) showed that …rms' competition generated by the common market enables regulators to reduce the informational rents captured by …rms, thereby reinforcing the need to open the markets to international trade. Using a static model with no investment possibility in cleaner production technology, Cremer and Gahvari (2004) showed that …rms switch to a less polluting but more costly production technique, under economic integration.
The fourth …eld deals with the timing of adoption of new technologies. The di¤usion of a new technology has been analyzed by Reinganum (1981). She considered an industry composed of two …rms which can adopt a cost reducing technology within a period of time. She showed that even in the case of identical …rms and complete information, there is di¤usion of innovation over time because one …rm innovates before the other and gains more. Fudenberg and Tirole (1985) made less strong conditions on the payo¤s of …rms and showed that under certain conditions there is di¤usion, whereas under other conditions …rms adopt this new technology simultaneously. Hoppe (2000) extended the work of Fudenberg and Tirole to include uncertainty regarding the pro…tability of the new technology. She showed that there may be second-mover advantages because of informational spillovers. Dutta et al. (1995) got a similar result in a context where the later innovator continues to develop the technology and eventually markets a higher-quality good. Riordan (1992) showed that price and entry regulations, in many cases, bene…cially slow down technology adoption and, in some other cases, change the order in which …rms adopt new technologies by speeding up one …rm's adoption date and slowing down the other's. Milliou and Petrakis (2011) showed that when goods are su¢ciently di¤erentiated, the adoption of a new technology occurs later than is socially-optimal.
Our paper di¤ers from the existing literature by the fact that we try to know how the adoption dates of clean technologies may be a¤ected when markets are opened to international competition, and how the regulator may change his behavior with respect to …rms he is regulating. Also, in the present paper, we study the relation between the adoption of clean technologies and transboundary pollution.
We consider a symmetric model composed of two countries and a monopolistic …rm operating in each country. Firms produce the same homogeneous good by using a polluting technology that uses fossil energy. However, these …rms can adopt a new and clean production technology by incurring an investment cost that decreases exponentially with the adoption date. This clean technology uses a renewable energy and therefore has a lower unit production cost. We study and compare the case where …rms are not regulated at all, and the case where each …rm is regulated at each period of time i.e. each non-cooperating regulator looks for static social optimality. In this latter case, a per-unit emission-tax is used when a …rm uses the polluting technology, and a per-unit production subsidy, than can be considered as a …scal incentive, is used when a …rm uses the clean technology. We also study and compare the case where each …rm operates in a separate home market, and the case where …rms compete in the same common market formed by the consumers of the two countries.
In autarky, since our model is symmetric, …rms adopt the clean technology simultaneously. However, in a common market, and because of the competition between …rms, we impose a condition on parameters to avoid the complicated case where …rms adopt at di¤erent dates, and we show that adoption is simultaneous.
When markets are opened to international competition, the per-unit emissiontax increases when the polluting technology is used, and the per-unit production subsidy decreases when the clean technology is used. These results are interesting and even surprising because one may think that, to give a competitive advantage to its domestic …rm, each regulator reduces the per-unit emissiontax and increases the per-unit production subsidy, when markets are opened to international trade. Ben Youssef (2009) found similar results with a di¤erent model where regulatory instruments are a per-unit emission-tax and a per-unit R&D subsidy.
Interestingly, the socially-optimal adoption date under a common market better internalizes transboundary pollution than that under autarky, and than the optimal adoption date of regulated …rms. However, the optimal adoption date of non-regulated …rms completely don't internalize transboundary pollution. Therefore, the regulator should know how to intervene to get …rms adopt at the socially-optimal dates. This result is of great interest because this paper is the …rst attempt linking the adoption of clean technologies with transboundary pollution. Notice that, using very di¤erent models than the present, Ben Youssef (2009) showed that R&D spillovers and the competition of …rms on the common market help non-cooperating countries to better internalize transboundary pollution, and Ben Youssef (2011) showed that the investment in absorptive R&D help non-cooperating countries to better internalize transboundary pollution.
The intervention of regulators on how to induce …rms adopting the clean technology at the socially-optimal adoption date completely changes when markets are opened to international competition. Indeed, in autarky (resp. common market), regulated …rms adopt earlier (resp. later) than what is socially-optimal, whereas non-regulated …rms adopt later than the socially-optimal adoption date and than the optimal adoption date of regulated …rms. Therefore, in autarky, regulators can induce …rms to adopt at the socially-optimal adoption date by giving them a postpone adoption subsidy. However, in common market, regulators can induce …rms to adopt at the socially-optimal adoption date by giving them a speed up adoption subsidy.
International competition reduces the instantaneous gain from using the clean technology of non-regulated and regulated …rms, with respect to autarky. Consequently, non-regulated and regulated …rms delay the adoption of the clean technology when markets are opened to international trade. However, the instantaneous social welfare gain from the adoption of the clean technology increases with market opening, leading to an early socially-optimal adoption date under a common market. These results are new and interesting because the impact of opening markets to international competition on the timing of adoption of clean technologies has not been previously studied. This paper is organized as follows. Section 2 deals with the autarky case. Section 3 deals with the common market case, and Section 4 compares the two market regimes. Section 5 concludes and an Appendix contains some proofs.

Autarky
We consider a symmetric model consisting of two countries and two …rms. Firm i located in country i is a regional monopoly and produces good i in quantity q i sold in the domestic market with the inverse demand function: p i = a 2q i ; a > 0:Thus, the market size of each country is a=2.
The consumption of q i engenders a consumers' surplus in country i equal to: At the beginning of the game i.e. at date 0, …rms produce goods by using an old and polluting production technology using fossil fuels and characterized by a positive emission/output ratio e > 0. The pollution emitted by …rm i is We suppose that pollution crosses the borders and that damages in country i are due to the domestic pollution and the foreign pollution: where > 0 is the marginal damage of domestic pollution and > 0 is the marginal damage of foreign pollution.
When …rm i uses the polluting technology, its unit production cost is d > 0 and its pro…t 1 is a id = p i (q i )q i dq i . Each …rm i behaves for an in…nite horizon of time and can adopt a clean production technology within a period of time i . This clean technology does not pollute at all, uses a renewable energy and therefore has a lower unit cost of production c verifying 0 < c < d. Thus, the pro…t of …rm i is a ic = p i (q i )q i cq i . We suppose that the marginal damage of production e is neither too small nor too high verifying the following condition: The instantaneous social welfare of country i is equal to the consumers' surplus, minus damages plus the pro…t of the domestic …rm: To get the new and clean production technology, an investment cost is necessary. This latter could comprise the R&D cost or the cost of acquisition and installation of the clean technology.
The cost of adopting the clean technology by …rm i at date i actualized at date 0 is: with > 0 is the cost of immediate adoption of the clean technology, r > 0 is the discount rate, and the parameter m denotes that the cost of adoption decreases more rapidly when it is greater. We assume that m > 1. 2 Function V is decreasing due to the existence of freely-available scienti…c research enabling a …rm to reduce the cost of adopting the clean technology when it delays its adoption, and is convex because the adoption cost increases more rapidly when a …rm tries to accelerate the adoption date.
Let's remark that i = +1 means that …rm i will never adopt the clean technology.

Non-regulated …rms
In this section, we study the case where, at each period of time, each …rm is not regulated even when it uses the polluting technology.
When both …rms use the polluting technology, then each one maximizes its pro…t na idd to get the optimal level of production: 3 When both …rms use the clean technology, then each one maximizes its pro…t na icc to get the optimal level of production: It is easy to verify that q na icc > q na idd meaning that …rms produce more with the clean technology because of its lower unit production cost: If only …rm 1 adopts the clean technology and …rm 2 still uses the polluting technology, then the pro…ts of …rms are denoted by na 1cd (q 1 ) and na 2cd (q 2 ), respectively. Optimal production quantities for …rms are given by: We can verify that q na 1cd = q na 1cc ; q na 2cd = q na 2dd and that q na 1cd > q na 2cd :Thus, the …rm using the clean technology produces more than that using the polluting technology.

Regulated …rms
In this section, we study the case where …rms are regulated at each period of time. First, we start by determining the socially-optimal production quantities for each regulator. Then, we determine the regulatory instruments inducing the socially-optimal production quantities in each country.
When both …rms use the polluting technology, the instantaneous social welfare of country i is: Maximizing the expression given by (7) with respect to q i gives the sociallyoptimal production level with the polluting technology for each regulator We assume the …rst inequality of the following condition such that production quantities are positive. Also, the second inequality is assumed to avoid studying the complicated case of non-simultaneous adoption of the clean technology in the common market case. Moreover, the second inequality of (1) assures that there is no contradiction in inequality (9) : Therefore, the maximum willingness to pay for the good must be higher than the marginal cost of production plus the marginal damage of production.
Since each …rm is a polluting monopoly, it is regulated. An emission-tax per-unit of pollution t a idd is su¢cient to induce the socially-optimal levels of production and pollution.
The instantaneous net pro…t of …rm i is: The socially-optimal per-unit emission-tax that induces …rm i to producê q a idd is: Using the expression ofq a idd , we can show that: 7 When e > d c 2 i.e. the marginal damage of pollution is high enough, the above condition is always satis…ed and the emission-tax is positive. When e < d c 2 and a < d+2 e, the emission-tax is positive. However, when e < d c 2 and a > d + 2 e i.e. the marginal damage of pollution is low enough, the emission-tax is negative meaning that each regulator subsidizes production to deal with monopoly distortion. If both …rms use the clean technology, the instantaneous social welfare of country i is: Maximizing the expression given by (13) with respect to q i gives the sociallyoptimal production level with the clean technology for regulator i: Using the second inequality of (1), we show thatq a icc >q a idd . Therefore, the clean technology enables to produce more and without polluting the environment.
We can establish that : When e > d c 2 i.e. the marginal damage of pollution is high enough, or when e < d c 2 and a < d + 2 e, the above condition is always satis…ed because regulators care about the environment whereas non-regulated …rms do not care about the environment. However, when e < d c 2 and a > d + 2 e i.e. the marginal damage of pollution is low enough, socially-optimal production is higher than the production of non-regulated monopolistic …rms.
With the clean technology, socially-optimal production is always higher than that of non-regulated …rms (q a icc > q na icc ). Since the production process is clean, each regulator gives his …rm a subsidy s a icc for each unit produced, which can be considered as a …scal incentive. One may think about production of electricity. A per-unit production subsidy can be given by a regulator when the production process is clean (using solar energy, for instance). This per-unit subsidy is chosen so that it induces the socially-optimal level of production. Indeed, the instantaneous net pro…t of …rms i is: The socially-optimal per-unit subsidy that induces …rm i to produceq a icc is: If we consider the case in which one of the two …rms, for instance …rm 1; has adopted the clean technology, whereas the other still produces using the polluting technology, then the pro…ts of …rms are a 1cd (q 1 ) and a 2cd (q 2 ), respectively. The instantaneous social welfare of regulator 1 and 2 are: Regulator i maximizes his social welfare function with respect to q i to get the socially-optimal production quantities: We can easily verify thatq a 1cd >q a 2cd meaning that it is socially preferred that the …rm using the clean technology produces more than that using the polluting technology.
Since q na 1cd = q na 1cc <q a 1cd =q a 1cc , regulator 1 can induce …rm 1 to produce the socially-optimal production quantities by an appropriate subsidy s a 1cd = s a 1cc . Since q na 2cd = q na 2dd >q a 2cd =q a 2dd , a per-unit emission-tax t a 2cd = t a 2dd is needed to induce …rm 2 to produce the socially-optimal quantity.
In the Appendix, we show that: 4 Thus, we can establish the following Proposition: Proposition 1 Under autarky, the instantaneous gain from using the clean technology is greater for the …rst adopter regulated …rm than for its regulator. This latter instantaneously bene…ts more from using the clean technology than its …rst adopter non-regulated …rm.
Indeed, when a regulated …rm adopts the clean technology, it no longer pays a pollution tax, receives production subsidies and its unit production costs decreases. This increases its instantaneous net pro…t signi…cantly. The instantaneous social welfare level increases due to the absence of local environmental damages and the lower production cost. However, this last increase is less important than that of the regulated …rm. The only bene…t of a non-regulated …rm from adopting the clean technology is the reduction of its unit production cost. Consequently, its instantaneous net pro…t increase is less important than that of the instantaneous social welfare.

Optimal adoption dates
In this section, we will determine the optimal adoption dates. We still suppose that, in case where …rms adopt at di¤erent dates, the …rst adopter is …rm 1 and the second adopter is …rm 2. Thus, in the following expressions, we suppose . This implies that the intertemporal net pro…t of non-regulated and regulated …rm i can be written as depending only on i :However, since S a 1cd 6 = S a 1cc and S a 2cd 6 = S a 2dd because of crossborder pollution, intertemporal social welfare of regulators 1 and 2 depend on 1 and 2 .
Each regulator chooses the socially-optimal adoption date that maximizes his intertemporal social welfare function. Each regulated and non-regulated …rm chooses the optimal adoption date that maximizes its intertemporal net pro…t.
The intertemporal social welfare of regulators 1 and 2, intertemporal net pro…ts of regulated and non-regulated …rm i are, respectively: In order to get positive adoption dates, we need the following condition, which can be always veri…ed by choosing and/or m high enough: 5 In the Appendix, we determine the optimal adoption dates which show that …rms adopt simultaneously the clean technology: Proposition 2 Because of symmetry, when markets are separated, …rms adopt the clean technology simultaneously. 5 Notice that the left expression of (26) is independent of parameters , m and r: Inequality (21) and the fact that m > 1, enable us to make the following ranking: We can state the following Proposition: The optimal adoption date of regulated …rms is earlier than that socially-optimal. However, the optimal adoption date of non-regulated …rms is later than that socially-optimal.
The above proposition shows that socially-optimal instantaneous regulation may not be dynamically optimal with respect to the adoption of clean technologies. They are due to the fact that, under autarky, the incentives to adopt are, in order, greater for regulated …rms, regulators and non-regulated …rms. This is clearly established by the inequalities in (21). This result is similar to the one established by Ben Youssef (2010) who used a model comprising one regulator and a monopolistic …rm.
Paradoxically, if regulators desire that regulated …rms delay their adoption to the socially-optimal adoption date, they must compensate …rms for the losses they incur by this adoption delay. If the intertemporal net pro…ts of the regulated …rm i are IU i ( a ) and IU i (^ a ) when the adoption dates are a and^ a , respectively, then the postpone adoption subsidy (compensation) is: Proposition 4 When markets are separated, each regulator can push his regulated …rm to delay its adoption of the clean technology by giving it a postpone adoption subsidy that compensates the …rm for the losses it incurs when the latter delays its optimal adoption date to the socially-optimal adoption date.

Common market
When markets are opened to competition, the inverse demand function of the perfect substitute goods produced by …rms becomes P = a (q i + q j ). The size of the integrated market is a. The total consumers' surplus is equally divided between the two symmetric countries: The emission-tax per-unit of pollution is t cm i and the per-unit production subsidy is s cm i :

11
When …rm i uses the polluting technology, its pro…t is given by cm id = p(q i ; q j )q i dq i , and when it uses the clean technology, its pro…t is given by The instantaneous social welfare of country i is equal to the consumers' surplus, minus damages plus the pro…t of the domestic …rm:

Non-regulated …rms
When both …rms use the polluting technology, each one maximizes its pro…t ncm idd to get the optimal level of production: When both …rms use the clean technology, each one maximizes its pro…t ncm icc to get the optimal level of production: As for the autarky case, the clean technology enables non-regulated …rms to produce more because of its lower unit production cost (q ncm icc > q ncm idd ) : If only …rm 1 uses the clean technology, whereas …rm 2 still uses the polluting technology, then the pro…t of each non-regulated …rm is ncm 1cd and ncm 2cd , respectively. The optimal productions are given by: The second inequality of condition (9) shows that q ncm 2cd < 0: Thus, the case where the two non-regulated …rms adopt at di¤erent dates is unrealistic. From now on, we will suppose that if non-regulated …rms adopt the clean technology, then this adoption is simultaneous.

Regulated …rms
When both …rms use the polluting technology, the instantaneous social welfare of regulator i is: Maximizing the expression given by (36) with respect to q i gives the sociallyoptimal production level with the polluting technology for regulator i: Since …rm i is a duopoly producing with pollution, it is regulated. A perunit emission-tax is su¢cient to induce the socially-optimal level of production. Indeed, the instantaneous net pro…t of …rm i is: The socially-optimal per-unit emission-tax that induces …rm i to producê q cm idd is: When both …rms use the clean technology, the instantaneous social welfare of country i is: Maximizing the expression given by (40) with respect to q i gives the sociallyoptimal production level with the clean technology for each regulator i: Let's notice that, because non-regulated …rms don't take into account environmental damages, they always produce more than what is socially-optimal (q ncm idd >q cm idd ): However, with the clean technology and because of the duopolistic distortion, non-regulated …rms always produce less than what is socially-optimal (q ncm icc <q cm icc ): Since the production process is clean, each regulator gives his …rm a per-unit production subsidy s cm icc ;which is chosen to induce the socially-optimal level of production. Indeed, the instantaneous net pro…t of …rms i is: The socially-optimal per-unit production subsidy that induces …rm i to produceq cm icc is: Consider the case where …rm 1 has adopted the clean technology, whereas …rm 2 still produces using the polluting technology. The instantaneous social welfare of regulator 1 and 2 are, respectively: Maximizing expressions given by (44) and (45) respectively with respect to q 1 and q 2 gives: Because of the second inequality of (9) and the …rst inequality of (1),q cm 2cd < 0:We conclude that considering the case where one …rm uses the clean technology and the other one uses the polluting technology is unrealistic. Let's notice that we have assumed the …rst inequality and the second inequality of conditions (1) and (9) to prevent the study of the complicated case where …rms adopt the clean technology at di¤erent dates. Indeed, even if it is possible to determine the optimal adoption dates, comparing them is very di¢cult to do in the common market case.
Proposition 5 Under common market, due to conditions assumed on parameters, …rms adopt the clean technology simultaneously.
In the Appendix, we show that: These inequalities enable us to establish the following Proposition: Proposition 6 Under common market, the instantaneous gains from using the clean technology are greater for regulators than for regulated …rms. These latter instantaneously bene…t more from the clean technology than non-regulated …rms.
The reasons explaining the bene…t from the clean technology are the same than for the autarky case. However, when regulated …rms compete in a common market, their instantaneous net pro…ts increase, due to the adoption of the clean technology, is less important than the increase of instantaneous social welfare levels.

Optimal adoption dates
When both …rms adopt the clean technology at the same date , the intertemporal social welfare of regulator i, intertemporal net pro…t of the regulated and non-regulated …rm i are, respectively: In the Appendix, we determine the socially-optimal adoption date for regulators, the optimal adoption date for regulated …rms and non-regulated …rms, which are respectively: Inequality (48) and the assumption m > 1, enable us to make the following ranking: Thus, we can state the following Proposition: Proposition 7 When markets are opened to competition, the socially-optimal adoption date is earlier than the optimal adoption date for regulated …rms. This latter is earlier than the optimal adoption date for non-regulated …rms.
The above proposition shows that, even in a common market, sociallyoptimal instantaneous regulation may not be dynamically optimal with respect to the adoption of clean technologies. They are due to the fact that, under a common market, the incentives to adopt the clean technology are, in order, greater for regulators, regulated …rms and non-regulated …rms. This is clearly demonstrated by the inequalities in (48).
If regulators desire that regulated …rms accelerate their adoption to the socially-optimal adoption date, they must compensate …rms for the losses they incur by an early adoption. If the intertemporal net pro…ts of the regulated …rm i are IU i ( cm ) and IU i (^ cm ) when the adoption dates are cm and^ cm , respectively, then the early adoption subsidy (compensation) is: Proposition 8 In a common market, each regulator can push his regulated …rm to accelerate its adoption of the clean technology by giving it an early adoption subsidy that compensates the …rm for the losses it incurs when this latter accelerates its optimal adoption date to the socially-optimal adoption date.

Autarky versus common market
Looking to expressions (27) and ( 52) The above expressions show that, under a common market, the sociallyoptimal adoption date internalizes transboundary pollution. However, under autarky, the socially-optimal adoption date does not internalize transboundary pollution. Moreover, under both market regimes, optimal adoption dates of regulated and non-regulated …rms completely don't internalize transboundary pollution. This is due to the fact that our damage function is linear with respect to the total pollution. Indeed, production for non-regulated …rms, socially-optimal production and net pro…t of …rms completely don't internalize transboundary pollution. 6 This result is of great interest because this paper is the …rst attempt linking adoption of clean technologies with transboundary pollution. Notice that, using a very di¤erent model, Ben Youssef (2009) showed that R&D spillovers and the competition of …rms on the common market help non-cooperating countries to better internalize transboundary pollution. Ben Youssef (2011) showed that the investment in absorptive R&D help non-cooperating countries to better internalize transboundary pollution. We can state the following Proposition:

Proposition 9
The socially-optimal adoption date under a common market better internalizes transboundary pollution than that under autarky, and than the optimal adoption date of regulated …rms. However, under both market regimes, the optimal adoption date of non-regulated …rms completely don't internalize transboundary pollution.
Let us notice that if there were no transfrontier pollution between countries, i.e. = 0;then from expressions (69) and (71), we deduce that the optimal adoption date for regulated …rms and the socially-optimal adoption date coincide under common market ( cm =^ cm ). Indeed, since the instantaneous social welfare gain from using the clean technology internalizes transboundary pollution causing a speedup in technology adoption, the absence of transboundary pollution delays the socially-optimal adoption date to the optimal adoption date for regulated …rms. Nonetheless, under autarky, the optimal adoption date of regulated …rms still remains earlier than that socially-optimal because this latter does not internalize transboundary pollution.
The comparison of optimal production quantities shows that the competition on the common market pushes non-regulated …rms to increase their production (q ncm idd > q na idd ; q ncm icc > q na icc ):However, socially-optimal productions are the same under the two market regimes (q cm idd =q a idd ;q cm icc =q a icc ):Consequently, when the polluting technology is used, the per-unit emission-tax is greater under common market (t cm idd > t a idd ). When the clean technology is used, the per-unit production subsidy is greater under autarky (s a icc > s cm icc ). These results are interesting and even surprising because one may think that, to give a competitive advantage to its domestic …rm, each regulator reduces the per-unit emission tax and increases the per-unit production subsidy, when markets are opened to international competition. Ben Youssef (2009) found a similar result with a di¤erent model where regulatory instruments are a per-unit emission-tax and a per-unit R&D subsidy.
Proposition 10 Opening markets to international competition increases the per-unit emission-tax when the polluting technology is used, and decreases the per-unit production subsidy when the clean technology is used.
In the Appendix, we show that, under a common market, the instantaneous social welfare gain from using the clean technology is greater than the instantaneous social welfare gain from using the clean technology of the …rst adopter under autarky. Thus, opening markets to international trade speeds up the socially-optimal adoption date (^ cm <^ a ). Let us notice that if there were no transfrontier pollution between countries, i.e. = 0;then from expressions (57) and (69), we deduce that the socially-optimal optimal adoption dates are the same under both market regimes (^ cm =^ a ).
We also deduce that the competition of regulated …rms on a common market reduces their instantaneous gain from using the clean technology with respect to the case where markets are separated. Thus, opening markets to international competition delays the adoption of the clean technology by regulated …rms ( a < cm ): Finally, we show that the competition of non-regulated …rms on a common market reduces their instantaneous gain from using the clean technology with respect to the case where markets are separated. Therefore, international competition delays the adoption of the clean technology by non-regulated …rms ( na < ncm ): Proposition 11 International competition reduces the instantaneous gain from using the clean technology by both non-regulated and regulated …rms, with respect to autarky. Consequently, non-regulated and regulated …rms delay the adoption of the clean technology when markets are opened to international trade. However, the instantaneous social welfare gain from using the clean technology increases with market opening, leading to an acceleration of the socially-optimal adoption date.
The above results are new and interesting because the impact of opening markets to international trade on the timing of adoption of clean technologies has not been previously studied.

Conclusion
In this paper, we consider two countries and a monopolistic …rm operating in each country. Firms produce the same homogeneous good by using a polluting technology that uses fossil energy. These …rms can adopt a new and clean production technology by incurring an investment cost that decreases with the adoption date. This clean technology uses a renewable energy and therefore has a lower per-unit production cost. We consider and compare the case where …rms are not regulated at all, and the case where each …rm is regulated at each period of time i.e. each regulator looks for static social optimality. When …rms are instantaneously regulated, a per-unit emission-tax is used when a …rm uses the polluting technology, and a per-unit production subsidy, that can be considered as a …scal incentive, is used when a …rm uses the clean production technology. We also study and compare the case where each …rm operates in a separate domestic market, and the case where …rms compete in the same common market formed by the consumers of the two countries.
Our results show that, contrary to what one may expect, international competition increases the per-unit emission-tax when the polluting technology is used, and decreases the per-unit production subsidy when the clean technology is used.
In autarky, because our model is symmetric, both …rms adopt the clean technology simultaneously. However, in a common market, because of the competition between …rms, non-simultaneous adoption may occur. We impose conditions on parameters to avoid the complicated case where …rms adopt at di¤erent dates, and we show that adoption is simultaneous. Indeed, even if it is possible to determine the optimal adoption dates, comparing them in the common market case is very di¢cult to do if adoption is not simultaneous.
Interestingly, the socially-optimal adoption date under a common market better internalizes transboundary pollution than that under autarky, and than the optimal adoption date of regulated …rms. However, the optimal adoption date of non-regulated …rms completely don't internalize transboundary pollution. Therefore, regulators should know how to intervene to get …rms adopting at the socially-optimal dates.
Under autarky, the instantaneous gain from using the clean technology is greater for regulated …rms than for regulators. These latter instantaneously bene…t more from using the clean technology than non-regulated …rms. Consequently, regulated …rms adopt earlier than what is socially-optimal, whereas non-regulated …rms adopt later than the socially-optimal adoption date. Therefore, in autarky, regulators can induce …rms to adopt at the socially-optimal adoption date by giving them postpone adoption subsidies. Interestingly, the behavior of regulators completely changes when markets are opened to international competition.
Indeed, under a common market, the instantaneous gain from using the clean technology is greater for regulators than for regulated …rms. These latter instantaneously bene…t more from using the clean technology than non-regulated …rms. Consequently, the socially-optimal adoption date is earlier than the op-timal adoption date for regulated …rms. This latter is earlier than the optimal adoption date for non-regulated …rms. Therefore, in a common market, regulators can induce regulated …rms to adopt at the socially-optimal adoption date by giving them speed up adoption subsidies.
Finally, international competition reduces the instantaneous bene…ts from using the clean technology of both non-regulated and regulated …rms, with respect to autarky. Consequently, non-regulated and regulated …rms delay the adoption of the clean technology when markets are opened to international trade. However, the instantaneous social welfare bene…t from the adoption of the clean technology is greater under common market, implying an early socially-optimal adoption date with respect to autarky. 6 Appendix 6.1 Autarky 6.1.1 Instantaneous gains from using the clean technology i) Social optimum *Using expressions (7) and (18): By using expressions ofq a 1dd andq a 1cd ; we get: *Using expressions (13) and (19): S a 2cc S a 2cd = [a (q a 2cd +q a 2cc ) c] (q a 2cc q a 2cd )+ (d c)q a 2cd + eq a 2cd By using expressions ofq a 2cc andq a 2cd ; we get: Given thatq a icc =q a 1cd andq a idd =q a 2cd , we have: ii) Non-regulated …rms *Since q na icc = q na 1cd , then: By replacing q na idd and q na icc between the above brackets by their values; we get: iii) Regulated …rms *Since q na icc = q na 1cd , then by using expressions (10) and (16): By changing the emission tax t a idd and the production subsidy s a icc by their expressions in function ofq a idd andq a icc , we obtain:

Comparison of instantaneous gains
*Using expressions (61) and (57), we have: By using expressions ofq a 1cd andq a 1dd in the above bracketed expression, we show that: *Using expressions (57) and (60), we obtain: By replacing the expression ofq a 1cc ,q a 1dd , q na 1cc , and q na 1dd by their values in the above brackets, we obtain: Using the …rst inequality of condition (9), we can prove that 2a c d 2 e > 0, then: Thus, we have the following ranking: The instantaneous gain from using the clean technology is higher for the …rst adopter regulated …rm than for its regulator, which bene…ts more than its non-regulated …rm. 20

Optimal adoption dates
We suppose that 1 2 , meaning that, in case of non-simultaneous adoption, …rm 1 is the …rst adopter and …rm 2 is the second.
i) Non-regulated …rms: Firm i maximizes its intertemporal net pro…t IU na i ( i ) given by (25) with respect to i : (mr) 2 e mr na i : Using the …rst order condition given by (65), we get: Thus, the second-order condition of optimality is veri…ed. ii) Regulated …rms: Firm i maximizes its intertemporal net pro…t IU a i ( i ) given by (24) with respect to i : Equation (66) is equivalent to: Because of m > 1 and condition (26), a > 0: We have: idd ) e r i (mr) 2 e mr i . Using the …rst-order condition given by (66), we get: The second-order condition of optimality is veri…ed.
iii) Social optimum Each regulator maximizes his intertemporal social welfare function IS a 1 ( 1 ; 2 ) and IS a 2 ( 1 ; 2 ); given by (22) and (23), with respect to 1 and 2 ; respectively: Equations (67) and (68) are respectively equivalent to: Because of m > 1, condition (26), inequalities (64), equalities (59) and (61), we get^ a 1 > 0 and^ a 2 > 0: We have: Using …rst-order conditions given by (67) and (68), we get: Thus, the second-order condition of optimality is veri…ed for each regulator. Because of equality (59), we have:^ a 1 =^ a 2 =^ a : Because of the …rst inequality of condition (9), we have 10a 5d 5c 9 e > 0, and: Under common market, the instantaneous gain from using the clean technology is more important for regulators than regulated …rms, which bene…t more than non-regulated …rms.

Optimal adoption dates i) Non-regulated …rms
Each non-regulated …rm i maximizes its intertemporal net pro…t given by (51) with respect to :  64) and (26), ncm > 0: We have: idd )e r (mr) 2 e mr : Using the …rst-order condition given by (73), we get: = (1 m)m r 2 e mr ncm < 0 Therefore, the second-order condition of optimality is veri…ed. ii) Regulated …rms Each regulated …rm i maximizes its intertemporal net pro…t IU cm i ( ) given by (50) with respect to :  (78) and (26), cm > 0: We have: idd )e r (mr) 2 e mr : Using the …rst-order condition given by (74), we get: Thus, the second-order condition of optimality is veri…ed. iii) Social optimum Each regulator i maximizes his intertemporal social welfare IS cm i ( ) given by (49) with respect to : (q cm icc +q cm idd )+ eq cm idd (d c+ e)(q a icc +q a idd ) Sinceq cm icc =q a icc andq cm idd =q a idd , then: Therefore: Because of m > 1, inequalities (76) and (26),^ cm > 0: We have: idd )e r (mr) 2 e mr : Using the …rst-order condition given by (75), we get @ 2 IS cm i (^ cm ) @ 2 = (1 m)m r 2 e mr^ cm < 0: Thus, the second-order condition of optimality is veri…ed.