Risk Aversion and Hydro-energy Reservoirs Management under Monopolistic and Oligopolistic Industrial Structures

This article analyses management of hydropower dams within monopolistic and oligopolistic competition and when hydroelectricity producers are risk averse and face demand uncertainty. In each type of market structure, we analytically determine the water release path at the closed-loop equilibrium. In the monopolistic case, we demonstrate that intertemporal allocation of hydro resources is done by additional pumping or storage depending on the relative abundance of water between different regions in order both to internalize the risk and to smooth the effect of uncertainty on electricity prices. In the oligopolistic case, we determine the conditions under which the relative scarcity (abundance) of water in the dam of a hydroelectric operator can favor additional strategic pumping (storage) in its competitor’s dams. Finally, we compare the effects of risk aversion on the output industry. We show, in monopolistic case, that the greater the generators’ risk aversion the larger the current water storage for future demands. In the oligopolistic case, however, the risk aversion increasing may be either in favor of reducing or increasing the total hydro output depending on demand uncertainty and market power parameters. We deduce that risk aversion coupled with market power may induce, under some condition, a misallocation of hydraulic resource across time.


INTRODUCTION
The electricity industry deregulation across the world has been motivated by several factors such as economic efficiency in production and investment, best resources allocation, benefits for consumer through lower prices and higher security.Some countries have liberalized only the high voltage generation and established a wholesale electricity market which matches supply and demand 1 .Following the liberalization of electricity industry, several issues in the wholesale market have been analyzed by the economic literature such as the inefficiency problem 2 due to the strategic 1.
Deregulation processes have engendered several market structures.The industry is ranked according to four models based on the degree of competition: vertically integrated monopoly, single buyer, competition on the wholesale electricity market and competition on the retail market (Joskow, 2006, Hunt, 2002).2. Borenstein et al. (2002) assessed the economic inefficiencies in California's electricity market caused by the exercise of market power, fluctuation input factor and scarcity of generation capacity.
behavior and the exercise of market power of electric producers in the purely thermal systems or mixed hydrothermal systems 3 .The problem of dynamic water allocation efficiency in deregulated purely hydroelectric industries 4 has attracted little attention.Ambec and Doucet (2002) examined the problem of managing run of river hydropower dams under a monopolistic and oligopolistic structure when water inflows are deterministic.They used a twoperiod model to show that the absence of a water market during hydroelectric production can engender two sources of loss of social welfare: Suboptimal management of water resources and the exercise of market power.Van Ackere and Ochoa (2010) used a stylized deterministic simulation model to evaluate the impact of the liberalization of the hydroelectric industry on the quantity and price of electricity produced.They show that total electricity production is clearly lower in a non-liberalized market.Haddad (2011) developed a two-period model that characterizes the effects of deterministic seasonal water inflows on storage capacity optimal water management.These studies have analyzed water allocation in deregulated electricity under a deterministic framework.However the impact of competition between hydroelectric generators that face both inflows and demand uncertainties can be analyzed through a stochastic dynamic game framework. 5In addition, the works above have not integrated the risk dimension, which is quite salient for suppliers of electricity from hydraulic structures.Water reserves are renewed randomly by precipitation.Thus, in the extreme case that precipitation is zero.The water stock is a temporarily finite resource because using a unit of water stored in the dam would constitute one unit less for the following period.Given the climate change phenomenon and the associated series of extreme events (IPCC, 2014), the challenge of optimal management of this resource over several periods of time has become more pressing.On the demand side, operators of hydroelectric plants are also facing several sources of uncertainty closely linked to different categories of electricity consumers.Residential demand strongly depends on climate conditions, which determine the intensity of use of home appliances, along with electricity prices (Reiss and White, 2005;Dergiades and Tsoulfidis, 2008).In contrast, commercial and industrial demand is strongly associated with economic conditions, and some macroeconomic policy changes (Dilaver and Hunt, 2011).Electricity demand (residential, industrial and commercial) therefore fluctuates over the short term, and operators may find it difficult to smooth prices (Genc and Thille, 2011).Climate change may also exacerbate water and energy tensions across sectors and regions, potentially impacting hydropower (either positively or negatively, depending on whether the potential climate-adaptation benefits of hydropower facilities are realized) and on other technologies that require water (Arent et al., 2011;Cisneros and Oki, 2014).Overall, fluctuations in water reserves between different hydroelectric sites coupled with electricity demand uncertainty may favor market inefficiencies added to strategic behavior by hydroelectric operators in the harnessing of this potential energy.Philpott et al. (2013) consider this dimension of risk in their analysis and show that risk hedging instruments can reduce losses of welfare associated with the presence of electricity supply uncertainty.However, these authors focus on equilibrium on a competitive market. 5Garcia et al. (2001), Dakhlaoui and Moreaux (2004) and Genc and Thille (2012) explored the implications of imperfect competition on electricity markets under uncertainty and without 5.Gen et al. (2007) studies several dynamic oligopolistic games under uncertainty by using a stochastic programming approach.
6. Philpott and Guan (2013) empirically analyze social welfare following the opening of a wholesale electricity market in New Zealand when the social planner is risk averse about uncertainty of water inflows.Aslo see Genc and Sen (2008) for an empirical analysis.
the risk aversion dimension.To the best of our knowledge, analytical works that examined imperfect competition when uncertainty is present did not consider hydropower producers' risk aversion.Our paper contributes to economic literature on electricity market by providing few answers to water resource allocation problems where all hydroelectric operators act strategically in the wholesale electricity market and facing both inflows and demand uncertainties.
To do so, we develop a stochastic and dynamic model in which the hydroelectric park comprise multiple mountain reservoir-type dams 7 , and posit two different industrial structures: monopolistic and oligopolistic.We analytically determine the water release path at closed-loop equilibrium.We show how a monopoly manages hydropower reservoirs by additional pumping or storing depending on the relative abundance of water between different regions to smooth the effect of demand uncertainty.In addition, risk aversion reduces variation of water pumping when the net flow of precipitation is either positive or negative.In the oligopolistic case with symmetric risk aversion, we determine the conditions under which the relative scarcity (abundance) of water in an operator's dam can favor strategic additional pumping (storage) at the competitor's dams.We compare the effects of risk aversion on the output industry.We show that, in monopolistic case, the greater the generators' risk aversion the larger the current water storage for future demands.In the oligopolistic case, however, the risk aversion increasing may be either in favor of reducing or increasing the total hydro output depending on demand uncertainty and market power parameters.
We deduce that risk aversion coupled to market power may induce, under some condition, a misallocation of hydraulic resource across time.Our results have practical economic implications.Indeed, the market inefficiencies due to uncertainties needed market regulation in order to reduce the possibility negative effects of uncertainty and market power on the intertemporal water resource allocation.The strategic water conservation by risky generators is due principally to the absence of market to share risk which can be solved by attribution of risk premium or implementing forward contracts.
The rest of the paper is organized as follows.Section 2 presents the dynamic and stochastic model of management of hydroelectric park with multiple dams.Section 3 analyzes the situation of a monopoly.Then, we examine in section 4 the case of oligopoly with risk symmetry.The last section concludes the paper.

Electricity Demand
Demand in period t is represented by an inverse demand function of linear form (Genc and Thille, 2012): Where p t represents the price of electricity, Q t the quantity demanded, b is a positive constant and the parameter at  is normally distributed with an expectation of a and a variance of σ 2 . 6 7. Pritchard et al. (2005), studies the optimization of hydroelectric reservoirs in a pool market by developing a dynamic programming model that groups trading periods into stages with the same stack offered at each trading period within a stage and using a normal distribution to represent the total amount of water used in each period

Electricity Production
Let us define by q it the electricity production of dam i (i=1,…,n) at time t and such that total electricity and production of different ) is totally consumed. 7Each facility i uses water stored in a hydropower dam of region i, denoted by s it .Without lost of generality, we assume that each unit released from the dam allows free generation of one unit of electricity.The dam of region i is regularly recharged by random flows of precipitation f it , which follow a random walk where E (f it ) = F i with F i > 0. 8 The evolution of the water available in the dam in region i is governed by the following recurrent equation on the dynamics of the stock: (2)

Hydropower Producer's Objective Function
At the start of each period t, the hydropower operator observes s it , the water available in region i, with certainty.He then decides on production q it which maximizes the discounted sum of expected utilities of the profit from operation of all hydropower facilities while meeting the n dynamic stochastic constraints on the evolution of water in various dams.
Following Hedfi et al. (2021), we assume that the cost of storage is a quadratic 9 function of the difference between the current stock, s it , and future storage at the end of period t+1, s it+1 .In other words, the electricity producer has to pay an environmental cost to minimize water level fluctuations.The regulator imposes a penalty on the difference between the current stock of water and that of the next period.The environmental cost is specified as for each hydropower dam, with γ > 0. The operator's instantaneous profit at period t is written as the difference between total revenues (RT t ) and the cost of storage in n reservoirs: We assume that at the beginning of each period, The operator decides the optimal electricity quantity that maximizes the expected utility of his profit.Utility function is characterized by The positive parameter A is the constant absolute risk aversion of Arrow-Pratt that measures the intensity of the player's risk aversion.U(.) is a continuous, strictly increasing and concave function. 10Given the 8.We assume that demand variations are mainly structural (affecting only the central tendency) and that there is no significant reason for the volatility to change across periods.9. To simplify the example, we assume that we never encounter the power transmission capacity constraint.Head loss during transmission is also ignored.10. Garcia et al. (2001) assume that natural filling of a dam follows a binomial distribution, whereas Genc and Thille (2011) assume a normal distribution.
In general, hydrologists posit a Markovian process (Karamous and Vasiliadis, 1992;Faber and Stedinger, 2001).11.Dakhlaoui and Moreaux (2004) presumed that the storage cost is a quadratic function of the difference between the current stock, s it , and an exogenous target storage s * .
12. The marginal utility is then a decreasing function, ie, for a given increase in exponential function, it can be shown that the maximization of the expected utility of the random profit of the producer at time t is equivalent to the maximization of the following equation: 11 We assume an electricity industry with a monopolistic structure.
Using the demand function defined by and the profit function given by , the operator's objective function defined by is: By assuming that the precipitation inflows to the different hydropower dams are independent, we can therefore write equation as: The monopoly chooses the electricity production path by n hydropower dams as a solution to the dynamic and stochastic optimization problem with an infinite horizon.The solution to the optimization problem of hydroelectric operator q it is (see Technical Appendix A1): Based on and the dynamic equation of stocks given by the expression in , the solution to the variation in stock in the hydro power dam in region i is: (8) and total production of the monopoly is:

Case of Abundance of Water in Region i and Shortage in Region j
In this case, there is an increase in the average precipitation inflows in region i (∆F i > 0), and an average decrease in flows in region j ((∆F j > 0).The average flow is presumed constant in regions other than i and j.
We assume that the gap between the variation of the rate of refilling of dam i and that of dam j is written as: ∆F j = −∆F i + profit, usefulness gets lower as earnings increase.13.Larue and Yapo (2000) considered the same utility function and expected profit specifications while assuming the response function of the risk-averse competitor as uncertain.
φ with ∆F j < φ < −∆F i .The total effect of fluctuations in water inflows in the two regions on production by the power plant of region i (∆q it ) is: The operator must therefore perform additional pumping from the dam of the region with abundant water and additional storage in the dam in the region with scarce water.The total impact of fluctuations in water inflows in the two regions i and j on production by the dam of region j is: Further, variation in total production of the two dams ' ' ' The variation in total production of the two dams depends on the sign of φ.
and , the impact of fluctuations of recharge flows on the amount of water available in the hydropower dam in region j is identical to that observed in region i: These results show that under its optimal solution, the monopoly have to keep the same change in stock in all dams by increasing its electricity production in the region with abundant water and decreasing production in the region with scarce water.
The variation in stock in the dams depends on the net variation in flows in the two regions.
To analyze the effect of risk aversion on total production, we compare the variation in total production with and without risk aversion.We have: 12. In the case where the average variation in flows of the two dams is not identical, ' ' F F i j z , the monopoly assumes additional storage costs in the two dams.The more φ increases, the higher these costs.In the particular case where I 0 ' ' F F i j , the total effect on hydropower production in regions i and j is ' ' q F i i and ' ' q F j i .In this case as well, the operator must satisfy electricity demand through additional pumping from the reservoir in the region with abundant water and reduce its production in the region with scarce water.To smooth the effects of this fluctuation of flows on electricity prices, it must ensure that this additional storage in the region with scarce water equalizes the additional discharge in the region with abundant water: This result shows that when the operator is risk neutral and the net flow of precipitation is positive (negative), the increase (decrease) in total production of the two regions is greater than when the operator is risk averse.

Case of Abundant Water in Regions i and j
We denote the gap between the variation in the refilling speed of dam j and that of region i by ω:

Proposition 1
In the case where the hydropower producer observes an increase in the average flow of precipitation in the two regions i and j (∆F i > 0 and ∆F j > 0), he decides to perform: • Additional pumping (storage) at the dam in region i (∆q it > (<) 0) if ω < (>)τ∆F j • Additional pumping (storage) at the dam in region j(∆q jt > (<) 0) if ω < (>)τ∆F i if • Additional pumping at the dams in regions i and j (∆q it > 0 et The proof of proposition 1follows from equations ( 13) and ( 14).
The additional discharge in the two regions due to abundant water is given as: Whereas the additional storage of water is evaluated at: In this case we have: |∆storage| < ∆discharge.In other words, in the case of abundant water in both regions, the additional interannual transfer of water from period t to period t+1 is less than its additional use at period t.Further, the analysis of the effects of the risk aversion coefficient show that w ' w storage A 0 and w ' w !discharge A 0 .These results are summarized by proposition 2. 13

Proposition 2
In the case of abundant water in regions i and j (∆F i > 0 and ∆F j > 0), the hydroelectricity producer uses two hydropower reservoirs to satisfy current demand for electricity.However, it should not fully use additional recharge to satisfy current demand for electricity energy in the form of water to satisfy future demand.Further, the greater the hydroelectric operator's risk aversion, the larger the quantity of water stored for future demand.

DAM MANAGEMENT UNDER AN OLIGOPOLISTIC STRUCTURE WITH SYMMETRIC ATTITUDE TOWARD RISK
In this section, we consider an electricity industry with an oligopolistic structure, with n firms that compete a la Cournot.The profit of hydroelectric firm i in period t is written as: At each period t, hydroelectric operator i maximizes the expected utility of its profit with a utility function characterized by constant absolute risk aversion identical for all hydroelectric operators (A i = A j = A).The objective function of firm I is therefore: The solution to the problem of maximization gives us equilibrium production of the dynamic Cournot closed-loop game of firm i (see Technical Appendix A2): In the case where the hydroelectric dams are independent and are in the same region (A i = A j = A), hydroelectric production of firm . Therefore, 13.In the case of a water shortage in regions i and j, we obtain the opposite results to those found in this section.
w w . This shows that the increase in storage costs motivates producer i to deviate from its equilibrium strategy by additional pumping of its current stock if and only if the average recharge speed of dam i is markedly higher than the water release rhythm from dam i in period t (case of strong abundant water), ∆s > 0. Conversely, the operator must perform additional storage in the case of low water levels ∆s ≤ 0 14 .Further, from (17): Any increase in risk level reduces production of firm i in period t when all hydroelectric operators have the same average flow of precipitation.Lastly, all increases in the risk aversion rate of firm I reduce the immediate use of water, that is it favors an intertemporal transfer of water from t to t+1.
Hydroelectricity producer i's production at period t depends positively on its inflow rates and negatively on the sum its competitor's flow of precipitation.Total equilibrium hydroelectric production at period t is written as:

Case of Water Abundance in Region i and Scarcity in Region j
In this case, the hydroelectric producer in region i observes an increase in its average flow of precipitation (∆F i > 0), whereas the hydroelectric producer in region j observes an average decrease in its flows (∆F j < 0).The average flow is presumed constant in regions other than i and j.Let κ be the gap between variation in the recharge rate of dam i and that in dam j: ∆F j = ∆F i + κ with ∆F j < κ < ∆F i .The total effect of the variation of average recharge inflow in the two regions on production of hydroelectric dams i and j is: 14 We have We therefore deduce that the hydroelectricity producer in the region with abundant water must do additional pumping, and inversely the hydroelectricity producer in the region with scarce water must do additional storage.In the case where ∆F j = ∆F i (κ = 0) additional pumping in the region with abundant water corresponds to additional storage in the region with a water shortage (∆q it = −∆q it ).Based on and , the impact of variation of the average water inflows on the variation of stock in the dams of regions i and j is:

Proposition 3
In the case where the hydroelectric operators in regions i and j observe an increase in the average flow of precipitation in region i and a decrease in average flow in region j (∆F i > 0 and ∆F j < 0), we have: (i) A decrease in the stock of water in the dams in regions i and j (d∆s it+1 ≤ 0 and d∆s jt+1 < 0 if ∆F j < κ ≤ η∆F j .(ii) A decrease in the stock of water in the dams in region j and an increase in the stock of water in the dam of region i (d∆s it+1 > 0 and d∆s jt+1 ≤ 0) if η∆F j < κ ≤ η∆F i (iii) A decrease in the stock of water in the dams of regions i and j. (d∆s it+1 > 0 and The proof of proposition 3 follows from equations ( 21) and ( 22).
Proposition 3 specifies the conditions under which, in a structure of imperfect competition, the possibility of storing electricity in the form of water may motivate hydroelectric operators to manage their water resources strategically according to the relative scarcity of water inflows in the dams.Case (i) is that of two dams situated in regions where the average abundance of water resources in the dam of hydroelectricity producer i does not compensate for the average scarcity in the dam of hydroelectricity producer j.In this case, the two hydroelectric operators benefit from deviating from their equilibrium strategy by performing additional pumping in the two regions.Despite the shortage of water in region j, hydroelectricity producer j knows that its competitor does not have sufficient additional water resources to play strategically against it on the electricity market.Hydroelectricity producer j consequently performs strategic pumping in its own dam despite its water shortage.
Condition (ii) is that of a dam of hydroelectricity producer j, which suffers from a shortage whereas the dam of hydroelectricity producer i experiences an increase in average water inflows.In this case, the abundance of inflows prompts the hydroelectricity producer to play strategically on the electricity market against hydroelectricity producer j, which suffers from a severe water shortage, by doing additional storage of this positive variation of its water inflows: this corresponds to strategic water storage.By additional pumping despite the scarcity of its water resources, producer j exacerbates its situation.
Under condition (iii), the hydroelectric operators increase additional storage in both dams when the gap between the variations in the average water inflows in region i and that in region j is less than the increase in average water in flows in the region with abundant water.In other words, in the case where the two dams are in two regions where the average abundance of water resources in the dam of hydroelectricity producer i can relatively compensate for the average scarcity in the dam of hydroelectricity producer j, the two players deviate from their equilibrium strategy by additionally reducing electricity production in both dams, i.e. the one with scarce water and the one with abundant water.The hydroelectricity producer with a water shortage, namely that in region j, knows that its competitor has not a sufficient increase in water inflows to play strategically against it.Because the water scarcity of hydroelectricity producer j is not severe relative to the water abundance of hydroelectricity producer i, then hydroelectricity producer j can respond to the strategic storage of hydroelectricity producer i by doing its own additional storage.
The solutions of ∆q it and ∆q jt (equations [19] and [20]) let us deduce the impact of the variation of precipitation inflows on total The variation in total production of the two dams depends on the sign of κ.
In the case where the increase in average water inflows in region i is less than the average scarcity in dam j, then total equilibrium production decreases.In other words, the relative scarcity of water resources in the whole stock favors additional storage of water.Conversely, relative abundance of water favors additional pumping at all hydropower reservoirs.In both cases, regardless of the hydroelectric operators' strategic behavior, the reservoirs will be operated with respect to merit order principle.
The effect of the risk aversion coefficient on ∆Q t C (the impact of the variation of precipitation inflows on quantity) is not monotone.
The sign of is determined by the sign of

Case of Abundant Water in Regions i and j
In this case, the hydroelectric operators in regions i and j observe an increase in the flow of precipitation (∆F j > 0 and ∆F i > 0).Let us denote by χ the gap between the variation in the recharge speed of dam j and that of region i: We then have:

Proposition 4
In the case where the hydroelectric operators in regions i and j observe an increase in the average flow of precipitation (∆F i > 0 and ∆F j > 0), we have: (i) Additional pumping from the dam of region i and additional storage in the dam of region j (∆q it > 0 and ∆q jt ≤ 0) if −∆F i < χ ≤ δ∆F i .(ii) Additional pumping in the dams of regions i and j (∆q it ≥ 0 and ∆q jt > 0) if −δ∆F i < χ ≤ δ∆F j .(iii) Additional pumping from the dam of region j and additional storage in the dam of region i (∆q it < 0 and ∆q jt > 0) if Proposition 4 states the conditions under which the possibility of storing electricity in the form of water, coupled with imperfect competition, can lead hydroelectric operators to manage their water resources strategically according to the relative abundance of water inflows in the dams.Under condition (i), hydropower operator i has a larger increase in average water inflows than that of hydroelectricity producer j.Because hydroelectricity producer i knows that the average increase in its water inflows can compensate for the gap in the variation of inflows in the two regions, he decides to do additional pumping on its dam.Consequently, hydroelectricity producer j can respond only by additional storage.In case (ii), hydropower operator i experiences an increase in its average water inflows that is slightly greater than that which occurs at the dam of hydroelectricity producer j.
In this case, both hydroelectric operators do additional pumping.
In case (iii), hydroelectricity producer j has a larger increase in its average inflows than that of hydroelectricity producer i.The relative abundance of water resources in the dam of hydroelectricity producer j compared with that of hydroelectricity producer i prompts operator i to increase its production, whereas hydroelectricity producer i will store its additional inflows.

CONCLUSION
This paper has analyzed the effect of uncertainties on the water resource management under two industrial structures, monopolistic and oligopolistic, when hydroelectricity producers are risk averse.
The analytic results differ for both cases.We show how a monopoly can manage its hydropower reservoirs through additional pumping or storage depending on the relative abundance of water between regions to smooth the effect of uncertainty on electricity prices.
In addition, risk aversion reduces the variation of water pumping when the net flow of precipitation is positive or negative.The intuition behind this result is that this water allocation policy may avoid economic inefficient due to uncertainty in the case of averse risk generator.However, under oligopolistic competition with symmetric risk aversion, we have specified the conditions under which relative scarcity (abundance) of water in an operator's dam can favor additional strategic pumping (storage) in its competitor's dams.When the average abundance of water resources in the dam of operator does not compensate for the average scarcity in the dam of the other one, the two operators benefit from deviating from their equilibrium strategy by performing additional pumping in the two regions.Conversely, when one producer suffers from a shortage whereas the other one experiences an increase in average water inflows, the second one plays strategically by doing additional storage.By comparing both cases, we deduce that risk aversion coupled to market power may induce, under some condition, a misallocation of hydraulic resource across time.Our results have practical economic implications.Indeed, the market inefficiencies due to uncertainties needed market regulation in order to reduce the possibility negative effects of uncertainty and market power on the intertemporal water resource allocation.
The strategic water conservation by risk averse generators is due principally to the absence of market to share risk which can be solved by attribution of risk premium or implementing forward contracts.Several others instruments can be used such as water trade market in particularly in the case a cascade on the same river system.Extensions to this analysis will include practical regulatory instruments into the wholesale electricity market in order to assess and compare their effects on intertemporal water allocation.Finally, we propose for future research to investigate the impact of imperfect competition and risk aversion not only on the water release but also on investment equilibrium policy and deduced the appropriate regulation instrument in order to mitigate the negative effects of market inefficiencies created by uncertainty in the electricity industry.

TECHNICAL APPENDIX A1. Derivation of Monopoly Solution
The objective function of hydroelectric operator is: With Q q t i t i ¦ , the total production of various hydroelectric dams.Assuming that precipitation flows of various dams are independent, we have: The monopoly chooses the electricity production path by the n dams' solution of dynamic and stochastic optimization problem with infinite horizon in following: given,

® °°°°°°°°0
Where E 0 (.) is the conditional expectation to information available on the stock of water in various dams at initial period.The initial stock of each dam (s i0 ) is known with certainty.The Lagrangian of this optimization problem is: Where {λ it } represents the state co-variables associated to each stochastic dynamic constraint on the storage of water in the dam i.This problem admits one solution of finite value of the objective function (Sargent, 2001).The First order conditions relative to the production of dam i at period t, q it , is: The first order conditions determining the level of water storage in the dam i at period t+1, s it+1 is: Using the delay operator L, Euler's equation is written: Where Cms it = γE t (s it − s it+1 ) is the marginal cost of storage in the dam i at period t.After simplification, the solution of λ it is: From (4), the in situ price of a unit of potential energy stored in the dam i at period t (λ it ) is equal to the discounted sum of the differences between the marginal cost at period t and the updated marginal cost at period t+1.Thus, in equilibrium, the producer must equalize the marginal value of water in stock at period t to the net marginal cost of inter-temporal transfer of electricity from period t to the following periods for later use.Thus, in each period t, hydroelectric operator does an inter-temporal tradeoff between conservation and exploitation of a water's unit in the dam of region i.If the marginal cost of storage is constant between periods (Cms it = Cms i ), the in situ price of a unit of potential energy stored in the dam i at period t also becomes constant (λ it = λ i ) and would be equal to the marginal cost of storage (λ i = Cms i ).
Taking into account the dynamic equation of storage, Euler's equation gives: Replacing λ it and λ it+1 by their expression, we obtain: for i =1,…,n.Making the sum for i, we obtain: Using the properties on the delay operator L and by replacing (6) in (5), we obtain: Using assumption of the random walk of the water inflows, the solution of q it is:

A2. Derivation of the Solution of the Oligopoly Model with an Identical Coefficients of Risk Aversion
The Lagrangian of duopoly problem is: Where {λ it } represents the state co-variable associated with the stochastic dynamic constraint on the storage of water in the dam of the hydro producer i.First order conditions are: Using recurrent equation on the dynamic of the stock, we have: Replacing ( 8) in(10), we obtain: with θ = γ + 2b + Aσ 2 .We denote by: Thereby, we obtain a system of reaction functions: Solving this system gives the following solution: Substituting dq it+1 , Z it+1 et Z jt+1 by theirs expressions in (15) and using the assumption of the random walk of the water inflows, we have, This implies: Using the delay operator L, we obtain: After simplification, we obtain: We replace H,ψ,θ by theirs expressions, we get the production of the equilibrium closed-loop Cournot dynamic game of the hydro producer i: with ψ = (θ−b) [θ + b (n−1)]