Exploring Interaction Effects of Climate Policies: A Model Analysis of the Power Market

The effectiveness of climate policy strongly depends on how these measures are implemented. National policy measures may have international spillover effects which partly neutralize domestic emission reduction, while different types of policy measures may offset each other as well. This paper explores the conditions for these interaction effects by using a concise partial-equilibrium two-country model of the electricity market which also includes a system for emissions trading. We find that the international spillover effects not only depend on the integration of electricity markets, but also on the tightness of the emissions-trading system. We show that this tightness is negatively related to the degree the supply of renewable energy is stimulated. We find that the more renewable energy is stimulated, the less domestic reduction in carbon emissions is offset by spillover effects. A more binding cap in the emissions-trading system makes national policies less effective. Hence, if climate-policy measures such as subsidies for renewable energy make the cap in the trading scheme less binding, these climate-policy measures become more effective.

In order to reduce carbon emissions in the power sector, governments are implementing a set of policy measures. These measures vary from subsidies for renewable-energy techniques to taxes on fossil-fuel electricity production and mechanisms for trading in emission rights. While some measures are taken on national level, others have an international character. Within the EU, coordination of climate policies is pursued by the European Commission. The Renewable Energy Directive (2009/28/EC), for instance, sets a binding target of 20 percent final energy consumption from renewable sources by 2020. Each EU Member State has to realize the renewable-energy target, but these countries are free to choose their own policies to stimulate deployment of renewable-energy sources. EU countries utilize different measures for this purpose, such as feedin-tariff subsidies and quota systems (Haas et al., 2010). In addition to this, several countries are considering to impose constraints on conventional power plants, in particular coal-fired power plants (EIA, 2014;EZ, 2015). These measures vary from implementing additional environmental standards (e.g. on fuel efficiency or emissions per unit) which make it complicated if not impossible for (old) coal-fired power plants to operate or to imposing a carbon tax which in particular raise the generation costs of coal-fired power plants. Besides this set of different national policy measures to reduce carbon emissions by the power sector, an emissions-trading system has been implemented on EU level. This EU Emission Trading System (ETS) is the largest cap and trade mechanism in the world in CO2 emissions. It sets up a cap on the total amount of CO2 emitted by installations of firms subject to this scheme. This cap is annually reduced in order to realize an overall reduction in carbon emissions. The initial allocation of the cap to participants was initially allocated by grandfathering, but more and more auctioning is used as allocation method (EC, 2012). In the secondary market, participants can trade in permits which results in a carbon price. Meanwhile, the European Commission is promoting the integration of national electricity markets to facilitate border-free trading across Europe, see Keay (2013). As a result, national power markets have become more closely integrated with each other, which may increase the international spillovers of national climate policies.
It is well established in economic literature that the coexistence of different types of climate policies may have counteracting effects (Schmalensee, 2012;Goulder, 2013; that case, theoretically, the level of emissions is only determined by the cap in the emissionstrading scheme (Tietenberg, 2006). If the cap remains the same, other instruments only affect the costs of reaching that target, but not the amount of emissions. If an emissions trading scheme is combined with subsidies for solar panels, for instance, it can be expected that the emissions within the power sector are reduced which lowers the overall demand for and, hence, the price of emissions permits, which in turn can stimulate other firms participating within the emissions trading scheme to raise their emissions since emitting has become cheaper (see e.g. Bergh et al., 2013;Böhringer and Rosendahl, 2011). This effect is called the waterbed effect of climate policy.
In this paper, we explore the conditions for the interaction effects to occur. For that purpose, we analyze the interaction of three types of policy measures to realize a transition of the electricity industry based on fossil fuels towards an industry with a lower level of carbon emissions. These policy measures are subsidies for renewable electricity, a fuel tax for fossil-fuel power plants and an international emissions trading scheme. In this paper we do not discuss the pros and cons of the individual climate-policy instruments as subsidies, taxes and emissions trading. Although one can discuss which instrument is best equipped to realise carbon reduction in a cost-effective way (see e.g. Aldy et al., 2010), in practice governments use packages of different types of instruments (Hughes and Urpelainen, 2015;Kautto et al., 2012;Del Rio and Mir-Artigues, 2014;Sijm, 2005).
Therefore it is also important to understand how they influence each other. In order to also analyze the international spillover effects of different national policies, we assume that the fuel tax is only implemented in one country. In order to analyze the interaction of different policy measures, we build a concise interconnected two region model with a large and small country in size. In this model, some producers are perceived as strategic players, hence they can exercise market power and influence the wholesale prices. Such a model is fairly well equipped to simulate the situation with a few centralized power producers, as it exists in several European countries such as the Dutch and German electricity market (see also Willems et al., 2009;Mulder et al., 2015;ten Cate and Lijesen, 2004). In our model, international trade is based on price-arbitrage opportunities. The energy trade is realized through the cross-border transmission lines. The size of the cross-border transmission capacity determines the magnitude of international trade and, hence, the potential cross-border spillover effects. Moreover, a carbon market is added to the electricity market, and consequently, the carbon price is part of the variable generation costs of fossil-fuel producers. In addition, we also take the stochastic nature of both supply and demand into account. Firms base their decisions regarding investments and the dispatch of plants on expected values for weather conditions, load levels and scarcity levels. Including probability distributions for wind and demand allows us to control for the volatility of market conditions in the power market.
Using a numerical application of our model, we find that combining the three different climatepolicy measures, including an emissions-trading system, may have a net effect on the level of carbon emissions, despite of the above-mentioned waterbed effect. This result comes from the fact that the carbon price in the trading scheme has a floor, i.e. it can never be lower than zero. This means that when other climate-policy measures are effective in reducing the demand for permits, they may also neutralize the waterbed effect. Our findings show that implementing national policies on top of an international emissions trading scheme can still be effective in reducing carbon emissions. As a matter of fact, although adding a carbon tax on top of an emissions trading scheme may result in more emissions reductions as the waterbed effect does not always work, this does not mean that such a policy is efficient.
The remainder of this paper is organized as follows. We review relevant literature in Section 2. In Section 3, we describe the key elements of the partial equilibrium model of the wholesale electricity market and define how the market equilibrium is determined. Section 4 presents the results for the policy variants. Section 5, finally, concludes.

Literature
This paper builds on and contributes to the literature of power market modeling and interaction effects of climate policies. A key question regarding the modelling of the electricity market is how to deal with strategic behaviour. Willems et al. (2009) compare two oligopolistic models of the electricity market: Cournot and Supply Function Equilibrium. 1 They show that both models explain roughly the same fraction of the observed price variations in the German electricity 1 Cournot equilibrium assumes that producers compete in the production quantity while the Supply Function Equilibrium assume that producers compete by bidding complete supply functions instead of one single quantity in an oligopolistic market with demand uncertainty. market. Furthermore, they suggest to use Cournot model for short-term model analysis as such a model can easily accommodate additional market conditions such as network constraints. Mulder et al. (2015) apply the Cournot model to the Dutch electricity market taking both the intermittent wind energy supply and fringe suppliers by combined heat and power (i.e., CHP) into account. As a result of the intermittent and fringe supplies, the wholesale prices tend to be lower. Using a competitive equilibrium model without strategic behavior among power generators, Saguan and Meeus (2014) investigate the interaction between cross-border transmission investments and renewable-energy policies. Their main conclusion is that renewable energy trade in order to comply with each member state targets is beneficial for both zones, but that an imperfect regulatory framework for transmission investment creates a significant cost for realising renewable-energy targets. In our model, some "big" producers are perceived as strategic players, hence they can exercise market power and influence the wholesale prices. Such a model fairly well resembles the situation with a few centralized power producers, such as in the Dutch and German electricity market.
Using several different models including partial equilibrium models and general equilibrium models, Calderón et al. (2016) find significant CO2 reductions through high carbon prices and abatement targets in Colombia. Benavente (2016) uses a computable general equilibrium model to examine the impact of a carbon tax in Chile. They conclude that such a policy is effective at reducing carbon emissions but at the cost of GDP losses. Ellerton and Fullteron (2014) find that a carbon tax on electricity in the U.S. can generate net negative domestic leakage as it raises the costs for other industries, which results in lower demand and, hence, lower production levels and carbon emissions by these industries. This impact of higher costs in the power industry on overall carbon emissions was also found by McKibbin et al. (2014). These authors conclude that the domestic carbon emissions outside the power sector decrease as higher electricity prices slow overall economic activity. Note that the above-mentioned papers only consider domestic carbon tax to reduce domestic carbon emissions. In a more than one country setting, Elliott et al. (2010) confirms that a uniform tax among all member countries is effective at reducing carbon emissions.
Moreover, in the above mentioned literature, the analysis of interaction of carbon taxes with other climate policies is not taken into account. From literature on emissions trading we know that the coexistence of different types of climate policies may have counteracting effects. When a cap-and-trade emissions scheme is implemented, the level of emissions is determined by the cap in the emissions-trading scheme (Tietenberg, 2006). If the cap remains the same, other instruments only affect the costs of reaching that target, but not necessarily the amount of emissions (see e.g. Bergh et al., 2013). As a result, the final level of emissions remains unchanged while the contribution of different emitters to this overall level has changed, which raises the costs of reaching the cap. Böhringer and Rosendahl (2011) find that the costs of realising a CO2 reduction target of 25% increase by more than 60% if the percentage renewable energy is stimulated by more than 10%. In other words: in case of an emissions-trading scheme, other measures directed at realising emission reduction merely affect the level as well as the allocation of costs of reaching the emission cap among the participants of the trading scheme without affecting the overall level of emissions (i.e. the benefits in terms of reductions of emissions remain the same). Because of this interaction effect, Böhringer (2014) concludes in his overview of two decades of European Climate policy, that renewable-energy subsidies and energy-efficiency mandates can result in higher costs for realising energy savings, energy efficiency improvements, and fuel switching that in case of a stand-alone cap-and-trade system.
From these papers, we learn that combining different types of climate-policy measures reduces the cost-effectiveness of climate policy. This strand of literature also states that adding other policy instruments to a system of emissions trading does not result in any additional emissions reduction (Sijm, 2005;Sorrel and Sijm, 2003). The arguments in favour of other policy measures, such as subsidies for renewable energy, are derived from the perceived benefits in terms of learning effects or security of energy supply. The contribution of our paper is that we analyse the conditions under which the interaction occurs or does not occur. In particular, we analyse in which circumstances climate-policy measures such as subsidies for renewables and taxes on fossilfuel use have an additional reducing effect on carbon emissions when also an emissions-trading scheme exists. In order to analyse the interaction of different types of climate-policy measures, we develop and apply a concise two-country model of the electricity market. The framework of this model is depicted in Figure 1. In the following sections, we introduce the corresponding components in detail.

Producers
On the supply side, the electricity market is composed of both centralized and decentralized power producers. The set of centralized power producers in country c is denoted as N c = {1, 2, · · · , n c }.
In general, n c is taken to be a small number. For example, in the Dutch electricity market, there are only a few major electricity producers (Electrabel (part of GDF SUEZ), E. ON Benelux, Essent (part of RWE) and Nuon (now subsidiary of Vattenfall)). In most cases, the power market is operated on a hourly basis. Therefore, we model the electricity market hourly and h ∈ {1, 2, · · · , 24} denotes hours in a day throughout the whole year. The years are indexed by y ∈ {1, · · · ,ȳ}. The model is simulated such that "1" represents the current situation and "ȳ" denotes the end year. Note that p cyh is the wholesale price per hour in country c year y. Note that the difference between fossil-fuel plants and wind turbines is that the costs on the margin for the wind turbines are almost zero, while the marginal costs for fossil-fuel plants are not zero and also include CO2 prices. We do not consider technology upgrades to reduce the marginal costs of fossil-fuel plants as we may assume that these are constant in the short term.
Assumption 3.1. Each centralized power producer i ∈ N c has the same constant marginal cost m c ∈ R + for fossil-fuel plants over year y in country c.
Note that we do allow different fossil-fuel production techniques in these two countries. Hence, the constant marginal costs might differ between them.
The deployment of wind energy mainly depends on the weather conditions and is stochastic, ex ante. Let w h denote the load factor at hour h to exploit the wind energy capacity. Because of the geographical proximity of neighoring countries, we assume that the production by wind turbines is subject to the same stochastic pattern in both countries.
Assumption 3.2. We assume that w h follows a certain discrete distribution, with realizations µ j h ∈ R + and each realization µ j h has a probability π j h ∈ R + . Note that j π j h = 1.
Note that q i cyh is composed of the production amount by fossil-fuel plants and also wind turbines, hence Note that q iF cyh and q iW cyh are the production part by fossil-fuel plants and wind turbines, respectively. And the realized wind energy production is calculated based on the realized load factor and generation capacity, At the beginning of a certain year, each centralized power producer's wind energy capacity Q iW cy is given and is assumed to be common knowledge. Note that the production amount is constrained by the generation capacity Q iF cy and Q iW cy , hence we have q iF cyh ≤ Q iF cy and q iW cyh ≤ Q iW cy . The aggregate fossil fuel generation capacity in country c year y is denoted as Q F cy and Q F cy = i∈Nc Q iF cy . The fossil-fuel generation capacities can be invested each year and we denote ∆Q F cy as the investment in fossil-fuel plants in country c year y.
Because of the large number of decentralized power producers, they are modeled as pricetakers which cannot exercise market power to influence wholesale market prices. Hence, the decentralized power producer equalizes their marginal benefits to marginal costs. The aggregate decentralized power production (D) mainly uses combined heat and power (DH), wind turbines (DW ) and solar cells (DS). Costs on the margin from wind and solar energies production are assumed to be zero while combined heat and power is a side product of the horticultural suppliers, whose main objective is to produce heat for their greenhouses. We assume that they have increasing marginal costs (see also Mulder et al. (2015)).
Assumption 3.3. The production amount by combined heat and power q DH cyh is assumed to be a linear function of electricity prices, where α D > 0 and β D > 0.
In addition, the expected production amount by solar cells is the product of the hourly load factor and installed capacities. Let u h be the realized load factor of solar cells at hour h. Hence, we have the following, where Q DS cy denotes the yearly generation capacity for solar cells. The sum of CHP and solar cells composes the aggregated production amount by fringe suppliers, where α cyh = α D + u h × Q DS cy and β = β D .

Demand side of the electricity market
The demand side of the wholesale electricity market consists of large electricity users (L) and retailers (R). Retailers sell electricity further to consumers and prosumers. We assume a linear demand function for large electricity users as follows, where a L h and b L h are parameters to be calculated, t L is the tax rate for large electricity users and τ h is the hourly network tariff paid by large electricity users. Hence, we implicitly assume that the tax rate and network tariffs do not change over time y.
The retail price is equal to the wholesale market price (p cyh ), plus a retail margin (r), taxes (or levies) t R and the dynamic network tariffs τ h . Hence, the demand function for consumers and prosumers can be specified as following, where a R h and b R h are parameters to be calculated. The aggregation of the demand from large users and retailers induces the total demand function faced by producers, where a h and b h are calculated from equations (4) and (5) and, Note that by introducing a dynamic network tariff τ h , we move the aggregate demand function upward or downward on a hourly basis, but the slope of the aggregate demand function does not change. Therefore, the aggregate demand function suppresses demand when there is a higher network tariff τ h and boosts demand when the network tariff is low.

Fossil fuel plants investment
When the expected production by RES is low or zero, the need for fossil fuel production might exceed the current generation capacity. As a result of this, electricity scarcity prices occur, see also ten Cate and Lijesen (2004). In the fossil-fuel investment decisions, we also take import and export into account. Electricity importing companies are modelled as price-takers. Let q I yh be the total electricity import. Following Mulder et al. (2015), the supply of the importers is approximated by a linear supply function, and the export amount by firm i is q iE cyh . We have the following equation for scarcity prices, Note that the first term Q F cy denotes the fossil-fuel generation capacity, the second term (δp cyh ) on the left-hand side of equation (8) denotes the import amount, 2 the third term (α cyh + βp cyh ) denotes the production amount by fringe suppliers and the fourth term i q iE cyh denotes the electricity export amount which is modelled exogenously. The right-hand side of equation (8) denotes the aggregate demand at a certain electricity price level.
The fossil-fuel plants investments ∆Q F cy are considered in a competitive setting in which firms cannot behave strategically and exercise market power. Assuming perfect foresight, expected long-run marginal revenues should be equal to long-run marginal costs. Following ten Cate and Lijesen (2004), we have the following: the price per MWh which is required to keep demand down to capacity (equation (8) where m c denotes the constant fossil fuel production costs. The expression {q F cyh = Q F cy + ∆Q F cy } denotes the set of hours when the capacity constraint is binding. Hence, the investment in fossil-fuel plants ∆Q F cy should be set at a level that equalizes expected marginal benefits (LHS of equation (9)) and marginal costs (RHS of equation (9)).

RES investment
We assume that the investments in RES depend on government subsidies. Suppose the RES subsidy budget for wind parks is B W cy and the budget for solar cells is B S cy . Moreover, we assume that the budget is financed by a tax on electricity consumption. The investment costs for wind parks and solar cells are denoted by c W y and c S y , respectively. The newly installed capacities for centralized power producers (∆Q W cy ) are calculated as follows, Similarly, the newly installed capacities for decentralized power producers (∆Q S cy ) are calculated as follows,

Market equilibrium
The wholesale electricity market is modeled as an imperfect market. Facing a certain demand curve, the producers compete in terms of quantities. The market reaches equilibrium when each producer's strategy is the best response to the strategies actually employed by its competitors.
Domestic electricity demand is met by centralized producers and the aggregate decentralized production, hence And the residual demand faced by i is given by, where q −i cyh denotes the sum of the other centralized producers' production amount except i. Note that in the above equation, we replace q D cyh by equation (3). Rearranging equation (11), we obtain, In practice, forward contracts (including long-term, day-ahead and intraday) also exist in the electricity wholesale market. We assume that centralized power producers are active in the forward market. Let q if cyh be the forward trading quantity by firm i and p f cyh be the forward price. Following Allaz and Vila (1993), "under perfect foresight, equilibrium requires the forward market to be efficient. This means that the forward price as a function of the forward positions must be equal to the price that will result from the Cournot competition on the spot market given these positions. Therefore, no arbitrage is possible." Similar idea can also be found in Mulder et al. (2015). Given the forward positions by each firm, firms compete over the production quantity. Hence, the production quantity is solved as a function of the forward positions. Then firms optimize their forward positions given the quantity solved from the production period, see Allaz and Vila (1993). Note that part of the production is met by wind energy.
All the derivations for the optimal production and forward positions are included in the Appendix A. We have the following results for the market equilibrium, Proposition 3.4. Under the following conditions, 1. The decentralized power production is given by equation (3); 2. The aggregate demand function is given by equation (6); 3. The demand is satisfied by centralized power producers and decentralized power production; 4. The ex post production from wind energy is calculated based on actual load factor µ j h and generation capacity (equation (2)); 5. Centralized power producers use both fossil fuels and wind energy (equation (1)); We have the following results: the optimal production amount using fossil fuels by firm i at hour h and time y is given as follows, subsequently, the optimal forward positions chosen by firm i can be solved from the following, rameter values α cyh = β = 0, m c = b, n c = 2, a h = a, b h = 1 and q iW cyh = 0, i ∈ N c , we obtain, The above results are in line with Proposition 2.3 in Allaz and Vila (1993). From equation (13), we can easily see that any production by RES will replace the energy production by fossil fuels for producer i. For each value of the load factor for wind turbines π j h , we would have a corresponding market equilibrium regarding fossil-fuels production (13) and wholesale prices (12). 3

Import/export and law of one price
In this section, we further investigate how import and export influence the domestic price, which is determined by (12). For country c, let IE cyh be the net export amount in hour h year y, i.e., export minus import. If there are price differences between these two countries, we expect that traders will profit from export from a lower price country to a higher price country. Note that in the first step of the calculation, we allow trades to equalize the prices between these two countries. Let p u cyh be the uniform prices between these two countries together with trading amount IE cyh , hence we have the following Note that q i cyh are solved from equation (13) together with the ex ante expected wind energy production. 4 In addition, we have Combining equations (15), (16) and (17), we solve for the corresponding p u cyh and IE cyh . In the second step, we check for the capacity constraint IU between these two countries. If IE N L,yh > IU , then the different prices p d cyh in both country are as follows, If IE N L,yh < −IU , then the prices in both countries are as follows,

Carbon market and the interaction with the electricity market
Finally, we add an international carbon market to the set of national electricity markets. Let cap y be the average daily carbon emission cap and P CO2 y be the daily CO2 price. For each fossilfuel production technique, the carbon emission coefficient is denoted as e c . In such a setting, the adjusted constant marginal costs ac cy for country c in year y are as follows, Given the constant marginal costs ac cy , we calculate the fossil-fuel production according to equation (13). Then we compare the actual daily carbon emissions summing over 24 hours with the daily emission cap. If the carbon emissions are above the cap, we keep increasing the carbon prices until the emissions are equal to or below the cap. Hence, the daily carbon price is determined by the daily cap and the daily aggregated demand for carbon permits.

Numerical analysis 4.1 Parameters and variants
In order to analyse the interaction effects between climate-policy measures, we conduct a numerical analysis with our model. We refer to a two interconnected region case where we have a large and small country. Differences in scale of countries are important in order to better assess international spillover effects from policies implemented in the larger country to the smaller country. The parameters for both countries are derived from the characteristics of Germany (large) and Netherlands (small), respectively, without the objective of fully representing these countries. Table 1 lists a brief summary of relevant parameters we have used in this paper.   In our simulation of these two countries, we approach each year by simulating 24 consecutive hours in different scenarios regarding wind speed and demand levels. For each wind speed level, we use a probability, based on empirical evidence. In order to define the hourly probability distribution we use actual hourly data on wind speed. We rank the hourly data from lowest to highest level and then determine the average value in three classes: the lowest 31%, the next 38% and the highest 31% of all observations. Table 3 gives the results for the wind speed. For the demand level, we scale up or down the intercept a h of the inverse demand function. For each scaling factor, we assign a corresponding probability and this holds for each hour. Table   2 reports the result for demand level in each hour. Note that for each hour, we have 3 discrete realizations of the wind energy load factor and 3 scalings of the demand. Therefore, we end up with 9 scenarios for each hour.Running the model for each hour for each scenario, we obtain for each hour a probability distribution of all results. This Monto-Carlo type of analysis enables us to deal with the impact of extreme circumstances, in particular regarding the impact of renewables on the electricity market and the emissions-trading scheme. According to the Statline database of Statistics Netherlands, the installed capacity for wind energy including onshore and offshore wind parks is roughly 2.9 GW in 2014. 5 Most wind energy production in the Netherlands is run by centralized power producers. The load factor per hour to employ the wind energy capacity is calculated based on the data from the Dutch Royal Meteorological Institute. 6 The installed capacities for wind energy production in Germany are taken to be about 57.5 GW. Because of the geographical proximity, the load factor of wind energy production in Germany is assumed to have the same discrete distribution per hour as in the Netherlands.
Decentralized power production mainly refers to CHP and solar energy. The minimum run load for CHP in the Netherlands and Germany is estimated to be 5 GWh. We roughly estimate We have taken hourly price elasticities and in general, a higher elasticity for off-peak hours and a lower elasticity for peak hours (9h -20 h). All hourly elasticities are in the range of -0.3 and -0.2.  Table 4. In the "Baseline" variant it is assumed that both countries annually increase the renewable-energy capacity while also a international cap-and-trade emissions trade system exists. The annual increase in RES capacity is based on the assumption that 10% of electricity tax revenues is used to finance the subsidies for these investments. The initial carbon emission cap level is chosen to be 1.04 Mtons per day, which is about equal to the aggregated daily emissions by the power industry in Germany and the Netherlands. In the "Baseline", we assume that the cap is reduced by 0.5% annually. In the policy variants "Lowercap" and "Highercap" the cap is annually reduced by 1% and 0.25% respectively. We are in particular

Results
We first present the numerical results for the "Baseline", which is the scenario where both countries stimulate RES by giving subsidies for investments, while also an international capand-trade system exists. We are interested in the following metrics: the wholesale electricity prices, the hourly RES production, the utilisation of fossil-fuel plants (defined as average hourly production in percentage of installed capacity), the CO2 prices and, finally, the CO2 emissions.
Then, we consider the variant of "Prodtax" which imposes a fossil-fuel tax in the large country on top of the "Baseline". Finally, we conduct a sensitivity analysis by changing the emission cap level 9 8 According to the CO2 emissions coefficients tons per MWh, we have taken 0.3 for gas-fired plants and 0.6 for coal-fired plants in the simulation. For a portofolio of 50% coal-fired and 50% gas-fired fossil-fuel plants with a carbon price of 25 euro per ton, we choose a level of 11.25 euro per MWh for the cabon taxes on top of the fossil-fuel production.
9 Note for figures 3 to 6: the thickest lines denote the variants with the default cap ("Baseline" and "Prodtax"), the thinnest lines denote the variants with the higher cap ("Baseline Highercap" and "Prodtax Highercap") As a result of an exogenous stimulation of investments in renewable energy capacity in the "Baseline" variant, this capacity increases strongly. As a consequence, the volatility in the supply by renewables increases strongly as well (Figure 2). This is related to the fact that the hourly production level by renewables is sometimes close to zero in case of unfavourable weather circumstances independent of the size of installed capacity. Hence, the lowest level of production by renewable energy capacity is hardly affected by the size of this capacity, while the maximum level is strongly related to this (Figure 2). On average, the hourly renewable energy production is much higher in 2030 compared with the level in 2016. This strong increase in RES capacity fairly well resembles the actual developments in many European countries. The utilisation of fossil-fuel plants in both countries goes down as a result of the increase in RES, see Figure 7. In addition, the annual reduction in the carbon emission cap raises the scarcity of carbon permits and, hence, the carbon price, see Figure 4 and Figure 5. Due to the different size of the initial installed generation capacities, the marginal production is more often run by the RES in the large country and less often in the small country. In the latter country, the upward price effect of the increasing carbon prices dominates the price-reducing effect of the increasing share of RES.
As a result, the strong increase of RES significantly reduces the price of electricity (as in the large country), but this appears not to be the case in the small country (see Figure 3). As the cross-border capacity has a limited size, traders are not able to fully benefit from these price differences. The remaining price differences indicate that this capacity is fully utilized.  Now, the question is what happens if the large country introduces a carbon tax on top of the measures stimulating the RES capacity and the international emissions trading system. The direct effect is that the generation costs of the fossil-fuel power plants in this country increase. As both countries are connected, the increase in generation costs in the large country implies that this country wants to import from the smaller country in those hours when RES capacity is not setting the price. As a result, production shifts to the smaller country. This shift of production by fossil-fuel plants implies a kind of carbon leakage. The introduction of a carbon tax in the large country raises the utilisation of fossil-fuel capacity in the small country (see Figure 7). This international spillover effect of national climate policies also raises the electricity price in the other country, as we observe a price increase in both countries compared with the "Baseline" (Figure 3).
The shift in the location of the production by fossil-fuel plants does, however, not mean that there is no effect on the price of carbon (Figure 4). The introduction of a carbon tax in one country results in a lower (average) CO2 price which implies that the overall demand for permits has been reduced. The negative impact of the carbon tax on the carbon price shows the existence of the waterbed effect: the emissions trading system becomes less effective if a carbon tax is introduced. However, we also see that the overall level of carbon emissions is lower when we have a carbon tax, which indicates that the waterbed effect does not fully neutralize the effect of the carbon tax. This result is related to the fact that the price of CO2 may be zero from time to time (Figure 6). If the price of CO2 is zero any other reduction in the demand for carbon permits cannot have any effect on the price anymore. Hence, we find that the combination of different policy measures to reduce carbon emissions may still be effective despite the interaction effects.
When we lower the emission cap, subsequently we observe a lower carbon emission level and a higher carbon price (Figure 4). Because of the higher carbon price, we find higher electricity wholesale prices in the small country: the price-reducing effect of the increase in RES capacity is completely neutralized by the price-increasing effect of the tighter carbon market (see Figure 3).
We also observe stronger spillover effect in fossil-fuel production: the utilisation of the fossil-fuel plants is more strongly increased when we have a lower cap in the emissions-trading system (see Figure 8). This implies that in case of tighter emissions-trading system, the international spillover effect of national policies are larger. More importantly, because of the stronger effect on the CO2 prices, there is less effect on CO2 emissions (see Figure 5). This is related to the fact that in case of a lower cap the prices are less often zero which makes it possible to be lowered and to obtain the waterbed effect. In this paper we have explored the conditions under which interaction effects occur between different types of climate-policy measures. Governments are combining different types of policy measures in order to realise their ambitious objectives regarding the reduction of carbon emissions. It is well established in the literature that the combined effect may be lower than the sum of the individual effects. Combining subsidies for renewable energy or taxes on fossil fuels together with a cap-and-trade system suffers from the waterbed effect. Moreover, national policies to reduce domestic emissions may be offset by international spillover effects. The question we have explored is whether this offsetting effect always occurs or whether it may be subject to specific conditions. This topic is relevant because in the EU, each country has the freedom to choose its own national energy policy despite of European climate-policy objectives. European countries apply a mixture of different types of policy measures which make it highly relevant to analyze the nature of and the conditions for the interaction effects.
Using a numerical partial two-country equilibrium model of the power market which also includes a cap-and-trade carbon system, we find spillover effects due to the integration of the two markets. Imposing a fossil-fuel tax in one country leads to a higher cost for fossil-fuel producers. Hence, this country imports more from the neighboring country. As a result of this, we observe a higher utilization of fossil-fuel capacity in the neighboring country. The lower the cap in the emissions-trading system, the stronger this effect appears to be. This result indeed shows that national policies to reduce carbon emissions may be offset by international spillover effects. Coordination of such policies may improve the effectiveness of such policies.
However, we find that the waterbed effect does not always hold. It appears that adding other climate-policy measures to an emissions-trading system may have a net effect on the level of carbon emissions. This result comes from the fact that the carbon price in the trading scheme has a floor, i.e. it can never be lower than zero. If subsidies for renewable energy result in a large amount of renewable-energy capacity this may sometimes, e.g. on sunny and windy days, result in an overall demand for carbon permits being below the supply of permits which brings the carbon price to zero. In such circumstances, giving more subsidies for renewables or imposing a tax of fossil fuel reduce the emissions by fossil-fuel plants without being neutralized by a waterbed effect. Hence, we find that the waterbed effect only holds if the cap-and-trade system is constantly binding, which means that there is always a positive price for the carbon permits.
The probability of a always binding emissions-trading system, however, reduces if countries keep increasing the size of installed RES capacity as is currently the case in many European countries.
The policy consequence of this finding is that national climate policies such as subsidy schemes for renewables may have a positive effect on reduction of carbon emissions, although the general literature says that such cannot be the case when an emissions-trading scheme exists.
These findings are based on a numerical analysis of a concise model of the electricity market.
The advantage of such a model analysis is that it gives insight in the interrelationships of a number of factors affecting the market. Because of its theoretical nature, this model analysis does not give precise estimates of the size of the relationships and the probability of the situations in which the interaction effect do not occur. Empirical research is needed to obtain precise estimates for the magnitude of actual interaction effects between current climate-change policies.
As we only focused on the occurrence and absence of interaction effects of different type of climate policies, we did not discuss the efficiency of these interaction effects. Although adding a carbon tax on top of an emissions trading scheme may result in more emissions reductions as the waterbed effect does not always work, this does not mean that such a policy is efficient. In order to analyse the efficiency effects of climate policies, a more general equilibrium approach is needed taking into account more kinds of interactions within the economy. A Optimal production amount by centralized power producers For notational convenience, we suppress the expectation sign for q iW cyh from now on. The production game for firm i ∈ N c is given by, where m c is the constant variable costs for firm i in country c to use the conventional resources to generate electricity and p cyh = a h −α cyh −q i cyh −q −i cyh b h +β . The first order conditions for firm i read, Equation (19) holds for every firm i and we can write the system of equations for the first order conditions of each producer into matrix form as follows, 1 · · · 1 1 2 · · · 1 1 1 · · · 1 · · · · · · · · · · · · 1 1 · · · 2 The above matrix solve q iF cyh , i ∈ N c as a function of q 1f cyh , q 2f cyh , · · · , q ncf cyh . Note that we also have q iW cyh = q jW cyh , i, j ∈ N c , i.e., the RES productions are also symmetric among all producers. We solve the above matrix and obtain the following solution for i, i ∈ N c , where i, j ∈ N c and j = i.
Now we move to the stage of firms choosing the optimal forward positions. According to Allaz and Vila (1993), we have the following maximization problem, According to the arbitrage condition, it should hold that p f cyh = p cyh . Hence, the above maximization problem can be simplified as, where p cyh = a h −q −i cyh −q i cyh −α cyh b h +β . Taking first order conditions with respect to q if cyh , we obtain the following equation for firm i,  (22) into (23), we obtain the following results, QED.

B Calculation of the aggregate demand function
Suppose we want to calculate the aggregate hourly dependent demand function, and the objective is to calculate parameters a h and b h . Given the price elasticities ε h in the literature and observed quantityp cyh and outputQ cyh in a load profile, we use the following formula to calculate a h and b h , Note that the above formula is implemented to calculate the aggregate demand function in the small country and the aggregate demand function for the large country is obtained by scaling up the demand function of the small country.