CER-ETH – Center of Economic Research at ETH Zurich Permit Markets, Carbon Prices and the Creation of Innovation Clusters Paper

Innovation clusters combining public and private eﬀort to develop breakthrough technologies promise greater technological advances to slow down climate change. We use a multi-country model with emissions permit trade to examine how international climate policy can incentivize countries to create such clusters. We ﬁnd that a minimal carbon price is needed to attract applied research ﬁrms, but countries may nevertheless fail to invest in com-plementary research infrastructure. We construct a mechanism that leads to innovation clusters. It is a combination of low permit endowments for the country with the lowest costs to build the needed infrastructure, compen-sation for this country by proﬁts from permit trade, and maximal possible permit endowments for the remaining countries.


Introduction
Motivation Most suggestions how to slow down climate change concern emissions reduction or technological advances. Technological advances could allow green(er) production or removal of CO 2 from the atmosphere. Ideally, the internalization of the externalities from carbon emissions with market-based policies-e.g. via permit markets or a carbon price-would both lower the emissions and foster the technological advances through R&D by private firms. While this may hold if small adjustments by private firms to existing technologies suffice, it is less clear for greater technological advances, which often require combining private and public effort (see e.g. Johnstone et al. (2010)).
Consider the development of the wind energy as an example. In the 1970s, the Danish government financed the development of blueprints for one-megawatt wind plants, based on the expectations that private firms would conduct the commercialization. 1 The government also financed the building of prototypes, allowing private firms to gain experience with the new technology and with production. Another example is Carbon Capture and Storage (CCS). Only if its costs could be reduced substantially, CCS would become a viable technology option. To achieve that, a combination of university-based research and more applied trials leading to actual cost estimates looks promising (see e.g. Economist, June 9th 2018, p. 71/72). A showcase is the firm Carbon Engineering, whose acting chief scientist is also a climate physicist at Harvard University. The firm recently published cost estimates of US$ 94-233 for pulling 1 tonne of carbon dioxin from the atmosphere, down from US$ 600 some years ago (see Tollefson (2018)).
Combining public and private effort to create local spillovers for innovation and to make greater technological advances leading to breakthrough technologies 2 works particularly well in innovation clusters, 3 , with prominent examples such as Silicon Valley and the Boston Area. In innovation clusters, publicly financed institutions, like universities, typically provides ideas, methods, prototypes, skilled labor and laboratories, which private applied research firms can use to commercialize blueprints. 4 Without technological breakthroughs, e.g. in carbon capture and storage technologies or energy storage, keeping global warming below the 2-degree target with a certain probability may be impossible to achieve. 5 Accordingly, several political agencies now push for green innovation clusters. 6 The major challenge in inducing innovation clusters for green technologies is the global public good property of slowing down climate change and the fact that costs of creating innovation clusters and technology development are borne at the local level. Accordingly, not all benefits are taken into account, and incentives for free-riding on other countries' effort exist, especially as the externalities from carbon emissions are not properly priced.
In this paper, we examine countries' incentives to create a green innovation cluster when they participate in an emissions trading system (ETS). An ETS is a structure that organizes permit trade across countries and involves international cooperation on permit issuance or carbon prices that should be achieved. We examine whether innovation clusters emerge in an ETS. Moreover, we suggest a set of rules for ETSs that can overcome barriers to innovation clusters. Our analysis can be applied to any ETS and in particular to the European trading system, EU-ETS. We use a static setting to model the interaction of the different agents in more detail while maintaining tractability.

Model and Results
We consider a multi-country model with an existing permit market administered by a trading agency. In each country, polluting production firms operate with an existing abatement technology. The detection of a new technology is only possible if an innovation cluster is in place. For an innovation cluster to form, two requirements have to be fulfilled. First, a country has to spend a fixed, country-specific amount to set-up research infrastructure. This may mean to laboratories, skilled workforce or blueprints. Second, applied research firms become active and use the research infrastructure. Our set-up captures the positive spillovers between publicly financed research infrastructure and private research effort. The detection of new technologies is uncertain, and as more applied research firms become active in the cluster, the probability of a new abatement technology being detected increases.
We capture the degree of cooperation between the countries on reducing greenhouse gas emissions by a carbon price target instead of an emissions target. 7 We focus on the carbon price for the following four reasons. First, it is a simple way to parameterize cooperation from pure decentralization to perfect cooperation. Second, recent political initiatives from Chancellor Merkel and President Macron have focused on a carbon price target for the EU-ETS. 8 Third, a carbon price represents a focal point that may foster 5 Many estimates of the transition path towards the 2-degree target rely heavily on carbon capture and storage.
6 Such cluster policy is implemented in the European Union, for instance. See https://ec.europa.eu/ growth/industry/policy/cluster de, last accessed: 13.11.2017 7 However, our mechanisms to induce an innovation cluster can also be applied to permit markets in which countries agree on some aggregate emissions target. 8 At the EU-ETS, a market stability reserve will operate, starting January 2019 (https://ec.europa. eu/clima/policies/ets/reform en, last accessed: 25.7.2018.), which is expected to stabilize prices. successful negotiations (Weitzman (2014), Cramton et al. (2017)). Fourth, the OECD focuses on a gradual approach to achieving uniform international carbon prices to realize some efficiency gains (OECD, 2016). 9 Our results may add insights on appropriate carbon price levels that take the creation of innovation clusters into account.
We note that a carbon price target in existing permit markets has to be implemented by an agreement on some aggregate emissions target, with a budget for the trading agency to ensure that the market price will not fall below the target. As part of our investigation, we will show how carbon price targets can be implemented by allowing a maximal permit endowment of countries, which local governments can then grandfather to local production firms.
If no other country creates an innovation cluster, a countries' decision whether to create one for green technologies depends on the benefits from lower abatement costs, net research infrastructure costs (expected gains, e.g. in terms of tax revenues, minus costs), as well as on the impact of a new (green) technology on costs or revenues from permit trade. For net-buyers on the permit market, detecting the new technology will lower trading costs. For net-sellers, the opposite is true. Effective costs for creating the research infrastructure may thus be higher or lower than the costs for research infrastructure itself.
We obtain the following insights regarding the creation of innovation clusters. First, a minimal carbon price target is needed to incentivize applied research firms to become active and use a created research infrastructure. Second, compared to the global social optimum, countries may lack incentives to invest in research infrastructure and make one innovation cluster possible, even if the permit price target in the ETS is high. Moreover, even if some countries have an incentive to invest in research infrastructure, the country with the lowest research infrastructure costs may not be among them. To overcome both potential inefficiencies within an ETS, 10 we introduce a mechanism-called "Innovationcluster-generation" procedure-that works as follows: According to the mechanism, the country with the lowest costs for research infrastructure is only endowed with few permits. It is compensated with profits generated by the trading agency. The remaining countries are given maximal possible permit endowments. The intuition for why the mechanism works is as follows. By giving only few free permits to the country with the lowest research infrastructure costs, this country's production firms become large net buyers of permits. Thus, this country has high incentives to invest in the research infrastructure and to attract applied research firms, such that an innovation cluster forms. This gives the chance to lower the abatement costs and the costs of buying large amounts of permits.
To motivate this country to participate, the mechanism compensates it by the profits of the trading agency. To generate sufficient profits for the trading agency, the permit endowments to the other countries have to be limited. This determines the maximal possible endowment of other countries. This mechanism does not only induce an innovation cluster, it also introduces a coordination on the country with the lowest research infrastructure costs.
Empirical studies on wind energy suggests that our mechanisms also applies in other areas. First, interaction between applied research from private firms and public research matters (Ek and Söderholm, 2010;Lindman and Söderholm, 2016). Second, environmental policy-feed-in tariffs in the case of wind energy-positively impact technology development (Lindman and Söderholm, 2016). With our paper, we provide an analytical framework to analyze several policy options to foster technological advances through innovation clusters.
Two remarks are in order with regard to common concerns related to the implementation of green innovation clusters. First, the crowding-out of growth effects is a concern. While investing in green innovation clusters generates opportunity costs with respect to investments in other types of innovation activities, there is no reason to assume that a green innovation cluster is less growth-enhancing than other types of clusters. Fostering the internalization of environmental externalities may even be more growth-enhancing in the longer-run (see Bretschger (2015); Bretschger et al. (2011);Riekhof et al. (2018)). The other concern is the possible "picking technology winners" by governments. While governments select the general direction of research and technology development by investing in research infrastructure for clean technologies, politicians cannot not pick "successful firms" or a specific technology in our model, as "picking" is replaced by patent races by private firms.

Relation to Literature
Our paper relates to the literature on environmental policy and innovation as well as on international carbon pricing. The literature on environmental policy and innovation focuses on applied research (see e.g. Vollebergh and Kemfert (2005) for a comprehensive overview). The importance for environmental policy to send the right price signals for applied research firms is established in Requate (2005), for instance. International carbon pricing is discussed e.g. in Weitzman (2014) and Cramton et al. (2017), 11 but generally not in connection with an emissions trading system. There is a literature on how to combine price floors with emissions trading, see e.g. Wood and Jotzo (2011). The discussion of hybrid approaches under uncertainty, with both price floor and price ceiling, goes back to Roberts and Spence (1976). Greaker and Hagem (2014) show that permit trade impacts strategic investment into abatement in a North-South model. In their model, permit 11 The discussion goes at least back to the famous paper of Weitzman (1974) trade always positively impacts investment, as only the North invests, who is a net-seller on the permit market per assumption. To our knowledge, climate policy with innovation clusters-with both applied research and publicly funded research infrastructure-in a multi-country model has not been discussed so far.
In the next section, we outline the details of our model. In Sections 3 to 5, we describe the different levels of the model in more detail and present results. In Section 6, we discuss how an innovation cluster can be induced by international climate policy and describe the Innovation-cluster-Generation procedure. We complete the paper with a discussion of the results and a conclusion in Section 7.

The Levels of the Model
We consider a multi-country model with greenhouse gas emissions and innovation clusters. We focus on carbon emissions leading to global damages and neglect potential additional local impacts of greenhouse gases. Innovation clusters consist of publicly funded research infrastructure and applied research activities. Individual countries host the innovation clusters. Countries are indexed by i ∈ {1, ..., m}, and in each country, there is a local government and a polluting production firm, which may abate emissions. 12 If a country hosts an innovation cluster, there are also applied research firms in that country. Initially, only an old abatement technology is available, but in an innovation cluster, a new abatement technology may be detected. A patent holder can than license the new technology to the production firms.
An innovation cluster forms if two actions are taken. First, a country invests in research infrastructure. Second, applied research firms become active and use the research infrastructure. Thus, an innovation cluster is in place. As a shortcut for both steps, we will say that research infrastructure investments of a county induce an innovation cluster. The idea is that a local government anticipates whether applied research firms will become active (see Condition (15) for details). An emissions trading system with an emissions target exists and is administered by an international trading agency. The price target to achieve the emissions target is conditional on the technological level and denoted by p T , with T ∈ O, N standing for whether only the old (O) or, additionally, the new (N ) technology is available in the economy. We directly focus on the price targets.
The model comprises two levels, namely a political and an economic level, embedded in an international context. The international context sets the organization of the emissions trading system and the strength of the climate policy. This includes a carbon price target and an initial amount of permits each country is endowed with. The political level consists of the decisions of the local government, who can decide about research infrastructure investments and on the number of permits to distribute freely to the local production firms. The economic level comprises the decisions taken by the local production firms and applied research firms. Figure 1 shows the different levels of the models-depicted by shaded areas-, the different actors-depicted by squares-, and their interaction-depicted by arrows. The exogenously given carbon price targets p T and countries' permit endowments i are both depicted by dashed arrows.  In the following, we describe the model in more detail. We first describe and analyze the economic level of the model for given political decisions. Then, we describe the international context before we discuss the political level.

The Economic Level
In this section, we set up the model's economic level and contrast the market outcome with the global socially-optimal solution.

The Set-up of the Economic Level
In each country i, the local production firm i produces an output, which leads to emissions e i , withē i ≥ 0. To simplify notation, the production firm in country i is also denoted by i. Keeping output fixed, production firm i can reduce emissions by abating an amount a i ≥ 0 at costs g t (a i ). The lower case "t", t ∈ {o, n}, refers to the technology actually used by a production firm. The old technology, indicated by the subscript "o", is available for free. We assume quadratic abatement costs of the form In an innovation cluster, the "new" abatement technology indicated by the subscript "n", may be detected. The new abatement technology-hence, "new technology"-lowers abatement costs, i.e. we assume b n < b o . In the following, "T ", T ∈ {O, N }, indicates the technology level available at the global scale, while "t", t ∈ {o, n}, refers to the technology actually used by a production firm.
The production firm is required to hold permits for the emissions it does not abate. It receives an amount of permits i for free from the local government, and can buy additional or sell superfluous permits at the prevailing carbon price p T to (from) the agency.
Each country suffers damages caused by aggregate emissions. In monetary terms, damages are given by with aggregate baseline emissionsĒ = m i=1ē i and marginal damages δ > 0. Countries face identical damages, but differ in their baseline emissions. Identical damages simplify the investigations and allow explicit solutions. The mechanism we propose can also be applied when damage functions differ across countries.
To set-up a research infrastructure, a country-represented by its local government-has to invest an amount F i in monetary terms (F i > 0). These costs consist of expenditures for research facilities, employment in these facilities and further costs for running them. We note that innovation clusters may generate benefits beyond developing new abatement technologies such as spillovers to other technologies or income generation. These benefits are deducted from the costs and thus F i has to be interpreted as net costs.
If a local government has invested F i , applied research firms decide whether to become active and use the research infrastructure to benefit from local spillovers. Specifically, if an applied research firm becomes active and invests the fixed amount x (x > 0), an innovation cluster is in place and the applied research firm can detect the new technology with probability π (0 < π < 1). 13 If there is no research infrastructure, the success probability of an applied research firm is zero. This is a stark assumption of local spillovers from research infrastructure to applied research. The innovation successes are stochastically independent across applied research firms. If k applied research firms become active in the innovation cluster, the overall probability of detecting the new technology is (1) Some remarks on the interpretation of the set-up are in order. First, we assume that costs F i are country-dependent, to reflect the fact that some countries have already invested into research infrastructure in the area considered or that some countries may be more productive in research infrastructure, e.g. have a university system that is better suited for developing new abatement technologies. Once an innovation cluster exists, the success probability π is the same across clusters. This simplifies the analysis, but is not essential. 14 Second, we assume that there is a large number of potential applied research firms that are mobile across countries. The idea is that a subset of individuals can start to innovate if conditions are sufficiently attractive. 15 If at least one applied research firm is successful, a patent holder is determined by fair randomization between all successful applied research firms. 16 The patent holder sets a licensing fee f (f > 0) and production firms decide whether to adopt the new technology at this fee. Production firms in all m countries have the same ability to adopt the new technology.

The Market Solution
We next provide the solution for the economic level of the model. For this purpose, we assume that a carbon price target p T , T ∈ {O, N }, has been set at the international level and will be enforced by the trading agency. Moreover, production firms have received grandfathered permits i from their local government. Finally, we assume that one country has invested in research infrastructure. 13 We focus on the interaction between research infrastructure and applied research. Alternatively, one could consider the interaction of many applied research firms and require the number of active applied research firms k to be above some threshold.
14 Indeed if the country with the best research infrastructure system can also produce the maximal success probability for applied research firms, no change in the analysis is required for this scenario. 15 We can assume that each country is populated by a set of agents and that a subset of these agents can become entrepreneurs (see e.g. Gersbach et al. (2018)). 16 The results are identical for joint patent holding of all successful applied research firms and profit sharing.
Then, the economic level of the model can be represented by a three-stage game: 1. Research activity of applied research firms, 2. Licensing of new technology, if detected, 3. Emissions abatement and permit trading of production firms.
We solve this three-stage process backwards.

Stage 3: Emissions Abatement and Permit Trading of the Production Firms
Suppose that a new technology has been detected and that each production firm has chosen the technology g t , t ∈ {o, n}. With grandfathered permits i , production firms choose abatement level a i by solving 17 which yields the first order condition taking the permit price p T , T ∈ {O, N }, as given. 18 For quadratic abatement costs, p T = b t a i . Since production firms that use the same technology abate an identical amount of emissions, we omit the index i and denote abatement decisions in the future by (3)

Stage 2: License Fee Setting by the Patent Holder and Technology Adoption by Production Firms
We next turn to the setting of the license fee and to the adoption of the new technology. 17 We assume price-taking behavior for the following reasons. In a country, every production firm can stand for many production firms. For example, on the EU-ETS, more than three thousand firms trade, and the three biggest emitting firms represent less than 8% of total emissions each (RWE: 7.1%, E.ON: 4.7% and Vattenfall: 4.2%.) (Nicolaï, 2015). With many trading firms, an equilibrium approximates the competitive equilibrium (Lange, 2012). In every country, the abatement technology could also be seen as the result of abatement efforts of a continuum of production firms. Each firm has the option to abate one unit of emissions at some costs. We also obtain price-taking behavior on the permit market when we consider a continuum of firms. 18 Global availability of the technology is depicted by T . t = n is thus only possible if T = N , i.e. if the new technology is available.
Given the new technology is detected and p T = p N prevails, production firm i's maximal willingness to pay for the new technology, denoted by W T P i , is The willingness to pay for the new technology is identical across production firms, and is independent of adoption decisions by other production firms. Hence, if the new technology is detected, the patent holder sets the license fee according to f = W T P , and all production firms adopt the new technology. Then, a i = a n,N for all production firms. Using Equation (3), we can write the fee that equals (4) as a function of p N . Proposition 1 summarizes the result.

Proposition 1
Suppose that the new technology has been detected. Then, all production firms adopt it at the feef Next, we turn to the decision of the applied research firms.

Stage 1: Research Activity of Applied Research Firms
Given that one local government has invested into research infrastructure, applied research firms decide whether to become active and use the created infrastructure, based on expected profits. An applied research firm only earns profits from licensing the new technology to all production firms if it is the patent holder. If several research firms have detected the new technology, the patent holder is determined by fair randomization. Then, using Equation (1), the probability for an applied research firm to detect a new technology and to become patent-holder is given by The probability can be approximated by the probability to detect the new technology divided by one plus the expected number of other successful active R&D firms, .
The approximation works well, especially for small π (see Gersbach and Riekhof (2017))which is a plausible assumption for the development of breakthrough abatement technolo-gies. It will be used in the subsequent analysis to calculate explicit formulas.
Applied research firms become active as long as expected profits are positive. Thus the number of active applied research firms k is based on the expected zero profit condition, We can re-arrange the equation and write the number of active applied research firms as a function of p N by using (5). This yields the next result.

Proposition 2
If one research infrastructure is created,k applied research firms witĥ become active and use the infrastructure such that an innovation cluster forms ifk(p N ) > 0.
It is convenient to treatk(p N ) as a real number in the analysis. Of course, the number of applied research firms becoming active is equal to the largest natural number smaller thank(p N ).
An important remark is in order. As the success of applied research firms is stochastically independent, and applied research firms are mobile, the number of active applied research firms would be the same and given in (6), distributed over all created research infrastructures if more than one country invests in research infrastructure. Hence, from a welfare perspective, one innovation cluster is sufficient to reap all possible benefits from innovation activities. 19 We assemble some intuitive properties ofk(p N ) in the following corollary.
Corollary 1 (i) Given a research infrastructure has been created, an innovation cluster is in place (ii) An increase of the carbon price target when the new technology is detected increases 19 We will comment in Section 7 on extensions of the model in which more than one cluster could be beneficial.
the number of applied research firms, (iii) A reduction in the marginal abatement costs of the new technology for a given level of emissions and a given carbon price target increases the number of applied research firms, The first property of Corollary 1 indicates that the number of applied research firms depends on the license income in relation to the research costs and the success probability.
If the licensing income increases due to a higher carbon price target p N or to lower abatement costs b n , the number of active applied research firms increases (Corollary 1 (ii) and (iii)).
Next, we consider which abatement level and what kind of research activity are optimal from the perspective of a global social planner.

The Global Optimal Solution
Before we proceed with the political level and the international context, it is useful to introduce the socially-optimal solution at the global level. We assume that a social planner can dictate abatement in all countries and that the social planner can determine in which countries research infrastructure investment takes place, as well as how many applied research firms become active. The social planner thus characterizes the optimal solution if countries cooperate perfectly to maximize aggregate welfare. 20 Let the superscript * denote variables describing the socially-optimal solution when research infrastructure is created and applied research firms become active such that an innovation cluster forms. Let Country i * denote the country with the lowest costs to create an innovation cluster, .., m. Then, Proposition 3 characterizes the socially-optimal solution.
Proposition 3 Suppose and a * given in (ii) below.
Then, the socially-optimal solution is characterized by (i) One innovation cluster with The proof is in Appendix A.
From a global perspective, it is optimal to establish an innovation cluster in the country with the lowest costs for research infrastructure, i.e. in Country i * with F i * = min{F i }. Still, it is only optimal to create an innovation cluster if total costs-i.e. costs for research infrastructure and applied research-are lower than the expected gains, i.e. if Expected gains consist of two terms. The first term is lower expected abatement costs from new abatement levels. The second term corresponds to the net effect of lower expected damages and more ambitious abatements. For a derivation, see Proof of Proposition 3.
In this paper, we examine under which conditions innovation clusters are (not) created, assuming the economy is in a situation in which it would be optimal from a global perspective to create an innovation cluster. Thus, we assume (8) holds for the remainder of the paper. The condition includes that k * > 0. 21 Proposition 4 summarizes results on how the socially-optimal abatement level can be induced in a market setting.
Proposition 4 The socially-optimal abatement level a * can be induced by setting carbon prices according to depending on the detection of the new technology.
The proof is in Appendix A.
The price menu p * N , p * O ensures the socially-optimal abatement level in a setting without a global social planner, independent from the detection of the new technology. This is not the classical result of the permit price being equal to marginal damages. The reason is that aggregate abatement levels (or aggregate permits) have to be chosen before technologies are realized. 22 While p * T can ensure the optimal abatement level, the price menu can neither simultaneously ensure k * nor ensure that the research infrastructure is created in Country i * . In Section 6, we present a mechanism that incentivizes Country i * to create an innovation cluster.

The International Context
We assume that an emissions trading system administered by a trading agency exists. The prevailing carbon price target reflects the strength of climate policy and depends on the countries' willingness to cooperate on abatement. For this paper we take the level of cooperation on abatement as a parameter and investigate how innovation clusters may emerge. There are many theories how a particular level of cooperation develops. A simple rationale is the likelihood of a complete breakdown of cooperation if one country withdraws from the agreement. If the likelihood is low (high), high (low) carbon price targets can be achieved. The trading agency ensures that the carbon price target prevails in the permit market.
The system has the following additional characteristics: First, countries face proportional burden-sharing, implying that each country can issue permits as a given fraction of its baseline emissions. Second, the sum of the initial amount of permits countries are endowed with should be set to guarantee a non-negative profit of the trading agency, i.e. the agency should be in a position to guarantee the carbon price target without needing additional funds from the countries. Third, the residual budget of the trading agency is equally distributed among all countries.
We how first present the carbon price as a measure of cooperation in abatement. We then discuss permit endowment and feasibility of the targeted price, taking the assumptions above into account.

Cooperation on Abatement
We measure the level of cooperation between the countries by a scalar λ that describes the agreed carbon price target relative to the carbon price needed to achieve the sociallyoptimal level. Countries cooperate to set a menu of carbon prices p O = λp * O and p N = λp * N (λ ≤ λ ≤ 1), with λ = λ denoting no cooperation, and λ = 1 denoting full cooperation. The level λ corresponds to the permit price when countries non-cooperatively decide about the amount of permits as best responses to each other, which results in the lowest possible but still positive carbon price. 23 The optimal prices p * O and p * N (see Proposition 4) can only be obtained under full cooperation. One remark is in order on how different cooperation levels impact local production firms' abatement. Given p N and p O from Equation (3), and Equations (9) and (10) from Proposition 4, we observe that abatement levels are the same under both scenarios, and we denote this level byâ(λ). It is given bŷ a(λ) := a n,N = a o,O = λa * .
While we have already assumed that parameters are such that an innovation cluster is socially optimal, we now make the following additional assumption. When research infrastructure is created under full cooperation, applied research firms will enter, i.e. k(p * N ) > 0. 24 We make this assumption to focus on incentives for investing in research infrastructure. If applied research firms did not even become active under full cooperation, one would have to focus more on incentivizing applied research to become active and not on the incentives for local governments to invest in research infrastructure. Later, we will discuss how applied research could be incentivized even ifk(p * N ) = 0. We now address initial permit endowment in relation to the feasibility of the carbon price 23 In our set-up, this solution is given by λ =p T p * T , withp T = g t (e e,T ) = b tĒ −Ẽ n , withẼ denoting the total amount of permits when each country decides on local permits issuance individually (see Gersbach and Riekhof (2017)). 24 This requireŝ target.

Initial Permit Endowment and A Carbon Price
For a given level of cooperation represented by p T = λp * T , we determine feasible initial permit endowments and describe the trading process. We also state how initial permit endowments determine the feasibility of a carbon price target under a non-negative profit of the trading agency.
With the initial allocation of permit endowments¯ i across countries determined by proportional burden sharing, we havē The parameter α represents the tightness of initial permit endowments. Local governments can buy additional permits or sell superfluous permits. Ultimately, they will grandfather an amount of i to their local production firms. Local governments and local production firms trade on the permit market.
The implementation of the initial permit endowment is feasible if the profit of the trading agency, is non-negative. Lemma 1 characterizes feasibility initial permit endowments related to an upper bound on α denoted byᾱ.

Lemma 1
Carbon price targets p T = λp * T , T ∈ {O, N }, are feasible for given initial permit endowments represented by α if and only if The proof is in Appendix B.
Some interpretations are in order. First, the condition of a non-negative residual budget of the trading agency determines an upper bound on α, i.e. on the initial permit endowments of the countries relative to their baseline emissions. Second, for the profit of the agency, it is irrelevant whether the local government buys more permits from the agency than αē i and grants them to their local production firms. The more the local government buys, the more permits the corresponding production firm will sell in the permit market. If a local government buys more permits, it simply redistributes revenues within the country, i.e. from the government to local production firms, without affecting the profit of the agency. Third, for every level of cooperation, there is an α that can guarantee a non-negative profit. The level of cooperation and the share of permit endowments relative to baseline emissions are inversely related: A high level of cooperation (high λ) means a high carbon price. For the trading agency to ensure the price target without an additional budget, the share α of permits needs to be relatively low. This is reflected in the property ∂ᾱ(λ) ∂λ < 0.
An example is illustrated in Figure 2.

The Political Level
Next, we consider the decision problem of local governments. Local governments want to minimize local costs when deciding about research infrastructure investments and on the amounts of permits grandfathered to the local production firms. The local costs K i are given as follows: Local costs include the licensing fee-if relevant-, abatement costs, permit trading costs (or revenues) of the local production firm, damages, trading costs (or revenues) of the local government, and refunding of the trading agency's profit, conditional on whether the new technology is detected or not. Local costs also include costs for research infrastructure-if the local government decides to make the investment. Costs for applied research do not need to be considered, as expected profits of applied research firms are zero.
In the following, we consider the amount of permits the local government grandfathers to the local production firm.

Grandfathering Permits to Local Production Firms
The local government's decision to grandfather i to the local production firm may imply buying (or selling) i − αē i from (to) the trading agency at the prevailing price. Lemma 2 shows that the grandfathering decision has no impact on total costs of a local government.

Lemma 2
The amount of permits grandfathered to the local production firm i is undetermined.
The proof is in Appendix B.
The reason for the result of Lemma 2 is that the local government considers total local costs. In such a set-up, it does not matter for a country whether the costs are paid by the local government or by the local production firm. If a local government buys additional permits at the given carbon price, the trading agency obtains revenues. As abatement a(λ) is independent of the amount of grandfathered permits, the production firm will sell the additional permits at the given price. The trading agency uses the revenues obtained from selling to the local government to buy the permits. The process is neutral for the agency's profit (see also Section 4.2).

Research Infrastructure Investments
Next, we consider the local government's decision on research infrastructure.
To consider the local government's decision on research infrastructure, it is useful to define total costs, denoted by K IC i , when some other country invested into research infrastructure to induce an innovation cluster, as well as to define total costs, denoted by K N IC i , when no country invested in research infrastructure and only the old technology is available. 25 As the choice of i does not influence the total costs considered by the local planner, and withĒ := m i=1¯ i , these total costs are given by If a country has invested in research infrastructure, it is not profitable for a second country to invest, as this would only add costs without adding benefits. Hence, a local government invests in research infrastructure if and only if no other country has undertaken these investments and if Building on Condition (14), Lemma 3 summarizes how the incentives of a local government to invest in research infrastructure depend on the level of cooperation λ and on permit endowments reflected by α. Indirectly, the incentive also depends on whether applied research firms will become active and use the infrastructure, i.e. on k > 0.

Lemma 3 The local government in country i invests in research infrastructure if no other country invests and if
Effective costs The proof is in Appendix B.
Condition (15) can be interpreted as follows. Expected gains from detecting the new technology, depicted on the right hand side (RHS), have to be at least as high as countryspecific effective costs to induce an innovation cluster, depicted on the left hand side (LHS). Let us discuss both terms in more detail.
The expected gains depicted on the RHS are the same for each country and consist of the license fee weighted by the relative reduction in marginal abatement costs with the new technology, multiplied with the overall probability of detecting the new technology.
As the licensing fee equals the production firms' willingness to pay to adopt the new technology, the RHS can be interpreted as the country's expected gains from the new technology, mainly in the form of reduced abatement costs.
On the LHS, we have the country-specific terms. The first term depicts the investment costs for research infrastructure. The second term depicts the change in costs or revenues from permit trading due to the detection of the new technology, net of the refunding of the trading agency's profit (see Equation (27) in Appendix B for details). The second term is driven by changes in the prevailing permit price, due to the detection of the new technology. Accordingly, we term it the "price-change effect".
The price-change effect can be positive or negative, depending on whether the country is a net-seller or a net-buyer on the permit market. In principle, one can classify the countries into three groups, namely net-buyers with above-average baseline emissions e i >Ē/m, countries with average baseline emissionsē i =Ē/m, and net-sellers with below-average baseline emissionsē i <Ē/m. Net-buyers gain from the detection of the new technology, as the price to buy additional permits on the permit market declines. They face a negative price-change effect and their effective costs to create an innovation cluster are reduced. Net-sellers lose from the detection of the new technology, as the selling price for superfluous permits on the permit market decreases. They face a positive pricechange effect, and effective costs to create an innovation cluster are increased. Countries with average baseline emissions are not affected.
Without loss of generality, we order the countries according to their baseline emissions e i , from lowest to highest. Let i = 1, ..., m s denote the net-sellers and i = m b , ..., m the net-buyers, with m s ≤ m b − 1, depending on the number of countries withē i =Ē/m.
Four points are important. First, although all countries initially receive a permit amount relative to their baseline emissions,¯ i = αē i , the refunding of the trading agency's profits leads to the separation of the countries into groups with above-average or below-average emissions. Second, if Condition (15) is fulfilled for several countries, local governments need to coordinate on who will invest in research infrastructure. Yet, local governments decide simultaneously, such that a coordination problem arises, with many possible solutions. A focal point 26 in this coordination game may be the following: The local government in the country with the lowest effective costs (LHS in Equation (15)) invests, because it will gain most from the detection of the new technology. This country may not be Country i * -the country with the lowest research infrastructure costs-, because of the price-change effect. Another country with F i > F i * , but with a negative price-change effect, may face the lowest effective costs to create an innovation cluster. As a result of permit trade, countries may not coordinate on the socially optimal Country i * to create an innovation cluster. 27 Third, if Condition (15) is not fulfilled for any country, all countries may still benefit from jointly financing research infrastructure investment in one country. In such a situation, additional mechanisms are needed. Here, permit trade may create opportunities for adding such mechanisms. We will develop one below. Fourth, Equation (15) cannot be fulfilled if k = 0. If upon investment in a research infrastructure, no applied firms are expected to enter-and Π(k = 0) = 0-, a local government would not invest in it in the first place.
We focus on how innovation clusters can be induced. We first discuss the impact of the level of cooperation and permit endowments on the emergence of innovation clusters. We then present a mechanism that solves the coordination and commitment problem and can thus help to create an innovation cluster, even in difficult settings.

Inducing Innovation Clusters
In this section, we explore different ways how the international community may induce an innovation cluster when Condition (15) is not fulfilled for any country. In addition, we explore a situation with different types of new technologies. We first discuss the type of solutions we explore to induce an innovation cluster.
In principle, many ways to incentivize an innovation cluster can be imagined. One possibility is to create a fund to which all countries contribute. Then, to be cost-effective, the money would be given to the country with the lowest costs to create the research infrastructure. A possible difficulty is that one cannot control to which extent this money 26 A focal point (also called a "Schelling Point") refers to a solution in games that all players are likely to choose in the absence of communication.
27 Without permit trade, a local planner would consider Based on (27), we can write will indeed be used for the development of new abatement technologies. Failure to detect a new technology may be due to lack of investment into research infrastructure or simply to bad luck. Hence, inducing innovation clusters has to rely on either additional costly monitoring devices, which will remain imperfect and require additional institutions, or on additional incentives for a country to invest in research infrastructure. We focus on another approach and stay within the given institutional set-up of a permit market, i.e. no further institutions are considered. As it will turn out, non are needed.

The Impact of Cooperation
When the level of cooperation is low, represented by a low value of λ, no local government may have incentives to invest in research infrastructure when endowed with αē i . This may be because no applied research firms would become active to use this infrastructure, or because costs to create this research infrastructure outweigh country-specific gains. It is useful to introduce the cooperation threshold λ • i , implicitly defined by We note that λ • i is uniquely determined and describes the level of cooperation at which country i is willing to invest in research infrastructure when countries are endowed with α(λ • i ) permits. We note that a particular value of λ • i may not be in [λ, 1] and hence no level of cooperation can induce an innovation cluster in country i. We also define λ • = min i=1,...,n {λ • i } as the minimal level of cooperation at which a first country is willing to create the research infrastructure if all countries are endowed with α(λ • ) permits. Moreover, λ • b = min i=m b ,...,m {λ • i } describes the minimal level of cooperation at which a first country that is a net-buyer on the permit market is willing to create the research infrastructure.
Proposition 5 states how changes in cooperation at the international level can induce the creation of an innovation cluster.
(ii) Consider countries i = m b , .., m that are net-buyers on the permit market. Suppose λ < λ • b < 1 and α =ᾱ(λ). Then, an increase in cooperation from λ to λ • b and a decrease of α toᾱ(λ • b ) induces an innovation cluster.
The proof is in Appendix B.
The relation between the cooperation level λ and incentives to invest in research infrastructure is as follows. First, higher carbon prices due to a higher level of cooperation increase the gains from detecting the new technology. Second, higher prices increase the license income of the patent holder, as production firms' willingness to pay to adopt the new technology increases. The higher license income attracts more applied research firms and the overall probability to detect the new technology increases. Thus, higher gains that will be obtained with a higher probability increase the local government's incentives to invest in research infrastructure. Moreover, only the higher income for the patent-holder may be sufficient to make applied research firms use the created research infrastructure, such that an innovation cluster will be in place.
If the share of permit endowments relative to baseline emissions α has to be lowered to ensure the feasibility of the carbon price targets at the higher cooperation level λ • , a potential countervailing force may arise. If α <ᾱ(λ • ), one can increase λ to λ • without the need to change α. If α =ᾱ(λ), an increase in the cooperation level has to be accompanied by a decrease in permit endowments. A lower permit endowment increases the incentives of net-buyers to set-up research infrastructure, but lowers incentives for net-sellers to do so. Next, we examine the impact of changes in α for the creation of an innovation cluster in more detail.

The Impact of Countries' Permit Endowments
The next proposition summarizes how the incentives to create an innovation cluster depend on the initial permit endowment.
Proposition 6 (i) Consider countries i = m b , .., m that are net-buyers on the permit market. Suppose α > α • , with andk(λp * N ) > 0, and no country faces incentives to induce an innovation cluster. Then, decreasing the share of grandfathered permits from α to α • for a given level of λ induces an innovation cluster in a country that is a net-buyer on the permit market.
(ii) Consider countries i = 1, .., m s that are net-sellers on the permit market. 18) andk(λp * N ) > 0, and no country faces incentives to induce an innovation cluster. Then, increasing the share of grandfathered permits from α to α • for a given level of λ induces an innovation cluster in a country that is a net-seller on the permit market.
The proof is in Appendix B.
The intuition is as follows. If the share of issued permits is too high, there are no incentives for net-buyers to invest in an innovation cluster. If α is lowered, net-buyers need to buy more permits. Therefore, inducing an innovation cluster to detect the new technology becomes more attractive. Net-sellers have the opposite incentives. An increase in α would make an innovation cluster more likely, as the gains from selling permits relative to lower abatement costs decrease. Scope for increasing permit issuance at a given cooperation level is only possible if α ≤ᾱ(λ). 28 Overall, inducing innovation clusters via increased cooperation (λ) or smaller permit endowments (α) entails several problems. First, countries are unlikely to coordinate on Country i * , which has the lowest costs to invest in research infrastructure. Second, αand thus the permit endowments-may have to be comparatively low if costs to create research infrastructure F i are large. Lastly and most importantly, an innovation cluster may simply be not viable, since no country faces sufficient incentives. This is particularly obvious for high-emissions country, as they face a double burden: They must buy a large amount of permits (to have incentives to create an innovation cluster), and they bear costs to invest in research infrastructure. Low-emissions countries may simply not expect sufficient gains from innovation clusters.
Next, we suggest how to solve these problems.

The Innovation-cluster-generation Procedure
We formulate three desiderata for inducing an innovation cluster. First, Country i * should invest in research infrastructure. Second, there should be a fair sharing of the overall burden. Third, countries should have maximal permit endowments for the given cooperation level. The justification for these three requirements is as follows. The first desideratum is the efficiency requirement with respect to investments in research infrastructure. With the second requirement, we capture the fairness requirement. A country that bears the cost of generating the innovation cluster should not fare worse than the other countries. Besides fairness as a requirement per se, this should help to foster cooperation on sufficiently high carbon price targets. The third requirement aims at easing political-economic constraints at the local level.
We next present the Innovation-cluster-generation procedure (henceforth "ICG-Procedure"), which can fulfill all described desiderata as best as possible. The procedure consists of the following three elements: • Country i * 's permit endowment is˜ , • Countries i = i * ' permit endowments areαē i , • Country i * receives the profitπ from the trading agency.
The procedure implements the desiderata in the following way. Permits˜ are set to ensure that Country i * has incentives to invest in research infrastructure. We call this the "Incentive Constraint". By investing in research infrastructure, Country i * faces expenditures the other countries do not have to bear. In addition, Country i * will only be endowed with relatively few permits, as˜ is set to incentivize research investments. To ensure overall fair sharing of the burden, Country i * is compensated by the profitπ from the trading agency. We term this the "Equally Well-off Condition" and explain it in more detail below. The remaining countries are endowed withαē i permits, withα set to ensure that the trading agency can financeπ and ensure that the targeted carbon price p T prevails. We term this the "Feasibility Constraint".
We interpret the Equally Well-off Condition as follows. Country i * should be indifferent between two situations: It creates the innovation cluster, is endowed with˜ permits and receives the profitπ; another country takes the role of creating the innovation cluster, is endowed with˜ permits and receivingπ. In the latter situation, Country i * would be endowed withαē i permits.
We make two additional assumptions that simplify the analysis. First, we assume that the hypothetical other country that creates the innovation cluster instead of Country i * has the same characteristics as Country i * , i.e. it has F i = F * i andē i =ē * i . Second, to ensure a non-negative profit for the trading agency in all circumstances and maximal permit endowment of the countries, we assume zero profit of the agency if the lower carbon price (p N ) prevails, i.e. if the new technology is detected.
Proposition 7 presents the results of the ICG-Procedure whenα,˜ ,π are determined according to the above criteria.

Proposition 7
Supposek(λp * N ) > 0 if investments into research infrastructure are made. Then, there exists a uniquely-determined combination ofα,˜ ,π defined by the following three conditions: • the Incentive Constraint to create an innovation cluster in Country i * , • the Feasibility Constraint of the trading agency, • and the Equally Well-off Condition which characterize the ICG-Procedure and implement one innovation cluster in Country i * .
The proof is in Appendix B.
Several remarks are in order. First, of course, ifπ < 0 were a solution, then the ICG-Procedure would be superfluous. We focus on constellations in which innovation clusters will not be created without the ICG-Procedure. Second, a binding constraint is that 0 ≤α ≤ 1. Ifα < 0 or˜ < 0, countries are endowed with a negative amount of permits. Both α < 0 or˜ < 0 are possible, but difficult to enforce. Lastly, besides being a way to ensure an innovation cluster, the ICG-Procedure is also a way for countries to coordinate on Country i * to invest in research infrastructure, and thus to increase social efficiency.
We now briefly discuss how the ICG-Procedure can be extended to include the case that k(λp * N ) = 0. Lower research costs are one way to incentivize the applied research firms to become active and use research infrastructure even when the carbon price is low. One could lower research costs x through subsidizing by the factor β, with 0 < β < 1. This research subsidy could be financed by adding the amount to the research infrastructure costs of a country, F i + k(1 − β)x. To ensurek(p N ) > 0, the scalar has to be sufficiently small, i.e.
which is derived from Equation (6). Then, the ICG-Procedure could be implemented based on the modified costs for research infrastructure Figure 3 illustrates an example for the ICG-Procedure and shows how the parameters of the procedure change with the level of cooperation. It starts with the cooperation level at which the investment into research infrastructure attracts applied research firms, such that an innovation cluster forms.

An Example
The lowest graph in Figure 3 illustrates how Country i * 's permit endowment˜ has to be changed in relation to the cooperation level. For the interpretation, remember that˜ is set such that the Incentive Constraint for Country i * to invest in research infrastructure holds with equality and that lower permit endowments increase the incentives to create an innovation cluster via the price-change effect.
Once the cooperation level and the carbon price are sufficiently high for the research infrastructure to attract applied research firms-which occurs at λ = 0.5-, a further increase in cooperation initially increases˜ . Higher carbon prices attract more applied research firms and the overall success probability increases without higher costs to create the innovation cluster, such that Country i * invests in research infrastructure even if it is endowed with more permits˜ .
Eventually, for higher levels of cooperation, a reduction of the number of permits given to Country i * is needed to incentivize investment in research infrastructure, as a second effect begins to dominate: Higher cooperation levels imply higher carbon prices, such that abatement a(λ) goes up. Without any changes in˜ , the price-change effect would become less relevant, making the creation of an innovation cluster less attractive for Country i * . Hence,˜ has to decline to restore the incentives to invest.
The graph in the middle illustrates how the paymentπ to Country i * connected to the Equally Well-off Condition reacts to the level of cooperation and thus to the carbon price target. It inversely mirrors the lower graph, asπ has to compensate additional expenditures of Country i * beyond research infrastructure costs F i * caused by buying permits as the permit endowment˜ is low.
The upper graph illustrates the negative relationship between the level of cooperation and the amount of initially-issued permits to all other countries (reflected byα). To ensure that a high carbon price target-due to a high level of cooperation-can be enforced with a non-negative profit of the trading agency, countries can only be endowed with few permits (cp. Lemma 1). We note that the change inπ is not mirrored by changes in α sinceα is determined by the requirement to ensure that the carbon price is feasible if no new technology is detected.

The Carbon Price and Its Impact on Cluster Type
So far, we have only considered one (given) type of new technology, b n , and have shown that the carbon price level matters for whether research infrastructure is paid for and whether applied research firms become active and use a research infrastructure. In this section, we consider the availability of several potential types of abatement technologies and discuss how a carbon price level may influence which type of technology is selected.
As before, new technologies can only be detected by an applied research firm in an innovation cluster. For technology type-specific innovation clusters, the local planners can choose which potential abatement technology type should be developed. We produce a simple extension of our analysis to multiple abatement technologies.
Consider several types of new technologies, depicted by b n (τ ). Let τ ∈ [τ , τ ] denote a technology index that ranks the different types of new technologies according to the resulting abatement costs, from lowest to highest. Country-specific costs to invest in research infrastructure are dependent on the technology index, with more efficient abatement technologies associated with higher research infrastructure costs. The ranking of countries in terms of research infrastructure costs is the same over all technologies, e.g. F i * (τ ) ≤ F i (τ ). Specifically, we assume with η F > η b > 0 and τ > 0. Furthermore, we assume that parameter constellations are such that it is never worthwhile to create more than one innovation cluster 29 and that applied research costs x and the probability of an applied research firm to detect b n (τ )-given the respective research infrastructure has been created-are the same across all technologies. 30 To examine the influence of the carbon price, we make a small adjustment to our framework and directly set the level of the carbon price target instead of the level of cooperation. This implies that when the new technology has been detected, abatement levels may differ compared to a situation when only the old technology is available. Note that 29 While an analysis of circumstances with several socially beneficial innovation clusters would be interesting, it goes beyond the scope of this paper. 30 We make this assumption to reflect the idea that success probabilities are hard to assess, and that there is no reason to assume a priori that a certain technology has a higher probability to be detected than another.
decisions at the economic level and the set-up at the international context-except for the representation of the carbon price target-remain the same. Now, instead of deciding whether to invest in research infrastructure or not, the local government has to decide for which technology type to create research infrastructure. We adjust Equation (14) to multiple technologies, i.e.
We assume that a local planner opts for the technology that results in the lowest abatement costs and still fulfills Equation (24). The upper graph in Figure 4 illustrates how the LHS and the RHS of Equation (24)with Equations (22) and (23) inserted-change with the technology index τ . As long as the RHS is equal to or above the LHS, it is worthwhile to create an innovation cluster for the new abatement technology depicted by the corresponding technology index. The lower graph in Figure 4 shows the technology index of the new abatement technology for which an innovation cluster would be created at a given carbon price level, i.e. for which Equation (24) holds with equality.
To sum up, the carbon price target does not only influence whether investments into research infrastructure take place, and whether and how many applied applied research firms become active to use the infrastructure-i.e. how probable the detection of a new technology becomes-, but also influences which type of new technology is developed. We note that the ICG-Procedure could be adapted to induce innovation clusters for more efficient new technologies.

Discussion and Conclusion
Innovation clusters appear to be crucial for technological advances to slow down climate change. Innovation clusters need research infrastructure, typically financed by the government-e.g. laboratories, skilled workforce, and blueprints-and applied research firms to detect a new technology and commercialize it. Public spending on research infrastructure is rationalized by positive spillovers to applied research. We discussed whether and how international cooperation on climate policy with carbon pricing and emissions trading systems can induce such innovation clusters.
Our analysis generates three major insights. First, the impact of carbon price targets goes beyond the pure internalization of externalities of emissions. We find that the level of carbon price targets influences whether and for which type of abatement technology research infrastructure is created, whether applied research firms become active and use the infrastructure such that an innovation cluster forms, and how high the overall probability to detect a new technology is. Second, permit trade impacts the creation of innovation clusters. This impact may increase the overall incentives to create an innovation cluster in a decentralized setting, but it may lead to the creation of an innovation cluster in a country with relatively high research infrastructure costs. Third, international climate policy can help to induce the creation of innovation clusters for green technologies, even under adverse conditions. The Innovation-cluster-Generation procedure we suggested combines the allocation of initial permit endowments with fair burden-sharing. Initial permit endowment ensures that the country with the lowest costs creates an innovation cluster. In addition, the ICG-Procedure aims for maximal permit endowment for the remaining countries, given that some revenues from permit trading are generated to financially support the country that invests in research infrastructure and creates the innovation cluster. While high carbon prices make the creation of an innovation cluster more likely, the ICG-Procedure can even induce the creation of an innovation cluster when carbon prices relative to the costs to create the corresponding research infrastructure are low.
Our model could be extended in different ways. One could explicitly model asymmetric information regarding abatement costs or damages, or include transaction costs of setting up the trading agency. Other variants of the ICG-Procedure are also conceivable. In the versions explored in this paper, the country that invests in research infrastructure is equally well-off in expected terms than other countries. However, if no technology is detected, it is ex-post worse-off, since it has to buy a large number of permits at a high price. Allowing for an additional insurance scheme among countries, based on the detection of new technologies, could mitigate this. Also, one could introduce several clusters operating simultaneously, which focus on different technologies. In such a situation, our proposed ICG-Procedure could be extended to serve as a coordination device that reduces duplication of research effort. Several countries may receive few permits and share the revenues from the trading agency. In a next step, dynamic incentives could also be considered explicitly.

A Proofs: Optimal Solution
Proof of Proposition 3. The global social optimum with innovation cluster is characterized by with first-order conditions leading to Note that Equation (25) implies that a i = a * .
All three conditions are fulfilled: The global social optimum without an innovation cluster is characterized by Below, we consider under which conditions optimal abatement in a situation without an innovation cluster is lower than in a situation with an innovation cluster: An innovation cluster is optimal if From a global perspective, costs are obviously minimized when the country with the lowest costs for research infrastructure makes the investment.
Proof of Proposition 4. Production firms' abatement is according to a t,T = p T /b t (see Equation (3)). With p * T , a * = p * T /b t , such that p * T = b t a * implements socially optimal abatement levels. Note that, as a i =ē i − i , an initial allocation of permits according to * i =ē i − a * can also implement the socially optimal solution in terms of abatement.

Proof of Lemma 2.
We assume that some other country invested into research infrastructure. Then, the costs faced by local government i given in (13) (with F i = 0) can be written as  (14), and subtract F i , we obtain Now, multiplying with −1, inserting for¯ i = αē i , using a n,N = a o,O =â (Equation (11)) and isolating country-specific terms yields Note thatf (p N )[b o − b n ]/b n > 0. Minor re-arrangements and inserting for p T = λp * T lead to (15).
Proof of Proposition 5. (i) As α =ᾱ(λ • ), the second term on the LHS of Equation (16)-the price-change effectis now independent of λ, and increasing the cooperation level up to λ • requires no change in α.
Accordingly, increasing λ to λ • will lead to a situation in which Condition (15) is fulfilled for at least one country, as λ • is defined as the minimum cooperation level for which at least one country has an incentive to create a research infrastructure.
(ii) If α =ᾱ(λ), the proof is as before, except that now a decrease in α has to be taken into account. When α =ᾱ(λ), an increase in λ needs to be accompanied by a decrease in α to ensure that the carbon price target remains feasible (non-negative profit of the trading agency, see Lemma 1). For net-buyers on the permit market withē i >Ē/m, the change in α reinforces the impact of an increase in λ.
In both cases, no research infrastructure investment may take place initially because local planners anticipate thatk(λp * N ) = 0, i.e. applied research firms would not become active. Only withk(λ • p * N ) > 0 and the resulting higher carbon prices, applied research firms will become active.
Proof of Proposition 6. The condition k(λp * N ) > 0 implies that applied research firms become active and use the research infrastructure once it is created and an innovation cluster is in place. At the given λ and α, no country has incentives to create a research infrastructure.
Based on (15), no country invests if Accordingly, the prevailing α is too high (low) for net-buyers (net-sellers), i.e.
and for Condition (15) to hold, α needs to be lowered (increased) for net-buyers (netsellers).
(i) For a given λ, lowering α is always possible (see Lemma 1).
(ii) An increase in α may violate the feasibility constraint (Lemma 1) such that additional constraints on α are imposed.
We discuss the possibility of α < 0 and α > 1 at the end of Section (6.3).

Proof of Proposition 7.
Based on (26), withẽ instead of αē i * , and without any share from the trading agency's profit, the Incentive Constraint to create an innovation cluster for Country i * is It determines˜ independent ofα andπ.
To ensure a non-negative profit of the trading agency under both carbon prices, we assumed that the agency has zero profit in case of lower carbon price (p N ), When we take into account that the agency's profit is zero when T = N , the Equally Well-off Condition requires Using (29), Equation (30) reduces to Π(k(p N ))p N + [1 − Π(k(p N ))]p O :=P We re-arrange (29) to obtainα We insert (32) into (31) to obtain .
Plugging this back into (32) yieldsα. Symbol Description m number of countries i country index 1, ..., i s country index for net-sellers i b , ..., m country index for net-buyers T index for technology level available in economy, with "O" for old and "N" for new t index for technology used by a production firm, with "o" for old and "n" for new p T price target conditional on the technology level e i baseline emissions of production firm ī E sum of baseline emissions over all production firms a i abated emissions of production firm i g o (a i ) abatement costs, old technology g n (a i ) abatement costs, new technology b o coefficient abatement costs, old technology b n coefficient abatement costs, new technology i permits issued in country i δ coefficient damages in a country F i research infrastructure costs of country i x research investment of applied research firm π probability of a successful innovation per active applied research firm k number of active applied research firms f license fee to use g n Π(k) overall probability of detection of the new technology a o,O emissions abated using the old technology a n,N emissions abated using the new technology a * socially optimal emissions abated by firm i F i * lowest research infrastructure costs i * country with lowest research infrastructure costs λ scalar to reflect cooperation level p * T price target to reach socially optimal abatement level α grandfathered permits over baseline emissions α upper bound on α to ensure feasibility of carbon pricē E total amount of grandfathered permits K IC i total costs when some other country invested into research infrastructure K N IC i total costs when no other country invested into research infrastructurẽ amount of grandfathered permits to Country i * under ICG-Procedurẽ α permits grandfathered over baseline emissions for countries i = i * under ICG-Procedurẽ π compensating amount under ICG-Procedure