# Direct Sourcing from Renewable Energy under Cost Uncertainty: A Bargaining Approach

Authors: Biao Tao; Gangcheng Cao; Tianxu Sun
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

Sourcing electricity from renewable energy suppliers becomes critical to improving renewable energy penetration. However, the common problem of bilateral bargaining between an electricity retailer and renewable energy supplier has rarely been studied in the literature. Considering the uncertainty of renewable generation cost, we introduce the incentive contract theory and signalling game to examine how the electricity procurement contract is designed in a Rubinstein bargaining framework. We derive the corresponding equilibrium outcomes depending on the renewable supplier’s generation cost uncertainty. The results show that if the possibility of a high generation cost is large, the retailer provides contracts for the renewable supplier of both high and low generation costs to achieve the simultaneous separating equilibrium; otherwise, he only proposes the contract for supplier with a low generation cost to achieve the sequential separating bargaining. Our work demonstrates that the proposed incentive contracts urge the renewable supplier to reveal the private information of generation cost and prevent the retailer’s profit deviation due to the adverse selection.


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## Body

## 1. Introduction

The high dependence on fossil fuels urges the energy sector to occupy the largest share of global carbon emissions, the major threat to global warming [1]. Facing this climate challenge, promoting the penetration of renewable energy prevails in different countries for its effectiveness in improving the energy supply structure and reducing the reliance on fossil fuels [2]. For instance, China commits that renewable energy will account for over 25% of its power consumption by 2030; the European Union also plans to boost renewable consumption by over 32% by 2030 [3].Generally, the electricity market plays an essential role in enhancing renewable energy consumption. The bilateral bargaining between electricity retailers and suppliers is one of two primary transaction methods in the electricity market [4]. In particular, the volume of bilateral electricity contracts traded has steadily increased and dominated as the global electricity system reform deepens. For example, over 90% of electricity is traded in the forward market of the United Kingdom [5]. During the negotiation, the wholesale electricity prices in the contracts are usually determined by the market participants’ hedging of risk exposures [6], which is quite relevant to the average generation costs. Note that the increasing renewable generation is also believed to reduce the electricity price since the marginal cost of the renewable generator is fewer than the conventional generator [7, 8]. This marginal cost-based view of renewable energy indeed impacts the price in the electricity market due to the merit-order clearance; however, it loses sight of the high intermittentness of renewable output; for example, the wind power’s capacity availability varies with daily standard deviations of up to 25% [9]. Therefore, as a crucial factor impressing the wholesale price, the uncertainty of the renewable average generation cost should be considered during contract negotiation between an electricity retailer and renewable supplier and worth studies but remains vague. Moreover, challenged by the electricity market balancing requirements (e.g., China establishes a strict electricity deviation penalty), the optimal energy procurement of electricity sales company needs to fully consider the purchasing costs of energy sources in order to maximize their own benefits [10]. Hence, it is intriguing to examine the problem of how the contract negotiation proceeds between an electricity retailer and renewable supplier in the context of the Rubinstein bargaining when faced with the implementation of market deviation assessment and the uncertainty of renewable energy generating costs.Nevertheless, most of the existing research on renewable energy power procurement has been conducted from the perspective of power generation companies, and the research on optimal energy procurement of power sales companies mainly focuses on the aspects of demand response [11], risk assessment [12], and market pricing [13], using research methods such as stochastic programming [14] and conditional value at risk [15], with less concentration on the bilateral bargaining between power sales companies and generation companies in the transaction process. Driven by the above discussion, our work explores the bilateral contracts between an electricity retailer and renewable supplier in the context of a Rubinstein bargaining game, where the decision variables include the quantity of electricity purchased and the wholesale price. The information asymmetry incurred by the uncertainty of renewable costs is considered, and different equilibrium outcomes of different parameter values are derived. We derive two equilibria, the simultaneous separating equilibrium and sequential separating equilibrium, and prove the dominated equilibrium under different conditions. The main contributions of our work can be summarized as follows:(i)
In the proposed bargaining mechanism, the information asymmetry of renewable generation cost can be disseminated through the designed incentive contracts, and the incurred adverse selection will also be prevented.(ii)
Our work provides a novel approach to solve the bilateral bargaining between an electricity retailer and renewable supplier, which bridges the gap in the literature of discussion on the bargaining in the electricity market. It may assist the electricity retailer when negotiating with the renewable energy supplier.The rest of our study is organized as follows. Section2 reviews the related literature, and Section 3 describes the model framework. We analyze the equilibrium outcomes under three scenarios, free screen, free signal, and cost signal, and provide the optimal contract menus for electricity retailer in Section 4. Section 5 concludes the paper.


## 2. Literature Review

In the electricity market literature, some attentions have been paid to studying the optimal energy procurement for the electricity retailer. Bahramara et al. [16] consider the strategic behavior of an electricity sales company in wholesale energy and reserve markets from the perspective of a price maker and model its energy procurement as a bi-level optimization problem. Moghimi and Barforoushi [17] construct a bi-level model to study the optimal sell and purchase strategies for a strategic power retailer in the energy and retail markets. Sun et al. [18] propose the optimal power procurement of power sales companies based on conditional value-at-risk assessment and evaluate the impact of proportional distribution of different trading models on the profits of power sales companies. Their results find the effectiveness of time of use and real-time pricing in affecting both the scale of electricity purchases and the expected profit of the electricity retailer. Do Prado and Qiao [19] establish a trading mechanism for short-term demand response (DR) between customers and an electricity retailer with self-production of renewable energy and illustrate its effectiveness through case studies. Do Prado and Qiao [20] also construct a short-term energy procurement model for an electricity retailer in a liberalized distributed renewable energy market through a bi-level stochastic programming. The stochastic day-ahead and real-time market prices and electricity demand are considered in the simulation cases to prove the effectiveness of their model. Simoglou et al. [21] formulate a novel optimization problem for the electricity retailer’s long-term electricity procurement problem. Their model considers different asset options and inherent technical and operating characteristics, and the retailer’s long-term strategy is formulated by a case study in the Greek market.Game theory is also customarily applied to study the complex purchasing process. Green [22] models the electricity retailing and competition in the wholesale market and points out the wholesale prices would be raised if the long-term contracts are undercut by the short-term contract. Azad and Ghotbi [23] model the participants’ strategic behaviors in the electricity market and calculate the clearing electricity price with a high penetration of small- and mid-sized renewable suppliers. Golpîra et al. [24] study an electricity supply chain coordination framework through the newsvendor model. Given retailers’ attitude towards the risk associated with the demand uncertainty and the aim of maximizing profits, they provide an optimal contract framework considering the overage and underage costs, and a discount policy to define the contract share and prices. Peura and Bunn [3] propose a game-theoretical model to study the impacts of intermittently wind power on prices in the forward market. They confirm that the contract prices can paradoxically increase with the generation capacity due to its uncertainty of output. However, the studies concentrating on how contracts are negotiated in the electricity market are limited, which is widely adopted as the default transaction method in the electricity market. Li et al. [25] formulate a cooperative Stackelberg game model in the retail market to coordinate peer-to-peer (P2P) energy trading among multiple prosumers. Anderson and Philpott [26] are the only publication that firstly considers the bilateral forward electricity contract negotiated between two firms with respective private information on the spot price. Their work defines equilibrium in the context of Nash bargaining and shows the influence of expectations of future prices on the contract outcomes.Our research also falls into the stream of literature discussing the bargaining issue over incentive contracts. Inderst [27] designs a bargaining game allowing both parties to propose menus. He proves that the bargaining game has a unique equilibrium for a subset of parameters where efficient contracts are implemented in the first period. Sen [28] analyzes the contract bargaining issue in the framework of alternative offers. His work characterizes a set of equilibrium outcomes under different parameter values. Yao [29] goes further based on Sen’s results. His paper introduces a strategic delay into the bargaining process and derives the least cost separating equilibrium by comparative analysis. Dittrich and Städter [30] further discuss a bargaining issue over an incentive-compatible contract in a moral hazard framework. Julien and Roger [31] design an ex-post bidding process during an bilateral meeting on an incentive contract. The equilibrium in their results is not constrained by welfare optimality due to the contracting risk. In line with this literature, we introduce the Rubinstein bargaining process to study the incentive contract over uncertain generation costs of the renewable supplier. To our best knowledge, our paper firstly introduces the approach of Rubinstein bargaining to evaluate the contract negotiation under information asymmetry in the electricity market.


## 3. Model

Consider a bilateral Rubinstein negotiation composed of an electricity retailer (ER, he) and a renewable supplier (RS, she). The ER and RS bargain over quantityq and the wholesale price w in the contract mq,w. Two companies make alternative offers, and ER makes the first offer by proposing a menu of contracts, mhqh,wh and mlql,wl, where mhqh,wh represents the contract designed for the high-type RS and mlql,wl represents the contract designed for the low-type RS The RS is free to choose one contract in the menu if she accepts or proposes her counteroffer of contracts if she rejects. The bargain proceeds until one participant commits an acceptance, and the other side cannot alter his or her offer until the next counteroffer is made. An illustration of the bargaining sequence is also presented in Figure 1. Both firms have the same discount factor δ, δ∈0,1, and wish to reach an agreement without delay. Following the literature, Admati and Perry [32] and Yao [29], we only consider pure strategies and equilibrium concept in our analysis is the perfect Bayesian equilibrium, in which the strategies are optimal under beliefs updated by Bayes’ rules.Figure 1
Illustration of ER and RS’s bargaining sequence.We denote the average generation costs of RS asc, which is her private information and cannot be accessed by ER. ER only holds a prior belief of c that c=ch with probability ε and small c=cl with probability 1−ε, where ch and cl represent a high generation cost and low generation cost, respectively. For ease of description, we call the RS with a high average cost as the high-type RS and the RS with a low average cost as the low-type RS. We also assume that the electricity retailing price of ER is a constant p, and the demand is a random variable X with a cumulative density function FX and a probability density function fx, where fx>0 for all x>0 and fx=0 otherwise. The demand distribution is assumed to have an increasing generalized failure rate (IGFR), which is a weak requirement satisfied by many distributions [33, 34]. Hence, we have hx=fx/1−Fx,h′x>0, and the demand for the electricity retailer is mq=Eminx,q.The real-time balance between electricity supply and demand is essential in the electricity system, and hence, a deviation penalty mechanism is usually applied to prompt the electricity sales company’s ability to forecast the actual electricity consumption. We denote the penalty coefficient asK, and the utility function of the electricity retailer is as follows:(1)πB=pq−∫0qFxdx−wq−K∫0qq−xfxdx.The payoffs for the RS are as follows:(2)πG=w−cq.Hence, the payoff function for the entire electricity supply chain is as follows:(3)π=pq−∫0qFxdx−cq−K∫0qq−xfxdx.Solving the first-order condition of (3), we obtain the first best quantity as follows:(4)q∗=F−1p−cp+k.To simplify the calculation, we assume that the demandX uniformly distributes between 0 and 1, thus:(5)q∗=p−cp+K.Substituting (5) into (3) and (1), we obtain the profits for the entire chain, and desired quantity in the contract is as follows:(6)mq∗=q∗−12q∗2,π¯=p−ch22p+K,π¯=p−cl22p+K.For ease of description, we also summarize all the parameters in Table1.Table 1
Parameter description.
ParameterDescriptionRemarkscThe production cost of the renewable supplierHigh cost expressed asch, and low cost expressed as cl.εProbability of the high-type renewable supplier1−εProbability of the low-type renewable supplierpElectricity retailing company’s marginal utility of purchasing one unit of electricityδBargaining patience coefficient (discount factor)KPunishment coefficient in the deviation punishment mechanismπBProfits of electricity retailing companyπGProfits of the renewable supplierπProfits of the entire supply chainThe profits are expressed asπ¯ when the production cost is high, and the profits are expressed as π¯ when the production cost is low.qPurchasing quantity in the contractThe quantity of the high-type contract isqh, and the quantity of the low-type contract purchases is ql.wThe wholesale price in the contractThe wholesale price of the high-type contract is expressed aswh, and the wholesale price of the low-type contract is expressed as wl.mq,wPower purchase contract representationThe high-type contract is expressed asmhqh,wh, and the low-type contract is expressed as mlql,wl.


## 4. Bargaining Process

### 4.1. Bargaining under Complete Information

According to Rubinstein [35], if two parties alternatively propose offers under complete information, the optimal contracts mq∗qh∗,wh∗ and ml∗ql∗,wl∗ can be directly derived in the following Table 2.Table 2
Optimal contractsmq∗qh∗,wh∗ and ml∗ql∗,wl∗.
ContractPayoffs of ERPayoffs of RSQuantityWholesale pricemq∗qh∗,wh∗πBh∗=1/1+δπ¯πGh∗=δ/1+δπ¯qh∗=p−ch/p+Kwh∗=δ/1+δπ¯/qh∗+chml∗ql∗,wl∗πBl∗=1/1+δπ¯πGl∗=δ/1+δπ¯ql∗=p−cl/p+Kwl∗=δ/1+δπ¯/ql∗+clSince this bargaining game is under complete information, the optimal quantities in the contracts are the first best results in (5) and the payoffs for the respective party are divided as the unique subgame perfect equilibrium in Rubinstein [35], where the ER gets 1/1+δ share and RS gets δ/1+δ share.


### 4.2. Bargaining under Asymmetric Information

We now investigate the scenario of asymmetric information. The uncertainty of the ER about the generation cost of RS leads to information asymmetry between buyers and sellers, and therefore, information screening and signalling problems arise in the bargaining process. Since ER does not have private information, he needs to design a set of incentive contracts to screen RS’s cost information when he makes an offer. In our framework, the ER may offer two different types of contract to induce the RS to make its own choice, or he may offer only one contract for the RS to expose her type. Next, in the counteroffer period, there are two options for RS that she may choose to send a true signal to ER by proposing a contract of her true type or muddle by sending a contract to mimic her type, thereby gaining a higher return.Intuitively, the high-type RS has no incentive to mimic her type because the imitation profits are negative,qlcl−ch<0. Hence, the remaining question is whether the low-type RS will mimic her type or not since the imitation profits are positive, qhch−cl>0, which may potentially increase her payoffs. In particular, for the low-type RS, the imitation profits are the accumulation of the share of the total surplus π¯ and qhch−cl if she mimics a high type. Note that the imitation payoffs for RS vary in different periods, and the variations are presented in Table 3.Table 3
Imitation payoffs for RS vary in different period.
Game stageThe actual profits of sending a false signalThe actual profits of sending a real signal (low-type profits)Imitation profitsHigh-type profits1st stageqhch−clδ/1+δπ¯δ/1+δπ¯2nd stageqhch−cl1/1+δπ¯1/1+δπ¯Therefore, comparing the profits for the low-type RS in the first stage, we can obtain that there is no incentive for the low-type RS to mimic, if the imitation profits in the first stage are inferior to the profits under complete information. We summarize the findings in Lemma1.Lemma 1.
If condition C1,qhch−cl<δ/1+δπ¯−π¯, is always satisfied, the game is naturally separated, in which the ER can screen the RS’s type by offering the optimal contracts in Table 2 and the RS will accept the contract based on her true type. The expected payoffs for ER are πB∗=ε1/1+δπ¯+1−ε1/1+δπ¯.Lemma1 indicates a free screen case, where the ER discerns the RS’s type without any profit deviation. This case is also discussed in the literature, such as Inderst [27] and Yao [29], which reveals that information asymmetry does not deteriorate both the total surplus and respective earnings since the outcome is the same as the complete information equilibrium.Then, if C1 is violated, the low-type RS earns more if she mimics, resulting in the free screen being impossible. However, the game moves to the next section where a case of free signalling takes place. Although the low-type RS earns more by imitating in the first stage, her type will be truthfully transited through contracts in the second stage. As such, we derive condition C2,δ/1+δπ¯−π¯≤qhch−cl<1/1+δπ¯−π¯, and characterize the equilibrium in the following Lemma 2 if C2 holds.Lemma 2.
When condition C2 holds, there exist two equilibria in the game, including a simultaneous separating equilibriumI1 and a sequential separating equilibrium E1. In the equilibrium I1, the ER proposes a set of contracts, mh1∗qh1∗,wh1∗ and ml1∗ql1∗,wl1∗, and the expected profits for ER are as follows:(7)πBI1=επh1∗−δ1+δπ¯+1−επ¯−δ1+δπ¯−qh1∗ch−cl,where πh1∗=Pqh1∗−1/2qh1∗2−chqh1∗−K1/2qh1∗2, and qh1∗ is the optimal quantity when C2 holds.In the equilibriumE1, the ER only proposes a contract ml∗ql∗,wl∗ in the first stage, and the low-type RS will accept it, but the high-type RS will send a counteroffer in the second stage. The expected payoffs for ER are as follows:(8)πBE1=εδ21+δπ¯+1−ε11+δπ¯.All the proofs are given in the appendix.Lemma2 illustrates two potential equilibria in the bargaining when C2 holds. The ER has two options once condition C1 is violated: one option is to costly screen the RS’s type through a menu of contracts in the first stage and the RS will choose a contract corresponding to her type; the other option is to only deliver a contract ml∗ql∗,wl∗, in which the low-type RS will accept in the first stage and wait for the high-type RS freely signal her type in the second stage. Thus, by comparing equations (7) and (8), we can obtain the dominated equilibrium in the game. Proposition 1 summarizes our findings.Proposition 1.
When condition C2 holds, there exists aε1,ε1∈0,1, where the simultaneous separating equilibrium I1 is dominated in the game if ε≥ε1, and the sequential separating equilibrium E1 is dominated if ε<ε1.The results in Proposition1 are intuitive since the ER’s decision is driven by the goal of payoff maximization. When ε is small, ER will design a contract only for the low-type RS to exclude the possibility of imitation and the high-type RS. Otherwise, ER will continue costly screen through a set of contracts in the first stage if the ε is significant enough. Note that no more refinements are required since alternative offers in bargaining provide continuation payoffs; the dominated equilibrium is a perfect Bayesian equilibrium in the game.Then, we concentrate on the case when both conditions C1 and C2 are violated. In this scenario, the low-type RS has an incentive to mimic high type in both stages, and the game moves to the costly signal situation, in which the high-type RS has to costly transit her type. To study this costly signalling game, we further assume that the ER holds his prior belief in the second stage. There exist typically two types of equilibria in the costly signal cases, the separating equilibrium and the pooling equilibrium. In the first, the RS proposes a distinct contract to reveal her generation cost truthfully; in the second, the RS proposes the same contract regardless of her type. However, according to the intuitive criterion, it is evident that only the low type wants to pool with the high type, the high-type RS does not want, and the ER would easily reject the pooling offer. We characterize the refinement of equilibrium in Lemma3.Lemma 3.
There is no pooling equilibrium, and only the least costly equilibrium survives in the game. (The least costly separating equilibrium describes the equilibrium where the firms distort their quantities in the contract to obtain the least deterioration in payoffs compared with complete information.)Moreover, since the pooling equilibrium cannot exist, the aforementioned sequential separating equilibriumE1 still exists, in which the ER offers a low-type contract. As such, another possible equilibrium remains how can the ER screen the RS type in the first stage.To examine this problem, we investigate the second stage. If it is the RS’s turn to make an offer, the low-type RS can easily reveal her generation cost by proposing a contactml∗ql∗,wl∗, but the high-type RS needs to distort the quantities to convince the ER that the low-type RS has no incentive to imitate. Therefore, the high-type RS’s problem is as follows:(9)MaxπG=wh−chqh.Subject to(10)qh−qh22−whqh−K12qh2≥δ1+δπ¯,PC′,11+δπ¯≥wh−clqh,IC′.Binding conditionsPC′ and IC′, we obtain the q^h∗ and π^G:(11)q^h∗=p−cl−δ/1+δp−cl2−p−ch2p+K,π^G=11+δπ¯−q^h∗ch−cl.We are now back to the first stage and solve the ER’s screen contract. The ER designs his purchasing contract by solving the following optimization problem:(12)MaxπB,πB=εpqh−qh22−whqh−K12qh2+1−εpql−ql22−wlql−K12ql2.Subject to(13)qhwh−cl≤qlwl−cl,IC1qlwl−ch≤qhwh−ch,IC2qhwh−ch≥δπ^GPC1′qlwl−cl≥δ1+δπ¯PC2.The incentive constrainsIC1, IC2, and PC2 are constant in equation (13), whereas the distortion in electricity quantity changes the high-type RS’s revenue in the second stage, thus affecting her payoffs in the first stage, which is also her participation constraint.The solutions to the optimization problem depend on the range ofqhch−cl≥δ/1+δπ¯−δπ^G. We summarize the domain of each condition in Figure 2 and characterize the potential equilibrium in Lemma 4.Figure 2
Domain of each condition.Lemma 4.
(1)
When condition C4 holds, there exists a simultaneous separating equilibriumI2, in which the ER offers a contract ml2∗ql2∗,wl2∗ to the low-type RS and proposes a contract mh2∗qh2∗,wh2∗ to the high-type RS. The ER’s expected profits are as follows:(14)πBI2=επh2∗−δπ^G+1−επ¯−δπ^G−qh2∗ch−cl,whereπh2∗=pqh2∗−1/2qh2∗2−chqh2∗−K1/2qh2∗2.(2)
hen condition C5,qhch−cl<δ/1+δπ¯−δπ^G, holds, there exists a sequential separating equilibrium I3, where the ER offers a contract ml3∗ql3∗,wl3∗ to the low-type RS and proposes a contract mh3∗qh3∗,wh3∗ to the high-type RS at the equilibrium. The ER’s expected profits are as follows:(15)πBI3=επh3∗−δπ^G+1−ε11+δπ¯,where πh3∗=pqh3∗−1/2qh3∗2−chqh3∗−K1/2qh3∗2.Lemma4 illustrates the sequential separating equilibria under different conditions. Intuitively, the ER’s expected profits increase with the low-type RS’s imitation payoffs since the ER pays the information rent. However, the RS’s expected profits do not. This scenario implies that private information always benefits the information holder in the Rubinstein bargaining. Moreover, we compare the ER’s expected profits in I2 and E1 and I3 and E1 to obtain the dominated equilibrium of the game under conditions C4 and C5. The results are characterized in Proposition 2 and Proposition 3.Proposition 2.
Under condition C4, there exists aε2,ε2=qh2∗−δq^h∗ch−cl/qh2∗ch−cl+πh2∗−δ/1+δπ¯−δ2/1+δπ¯. If ε≥ε2, the simultaneous equilibrium I2 dominates the game; otherwise, the sequential equilibrium E1 dominates the game.Proposition2 summarizes the dominated equilibrium under condition C4. The ER still trade-offs between paying an information rent to the low-type RS or offering only one contract to exclude the high-type RS in the first stage. Hence, if the possibility of a high-type RS is minor, the sequential equilibrium E1 dominates, but if the possibility of a high-type RS is large, the ER chooses a menu of screen contracts.Proposition 3.
Under condition C5, only the simultaneous equilibriumI3 dominates the game.In Proposition3, the situation differs if C5 holds because the simultaneous equilibrium I3 is strictly larger than the sequential equilibrium E1. Meanwhile, the contract in equilibrium E1 is included in the menu of I3, which indicates that I3 is the unique equilibrium in the bargaining.


## 4.1. Bargaining under Complete Information

According to Rubinstein [35], if two parties alternatively propose offers under complete information, the optimal contracts mq∗qh∗,wh∗ and ml∗ql∗,wl∗ can be directly derived in the following Table 2.Table 2
Optimal contractsmq∗qh∗,wh∗ and ml∗ql∗,wl∗.
ContractPayoffs of ERPayoffs of RSQuantityWholesale pricemq∗qh∗,wh∗πBh∗=1/1+δπ¯πGh∗=δ/1+δπ¯qh∗=p−ch/p+Kwh∗=δ/1+δπ¯/qh∗+chml∗ql∗,wl∗πBl∗=1/1+δπ¯πGl∗=δ/1+δπ¯ql∗=p−cl/p+Kwl∗=δ/1+δπ¯/ql∗+clSince this bargaining game is under complete information, the optimal quantities in the contracts are the first best results in (5) and the payoffs for the respective party are divided as the unique subgame perfect equilibrium in Rubinstein [35], where the ER gets 1/1+δ share and RS gets δ/1+δ share.


## 4.2. Bargaining under Asymmetric Information

We now investigate the scenario of asymmetric information. The uncertainty of the ER about the generation cost of RS leads to information asymmetry between buyers and sellers, and therefore, information screening and signalling problems arise in the bargaining process. Since ER does not have private information, he needs to design a set of incentive contracts to screen RS’s cost information when he makes an offer. In our framework, the ER may offer two different types of contract to induce the RS to make its own choice, or he may offer only one contract for the RS to expose her type. Next, in the counteroffer period, there are two options for RS that she may choose to send a true signal to ER by proposing a contract of her true type or muddle by sending a contract to mimic her type, thereby gaining a higher return.Intuitively, the high-type RS has no incentive to mimic her type because the imitation profits are negative,qlcl−ch<0. Hence, the remaining question is whether the low-type RS will mimic her type or not since the imitation profits are positive, qhch−cl>0, which may potentially increase her payoffs. In particular, for the low-type RS, the imitation profits are the accumulation of the share of the total surplus π¯ and qhch−cl if she mimics a high type. Note that the imitation payoffs for RS vary in different periods, and the variations are presented in Table 3.Table 3
Imitation payoffs for RS vary in different period.
Game stageThe actual profits of sending a false signalThe actual profits of sending a real signal (low-type profits)Imitation profitsHigh-type profits1st stageqhch−clδ/1+δπ¯δ/1+δπ¯2nd stageqhch−cl1/1+δπ¯1/1+δπ¯Therefore, comparing the profits for the low-type RS in the first stage, we can obtain that there is no incentive for the low-type RS to mimic, if the imitation profits in the first stage are inferior to the profits under complete information. We summarize the findings in Lemma1.Lemma 1.
If condition C1,qhch−cl<δ/1+δπ¯−π¯, is always satisfied, the game is naturally separated, in which the ER can screen the RS’s type by offering the optimal contracts in Table 2 and the RS will accept the contract based on her true type. The expected payoffs for ER are πB∗=ε1/1+δπ¯+1−ε1/1+δπ¯.Lemma1 indicates a free screen case, where the ER discerns the RS’s type without any profit deviation. This case is also discussed in the literature, such as Inderst [27] and Yao [29], which reveals that information asymmetry does not deteriorate both the total surplus and respective earnings since the outcome is the same as the complete information equilibrium.Then, if C1 is violated, the low-type RS earns more if she mimics, resulting in the free screen being impossible. However, the game moves to the next section where a case of free signalling takes place. Although the low-type RS earns more by imitating in the first stage, her type will be truthfully transited through contracts in the second stage. As such, we derive condition C2,δ/1+δπ¯−π¯≤qhch−cl<1/1+δπ¯−π¯, and characterize the equilibrium in the following Lemma 2 if C2 holds.Lemma 2.
When condition C2 holds, there exist two equilibria in the game, including a simultaneous separating equilibriumI1 and a sequential separating equilibrium E1. In the equilibrium I1, the ER proposes a set of contracts, mh1∗qh1∗,wh1∗ and ml1∗ql1∗,wl1∗, and the expected profits for ER are as follows:(7)πBI1=επh1∗−δ1+δπ¯+1−επ¯−δ1+δπ¯−qh1∗ch−cl,where πh1∗=Pqh1∗−1/2qh1∗2−chqh1∗−K1/2qh1∗2, and qh1∗ is the optimal quantity when C2 holds.In the equilibriumE1, the ER only proposes a contract ml∗ql∗,wl∗ in the first stage, and the low-type RS will accept it, but the high-type RS will send a counteroffer in the second stage. The expected payoffs for ER are as follows:(8)πBE1=εδ21+δπ¯+1−ε11+δπ¯.All the proofs are given in the appendix.Lemma2 illustrates two potential equilibria in the bargaining when C2 holds. The ER has two options once condition C1 is violated: one option is to costly screen the RS’s type through a menu of contracts in the first stage and the RS will choose a contract corresponding to her type; the other option is to only deliver a contract ml∗ql∗,wl∗, in which the low-type RS will accept in the first stage and wait for the high-type RS freely signal her type in the second stage. Thus, by comparing equations (7) and (8), we can obtain the dominated equilibrium in the game. Proposition 1 summarizes our findings.Proposition 1.
When condition C2 holds, there exists aε1,ε1∈0,1, where the simultaneous separating equilibrium I1 is dominated in the game if ε≥ε1, and the sequential separating equilibrium E1 is dominated if ε<ε1.The results in Proposition1 are intuitive since the ER’s decision is driven by the goal of payoff maximization. When ε is small, ER will design a contract only for the low-type RS to exclude the possibility of imitation and the high-type RS. Otherwise, ER will continue costly screen through a set of contracts in the first stage if the ε is significant enough. Note that no more refinements are required since alternative offers in bargaining provide continuation payoffs; the dominated equilibrium is a perfect Bayesian equilibrium in the game.Then, we concentrate on the case when both conditions C1 and C2 are violated. In this scenario, the low-type RS has an incentive to mimic high type in both stages, and the game moves to the costly signal situation, in which the high-type RS has to costly transit her type. To study this costly signalling game, we further assume that the ER holds his prior belief in the second stage. There exist typically two types of equilibria in the costly signal cases, the separating equilibrium and the pooling equilibrium. In the first, the RS proposes a distinct contract to reveal her generation cost truthfully; in the second, the RS proposes the same contract regardless of her type. However, according to the intuitive criterion, it is evident that only the low type wants to pool with the high type, the high-type RS does not want, and the ER would easily reject the pooling offer. We characterize the refinement of equilibrium in Lemma3.Lemma 3.
There is no pooling equilibrium, and only the least costly equilibrium survives in the game. (The least costly separating equilibrium describes the equilibrium where the firms distort their quantities in the contract to obtain the least deterioration in payoffs compared with complete information.)Moreover, since the pooling equilibrium cannot exist, the aforementioned sequential separating equilibriumE1 still exists, in which the ER offers a low-type contract. As such, another possible equilibrium remains how can the ER screen the RS type in the first stage.To examine this problem, we investigate the second stage. If it is the RS’s turn to make an offer, the low-type RS can easily reveal her generation cost by proposing a contactml∗ql∗,wl∗, but the high-type RS needs to distort the quantities to convince the ER that the low-type RS has no incentive to imitate. Therefore, the high-type RS’s problem is as follows:(9)MaxπG=wh−chqh.Subject to(10)qh−qh22−whqh−K12qh2≥δ1+δπ¯,PC′,11+δπ¯≥wh−clqh,IC′.Binding conditionsPC′ and IC′, we obtain the q^h∗ and π^G:(11)q^h∗=p−cl−δ/1+δp−cl2−p−ch2p+K,π^G=11+δπ¯−q^h∗ch−cl.We are now back to the first stage and solve the ER’s screen contract. The ER designs his purchasing contract by solving the following optimization problem:(12)MaxπB,πB=εpqh−qh22−whqh−K12qh2+1−εpql−ql22−wlql−K12ql2.Subject to(13)qhwh−cl≤qlwl−cl,IC1qlwl−ch≤qhwh−ch,IC2qhwh−ch≥δπ^GPC1′qlwl−cl≥δ1+δπ¯PC2.The incentive constrainsIC1, IC2, and PC2 are constant in equation (13), whereas the distortion in electricity quantity changes the high-type RS’s revenue in the second stage, thus affecting her payoffs in the first stage, which is also her participation constraint.The solutions to the optimization problem depend on the range ofqhch−cl≥δ/1+δπ¯−δπ^G. We summarize the domain of each condition in Figure 2 and characterize the potential equilibrium in Lemma 4.Figure 2
Domain of each condition.Lemma 4.
(1)
When condition C4 holds, there exists a simultaneous separating equilibriumI2, in which the ER offers a contract ml2∗ql2∗,wl2∗ to the low-type RS and proposes a contract mh2∗qh2∗,wh2∗ to the high-type RS. The ER’s expected profits are as follows:(14)πBI2=επh2∗−δπ^G+1−επ¯−δπ^G−qh2∗ch−cl,whereπh2∗=pqh2∗−1/2qh2∗2−chqh2∗−K1/2qh2∗2.(2)
hen condition C5,qhch−cl<δ/1+δπ¯−δπ^G, holds, there exists a sequential separating equilibrium I3, where the ER offers a contract ml3∗ql3∗,wl3∗ to the low-type RS and proposes a contract mh3∗qh3∗,wh3∗ to the high-type RS at the equilibrium. The ER’s expected profits are as follows:(15)πBI3=επh3∗−δπ^G+1−ε11+δπ¯,where πh3∗=pqh3∗−1/2qh3∗2−chqh3∗−K1/2qh3∗2.Lemma4 illustrates the sequential separating equilibria under different conditions. Intuitively, the ER’s expected profits increase with the low-type RS’s imitation payoffs since the ER pays the information rent. However, the RS’s expected profits do not. This scenario implies that private information always benefits the information holder in the Rubinstein bargaining. Moreover, we compare the ER’s expected profits in I2 and E1 and I3 and E1 to obtain the dominated equilibrium of the game under conditions C4 and C5. The results are characterized in Proposition 2 and Proposition 3.Proposition 2.
Under condition C4, there exists aε2,ε2=qh2∗−δq^h∗ch−cl/qh2∗ch−cl+πh2∗−δ/1+δπ¯−δ2/1+δπ¯. If ε≥ε2, the simultaneous equilibrium I2 dominates the game; otherwise, the sequential equilibrium E1 dominates the game.Proposition2 summarizes the dominated equilibrium under condition C4. The ER still trade-offs between paying an information rent to the low-type RS or offering only one contract to exclude the high-type RS in the first stage. Hence, if the possibility of a high-type RS is minor, the sequential equilibrium E1 dominates, but if the possibility of a high-type RS is large, the ER chooses a menu of screen contracts.Proposition 3.
Under condition C5, only the simultaneous equilibriumI3 dominates the game.In Proposition3, the situation differs if C5 holds because the simultaneous equilibrium I3 is strictly larger than the sequential equilibrium E1. Meanwhile, the contract in equilibrium E1 is included in the menu of I3, which indicates that I3 is the unique equilibrium in the bargaining.


## 5. Numerical Study

To intuitively reveal the results in the model, we propose a numerical case in this section to further examine the parameters’ impacts on the incentive contracts. We firstly summarize some example contracts with different parameters of negotiation in Table4.Table 4
Optimal contracts with different parameters.
pchclKεδConditionOptimal contractπB1.110.10.020.50.9C1mh0.09,1.02,ml0.89,0.34πB∗=0.123.110.10.020.50.1C2mh0.38,1.17,ml0.96,0.53πBI1=0.773.910.10.020.90.9C3ml0.97,1.69πBE1=0.51Then, we evaluate how the equilibrium varied with different parameters. We show the equilibriumI1 with parameters p=4,ch=0.5,cl=0.3,K=0.5, and ε=0.5 and stimulate the sensitivity analysis on each parameter.The results in Figure3 clearly demonstrate the negative relations between these four parameters and the expected πb. Then, we also show the sensitivity analysis on the threshold of possibility ε in Proposition 1 and Proposition 2.Figure 3
(a) Relations betweenπb and ε and δ, (b) relations between πb and K, (c) relations between πb and ch, and (d) relations between πb and cl.
(a)(b)(c)(d)As presented in Figure4(a), the red line states the threshold value of ε1 when πBI1ε1=πBE1ε1. The results in Figures 4(b) to 4(d) reveal how ε1 varies with other parameters. The threshold ε1 increases with the ch,cl, and K, indicating that the larger differential of profits generated by different generation costs would lead the ER to be more likely choosing the sequential equilibrium. However, the relation between ε1 and δ is complex, and ε1δ ramps fast as δ increases to 0.2 and then decreases with δ increasing from 0.2 to 1.Figure 4
(a) Relations betweenπb and ε and δ, (b) relations between ε and ch and δ, (c) relations between ε and K and δ, and (d) relations between ε and cl and δ.
(a)(b)(c)(d)Then, the same simulation onε2 is also analyzed in Figure 5. The red line in Figure 5(a) is the solution ε2 to the equation πBI2ε2=πBE1ε2. The results in Figures 5(b) to 5(d) illustrate how ε2 varies with other parameters. The threshold ε2 increases with δ. However, the relations between ε1 and ch,cl, and K are complex, which are highly dependent on δ.Figure 5
(a) Relations betweenπb and ε and δ, (b) relations between ε and ch and δ, (c) relations between ε and cl and δ, and (d) relations between ε and K and δ.
(a)(b)(c)(d)


## 6. Conclusion

The motivation of our work is an observation that the study of a bilateral bargaining process between a power retailer and renewable supplier over electricity contracts when facing the uncertainty of renewable generation cost is blank. At present, the power retailer plays an essential role in promoting the consumption of renewable electricity and renewable energy penetration. Hence, it is meaningful to study the bargaining process over incentive contracts in the electricity market and provide some managerial insights for the electricity retailer. Our main conclusions can be summarized as follows:(1)
The electricity retailer can screen the generation cost of the renewable supplier through a menu of incentive contracts, while the renewable energy generators can also send cost signals through their contracts to actively expose the true cost information. As such, to avoid adverse selection in a framework of Rubinstein’s bargaining, there are two possible equilibria in designing the incentive contracts for the electricity retailer, either to offer a set of different types of high-cost and low-cost contracts to induce the seller to choose autonomously, or to propose only low-cost contracts for the supplier and wait for the supplier to be actively exposed at the next stage.(2)
According to the volume of imitation returns for the low-cost type renewable supplier, we summarize three ranges for natural information screening, natural signalling, and costly signalling, solve for equilibrium, and specify the quantities and wholesale price in the contracts under different conditions. On the contrary, when the imitation revenue is high, high-cost generators need to pass on their true cost information and convince the electricity sales company to protect their own revenue. The high-cost generators will experience quantity distortion during bargaining. This can lead to a profit distortion compared with their first best output, which is the cost paid by the high-type RS to transmit her true signal.(3)
Although both parties in the bargaining process want to reach an agreement as soon as possible, equilibrium is not always achieved in the first stage, depending on the probability that the renewable supplier has a high generation cost. When the likelihood of being of a high-type renewable supplier is sufficiently high, the power retailer’s strategy is to offer a menu of screen contracts to RS, so that the agreement is reached as soon as possible in the first stage, achieving instantaneous separation equilibrium; conversely, when this being a high-type RS probability is low, the ER will not offer a contract for the high-type RS in the first stage to avoid adverse selection. His strategy is only to propose a contract for the low-type RS and to judge the true cost information of the RS based on her decision. If the RS has a high generation cost, they will reach an agreement to achieve sequential separation equilibrium in the second stage.Although our model comprehensively studies the bilateral bargaining process between RS and ER, some directions can still be extended in the future. First, the electricity market consists of many power retailers, and thus, it is interesting to study the renewable supplier either prices a “take it or leave it” contract or applies the bargaining method over electricity contracts in a sequential selling process. Second, the different risk preferences of power retailers and renewable suppliers can be another direction, directly affecting the contracts’ purchasing quantities and price. Third, the contract for difference is also considered in electricity market literature, and hence, it is also intriguing to research the variation of the market outcome if the contract for difference is applied. We leave all these issues in the future works.



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