Optimal management of a renewable energy source under uncertainty

Transcription

1 Optimal management of a renewable energy source under uncertainty Catherine Bobtcheff University of Toulouse I, LERNA, CEA 21 Allée de Brienne, Toulouse, France catherine.bobtcheff@univ-tlse1.fr May 5, 2006 Abstract This paper deals with an economy in which a representative agent has access to different energy sources to produce electricity. The first model focuses on the optimal use of a dam from which water can be extracted and which is supplied with a random flow of water. The presence of constraints on a minimal and on a maximal storage capacity makes consumption smoothing possible only when the quantity of water stored in the dam lies in a certain range that we determine. In a second stage, we introduce a second energy source with unlimited supply at some exogenous competitive price. Electricity consumption smoothing is possible even when the dam is almost empty thanks to this alternative costly energy source. Moreover a comparative static analysis reveals that the marginal propensity to produce hydroelectricity is an increasing function of the price of the second technology. Therefore, the availability at a low price of the fossil source improves time diversification. Finally, the optimal electric park is composed of a number of dams that is increasing with the price of the second technology. Keywords: intertemporal expected utility maximization, JEL Classification: C61, Q25, Q42. Acknowledgements: I am particularly grateful to my supervisor Professor Christian Gollier for his help and his advice. I also would like to thank Claude Crampes, Juan Miguel Gallego, Michel Moreaux and Johannes Voget for helpful comments. 1

2 1 Introduction The aim of this work is to determine the optimal structure of an electric park that generates power with different energy sources. Indeed, governments are increasingly concerned with the use of renewable energy sources besides classic thermal power sources because they want to preserve the environment and the energy sources that are exhaustible. However, renewable energy sources are not easy to use as their availability is not constant over time. Consumers indeed do not have to support the risk of a shortage of electricity. Therefore an electric park must be determined by taking into account this random availability and its management has to solve the problem of providing enough electricity even when renewable sources are not available in the short run. Norway, for instance, is the sixth largest hydropower generator in the world and the biggest in Europe. Hydropower accounts for 99% of the electricity generated and annual production varies to a great extend in line with precipitation levels. Thus, when the country faces dry periods as it was the case in 2002 and 2003, hydropower reservoirs are very important as buffers between output and consumption. Besides hydropower, electricity is also generated from sources such as natural gas and wind. Indeed, gas-fired power station can be started up and closed down at short notice. They are suitable for providing peak-load power but have a relatively high cost. In fact, during dry periods, the loss of hydropower output is offset by increasing thermal power generation 1. In this work, we mainly focus on two energy sources: the renewable energy source has a random availability whereas the thermal power source is available at an exogenous market price. The possibility to store the renewable energy source in a dam allows to smooth consumption over time. During a dry episode, some of the water stored in the dam is consumed and the water reserve goes down, potentially to the lower limit of the reservoir. In that case, it may be possible that electricity consumption be limited or rationed. On the contrary, when the water inflow is higher, the dam is replenished potentially up to the maximum capacity of the reservoir. Therefore, the capacity of the dam is a key factor of the optimal management policy as the Norwegian example showed us. Besides this renewable energy source, the permanent availability of the thermal power source softens the effect of the uncertainty of the water inflow. There is a large body of literature concerning commodity storage that presents meaningful results. Williams and Wright [22, 23] developed a model where the supply is stochastic and where production and storage are performed by competitive profit maximizers. They found that storage is much more effective in eliminating excessive levels of consumption and low prices than in preventing low levels of production and high prices. They explained this result by evoking the non symmetry of storage. Indeed, storage has to be non negative. When Williams and Wright looked at the impact of storage on the welfare, the global effect is not 1 All these facts have been extracted from a report of the Ministry of Petroleum and Energy of Norway [18]. 2

3 clear. But it seems that storage is better for producers. Indeed, from the consumers point of view, storage is favorable when price stabilization is partial. Deaton and Laroque [4, 5] also worked on the topic of commodity prices and commodity storage. In the second paper, they tried to explain some stylized facts of commodity price behavior by fitting a competitive storage model directly to the data. They proposed two ways to model productivity shocks: either iid shocks or time dependent (autoregressive) shocks. But finally none of the two models fits the data well. Deaton and Laroque explained this failure as follows: all the autocorrelation in the data has to be attributed to the underlying processes. Although speculation is capable of increasing the autocorrelation that would otherwise exist in an unmoderated price series, it cannot raise it to the levels that we observe. Concerning the analysis of the use of energy sources, many directions have been explored. On the one hand, equilibrium models have been developed. So, Garcia, Reitzes and Stacchetti [7] analyzed the price formation process and its policy implications in an infinite horizon duopoly model. They focused on two hydroelectricity producers who engage in dynamic Bertrand competition. At each date, water reservoirs are replenished with some strictly positive probability and a price cap affects the opportunity cost of producing electric power. They found that hydroelectricity producers might sell less today to have more capacity tomorrow: they adopt a strategic pricing behavior. They explained that the introduction of a price cap may shift down the entire price distribution. Crampes and Moreaux [2] studied a model where two energy sources are available: hydroelectricity and thermal electricity. They focused on a model of two time periods and did not introduce uncertainty in the hydroelectric technology. They successively studied the case of a central planner, of a monopoly that is regulated or not and the case of Cournot competition. They concluded that in the presence of hydroelectricity, thermal plants have to be dynamically planned. Moreover the optimized output for the thermal station is determined by the intertemporal specification of costs and utility. On the other hand, Hotelling [15] did not focus on equilibrium situations but rather analyzed the optimal rate of depletion of an exhaustible resource. He found that the rate at which consumption falls over time is equal to the ratio of the discount rate and the elasticity of marginal utility of consumption. As Heal [12] noted in his review on the optimal use of exhaustible resource, many extensions to this initial model have been developed. Hoel [13], for instance, introduced uncertainty in a setting with two energy sources: the date when the substitute will become available is known, but its unit cost is uncertain. He found that an increase in uncertainty may increase the consumption depending on the shape of the utility function. A parallel can be drawn between a dam that contains water and the savings of an agent and between the water flow that enters a dam and the random revenue of the agent. In models used to study the consumption/saving behavior of agents, one usually takes into account the fact that the agent is not allowed to borrow at each period of time. Without such liquidity constraints, agents would smooth their consumption perfectly over time. But with liquidity constraints, agents are not able any more to use a potential increase of their random revenue in the future 3

4 by increasing today the amount they are allowed to borrow. Therefore the consumption level decreases even if the constraints are not binding because of this threat of not being able to borrow. Such models have been studied by Deaton [3], Zeldes [25], Carroll [1] and Gollier [10]. In this paper, we aim at determining the optimal electric park that generates power with two energy sources. This model is an extension of the initial Hotelling model in which uncertainty has been added. Indeed, it is assumed here that the level of the stock of one of the energy sources is uncertain and the analysis concerns the optimal management of random stocks. We are in a setting close to the one of Crampes and Moreaux (when they study the case of a central planner) as far as we consider the optimal allocation between two energy sources. However, two main features have been added: not only do we consider an infinite horizon model, but we also introduce uncertainty on the water inflow. A social planner chooses his energy consumption at each period according to the state of the system and his expectations. He maximizes the discounted sum of the expected utility he gets from the use of different energy sources. We first study the optimal consumption flow of the agent when only hydroelectricity is available. He can consume water stored in a dam that is supplied by a random water inflow. The setting is also close to the one of Deaton except that there is a second kind of liquidity constraint : not only are consumers unable to consume water not yet fallen in the reservoir, but it is also not possible to store more water in the dam than its capacity. We find that the use of the dam allows electricity smoothing when the quantity of available water is in a given range [z, z ]: indeed in this region, the agent prefers cutting down on his consumption today to let enough water in the dam for the future. But when the quantity of available water is too low (lower than some threshold z ), the agent consumes it completely expecting that future rainfalls will replenish the dam in the future. A second energy source is then added and the optimal combination between the two energy sources allows a better smoothing of electricity consumption even when the costly energy source is not consumed. Moreover the introduction of the second energy source shifts up the water consumption of the agent. Besides the study of the allocation between the two sources, we also obtain a surprising result when we consider the effect of an increase in uncertainty. For some values of the quantity of available water, the agent consumes more water under uncertainty than under certainty. Once the optimal allocation has been determined, we consider a long term situation where the characteristics of the electric park have to be determined. We study the optimal number of dams that is increasing with the price of the second energy source. In the next section of this paper, as a benchmark, we study the optimal allocation between the two energy sources when there is no uncertainty. Section 3 describes the model in an uncertain setting when only hydroelectricity is available and obtains the optimal consumption flow in this case. In section 4, thermal energy is added and the optimal allocation between the two energy sources is described. Section 5 is devoted to the characteristics of an optimal electric park in the long term. In section 6, we propose as an extension to introduce a third energy 4

5 source. Section 7 concludes. 2 Model under certainty Before focusing on the model with uncertainty, we present as a benchmark the case where there is no uncertainty on the flow of water that supplies the dam. 2.1 Basic model We consider a small economy in which a representative agent consumes electricity produced using two different technologies: hydroelectric energy and thermal power. Hydroelectricity is obtained from water extracted from a dam. The dam is supplied by a constant flow of water and is characterized by its capacity Z. Thermal power is available at any time at a constant exogenous market price. In this setting, hydroelectricity is a renewable resource whereas the relative scarcity of the thermal power is expressed in its price: there is no constraint on its availability given its market price. Therefore, thermal power is a backstop technology of hydroelectricity in the sense used by Nordhaus [19] and Heal [12]: a technology that can provide substitutes for the resource once it is fully depleted, and can provide these substitutes on a very large scale indeed. We consider the standard setting where a representative agent chooses at each period his energy consumption according to the state of the system. He aims at maximizing the intertemporal utility he gets from his consumption. The use of the dam introduces some constraints. On the one hand, the representative agent cannot consume more water than the quantity stored in the dam, but on the other hand, the quantity of water consumed has to be high enough since the amount of water stored is limited by the capacity of the dam Z. Concerning thermal power, the only constraint is a non-negativity one. Let us introduce: w t the amount of water in the dam at the beginning of period t, µ the constant flow of water that enters the reservoir in period t, z t the total amount of water that is available to the agent in period t, that is z t = w t + µ. This implies that z t µ t, c t the amount of water that is extracted from the dam in period t, Z the capacity of the dam. w t, µ, z t, c t and Z are measured in meters and refer to heights of water in the reservoir, x t the quantity of electricity produced with thermal power consumed in period t and measured in kwh, p the unit price of thermal power, 5

6 R the coefficient that allows to convert a height of water into a quantity of electricity produced in kwh. It depends on the characteristics of the dam: its height, its flow, β the discount factor with β < 1, u the utility function: it is increasing and concave. The timing is represented on Figure 1. Figure 1: Timing The dynamics of the water that is available to the agent is equal to z t+1 = z t c t + µ. The agent cannot consume more than what is available, therefore the first constraint reduces to c t z t. Finally, the remaining stock of water has to be lower than the capacity of the dam: w t+1 Z. But as the dynamics of the quantity of water stored in the dam is w t+1 = w t + µ c t = z t c t, the constraint comes down to z t c t Z. With a quasi linear utility function, the program of the agent is: subject to max {c t,x t } t 0 β t [u (Rc t + x t ) px t ] (1) t=0 z t+1 = z t c t + µ (2) c t z t (3) c t z t Z (4) c t 0 (5) x t 0 (6) z 0 given (7) Concerning the utility function of this program, we decide to choose a utility function such that the price elasticity of the demand for thermal power x (p) is constant and equal to ε: ε = dx(p) dp x(p) p = u (x) xu (x). 6

7 u (x) = C x 1 ε +1 1 ε +1 + cte is the solution of this first order differential equation in u. Indeed, a constant has to be added to obtain a utility function. However we do not work with elasticities but with the concavity coefficient γ = 1 ε : u (x) = x1 γ 1 γ + cte. At each period, the dam is replenished with the certain water inflow µ > 0. Therefore, the agent s consumption is at each period greater or equal than µ: t, c t µ and the constraint c t 0 is always satisfied. For the rest of the section, this constraint will be skipped. 2.2 Shape of optimal consumption paths We begin this paragraph with a useful result that characterizes the allocation between the two energy sources. Proposition 1 Thermal power is consumed after having consuming all available water in the reservoir: x t > 0 c t = z t. This result means that the electricity consumption path can be decomposed into two phases. As in Hotelling, the cheapest energy source is consumed first. Once the water reserve is fully depleted, the thermal power is used in combination with the constant inflow of water. driving force for this result is the willingness of the social planner to postpone energy expenditure because β < 1. The Proof: Suppose the proposition does not hold: x t > 0 and c t < z t. Consider the following strategy: {ĉ t, x t } with ĉ t = c t + ε R and x t = x t ε, {ĉ t+1, x t+1 } with ĉ t+1 = c t+1 ε R and x t+1 = x t+1 + ε. What is the change in the utility of the agent in each period? In period t, u t = p, in period t + 1, u t+1 = p. Therefore, the total effect, u = β t p β t+1 p = β t p (1 β) is strictly positive, and strategy {ĉ, x} is strictly preferred to strategy {c, x} that is the optimal one. This leads to a contradiction. We solve the problem by backward induction. We first determine the optimal consumption of thermal power in the second stage of the process, i.e. when the dam is empty. We then use this information to determine the optimal consumption of hydroelectricity in the first stage. To do so, let us introduce the function: û (c t ; R, p) = max x t 0 u (Rc t + x t ) px t If ν is the Lagrangian multiplier associated with the constraint x t u (Rc + x) p + ν = 0. With e = u 1 (p), two cases occur: 0, the FOC reads 7

8 either ν = 0, what implies x 0. It follows that u (Rc + x) = p = u (e ), x = e Rc 0, and Rc e, or ν > 0, what implies x = 0 and u (Rc + x) = p ν < u (e ), x = 0, and Rc > e. Therefore, this function û is equal to: where { u (e û (c; R, p) = (p)) p (e (p) Rc) if Rc e (p) u (Rc) if Rc e (p) e (p) = u 1 (p). We present the shape of û on Figure 2. When c e (p) /R, it is a straight line and once this threshold is crossed, it is equal to the utility function u. Figure 2: Indirect utility function û The program of the agent reduces to: subject to max {c t } t 0 β t û (c t ; R, p) (8) t=0 z t+1 = z t c t + µ (9) c t z t (10) c t z t Z (11) z 0 given (12) The introduction of this indirect utility function û allows us to note that a minimum amount of electricity is consumed at each period. This minimum consumption equals e = u 1 (p). 8

9 Indeed, t, x (t) + Rc (t) e. x (t) + Rc (t) = e can only be the case if the constant inflow of water is not sufficient to satisfy all the demand, i.e. µ < e /R. Two cases may occur. Either µ < z t < e /R: there remains a strictly positive quantity of water in the dam before the precipitation replenishes the reservoir (w t > 0). Or z t = µ < e /R: there is no water stock in the dam anymore and at each time period, water consumption is equal to the rainfall. x (t) + Rc (t) > e implies that no thermal power is consumed (x (t) = 0) and electricity is produced using only the hydroelectric technology. path is therefore equal to x (t) = max (e (p) Rc (t), 0). Recalling that t, z t µ, three cases may occur: The optimal thermal power consumption 1. z 0 > e R > µ: the water consumption flow is decreasing until it reaches e R. Thereafter, it remains constant that is equal to µ and thermal power is consumed in quantity e Rµ, 2. e R > z 0 > µ : water consumption is decreasing until it reaches µ. thermal power consumption is strictly positive from the first period on and is equal to e R c t, 3. z 0 > µ > e R : the water consumption flow is decreasing until it reaches µ and it is equal to µ thereafter. No thermal power is consumed. We state the problem for case 1. Denoting by T the last time period for which no thermal power is consumed, the program of the agent reads: max c(t) T β t u (Rc (t)) + β T +1 [u (e ) p (e Rc (T + 1))] t=0 subject to z t+1 = z t c t + µ c (T ) > e R t=t c (T + 1) = z 0 + (T + 1) µ c (T + 1) < e R c (T + 1) > µ z 0 given + t=t +2 t=0 c t β t (u (e )) p (e Rµ)) The first constraint is the usual one on the dynamics of available water, the second inequality means that at time T, no thermal power is consumed yet. The third equation means that at time T + 1, all the available water increased by the flow of water is consumed. Indeed, if it was not the case, water would remain in the dam in period T + 1. Therefore no thermal power would be consumed in this time period what would not be consistent with the definition of T. The fourth inequality means that at time T + 1, thermal power is consumed and the last 9

10 equality ensures that until T, no more water than the quantity stored in the dam has been consumed. The resolution of the program is in the appendix. We present the shape of the optimal water consumption path on Figure 3. γ is taken to be equal to 5. This is consistent with the estimation of the price elasticity for different European countries found in Söderholm [20] whose mean amounts to The constant inflow of water is equal to 2. Concerning the other parameters, the values chosen are β = 0.95, Z = 10, z 0 = 10 and p = e (p) is therefore equal to These values will be used for all graphical representations if nothing else is mentioned. Figure 3: Optimal power consumption path Hydroelectricity consumption is decreasing until the dam is empty. Afterwards, it is constant and equal to Rµ. Thermal power is equal to max (e (p) Rc (t), 0). When there is still some water in the dam, total electricity is greater then e (p). On the contrary, once thermal power is consumed, the total electricity consumption amounts to e (p). In order to be able to draw comparisons with the case where the inflow of water is uncertain (see following section), we give the shape of the water consumption flow with respect to the quantity of available water on Figure 4. This actually amounts to studying c 0 with respect to z 0. Note that there is a kink: indeed, for low levels of stored water, all the available water is consumed. But afterwards, this is a step function because of the definition of the time T from which on thermal power is consumed. As we work in a discrete time model, T has to be integer. It is meaningful to study the marginal propensity to consume hydroelectricity c (z). In case 1, c z = 1 if z < e R β 1 γ 1 β 1 γ 1 β 1+T γ 10 otherwise

11 Figure 4: Optimal water consumption path in function of the quantity of water available First of all, as β < 1, the marginal propensity to consume hydroelectricity is strictly positive. Moreover it is a decreasing function of z: c is a concave function. In fact, the marginal propensity to consume is a measure of the efficiency of intertemporal smoothing: if smoothing were perfect, c (z) would be equal to zero. On the contrary, when c (z) = 1, there is no smoothing at all since all the water stored in the reservoir is consumed. As the marginal propensity to consume is strictly less than 1 if z < e R β 1 γ, time diversification is possible when z is higher than this threshold. A comparative statics analysis leads to interesting results: 1. As 1 β 1 γ 1 β 1+T γ is decreasing with T, the date from which on coal is consumed, a higher T improves time diversification. This result is very intuitive: when T is large, more water is available at the beginning (z 0 is greater) and consumption smoothing is made easier. 2. On the contrary, as 1 β 1 γ 1 β 1+T γ is an increasing function of γ (and of ε since γ = 1/ε), a higher elasticity deteriorates time diversification. Once more, this is quite intuitive. If γ increases, e (p) decreases and consumption smoothing is more difficult. Optimal policies under certainty have been carefully described in this part. From now on, we consider the case of a random inflow of water. It is not possible any more to obtain an analytical solution for water consumption in that case, therefore a numerical approach will be adopted. 3 Model with hydroelectricity under uncertainty When uncertainty is included in the model, the program of the representative agent can be written as: 11

12 subject to max {c t,x t } t 0 E 0 β t [u (Rc t + x t ) px t ] (13) t=0 z t+1 = z t c t + ỹ t+1 (14) c t z t (15) c t z t Z (16) x t 0 (17) z 0 given (18) where ỹ t is the random water inflow of period t. At each time period, we have that z t > min ỹ. In order to see the impact of the introduction of uncertainty, we first focus on the case where only hydroelectricity is available. In this case, the program of the agent is as in (13)-(18) except that x is constrained to be zero. The resolution of such a problem is made more convenient by using the Bellman equation: subject to v (z) = max {u (Rc) + βev (z c + ỹ)} (19) c c z (20) c z Z (21) The value function v is strictly increasing and strictly concave. 2 If λ and η are the Lagrangian multipliers associated with constraints (20) and (21), the FOC is: L/ c = Ru (Rc) βev (z + ỹ c) λ + η = 0. This can be summarized into the three following cases: βev (z + ỹ c) if (20) is binding Ru (Rc) βev (z + ỹ c) if (21) is binding = βev (z + ỹ c) otherwise. Concerning the second order conditions 2 L c 2 satisfied because of the concavity of u and v. = R 2 u (Rc) + βev (z + ỹ c) 0, they are The numerical resolution of the problem is represented on Figure 5 with the usual parameters values. Moreover we consider a random inflow ỹ that takes the values 0, 3 and 6 with equal probabilities this implies that t, z t 0. We denote by µ its mean value. We have a first result that characterizes the consumption flow: Lemma 1 The agent s consumption is strictly positive when the quantity of water that is available to the agent is strictly positive: z > 0, c (z) > 0. 2 See for example Stokey and Lucas [21] for a proof of this result under certainty. 12

13 Figure 5: Consumption of water in function of the quantity of available water Proof: Let us suppose that there exists z 0 > 0 such that c (z 0 ) = 0. In this case, constraint (20) does not bind and the FOC reads: Ru (Rc (z 0 )) = βev (z 0 + ỹ) βv (z 0 ) because v is concave < v (z 0 ) because β < 1 = Ru (Rc (z 0 )) because of the envelope theorem. Therefore it cannot be the case that Ru (Rc (z 0 )) = βev (z 0 + ỹ) and constraint (20) binds leading to c (z 0 ) = z 0 > 0. Thanks to this result, it is not necessary to add the constraint c 0 to program (19) because it is never binding when z > 0. According to Figure 5, there exist two thresholds 3 z and z such that z z, c(z) = z, z z, c(z) = z Z. In order to understand how hydroelectricity consumption smoothing is possible for different levels of stored water, let us compute the marginal propensity to consume c/ z from the first order conditions. We already noted in the previous section that it is a good measure of the 3 The existence of z has already been proven by Deaton [3] and Deaton and Laroque [4, 5]. In Deaton s model studying the consumption/saving behavior of agents, there is only one constraint on the maximal amount that can be borrowed which corresponds to (20). 13

14 efficiency of smoothing. c z = 1 if z z βev (z+ỹ c(z)) u (c(z))+βev (z+ỹ c(z)) if z < z < z 1 if z z First of all, c (z) is strictly positive implying that c is a strictly increasing function. Next, note that when neither (20) nor (21) is binding, the marginal propensity to consume is strictly less than 1 and time diversification is possible. On [z, z ], the dam plays the role of intertemporal consumption smoothing. As the agent knows that in the future he could not consume the quantity of water he would like, he prefers cutting down on his consumption today to let enough water in the dam for the future. When c (z) z, the agent modifies his consumption behavior and there is no consumption smoothing. Indeed, the agent knows that, in the next period, the flow of water that will replenish the dam will be greater or equal to min ỹ. Therefore, he empties the water out of the reservoir. In our illustration, as min ỹ=0, there might be no water the following period. Interval [0, z ] is thus tiny. When c (z) z, there is no consumption smoothing neither. The agent consumes more water than what he would have consumed if the capacity of the dam had been higher: indeed he consumes the surplus of water that comes from the precipitation and that he does not want to waste. If min ỹ > 0, z would have been higher. Indeed, knowing that the precipitation will be strictly positive the following time period, the agent is going to consume a bit more for very low levels of available water. Note that the marginal propensity to consume is first decreasing and then increasing. At the approach of z, the agent realizes that consumption smoothing is not possible anymore. Therefore, he cuts down on his consumption in the face of the risk of a dry period that lasts for many periods. Similarly, at the approach of z, the agent suddenly increases his hydroelectricity consumption because of the risk of a very rainy period for many successive time periods. Therefore, function c is successively concave then convex. It is also meaningful to look at the evolution of the consumption of the agent over time. On Figure 6, the consumption flow for 100 time periods is represented together with the water stock. It has been obtained through a simulation of the random variable ỹ. The path of water consumption has a completely different shape than in the certainty case where it was a decreasing function of time. Here we see how the dam allows water consumption smoothing. Indeed, the variations in the consumption flow are smaller than the stock variations. When the stock of water approaches the capacity of the dam (i.e. the constraint on the lower bound is binding), consumption smoothing is less efficient. This is also the case when the dam is almost empty (i.e. the constraint on the upper bound is binding). In this two cases, the dam does not play its smoothing role because of the technical constraints. This section concerning the model with one energy source allows us to have a first idea of the shape of the optimal consumption function. We have studied the role of the dam and have observed that it allows to smooth consumption only when the stock of water in the dam is in the range [z, z ]. 14

15 Figure 6: Evolution of the stock and of the consumption with the time 4 Model with two energy sources We introduce a second energy source and study the optimal allocation between both energy sources. What is the impact of the existence of a fossil source the management of the water resource and on the optimal consumption of electricity? The problem of the representative agent now reads 4 : subject to v (z; p) = max {u (Rc + x) px + βev (z c + ỹ; p)} (22) c,x c z (23) c z Z (24) x 0 (25) As in the first section, we can state the problem with the utility function û: subject to v (z; p) = max û (c; R, p) + βev (z c + ỹ; p) (26) c c z (27) c z Z (28) where 4 We retrieve the result of the previous section when p +. 15

16 and e is defined by u (e ) = p. û (c; R, p) = { u (e ) p (e Rc) if c e R u (Rc) if c e R The problem is now stated in the same way than in the previous section where only one energy source is available. The difference arises from the shape of the utility function. Whereas it was strictly concave in the previous section, here û is linear for c ] 0, e R [ and strictly concave for c ] e R, + [ (see Figure 2). As in the previous section, c (z) = 0 z = 0. Moreover, we retrieve the result of section 2 concerning thermal power consumption: Thermal power is consumed only if there is no water available in the dam. (the proof is the same than in the certainty case). With two energy sources, there are two means for the agent to smooth his electricity consumption: consuming thermal power when water extraction is low or storing water in the dam when precipitation is large. This result shows that he never uses both means in the same time. Indeed, when there is little water available, the agent does not use the dam to store water. He prefers consuming all the water available and smoothing his electricity consumption with the consumption of thermal power. On the contrary, when there is more water in the dam, the agent does not consume thermal power any more, but he uses the dam to smooth his electricity consumption. Analytically, we have: z such that Rz e, c (z) = z, and x (z) = p 1 γ Rz z such that Rz e, x (z) = 0, and c (z) is the solution of Ru (Rc) = βev (z c + ỹ)+λ η, where λ and η are the Lagrange multipliers associated with the constraints (27) and (28). As in the certain case with two energy sources, a minimum amount of electricity is consumed at each period. It is equal to e = u 1 (p): z, x (z) + Rc (z) e. If x (z) + Rc (z) = e, electricity is produced through a combination of the thermal power and water. In this case, the role of thermal power is to complement the level of electricity consumed up to e because there is not enough water in the dam. Note that we are in a region where the marginal utility û is constant, meaning that the price of thermal power p is strictly less than the marginal utility of electricity consumption u (Rc + x). Moreover, the marginal value of water λ is strictly positive and equal to Rp βev (ỹ). When x (z)+rc (z) > e, no thermal power is consumed (x (z) = 0) and electricity is only produced with hydroelectric technology. It can be the case that the agent consumes all the water stored (c (z) = z), but in most cases, the dam plays here its smoothing role (except if z is too high and we are in the case where the constraint on the lower bound is binding). In fact, when c (z) = z, the marginal value of water λ is strictly positive and equal to Ru (Rz) βev (ỹ). On the contrary, when neither constraint is binding, λ = 0 and the marginal value of water is equal to zero since there remains water in the dam for the following period. 16

17 4.1 Comparative Statics with respect to Price On Figures 7 and 8, we present the consumption of thermal power and water in function of the stock of water in the dam for different values for p (for p = 0.1 to p = 0.01). Figure 7: Consumption of thermal power in function of the quantity of available water for different p Concerning the consumption of thermal power, recall that with our specification e = p 1 γ ( ) and x (z) = max p 1 γ Rz, 0 : this is a straight line before being equal to zero. As we see on the graph, x (z; p) is decreasing with z until it reaches 0 and is decreasing with p. The numerical resolution reveals that c (z; p) is decreasing with p. When p increases, the agent knows that he is going to consume less thermal power since it is more expensive. That is why, for a given value of z, he prefers consuming less water too, in order to keep stock for the future. This shift of the consumption flow when the price of the second energy source increases expresses a precautionary behavior of the agent. Note that as p decreases, c/ z decreases: the existence of the thermal power at a low price improves the intertemporal diversification even when the fossil source is not consumed (for values of z such that e (p) Rz > 0). Moreover we have the following property concerning the value function v. Lemma 2 v (z, p) is a decreasing function of p. Proof: Consider p 1 < p 0 and recall that + v (z t ; p) = max E β t (u (Rc t + x t ) px t ) c t,x t t=0 17

18 Figure 8: Consumption of water in function of the quantity of available water for different p subject to z t+1 = z t c t + ỹ t c t z t c t z t Z x t 0 Suppose that we have the optimal policy c (z; p 0 ) and x (z; p 0 ) at price p 0. This allocation is feasible at price p 1. Therefore: + v (z, p 0 ) E β t (u (Rc (z, p 0 ) + x (z, p 0 )) p 1 x (z, p 0 )) E t=0 + t=0 = v (z, p 1 ) β t (u (Rc (z, p 1 ) + x (z, p 1 )) p 1 x (z, p 1 )) Table 1 presents the proportion of thermal power and water in the total amount of electricity consumed for different values for p. These values have been obtained by simulating the random variable ỹ times: one obtains a path for the stock of water in the dam for periods and consequently the consumption of water and of thermal power. Then, we compute the proportion of electricity from water and from thermal power for those periods. way. As p decreases, the proportion of thermal power increases and this happens in an exponential 18

19 p electricity (from water) electricity (from thermal power) % 1.49% % 2.27% % 4.11% % 7.58% % 13.23% Table 1: Proportion of hydroelectricity and thermal power in the total consumption for different values for p 4.2 Comparative Statics with respect to Uncertainty Let us study the impact of increasing uncertainty of the water inflow on the electricity consumption. We consider three mean preserving spreads of random inflows of water: where there is no uncertainty ( ỹ = { 2, 2, 2; 1 3, 1 3, 1 (ỹ { 3}), when there is a low level of uncertainty = 1, 2, 3; 1 3, 1 3, 1 }) 3 and when there is a higher level of uncertainty ( ỹ = { 0, 2, 4; 1 3, 1 3, 1 3}). In Figure 9, we zoom on the low levels of available water. Figure 9: Water consumption for different levels of uncertainty when the dam is almost empty According to the result stated by Leland [17], an agent is prudent (in the sense where he consumes less today and saves more) if and only if the marginal utility of future consumption is convex. With the Bellman formulation, the maximization problem reads max c û (c) + βev (z c + ỹ). As the maximization operator preserves prudence 5, in this case an agent is prudent if û is convex. Recall that û that is represented on Figure 10 is equal to: { û Rp if Rc e (p) (c; R, p) = Ru (Rc) < Rp if Rc e (p) 5 See Gollier [10], chapter

20 Note first that û is neither concave nor convex. Therefore, no general conclusion can be drawn about the effect of uncertainty on the optimal extraction. However, the marginal utility function is plotted on the following graph and we observe that it is locally concave for values of c close to e R. For higher values of c, it is convex. Figure 10: The marginal value of the indirect utility function û As we focus on the effect of an increase in uncertainty at date t + 1 on consumption at date t, we look at values of z satisfying the following equation: c (z c (z) + µ) = e R. The numerical resolution of this equation is z = For values of z around 2.55, the marginal utility of future consumption is concave, therefore the agent consumes more under uncertainty than under certainty. Intuitively, for values of z around the kink of the marginal utility function, if there is an increase in uncertainty, two cases occur: either the random flow is very low. In this case, the marginal utility is constant and the agent electricity consumption is equal to the minimum level e (p), or the random flow is very high and the agent consumes more than if uncertainty had been lower. But this increase in consumption does not hold any more for values of z higher than this threshold because, in this case, the marginal utility of the future consumption is convex: the agent consumes less under uncertainty. And for values of z lower than this threshold, consumption in both cases are equal since the marginal utility of future consumption is linear. We recover all these results on the graphs. Let us just note that this result on the prudence of the representative agent is close to the one obtained by Hoel [13] when he studied the optimal exhaustible resource extraction when the future substitute has an uncertain cost. Indeed, he found that if 20

21 the marginal utility of future consumption is concave, the resource extraction will be increased by a mean cost preserving increase in uncertainty. In this section, given the structure of the electric installations, we have determined the optimal consumption of water and thermal power. The introduction of the second energy source allows to improve electricity consumption smoothing both when there is not many water in the reservoir and when thermal power is not consumed. Therefore, the welfare of the agent increases when a second energy source is introduced. This is equivalent to saying that the value function v is decreasing with p (indeed, the model without thermal power plant is equivalent to let p tend to + ). The next step is to determine the optimal size of an electric park that generates power with two kinds of power plants: a hydroelectric power plant and a thermal power plant. 5 What is the optimal infrastructure? With the previous section, we are able to compute the value that the agent gets from the use of the dam and from the consumption of thermal power. Now, given the value extracted from a dam, the next step is to compute the optimal number of dams. Let us first introduce the number of dams as a parameter in the program (26): subject to v (z; α, p) = max u (Rc + x) px + βev (z c + αỹ; α, p) (29) c,x c z (30) c z αz (31) x 0 (32) α stands for the number of dams. If α dams are used, the total flow of water that fills then amounts to αỹ and the total capacity is equal to αz. ỹ and Z are exogenous parameters. To see the effect of the introduction of the coefficient α, we present water consumption for different values for α. We slightly modify the values taken until now: we chose ỹ = {0.25, 0.75, 1.25} and Z = 2.5. We recover the former values for α = 4. As α increases, two opposite effects appear: as α increases, it is as if the size of the dam increased. If there were no increase of the flow of water in the same time, this would tend to decrease the consumption flow. Indeed, with a larger dam, the agent can smooth his consumption in a more efficient way and therefore is tempted to consume less today and to store more water for the future. as α increases, the quantity of rainfalls that fills the dams increases, also. If this increase was taken independently without considering the increases in the size of the dam, it would clearly increase the consumption of the agent. Indeed, he knows that at each period, there will be more water. 21

22 The second effect dominates for value of z comprised between 0 and αz, therefore c (z, α) increases as α increases z [ 0, αz ]. Once v (z; α, p) has been obtained, the agent is going to focus on the utility retired from α empty dams v (0; α, p). Figure 11: Utility retired from empty dams: v (0; α, p) for different thermal power prices It is obvious that v (0; α, p) is an increasing function of α and as already proven v (0; α, p) is a decreasing function of p (see on Figure 11). In order to obtain the optimal number of dams, the agent is going to maximize the utility he obtains from α empty dams from which he subtracts the cost of building α dams. This maximization program reads: max v (0; α, p) C (α) (33) α We choose a cost function that is linear in the number of dams: C (α) = kα + K where K is a constant. Note that as v is concave with respect to α, the objective function is concave, too. The numerical resolution whose results are reported on Table 2 reveals that α (p) is an increasing function of p. p number of dams Table 2: Optimal number of dams for different values for p 22

23 When the thermal power price increases, it is optimal to build more dams. Indeed when p increases, the agent uses less thermal power. In order to smooth his consumption, the second way of smoothing (the dam) will be more used. That is why he prefers having more dams. In this section where we ask ourselves on the characteristics of an optimal electric park, we have studied the utility retired from α empty dams net from the cost of building these α dams. The optimal number of dams is comprised between 3 and 5 for the values of p we have chosen. Moreover, as p increases, the optimal electric park is composed of more dams. 6 Extensions In this part, we suppose that a third energy source is available to the agent, wind energy for instance. In a first step, we focus on hydro power and wind power. Indeed, we know from section 2 that introducing thermal power will be equivalent to using the utility function û instead of the utility function u. Suppose the agent can consume hydroelectricity and wind electricity. The dam is supplied with a random inflow of water. Wind energy is also random. Firstly, let us suppose that two cases can occur: either there is no wind, in which case no wind electricity can be consumed (what happens with probability q), or there is wind and the quantity of wind electricity amounts to W (what happens with probability (1 q)). The program of the agent is then the following: v 0 (z) = max c0 u (Rc 0 ) + β [qev 0 (z c 0 + ỹ) + (1 q) Ev 1 (z c 0 + ỹ)] subject to c 0 z, c 0 0, c 0 z Z v 1 (z) = max c1 u ( Rc 1 + W ) + β [qev 0 (z c 1 + ỹ) + (1 q) Ev 1 (z c 1 + ỹ)] subject to c 1 z, c 1 0, c 1 z Z Let us denote λ 0, µ 0 and ν 0 (resp. λ 1, µ 1 and ν 1 ) the three Lagrange multipliers associated with the constraints defining c 0 (resp. c 1 ). We first have some results on the shape of the consumption flows: Lemma 3 Consumption flows c 0 (z) and c 1 (z) have the following properties: 1. z > 0, c 0 (z) > 0, 2. if c 1 (z) = 0, then Rc 0 (z) < W, 3. if c 1 (z) = z, then c 0 (z) = z. The opposite is not true, 4. if c 0 (z) = z Z, then c 1 (z) = z Z. Once more, the opposite is not true. Proof: We are going to prove successively the four assertions. 1. Suppose there exists z 0 > 0 such that c 0 (z 0 ) = 0. This implies that µ 0 0. The FOC of the maximization program leads to: [ ] Ru (Rc 0 ) = β qev 0 (z 0 c 0 + ỹ) + (1 q) Ev 1 (z 0 c 0 + ỹ) 23

24 < qv 0 (z 0 ) + (1 q) v 1 (z 0 ) because v is concave = qru (c 0 ) + (1 q) Ru ( Rc1 + W ) because of the envelope theorem. This leads to Rc 0 > Rc 1 + W what is not possible since c 1 0. Therefore, the constraint c 0 (z) 0 never binds for strictly positive z and µ 0 = Suppose z 0 is strictly positive and c 1 (z 0 )=0. The FOC of the maximization program leads to: This leads to Rc 0 (z 0 ) < W. Ru ( Rc1 + W ) < qev 0 (z 0 + ỹ) + (1 q) Ev 1 (z 0 + ỹ) < qru (Rc 0 ) + (1 q) Ru ( Rc1 + W ). 3. We suppose c 1 (z) = z and c 0 (z) < z. c 1 (z) = z implies that Ru ( Rz + W ) = β [qev 0 (ỹ) + (1 q) Ev 1 (ỹ λ 1. c 0 (z) < z implies that [ ] Ru (Rc 0 ) = β qev 0 (z c 0 + ỹ) + (1 q) Ev 1 (z c 0 + ỹ) [ ] β qev 0 (ỹ) + (1 q) Ev 1 (ỹ) = Ru ( Rz + W ) λ Ru ( Rz + W ). The concavity of function u implies that Rc 0 Rz + W, what cannot happen. Therefore there is a contradiction and c 0 (z) = z. 4. As for the previous result, let us suppose that c 0 (z) = z Z and c 1 (z) > z Z. The ( ) ( ) equality implies that Ru Rz RZ = β [qev ( ) ] 0 Z + ỹ + (1 q) v 1 Z + ỹ ν 0. The inequality implies that: ( Ru Rc1 + W ) [ ] = β qev 0 (z c 1 + ỹ) + (1 q) v 1 (z c 1 + ỹ) [ ( ) > β qev ( ) ] 0 Z + ỹ + (1 q) v 1 Z + ỹ = Ru ( Rz RZ ) + ν0 > Ru ( Rz RZ ). This leads to W < 0, a contradiction. Therefore c 1 (z) = z Z. Thanks to this lemma, we know that z > 0, c 0 (z) > 0, therefore the constraint c 0 (z) z can be omitted. On the contrary and unlike the case where wind power was not available, c 1 (z) might be equal to zero even when the quantity of water that is available to the agent is strictly positive.indeed, when there is wind power, the agent is tempted to keep water in stock for the future in case wind power and/or precipitation are low the following time period. Thus, when z is very low, he prefers not consuming hydropower. represent a first step competed with the following proposition: 24 Concerning results 3 and 4, they only

25 Proposition 2 For any level of available water in the dam z: the agent consumes more hydroelectricity when there is no wind: c 1 (z) c 0 (z), the agent consumes more electricity when there is wind: Rc 0 (z) Rc 1 (z) + W. c 1 (z 0 ). Proof: Concerning the first result, suppose by contradiction there exists z 0 such that c 0 (z 0 ) < When neither constraint is binding and by the concavity of u, this implies that: Ru (Rc 0 (z 0 )) > Ru ( Rc 1 (z 0 ) + W ). According to the first order conditions, we have then: q [Ev 0 (z 0 c 0 (z 0 ) + ỹ) Ev 0 (z 0 c 1 (z 0 ) + ỹ)] > (1 q) [Ev 1 (z 0 c 1 (z 0 ) + ỹ) Ev 1 (z 0 c 0 (z 0 ) + ỹ)]. As c 0 (z 0 ) < c 1 (z 0 ) and as z v (z; 0) is a decreasing function, the left hand side is strictly negative. Similarly, the right hand side is strictly positive and we have a contradiction. When one of the constraint is binding, we know according to the results of the previous lemma that c 1 (z) c 0 (z). Concerning the second point, suppose there exists z 1 such that Rc 0 (z 1 ) > Rc 1 (z 1 ) + W. When neither constraint is binding, this implies that: Ru (Rc 1 (z 1 ) + W ) > Ru (Rc 0 (z 1 )), and therefore: (1 q) [Ev 0 (z 1 c 1 (z 1 ) + ỹ) Ev 0 (z 1 c 0 (z 1 ) + ỹ)] > q [Ev 1 (z 1 c 0 (z 1 ) + ỹ) Ev 1 (z 1 c 1 (z 1 ) + ỹ)]. Once more, the assumptions imply that the left hand side is strictly negative and the right hand side strictly positive, we have a contradiction. Now we consider rapidly the cases where one of the constraint is binding: If c 1 (z) = 0, then according to the previous lemma, Rc 0 (z) < W. If c 0 (z) = z and c 1 (z) < z, then the FOC lead to ( Ru Rc1 + W ) [ ] < β qev 0 (ỹ) + (1 q) Ev 1 (ỹ) = Ru (Rc 0 ) λ 0 < Ru (Rc 0 ) Therefore, Rc 1 + W > Rc 0. If c 1 (z) = z Z and c 0 (z) > z Z, a similar reasoning leads to the result. When wind power is available, the agent prefers saving water and taking advantage of wind power. However, when looking at the total electricity consumption, it is greater when there is wind power. Let us now introduce an auxiliary problem: the agent s problem when a quantity W of wind power is available to the agent with certainty at each period: v (z; W ) = max {u (Rc + W ) + βev (z c + ỹ; W )} (34) c 25

26 subject to c z (35) c z Z (36) Let us denote by c (z; W ) the solution to this program. The consumption flow c (z; W ) is represented for different values of W on Figure 12. Figure 12: Consumption flows c(z; W ) for different values of W We observe that as the quantity of available wind power increases, the consumption of hydro power also increases. The agent adopts a precautionary behavior. Knowing that less wind power is available to him, he prefers cutting downs today on his hydro power consumption. But contrary to the case where hydroelectricity and thermal power were available in section 4, here the introduction of wind power does not modify a lot the marginal propensity to consume c/ z. Note that when W = 0, we have that c 0 (z) = c 1 (z) = c (z; 0). Moreover, we have the following result concerning the indirect utility functions: Proposition 3 The utility the agent has from a quantity z of water in the dam is larger when there is wind than when there is no wind, and is larger when there is no wind than when there are no windmills: v 1 (z) v 0 (z) v (z; 0). Proof: By formulating the problem without the Bellman equation: + [ ( v 1 (z) = u (Rc 1 (z) + W ) + E β t u Rc 1,t + ξ )] t W 26 t=1