metadata: authors: 0: author: Nikandrova,Arina
1: author: Steinbuks,Jevgenijs
count: World
docna: 0: docna: Contracting for the second best in
            dysfunctional electricity markets
docty: Policy Research Working Paper
theme: Infrastructure services for private sector development,Regulation and competition policy,Regional integration,Environmental policies and institutions
majtheme: Financial and private sector development,Trade and integration,Environment and natural resource management
subtopic: Private Sector Economics,Power & Energy Conversion,Energy Policies & Economics,Private Sector Development Law,International Trade and Trade Rules,Marketing,Energy Demand,Energy and Environment,Energy and Mining
teratopic: Energy,Infrastructure Economics and Finance,Macroeconomics and Economic Growth
abstracts: cdata!: Power pools constitute a set of
            sometimes complex institutional arrangements for
            efficiency-enhancing coordination among power systems. Where
            such institutional arrangements do not exist, there still
            can be scope for voluntary electricity-sharing agreements
            among power systems. This paper uses a particular type of
            efficient risk-sharing model with limited commitment to
            demonstrate that second-best coordination improvements can
            be achieved with low to moderate risks of participants
            leaving the agreement. In the absence of an impartial market
            operator who can observe fluctuations in connected power
            systems, establishing quasi-markets for trading excess
            electricity through the kind of mechanism described here
            helps achieve sustainable cooperation in mutually beneficial
            electricity sharing.
repnme: repnme: Contracting for the second best in
            dysfunctional electricity markets
display_title: Contracting for the second best in
            dysfunctional electricity markets
listing_relative_url: /research/2014/06/19747558/contracting-second-best-dysfunctional-electricity-markets
topicv3: Private Sector Development m1326060 1921,Energy and Extractives m1326595 507,Trade m1326924 698,Power Systems m1327093 508,Power and Electricity Sector m1327094 515
publishtoextweb_dt: 2014-06-01T00:00:00Z
historic_topic: Energy,Infrastructure Economics and Finance,Macroeconomics and Economic Growth
no_of_pages: 40
author_keys: 0: author_key: 1293874
1: author_key: 666646
model_extractions: WPS6955 Policy Research Working Paper 6955 Contracting for the Second Best in Dysfunctional Electricity Markets _Arina Nikandrova Jevgenijs Steinbuks_ The World Bank Development Research Group Environment and Energy Team June 2014
Policy Research Working Paper 6955

# **Abstract**

Power pools constitute a set of sometimes complex institutional arrangements for efficiency-enhancing coordination among power systems. Where such institutional arrangements do not exist, there still can be scope for voluntary electricity-sharing agreements among power systems. This paper uses a particular type of efficient risk-sharing model with limited commitment to demonstrate that second-best coordination improvements can be achieved with low to moderate risks of participants leaving the agreement. In the absence of an impartial market operator who can observe fluctuations in connected power systems, establishing quasi-markets for trading excess electricity through the kind of mechanism described here helps achieve sustainable cooperation in mutually beneficial electricity sharing.

This paper is a product of the Environment and Energy Team, Development Research Group. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at jsteinbuks@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Contracting for the Second Best in Dysfunctional Electricity Markets<sup>_∗_</sup>

Arina Nikandrova and Jevgenijs Steinbuks JEL: C73, L94, O13 Keywords: Electricity Trade, Risk Sharing Arrangements, Self-Enforcing Contracts Sectors: Energy, Public Sector and Governance

> _∗_ Nikandrova: Birkbeck, University of London, a.nikandrova@bbk.ac.uk. Steinbuks: Development Research Group, The World Bank, jsteinbuks@worldbank.org. Acknowledgements: We would like to thank Javier Bustos Salvagno, Avinash Dixit, Romans Pancs, Vladimir Parail, Michael Pollitt, Michael Toman, and the participants of the 12th Annual International Industrial Organization Conference for their helpful suggestions and comments. Responsibility for the content of the paper is the authors’ alone and does not necessarily reflect the views of their institutions, or member countries of the World Bank. Financial support from the Australian Agency for International Development and from the Partnership for South Asia Regional Integration Programmatic Trust Fund is greatly acknowledged.
# **1 Introduction**

Availability of modern electric power infrastructure is widely recognized as one of the key driving forces for accelerated economic development (Jorgenson 1984, Toman and Jemelkova 2003, Briceno-Garmendia et al. 2004, Chakravorty et al. 2014). Yet in many developing countries, the electric power systems are dysfunctional, characterized by poor access to electric grid, unreliable power supply, inefficient generating capacities, deficient maintenance, and losses in transmission and distribution (Eberhard et al. 2008, Foster and Steinbuks 2009, Sen and Jamasb 2012). Seeking to improve the performance of their electric power systems, many developing countries have undertaken deregulation, privatization, and interconnection processes, with the final goal of creating functioning wholesale and retail electricity markets (Joskow 2008). While these reforms have led to positive and globally widespread, though modest, efficiency gains (Pollitt 2012b), the reform process in most developing countries has appeared to be slow and difficult, even after more than two decades of reforms (Besant-Jones 2006, Kessides 2012). In light of the unique institutional characteristics of many developing countries, increasing doubts have been recently expressed about the applicability of the standard textbook model of utility reform to these countries. In many parts of the world, electricity markets have evolved or are evolving into hybrid forms, adapting some elements of the standard market-based model, but not completely unbundled, privatized, or competitive (Gratwick and Eberhard 2008).

The integration of electricity markets is frequently seen as a low-hanging fruit of improving the performance of electric power systems. A number of regions, including Northern Europe, regions in the United States, Western and Southern Africa, and Latin America moved in the direction of integrating their electricity systems (Ochoa et al. 2013). The potential economic benefits from connecting power systems are widely documented (Gately 1974, Gnansounou and Dong 2004, Pierce et al. 2007) and, among others, include the benefits of diversifying generation mix, achieving economies of scale, promoting competition, and reducing the need for new generation capacity. The latter reason for integration is especially compelling. Electricity is a largely non-storable commodity, with uncertain demand and supply. In particular, demand for electricity varies greatly with factors such as the weather, time of day, and season, while electricity supply can be subject to unexpected failure of generation units and unanticipated changes in the output of intermittent renewable generation such as wind.

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Consequently, to balance demand and supply of electricity in real time, it is necessary to maintain a large safety margin of flexible generation capacity that remains idle in the vast majority of periods. This capacity is associated with a large economic cost, given by the fixed costs of generation capacity. Connecting different power systems can reduce the need for such idle capacity, and hence realize the economic benefits of trade in electricity, especially in countries with a non-trivial share of intermittent electricity generation capacity. The empirical evidence indicates that regional integration of electricity markets has been largely successful in developed countries, such as North European countries (Nord Pool) or Pennsylvania-New Jersey-Maryland (PJM) interconnection in the United States (Douglas 2006, Pollitt 2008), but less so in some developing countries, e.g., in West Africa and Central America, where despite significant potential benefits traded quantities are still extremely small (Eberhard et al. 2008, Pineau 2008, Reinstein et al. 2011).

One explanation for poor performance of regionally integrated electric systems in the developing countries is that weak institutional arrangements are not capable of enforcing “contracts” among member states.<sup>1</sup> In developed countries electricity trading is facilitated and enforced through complex institutional mechanisms. These arrangements frequently feature an independent system operator or independent transmission system operator, which has responsibility for controlling the access to and use of the transmission grid by competing generators and retailers (Pollitt 2012a). Electricity trading could take place in day ahead, hour ahead, and real time energy markets. Other institutions aimed at correcting different market and coordination failures include separate markets for locational marginal pricing, ancillary services, and financial transmission rights (Sioshansi 2008). On the contrary, in developing countries electricity trading is dominated by bilateral contracts. Even in the Southern African Power Pool, the most advanced electricity sharing institution in developing countries, bilateral contracts account for more than 90 percent of electricity trade (Musaba 2009). These bilateral electricity trading contracts are often subject to a “hold-up” problem (Williamson 1979, Tirole 1986), whereby large sunk costs of establishing electricity trading arrangements create incentives for not honoring the terms of the contract by either of the trading sides.

> 1Interestingly enough, the opposite case of excess rigidity in renegotiating contracts was found to be a bottleneck in establishing independent transmission system operator and achieving greater regional integration of electric system in Netherlands (Mulder and Schestalova 2006). After these issues were resolved, a highly successful Norway–Netherlands interconnector was established.

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Thus, for electricity exchange between developing countries to take place, any contractual relationship between utilities located in different countries should be self-enforcing. That is at each moment in time, it should be in the interest of each electric power provider to participate in an electricity sharing arrangement. Every time a participating provider is called upon to transfer electricity to a neighboring country, it weighs the cost of fulfilling this obligation against the benefit from future cooperation. The electricity sharing arrangement is self-enforcing as long as the discounted present value of the future benefit flow outweighs the current cost. The possibility that a power system may abandon the electricity sharing arrangement at any moment of time limits the scope for cooperation. In particular, electricity sharing arrangements that exploit unpredictable fluctuations in supply/demand tightness conditions are difficult to implement.

The aim of this paper is to study the scope for voluntary electricity sharing arrangements among power systems where functioning institutional arrangements for more formal cooperation do not exist. We start with a premise that an interconnector can be a substitute for reserve generation capacity in the connected power systems, provided that stochastic variations in demand and supply are not perfectly positively correlated across these systems. A power system that is subject to a negative supply shock (e.g. drought or low wind) can import power from a neighboring system that is not subject to the same shock. Thus connecting power systems could bring welfare gains through reduction in volatility of demand/supply imbalances even without a consistent difference in the timing of peak demand periods.

As a base framework, we adapt the efficient risk sharing model with limited commitment to electricity sharing arrangements. The model has been developed theoretically, among others, by Thomas and Worrall (1988), Kocherlakota (1996), and Kletzer and Wright (2000), and subsequently applied to understanding informal contracts in developing countries (Foster and Rosenzweig 2001, Albarran and Attanasio 2003, Ligon et al. 2002, Schechter 2007, Dubois et al. 2008). To our knowledge this is the first time this framework is applied to study electricity markets.

We assume that two developing-country power systems are connected via an electricity interconnector. As many developing countries experience power shortages, with their energy demand needs vastly outstripping supply, we assume that each power system operates close to its full capacity and the electricity supply is perfectly inelastic. Furthermore, electricity 4
supply is subject to random production shocks, which reflect unexpected failures of generation units or unanticipated changes in the output of intermittent renewable generation. By entering a cooperative arrangement, each power system promises to transfer some of the electricity it produces to the other power system, whenever such transfer is required. To capture the need for the electricity contracts to be self-enforcing, the model features forward-looking intertemporal participation constraints. Thereby at any moment of time and for any realization of electricity production shocks, the discounted expected utility from participating in an electricity sharing arrangement is greater than or equal to the autarky payoff. Consistent with game-theoretic literature the implicit assumption in the formulation of the participation constraint is that a deviation from an electricity sharing arrangement triggers the most severe punishment, i.e, permanent termination of the arrangement.<sup>2</sup> The model’s main insight is that the scope for cooperation crucially depends on predetermined risks of terminating an ongoing relationship due to, e.g., political economy issues in one or both countries that host the interconnector. These risks are not trivial in many developing countries. For example, it is widely documented that during election season state- or municipal-run utilities may have strong incentives to allocate their excess capacities for the needs of local agricultural consumers (who are highly organized, active, and carry significant weight in state elections) instead of selling them across the border (see e.g., Joseph 2010, Sen and Jamasb 2012). If the risks of terminating contractual agreements are small, the intertemporal participation constraints are never binding, and the first-best efficient contracting is self-enforcing. As the likelihood of terminating the ongoing relationship increases, the intertemporal participation constraints are binding infinitely often, and the first-best contract is no longer feasible. Nonetheless cooperation still yields an improvement over autarkic outcome as it allows some intertemporal smoothing of electricity consumption. If the expected probability of terminating an ongoing relationship is high (e.g., in cases of very substantial political risks), autarky is the only self-enforcing outcome.

We start with the assumption that implementation of the voluntary electricity sharing arrangements is facilitated by an independent system operator, which can perfectly observe production shocks in the connected power systems and on the basis of this information, recommend (in a non-binding manner) the appropriate dispatch. In this setting, the opti-

> 2This assumption is made for computational convenience, but results would be qualitatively the same with less severe, but more realistic punishments (see footnote 11 for details).

5
mal electricity sharing arrangement entices a power provider with the binding intertemporal participation constraint to stay in the arrangement by promising an increase in its current and the future expected electricity consumption. The promised increase is just sufficient to prevent the exit of the constrained power system and the new higher expected consumption level remains unchanged as long as there are no other binding participation constraints.

In practice, even in developed countries, establishing an independent system operator is associated with significant costs (Federal Energy Regulatory Commission 2004), and these costs are likely to be even higher in developing countries. We show that if production shocks in the connected power systems are not perfectly observable by all parties, some welfare gains can still be realized through an electricity sharing arrangement, but these gains are lower relative to the previous case of fully observable electricity production. This happens because, to increase domestic electricity consumption, each power system is tempted to understate its current supply conditions. As shown more generally by Hertel (2004), to create sufficient incentives for truthful revelation of the production shocks, demanding an electricity transfer from the other party in the current period should be punished by lower expected future electricity imports, or, equivalently, exporting electricity in the current period should be rewarded by higher expected electricity imports in the future.<sup>3</sup> We argue that with imperfect monitoring of production shocks, the feasible cooperative arrangements can be implemented via a quasi-market for trading excess electricity. For a quasi-market to operate, the connected power systems need to introduce notional coins (e.g., megawatt chips), which are worthless outside their cooperative arrangement. These chips can be used to keep track of electricity import/export imbalance between the parties to the contract; the party that runs out of chips needs to earn some back through electricity exports before it can receive further imports.

Some form of cooperative arrangements either through establishing an independent system operator or through quasi-markets enable realizing gains from electricity trade when institutional characteristics of developing countries make establishing more complex market arrangements difficult. Of course, our results should not be interpreted as a substitute for the standard reform model. While in principle, restructuring and deregulation of electricity

> 3Atkeson and Lucas (1992) and Thomas and Worrall (1990) also study dynamic contracts in settings with private information. However, these papers assume full commitment, i.e., players are committed to the contract signed at the beginning and cannot walk away from it at the later date.

6
markets are not pre-requisites for successful international power sector cooperation, in many power systems, the processes of deregulation and integration went hand in hand since markets provided a framework within which mutually beneficial arrangements could be established without the need for bilateral negotiation. Hence in the longer term, as their institutions strengthen, developing countries may wish to move towards a more market-based approach to organizing their electricity systems in order to realize a greater share of the potential gains from trade in electricity.

# **2 Model**

Consider a version of a dynamic risk sharing model with limited commitment introduced by Kocherlakota (1996). Two power systems, or players 1 and 2, are connected via an electricity interconnector with exogenous capacity _K_ . Both systems have identical marginal costs of generation, which are normalized to zero.<sup>4</sup> Each power system operates close to full capacity with perfectly inelastic supply, which is subject to random production shocks. These shocks capture fluctuations in electricity production due to unexpected failures of generation units or unanticipated changes in the output of intermittent renewable generation such as wind or solar photovoltaics.<sup>5</sup> Thus, at time _t,_ the domestic electricity production in power system _i_ is _yi,t_ = _y_ ̄ + _εi,t,_ where _y_ ̄ is a constant and _εi,t_ are random variables with distribution that is

> 4This assumption allows focusing on reducing the need for idle capacity as opposed to other potential benefits from power system integration. In real life situations, one would expect power systems to have different marginal costs of generation. These differences could arise because of idiosyncrasies in generation assets (e.g., fossil fuel fired power plants have higher marginal costs of generation than hydroelectric or nuclear power plants) or input subsidies, which are not uncommon in many developing countries. Billette de Villemeur and Pineau (2012) have shown that if marginal costs to generation are different, integrating power systems would still achieve the first best outcome absent other market distortions (e.g., environmental externalities). If one of the industries is regulated, productive inefficiencies could arise.

> 5The focus of the paper is on intraday exchange of electricity with an interconnector being used as a substitute for peaking generation capacity in the connected power systems. Examining the scope for cooperation on intraday basis is particularly important for developing countries, where traditionally shortterm trade contract were difficult to establish. For example, in the Southern African Power pool, bilateral trading agreements are long-term and are not flexible enough to accommodate varying demand profiles (Musaba 2009). Nevertheless, our model can also be applied to seasonal electricity exchange between power systems with different generation assets.

7
symmetric around zero and has a discrete support where _ε_<sup>_s_</sup> _< ε_<sup>_s_+1</sup> and _y_ ̄ + _ε_<sup>1</sup> _≥_ 0. Shocks _εi,t_ are i.i.d. across time with _p_<sup>_s_</sup> := P ( _εi,t_ = _ε_<sup>_s_</sup> ). The aggregate supply _y_ 1 _,t_ + _y_ 2 _,t_ is denoted by _Yt_ which may or may not be constant over time.

The random variable _εt_ = ( _ε_ 1 _,t, ε_ 2 _,t_ ) captures the state of electricity production in period _t_ . All variables referring to state � _ε_<sup>_s_</sup> _, ε_<sup>_s′_�</sup> have a superscript _s, s_<sup>_′_</sup> , where _s, s_<sup>_′_</sup> _∈{_ 1 _, ..., S} ._ Thus, _π_<sup>_s,s′_</sup> := P � _ε_ 1 _,t_ = _ε_<sup>_s_</sup> _∩ ε_ 2 _,t_ = _ε_<sup>_s′_�</sup> is the probability that state � _ε_<sup>_s_</sup> _, ε_<sup>_s′_�</sup> is realized in period _t_ . We assume that the joint distribution of production shocks is symmetric so that the probability of state ( _ε, ε_<sup>_′_</sup> ) is equal to the probability of ( _ε_<sup>_′_</sup> _, ε_ ) _._ This formulation of production shocks is capable of capturing various dependence structures.<sup>6</sup>

**Example 1.** _Independent production shocks:_ If _π_<sup>_s,s′_</sup> = _p_<sup>_s_</sup> _p_<sup>_s′_</sup> , the production shocks are independent across the power systems and aggregate production varies over time.

Alternatively, the production shocks could be perfectly negatively correlated.<sup>7</sup>

**Example 2.** _No aggregate uncertainty:_ If player 1 produces _y_ 1 _,t_ , and player 2 produces _Y − y_ 1 _,t_ , there is no aggregate uncertainty, but the distribution of output varies over time. As aggregate production is _Y_ in every period, _yi_<sup>_s_=</sup><sup>_Y−y_</sup> _i_<sup>_S−s_+1</sup> , where _yi_<sup>_s_:=</sup><sup>_y_ ̄+</sup><sup>_εs_and</sup> _yi_<sup>_S−s_+1</sup> := _y_  ̄ + _ε_<sup>_S−s_+1</sup> is production of player _i_ experiencing production shock _ε_<sup>_s_</sup> and _ε_<sup>_S−s_+1</sup> _,_ respectively. Consequently, _p_<sup>_s_</sup> = _p_<sup>_S−s_+1</sup> and _π_<sup>_s,s′_</sup> = _p_<sup>_s_</sup> whenever _s_<sup>_′_</sup> = _S − s_ + 1 and _π_<sup>_s,s′_</sup> = 0 otherwise.

Power system _i_ has the instantaneous utility function _ui_ ( _c_ ) _,_ which is increasing, strictly concave and continuously differentiable in domestic electricity consumption with lim _c→_ 0 _u_<sup>_′_</sup> _i_<sup>(</sup><sup>_c_) =</sup> + _∞._<sup>8</sup>

> 6It is important, however, that the production shocks in the connected power systems are not perfectly positively correlated; otherwise whenever one power system experiences relatively low production, the other system also experiences low production and there is no scope for mutually beneficial electricity exchange. In such case of perfect positive correlation, welfare may be enhanced only through installing additional generation capacity.

> 7See Ljungqvist and Sargent (2004), for a textbook treatment of this example.

> 8In addition, it is possible to normalize the utility function so that _ui_ ( ̄ _y_ ) = 0 _, ui_ ( _c_ ) _<_ 0 for all _c < y_  ̄

8
From the perspective of a planner, at time _t,_ the overall utility of a power system from the stream of future electricity consumption is given by where _δ ∈_ (0 _,_ 1) is a discount factor and E _t_ [ _·_ ] is expectation operator conditional on information publicly available at _t_ . The discount factor _δ_ can be viewed as an exogenous probability of terminating an ongoing cooperative relationship due to, e.g., political economy reasons in one of the countries that host the connected power systems. In this interpretation, each power system prefers early payoffs due the possibility that the cooperation may end before later payoffs can be collected. Alternatively, _δ_ can also be viewed as being implied by the interest rate used to discount investment in the connected power systems. In either case, the magnitude of _δ_ crucially depends on the frequency of interaction, with more frequent electricity exchange commanding higher _δ_ .

We consider two alternative information structures. First, we assume that there is an independent cross-border system operator who observes and truthfully reveals production shocks in the connected power systems to all interested parties. Under the alternative scenario, each power system’s production shocks are observed exclusively by that power system. This latter assumption is motivated by the fact that in practice, establishing a reliable monitoring institution is associated with significant costs and may not be feasible.

By entering an electricity sharing arrangement the two power systems engage in an electricity exchange game with the following interaction in each period _t_ . At the beginning of period _t,_ the state variable _εt_ = ( _ε_ 1 _,t, ε_ 2 _,t_ ) is realized. Then either the system operator reveals _εt_ to both players or in the absence of the cross-border system operator, both power systems simultaneously announce their production shocks and their announcements result in the ˆ revealed state _εt_ = (ˆ _ε_ 1 _,t,_ ˆ _ε_ 2 _,t_ ). Finally, one of the power systems voluntarily transfers some non-negative amount of electricity that it generates to the other system;<sup>9</sup> the transfer results and _ui_ ( _c_ ) _>_ 0 for all _c > y_ ̄, i.e., only consumption that exceeds the level _y_ ̄ creates well-being, but when consumption falls below _y_ ̄, the power system experiences disutility due to involuntary demand interruption. This normalization reflects the empirical evidence that overstretched power systems in many developing countries hardly produce enough electricity to prevent involuntary interruption of demand.

> 9Assuming that one player makes a transfer, while the other player makes no transfer is without loss of

9
in the electricity consumption profile ( _c_ 1 _,t, c_ 2 _,t_ ).

Let _h_<sup>_t_</sup> = _{ετ }_<sup>_t_</sup> _τ_ =1<sup>or</sup><sup>_ht_=</sup><sup>_{ε_ˆ</sup><sup>_τ}t_</sup> _τ_ =1<sup>betheperiod-</sup><sup>_t_historyofstaterealizations(inpresence</sup> of a system operator capable of monitoring production shocks) or revealed state realizations (when there is no system operator capable of monitoring), respectively. An electricity sharing arrangement, or a **contract,** is a sequence of functions ( _c_ 1 ( _h_<sup>_t_</sup> ) _, c_ 2 ( _h_<sup>_t_</sup> )) _,_ where _ci_ ( _h_<sup>_t_</sup> ) is player _i_ ’s electricity consumption after the period- _t_ history _h_<sup>_t_</sup> . A contract induces an **allocation** , which is a stochastic process _{_ ( _c_ 1 _,t, c_ 2 _,t_ ) _}_<sup>_∞_</sup> _t_ =1<sup>_,_where</sup><sup>_ci,t_isperiod-</sup><sup>_t_consumptionofplayer</sup><sup>_i_,</sup> _ci,t_ = _ci_ ( _h_<sup>_t_</sup> ) _._ A **feasible allocation** must satisfy the resource constraints, for all _t_ , according to which aggregate consumption cannot exceed aggregate production of electricity and electricity transfers cannot exceed the interconnector’s capacity.

# **3 Two Benchmarks**

## **3.1 Autarky Outcome**

The **autarkic allocation** _{_ ( _y_ 1 _,t, y_ 2 _,t_ ) _}_<sup>_∞_</sup> _t_ =1<sup>,inwhichnopowersystemevermakespositive</sup> transfers, results in a payoff of to player _i_ .

Payoff _<u>vi</u>_<sup>definesthereservationutilityofpowersystem</sup><sup>_i_inanyvoluntarycontract.The</sup> assumption of concave instantaneous utility functions implies that non-trivial contracts with strictly positive electricity transfers may yield higher lifetime utility to each power system whenever they enable electricity consumption smoothing over time as well as over states.

generality. Since smaller transfers pose less stringent incentive constraints than do larger ones, there is no need to consider cases in which the players make simultaneous transfers.

10
# **3.2 The First-Best Allocation**

Now consider a planner who aims to maximize the social surplus defined here as the discounted sum of the utilities of two power systems. It is assumed that the planner observes the realization of the state variable _εt_ = ( _ε_ 1 _,t, ε_ 2 _,t_ ) in each period _t._<sup>10</sup> The **first-best efficient contract** is a solution to the following optimization problem: where 0 _≤ λ_ 0 _≤∞_ is the relative Pareto weight of player 2 in the planner’s objective. Weight _λ_ 0 determines the distribution of payoffs across the two power systems. A utilitarian planner puts equal weight on the well-being of both players and sets _λ_ 0 = 1 _,_ but asymmetric payoff distributions are also consistent with efficiency. In particular, by changing _λ_ 0 from 0 to _∞_ it is possible to trace the entire Pareto frontier, with higher values of _λ_ 0 corresponding to contracts in which player 2 gets more of the potential surplus from electricity exchange.

In the planner’s problem, constraints (2) and (3) are feasibility constraints and constraint (4) ensures that ex ante each power system (weakly) prefers cooperation to autarky.

The planner’s problem is well-behaved and thus has a solution. The planner can always choose the autarkic allocation, but in general, he can do better than that, as risk-averse players strictly prefer to smooth consumption over time as well as over states.

The solution to the planner’s problem can be characterized as follows. Since the utility function of each power system is strictly increasing, the resource constraint (2) must be binding and at the optimum, _c_ 1 _,t_ + _c_ 2 _,t_ = _Yt_ in every period. Moreover, since shocks _εi,t_

> 10As we are primarily concerned with finding the first-best efficient contracting solutions, we ignore possible market failures, such as e.g., externalities associated with electric power generation, in the social planner’s problem. Billette de Villemeur and Pineau (2010) have shown that if these externalities are present connecting electricity systems can result in welfare losses.

11
are i.i.d. across time, the first-best efficient allocation must feature stationary consumption profiles. Then, the first order conditions of the planner’s problem imply that in every period, the state-� _ε_<sup>_s_ˆ</sup> _, ε_<sup>_s_ˆ</sup><sup>_′_�</sup> consumption of player 2, _c_<sup>_s,_ˆ</sup><sup>_s_ˆ</sup><sup>_′_</sup> _,_ satisfies where _<u>μs,</u>_ ˆ _s_ ˆ<sup>_′_</sup> is the Lagrange multiplier associated with constraint and _μ_ ̄<sup>_s,_ˆ</sup><sup>_s_ˆ</sup><sup>_′_</sup> is the Lagrange multiplier associated with constraint Thus, the first-best efficient allocation features maximal insurance for both power systems against fluctuations in their electricity output. Consumption of each power system varies across states only insofar as there is uncertainty about the aggregate supply of electricity or the flows through the interconnector are capacity constrained. If there is no aggregate uncertainty and flows through the interconnector are never constrained, the efficient consumption profile features full-insurance with constant electricity consumption over time and across states. If there is aggregate uncertainty, but the interconnector capacity constraint is never binding, players smooth their consumption so as to keep the ratio of their marginal utilities constant and equal to _λ_ 0 over time and across states. If the capacity constraint of the interconnector between the power systems is binding in some state � _ε_<sup>_s_ˆ</sup> _, ε_<sup>_s_ˆ</sup><sup>_′_�</sup> , keeping the ratio of marginal utilities constant across states is no longer possible. In this case, the first-best efficient consumption profiles are still stationary, but the marginal utility ratio satisfies whenever the interconnector capacity constraint restricts the electricity flows from player 2 12
to player 1 (i.e., constraint 6 binds), and and whenever the interconnector capacity restricts the flows in the other direction (i.e., constraint 7 binds).

# **4 Optimal Self-Enforcing Contracts**

The planer’s problem ensures that each power system participating in an electricity sharing arrangement on average is at least as well off as under autarky. However, the solution to planner’s problem does not guarantee that after _every_ history of production shocks, it is in the interest of each power system to participate in the arrangement, rather than to renege on it. In reality, any arrangement between power systems is limited by the two-sided lack of commitment, as at any moment, each power system may decide to default on its current electricity export obligations. In particular, every time a power system is called upon to transfer electricity to a neighboring power system, it weighs the cost of fulfilling this obligation against the benefit from future cooperation. The electricity sharing contract is voluntary or self-enforcing if after any history of production shocks, the discounted present value of the contract’s future benefit flow outweighs the current (welfare) cost of export. This section demonstrates that the history-by-history participation constraints limit the scope for cooperation among the power systems.

It will be assumed that, if either party violates the contract, both power systems irrevocably revert to autarky. This assumption is consistent with viewing the electricity sharing arrangement as a subgame perfect equilibrium of an infinitely repeated game. In a one-shot interaction, making no transfer to the other system is a best response to any transfer choice of the opponent. Thus, the threat of reverting to autarky and never making any transfer is credible for any discount rate _δ_ . Moreover, since autarky provides less utility than any other feasible allocation that does not involve disposing of the produced electricity, reverting to autarky constitutes the most severe punishment that can be imposed on a deviant from an electricity sharing contract, i.e., it constitutes an optimal penal code in the sense of Abreu 13
(1988). Thus, in the context of this paper, only those electricity sharing arrangements are **sustainable** which after any history of production shocks, provide each power system with a lifetime utility at least as high as its lifetime utility under autarky.<sup>11</sup> An allocation is **(Pareto) optimal** if there exists no other feasible sustainable allocation which offers both power systems at least as much expected utility and one power system strictly more. Equivalently, an allocation is Pareto optimal if among all feasible allocations, it delivers to player 1 the largest expected utility for a given player 2’s expected utility.

# **4.1 Cooperative Outcomes with Perfectly Observable Shocks**

We first consider the situation when the implementation of the voluntary electricity sharing arrangements is facilitated by an independent system operator, which can perfectly observe production shocks in the connected power systems and on the basis of this information, recommend (in a non-binding manner) the appropriate dispatch.<sup>12</sup> Following Thomas and Worrall (1988) approach, let _V_<sup>_s,_ˆ</sup><sup>_s_ˆ</sup><sup>_′_</sup> ( _U_ ) be the maximal utility of player 1 when the continuation utility promised to player 2 is _U. V_<sup>_s,_ˆ</sup><sup>_s_ˆ</sup><sup>_′_</sup> ( _U_ ) represents the Pareto frontier conditional on state � _ε_<sup>_s_ˆ</sup> _, ε_<sup>_s_ˆ</sup><sup>_′_�</sup> _._ To characterize the frontier suppose that after observing the realization of the current-period production shocks, the planner chooses a consumption level _c_ for player 2 and for every possible realization of production shocks in the next period, � _ε_<sup>_s_</sup> _, ε_<sup>_s′_�</sup> , a continuation utility _U_<sup>_s,s′_</sup> for player 2. Thus the system enters next period carrying a vector <u>�</u> _U_<sup>1</sup><sup>_,_1</sup> _, U_<sup>1</sup><sup>_,_2</sup> _, ..., U_<sup>1</sup><sup>_,S_</sup> _, U_<sup>2</sup><sup>_,_1</sup> _, U_<sup>2</sup><sup>_,_2</sup> _, ..., U_<sup>_S,S_�</sup> of contingent continuation utilities for player 11In real world situations, it might be difficult or undesirable to commit to irrevocably break off cooperation upon the first non-compliance with the arrangement, particularly if setting up the infrastructure for cooperation requires significant pecuniary investment, such as investment in establishing an independent system operator, and non-pecuniary investment, such as investment in good-will building. Hence, the assumption that a breach of the contract triggers the permanent return to autarky may seem extreme. However, the results in this paper would not change qualitatively if a non-compliance was punished by a temporary reversion to autarky followed by an eventual return to electricity sharing (e.g., tit-for-tat strategy). It is even possible to design a contract such that, instead of the reversion to autarky, the punishment involves the point on the Pareto frontier for the current state which gives the autarky utility to the deviant power system (see, for example, Kletzer and Wright (2000)). Such contract would be weakly renegotiation-proof in the sense of Farrell and Maskin (1989).

> 12In our model, the only function of the independent system operator is coordination of cross-border electricity flows. In real world, independent system operators are more complex systems, which perform many other functions, such as e.g., maintaining transmission capacity and ensuring non-discriminatory access to the grid for individual generators (Sioshansi 2008, Pollitt 2012a).

14
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surpluses for each power system must be an interval. Let _<u>V</u> s,_ ˆ _s_ ˆ<sup>_′_</sup> _, V_ ̄ _s,_ ˆ _s_ ˆ<sup>_′_�</sup> and _<u>U</u> s,_ ˆ _s_ ˆ<sup>_′_</sup> _, U_ ̄ ˆ _s,s_ ˆ<sup>_′_�</sup> � � denote such an interval for player 1 and player 2, respectively. The participation constraints of players imply that _<u>V</u> s,_ ˆ _s_ ˆ<sup>_′_</sup> = _u_ 1 _y_ 1<sup>_s,_ˆ</sup><sup>_s_ˆ</sup><sup>_′_</sup> + _δv_ ~~1~~<sup>and</sup><sup>_<u>U</u>_</sup> _s,_ ˆ _s_ ˆ<sup>_′_</sup> = _u_ 2 _y_ 2<sup>_s,_ˆ</sup><sup>_s_ˆ</sup><sup>_′_</sup> + _δv_ ~~2~~<sup>_._Furthermore,the</sup> � � � � fact the total production in every period is limited and preferences are non-satiated and represented by a strictly concave utility function implies that the Pareto-frontier _V_<sup>_s,_ˆ</sup><sup>_s_ˆ</sup><sup>_′_</sup> ( _U_ ) is decreasing, strictly concave and continuously differentiable on _<u>U</u> s,_ ˆ _s_ ˆ<sup>_′_</sup> _, U_ ̄ ˆ _s,s_ ˆ<sup>_′_�</sup> _._ � The first order conditions in conjunction with the Envelope theorem imply: According to the envelope condition (15), multiplier _λ_ measures the rate at which player 1’s utility can be traded off against the utility of player 2, conditional on the current state, i.e., _λ_ is the relative Pareto weight of player 2. Once the next period’s state is realized, the new value of _λ_ is determined by (14). The current consumption profile of players is pinned down by equation (13), according to which _λ_ is also equal to the ratio of the marginal utilities of consumption, subject to the interconnector capacity constraints being satisfied. Thus either there is a unique interior solution with the ratio of the marginal utilities equal to _λ_ , or _λ_ lies outside the set of marginal utility ratios which can be generated by feasible transfers in state � _ε_<sup>_s_</sup> _, ε_<sup>_s′_�</sup> . In the latter case, there is a corner solution with the entire interconnector capacity being utilized for transferring the maximal amount of electricity to one of the power systems. Hence, as in Ligon et al. (2002), the optimal contract is fully characterized by the evolution of _λ_ .

Let _λ_ ( _h_<sup>_t_</sup> ) be the value of _λ_ at date _t_ after the history _h_<sup>_t_</sup> . Ligon et al. (2002) shows that _λ_ ( _h_<sup>_t_</sup> ) satisfies a simple updating rule.

**Proposition 1.** _The optimal contract is fully characterized as follows: There exist S × S state dependent intervals_ _<u>λs,s</u>_<sup>_′_</sup> _,_  ̄ _λs,s_<sup>_′_�</sup> _, s, s_<sup>_′_</sup> = 1 _,_ 2 _, ..., S, such that λ_ ( _h_<sup>_t_</sup> ) _evolves according to_ �

16
_the following rule. Let h_<sup>_t_</sup> _be period-t history and let_ � _ε_<sup>_s_</sup> _, ε_<sup>_s′_�</sup> _be the state in period t_ + 1 _, then_ _where λ_ ( _h_<sup>0</sup> ) = _λ_ 0 _is the initial value for the relative Pareto weight λ._ _Proof._ See proof of Proposition 1 in Ligon et al. (2002) Note that equation (13) is reminiscent of (5), i.e., if _λ_ were not changing over time, the optimal contract would correspond to the first-best efficient contract. However, (14) shows that whether _λ_ changes over time depends on whether or not the players’ participation constraints are binding. In particular, every time that the participation constraint for player 2 is binding, his relative Pareto weight _λ_ is increased; every time that the participation constraint of player 1 is binding, player 2’s weight is reduced. In fact, for every state � _ε_<sup>_s_</sup> _, ε_<sup>_s′_�</sup> , there is an interval of possible relative Pareto weights _<u>λs,s</u>_<sup>_′_</sup> _,_ ̄ _λs,s_<sup>_′_�</sup> _._ If previous period relative � Pareto weight lies outside the current interval, _λ_ must be changed by the smallest possible _s,s_<sup>_′_</sup> amount to get it into the new interval. In the updating rule (16), weight _<u>λ</u>_ corresponds to player 2 being held down to its minimum surplus _u_ 2 _y_ 2<sup>_s,s′_</sup> + _δv_ ~~2~~<sup>andbeingconstrained;</sup> � � weight _λ_<sup> ̄</sup><sup>_s,s′_</sup> corresponds to player 1 being held down to its minimum surplus _u_ 1 _y_ 1<sup>_s,s′_</sup> + _δv_ 1 � � and being constrained.

The time variation of _λ_ ensures that the power system with the binding participation constraint is promised a higher future electricity consumption stream. The increase in the expected future consumption is just sufficient to prevent the default of the constrained player. Moreover, the resulting higher expected consumption level will remain unchanged as long as there are no binding participation constraints. Thus in the optimal contract, a power system is induced not to abandon the electricity sharing contract by increasing its future expected consumption of electricity permanently, not only in the period where the power system is tempted to default.

Proposition 2 shows that the properties of the solution are such that for high discount factors first-best efficient allocations are sustainable, but for low discount factors the only sustain17
able allocation is autarkic. Moreover, for intermediate discount factors the optimal contract improves on autarky.

**Proposition 2.** _(i) There exists a critical_ 0 _< δ_<sup> ̄</sup> _<_ 1 _such that for all δ ∈_ � _δ,_  ̄ 1� _, there is some first-best efficient contract which is sustainable; (ii) there exists a critical_ 1 _>_ _<u>δ</u> >_ 0 _such that for all δ ∈_ (0 _, δ_ ) _, there is no non-autarkic sustainable contract; (iii) for δ ∈_ <u>�</u> _<u>δ,</u> δ_<sup> ̄</sup> � _, the optimal contract improves on autarky, but is not first-best efficient. Moreover, welfare improvement of an optimal contract is continuous in δ._

_Proof._ (i) The critical _δ_<sup> ̄</sup> _<_ 1 exists by the Folk theorem for infinitely repeated games with varying states.

(ii) It is clear that when _δ_ = 0 _,_ the only sustainable contract is autarkic. Proposition 2 (v) in Ligon et al. (2002) shows that also for a sufficiently small, but strictly positive _δ_ , power systems make no transfers to each other in an optimal contract. (iii) Let _ζ_<sup>_s,s′_</sup> _≡ u_<sup>_′_</sup> 1<sup>( ̄</sup><sup>_y_+</sup><sup>_εs_)</sup><sup>_/u′_</sup> 2 � _y_ ̄ + _ε_<sup>_s′_�</sup> be the autarkic ratio of marginal utilities in state � _ε_<sup>_s_</sup> _, ε_<sup>_s′_�</sup> _,_ where each power system consumes all electricity it produces. By Proposition 2 (iv) in Ligon et al. (2002), for each state � _ε_<sup>_s_</sup> _, ε_<sup>_s′_�</sup> _, ζ_<sup>_s,s′_</sup> _∈_ _<u>λs,s</u>_<sup>_′_</sup> _,_ ̄ _λs,s_<sup>_′_�</sup> _,_ i.e., each _λ_ interval contains � the associated autarkic ratio of marginal utilities. Moreover, min _s,s′_ _<u>λs,s</u>_<sup>_′_�</sup> = min _s,s′_<sup>�</sup> _ζ_<sup>_s,s′_�</sup> <u>�</u> and max _s,s′_<sup>�</sup> _λ_<sup> ̄</sup><sup>_s,s′_�</sup> = max _s,s′_<sup>�</sup> _ζ_<sup>_s,s′_�</sup> and, for any state (ˆ _s,_ ˆ _s_<sup>_′_</sup> ) that is not associated with the extremal marginal utility ratios, min _s,s′_<sup>�</sup> _ζ_<sup>_s,s′_�</sup> or max _s,s′_<sup>�</sup> _ζ_<sup>_s,s′_�</sup> _,_ the autarkic marginal utility ratio _ζ_<sup>_s,_ˆ</sup><sup>_s_ˆ</sup><sup>_′_</sup> is contained in the interior of the _λ_ interval _<u>λs,</u>_ ˆ _s_ ˆ<sup>_′_</sup> _,_ ̄ _λs,_ ˆ _s_ ˆ<sup>_′_�</sup> _._ This implies that when _δ_ ̄ _> δ >_ _<u>δ</u>_ and the _λ_ intervals contain more than a single� point, but do not all overlap, a power system with the binding participation constraint is making an electricity transfer. In an optimal contract, _λ_ follows updating rule (16) and thus in the long-run, it takes values only in the set _<u>λs,s</u>_<sup>_′_</sup> _,_ ̄ _λs,s_<sup>_′_�</sup><sup>_S_Thisimpliesthatparticipationconstraintsarebindinginfinitely</sup> <u>�</u> _s,s_<sup>_′_</sup> =1<sup>_._</sup> often and consequently also electricity transfers are made infinitely often in any optimal contract. Since making no transfers is always possible, the optimal electricity transfers must be weakly improving on autarky. Furthermore, by Proposition 2 (vi) in Ligon et al. (2002), each _<u>λs,s</u>_<sup>_′_</sup> and _λ_ ̄ _s,s_<sup>_′_</sup> is continuous in _δ._ Thus also the electricity transfers are continuous in _δ,_ leading to a continuous welfare improvement.

When _δ ≥ δ,_<sup> ̄</sup> some first-best contract is sustainable. By (13), in a first-best contract, the ratio of the marginal utilities of power systems varies only insofar as the capacity constraint 18
of the interconnector is binding. Hence, to satisfy (16) it must be the case that all _λ_ intervals characterizing the optimal contract have some points in common. If _λ_ 0 belongs to the overlap region, _λ_ never changes. Moreover, starting from any _λ_ 0, the updating rule (16) leads _λ_ towards the region where the _λ_ intervals overlap and any contract converges to some firstbest contract in the long-run.

When _δ ≤_ _<u>δ,</u>_ the future benefit from continuing cooperation is heavily discounted and as such is smaller that the immediate benefit from reneging on the required electricity transfer. Consequently, no transfers are ever made and autarky is the only sustainable outcome. In this case, each _λ_ interval is degenerate and contains only the associated autarkic ratio of marginal utilities.

For intermediate _δ ∈_ � _<u>δ,</u> δ_<sup> ̄</sup> � _, λ_ intervals are non-degenerate and power systems make electricity transfers to each other. These transfers achieve some consumption smoothing and thus are welfare improving. However, attaining full consumption smoothing is not possible as transfers are limited by the binding participation constraints. To induce higher transfers, whenever the exporting power system’s participation constraint binds, this power system is promised a higher consumption stream in the future, i.e., _λ_ = _λ_ 0. As _δ_ increases, participation constraints in more states get relaxed and more extensive consumption smoothing becomes feasible.

Note that by the Envelope theorem where inequality is strict when either constraint (11) or constraint (12) binds. Thus, the model implies that the social surplus generated by an optimal sustainable contract is strictly increasing in _K,_ the interconnector’s capacity, only when interconnector’s capacity constraint binds in some state. On its own this observation is trivial, but whether the interconnector’s capacity constraint is sometimes biding in an optimal contract depends on the discount factor _δ._ In particular, as discussed above, for low values of _δ,_ the binding participation constraints of power systems limit the optimal sustainable electricity transfers, but as _δ_ increases, the participation constraints get relaxed and optimal electricity transfers increase. Consequently, the threshold capacity at which the interconnector’s capacity constraint ceases to bind in any 19
state also increases with _δ._ This suggests that in a model where in the first stage, a social planner chooses the interconnector’s capacity (at some increasing cost) and in the second stage, the connected power systems engage in a sustainable electricity sharing arrangement, the optimal interconnector’s capacity would be increasing in _δ._ Intuitively, the predetermined risks of terminating the cooperative arrangement should be sufficiently low to justify an investment in interconnection; establishing physical connections between power systems does not suffice for establishing successful cross-border cooperation in environments with weak or non-existent contract enforcement institutions.

# **Examples**

To illustrate some properties of the model we consider two simple examples constructed along the lines of Examples 1-2 in Section 2.<sup>14</sup> In the first example, production shocks of the two power systems are perfectly negatively correlated and there is no aggregate uncertainty; in the second example, power systems are subject to independent production shocks. In both examples, it is assumed that the power systems have identical preferences represented by the logarithmic utility function _ui_ ( _·_ ) = ln ( _·_ ) for _i_ = 1 _,_ 2 and that the capacity constraint of the interconnector is never binding.

## **No aggregate uncertainty**

Suppose that the production shocks of each power system are independent across time and can take three values _{−ε,_ 0 _, ε} ._ In every period, the production of the two power systems sums up to _Y_ = 2 ̄ _y,_ which implies that there are three states _hl_ , _lh_ and _mm_ with production levels ( ̄ _y_ + _ε,_ ̄ _y − ε_ ) _,_ ( ̄ _y − ε,_ ̄ _y_ + _ε_ ) and ( ̄ _y,_ ̄ _y_ ) _,_ respectively. Suppose that states _hl_ and _lh_ occur with probability _p_ (1 _− p_ ) each and state _mm_ occurs with probability 1 _−_ 2 _p_ (1 _− p_ ) _,_ where _p ∈_ (0 _,_ 1) _._ In this setting the first-best contract results in full insurance for both players with electricity consumption not varying across time and states. Thus, if ex ante the two players have equal weights in the social planner’s objective, i.e., _λ_ 0 = 1 _,_ the first-best outcome involves electricity

> 14Both examples are very close to the example in Ligon et al. (2002).

20
consumption of _y_ ̄ in every period for every player. This outcome can be achieved through a transfer of _ε_ from player 1 to player 2 in state _hl_ and a transfer of the same magnitude in opposite direction in state _lh_ . Such transfers are self-enforcing if the participation constraint of the power system that is required to make an electricity transfer is satisfied, i.e., if where _<u>v</u>_ is the autarkic lifetime utility defined in (1).

In terms of characterization of the optimal contract described in Proposition 1, there is an interval of relative Pareto weights _λ_ corresponding to each state _hl_ , _lh_ and _mm_ . Proof of Propo- _hl lh_ sition 2 implies that with logarithmic utility _<u>λ</u>_ = ( ̄ _y − ε_ ) _/_ ( ̄ _y_ + _ε_ ) and ̄ _λ_ = ( ̄ _y_ + _ε_ ) _/_ ( ̄ _y − ε_ ) _._ Moreover, since the setup is symmetric, it must be the case that _λ_<sup> ̄</sup><sup>_hl_</sup> = 1 _<u>/λlh</u>_ and _λ_ ̄ _mm_ = _mm_ 1 _<u>/λ</u>_ . Thus, to obtain the complete characterization of the optimal contract, we only need _lh mm_ to determine _<u>λ</u>_ and _<u>λ</u> ._ There are three cases that need to be considered separately: (1) the _λ_ intervals are disjoint; (2) intervals <u>�</u> _<u>λhl,</u>_ ̄ _λhl_<sup>�</sup> and <u>�</u> _<u>λlh,</u>_ ̄ _λlh_<sup>�</sup> each overlap with <u>�</u> _<u>λmm,</u>_ ̄ _λmm_<sup>�</sup> _,_ but not with each other; and (3) all three intervals have some points in common (see Figure 1). In each of these cases, (16) determines the evolution of _λ_ .<sup>15</sup> Participation constraint (17) defines a critical discount factor _δ_<sup> ̄</sup> such that for all _δ ≥ δ,_<sup> ̄</sup> all three _λ_ intervals have at least one point in common. In particular, _λ_ 0 = 1 for sure belongs to the overlap region, and thus _λ_ never changes from its initial value and electricity transfers achieve full insurance with each power system consuming _y_ ̄ in every period and every state.<sup>16</sup> By proposition 2, when _δ ≤_ _<u>δ</u>_ <u>,</u> no non-autarkic contract is sustainable. Hence, in each state

> 15In each of these cases, _<u>λlh</u>_ and _<u>λmm</u>_ can be obtained through the following three-step procedure. First, starting from an end-point of a _λ_ interval, the evolution of _λ_ described in Proposition 1 allows expressing the discounted life-time utility of each power system in terms of current period utility and continuation discounted life-time utilities starting from various end-points of _λ_ intervals. Solving the system of thus obtained simultaneous equations, it is possible to obtain all relevant discounted life-time utilities in terms of current period utilities in different states. Then the first order condition (13) can be used to express these current period utilities as functions of _λ_ ’s. Finally, binding participation constraints of power system 2 in states _lh_ and _mm_ can be used to solve for _<u>λlh</u>_ and _<u>λmm</u>_ as required.

> 16Since in any asymmetric efficient allocation one of the power systems is strictly worse off than in the symmetric efficient allocation, the participation constraint of the exporting power system is satisfied for the largest set of discount factors when _c_<sup>_s,s_</sup> 1<sup>_′_</sup> = _c_<sup>_s,s_</sup> 2<sup>_′_</sup> = _Y_<sup>_s,s′_</sup> _/_ 2 _,_ where _Y_<sup>_s,s′_</sup> is the aggregate production in state _ε_<sup>_s_</sup> _, ε_<sup>_s′_�</sup> _._ Hence, when _δ_ = _δ,_<sup> ̄</sup> the equal-payoff contract is the only sustainable first-best efficient contract. � The set of sustainable first-best efficient electricity sharing arrangements expands as _δ_ increases.

21
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## **Independent shocks**

Suppose that production shocks are independent across the power systems and across time and can take two values _{−ε, ε}_ with P ( _εt_ = _−ε_ ) = _p._ There are then four states, labeled _hh_ , _hl_ , _lh_ and _ll_ , where _hh_ indicates that both power systems produce _y_ ̄ + _ε, hl_ indicates ̄ ̄ that player 1 produces _y_ + _ε,_ while player 2 produces _y − ε,_ and so on. In this example the aggregate production of electricity varies across states. Nevertheless, the formal analysis is almost identical to the example with perfectly negatively correlated production shocks. This is because with logarithmic utility, the _λ_ intervals for the states _ll_ and _hh_ coincide and by symmetry there are again just three cases to consider: (1) _λ_ intervals are disjoint; (2) intervals partially overlap; and (3) all _λ_ intervals have some points in common.<sup>18</sup> Given the parametrization, Figures 1, 3 and 2 are also applicable to this example, if interval � _<u>λmm,</u>_ ̄ _λmm_<sup>�</sup> is understood to apply in states _hh_ and _ll_ .

As before, also in this example, in a state with asymmetric production distribution, the disadvantaged power system receives an electricity transfer and becomes a debtor. The current debtor repays the debt in states with symmetric production distribution, but now the repayments are scaled up or down, proportionately to the changes in aggregate electricity production.

# **4.2 Cooperative Outcomes with Privately Observable Shocks**

In Section 4 we showed that long-term cooperative arrangements can deliver some benefits of connecting power systems when power systems are sufficiently patient. However, the analysis relied crucially on the assumption that there is an independent system operator who can perfectly observe production shocks in the connected regions. In practice, establishing such a monitoring institution is associated with significant costs, especially in developing countries. Moreover, even if such operator exists, it cannot always discover the true nature of production shocks. For instance, it is very difficult to establish whether a decrease in electricity production is due to a genuine outage or is a result of strategic manipulation of generators (Fogelberg and Lazarczyk 2014). This suggests that in reality power system’s production shocks are likely to be observed exclusively by that power system.

> 18See Ligon et al. (2002) for details.

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If production shocks in the connected power systems are not perfectly observable by all parties, welfare gains that could be realized through an electricity sharing arrangement are lower relative to the case of no information asymmetries. This is because to avoid exporting electricity, each power system is tempted to misreport its supply conditions when current production is relatively high, but this cannot be observed by the other party. To create sufficient incentives for truthful revelation of the production shocks, in addition to participation constraints, also incentive compatibility constraints must be satisfied in every period. These additional constraints should ensure that demanding an electricity transfer from the other party in the current period is punished by lower expected future electricity imports, or, equivalently, exporting electricity in the current period is rewarded by higher expected electricity imports in future.<sup>19</sup> The importance of incentive compatibility constraints in models with privately observed random shocks has been long recognized. For example, Atkeson and Lucas (1992) study optimal consumption smoothing in an economy with large number of consumers who privately observe taste shocks affecting their marginal utility of current consumption, while Thomas and Worrall (1990) study optimal lending agreements in a model where a risk averse borrower is privately informed of his income shocks. However, these papers assume full commitment, i.e., players are committed to the contract signed at the beginning and cannot walk away from it at the later date. More recently Hertel (2004) characterizes optimal risk sharing in a model with two players, who can opt out of the agreement at any time and where one agent’s stochastic income realizations are his private information. While Hertel (2004) is relevant for modeling cooperation between connected power systems, deriving full characterization of an electricity sharing contract with unobservable production shocks is beyond the scope of our paper. Instead, we argue that a quasi-market for excess electricity is a simple way to achieve some cooperation when production shocks are privately observed by each power system.

> 19Under perfect monitoring of shocks, a negative production shock also results in a reduction of current and future electricity consumption, when discount factor _δ_ is relatively low and the participation constraint of the exporting power system is binding. However, the optimal electricity sharing arrangement is always forward looking as each power system is willing to fulfill its current export obligation as long as it expects sufficient benefits from continuing cooperation in the future. In contrast, with unobservable production shocks, the reduction in future expected electricity consumption following a run of low production reports is a _retrospective_ disciplining device that ensures truthful reporting of shocks. For example, results from Hertel (2004) suggest that in an optimal electricity sharing arrangement with unobservable production shocks, electricity consumption should strictly decrease over time for a power system that is repeatedly hit by an unfavorable production shock even when no participation constraints are binding.

27
The quasi-market relies on a “chip mechanism”. In this mechanism, the connected power systems are endowed with a certain number of megawatt chips. The chips can only be used to keep track of electricity import/export imbalance between the parties to the contract and are worthless outside the cooperative relationship. When power system _i_ exports electricity to power system _j_ and _j_ owns some chips, _j_ gives chips to _i_ in exchange for electricity. When one of the power systems has all of the available chips, it stops exporting electricity to the other system until the other system has earned back some chips.<sup>20</sup> By the way of illustration, consider the example in Section 4.1 with independent shocks taking two values _−ε_ and + _ε_ . In this context, suppose that each player is privately informed of the realization of its current production shock and the interconnector’s capacity is _K ≤ ε_ . Consider the chip mechanism with _N_ chips that fully utilizes the interconnector’s capacity, the _N_ **-chip** _K_ **-capacity mechanism** . In this mechanism, whenever a power system holding some chips experiences a negative production shock, but the other power system has high production, the power system with low current production receives an electricity transfer of _K_ in exchange for a chip.

Let _Vn_<sup>_N_,</sup><sup>_n_= 0</sup><sup>_,_1</sup><sup>_, ..., N_denote a player’s expected discounted payoff in the</sup><sup>_N_-chip mechanism</sup> when he holds _n_ chips. Then the _N_ -chip mechanism delivers the following payoffs: for _n_ = 1 _, ..., N −_ 1 and at the boundary

> 20M ̈obius (2001) and Hauser and Hopenhayn (2008) study chip mechanism in the context of continuoustime repeated favor-exchange game; there is also a rich literature on favor exchange in discrete time (see e.g., Nayyar 2009, Kalla 2010, Abdulkadiroglu 2013, Abdulkadiroglu and Bagwell 2012). Olszewski and Safronov (2012) study chip strategies in repeated oligopoly games, in which firms privately observe their costs of production.

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By Lemma 1 in Abdulkadiroglu and Bagwell (2012), this system of equations has a unique solution � _Vn_<sup>_N_</sup> � _Nn_ =0<sup>.</sup> Each power system wants to participate in the mechanism if its participation constraint is satisfied, i.e., it must be the case that for _n_ = 0 _, ..., N_ , _Vn_<sup>_N≥_</sup><sup>_<u>v</u>_</sup> _<u>,</u>_ where _<u>v</u>_ is defined in (1) and represents the minimum lifetime utility that each player can guarantee himself in autarky. In addition, to induce the power system with positive production shock to report its production truthfully, the mechanism must satisfy the following incentive compatibility constraints: for _n_ = 1 _, ..., N −_ 1 and at the boundary The constraint (19) ensures that the player holding no chips wants to reveal high production levels, despite the possibility that with probability (1 _− p_ ) the opponent will request an electricity transfer from him in exchange for a chip. The constraint (20) ensures that the current holder of all chips will want to reveal high production levels, forgoing an electricity transfer from the opponent that would increase the current electricity consumption to ̄ _y_ + _ε_ + _K_ if the opponent’s production level also turns out to be high. The constraint (18) ensures that the player holding _n_ chips wants to reveal high production levels, despite facing both the temptation of a player with no chips and the temptation of a player with all the chips. A chip mechanism satisfying constraints (18)-(20) is called _incentive compatible._ Proposition 3 shows that for high discount factors, there is an incentive compatible _N_ -chip _K_ -capacity mechanism that improves on autarky, but even the best incentive compatible _K_ -capacity chip mechanism delivers payoffs lower than in the case of publicly observable production shocks. When _δ_ is high, the main cause of inefficiency in the chip mechanism is accumulation of chips in the hands of one party. The power system holding all chips stops exporting electricity until it receives some electricity imports. This implies that cooperation 29
partially breaks down when one of the power systems is subject to a prolonged run of negative production shocks and is unable to export any electricity. For lower values of _δ,_ a _K_ -capacity chip mechanism also fails to adjust the magnitude of the required electricity transfers in an optimal manner.

**Proposition 3.** _There exists a critical_ 0 _< δ_<sup>_∗_</sup> _<_ 1 _such that for all δ ∈_ ( _δ_<sup>_∗_</sup> _,_ 1) _, there exists an incentive compatible K-capacity chip mechanism. Any incentive compatible K-capacity chip mechanism improves on autarky, but yields a payoff that is lower than the payoff in the optimal symmetric contract with observable production shocks (i.e., the contract with λ_ 0 = 1 _)._

_Proof._ Consider the chip mechanism with one chip.<sup>21</sup> In this mechanism, the only binding constraint is the incentive constraint of the player with no chips, (19), as this constraint implies (20) (by strict concavity of the utility function _u_ ( _·_ )) as well as the participation constraints of the power systems. Hence, by rearranging (19) it is possible to obtain a critical discount factor _δ_<sup>_∗_</sup> _<_ 1 such that for all _δ ≥ δ_<sup>_∗_</sup> the one-chip _K_ -capacity mechanism is incentive compatible. This mechanism improves upon autarky as players make positive electricity transfers thereby achieving some electricity consumption smoothing.

Consider an incentive compatible _K_ -capacity chip mechanism and compare it to the optimal contract with observable production shocks described in Section 4.1. If _<u>δ</u> < δ < δ_<sup> ̄</sup> , the optimal contract requires transfers in states with symmetric as well as asymmetric production distribution, even though there are no a priori restrictions on admissible transfers. In any _K_ -capacity chip mechanism, however, players make transfers only in asymmetric production states. Consequently, in this range of _δ,_ any incentive compatible chip mechanism must yield lower life-time utility to players than the optimal contract. Suppose _δ ≥ δ._<sup> ̄</sup> For this range of _δ_ , the optimal contract with _λ_ 0 = 1 requires transfers of size _ε_ only in asymmetric production states. Moreover, since the contract is forward looking, there is no upper bound on the number of transfers a player makes before receiving a transfer back. In any incentive compatible chip mechanism, however, the number of chips must be finite. Moreover, since production shocks are iid, a history of shocks which requires a player to make more than _N_ consecutive transfers can occur with a strictly positive probability. In this case, one of the players accumulates all the chips, upon which he makes no more electricity transfers until

> 21This mechanism corresponds to the _simple EM relationship_ of Abdulkadiroglu and Bagwell (2012).

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the other player earns some chips back by exporting electricity. Thus, even if _K_ = _ε,_ an incentive compatible chip mechanism delivers less than maximal insurance and yields payoffs lower than the payoffs in the optimal contract with observable production shocks.

Note that in an _N_ -chip mechanism, condition is sufficient for incentive compatibility as it implies all the incentive compatibility as well as participation constraints of the power systems. Given this, it is possible to apply the algorithm of Abdulkadiroglu and Bagwell (2012) to find the best _K_ -capacity chip mechanism satisfying (21):

- Start with _N_ = 1: Given _N_ -chips, write the expected payoffs recursively and calculate the unique solution of the resulting system of equations.

- Check if (21) is violated. If it is violated, then the optimal number of chips is _N −_ 1; otherwise repeat the two steps with _N_ + 1 chips.

Figure 5 illustrates the scope for thus found _ε_ -capacity and _ε/_ 2-capacity chip mechanisms to improve on autarky, when _y_ ̄ = 100 _, ε_ = 15 and _p_ = 0 _._ 2 _._ The figure demonstrates that as _δ_ increases, the number of chips consistent with incentive compatibility increases, thereby increasing the welfare benefit of the mechanism. Reducing _K,_ the capacity of the chip mechanism, reduces its welfare gains for high _δ,_ at the same time relaxing the binding incentive compatibility constraints for lower _δ._ Thus reducing _K_ can increase the range of the discount factors for which some cooperation is feasible. This is consistent with findings in Section 4: in an optimal sustainable contract, electricity transfers in asymmetric production states are lower than _ε_ when discount factor _δ_ takes intermediate values. For comparison purposes, Figure 5 also depicts the per-period utility of a power system participating in an optimal sustainable contract. It is clear that establishing an independent system operator capable of monitoring production shocks in the connected power systems brings much higher welfare gains than a simple chip mechanism considered in this section.

The chip mechanism considered here effectively is a flexible electricity swap agreement that breaks even on average (i.e. where power flows between the parties to the agreement are 31
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balanced in the long-run). A more sophisticated chip mechanism could feature more flexible use of volumes or price adjustments. For example, it could feature changing terms of trade whenever a power system becomes constrained, so that each additional MWh of electricity provided by the constrained power system commands a higher price in terms of future electricity transfers. This could induce power systems to provide more electricity export than in the case where the exchange rate between a MWh today and MWh in future is always one.

# **5 Conclusions**

This paper proposes a theoretical framework for analyzing the scope for voluntary electricity sharing arrangements among power systems where functioning contract enforcement institutions do not exist and are difficult to establish. Drawing from the game-theoretic literature, we adapt the efficient risk sharing model with limited commitment to electricity sharing arrangements, where each power system can terminate ongoing cooperation at any moment. The paper’s main insight is that a voluntary electricity sharing arrangement is capable of bringing welfare gains when electricity production shocks are perfectly observable by an independent system operator as well as when production shocks are only privately observable by each connected power system.

Introducing this theoretical framework to a new, highly idiosyncratic, electricity sector comes at a cost; in this early stage we have to assume that the transmission capacity is exogenous to derive key analytical results. In the real world, power systems, which have not been previously involved in cross-border electricity exchange, are unlikely to have extensive transmission lines across borders. In most developing countries, transmission constraints are a serious setback to electricity trading, as the existing transmission capacity is insufficient for clearing most of the requested cross-border trades. In future research we plan to explicitly model investment in new transmission lines, thereby endogenizing the transmission capacity. We also aim to account for the possibility that the transmission capacity is subject to failures due to unexpected physical damage or routine maintenance work on transmission lines.

Our theoretical results indicate that for successful cross-border cooperation, the predetermined risk of terminating an ongoing relationship should be sufficiently small. Hence, a meaningful quantitative evaluation of the scope for electricity sharing requires an estimate of 33
this risk. In practice, the risk of termination may stem from, e.g., political economy issues in one or both countries that host the interconnector, but further research is required to analyze carefully the determinants and magnitude of this risk in specific regions of interest.

Another important dimension for further research is to deploy the theoretical framework to quantify feasible welfare gains from electricity exchange in regions contemplating crossborder cooperation, such as e.g., the South Asia region. In its current form, the model is well suited for studying asymmetric bilateral relationships. The analysis of multilateral pools will be qualitatively similar to that of bilateral contracts with participation and incentive compatibility constraints limiting the extent of cooperation. However, calibration and simulation of numerical models to fit actual grids and interconnector capacities in the regions of interest will be computationally more challenging because of extending the framework to power pools consisting of more than two parties, and using more complex functional forms for characterizing electricity supply and demand relationships.

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