Endogenizing the Cap in a Cap-and-Trade System: Assessing the Agreement on EU ETS Phase 4


In early 2018, a reform of the world’s largest functioning greenhouse gas emissions cap-and-trade system, the EU Emissions Trading System (ETS), was formally approved. The reform changed the main principles of the system by endogenizing the emissions cap. We show that the emissions cap is now affected by the allowance demand and is therefore no longer set directly by EU policymakers. As a consequence, national policies that reduce allowance demand can now reduce long-run cumulative emissions, which is not possible in a standard cap-and-trade system. Using a newly developed dynamic model of the EU ETS, we show that policies that reduce allowance demand can have substantial effects on cumulative emissions after the reform. Model simulations also suggest that the reform reduces long-run cumulative emissions and, to a lesser extent, reduces emissions in the short run. Even so, the reform has a small short-run impact on the currently large allowance surplus.

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  1. 1.

    This effect is often referred to as the waterbed effect, and it is well established in the cap-and-trade literature. The waterbed effect implies a carbon leakage rate of 100 pct. within the cap-and-trade system. There might also be carbon leakage between sectors covered by the cap-and-trade system and the remaining part of the economy (see Jarke and Perino 2018). In addition, there might be carbon leakage between the region implementing the policy and other parts of the world (e.g., Böhringer et al. 2017). Our focus is on emissions within the cap-and-trade system.

  2. 2.

    National policies might also affect long-run emissions through the EU ETS negotiations. Yet, it is difficult to verify this effect, and it is especially difficult to establish a causal link between a particular national policy and an EU ETS negotiation outcome. In addition, small changes in allowance demand might not affect the negotiation outcome.

  3. 3.

    See Ellerman et al. (2016) for an overview of the history and structure of the pre-reform EU ETS.

  4. 4.

    The allowance surplus is basically the amount of banked allowances available to the market. More precisely, the allowance surplus is given by the allowance supply minus the allowance demand (primarily surrendered allowances) minus the stock of allowances in the MSR.

  5. 5.

    We formulate the problem of the representative firm as a profit maximization problem, while Rubin (1996) and others (e.g., Schennach 2000; Perino and Willner 2017; Silbye and Sørensen 2019; Quemin and Trotignon 2019) formulate it as an abatement cost minimization problem. Yet, the two approaches are basically two sides of the same coin. As we formulate the objective function as a power function, we find that the profit maximization formulation is more in line with typical formulations in the literature.

  6. 6.

    In addition, Gerlagh and Heijmans (2018) and Kollenberg and Taschini (2019) discuss the latest EU ETS reform based on their analytical models.

  7. 7.

    Perino and Willner (2017) investigate three reform proposals for the EU ETS, one of which includes the key change to the MSR that this paper focuses on.

  8. 8.

    Rootzén and Johnsson (2013, 2015) find that it will be difficult to meet ambitious emission targets for petroleum refineries, iron and steel plants, and cement producers in the absence of major breakthrough technologies like effective carbon capture and storage. Thus, it might be hard to bring down emissions in this part of the ETS sector.

  9. 9.

    From here, we use the shorthand “CO2” to refer to CO2-equivalents.

  10. 10.

    This is often referred to as an increase in the linear reduction factor from 1.74 to 2.2 pct. The amount of allowances reduced is calculated as the linear reduction factor multiplied with the average yearly amount of newly issued allowances over the period 2008–2012.

  11. 11.

    The representative firm assumption is motivated by Rubin (1996). Specifically, Rubin shows that firms within a cap-and-trade system that can trade allowances with each other behave like a central planner that efficiently allocates allowances to each unit to minimize costs. This representative firm approach is taken in many other analyses of cap-and-trade systems (e.g., Schennach 2000; Perino and Willner 2017).

  12. 12.

    Note that \({B}_{t}\) is the allowance surplus at the end of period \(t-1\). Hence, it is the allowance surplus at the end of period \(t-1\) that determines the MSR allowance allocations. These allocations will, however, take place in period \(t+1\). In reality, the allowance surplus at the end of year \(t-1\) is published in May of year \(t\), and the allocations to or from the MSR take place over 12 months from September 1st of year \(t\) (EU, 2015). As the model is formulated in years, it cannot capture this timing perfectly. However, the majority of the allowance allocations to or from the MSR takes place in year \(t+1\), and thus, the modeling approach taken here captures the main mechanism.

  13. 13.

    A cost the firm must pay as long as production occurs, but which is independent of the production scale. A typical example is lights in a factory.

  14. 14.

    The method is described by Adda and Cooper (2003, p. 34–40). The value function is computed from the Bellman equation by backward iteration. Starting from the period \(T\) and moving backward in time, the value function is maximized with respect to emissions for a tight grid of states, i.e. allowance surplus values. Through interpolation methods, the procedure leads to policy functions: the optimal emission level at time \(t\) is a function of the allowance surplus at time \(t\). The optimal emission path is simulated from the initial allowance surplus and the policy functions. As the first-order conditions derived in the main text are not used in the algorithm, they can be employed to check that the solution is consistent with the maximization problem of the representative firm.

  15. 15.

    This number is computed from the expected number of unallocated allowances in 2020, and the 900 m allowances back-loaded into the MSR according to EU (2015) and the European Commission (2017a). According to European Commission (2015b), market participants expected that between 550 and 700 m allowances were transferred to the MSR in 2020. There are currently more than 300 m unallocated allowances in the New Entrants Reserve (European Commission, 2018) and about 300 m unallocated allowances due to closures and changes in production capacity (European Commission, 2017c). Based on European Commission (2015a, 2017c, 2018), the latter has historically increased faster than the former. Thus, it seems fair to assume that more than 600 m unallocated allowances are placed in the MSR by 2020. In this paper, we assume that 650 m unallocated allowances are placed in the MSR, but the exact number is not important for the results presented below.

  16. 16.

    The fossil fuel price is based on the average projected coal price from the Danish Energy Agency (2017) over the period 2015 to 2040.

  17. 17.

    In general, risky assets should yield an annual return of about 7 pct. (Jordà et al. 2017). However, firms can also hold allowances to hedge against future price movements. This reduces the required return below that of a standard risky asset. Neuhoff et al. (2012) find EU ETS coved firms hedging their risk on allowance price movements require a return of 5 pct. The discount rate is discussed further in Sect. 5, and it is shown that the main results can also be obtained using a lower or higher return requirement.

  18. 18.

    Emissions are based on the European Environment Agency (2017a) and the allowance price is computed based on EEX (2018). The historical emissions reported by the European Environment Agency (2017a) reflects the current scope of the EU ETS.

  19. 19.

    Silbye and Sørensen (2019) and Quemin and Trotignon (2019) also find approximately linear emission paths.

  20. 20.

    The net allowance demand effect considered in this paper corresponds to the demand effect after internal leakage as defined by Perino et al. (2019). The inter-ETS leakage computed in the present paper therefore corresponds to the waterbed effect as defined by Perino et al. (2019).

  21. 21.

    Salant (2016) documents that several media were questioning the legitimacy of the EU ETS in the beginning of 2013. For instance, The Economist (2013) wrote a critical article on the EU ETS in April 2013 with the title “ETS, RIP”.


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We would like to thank Lars Gårn Hansen, Thomas Bue Bjørner, Louis Birk Stewart, Peter Birch Sørensen, and Frederik Silbye as well as seminar and conference participants at the Technical University of Denmark; the Danish Ministry of Energy, Utilities and Climate; the Danish Energy Agency; the University of Copenhagen; the Danish Environmental Economic Conference 2018; and the conference “Emissions Trading: Which Way Forward” 2019, for comments, discussions, and suggestions. Finally, we wish to thank our anonymous referees, whose critical comments greatly improved the paper. Any remaining errors are our own.

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Correspondence to Peter K. Kruse-Andersen.

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Appendix 1: Model with Explicit Allowance Market

In the main text, we abstract from modeling the allowance market. This appendix shows that an inclusion of the allowance market would not affect the simulation outcomes presented in the main text.

Let the profit function of the representative firm, \(F(\cdot )\), be given by:

$$F(e_{t} ,A_{t} ,y_{t}^{All} ,y_{t} ) = f(e_{t} ,A_{t} ) - p_{t} (y_{t} - y_{t}^{All} ),$$

where \({p}_{t}\) is the market price for emission allowances, \({y}_{t}^{All}\) is the amount of freely allocated allowances, and \({y}_{t}\) is the net allowance demand of the firm: the sum of freely allocated allowances and allowances purchased by the representative firm (the latter may be negative). A net purchase of allowances (\({y}_{t}>{y}_{t}^{All}\)) has a negative effect on the firm’s profits and vice versa. Note that the firm takes both \({p}_{t}\) and \({y}_{t}^{All}\) as given.

The representative firm maximizes \(F(\cdot )\) subject to the dynamic allowance constraint with respect to emissions and the net allowance intake. Assuming an interior solution, the first-order conditions associated with the problem imply:

$$\lambda_{t} = p_{t}$$
$$f_{e}^{^{\prime}} (e_{t} ,A_{t} ) = \lambda_{t}$$
$$\lambda_{t + 1} = (1 + r)\lambda_{t}$$

where \({\lambda }_{t}\) is the shadow price of emission allowances. It is easy to derive the optimality conditions from Sect. 3.2 from these equations.

The first condition shows that if there is allowance trading, the price of allowances equals the shadow price of emission allowances, which equals the marginal profit of emissions as stated in the main text.

The arbitrage condition \({p}_{t}={f}_{e}^{^{\prime}}({e}_{t},{A}_{t})\) ensures an equilibrium on the market for allowances. The condition ensures that along the optimal emission path, the firm is indifferent between selling an allowance, earning \({p}_{t}\), and using the allowance, earning \({f}_{e}^{^{\prime}}({e}_{t},{A}_{t})\). If it was the case that \({p}_{t}>{f}_{e}^{^{\prime}}({e}_{t},{A}_{t})\), the firm would sell as many allowances as possible, while the firm would demand more allowances if \({p}_{t}<{f}_{e}^{^{\prime}}({e}_{t},{A}_{t})\). As there is only one agent demanding allowances in the model – the representative firm – and since the amount of actioned allowances, \({y}_{t}^{Auc}={y}_{t}-{y}_{t}^{All}\), is determined by the trading system (and unaffected by the firm’s behavior in period \(t\)), it must be such that in equilibrium \({p}_{t}={f}_{e}^{^{\prime}}({e}_{t},{A}_{t})\).

As the first-order conditions boil down to the same optimality condition as in the model presented in the main text, the solution to the firm’s problem is unchanged by the inclusion of the allowance market. Accordingly, the extended model leads to the same simulation outcomes as the model presented in the main text.

One potential concern is that when the firm stops saving allowances, \({B}_{t+1}=0\), one could end in a situation where \(F(\cdot )\) is negative while \(f(\cdot )\) is positive. In this case, the two models would have different solutions: the firm would shut down production in the extended model, while it would continue production in the model presented in the main case.

However, one can show that this situation will never occur. Plotting the equilibrium allowance price into the per period profit function:

$$F(e_{t} ,A_{t} ,y_{t}^{All} ,y_{t} ) = f(e_{t} ,A_{t} ) - f_{e}^{^{\prime}} (e_{t} ,A_{t} )y_{t}^{Auc} .$$

The amount of auctioned allowances can be expressed as: \({y}_{t}^{Auc}={\eta }_{t}{y}_{t}\), where \({\eta }_{t}\in \left[\mathrm{0,1}\right].\) When there is no allowance saving, \({B}_{t+1}=0\), then \({e}_{t}\ge {y}_{t}\). Since \(f(\cdot )\) is a concave function and \(f\left(0,{A}_{t}\right)=0\) it must hold that: \(f\left({e}_{t},{A}_{t}\right)-{f}_{e}^{^{\prime}}\left({e}_{t},{A}_{t}\right){e}_{t}>0.\) It then follows that:

$$F(e_{t} ,A_{t} ,y_{t}^{All}, y_t ) = f(e_{t} ,A_{t} ) - f_{e}^{^{\prime}} (e_{t} ,A_{t} )\eta_{t} y_{t} > 0.$$

Accordingly, when \({B}_{t+1}=0\) then \(F(\cdot )\) is positive whenever \(f\left(\cdot \right)\) is positive.

We also note that the concavity of \(f(\cdot )\) ensures that the firm can always ensure that \(F(\cdot )\) is positive. This follows from the fact that the firm can always set \(e_{t} = y_{t}^{Auc}\) in which case:

$$F(y_{t}^{Auc} ,A_{t} ,y_{t}^{All} ,y_{t} ) = f(y_{t}^{Auc} ,A_{t} ) - f_{e}^{^{\prime}} (y_{t}^{Auc} ,A_{t} )y_{t}^{Auc} > 0.$$

Appendix 2: Actual and simulated emissions for the period 2008–2017

This appendix shows that the calibrated model leads to a good fit between simulated and historical EU ETS emissions. Specifically, we focus on the period 2008–2017, as the first phase of the EU ETS (2005–2007) was a test phase and emission allowances from this phase could not be saved for future phases. The simulation, therefore, starts in 2008, where the allowance surplus was zero, and it ends in 2110. The MSR is absent in this simulation, as the introduction of the MSR was unexpected in 2008. We also disregard the change to the linear reduction factor from 2021, as this revision is part of the latest reform and probably unexpected in 2008, although this assumption has little effect on the emission levels for the period 2008–2017 found here. One limitation of this exercise is that we ignore the impact of actual and expected revisions to the ETS through the first part of the simulated period like the MSR reform from 2015.

Figure 4 shows the simulated emission path and actual emissions for the period 2008–2017. The two emission paths follow each other closely over the period. The simulation overestimates emissions toward the end of the period. This is expected, as the simulation does not take the MSR reform from 2015 and the expectations about the latest ETS reform into account. Both reforms reduce the available amount of allowances in the short and medium run, resulting in lower short-run emissions.

Fig. 4

Source: Own calculations and European Environment Agency (2018)

Actual and simulated CO2 emissions, 2008–2017

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Beck, U., Kruse-Andersen, P.K. Endogenizing the Cap in a Cap-and-Trade System: Assessing the Agreement on EU ETS Phase 4. Environ Resource Econ 77, 781–811 (2020). https://doi.org/10.1007/s10640-020-00518-w

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  • Cap-and-trade
  • EU ETS
  • Market Stability Reserve
  • Overlapping policies


  • Q48
  • Q54
  • Q58