We propose and explore financial instruments supporting programs for reducing emissions from deforestation and forest degradation (FI-REDD). Within a microeconomic framework we model interactions between an electricity producer (EP), electricity consumer (EC), and forest owner (FO). To keep their profit at a maximum, the EP responds to increasing \(\hbox {CO}_2\) prices by adjusting electricity quantities generated by different technologies and charging a higher electricity price to the EC. The EP can prepare for future high (uncertain) \(\hbox {CO}_2\) prices by employing FI-REDD: they can purchase an amount of offsets under an unknown future \(\hbox {CO}_2\) price and later, when the \(\hbox {CO}_2\) price is discovered, decide how many of these offsets to use for actually offsetting emissions and sell the rest on the market, sharing the revenue with the FO. FI-REDD allows for optional consumption of emission offsets by the EP (any amount up to the initially contracted volume is allowed), and includes a benefit-sharing mechanism between the EP and FO as it regards unused offsets. The modeling results indicate that FI-REDD might help avoid bankruptcy of \(\hbox {CO}_2\)-intensive producers at high levels of \(\hbox {CO}_2\) prices and therefore serve as a stabilizing mechanism during the transition of energy systems to greener technologies. The analytical results demonstrate the limits for potential market size explained by existing uncertainties. We illustrated that when suppliers and consumers of REDD offsets have asymmetric information on future \(\hbox {CO}_2\) prices, benefit-sharing increases the contracted REDD offsets quantity.
1 Introduction
This paper is devoted to the elaboration of financial instruments supporting the reduced emissions from deforestation and degradation (REDD+) mechanisms [24, 30]. Over 500 REDD+ pilot projects were initiated worldwide in the decade following the United Nations framework convention on climate change (UNFCCC) negotiations [12]. The total global pledge for REDD+ between 2006 and 2014 was estimated at USD 9.8 billion [11]. Only 10 % of this amount comes from voluntary carbon market with the majority funded by multilateral and bilateral donors (public sector) including the World Bank, the UN REDD initiative, and the Norwegian government [12, 29]. The EU and its Member States provided EUR 3 billion in support of REDD+ activities, through a wide range of thematic areas, countries and funding channels [11]. Most of this donor funding is intended for development of governing capacity, which is necessary to properly manage REDD+ funding [12].
REDD is a relatively low-cost mitigation option [6, 24], and its integration in the global mitigation strategy has a potential to allow for larger emissions reductions and a lower overall abatement cost [18]. This integration can be done by linking REDD as an emission reduction credit program to major cap-and-trade programs [2]. For example, [34] marks the growing linkages between the world’s cap-and-trade systems for GHGs, both directly between systems, and indirectly via connections to credit systems such as the clean development mechanism. However, there is still an ongoing discussion related to uncertainties and risks in REDD implementation. It is difficult to anticipate the combined effects on carbon and other co-benefits owing to the disparity between the activities available under the REDD program [7, 22]. Accepting this uncertainty, we explore the relation between REDD supplier and GHG-emitting energy producer in the context of a potentially emerging REDD offsets market.
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In the papers [13, 39] the price-taking electricity producer’s decision-making consists of choosing between investing in research and development (R&D) to implement new technologies (carbon capture and storage (CCS) modules) and buying REDD options. Our approach differs in several ways. Firstly, we consider the case when the energy producer has market power (see, e.g. [16, 17])—the ability to reduce the production output and charge higher electricity prices to consumers. Thus, in the face of uncertain \(\hbox {CO}_2\) prices, the electricity producer with market power has more flexibility compared to the price-taking energy producer. We, therefore, consider a homogenous product oligopoly model, in which each firm is maximizing its unilateral profits against the residual demand curve constructed using the equilibrium actions chosen by its competitors [42]. In this setting, the residual demand function faced by a single firm gives the difference between the market demand and the amount that firm’s competitors are willing to supply at each price. Secondly, the electricity producer in our model is a medium-term decision maker: they do not change their technology portfolio by decommissioning \(\hbox {CO}_2\)-intensive plants and building new power plants (which would be a long-term investment). The optimization model works with two time steps: initial (low) \(\hbox {CO}_2\) price and future (uncertain) \(\hbox {CO}_2\) price. Generally, both investments in power generation capacity and forest management are long-term issues. However, in our analysis we focus specifically on medium-term, where the transition phase to higher \(\hbox {CO}_2\) prices and REDD offsets acceptance might happen. The problem of REDD in the long-term forest management is very complex due to uncertainties in estimating the baseline and sustainable forest management impacts. For example, there are many methodological challenges related to forest definition, additionality, leakage and permanence [35, 40]. In our study we do not focus on these technical aspects of REDD implementation. In our model REDD offsets are traded bilaterally between the REDD supplier (forest owner) and consumer (electricity producer) in the “first” period when the future \(\hbox {CO}_2\) prices are uncertain. We assume the acceptance of REDD credits for emission offsetting in the future (“second” period when the \(\hbox {CO}_2\) price reveals), so that the contracted amount of REDD offsets (in tons of \(\hbox {CO}_2\)) can be used for emissions offsetting. We assume that in the “first” period the forest owner can supply the amount of REDD offsets sufficient to cover potential emissions of the electricity producer in the “second” period. In this setup the parties choose their fair prices of REDD offsets in the “first” period—before the \(\hbox {CO}_2\) price reveals in the “second” period.
This simplified rather conceptual modeling approach is justified because a dynamic model would require additional information about the future which is not available at the moment: \(\hbox {CO}_2\) price formation process, REDD offsets acceptance on the market, etc. For this same reason we focus on the direct contracting of REDD offsets between the forest owner and electricity producer, and do not consider market modeling. We constructed a microeconomic model of interaction between the forest owner (REDD-supplier), electricity producer, and electricity consumer. \(\hbox {CO}_2\) prices are exogenous in the proposed partial equilibrium modeling framework. The decision-making process of the electricity producer (under condition of existing or absent \(\hbox {CO}_2\) tax/price) consists of: (1) choosing power plant load factors to minimize the cost given the hourly electricity demand profile and installed capacities of particular power generation technologies; and (2) choosing an electricity price to maximize the profit based on the demand function indicating consumer sensitivity to electricity prices (see, e.g., [25, 38]). Here we apply a constant elasticity demand function [4] as a residual demand function in the oligopoly model.
The elevating \(\hbox {CO}_2\) price might impact not only the profits of the electricity producer (which decrease), but also the electricity prices for the consumer (which increase) [20]. Hence, some financial instruments may be implemented today in order to be prepared for uncertain \(\hbox {CO}_2\) prices in the future. We propose and explore financial instruments supporting the REDD program. On the supply side of the REDD-based emission offsets we model a forest owner who decides to preserve the forest and sell respectively generated REDD-based emission offsets (further—REDD offsets). The focus of our analysis is how the forest owner and the electricity producer evaluate their fair prices for different amounts of REDD offsets. In the paper, the “fairness” of the price is understood in the sense of each parties’ indifference to engaging in contracting a given amount of REDD offsets. The fair price of the electricity producer (forest owner) means that for higher (lower) prices the electricity producer (forest owner) will not want to engage in the contract. Where both parties can agree on the fair price, the problem is then to find the maximum amount of REDD offsets which can be contracted. A similar approach in a different problem setup is considered in the paper [37], where the authors developed a newsvendor model to determine the optimal price and volume of CCS contracts to maximize the expected profit of a storage operator. Forest carbon credits come with inherent risk—carbon storage activities may fail due to leakage, forest fires, and increasing profitability of using the land for alternative activities [35]. Here we do not discuss these risks in detail, assuming that they are either reflected in the offsets price uncertainty or/and are insured.
The general idea of benefit-sharing is important within the REDD context [9, 23]. Here we consider specifically a situation where benefits are shared between the REDD supplier and consumer. Technically, a similar mechanism design, including the sharing ratio, is implemented in various studies applied to problems other than REDD and based on different modeling approaches, see e.g. parameter \(\lambda \) in [5] applied to the problem of social control of government expenditures and parameter \(\alpha \) in [28] applied to the contracts in nuclear industry. The benefit-sharing scenarios in which certain shares of the project’s revenues are earmarked for the local government and injected back into the regional economy are considered in [10]. The benefit-sharing concept is also relevant in the international law context [27].
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We propose a benefit-sharing mechanism that is activated when the electricity producer emits less than the amount of REDD offsets contracted in the first period (without \(\hbox {CO}_2\) price); in this case the unused amount of REDD offsets is shared with the forest owner in the second period. We show that for this benefit-sharing mechanism there is an equilibrium amount of REDD offsets up to which the fair prices coincide, meaning that the deal takes place. We prove that for larger amounts of REDD offsets the desired price of the electricity producer (buyer) is lower than the price of the forest owner (seller), meaning that for these larger amounts the deal is not possible. The paper considers mathematical constructions and properties of the proposed financial instrument. Analytical results presented in the paper are illustrated by a numerical case study based on realistic data for regional electricity production. The modeling results indicate that financial instruments supporting REDD might help avoid bankruptcy of \(\hbox {CO}_2\)-intensive producers at high \(\hbox {CO}_2\) price levels and, therefore, serve as a stabilizing mechanism during the transition of energy systems to greener technologies.
2 Modeling framework
In this section, firstly, we present a model of an electricity producer with market power operating without contracting REDD offsets. The decision-making of the electricity producer consists in choosing a technological mix in order to meet the hourly demand and to maximize profit. In this framework the optimal response in terms of the emissions reduction and raising electricity price can be constructed for any \(\hbox {CO}_2\) price. Secondly, we introduce a two-period model for REDD offsets contracting. In the first period (“today”) there is no \(\hbox {CO}_2\) price and the electricity producer sells the optimal amount of electricity (maximizing their profit). This quantity is generated by an optimal technological mix. A corresponding \(\hbox {CO}_2\) quantity is emitted, and an equilibrium electricity price is charged to consumers. In the second period (“tomorrow”), the uncertain \(\hbox {CO}_2\) price appears. Given the electricity producer’s technological capacity and market power, they can solve an optimization problem for potential \(\hbox {CO}_2\) prices. In [20] we show that in response to growing \(\hbox {CO}_2\) price, the profit decreases and electricity price increases. Therefore uncertain \(\hbox {CO}_2\) prices bear risk of profit loss for the electricity producer. Profit losses are higher for higher \(\hbox {CO}_2\) prices. This leads to a consideration of the possibility to contract REDD offsets today in order to offset emissions tomorrow. Assuming a fixed amount of contracted REDD offsets in the first period, and given distribution of uncertain \(\hbox {CO}_2\) price in the second period, the electricity producer solves the optimization problem (choosing technological mix for each \(\hbox {CO}_2\) price realization in the second period) with two options: either (1) to emit more than the available REDD offsets, or (2) to emit less and, hence, sell the excess of offsets on the market, sharing the benefit with the forest owner. The second option is considered under a benefit-sharing ratio given in the first period. Based on the comparison of the second-period expected profits with and without contracting REDD offsets, the electricity producer evaluates their fair (indifference) price for each amount of offsets in the first period. Thus, our study deals with the utility indifference pricing similar to [14].
The forest owner—the seller of REDD offsets—calculates their fair price in the first period by comparing their expected profits with and without selling REDD offsets to the electricity producer in the first period. In the latter case the forest owner sells the offsets only in the second period at the realized market price. The forest owner knows the electricity producer’s best response in the second period in terms of their utilization of REDD offsets for actual offsetting versus the option of partly reselling them on the market. This best response is expressed in the offsets amounts shared in the second period under a given amount of REDD offsets contracted in the first period and benefit-sharing ratio fixed in the first period. The focus of the study is on construction of the fair prices for a range of REDD-based offsets in order to find the quantities which can be contracted (i.e. those quantities for which the seller’s price does not exceed the buyer’s price). We also look at how the benefit-sharing ratio can impact contracted amounts.
2.1 Notations
In our model the electricity producer uses n technologies that vary in costs (US$/MWh, excluding emission costs) and emission factors (ton of \(\hbox {CO}_2\)/MWh). Let us introduce the following notations:
\(a_i, i=1,\ldots ,n\) are installed capacities (MW);
\(v_i\) are variable costs (US$/MWh);
\(d_j, j=1,\ldots ,24\) is hourly average demand (MW);
\(\varvec{x}=\{x_{ij}\}, i= 1,\ldots ,n, \, j=1,\ldots ,24\), is a matrix of hourly load factors (controls, ratio between 0 and 1);
\(\varvec{q}(\varvec{x})=(q_1,\ldots , q_{24})=\{ \sum _{i=1}^{n} a_i x_{ij} \}\) is a vector of hourly outputs (MWh);
\(Q =Q(\varvec{x})= \sum _{i=1}^n a_i \sum _{j=1}^{24} x_{ij}\) is aggregate daily production (MWh);
\(P^e\) is electricity price (US$/MWh);
\(D^{-1}: P^e=D^{-1}(Q)\) is inverse demand function (see Sect. 4.1);
\(\varepsilon _i\) are emission factors (ton of \(\hbox {CO}_2\)/MWh);
p is \(\hbox {CO}_2\) price (US$/ton of \(\hbox {CO}_2\)).
For each matrix of load factors \(\varvec{x}\) the profit of the electricity producer in the absence of \(\hbox {CO}_2\) price is calculated as follows:
We will assume that the \(\hbox {CO}_2\) price belongs to a segment \(p \in [0, \tilde{p}]\). Let us note that the profit component \(\Pi _e\) and emissions E do not directly depend on price p, however, they are indirectly determined by the technological possibilities of the electricity producer.
We assume that the hourly profile changes proportionally to the aggregate demand (see [20] and Sect. 4.1 for details) and introduce the feasibility domain \(\varvec{X}\), which contains all technological mixes (controls) satisfying the hourly demand:
where \(\varvec{d^0}=(d^0_1,\ldots ,d^0_{24})\) and \(Q^0\) are, respectively, the initial hourly and daily aggregate demands (at zero \(\hbox {CO}_2\) price).
In our modeling framework we consider the electricity producer as a profit-maximizing decision maker. The profit maximization problem is formulated as follows.
Problem 1
(without REDD offsets) Given the feasibility domain \(\varvec{X}\) (6), the electricity producer chooses technological mix \(\varvec{x}\) maximizing their profit (5) at a \(\hbox {CO}_2\) price p:
Let us denote a solution to Problem 1—the optimal technological mix—by the symbol \(\varvec{x^*_1}=\varvec{x^*_1}(p)\) for any price \(p \in [0, \tilde{p}]\). Then, by definition of \(\varvec{x^*_1}\) for any \(\varvec{x} \in \varvec{X}\) (6) the following inequality holds:
The corresponding electricity price is calculated as \(P^e(Q(\varvec{x^*_1}(p))\).
2.2 First period: no carbon price
We assume, that in the first period (“today”) there is no \(\hbox {CO}_2\) price. Hence, Problem 1 is solved for \(p=0\). The electricity producer chooses an optimal technological mix \(\varvec{x^*_1}(0)\), and gets the corresponding profit \(\hat{\Pi }(0)\), by charging electricity price \(P^e=D(Q^0)\) (see (6)). Simultaneously, they emit the amount \(E^0=E(\varvec{x^*_1}(0))\) of \(\hbox {CO}_2\). The electricity producer envisions uncertain \(\hbox {CO}_2\) prices in the second period (“tomorrow”).
2.3 Second period without REDD offsets: assumptions for modeling
In the second period the electricity producer can solve Problem 1 for any potential \(\hbox {CO}_2\) price. In our study we assume the following properties of optimal profit \(\hat{\Pi }(p)\) (9) and emissions \(\hat{E}(p)=E(\varvec{x^*_1}(p))\) with respect to \(\hbox {CO}_2\) price.
Assumption 1
The optimal profit and optimal emissions achieve their maxima at zero \(\hbox {CO}_2\) price, \(p=0\), and are continuous strictly declining functions with respect to growing p:
This assumption is straightforward in the provided modeling framework [20]. It is also consistent with results of larger scale modeling [31].
Remark 1
Under assumption 1 for every \(\hbox {CO}_2\) price \(p \in [0, \tilde{p}]\) there exists a unique emissions level \(\hat{E}(p) = E(\varvec{x^*_1}(p))\) corresponding to maximum profit \(\hat{\Pi }(p)\).
Remark 2
Assumption 1 basically restricts the consideration of electricity producers to those unfavorably (negatively) affected by an emerging \(\hbox {CO}_2\) price. Those who can potentially benefit from it, e.g. due to a competitive advantage, are not considered here. This situation is beyond the scope of this paper, which is focused on the problem of \(\hbox {CO}_2\)-intensive power generation.
Based on Assumption 1 we prove the following lemma.
Lemma 1
For any \(\varvec{x} \in \varvec{X}\) (6), such that \(E(\varvec{x}) \ne E(\varvec{x^*_1}(p))\), the following inequality holds for all \(p \in (0, \tilde{p}]\):
Lemma 1 has the following meaning. If we fix \(\hbox {CO}_2\) price p and select an arbitrary mix of technologies \(\varvec{x}\) satisfying the hourly demand, such that the corresponding emissions differ from optimal emissions for the price p, then this mix \(\varvec{x}\) is not optimal for the electricity producer in the sense of profit maximization.
2.3.1 Modeling \(\hbox {CO}_2\) price uncertainty
Let the future \(\hbox {CO}_2\) price be an uncertain variable [33] following a discrete probability distribution:
where \(w_l\) stands for probability, and realizations of possible prices \(p_i \ne p_j\), if \(i\ne j\).
In our model we assume that the electricity producer and forest owner are both risk neutral and, therefore, we deal with expected values. Given the distribution (12), the electricity producer calculates their expected utility in the second period (without REDD):
High \(\hbox {CO}_2\) price decreases the profit of the electricity producer. The emitter can prepare for possibly high future \(\hbox {CO}_2\) price by contracting REDD offsets before the information about \(\hbox {CO}_2\) price is revealed; contracted REDD offsets would allow offsetting \(\hbox {CO}_2\) emissions in the future with forest owners supplying REDD offsets. Let us note that we are not taking into account additional factors in the payoff of the forest owner, e.g. the opportunity of deforesting and selling the wood. We assume that the forest owner decided to keep the forest for generating REDD offsets.
A problem is divided into two stages: at the first stage forest owner and electricity producer assign their prices for an amount \(\mathcal {E}\in (0, E^0]\) of REDD offsets. Here \(E^0\) is the maximum amount of emissions—generated by the electricity producer at zero \(\hbox {CO}_2\) price, i.e. \(E^0=\hat{E}(0)\). If the forest owner does not make contracts with the electricity producer in the first period, for any amount \(\mathcal {E}\) their expected utility in the second period is calculated as follows:
where \(\Pi ^0_F=\Pi ^0_F(p_l)\) is their profit in the second without contracting the electricity producer in the first period, and \(\bar{p}\) is the mean of distribution (12). Expression (14) means keeping all the offsets in the first period, and selling them in the second period at a market price.
2.4 Second period with REDD offsets
At the second stage they face a realization of uncertain \(\hbox {CO}_2\) price. At each realization of the \(\hbox {CO}_2\) price the electricity producer can either use all REDD offsets (by emitting more or equal to the previously contracted amount \(\mathcal {E}\)), or emit less than \(\mathcal {E}\) and share the benefit with the forest owner from selling the rest (unused offsets) in the market (at a market price p).
Benefit-sharing mechanism The electricity producer and forest owner, when selling offsets on the market, get shares of the market price \(\delta \) and \((1 - \delta )\) respectively, so that:
If \(\delta =1\), the electricity producer has the right to sell the offsets in the second period at a market price without sharing the profit with forest owner.
If \(\delta =0\), the electricity producer can only use the contracted REDD credits to offset the factual amount of their emissions and the unused credits are returned (without compensation) back to the forest owner, i.e. no resale by the electricity producer is possible on the market. The profit from unused offsets goes entirely to the forest owner.
If \(0<\delta <1\), the electricity producer faces a trade-off between emitting more and, hence, using more of the contracted REDD credits for offsetting their emissions versus sharing the profit with the forest owner from selling the offsets at the market price.
The benefit-sharing ratio\(\delta \) is included in the evaluation process of offsets amount \(\mathcal {E}\) by the REDD offsets supplier (forest owner) and consumer (electricity producer) in the first period. Namely, for a given \(\delta \) and \(\mathcal {E}\) in the first period, the forest owner and electricity producer choose their fair prices, based on the electricity producer’s decision-making in the second period with REDD. This is illustrated in Fig. 1.
Fig. 1
Illustration of the problem setting: potentially contracted amount of offsets \(\mathcal {E}\) and realization of the uncertain \(\hbox {CO}_2\) price. Given \(\mathcal {E}\) and benefit-sharing ratio \(\delta \) the electricity producer optimizes their technological mix in order to maximize their profits in the second period. This optimization determines whether they share REDD offsets with the forest owner, and, if yes, the amount of shared offsets. This best response in the second period is used by parties to evaluate the fair prices (23), (24) for the amount \(\mathcal {E}\) (under a given \(\delta \)) in the first period
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We assume that the forest owner and electricity producer face the same \(\hbox {CO}_2\) price distribution. The presence of REDD offsets at the second stage of the model leads to the following modification in the decision-making problem of the electricity producer compared to the case without REDD (see Problem 1).
Problem 2
(with REDD offsets) Given the feasibility domain \(\varvec{X}\) (6), \(\hbox {CO}_2\) price distribution \(\{p_l,w_l\}\) (12), benefit-sharing ratio \(\delta \in [0,1]\) and amount of REDD offsets \(\mathcal {E} \in (0,E^0]\) contracted in the first time period the electricity producer chooses technological mix \(\varvec{x}\) maximizing their profit at a \(\hbox {CO}_2\) price realization \(p_l\) in the second time period:
Here \([y]_+=\max \{y,0 \}\), meaning that the electricity producer can offset their emissions up to the amount \(\mathcal {E}\) by using REDD offsets, the rest is sold on the market and the profit is shared with the forest owner.
The optimal technological mix \(\varvec{x^*_2}(p_l)\)—solution to (15)—generates the maximum profit with REDD: \(\hat{\Pi }^R(p_l)=\Pi ^R(\varvec{x^*_2}(p_l), p_l)\) at a particular \(\hbox {CO}_2\) price \(p_l\). We denote by the symbol \(E^R (p_l)=\big [\mathcal {E}- E(\varvec{x^*_2}(p_l)) \big ]_{+}\) the corresponding amount of emission offsets, shared with the forest owner in the second period. The corresponding expected values are calculated as follows:
Here \(E^R (p_l)\) is the best response of the electricity producer in the second period, which is known to the forest owner in the first period. This allows to calculate their expected profit, when contracting REDD:
Here \(\Pi ^R_F\) denotes the forest owner’s profit, when dealing with the electricity producer in the first period. Note that profit of the electricity producer, \(\hat{\Pi }^R\), with REDD in the second period does not include the cost of offsets paid to the forest owner in the first period. The profit of the forest owner, \(\Pi ^R_F\), with REDD in the second period also does not include the benefit from selling offsets to the electricity producer in the first period. We will introduce these costs (benefits) below, when defining the fair prices for REDD offsets.
2.5 Fair prices of REDD offsets
The discussion below is devoted to the valuation of various amounts of REDD offsets contracted in the first time period under unknown \(\hbox {CO}_2\) price assuming the given distribution (12) and benefit-sharing ratio \(\delta \in [0,1]\). The forest owner and electricity producer evaluate their fair (indifference) prices for the given amount of options. Thus, the forest owner chooses their fair price \(p_F\) at which they can sell the amount of offsets \(\mathcal {E}\) in the first period using the utility indifference condition:
meaning that the forest owner can either sell the emission offsets to the electricity producer in the first period and, possibly, get a profit share in the second period, or keep the offsets and sell them in the second period at the market \(\hbox {CO}_2\) price without engaging in a deal with the electricity producer. Substituting (18) and (14) to (19), we obtain the following equation with respect to \(p_F\):
For the given \(\hbox {CO}_2\) price distribution \(\{p_l, w_l\}, l=1,\ldots ,m\) (12), benefit-sharing ratio \(\delta \in [0,1]\) and amount of offsets \(\mathcal {E} \in (0, E^0]\) based on Eqs. (20), (22) we derive the fair prices of the forest owner \(p_F\) and the electricity producer \(p_E\):
For a fixed parameter \(\delta \in [0,1]\) Eqs. (23), (24) represent supply and demand curves for REDD offsets within the suggested benefit-sharing approach. The following analysis is based on the decision-making of the electricity producer with REDD offsets in the second period. We find their fair price \(p_E\) (24) in the first period for a given \(\delta \) and \(\mathcal {E}\). The forest owner knows the best response of the electricity producer in terms of \(E^R(p_l)\). Based on this information, they calculate their fair price, \(p_F\), in the first period according to Eq. (23).
2.6 Decision-making with REDD offsets, benefit-sharing mechanism, and known realization of \(\hbox {CO}_2\) price in the second time period
In order to analyze the behavior of the electricity producer let us consider a certain realization of \(\hbox {CO}_2\) price \(p=p_l\) in the second period. Technically, Problem 2 can be split into two alternative profit-maximizing tasks. In short, they represent two alternatives: to share, or not to share.
Problem 3
(\(E(\varvec{x}) \ge \mathcal {E}\)) Given the feasibility domain \(\varvec{X}\) (6) and the amount of REDD offsets \(\mathcal {E} \in (0, E^0]\), find technological mix \(\varvec{x}\) maximizing the profit at a \(\hbox {CO}_2\) price realization p:
Let us denote the solution to Problem 3 by the symbol \(\varvec{x^*_3} = \varvec{x^*_3}(p) \in \varvec{X_3}\). The corresponding maximum profit is given by the relation:
(\(E(\varvec{x}) \le \mathcal {E}\)) Given the feasibility domain \(\varvec{X}\) (6), benefit-sharing ratio \(\delta \in [0,1]\) and the amount of REDD offsets \(\mathcal {E} \in (0, E^0]\), find technological mix \(\varvec{x}\) maximizing the profit at a \(\hbox {CO}_2\) price realization p:
Let us denote the solution to Problem 4 by the symbol \(\varvec{x^*_4}=\varvec{x^*_4}(p) \in \varvec{X_4}\). The corresponding maximum profit is given by the relation:
Thus, for any fixed amount \(\mathcal {E} \in (0, E^0]\) available in the second period, the electricity producer chooses the best response to \(\hbox {CO}_2\) price \(p=p_l\) in terms of profit-maximization—between \(\hat{\Pi }_3^R\) (28) and \(\hat{\Pi }_4^R\) (32):
The described optimization alternatives as possibilities for the electricity producer in our two-stage model are illustrated in Fig. 1.
3 Analytical results
In this section we analytically find the maximum profit (33) of the electricity producer depending on the amount of REDD offsets \(\mathcal {E} \in (0,E^0]\) and determine the corresponding fair prices of the forest owner and electricity producer. This allows us to obtain an estimate of the amount of REDD offsets that can be contracted.
We introduce the following function of \(\hbox {CO}_2\) price p:
For any \(\hbox {CO}_2\) price realization \(p \in [0, \tilde{p}]\) in the second period, any benefit-sharing ratio \(\delta \in [0,1]\), and any fixed amount \(\mathcal {E}\) of offsets contracted in the first period such that \(\mathcal {E} \in (0, E(\varvec{x^*_1}(p))]\), the maximum profit with REDD (33) in the second period is calculated as follows:
$$\begin{aligned} \hat{\Pi }^R(p) = \hat{\Pi }^R_3 = \Pi _e(\varvec{x^*_1}(p)) - p E(\varvec{x^*_1}(p)) + p \mathcal {E}. \end{aligned}$$
The definition of Lemma 2 is that for any realization of future \(\hbox {CO}_2\) price p in the second period the optimal technological mix, solving Problem 2 with REDD, is the same as in Problem 1 without REDD, i.e. \(\varvec{x^*_2}(p) =\varvec{x^*_1}(p)\), provided that the offsets amount (contracted in the first period) does not exceed the optimal quantity of emissions (without REDD) for that \(\hbox {CO}_2\) price p (in the second period), \(\mathcal {E} \le E(\varvec{x^*_1}(p))\).
Corollary 1
If conditions of Lemma 2 are satisfied at a \(\hbox {CO}_2\) price realization \(p=p_l\) in the distribution (12), then \(E^R(p)=0\) and according to \(\hat{\Pi }(p)\) (9), \(\hat{\Pi }^R(p)\) (39), \(\phi (p)\) (37), \(\xi (p)\) (35), the equality takes place:
For any \(\hbox {CO}_2\) price realization \(p \in [0, \tilde{p}]\) in the second period, any benefit-sharing \(\delta \in [0, 1)\), and any amount \(\mathcal {E}\) of offsets contracted in the first period such that \(\mathcal {E} > E(\varvec{x_1^*}(p))\), the following inequality takes place:
Finally, let us formulate and prove the following theorem.
Theorem 1
For a given \(\hbox {CO}_2\) price distribution \(\{p_l, w_l\}, l=1,\ldots ,m\) (12) and for any benefit-sharing ratio \(\delta \in [0, 1)\) there exists an amount \(\tilde{\mathcal {E}} \in (0, E^0]\) of REDD offsets up to which the fair prices of the forest owner \(p_F\) (23) and of the electricity producer \(p_E\) (24) coincide and are equal to the expected \(\hbox {CO}_2\) price \(\bar{p}\). This amount equals the minimum optimal quantity of emissions generated by the electricity producer (in the second period without REDD offsets) at the maximum possible \(\hbox {CO}_2\) price \(\tilde{p} = \max \{p_l\}\):
For any amount of REDD offsets larger than \(\tilde{\mathcal {E}}\) (43) the fair price of the forest owner \(p_F\) is higher than the fair price of the electricity producer \(p_E\):
Theorem 1 shows that in the case of a bounded \(\hbox {CO}_2\) price distribution, the forest owner and electricity producer can contract any amount \(\mathcal {E}\in (0, \tilde{\mathcal {E}}]\) of REDD offsets for the fair price \(\bar{p}\). Thus, in the considered risk-neutral case, only two characteristics of distribution fully determine the solution to the problem: the mean and the highest price.
The practical consequence following from this main result is that—on one hand—the contracted amount is limited by the potentially high future \(\hbox {CO}_2\) price (the higher the price, the lower is the contracted amount). On the other hand, even with a potentially high \(\hbox {CO}_2\) price, the contracted amount is non-zero, hinting at possible implementation of the REDD-based offset instrument featuring a benefit-sharing approach as considered in this paper.
4 Modeling results
Analytical results obtained in the previous section are valid for a broad range of possible model setups in our modeling framework. In order to provide a numerical example and illustrate the impact of a contracted amount of REDD offsets on the profit distribution of the electricity producer, we calibrate the model for a realistic case-study, and carry out numeric optimization.
4.1 Data and calibration
Technologies in the model In our illustrative case study a regional electricity producer is operating power plants with the following technologies: coal (pulverized coal steam), combustion turbine (natural gas-fired) and combined cycle gas turbine (CCGT) (see [25]). The corresponding fixed and variable costs, as well as the installed capacities are given in Table 1. The total size of installed capacity (7900 MW) is chosen to illustrate a model at a regional scale, and is roughly equivalent to the installed capacity of Belarus.1
Average hourly electricity demand To construct an economically efficient production plan the electricity producer has to determine the combination of technologies to be used hourly during the day in order to satisfy the hourly demand profile. A hypothetical demand profile for an average day of the year is depicted in Fig. 2. It features the same shape (peaks) as the regional profiles provided in the literature [1, 3]. The hourly demand values are scaled to match the installed capacity of the electricity producer (as in Table 1). Similar to [1] we use the hourly average demand for each day over a longer period, e.g. one year. This simplification allows us to link the hourly profile with the aggregate demand. We estimate the hourly profile change assuming that a change in aggregate demand leads to the proportional shifts in every hour of the profile for an average day. Our model is working with an average demand profile at the annual scale and provides a higher level of abstraction than the unit commitment (UC) problem (see e.g. [32]).
Fig. 2
Average hourly electricity demand
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Demand function We assume that the electricity producer has market power in the region, and use a constant elasticity demand curve, that is commonly employed in aggregate energy demand studies [4, 21]. The consumers respond to the change in electricity price \(P^e\) by changing the consumption Q according to an aggregate demand function \(D(P^e)\), i.e.:
where \(A>0\) is a constant, and \(\alpha <0\) is the constant elasticity of demand. The coefficients of the aggregate demand function in our model are calibrated in such a way that a realistic electricity price (close to European2 electricity price) is achieved as a solution to Problem 1. The estimated parameters of the demand function (45) are \(A = 1.05 \times 10^5\), \(\alpha = -0.612\). These values are consistent with \(P^e = 90.5 \, \hbox {US} \$ { / } \mathrm{MWh}\) at profit’s maximum without \(\hbox {CO}_2\) price. The value of elasticity coefficient \(\epsilon _d=\frac{1}{\alpha }= -1.63\) is within a plausible range as estimated in the literature (for a set of OECD countries it was found to be within the confidence interval of \(-2.3,\ldots ,-0.1\), see, e.g. [21]). In our example the profit maximizing quantity is \(Q^0 = 100.47 \, \hbox {GWh}\) (which is approximately equal to the average daily electricity consumption in Belarus3), and the corresponding profit is \(\hat{\Pi }(Q^0) = 3.56 \, \hbox {mln. US} { \$} \).
Emissions factors For presently operating, coal-fired power plants the cummulative emissions range between 950 and 1250 g\(\hbox {CO}_2\) eq/kWh [41]. In our study we use a value from an indicated interval given in Table 1. Emissions factors for gas powered plants are taken from [36].
4.2 Numerical results
Simulations were carried out for the discrete (nine points) approximation of a uniform price distribution within the range 0–80 US$/ton of \(\hbox {CO}_2\):
Fair prices of the electricity producer (EP) and forest owner (FO) depending on the volume of REDD offsets. Benefit-sharing ratio is \(\delta =0.5\), and future \(\hbox {CO}_2\) price distribution is uniform within the range 0–80 US$/ton \(\hbox {CO}_2\)
×
Sizes of REDD-based offset contracts used in the model are within the range \([0, E^0]\), where \(E^0\) is the optimal emissions without \(\hbox {CO}_2\) price. In Fig. 3 the fair prices (23), (24) with respect to the contracted amount of offsets \(\mathcal {E} \le E^0\) are depicted for the benefit-sharing ratio \(\delta =0.5\). The plot demonstrates that the maximum amount of emissions offsets for which the deal can take place is \(\tilde{\mathcal {E}}=E(\varvec{x^*}(p_9))=10.93~\hbox {MtCO}_2\). That amount the electricity producer emits at the maximum \(\hbox {CO}_2\) price \(p_9=80\) US$/ton \(\hbox {CO}_2\), while maximizing their profit. For amounts larger than \(\tilde{\mathcal {E}}\) the fair price of the forest owner is higher than the fair price of the electricity producer. This is consistent with analytical results of the paper.
Fig. 4
Profit of the electricity producer (distribution) without contracting REDD offsets and with contracting REDD offsets for the optimal volume (\(\tilde{\mathcal {E}}=10.93~\hbox {MtCO}_2\)) and benefit-sharing ratio \(\delta =0.5\)
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In Fig. 4 we show how the contracted amount \(\tilde{\mathcal {E}}\) impacts the profit distribution of the electricity producer. The plot shows that the mean profit determined by the considered \(\hbox {CO}_2\) price distribution stays the same with REDD offsets (compared to the case without them), according to indifference condition (22). However, entering into a REDD offsets contract does impact the profit distribution. Firstly, according to Theorem 1, the fair price of the amount \(\tilde{\mathcal {E}}\) is \(p_E=\bar{p}\). Secondly, at each price realization, Lemma 2 is valid, meaning that, profits with REDD and without REDD are calculated according to Eq. (39). Including the cost of offsets, \(p_E \tilde{\mathcal {E}}\), in the first period, we get the net profit with REDD at each price realization in the second period:
This analytical expression is reflected in Fig. 4. The profits coincide at the mean price, which is \(\bar{p}=40\) US$/ton \(\hbox {CO}_2\); the profit with REDD is lower than the profit without REDD for price realizations \(p<\bar{p}\), and higher—for price realizations \(p>\bar{p}\). Notably, REDD offsets help the electricity producer to be better off for higher \(\hbox {CO}_2\) prices and almost double their profit in this case.
4.3 The role of the benefit-sharing mechanism
In this section we provide a numerical illustration of how the benefit-sharing mechanism determined by the parameter \(\delta \) can impact the amount of contracted REDD offsets. Even though the benefit-sharing concept is at the core of the suggested construction, it remains inactive if electricity producer and forest owner use one common \(\hbox {CO}_2\) price distribution in their decision-making. Let us explore a situation when the electricity producer and forest owner perceive the \(\hbox {CO}_2\) price distribution asymmetrically. For instance, they could put different weights \(w_l\) in (12) for the same values of \(\hbox {CO}_2\) prices \(p_l\). For example, the forest owner may expect the distribution as in (46), while the electricity producer may put more weight on higher prices, i.e. their distribution can be:
The difference between (46) and (48) can be interpreted as the electricity producer is more sensitive to larger profit losses (risk-averse). In the case of price distribution (48) (risk-averseness), the contracted amount \(\tilde{\mathcal {E}}\) of the electricity producer is higher as they are willing to pay a higher fair price compared to the case with the distribution (46). Figure 5 demonstrates how the benefit-sharing ratio \(\delta \) impacts the amount of contracted offsets \(\tilde{\mathcal {E}}\): for larger parameter \(\delta \), the contracted amount \(\tilde{\mathcal {E}}(\delta )\) is larger. In this example for \(\delta =0.5\) the contracted amount of REDD offsets has increased considerably compared to no benefit-sharing, \(\delta =0\). This preliminary analysis shows that the benefit-sharing mechanism has a potential to increase the volume of REDD contracts. Note that in Fig. 5 the total sum of the deal is growing despite the lower price, i.e. about 30 % increase in REDD financing can be achieved.
Fig. 5
Impacts of benefit-sharing on the contracted amounts of REDD-based offsets under asymmetric information on the \(\hbox {CO}_2\) price distribution. Dashed lines are fair prices of the forest owner (FO) and electricity producer (EP) for benefit-sharing ratio \(\delta =0.5\), solid lines—for \(\delta =0\) (no benefit-sharing)
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5 Discussion and conclusions
A considerable total share of emissions is derived from the energy sector (see, e.g., [8]). Therefore in developing a fair mechanism for REDD it is important to understand the decision-making process (optimal behavior) of energy producers. Our model deals with medium-term planning of the electricity producer who possesses flexibility in their responses to uncertain \(\hbox {CO}_2\) prices. The electricity producer in the model is restricted in exercising market power (charging the electricity price) by the elasticity of residual demand and maximizes their profit by optimizing technological mixes in the production. The analytical results provided in the paper are based only on the assumption that with growing \(\hbox {CO}_2\) price, optimal profit and corresponding emissions are strictly declining functions. The problem of optimal usage of REDD offsets by the electricity producer is formalized in the two-period model with an uncertain \(\hbox {CO}_2\) price.
The valuation of REDD offsets by the forest owner (seller) and electricity producer (buyer) based on their fair (indifference) conditions has several important implications. We show that when there is no profit sharing mechanism and the forest owner can use and resell all the offsets traded at the first stage, then the agreement can be made for any amount of REDD offsets at the mean \(\hbox {CO}_2\) price. This, however, would imply a high level of certainty about the future \(\hbox {CO}_2\) price, so that the risk for the buyer (possible lower price) and risk for the seller (possible higher price) are acceptable in the sense that the risk-neutral approach is applicable. In the case of benefit-sharing coming from selling the unused offsets on the market and sharing the profit between the forest owner and electricity producer, we analytically prove two results. Firstly, there exist amounts of REDD offsets for which the fair prices coincide and are equal to the mean \(\hbox {CO}_2\) price. The maximum contracted amount corresponds to the minimum amount of emissions generated by the electricity producer—at the maximum expected \(\hbox {CO}_2\) price. Secondly, for larger amounts the fair price of the electricity producer is lower than the fair price of the forest owner and, therefore, these amounts are not contracted. This fact is not straightforward in the scenario when the electricity producer can share the profit with the forest owner, and the analytical proof is given in Lemma 3 (see Case 2 in Appendix A.3). This means that someone will lose if the forest owner agrees to sell a “larger” amount of REDD offsets at the price suggested by the electricity producer compared to a situation when not making this deal. In our problem setting two characteristics of the expected \(\hbox {CO}_2\) price distribution are important for contracting REDD offsets—the mean \(\hbox {CO}_2\) price, and the maximum expected \(\hbox {CO}_2\) price.
We illustrated the impact of contracted REDD offsets on profit distribution of the electricity producer in the numerical example based on realistic data. The modeling results indicate that contracted REDD offsets might help avoid bankruptcy of \(\hbox {CO}_2\)-intensive producers at high levels of \(\hbox {CO}_2\) price and therefore serve as a stabilizing mechanism during the transition of energy systems to greener technologies.
The idea behind the benefit-sharing mechanism that we suggested to use, is to reduce the offsets price in the first period providing more favorable conditions for the electricity producer and keeping the flexibility for additional income for the forest owner (as additional benefits might come later in the second period). As we basically explore a contract between two parties, from the legal perspective, we could imagine a trustee keeping the offsets and making sure the benefits are shared. There are also other challenges related to institutional options, e.g. monitoring, reporting and verification for REDD+ programs [15]. However these aspects are clearly beyond the scope of our study.
The current partial equilibrium model connecting \(\hbox {CO}_2\)-intensive energy sector with REDD creates the necessary prerequisites for modeling REDD in a wider context. A possible model expansion could be connected with merging the current framework with the detailed analysis of the supply side for REDD, i.e. a sustainable forest management model, where risks associated with ecological hazards, e.g. forest fires [19, 26], could be adequately implemented. Finally, institutional grounds could be defined in connection with these research studies.
In the presented modeling exercise, we would like to highlight a situation of asymmetric \(\hbox {CO}_2\) price distributions. It provides a preliminary illustration of the positive role of a benefit-sharing mechanism by increasing contracted amounts of REDD offsets, as well as the volume of a deal. This mechanism has a potential for initiating the direct contracting of REDD offsets, as it provides more flexibility to both parties: energy sector and forestry (lowering initial price for the former and keeping the option for additional benefits for the latter). Let us note, that there is an actual possibility for small-scale implementation of such mechanism, e.g. through the currently active Verified Carbon Standard (VCS) system,4 which is a known developer of standards for REDD projects.
Acknowledgments
Open access funding provided by International Institute for Applied Systems Analysis (IIASA). This research has received financial support from the project “Options Market and Risk-Reduction Tools for REDD+” funded by the Norwegian Agency for Development Cooperation under agreement number QZA-0464 QZA-13/0074, by the project “Delivering Incentives to End Deforestation: Global Ambition, Private/Public Finance, and Zero-Deforestation Supply Chains” funded by the Norwegian Agency for Development Cooperation under agreement number QZA-0464 QZA-16/0218, and was also supported by the EU’s Seventh Framework Programme (FP7) under Grant Agreement No. 603906 (ECONADAPT). We thank anonymous reviewers for their useful comments and suggestions as well as Ruben Lubowski, Alexander Golub, Matthew Cantele, Cynthia Festin, Johannes Pirker, and Hugo Valin for discussions, reviews, and helpful comments.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Secondly, let us assume on the contrary that for some \(\bar{\varvec{x}} \in \varvec{X}\), such that \(E(\bar{\varvec{x}}) \ne E(\varvec{x^*_1}(p))\), relation (49) is equality. Then we have:
Firstly, by the condition of the lemma \(E(\varvec{x^*_1}(p)) \ge \mathcal {E}\), meaning that \(\varvec{x^*_1}(p) \in \varvec{X_3}\) (27). Secondly, (25) is equivalent to (7) and, hence, solution to Problem 3 coincides with the solution to Problem 1: \(\varvec{x^*_3}=\varvec{x^*_1}(p)\). To complete the proof we need to show that: \(\hat{\Pi }^R_3 \ge \hat{\Pi }^R_4\) in (33).
Indeed, using (28), (32) and definition of maximum at price p (8), for \(\varvec{x^*_4} \in \varvec{X_4} \subset \varvec{X}\) we come to the following chain of inequalities:
$$\begin{aligned} \hat{\Pi }^R _3 (p)= & {} \Pi _e(\varvec{x^*_3}) - p (E(\varvec{x^*_3})-\mathcal {E}) = \Pi _e(\varvec{x^*_1}(p)) - p E(\varvec{x^*_1}(p)) + p \mathcal {E} \nonumber \\\ge & {} \Pi _e(\varvec{x^*_4}) - p E(\varvec{x^*_4}) + p \mathcal {E} = \Pi _e(\varvec{x^*_4}) - p (E(\varvec{x^*_4}) - \mathcal {E}) \nonumber \\\ge & {} \Pi _e(\varvec{x^*_4}) - \delta p (E(\varvec{x^*_4}) -\mathcal {E}) = \hat{\Pi }_4^R (p). \end{aligned}$$
(52)
This relation means that \(\hat{\Pi }^R _3 (p) \ge \hat{\Pi }^R _4 (p)\) if \(\delta =1\), and \(\hat{\Pi }^R _3 (p) > \hat{\Pi }^R _4 (p)\) if \(\delta \in [0,1)\). Thus,
Let us consider two cases depending on the optimal profit in (33).
Case 1\(\hat{\Pi }^R(p) = \hat{\Pi }_3^R\), meaning that the electricity producer does not share emission offsets with the forest owner and emits \(E(\varvec{x^*_3}) \ge \mathcal {E}\). In this case \(E^R(p)=0\), meaning that:
For \(\varvec{x^*_3} \in \varvec{X}\) such that \(E(\varvec{x^*_3}) = \mathcal {E} > E(\varvec{x^*_1}(p))\) we can apply Lemma 1. Hence, (55) leads to the required inequality:
Case 2\(\hat{\Pi }^R(p) = \hat{\Pi }_4^R\), meaning that the electricity producer can share emission offsets. Let us find the optimal technological mix \(\varvec{x^*_4}\)—solution to Problem 4. Problem (29)–(31) is equivalent to the following problem (see (5), (7)):
Case 2a\(\mathcal {E} \ge E(\varvec{x^*_1}(\delta p))\), meaning that the contracted amount of offsets is larger than optimal emissions at the \(\hbox {CO}_2\) price \(\delta p\). In this case \(\varvec{x^*_1}(\delta p) \in \varvec{X_4}\) (31), and as it is the solution to (57), we have:
For the optimal mix \(\varvec{x^*_4} \in \varvec{X}\) such that (59) is true one can apply Lemma 1:
$$\begin{aligned} \Pi _e(\varvec{x^*_1}(p))-E(\varvec{x^*_1} (p))p > \Pi _e(\varvec{x^*_4})- E(\varvec{x^*_4}) p. \end{aligned}$$
(62)
Substitution of (62) to (61) gives the required inequality:
$$\begin{aligned} \xi (p)= & {} \delta p + \frac{\Pi _e(\varvec{x^*_4})- E(\varvec{x^*_4}) \delta p - \big ( \Pi _e(\varvec{x^*_1}(p))-E(\varvec{x^*_1}(p))p \big )}{\mathcal {E}} \nonumber \\< & {} \delta p + \frac{\Pi _e(\varvec{x^*_4})- E(\varvec{x^*_4}) \delta p - \big ( \Pi _e(\varvec{x^*_4})-E(\varvec{x^*_4})p \big )}{\mathcal {E}}\nonumber \\= & {} \delta p + \frac{(1-\delta ) p E(\varvec{x^*_4})}{\mathcal {E}} = \phi (p). \end{aligned}$$
(63)
Case 2b\(\mathcal {E} < E(\varvec{x^*_1}(\delta p))\), meaning that the contracted amount is less than optimal emissions at price \(\delta p\). In this case \(\varvec{x^*_1}(\delta p) \notin \varvec{X_4}\) (31). According to Assumption 1 there exist price \(\hat{p}\), \(\delta p< \hat{p} < p\), and technological mix \(\varvec{x^*_1}(\hat{p})\) (see Fig. 6), such that:
Below we show that technological mix \(\varvec{x^*_4}=\varvec{x^*_1}(\hat{p})\) is the solution to Problem 4 in this case. Let us take a technological mix \(\varvec{\tilde{x}_4} \in \varvec{X_4}\) (31) different from \(\varvec{x^*_1}(\hat{p})\) (64). As \(E(\varvec{\tilde{x}_4}) \le E(\varvec{x^*_1}(\hat{p}))\), the following inequality holds:
We have proved that in Case 2b the electricity producer does not return any offsets to the forest owner and uses the whole amount \(\mathcal {E}\). Thus, in this case \(\hat{\Pi }^R(p)=\hat{\Pi }^R_3=\hat{\Pi }^R_4\) and this situation is the same as in Case 1. Thus, we have proved that in all cases:
Optimal emissions of the electricity producer with respect to \(\hbox {CO}_2\) price (horizontal axis) without REDD offsets. A conceptual graph of \(\hat{E}(p)\), satisfying Assumption 1
According to Assumption 1 the amount \(\tilde{\mathcal {E}}\) (43) is emitted by the electricity producer at any price \(p_l\) in the distribution (12). Hence, for every \(p=p_l\) in the distribution the conditions of Lemma 2 are true, meaning that according to Corollary 1:
$$\begin{aligned} \phi (p)=\xi (p)= p. \end{aligned}$$
(71)
Substituting (71) to definitions of fair prices (36), (38) we get:
The same reasoning is valid for any \(\mathcal {E} \in (0, \tilde{\mathcal {E}}]\), and, hence, (42) is true.
For the amount of REDD offsets \(\mathcal {E} \in (\tilde{\mathcal {E}}, E^0]\) for some \(\hbox {CO}_2\) price realizations \(p=p_l\) in distribution (12) the conditions of Lemma 2 are satisfied and, hence, \(\phi (p_l)=\xi (p_l)\). At the same time, there are price realizations in distribution (12), at which conditions of Lemma 3 are satisfied (at least for the price \(\tilde{p}=\max \{p_l\}\)), meaning that \(\phi (p_l=\tilde{p})>\xi (p_l=\tilde{p})\), and hence:
