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Simulating emissions intensity targets with energy economic models: algorithm and application

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Abstract

Pressure on developing economies to make quantifiable emissions reduction commitments has led to the introduction of intensity based emissions targets, where reductions in emissions are specified with reference to some measure of economic output. The Copenhagen commitments of China and India are two prominent examples. Intensity targets substantially increase the complexity of policy simulation and analysis, because a given emissions intensity target could be satisfied with a range of emissions and output combinations. Here, a simple algorithm, the Iterative Method, is proposed for an energy economic model to find a unique policy solution that achieves an emissions intensity target at minimum economic loss. We prove the mathematical properties of the algorithm, and compare its numerical performance with other methods’ in the existing literature.

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Notes

  1. See e.g., Lu et al. (2013) for a thorough literature and policy review.

  2. In economic modelling, closure is the decision of which variables are exogenous and which are endogenous.

  3. See http://www.gams.com.

  4. See http://www.monash.edu.au/policy/gempack.htm.

  5. Please see Table 1 on page 1166 and Figure 2 on page 1168 of Lu et al. (2013) for a derivation of this using the EIA International Energy Statistics.

  6. This is more than three times of the carbon tax in Australia from July 2012 to June 2014 or about ten times of the current EU Allowance price.

  7. This target is derived in Table 1 on page 1166 and Figure 2 on page 1168 of Lu et al. (2013) using the EIA International Energy Statistics.

  8. These carbon prices are within range found by Lu et al. (2013).

  9. The source codes of the script are available on request.

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Authors

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Correspondence to Yiyong Cai or Yingying Lu.

Additional information

This study was undertaken while Yiyong Cai worked with the Australian National University.

Appendix: Algorithm properties

Appendix: Algorithm properties

In this appendix, we will show how our Iterative Method works to find that achieves the emissions intensity target while minimizing the economic cost. For future reference, let us define the auxiliary function \(\hat{{E}}\) such that \(\hat{{E}}(\tau )=\varepsilon \times G(\tau )\), and note that a policy solution exists where \(\hat{{E}}(\tau )=\varepsilon \times G(\tau )=E(\tau )\). As the auxilary function \(\hat{{E}}\) only differes from G by a scaler, \(\hat{{E}}\) has the same properties as G does. In other words, Assumption 3 is equivalent to

Assumption 5

The auxilary function \(\hat{{E}}\) is continuous. It satisfies: (1) \(\hat{{E}}(\tau )>\hat{{E}}(\tau _m )\) for all \(\tau \in [0,\tau _m )\) ; and (2) \(\hat{{E}}\) is weakly decreasing on \([\tau _m ,\infty )\).

Similarly, Assumption 4 is equivalent to

Assumption 6

\(E(0)>\hat{{E}}(0).\)

Figure 6 contains examples where Assumptions 1, 2, 5 and 6 are satisfied and Fig. 7 contains examples where at least one of the assumptions are violated. More specifically, in Fig. 6, Example a corresponds to the scenario where only one policy solution to the emissions intensity target exists; Example b corrresponds to the scenario where there are two solutions; Example c corresponds to the scenario in which the rebound effects discussed above are prominent and the carbon price is sufficiently low. It shall be shown that the algorithm works for all three examples. In contrast, in Fig. 7, Example a violates Assumption 1 as there is no policy solution to the emissions intensity target; Example b violates Assumption 5 which unrealistically states that the carbon price will continue to increase GDP over quite a wide range; and Example c violates Assumption 6 because the BAU emission intensity is lower than the target and thus no mitigation is needed.

Fig. 6
figure 6

Examples where the algorithm can be applied

Fig. 7
figure 7

Examples where the algorithm cannot be applied

Our proposed algorithm is characterised by the following properties:

Property 1

For any \(k\ge 0\) and \(0\le \tau _k <\tau _m ,\, \tau _{k+1} \ge \tau _k\).

Proof

Suppose for a contradiction that \(\tau _{k+1} <\tau _k \). Then by Assumption 2,

$$\begin{aligned} \hat{{E}}(\tau _k )=E(\tau _{k+1} )\ge E(\tau _k ) \end{aligned}$$

Since \(E(0)>\hat{{E}}(0)\), if \(\hat{{E}}(\tau _k )>E(\tau _k )\), then by continuity there exists \(\tau _a \in [0,\tau _m )\) such that \(E(\tau _a )=\hat{{E}}(\tau _a )\), contradicting the definition of \(\tau _m \); if \(\hat{{E}}(\tau _k )=E(\tau _k )\), then \(\tau _k \) is a solution and by definition of \(\tau _m \) we have \(\tau _k \ge \tau _m \), again a contradiction. \(\square \)

Property 2

For any \(k\ge 0 \)   and    \(0\le \tau _k <\tau _m,\,\tau _{k+1} \le \tau _m\).

Proof

Suppose for a contradiction that \(\tau _{k+1} >\tau _m \). Then by Assumptions 2 and 5,

$$\begin{aligned} \hat{{E}}(\tau _k )=E(\tau _{k+1} )\le E(\tau _m )=\hat{{E}}(\tau _m )<\hat{{E}}(\tau _k ) \end{aligned}$$

which is a contradiction. \(\square \)

Property 3

Let \(\tau _0 =0\). The sequence \((\tau _k )\) converges to \(\tau _m\).

Proof

This follows readily from Properties 1 and 2, and the Monotone Convergence Theorem. \(\square \)

Altogether, Properties 13 ensures that our proposed algorithm will solve for \(\tau _m \). By Assumption 3, \(G(\tau _m )\) is the highest possible GDP outcome that is compatible with the emissions intensity target. In other words, the economic loss, as measured by the deviation in GDP from the BAU projection, is minimized.

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Cai, Y., Lu, Y., Stegman, A. et al. Simulating emissions intensity targets with energy economic models: algorithm and application. Ann Oper Res 255, 141–155 (2017). https://doi.org/10.1007/s10479-015-1927-0

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