Forward Contracts and Allocation of Water

Resources in Deregulated Hydrothermal

Electricity Markets

Luiz Fernando Rangel

University of Auckland Energy Centre

This version: 16 July 2008

Abstract

This article studies the e¤ects of long-term contracts on the management of

hydro reservoirs in a liberalized electricity market. It is divided in two parts.

The �rst part analyzes the strategic incentives of generators to participate

in forward markets under a deterministic setting. It is demonstrated that

forward energy contracts not only induce an increase in total output, but also

an intertemporal allocation of hydro resources closer to the social optimum.

The second part of the article switches to an uncertainty setting and considers

the e¤ects of risk-aversion on hydro reservoir operation. It is shown that, in the

case of stochastic demand, the generator reduces risk by allocating less hydro

output to the period in which uncertainty is present. In the case of stochastic

in�ows, however, risk may be either reduced or increased by allocating less

hydro output to the period in which uncertainty is present, depending on the

initial storage level. Forward contracts allow a risk-averse generator to move

hydro allocation closer to the risk neutral benchmark.

JEL Classi�cation: L13, L50, L94, Q25, Q40

Keywords: Forward markets, Long-term contracts, Water resources, Hy-

dropower, Electricity

1 Introduction

Following electricity industry deregulation across the world, several issues concerning

the functioning of the market have been raised, together with suggestions about

1

how to address them. Speci�cally, energy forward markets and long-term contracts

have been proposed as a solution to two observed problems: market power abuse by

the generators and ine¢ cient allocation of risk where industry segments have been

vertically separated. The present article analyses these points in the context of a

hydro-based electricity system.

The theoretical foundations for the use of forward contracts to mitigate market

power in commodity spot markets can be traced back to the seminal work of Allaz

and Vila (1993). They show that forward contracts have the e¤ect of inducing �rms

to increase their spot market output levels, since the share of output that has been

traded in the forward market at a �xed price will not be a¤ected by the corresponding

spot price reduction. In other words, price reductions brought about by a marginal

increase in production a¤ect a smaller share of infra-marginal units. Collectively, �rms

are worse o¤ with the existence of these forward contracts, since industry pro�ts are

reduced, but individually each �rm has incentives to sign such contracts in order to

�preempt�the spot market and increase its own market share at the expense of its

rivals. In game-theoretical terms, this can be seen as a typical prisoner�s dilemma.

Even though the Allaz and Vila (1993) model is not speci�c to energy markets,

being applicable instead to commodity markets in general, several subsequent theo-

retical and empirical studies have built upon their framework to study the electricity

industry. Green (1999) adopts a theoretical model that di¤ers from the one of Allaz

and Vila (1993), in that the spot market operates under supply function competition

instead of Cournot competition, but arrives at the same conclusion: a higher level

of contracts lowers spot prices. Wolak (2000), using data from the Australian Na-

tional Electricity Market, presents empirical evidence that �nancial hedge contracts

do a¤ect generators�bidding behaviour and mitigates market power. Willems (2005)

shows that the results of Allaz and Vila (1993) still hold, although less strongly, if

the contracts take the form of options instead of futures. Bushnell (2007) extends the

Allaz and Vila (1993) model to incorporate multiple �rms and a more general cost

function. Bushnell, Mansur and Saravia (2007) compare three local markets within

the U.S.A.: California, New England and PJM. They show that California is the least

competitive of the three even though it is also the least horizontally concentrated,

while PJM is fairly competitive even though it is by far the most horizontally concen-

trated. The authors attribute this result to the higher degree of vertical arrangements,

such as bilateral contracts, in PJM.

Most of these articles, as well as the majority of analyses of competition in the

electricity industry, are based on static models. This type of model is well suited for

2

studying thermal-based systems, but not for hydro-based systems, since the operation

of hydro reservoirs is an intrinsically dynamic problem.1 Among the �rst papers in

the economic literature to take into account the speci�c considerations raised by

hydropower are Crampes and Moreaux (2001), Johnsen (2001) and Bushnell (2003).

They show that market power in hydro-based systems has the e¤ect of inducing a

misallocation of water resources across time. Indeed, in hydro-based markets, the key

issue is the strategic allocation of a given amount of output across periods, rather

than a straightforward reduction in total output, as in thermal-based systems. The

reason is that, in order to reduce total output, hydro generators would need to spill

water, which may be easily detected by regulators. The strategy of reallocating water

resources across time periods without any spillage, on the other hand, is much subtler

and therefore harder to be monitored by third-parties. In short, the �rm exploits

market power by producing more hydropower in periods when demand is relatively

elastic, therefore having less water available in periods when demand is relatively

inelastic.

The �rst part of the present article puts together the topics discussed in the

last two paragraphs, considering the e¤ects of long-term energy contracts on the

management of hydro reservoirs by strategic generators. We examine a two-period

model where the total amount of hydro output across the two periods is assumed to

be given, that is, it is the same under perfect competition or under market power. As

in Crampes and Moreaux (2001) and Bushnell (2003), we show that strategic �rms

will allocate production di¤erently across the two periods. We then introduce a round

of forward trading before the output decisions are made. It is shown that under this

framework forward contracts have an additional positive e¤ect on e¢ ciency: besides

inducing higher total output, as in the model of Allaz and Vila (1993), they induce an

intertemporal allocation of hydro production which is closer to the social optimum.

Besides the bene�cial e¤ect of reducing incentives to exercise market power, an-

other argument often put forth in support of forward contracts regards their role in

protecting risk-averse agents under an environment characterized by a high degree of

uncertainty. This issue is especially relevant in those markets where the electricity

industry has been vertically separated, since under vertical integartion most of the

risks faced by the di¤erent segments of the industry was internalized by the integrated

entity, but this feature disappears under vertical separation. It is usually argued that

1To be sure, the so-called unit commitment problem introduces a dynamic aspect to thermalplant operation as well. However, the time scope for this e¤ect is just a few hours at most, whereasfor hydro plants the time scope of the decision making process is several months, seasons or evenyears, depending on the size of the reservoirs.

3

the resulting allocation of risk is ine¢ cient, but long-term contracts and other vertical

arrangements can mimic the risk allocation of an integrated industry. Uncertainty

and risk aversion are characteristics clearly present in all electricity markets, but even

more so in hydro-based than in thermal-based systems, due to the stochasticity of

in�ows: while the inputs for thermal generation (oil, gas, coal) can in general be

easily acquired in commodity markets, the availability of input for hydro generation

(water) depends mostly on the goodwill of nature.

Several authors have included uncertainty in their models of hydroelectricity mar-

kets, such as Johnsen (2001) and Mathiesen, Skaar and Sorgard (2003), but in these

papers the framework is one of risk-neutrality. The e¤ects of uncertainty and risk-

aversion over the incentives of electricity �rms to trade forward contracts and ver-

tically integrate are analysed in Powell (1993), Bessembinder and Lemmon (2002)

and Baldursson and von der Fehr (2007); however, these papers implicitly assume a

thermal-based market as their models contain only one production period, so that the

issue of intertemporal output allocation is not present. To my knowledge there has

been no previous work in the economic literature dealing with the role of generator�s

risk-aversion on the intertemporal allocation of hydro output.

The second part of this article will address this issue. We will modify the model

presented in the �rst part of the article by incorporating uncertainty about demand

and water in�ows. We will start by examining a two-period model in which uncer-

tainty is present in the second period only, and show how the output decision of a

risk-averse generator di¤ers from a benchmark of risk-neutrality. We then proceed

to introduce a round of forward contract trading before the two production periods.

We will argue that under demand uncertainty the generator always chooses to be a

net seller in the forward market, while under in�ow uncertainty the generator can

be either a net buyer or seller of forward contracts, i.e., take either a long or a short

position, depending on the intial storage levels and on the expectation about in�ows.

The remainder of the paper is organized as follows. Section 2 analyzes the strate-

gic incentives of generators to participate in forward markets under a deterministic

setting. Section 3 switches to an uncertainty setting and shows how forward con-

tracts o¤set risk and change the intertemporal allocation of water resources. Section

4 summarizes the main conclusions.

4

2 Forward Contracts, Cournot Competition and

Hydro Allocation

We begin by analysing a single-stage game where �rms trade in a spot market only,

and no forward markets exist. There are two time periods, denoted 1 and 2, which

could be interpreted as summer and winter. Two power producers control both hydro

and thermal capacity. Hydro and thermal outputs of �rm i in period t are denoted qHtiand qTti , respectively. The thermal cost function is assumed identical for both �rms

and both periods: Cti(q) = C(q) = 12cq2, where c is a given constant.

We make four assumptions regarding hydro technology. First, the quantity of

water stored in generator i�s reservoirs is Si = S for both producers. Second, water

units are normalized in terms of energy, i.e., one unit of water produces one unit of

electricity. Third, there is no water spillage, so that available water is fully used at

the end of period 2, and the generator�s decision regarding its hydro resources boils

down to allocating its stock of water across periods. In practice, water spillage is

easily observed by regulators, so it is not a viable strategy for abusing market power.

These three assumptions taken together translate into the following hydro resource

constraint for each �rm: qH1i+qH2i = S. The fourth assumption about hydro technology

is that variable production costs are equal to zero. This is a standard assumption

in the hydropower economics literature and re�ects the fact that, in reality, variable

costs of hydro plants are very small, or even negligible, compared to thermal variable

costs.

Aggregate hydro output in period t is de�ned as qHt �P

i qHti . Similarly, aggregate

thermal output is qTt �P

i qTti . Demand functions are assumed to be linear, and

we allow for di¤erent demand levels across periods: P1(qH1 ; qT1 ) = 1 � qH1 � qT1 and

P2(qH2 ; q

T2 ) = ��qH2 �qT2 , where � may be less or greater than 1. The period with the

higher intercept will be called the peak period, and the one with the lower intercept

will be called the o¤-peak period.

2.1 The Social Optimum

The net social surplus function can be written as:Z qH1 +qT1

0

(1� x)dx�X

i

1

2c(qT1i)

2 +

Z qH2 +qT2

0

(�� x)dx�X

i

1

2c(qT2i)

2 (1)

The socially optimal solution is found by maximizing this function with respect to

each generator�s thermal and hydro outputs at each period, subject to both hydro

5

resource constraints.

It is easy to check that the solution entails prices being equal across time, 1 �qH1 � qT1 = �� qH2 � qT2 , and this common price being equal to thermal marginal costsin both periods, cqTti . Algebraically, the socially optimal aggregate hydro and thermal

outputs, denoted by the superscript o, can be shown to be:

qHo1 = S +1� �2

; qHo2 = S � 1� �2

; qTo1 = qTo2 =1 + �� 2S2 + c

: (2)

We note that thermal production is the same in both periods and that hydro

production is higher in the peak period. In this sense, thermal plants provide baseload

power, while hydro plants are used to shave-o¤ the demand peaks. This �peak-

shaving�property of hydro resources has been previously identi�ed in Crampes and

Moreaux (2001), Johnsen (2001) and Bushnell (2003). Of course, the fact that thermal

outputs are constant comes from the assumption of thermal cost functions being the

same in both periods. If cost functions were di¤erent, thermal production would

accordingly change over time, but shifts of the demand curve would still be addressed

by the hydro resource.

2.2 Cournot Competition

We now consider an open-loop oligopolistic equilibrium, where producers decide on

production levels for both periods at the beginning of the �rst period. That is, �rm�s

strategy is a set of hydro and thermal outputs chosen at the outset, and period 2

quantities cannot be conditioned on any information observed after period 1. The

qualitative results of this paper would not change if instead a closed-loop equilibrium

concept were adopted, where period 2 production decisions could be made contingent

on observed period 1 outcomes. This problem can be stated as follows:

maxqH1i ;q

T1i;q

H2i ;q

T2i

(1� qH1 � qT1 )(qHti + qTti)�1

2c(qT1i)

2 + (�� qH2 � qT2 )(qHti + qTti)�1

2c(qT2i)

2

s:t: qH1i + qH2i = S: (3)

After computing the �rst-order conditions, we can solve the resulting system of

equations to obtain the best-response functions. Letting the �rm�s total output in

period t be de�ned as qti � qHti + qTti , these functions can be written as:

6

QH1i(q1j; q2j) =S

2+1� q1j4

� �� q2j4

; (4)

QT1i(q1j; q2j) = QT2i(q1j; q2j) =

1 + �� 2S � q1j � q2j2(2 + c)

: (5)

Firm i�s best-responses depend only on its competitor�s total production on each

period (q1j and q2j), and not on the distribution of this production among hydro

and thermal within each period. Note that management of the reservoir is done in

such a way that all output variation across periods comes from the hydro resource.

In fact, hydro generation is adjusted to residual demand: if residual demand facing

�rm i is the same in both periods (1 � q1j = � � q2j), then its hydro best-responseswill also be the same in both periods; however, if residual demand is higher in one

of the periods, the corresponding hydro best-response will be higher in that period.

Meanwhile, thermal production is the same in both periods. It is interesting to note

that, even though thermal technology is intrinsically static, its operation takes into

account dynamic considerations: thermal best-response in, say, period 1 depends not

only on the competitor�s output in that period, but also on the competitor�s output

in period 2, since this latter quantity a¤ects the residual demand in period 2, which

feedbacks to period 1 through the hydro best-responses.

At the equilibrium, each �rm�s hydro and thermal production levels turn out to

be:

qH�1i =S

2+1� �6

; qH�2i =S

2� 1� �

6; qT�1i = q

T�2i =

1 + �� 3S2(3 + c)

: (6)

We can see that, as in the socially optimal soutionn, the oligopolistic equilibrium

also displays the �peak-shaving� property: hydro production will be higher in the

peak demand period, i.e., qH�1i > qH�2i if and only if 1 > �. The di¤erence between

the socially optimal and the oligopolistic solutions is that, in the former case, hydro

resources are used to shave-o¤ prices while, in the latter case, they are used to shave-

o¤marginal revenues. It is also interesting to note that hydro output depends only on

demand and reservoir storage levels, but not on thermal costs, while thermal output

does depend on hydro storage levels, besides demand and thermal costs.

Industry hydro and thermal outputs in each period, denoted respectively by qH�t �XiqH�ti and qT�t �

XiqT�ti , are then:

qH�1 = S +1� �3

; qH�2 = S � 1� �3

; qT�1 = qT�2 =1 + �� 3S3 + c

: (7)

This leads to the �rst proposition:

7

Proposition 1 Thermal outputs in both periods are higher under the socially optimal

than under the oligopolistic solution (qTot > qT�t ; t = 1; 2). Hydro outputs are such

that more water is allocated to the peak period in the socially optimal solution than in

the oligopolistic solution (qHot > qH�t in the peak period and qH�t > qHot in the o¤-peak

period).

Proof. The proof is straightforward by comparing the expressions in (2) and (7).

The �rst statement of the proposition, regarding thermal generation, re�ects the

traditional Cournot e¤ect of strategic �rms restricting production in order to obtain

a higher price. The second statement says that strategic �rms will withhold some

hydro production in the peak period, thereby allocating more water to the o¤-peak

period. By doing so, they force prices up in the period when demand is largest, which

more than compensates the price reduction in the period when demand is low.2 This

second e¤ect replicates the results derived in Crampes and Moreaux (2001), Johnsen

(2001) and Bushnell (2003), and it is illustrated in Figure 1. The �gure has been

drawn assuming that period 1 is the peak. The width of the central �box�is equal

to S, the amount of water the �rm has stored. Hydro output in period 1 is measured

from left to right from the leftmost vertical axis, and in perid 2 from right to left

from the rightmost vertical axis. The therrmal marginal cost function and thermal

output are measured from right to left from the leftmost vertical axis for period 1,

and from left to right from the rightmost vertical axis in period 2. The full lines in

the central box correspond to the residual demand functions for hydro output faced

by generator i in each period, which are equal to the market demand, minus the

generator�s own thermal production, minus the other generator�s total output at each

price. The dashed lines represent the marginal revenue of hydro output. The socially

optimal allocation of the water resource would be at point A, where residual demand

functions for hydro output meet. At this point, prices are the same in both periods.

The solution for a strategic generator corresponds to point B, where marginal revenues

of hydro output meet. The �gure illustrates that point B is to the left of point A,

meaning that, in the high-demand period 1, the duopolist�s hydro production is lower

and the price is higher than the socially optimal.

Let us now show that introducing a round of forward trading reduces this distor-

tion.2More generally, Mathiesen, Skaar and Sorgard (2003) point out that the allocation of water

depends on demand elasticities rather than demand levels: strategic �rms will withhold hydroproduction in periods of lower elasticity. In our linear model, demand levels and elasticities aredirectly related, with the elasticity being lower (at given prices) in the period with the higherintercept.

8

Figure 1: Misallocation of hydro output by a strategic generator

2.3 The Role of Forward Contracts

In this subsection, we analyse a two-stage game. The �rst stage consists in a for-

ward market, where generators and consumers can sign long-term contracts for the

electricity that will be produced and consumed in the second stage. We assume that

contracts for both periods 1 and 2 are signed simultaneously, i.e., there is only one

time period in the contracting stage. The second stage, as before, consists of two time

periods and outputs for both periods are decided simultaneously at the beginning of

the period 1. As usual this game is solved by backward induction.

2.3.1 The production stage

In the beginning of the second stage �rm i decides on its output levels {qH1i ; qT1i; q

H2i ; q

T2i},

realizing that it will receive a �xed price of Ft for the amount of output that was sold

forward in the contracting stage, kti, and the spot price Pt(qHt ; qTt ) for any additional

output, qHti +qTti�kti. Note that we allow this last value to be positive or negative, that

is, the generator may in principle have long or short forward positions. Therefore,

the �rm�s pro�t maximization problem may be written as follows:

9

maxqH1i ;q

T1i;q

H2i ;q

T2i

(1� qH1 � qT1 )(qH1i + qT1i � k1i) + F1k1i �1

2c(qT1i)

2

+(�� qH2 � qT2 )(qH2i + qT2i � k2i) + F2k2i �1

2c(qT2i)

2 (8)

s:t: qH1i + qH2i = S:

After computing the �rst-order conditions and solving the resulting system of

equations, the best-response functions turn out to be:

QH

1i(q1j; q2j) =S

2+1� q1j4

� �� q2j4

+k1i � k2i

4; (9)

QT

1i(q1j; q2j) = QT

2i(q1j; q2j) =1 + �� 2S � q1j � q2j + k1i + k2i

2(2 + c): (10)

Note that �rm i�s best-responses in any given period depend on the amount of forward

contracts it has signed for both periods. De�ning the vector k � (k1i; k2i; k1j; k2j),

the equilibrium outputs of the second stage subgame as functions of k are:

qH1i (k) =S

2+1� �6

+2(k1i � k2i)� (k1j � k2j)

6; (11)

qH2i (k) =S

2� 1� �

6� 2(k1i � k2i)� (k1j � k2j)

6; (12)

qT1i(k) = qT2i(k) =

1 + �� 3S2(3 + c)

+(2 + c)(k1i + k2i)� (k1j + k2j)

2(1 + c)(3 + c): (13)

It is interesting to note that thermal outputs depend only on the total amount

of contracts signed for the two time periods, k1i + k2i, but not on the distribution

of contracts across time. Hydro outputs, however, depend on the temporal distrib-

ution of those contracts or, more precisely, on the di¤erence between both periods�

contracted amounts, k1i � k2i.

2.3.2 The contracting stage

Proceeding backwards to the �rst stage, �rms choose the amount of output to be

sold on the forward market for each one of the second stage�s time periods. Denoting

aggregate hydro and thermal outputs respectively as qHt (k) =P

i qHti (k) and q

Tt (k) =

10

Pi qTti(k), the �rm�s problem in the �rst stage can be stated as:

maxk1i;k2i

(1� qH1 (k)� qT1 (k))(qH1i (k) + qT1i(k)� k1i) + F1k1i �1

2c(qT1i(k))

2

+(�� qH2 (k)� qT2 (k))(qH2i (k) + qT2i(k)� k2i) + F2k2i �1

2c(qT2i(k))

2: (14)

These contracts are sold either to consumers and retailers directly or to traders and

speculators who then resell this energy to consumers in the second stage. Following

Allaz and Villa (1993), forward prices F1 and F2 are determined by an assumption

that there are no opportunities for arbitrage between forward and spot markets. In a

setting with risk neutrality and no uncertainty, this translates into the forward price

being equal to the spot price for each period, i.e., Ft = Pt. Under this assumption,

the above maximization problem reduces to:

maxk1i;k2i

(1� qH1 (k)� qT1 (k))(qH1i (k) + qT1i(k))�1

2c(qT1i(k))

2

+(�� qH2 (k)� qT2 (k))(qH2i (k) + qT2i(k))�1

2c(qT2i(k))

2: (15)

Letting the superscript f denote the equilibrium in the game with forward con-

tracts, the next proposition characterizes the equilibrium in the contracting stage.

Proposition 2 The equilibrium amount of output sold through forward contracts by

�rm i for periods 1 and 2 satisfy:

kf1i � kf2i =

1� �5

and

kf1i + kf2i =

1 + �+ cS

5 + 5c+ c2:

Proof. See appendix.

It is interesting to note that the total amount of contracts traded depend on

several factors: demand, thermal marginal costs and the level of water stocks. The

distribution of these contracts across periods, on the other hand, depend only on

demand, being proportional to the di¤erence between both periods�demands para-

meters (1��). Firms will thus sell more contracts for the peak than for the o¤-peakperiod, i.e., kf1i > k

f2i if and only if 1 > �.

De�ne the vector kf � (kf1i; kf2i; k

f1j; k

f2j). We obtain the equilibrium outputs in

the production stage by substituting the expressions given in Propositon 2 into (11),

11

(12) and (13). Finally, aggregate hydro and thermal outputs, denoted respectively

qHft �X

iqHfti (k

f ) and qTft �X

iqTfti (k

f ), turn out to be:

qHf1 = S +2(1� �)

5; qHf2 = S � 2(1� �)

5; (16)

qTf1 = qTf2 =1 + �� 3S3 + c

+1 + �+ cS

(3 + c)(5 + 5c+ c2): (17)

We can now state the following proposition:

Proposition 3 Oligopolistic outputs with forward contracts fall inbetween the val-

ues corresponding to the social optimum and the oligopolistic equilibrium without con-

tracts:

Thermal outputs are such that qTot > qTft > qT�t ; t = 1; 2;

Hydro outputs are such that qHot > qHft > qH�t in the peak period, and conversely

qH�t > qHft > qHot in the o¤-peak period.

Proof. The proof is straightforward by comparing the expressions in (2), (7), (16)and (17).

The part of the proposition regarding thermal outputs corresponds to the Allaz

and Vila (1993) e¤ect of forward contracts: they lead to an overall increase in thermal

output. The second part of the proposition shows that forward contracts also have

the e¤ect of reducing the misallocation of water due to market power: strategic �rms

will withhold less production in peak periods compared to the setting without a

forward market. The intuition is that each �rm individually has more incentives to

preempt its rival and raise its market share in the peak period, which more than

compensates the corresponding reduction of its market share in the o¤-peak period.

In other words, each �rm wants to sign more long-term contracts than its rival for the

period when prices and demand are higher, in order to grab a larger piece of the �big

cake�, even if this means getting a smaller piece of the �little cake�. As both �rms

do this simultaneously, the result is increased output in the high demand period.

This is illustrated by Figure 2, which replicates the central box of Figure 1, de-

picting the original socially optimal and strategic hydro allocations without contracts,

points A and B respectivelly, plus the allocation with forward contracts, denoted here

by the point C. The e¤ect of contracts is to move the marginal revenue curves up-

wards. In fact, in our model with linear demand functions with slope equal to 1,

the marginal revenue line for �rm i in period t will shift parallelly by exactly kti.

Since, as explained above, the generator wishes to sell more forward contracts for the

12

high demand period (which is assumed to be period 1 in the �gure), the marginal

revenue line for period 1 will shift up by more than the corresponding line for period

2. Therefore, the point where these two new marginal revenue lines meet, point C, is

located to the right of point B: the strategic �rm will withhold less hydro output in

the period 1, so that hydro allocation will be closer to the social optimum.

Figure 2: Allocation of hydro output with forward contracts

3 Forward Contracts, Uncertainty and Risk Aver-

sion

The previous section has shown how contracts a¤ect strategic behaviour with two

�rms under Cournot-type competition. In this section, we will ignore the issue of

strategic interactions among �rms and adopt a monopolistic setting, in order to isolate

the e¤ects of uncertainty and risk aversion. This simpli�es the analysis, while the

qualitative results below should carry on to the competitive case since, with a proper

reinterpretation of the demand functions as residual demand functions, the overall

13

e¤ects of risk aversion on output levels under monopoly would also apply to the best-

response functions under oligopoly. We will study two types of uncertainty: with

respect to demand and with respect to in�ows.

3.1 Demand Uncertainty

3.1.1 Demand uncertainty without contracts

In the monopolistic framework, inverse demand functions are p1 = 1 � qH1 � qT1 andp2 = � � qH2 � qT2 . Let us assume that the second period demand parameter � isstochastic. When choosing its production pro�le fqH1 ; qT1 ; qH2 ; qT2 g, the generator doesnot know the value of �, but it recognizes its subjective probability distribution. In

this uncertain environment, the objective of the risk-averse �rm is to maximize the

expected utility of pro�t, E[U(�)], where E denotes expectation with respect to � and

U is an increasing and concave function of pro�t �. The generator�s maximization

problem becomes:

maxqH1 ;q

T1 ;q

H2 ;q

T2

E[U((1� qH1 � qT1 )(qH1 + qT1 )�1

2c(qT1 )

2+(�� qH2 � qT2 )(qH2 + qT2 )�1

2c(qT2 )

2)]

s:t: qH1 + qH2 = S: (18)

Let us denote marginal revenue in period t by MRt. It is clear thar MR1 is

deterministic, whileMR2 depends on the realization of the random variable �. From

the �rst-order conditions for pro�t maximization, the following equations hold:

E[U 0(�):(MR1 �MR2(�))] = 0E[U 0(�):(MR1 � cqT1 )] = 0 (19)

E[U 0(�):(MR2(�)� cqT2 )] = 0

These conditions imply that, as before, optimal thermal outputs will be the same

in both periods. The �rst equation is essentially the stochastic equivalent of the

�marginal-revenue-shaving�result obtained in the previous section for the determin-

istic case. Under risk-neutrality U 0(:) would be constant, so this equation would imply

that hydro reservoir management leads to the intertemporal equalisation of expected

marginal revenues. Denoting the risk-neutrality benchmark by the superscript n, the

14

system (19) can be solved to obtain the risk-neutral outputs:

qHn1 =S

2+1� E(�)

4; (20)

qTn1 = qTn2 =1 + E(�)� 2S2(2 + c)

: (21)

It is interesting to note that these expressions are similar to the best-response

functions in (4) and (5), except that we are now in a monopolistic setting and thus

the qtj�s are identically zero.3 The behaviour of a risk-neutral �rm under uncertainty

is the same as it would be under certainty if � were known to be equal to the expected

value of its original distribution.

The question we are interested in is: how does the output of a risk-averse generator

compares to this risk-neutrality benchmark. Denoting the risk-averse outputs by the

superscript a, the algebraic solutions of (19) for a concave U(:) are:

qHa1 =S

2+1� E(�)

4� Cov(U

0(�); �)

4E[U 0(�)]; (22)

qTa1 = qTa2 =1

2(2 + c)

�1 + E(�)� 2S + Cov(U

0(�); �)

E[U 0(�)]

�: (23)

We are now ready to state the following proposition.

Proposition 4 When second period demand is stochastic, a risk-averse �rm will

allocate more hydro output to the �rst period, and less to the second period, than a

risk-neutral �rm : qHa1 > qHn1 and qHa2 < qHn2 . Thermal outputs are reduced in both

periods: qTat < qTnt , for t = 1; 2.

Proof. See appendix.

This result is fairly intuitive, since in our setup the variance of the �rm�s pro�ts

is just equal to the variance of its period 2 revenue, V ar(p2:q2), and as q2 is non-

stochastic and p2 is linear, this is equal to (q2)2V ar(�). By allocating more hydro

output to period 1 (when the market price is deterministic) and less to period 2 (when

the price is uncertain), and moreover reducing thermal outputs, the �rm decreases

its stake in period 2 and therefore the variability of its pro�ts, up to the point in

which pro�ts and exposure to risk are balanced. In a more general setting with

several periods, if demand is stochastic in every period but its variance di¤ers from

3This suggests that, as mentioned earlier, the qualitative results obtained in this monopolisticsetting would carry on to the oligopolistic case if we reinterpret demand to be residual demand.

15

one period to another, we would expect the generator to move relatively more hydro

output away from periods with higher demand uncertainty.

3.1.2 Demand uncertainty with forward contracts

The question we are interested in is: if given the opportunity of locking in part of

its sales at a pre-arranged �xed price through forward contracts, to what extent the

monopolist would do so? We analyse this through a two stage game similar to the

one used in the previous section: in the �rst stage, the generator can sign forward

contracts, and in the second stage production takes place across two time periods.

The production problem may be written as follows:

maxqH1 ;q

T1 ;q

T2

E[U((1� qH1 � qT1 )(qH1 + qT1 � k1) + F1k1 � 12c(qT1 )

2

+(�� S + qH1 � qT2 )(S � qH1 + qT2 � k2) + F2k2 � 12c(qT2 )

2)]:(24)

The solution of this problem, parameterized by k � (k1; k2), is denoted by qH1 (k),qT1 (k) and q

T2 (k):

qH1 (k) =S

2+1� E(�)

4� Cov(U

0(�); �)

4E[U 0(�)]+k1 � k24

; (25)

qT1 (k) = qT2 (k) =

1

2(2 + c)

�1 + E(�)� 2S + Cov(U

0(�); �)

E[U 0(�)]+ k1 + k2

�: (26)

Once again we assume the presence of traders/speculators who perfectly arbitrage

between forward and spot markets. Furthermore, we assume that these arbitrageurs

are risk-neutral, so that F1 = P1 = 1�qH1 �qT1 and F2 = E(P2) = E(�)�S+qH1 �qT2 .Proceeding backwards to the �rst stage of the game, we can plug these no-arbitrage

conditions together with (25) and (26) into the generator�s pro�t function, and state

its problem at the contracting stage as:

maxk1;k2

E[U((1� qH1 (k)� qT1 (k))(qH1 (k) + qT1 (k))�1

2c(qT1 (k))

2

+(�� S + qH1 (k)� qT2 (k))(S � qH1 (k) + qT2 (k)) (27)

+(E(�)� �)k2 �1

2c(qT2 (k))

2)]:

By substituting the optimal values of k1 and k2 into (25) and (26), we obtain

the outputs of a risk-averse generator when forward contracts are available. Let us

denote these outputs by qHaft and qTaft . Note that we do not impose non-negativity

constraints on k1 and k2, so that we leave open the possibility of generators adopting

16

either short or long forward positions. However the next proposition shows that,

under demand uncertainty and risk aversion, k1 and k2 are always positive, i.e., the

generator will always sell forward contracts.

Proposition 5 At the equilibrium of the forward contract market, the risk-averse

generator sells a positive amount of contracts for both periods, with the amount sold

for period 2 being higher: k2 > k1 > 0. Compared to the outcome without contracts,

the existence of a forward market lowers hydro output in period 1 and, correspond-

ingly, raises hydro output in period 2: qHa1 > qHaf1 and qHa2 < qHaf2 . Thermal outputs

are higher in both periods with forward contracts: qTat < qTaft , for t = 1; 2.

Proof. See appendix.

In the case without contracts, reallocation of water was the only risk-management

mechanism available. We have seen that the risk-averse generator would reduce its

exposure to risk by increasing hydro output in period 1 and lowering it in period 2,

compared to the risk-neutral allocation, i.e., qHa1 > qHn1 and qHa2 < qHn2 . When a

forward market arises, risk can also be reduced by locking in the prices for a part of

period 2 sales, so the generator does not need do move as much hydro output away

from period 2. The reduction in risk exposure brought about by forward markets also

induces the generator to increase thermal production.

As the proof of proposition 5 shows, the optimal values of k1 and k2 are given by

the expressions:

k1 = �c

2

Cov(U 0(�); �)

E[U 0(�)]and k2 = �

c+ 4

2

Cov(U 0(�); �)

E[U 0(�)]: (28)

These equations con�rm that, in a monopolistic setting, uncertainty and risk aversion

combined are the only reason why the �rm would choose to trade in the forward mar-

kets. If either there was no uncertainty (constant �) or the producer was risk-neutral

(constant U 0), the covariance above would be zero and the optimality conditions

would imply k1 = k2 = 0.

3.2 In�ow Uncertainty

3.2.1 In�ow uncertainty without contracts

Let us now go back to the case of deterministic �. Let us assume instead that, in

addition to the stock of water available at the beginning of period 1, denoted S,

there are additional in�ows that arrive at the beginning of period 2, denoted I2.

17

This additional in�ow is stochastic and at the beginning of period 1, when the �rm

must decide its output levels, it does not know its realization, although it knows the

distribution. We will again assume that there is no water spillage, so that all available

water must be used by the end of period 2. This means that the generator decides

on its period 1 hydro output, qH1 , while period 2 hydro output is given residually by

qH2 = S + I2 � qH1 . As before, thermal outputs qT1 and qT2 are set at the beginning ofperiod 1, so that qT2 is not allowed to depend on observed realization of I2.

4 Formally,

we would also have to take into account the constraint 0 � qH1 � S, but to simplifythe analysis we will just assume that the model parameters are such that this holds.

The �rm maximizes the expected utility of pro�ts, so the problem can be formulated

as follows:

maxqH1 ;q

T1 ;q

T2

E[U((1� qH1 � qT1 )(qH1 + qT1 )�1

2c(qT1 )

2 (29)

+(�� S � I2 + qH1 � qT2 )(S + I2 � qH1 + qT2 )�1

2c(qT2 )

2)]:

The pro�t-maximizing conditions can be stated as E[U 0(�):(MR1 �MR2)] = 0

and E[U 0(�):(MRt � cqTt )] = 0. One can solve the resulting system of equations to

obtain the optimal outputs under risk-aversion :

qHa1 =S + E(I2)

2+1� �4

+Cov(U 0(�); I2)

2E[U 0(�)]; (30)

qTa1 = qTa2 =1 + �� 2S � 2E(I2)

2(2 + c)� Cov(U 0(�); I2)

(2 + c)E[U 0(�)]: (31)

In the risk-neutrality benchmark, where U 0(:) is a constant, the optimality condi-

tions reduce to MR1 = cqT1 = cqT2 = E(MR2): once again, we can see that optimal

operation hydro reservoirs leads to an intertemporal smoothing of marginal revenues.

Denoting this benchmark by the superscript n, equations (30) and (31) reduce to:

qHn1 =S + E(I2)

2+1� �4

; (32)

qTn1 = qTn2 =1 + �� 2S � 2E(I2)

2(2 + c): (33)

4The qualitative results of the analysis that follows would not change if period 2 thermal outputcould be made contingent on I2.

18

Again, we note the similarity between these two equations and the expressions in

(4) and (5) with qtj = 0. Firm�s behaviour under uncertainty and risk-neutrality is

equivalent to a certainty setting where the total stock of water is known to be equal

to S + E(I2).

We are interested in how risk-aversion changes the generator�s behaviour. It turns

out that this is dependent on the distribuition of I2, in contrast with the case of

demand uncertainty, where the results in Proposition 4 rest solely on pro�ts being

increasing in �, but the exact shape of the distribution of � is largely irrelevant.

The next proposition shows what can be said in the case of in�ow uncertainty if the

distribution of I2 is symmetric.

Proposition 6 Suppose that the distribution of I2 is symmetric. If S+E(I2) > 1+�2,

then qHa1 > qHn1 and qHa2 < qHn2 : a risk-averse �rm will produce more hydro output

in period 1 (and less in period 2) than a risk-neutral �rm. If S + E(I2) < 1+�2, then

qHa1 < qHn1 and qHa2 > qHn2 : a risk-averse �rm will produce less hydro output in period

1 (and more in period 2) than a risk-neutral �rm.

Proof. See appendix.

We have seen earlier that, in the case of stochastic demand, exposure to risk is

always reduced by moving hydro output away from the period in which uncertainty

is present, which is fairly intuitive. The above proposition shows that in the case

of stochastic in�ows, surprisingly, the generator may actually reduce its exposure to

risk by moving more output to the period in which uncertainty exists. This result is

related to the fact that under demand uncertainty a higher realization of the random

variable always bene�ts the producer, while under in�ow uncertainty it is not clear

whether a higher realization of the random variable leads to an increase or a decrease

of pro�ts.

Figure 3 illustrates this argument and lays out the intuition for the proposition.

The function plotted in the �gure, labelled Rev2(q), shows the generator�s revenue

from a given amount of hydro output in period 2. For the sake of illustration, suppose

that the distribution of I2 is binary and symmetric, so that I2 � E(I2) may be, withequal probability, either equal to c or to �c, where c is some constant. Under theno-spillage assumption, if expected hydro output in period 2 is q, its actual realization

can be either q � c or q + c with equal probability. Let us now see that the e¤ecton risk of a small increase in expected period 2 hydro output depends on its initial

value. If q is low enough, expected marginal revenue in period 2 at point q will be

positive, as shown in the �gure. The di¤erence between realized period 2 revenues if

19

I2 turns out high and if it turns out low is labelled �Rev2(q) in the picture. Now,

if the generator reduces hydro output in period 1, so that expected hydro output in

period 2 increases to q, then the di¤erence in realized revenues becomes �Rev2(q),

which is, due to concavity of the revenue function, lower than �Rev2(q), as the

�gure illustrates. In other words, increasing the expected value of period 2 hydro

output decreases the dispersion of period 2 revenues, thus reducing risk. The reverse

argument would show that, if q is initially high enough so that it is on the decreasing

part of the revenue function, then the dispersion of revenues could be reduced by

setting q lower than q.

Figure 3: Reducing revenue dispersion

What Proposition 6 says is that, if S + E(I2) is low enough, the expected value

of qHn2 is on the increasing part of period 2�s revenue function, so that a risk-averse

generator will set its output in such a way that qHa2 > qHn2 . Conversely, if S+E(I2) is

high enough, then risk-neutral outputs are such that the expected marginal revenue

of hydro output in period 2 is negative, and exposure to risk can be reduced by setting

qHa2 < qHn2 .

The above discussion suggests that the results in the proposition are valid not

only for our linear demand setting, but also for any shape of the demand function

provided that the corresponding revenue function is concave.

One may wonder why Proposition 6 requires the distribution of I2 to be symmetric.

Figure 4 shows an example in which the marginal revenues at q and q are both positive,

20

as in �gure 3, but �Rev2(q) is larger than �Rev2(q). This situation might arise when

the distribution of I2 is positively skewed. In our binary distribution example, positive

skewness means that I2 � E(I2) may be either equal to �c with a high probabilityor equal to d with a low probability, where 0 < c < d. The �gure illustrates that if d

large enough and if the probability of its occurrence is low enough, then the revenue

function may be increasing at a point such as q, but actual revenues for an output

realization of q � c are higher than for a realization of q + d. In other words, a highrealization of in�ow actually decreases revenues, even though the generator might

locally bene�t from an in�ow slightly above the average. Furthermore, the �gure

illustrates that the dispersion of revenue realizations increases as we move from q to

q. Thus, with an asymmetric distribution, contrary to Proposition 6, exposure to risk

may actually increase by setting qHa2 > qHn2 , or equivalently qHa1 < qHn1 , even though

the initial value of S + E(I2) is low.

Figure 4: An example of asymmetric distribution

21

3.2.2 In�ow uncertainty with forward contracts

Let us now introduce a round of forward contracting before the production stage. For

the case of in�ow uncertainty, the production problem may be written as follows:

maxqH1 ;q

T1 ;q

T2

E[U((1� qH1 � qT1 )(qH1 + qT1 � k1) + F1k1 �1

2c(qT1 )

2 (34)

+(�� S � I2 + qH1 � qT2 ):(S + I2 � qH1 + qT2 � k2) + F2k2 �1

2c(qT2 )

2)]:

Denote the solution by qH1 (k), qT1 (k), q

T2 (k):

qH1 (k) =S + E(I2)

2+1� �4

+Cov(U 0(�); I2)

2E[U 0(�)]+k1 � k24

; (35)

qT1 (k) = qT2 (k) =

1 + �� 2S � 2E(I2)2(2 + c)

� Cov(U 0(�); I2)

(2 + c)E[U 0(�)]+k1 + k22(2 + c)

: (36)

Once again we assume the existence of risk-neutral arbitrageurs who ensure that

F1 = P1 = 1 � qH1 � qT1 and F2 = E(P2) = � � S � E(I2) + qH1 � qT2 . Substitutingthese no-arbitrage conditions together with (35) and (36) into the generator�s pro�t

function, the problem in the contracting stage may be written as:

maxk1;k2

E[U((1� qH1 (k)� qT1 (k))(qH1 (k) + qT1 (k))�1

2c(qT1 (k))

2

+(�� S � I2 + qH1 (k)� qT2 (k))(S + I2 � qH1 (k) + qT2 (k)) (37)

+(I2 � E(I2))k2 �1

2c(qT2 (k))

2)]:

The next propostion shows that under in�ow uncertainty, contrary to the case of

demand uncertainty, the generator may actually choose to adopt a long position in

the forward market, i.e., to buy contracts instead of selling them.

Proposition 7 If S+E(I2) > 1+�2, at the equilibrium of the forward contract market,

k2 > k1 > 0, i.e., the generator sells electricity in the forward market. Forward

contracts induce the risk-averse generator to decrease hydro output in period 1 and

increase hydro output in period 2. Thermal outputs in both periods are higher with

contracts.

If S+E(I2) < 1+�2, at the equilibrium of the forward contract market, k2 < k1 < 0,

i.e., the generator buys electricity in the forward market. Forward contracts induce

the risk-averse generator to increase hydro output in period 1 and decrease hydro

output in period 2. Thermal outputs in both periods are lower with contracts.

22

Proof. See appendix.

We again observe that the incentives depend on the expectation about available

water resources, S + E(I2). If total water resources are expected to be high, the

outcome is similar to the case of demand uncertainty: the generator sells electricity

forward and, since water allocation is not anymore the only risk management instru-

ment, it moves more hydro output to period 2 than in the case without contracts,

while increasing thermal outputs in both periods. Conversely, if total water resources

are expected to be low, then the generator will buy electricity in the forward market,

which is a fairly intuitive result: in many countries, hydro generators indeed become

net buyers of electricity during dry years. But, since the amount of output subject to

spot price �uctuations in period 2 is qH2 + qT2 �k2, buying electricity forward (k2 < 0)

has the side e¤ect of increasing the generator�s exposure to spot price risk. To o¤set

this, the generator will move some hydro output away from period 2 and decrease

thermal production, compared to the case without contracts.

4 Conclusion

This paper studies the e¤ects of contracts on the behaviour of hydropower producers.

Previous literature has shown that strategic behaviour by hydro generators induces a

distortion on the intertemporal allocation of output. This paper has shown that the

existence of a forward electricity market reduces this distortion. This is similar to

the well-known result of Allaz and Vila (1993) that forward contracts curtail market

power, with one di¤erence: in the traditional Cournot-type model of Allaz and Vila

(1993) market power translates into a straighforward reduction in output, while in

our setting, under the assumption of no spillage of water, total output is �xed and

market power translates into a intertemporal misallocation of this output.

This paper also provides an analysis of the e¤ects of uncertainty and risk aversion

on the intertemporal allocation of water resources and hydro generation. It was shown

that, in the case of demand uncertainty, when a forward market does not exist, the

generator reduces its exposure to risk by moving hydro output away from the period

in which uncertainty is present. The introduction of a forward market reduces this

distortion. In the case of in�ow uncertainty, it has been shown that, surprinsingly,

exposure to risk may be reduced by moving more hydro output to the period in which

uncertainty is present; this will be the case when total water in�ows are expected to

be low. The e¤ect of forward markets on the allocation of hydro output also depends

on whether expected in�ows are high or low.

23

In future research, it would be interesting to study how the existence of forward

contracts a¤ect the incentive of existing �rms to invest in new capacity or new �rms

to enter the market. This could be done by introducing another stage in which �rms

decide on reservoir size, or in which the number of �rms participating in the forward

and spot markets is determined endogenously.

References

[1] Allaz, B., Vila, J.L. (1993). �Cournot Competition, Forward Markets and E¢ -

ciency�. Journal of Economic Theory, 59, 1-16.

[2] Baldursson, F., von der Fehr, N.H. (2007) �Vertical Integration and Long-Term

Contracts in Risky Markets�. IoES Working Paper Series W06:08, University of

Iceland.

[3] Bessembinder, H., Lemmon, M. (2002). �Equilibrium Pricing and Optimal Hedg-

ing in Electricity Forward Markets�. Journal of Finance, 57(3), 1347-1382.

[4] Bushnell, J. (2003). �A Mixed Complementarity Model of Hydrothermal Elec-

tricity Competition in the Western United States�. Operations Research, 51(1),

80-93.

[5] Bushnell, J., Mansur, E., Saravia, C. (2007). �Vertical Arrangements, Market

Structure and Competition: An Analysis of Restructured U.S. Electricity Mar-

kets�. American Economic Review, forthcoming.

[6] Bushnell, J. (2007). �Oligopoly Equilibria in Electricity Contract Markets�. Jour-

nal of Regulatory Economics, 32, 225-245.

[7] Crampes, C., Moreaux, M. (2001). �Water resource and Power Generation�.

International Journal of Industrial Organization, 19, 975-997.

[8] Green, R. (1999). �The Electricity Contract Market in England and Wales�.

Journal of Industrial Economics, 47(1), 107-124.

[9] Johnsen, T.A. (2001). �Hydropower Generation and Storage, Transmission Con-

straints and Market Power�. Utilities Policy, 10, 63-73.

[10] Mathiesen, L., Skaar, J., Sorgard, L. (2003). �Water With Power: Market Power

and Supply Shortage in Dry Years�. Discussion Paper 24/2003, Norwegian School

of Economics and Business Administration.

24

[11] Powell, A. (1993). �Trading Forward in an Imperfect Market: The Case of Elec-

tricity in Britain�Economic Journal, 103, 444-453.

[12] Willems, B. (2005). �Cournot Competition, Financial Option Markets and E¢ -

ciency�. CSEM Working Paper 139, University of California Energy Institute.

[13] Wolak, F. (2000). �An Empirical Analysis of the Impact of Hedge Contracts on

Bidding Behavior in a Competitive Electricity Market�. International Economic

Journal, 14(2), 1-39.

A Appendix

Proof of Proposition 2. First note that at the production stage the �rst order

conditions of (8) can be stated as, respectively:

(1� qH1 � qT1 � qH1i � qT1i)� (�� qH2 � qT2 � qH2i � qT2i) = k2i � k1i1� qH1 � qT1 � qH1i � qT1i � cqT1i = �k1i (38)

�� qH2 � qT2 � qH2i � qT2i � cqT2i = �k2i

These expressions will be used later in the proof. Proceeding to the contracting stage,

the �rst-order condition of (15) with respect to k1i is:

(1� qH1 � qT1 � qH1i � qT1i � cqT1i)@qT1i@k1i

+ (�� qH2 � qT2 � qH2i � qT2i � cqT2i)@qT2i@k1i

+(1� qH1 � qT1 � qH1i � qT1i)@qH1i@k1i

+ (�� qH2 � qT2 � qH2i � qT2i)@qH2i@k1i

(39)

�@qT1j@k1i

(qH1i + qT1i)�

@qH1j@k1i

(qH1i + qT1i)�

@qT2j@k1i

(qH2i + qT2i)�

@qH2j@k1i

(qH2i + qT2i) = 0:

For notational clarity, the function arguments k have been omitted. Using expressions

in (11), (12) and (13) to obtain the above partial derivatives with respect to k1i,

together with the �rst-order conditions from the production stage laid out in (38),

the last equation can be rewritten as:

(�k1i)2 + c

2(1 + c)(3 + c)+ (�k2i)

2 + c

2(1 + c)(3 + c)+ (k2i � k1i)

2

6(40)

+1

2(1 + c)(3 + c)(qH1i + q

T1i + q

H2i + q

T2i) +

1

6(qH1i + q

T1i � qH2i � qT2i) = 0:

25

Noting that qH1i + qT1i+ q

H2i + q

T2i = 2q

T1i+ S and q

H1i + q

T1i� qH2i � qT2i = 2qH1i � S, we can

once again use (11), (12) and (13) to rewrite equation (40) as:

1

2(1 + c)(3 + c)

�S +

1 + �� 3S(3 + c)

+(2 + c)(k1i + k2i)� (k1j + k2j)

(1 + c)(3 + c)� (k1i + k2i)(2 + c)

�+1

6

�1� �3

� 4(k1i � k2i)3

� (k1j � k2j)3

�= 0 (41)

Following analogous computations, the �rst-order condition of (15) with respect

to k2i can be shown to yield:

1

2(1 + c)(3 + c)

�S +

1 + �� 3S(3 + c)

+(2 + c)(k1i + k2i)� (k1j + k2j)

(1 + c)(3 + c)� (k1i + k2i)(2 + c)

��16

�1� �3

� 4(k1i � k2i)3

� (k1j � k2j)3

�= 0 (42)

Note that in the last two equations, the �rst term in the respective lefthand sides

are equal and the second terms are the inverse of one another. Therefore (41) and

(42) can hold simultaneously if and only if the terms is square brackets are both zero:

S +1 + �� 3S(3 + c)

+(2 + c)(k1i + k2i)� (k1j + k2j)

(1 + c)(3 + c)� (k1i + k2i)(2 + c) = 0 (43)

and1� �3

� 4(k1i � k2i)3

� (k1j � k2j)3

= 0: (44)

It is easy to check that the above equations are solved by:

k1j � k2j = k1i � k2i =1� �5

(45)

and

k1j + k2j = k1i + k2i =1 + �+ cS

(3 + c)(2 + c)� 1 : (46)

These are the expressions given in the proposition.

Proof of Proposition 4. The equation for qHa1 can be rewritten as:

qHa1 =S

2+1� E(�)

4� Cov(U

0(�); �)

4E[U 0(�)]: (47)

Under risk aversion, U(:) is an increasing and concave function. Therefore, Cov(U0(�);�)

E[U 0(�)]

has the opposite sign as Cov(�; �). Since, once output levels are set, pro�ts are

26

increasing in �, Cov(�; �) is positive and thus qHa1 > S2+ 1�E(�)

4= qHn1 .

An analogous argument holds for thermal outputs.

Proof of Proposition 5. The �rst order conditions of (24) with respect to qH1 , qT1

and qT2 , respectively, are:

E[U 0(�):(1� 4qH1 � 2qT1 � �+ 2S + 2qT2 + k1 � k2)] = 0E[U 0(�):(1� 2qH1 � 2qT1 � cqT1 + k1)] = 0 (48)

E[U 0(�):(�� 2S + 2qH1 � 2qT2 � cqT2 + k2)] = 0:

Going back to the contracting stage, the �rst-order condition of (27) with respect

to k1 may be written as:

E

�U 0(�):

�(1� 4qH1 � 2qT1 � �+ 2S + 2qT2 )

@qH1@k1

+ (1� 2qH1 � 2qT1 � cqT1 )@qT1@k1

+(�� 2S + 2qH1 � 2qT2 � cqT2 )@qT2@k1

��= 0: (49)

From the expressions for qH1 (k), qT1 (k) and q

T2 (k) we know that

@qH1@k1

= 14and @qT1

@k1=

@qT2@k1

= 12(2+c)

. Together with the �rst-order conditions laid out in (48) , we can state

equation (49) as:

1

4E[U 0(�):(k2 � k1)]�

1

2(2 + c)E[U 0(�):k1]�

1

2(2 + c)E[U 0(�):k2)] = 0: (50)

Since k1 and k2 are deterministic, this is equivalent to:

(k2 � k1)�2

2 + c(k1 + k2) = 0: (51)

Following similar computations, one can verify that the �rst-order condition of

(27) with respect to k2 yields the following condition:

(k2 � k1) +2

2 + c(k1 + k2) + 4

E[U 0(�):(�� E(�))]E[U 0(�)]

= 0: (52)

Noting that E[U 0(�):(��E(�))] = Cov(U 0(�); �), it is easy to check that the systemformed by equations (51) and (52) is solved by:

k1 = �c

2

Cov(U 0(�); �)

E[U 0(�)]and k2 = �

c+ 4

2

Cov(U 0(�); �)

E[U 0(�)]: (53)

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Since U is concave and � is increasing in �, the covariance is negative, which

implies that k2 > k1 > 0.

The rest of the proposition follows by comparing (22) with (25) and (23) with

(26).

Proof of Proposition 6. According to equations (30) and (31):

qHa1 = qHn1 +Cov(U 0(�); I2)

2E[U 0(�)]; (54)

and

qTat = qTnt � Cov(U 0(�); I2)

(2 + c)E[U 0(�)]; for t = 1; 2: (55)

In the above expressions, � is the pro�t computed at point (qHa1 ; qHa2 ; qTa1 ; qTa2 ). Under

risk-aversion, U 0 is a positive and decreasing function, so Cov(U 0(�);I2)E[U 0(�)] has the opposite

sign as Cov(�; I2). For our speci�cation of the demand function, we have:

Cov(�; I2) = Cov((�� qHa2 � qTa2 )(qHa2 + qTa2 ); I2)

= Cov((�� S � I2 + qHa1 � qTa2 )(S + I2 � qHa1 + qTa2 ); I2)

= Cov((�� 2S + 2qHa1 � 2qTa2 )I2 � I22 ; I2) (56)

= (�� 2S + 2qHa1 � 2qTa2 ):V ar(I2)� Cov(I22 ; I2)

= (�� 2S � 2E(I2) + 2qHa1 � 2qTa2 ):V ar(I2)� E[(I2 � E(I2))3]:

The last equality comes from the fact that, for any random variable x, Cov(x2; x) =

E[(x � E(x))3] + 2E(x)V ar(x). The distribution of I2 is assumed to be symmetric,thus E[(I2 � E(I2))3] = 0. Since the variance is always positive, Cov(�; I2) has thesame sign as �� 2S � 2E(I2) + 2qHa1 � 2qTa2 :Let us begin by analysing the case of S + E(I2) < 1+�

2. Assume that qHa1 � qHn1 .

In this case, (54) and (55) imply that qTa2 � qTn2 must also hold. Therefore:

�� 2S � 2E(I2) + 2qHa1 � 2qTa2

� �� 2S � 2E(I2) + 2qHn1 � 2qTn2 (57)

= �� 2S � 2E(I2) + S + E(I2) +1� �2

� 1 + �� 2S � 2E(I2)(2 + c)

=c

2 + c

�1 + �

2� S � E(I2)

�> 0:

28

Therefore, Cov(�; I2) > 0 andCov(U 0(�);I2)E[U 0(�)] < 0. But then the assumption qHa1 � qHn1

would contradict equation (54). It follows that if S + E(I2) < 1+�2, then qHa1 � qHn1

cannot be true.

The reverse argument shows that if S+E(I2) > 1+�2, then qHa1 � qHn1 would result

in a contradiction.

Proof of Proposition 7. Following a derivation analogous to the one in the proof

of Proposition 5, the equilibrium at the contracting stage can be characterized by the

equations:

(k2 � k1)�2

2 + c(k1 + k2) = 0 (58)

and

(k2 � k1) +2

2 + c(k1 + k2)� 4

E[U 0(�):(I2 � E(I2))]E[U 0(�)]

= 0: (59)

The solution is:

k1 =c

2

Cov(U 0(�); I2)

E[U 0(�)]and k2 =

c+ 4

2

Cov(U 0(�); I2)

E[U 0(�)]: (60)

We know from the proof of Proposition 6 that S+E(I2) < 1+�2implies qHa1 < qHn1

and Cov(U 0(�);I2)E[U 0(�)] < 0. In this case, we have that k2 < k1 < 0. Conversely, S+E(I2) >

1+�2implies Cov(U 0(�);I2)

E[U 0(�)] > 0, and thus k2 > k1 > 0. The remaining statements in the

proposition follow by comparing (30) with (35) and (31) with (36) in each of the two

cases.

29