The strategic effects of forward markets on oligopolistic ... fileThe strategic effects of forward markets on oligopolistic electricity equilibria-2 Introduction •Weask: – Do long-term

The strategic effects of forward marketson oligopolistic electricity equilibria

Francisco Amaya (Universidad Pontificia Comillas de Madrid)

[email protected]

ENERDAY, Dresden 21th March 2006

The strategic effects of forward markets on oligopolistic electricity equilibria - 2

Introduction

• We ask :– Do long-term contracts provide additional tools for market players to

exercise market power?Or, on the contrary

– Do long-term contracts present any pro-competitive effects?

• We answer :– Voluntary forward trading will not limit the players´ability to exercise

market power– If long-term contracts limit market power then forward trading would not

happen


Outline

• The basic model

• Commitment strategies in the spot market as analternative to forward trading

• The pro-competitive effect in a successive marketsframework


• Firms produce and at the same marginal cost

• Firms payoff is given by

• Spot price is given by the inverse demand function

• Firms contract and at forward price

• Assume for sales and for purchases

The basic model: Allaz and Vila (i)

• Two-period model. Two symmetric Cournot duopolist firms and .

• Firms produce at the same marginal cost

if

ji

First period: Forward market

Second period: Spot market

jf

i j

c

fp

iq

s ii

p a q= - å( )i

s s i j i i ip q q q f c qé ùP = + × - - ×ê úë û

0if >

jq

0jf <


• Firms payoff (considering the no-arbitrage condition, )is given by

The basic model: Allaz and Vila (ii)ji

• We proceed backwards to obtain the spot and forward equilibrium:

First period: Forward market equilibrium

Second period: Spot market equilibrium

2 2 23 3 3

i j j i i ji j s

a c f f a c f f a c f fq q p

- + - - + - + - -= = =

/ 0if if¶ P ¶ = / 0j

jf f¶P ¶ =

/ 0is iq¶ P ¶ = / 0j

s jq¶ P ¶ =

, , ,if s i j i i j i i jp f f q f f c q f fé ù é ù é ùP = × - ×ê ú ê ú ê úë û ë û ë û

f sp p=

2( ) 45 5 5i i s f

a ca c a cf q p p-- += = = =







2 2 23 3 3

i j j i i ji j s


- + - - + - + - -= = =

/ 0if if¶ P ¶ = / 0j

jf f¶P ¶ =

2( ) 45 5 5i i s f

a ca c a cf q p p-- += = = =

/ 0is iq¶ P ¶ = / 0j

s jq¶ P ¶ =


f sp p=







2 2 23 3 3

i j j i i ji j s


- + - - + - + - -= = =

/ 0if if¶ P ¶ = / 0j

jf f¶P ¶ =

/ 0is iq¶ P ¶ = / 0j

s jq¶ P ¶ =


f sp p=

2( ) 45 5 5i i s f

a ca c a cf q p p-- += = = =







2 2 23 3 3

i j j i i ji j s


- + - - + - + - -= = =

/ 0if if¶ P ¶ = / 0j

jf f¶P ¶ =

/ 0is iq¶ P ¶ = / 0j

s jq¶ P ¶ =


f sp p=

2( ) 45 5 5i i s f

a ca c a cf q p p-- += = = =







2 223 3 3

j j i ji ii j s

a c f f a c f fa c f fq q p- + - + - -- + -= = =

/ 0if if¶ P ¶ = / 0j

jf f¶P ¶ =

/ 0is iq¶ P ¶ = / 0j

s jq¶ P ¶ =


f sp p=

2( ) 45 5 5i i s f

a ca c a cf q p p-- += = = =


The basic model: Allaz and Vila (iii)ji

• The pro-competitive effect– Companies optimize their total income when selling part of their energy in

the forward market (a>c). – Firms, find profitable to increase his production in the spot market, and

thus prices are reduced.

• The emergence of forward market– When one of the two duopolist does not trade in the forward market, the

equilibrium outcome is the Stackelberg result and thus the forward-contracted producer greatly benefits from his strategy.

– However, this implies somehow that at the spot market the non-forward-contracted producer will assume the responsibility of withdrawing some quantity to increase spot prices.


Outline

• The basic model




Commitment strategies in the spot market (i)

• Committing to certain out-of-Cournot strategies in the subsequent spot market (non-forward-contracted producer does not assume theresponsability of withdrawing some quantity to increase spot price) allowthe non-forward-contracted producer to punish the forward-contractedproducer.

• To model these out-of-Cournot committed strategies we state a dynamiccompetition model.


• Repeated two-stage model. Two symmetric Cournot duopolist firmsand .

• Firms produce and at the same marginal cost

• Firms choose their strategic behaviour

• Firms payoff is given by

• Spot price is given by the inverse demand function

• Firms contract and at forward price

• Assume for sales and for purchases

Commitment strategies in the spot market(ii)

if

ji

First period: Forward market

Second period: Spot market

jf

i j

c

fp

iq

s ii

p a q= - å( )i

s s i j i i ip q q q f c qé ùP = + × - - ×ê úë û

0if >

jq

0jf <

/i j idq dqq =/j i jdq dqq =

, [ 1, 0]i jq q Î -

• Firms payoff (considering the no-arbitrage condition, )is given by, , ,i

f s i j i i j i i jp f f q f f c q f fé ù é ù é ùP = × - ×ê ú ê ú ê úë û ë û ë û

f sp p=


Commitment strategies in the spot market(iii)


ji

Second period: Spot market equilibrium / 0is iq¶ P ¶ = / 0j

s jq¶ P ¶ =

1 1 2 2

1 2 2

2 3 2 2

j i i j i j

i j i j j i i j

i j i j i j

a q b q d l b q q q q

l q q q q r l a q q q q

m a b q q g l m q q q q

= + = + = + = + + + ×

= + + + × = + = + + + ×

= + = + + = + = + + + ×

i ji

a c f fq

a a d ag

× - × + × - ×=j i

ja c f f

qb b r b

g× - × + × - ×

= i js

a c f fp

l m l lg

× + × - × - ×=

First period: Forward market equilibrium / 0if if¶P ¶ = / 0j

jf f¶P ¶ =

[ ] [ ]( ) ( )(3 )i j

a cf f

l d a d g m l al d a

× - - × - -= =

× -


Commitment strategies in the spot market (iv)

• Consider a sequence of market interaction between the two duopolists which unfolds as follows:

1. Only spot market exists Nash-Cournot equilibrium.

2. A forward market is introduced. Producer trades forward and does not trade forward. Both producers behave according to Cournot competition in the spot market. Stackelberg equilibrium

3. Both producers trade forward the same quantities and behave according to Cournot competition in the spot market. Allaz&Vila equilibrium

4. Producer does not trade forward and plays out-of-Cournot commitment strategies in the spot market. Producer assume as certain the out-of-Cournot strategies of producer but he operates according to Cournot competition in the spot market. Producer punishes heavily.

5. Producer does not trade forward and plays out-of-Cournot commitment strategies in the spot market. Producer assume as certain the out-of-Cournot commitment strategies of producer and also plays out-of-Cournotstrategies in the spot market. Producer punishes to a less extent.

i j

ij

j

j

j

ji

j

ji

i

i



• Comparation between the previous cases

Case θi θj fi fj qi qj ps πi πj

1,0 0,0 0,0 0,0 0,0 5,7 5,7 8,7 49,1 49,12,0 0,0 0,0 5,0 0,0 9,0 4,0 7,0 63,0 28,03,0 0,0 0,0 5,0 5,0 7,3 7,3 5,3 39,1 39,14.1 0,0 -0,4 5,0 0,0 8,3 5,5 6,3 51,9 34,24.2 0,0 -0,5 5,0 0,0 8,0 6,0 6,0 48,0 36,04.3 0,0 -0,6 5,0 0,0 7,7 6,7 5,7 43,4 37,84.4 0,0 -0,7 5,0 0,0 7,3 7,5 5,3 38,1 39,44.5 0,0 -0,8 5,0 0,0 6,7 8,6 4,7 31,7 40,44.6 0,0 -0,9 5,0 0,0 6,0 10,0 4,0 24,0 40,04.7 0,0 -1,0 5,0 0,0 5,1 11,8 3,1 16,0 36,75.1 -0,4 -0,4 5,0 0,0 9,6 4,6 5,8 55,5 26,65.2 -0,5 -0,5 5,0 0,0 9,8 4,8 5,4 52,9 25,95.3 -0,6 -0,6 5,0 0,0 10,0 5,0 5,0 50,0 25,05.4 -0,7 -0,7 5,0 0,0 10,2 5,2 4,6 46,6 23,85.5 -0,8 -0,8 5,0 0,0 10,5 5,5 4,1 42,8 22,35.6 -0,9 -0,9 5,0 0,0 10,7 5,7 3,6 38,3 20,45.7 -1,0 -1,0 5,0 0,0 11,0 6,0 3,1 33,6 18,3








5. Producer does not trade forward and plays out-of-Cournot commitment strategies in the spot market. Producer assume as certain the out-of-Cournot commitment strategies of producer and also plays out-of-Cournot strategies in the spot market. Producer punishes to a less extent.

i j

ij

j

j

j

ji

j

ji

i

i





1,0 0,0 0,0 0,0 0,0 5,7 5,7 8,7 49,1 49,12,0 0,0 0,0 5,0 0,0 9,0 4,0 7,0 63,0 28,03,0 0,0 0,0 5,0 5,0 7,3 7,3 5,3 39,1 39,14.1 0,0 -0,4 5,0 0,0 8,3 5,5 6,3 51,9 34,24.2 0,0 -0,5 5,0 0,0 8,0 6,0 6,0 48,0 36,04.3 0,0 -0,6 5,0 0,0 7,7 6,7 5,7 43,4 37,84.4 0,0 -0,7 5,0 0,0 7,3 7,5 5,3 38,1 39,44.5 0,0 -0,8 5,0 0,0 6,7 8,6 4,7 31,7 40,44.6 0,0 -0,9 5,0 0,0 6,0 10,0 4,0 24,0 40,04.7 0,0 -1,0 5,0 0,0 5,1 11,8 3,1 16,0 36,75.1 -0,4 -0,4 5,0 0,0 9,6 4,6 5,8 55,5 26,65.2 -0,5 -0,5 5,0 0,0 9,8 4,8 5,4 52,9 25,95.3 -0,6 -0,6 5,0 0,0 10,0 5,0 5,0 50,0 25,05.4 -0,7 -0,7 5,0 0,0 10,2 5,2 4,6 46,6 23,85.5 -0,8 -0,8 5,0 0,0 10,5 5,5 4,1 42,8 22,35.6 -0,9 -0,9 5,0 0,0 10,7 5,7 3,6 38,3 20,45.7 -1,0 -1,0 5,0 0,0 11,0 6,0 3,1 33,6 18,3








5. Producer does not trade forward and plays out-of-Cournot commitment strategies in the spot market. Producer assume as certain the out-of-Cournot commitment strategies of producer and also plays out-of-Cournotstrategies in the spot market. Producer punishes to a less extent.

i j

ij

j

j

j

ji

j

ji

i

i





1,0 0,0 0,0 0,0 0,0 5,7 5,7 8,7 49,1 49,12,0 0,0 0,0 5,0 0,0 9,0 4,0 7,0 63,0 28,03,0 0,0 0,0 5,0 5,0 7,3 7,3 5,3 39,1 39,14.1 0,0 -0,4 5,0 0,0 8,3 5,5 6,3 51,9 34,24.2 0,0 -0,5 5,0 0,0 8,0 6,0 6,0 48,0 36,04.3 0,0 -0,6 5,0 0,0 7,7 6,7 5,7 43,4 37,84.4 0,0 -0,7 5,0 0,0 7,3 7,5 5,3 38,1 39,44.5 0,0 -0,8 5,0 0,0 6,7 8,6 4,7 31,7 40,44.6 0,0 -0,9 5,0 0,0 6,0 10,0 4,0 24,0 40,04.7 0,0 -1,0 5,0 0,0 5,1 11,8 3,1 16,0 36,75.1 -0,4 -0,4 5,0 0,0 9,6 4,6 5,8 55,5 26,65.2 -0,5 -0,5 5,0 0,0 9,8 4,8 5,4 52,9 25,95.3 -0,6 -0,6 5,0 0,0 10,0 5,0 5,0 50,0 25,05.4 -0,7 -0,7 5,0 0,0 10,2 5,2 4,6 46,6 23,85.5 -0,8 -0,8 5,0 0,0 10,5 5,5 4,1 42,8 22,35.6 -0,9 -0,9 5,0 0,0 10,7 5,7 3,6 38,3 20,45.7 -1,0 -1,0 5,0 0,0 11,0 6,0 3,1 33,6 18,3









i j

ij

j

j

j

ji

j

ji

i

i





1,0 0,0 0,0 0,0 0,0 5,7 5,7 8,7 49,1 49,12,0 0,0 0,0 5,0 0,0 9,0 4,0 7,0 63,0 28,03,0 0,0 0,0 5,0 5,0 7,3 7,3 5,3 39,1 39,14.1 0,0 -0,4 5,0 0,0 8,3 5,5 6,3 51,9 34,24.2 0,0 -0,5 5,0 0,0 8,0 6,0 6,0 48,0 36,04.3 0,0 -0,6 5,0 0,0 7,7 6,7 5,7 43,4 37,84.4 0,0 -0,7 5,0 0,0 7,3 7,5 5,3 38,1 39,44.5 0,0 -0,8 5,0 0,0 6,7 8,6 4,7 31,7 40,44.6 0,0 -0,9 5,0 0,0 6,0 10,0 4,0 24,0 40,04.7 0,0 -1,0 5,0 0,0 5,1 11,8 3,1 16,0 36,75.1 -0,4 -0,4 5,0 0,0 9,6 4,6 5,8 55,5 26,65.2 -0,5 -0,5 5,0 0,0 9,8 4,8 5,4 52,9 25,95.3 -0,6 -0,6 5,0 0,0 10,0 5,0 5,0 50,0 25,05.4 -0,7 -0,7 5,0 0,0 10,2 5,2 4,6 46,6 23,85.5 -0,8 -0,8 5,0 0,0 10,5 5,5 4,1 42,8 22,35.6 -0,9 -0,9 5,0 0,0 10,7 5,7 3,6 38,3 20,45.7 -1,0 -1,0 5,0 0,0 11,0 6,0 3,1 33,6 18,3









i j

ij

j

j

j

ji

j

ji

i

i





1,0 0,0 0,0 0,0 0,0 5,7 5,7 8,7 49,1 49,12,0 0,0 0,0 5,0 0,0 9,0 4,0 7,0 63,0 28,03,0 0,0 0,0 5,0 5,0 7,3 7,3 5,3 39,1 39,14.1 0,0 -0,4 5,0 0,0 8,3 5,5 6,3 51,9 34,24.2 0,0 -0,5 5,0 0,0 8,0 6,0 6,0 48,0 36,04.3 0,0 -0,6 5,0 0,0 7,7 6,7 5,7 43,4 37,84.4 0,0 -0,7 5,0 0,0 7,3 7,5 5,3 38,1 39,44.5 0,0 -0,8 5,0 0,0 6,7 8,6 4,7 31,7 40,44.6 0,0 -0,9 5,0 0,0 6,0 10,0 4,0 24,0 40,04.7 0,0 -1,0 5,0 0,0 5,1 11,8 3,1 16,0 36,75.1 -0,4 -0,4 5,0 0,0 9,6 4,6 5,8 55,5 26,65.2 -0,5 -0,5 5,0 0,0 9,8 4,8 5,4 52,9 25,95.3 -0,6 -0,6 5,0 0,0 10,0 5,0 5,0 50,0 25,05.4 -0,7 -0,7 5,0 0,0 10,2 5,2 4,6 46,6 23,85.5 -0,8 -0,8 5,0 0,0 10,5 5,5 4,1 42,8 22,35.6 -0,9 -0,9 5,0 0,0 10,7 5,7 3,6 38,3 20,45.7 -1,0 -1,0 5,0 0,0 11,0 6,0 3,1 33,6 18,3


Outline

• The basic model




The pro-competitive effect in successive markets (i)

• Successive markets may also explain why the Allaz and Vila results do not seem to be happening in real power markets.

• The price of the following forward market can be affected by the decisions taken in the previous spot market.

• The production contracted in the following forward market can profit from an increase in the prices of the present spot market.

• The pro-competitive effect of forward trading is compensated by the opportunity to influence on the next forward prices, and the market tends to move back to Cournot.


The pro-competitive effect in successive markets (ii)

• Game similar to section 2 but includes four stages: forward market 1 (FM1), spot market 1 (SM1), forward market 2 (FM2) and spot market 2 (SM2).

The game unfolds as follows:

1. Firms simultaneously choose the amount of forward contracts they want to sell or buy in FM1 that calls for delivery of the good in SM1.

2. Firms simultaneously choose quantities for production that cover their SM1 sales and their FM1 obligations.

3. Firms simultaneously choose the amount of forward contracts they want to sell or buy in FM2 that call for delivery of the good in SM2.

4. Firms simultaneously choose quantities for production that cover their SM2 sales and their FM2 obligations

5. We are interested in the study of the interaction between FM1 and FM2 with SM1.

1if

1iq

2if

2iq


The pro-competitive effect in successive markets (iii)

1if

1iq 2

if2iq

FM1 FM2SM1

Delivery

_ __

_ _

_ _

_ _

_ _

_ _

_ _

_ _

_ _

_ _

_ _

_ _

__

2fp1

sp SM2

Delivery


The pro-competitive effect in successive markets (iv)

Firm i´s payoff in SM1 is as follows:

Considering

1 1 1 1 1 1 1

2 2 2 2 2 2

2 2

, ( )

, ( )

is s i j i i i i

s i j i i i i

i f

p q q q f c q

p q q q f c q

f p

é ùP = × - - × +ê úë ûé ù× - - × +ê úë û

2 2 2

1

,0

s i j

i

p q q

q

é ù¶ ê úë û=¶

2 1

1 0i i

i

q q

q

é ù¶ ê úë û=¶

2

1 0i

i

fq

¶ =¶

1 1/ 0is iq¶ P ¶ =

211 1 1 2

1 1( ) 0fsi i s i i

i i

pp q f p c fq q

¶¶ × - + - + × =¶ ¶

If we ignore the influence of the spot market in the subsequent forward price

2 1/ 0f ip qº ¶ ¶ = Allaz and Vila


The pro-competitive effect in successive markets (v)

212 1

1 1fs

f si i

ppp pq q

¶¶= ® =¶ ¶

The best estimator for the contract price in the FM2 is the spot price in the previous market

1 11 1 2 1

1 1 ( )s ss i i i i

i i

p pp c q f fq q

¶ ¶= - × - × -¶ ¶

1 11 1 2 1

1 1 ( ) 0s si s i i i

i i

p pq p c f fq q

¶ ¶× + - + × - =¶ ¶

If we consider a discount factor the optimality condition is given by

2 1i if f=

1d <

Note that if Cournot

1 11 1 2 1

1 1 ( ) 0s si s i i i

i i

p pq p c f fq q

d¶ ¶× + - + × × - =¶ ¶

The anti-competitive effect due to the opportunity to influence in the spot market on the next forward prices becomes smooth when the subsequent forward contract is signed a long time after spot market takes place –i.e. discount factor is nearby zero-. In this particular case the pro-competitive effect remains.

The strategic effects of forward marketson oligopolistic electricity equilibria

Francisco Amaya Gordillo(Universidad Pontificia Comillas de Madrid)

[email protected]

ENERDAY, Dresden 21th March 2006

Documents

The strategic effects of forward markets on oligopolistic ... fileThe strategic effects of forward markets on oligopolistic electricity equilibria-2 Introduction •Weask: – Do long-term