ECON 4930 Spring 2013Electricity EconomicsLecture 8 (Chapter 8)

Lecturer:Finn R. Førsund

Market power 1

Background

• The use of market power is a potential problem of the deregulated electricity sector

• 20 % of the world’s electricity is produced by hydropower and 1/3 of the countries have more than 50%

• Hydro power is a special case due to zero variable cost, water storage, high power capacity and maximal flexible adjustment

Market power 2

Difference between objective functions for social planning and monopoly

• Objective function social planning– Area under the demand function; consumer + producer surplus (zero variable costs)

• Objective function monopoly– Producer surplus: total revenue (zero variable costs)

Market power 3

1 0( )

HteT

tt zp z dz

1( )T H H

t t ttp e e

Monopoly with reservoir constraint

Market power 4

1

1

max ( )

subject to

, 0 , 1,..,, , , given, free

TH H

t t tt

Ht t t t

t

Ht t

t o T

p e e

R R w eR R

R e t TT w R R R

Solving the monopoly optimisation problem

• The Lagrangian function

Market power 5

1

11

1

( )

( )

( )

TH H

t t tt

TH

t t t t tt

T

t tt

L p e e

R R w e

R R

Solving, cont.

• The Kuhn – Tucker conditions

Market power 6

1

1

( ) ( ) 0 ( 0 for 0)

0 ( 0 for 0)

0 (0 for )0 (0 for ) , 1,..,

H H H Ht t t t t t tH

t

t t t tt

Ht t t t t

t t

L p e e p e eeL RR

R R w eR R t T

Interpretation of conditions• Assuming positive production in all periods• Using flexibility‐corrected prices

– Flexibility is the inverse of elasticity• Assuming positive flexibility‐corrected prices is a stronger assumption than non‐satiation

• Market prices will vary with relative elastic periods having lower prices than relative inelastic periods

• A strategy of shifting water between periods

Market power 7

1

( ) ( ) ( )(1 ) 0( 0 for 0), 1,..,

H H H Ht t t t t t t t t t

t t t t

p e e p e p eR t T

0H

t tt H

t t

p ee p

Limited reservoir and monopoly

• The flexibility‐corrected price substitute for the optimal price

• Flexibility‐corrected prices may differ even if reservoir constraints are not binding

• Social solution may be optimal for the monopolist if constraints are binding

• Differences with social solution depend on the demand elasticities

• Spilling may be optimal for the monopolist

Market power 8

Illustration of monopoly solutionwith reservoir constraint not binding

Market power 9

Period 1

p1M

p2M

A B C D

Period 2

p1S

p2S

λM

Illustration of monopoly solutionwith reservoir constraint binding

Market power 10

Period 1

=p1M

= p2M

A B C D

Period 2

p1S

p2S

2M

1M 1

Illustration of monopoly solution withreservoir constraint binding and spill

Market power 11

Period 1

p1M

=p2M

A B C D

Period 2

p2S

Spill

2M

1M

1

p1S

Spilling and monopoly

• Depending on the characteristics of the demand functions it may be optimal for the monopoly to spill water

• Spilling can be regulated by prohibition– Making the water accumulation equation  an equality constraint

• Zero‐spilling regulation will reduce the monopoly profit

Market power 12

1H

t t t tR R w e

Restricting spill

Market power 13

1

1

max ( )

subject to

, 0 , 1,..,, , , given, free

TH H

t t tt

Ht t t t

tH

t t

t o T

p e e

R R w eR R

R e t TT w R R R

• The Kuhn – Tucker conditions

• The reservoir constraint is now an equality constraint and λt is free in sign

• The price flexibility becomes less than ‐1, and the water value negative for period 1

• We get the social solution for water use and prices

Market power 14

1

( ) ( ) 0 ( 0 for 0)

0

0 (0 for ) , 1,..,

H H H Ht t t t t t tH

t

t t tt

t t

L p e e p e eeLR

R R t T

Illustration of monopoly solutionwith reservoir and spill

Market power 15

Period 1

p1M

=p2M

A B C D

Period 2

p2S

Spill

2

1

1

p1S

1<0

Monopoly with hydro and thermal plants

• Optimisation problem

Market power 16

1

1

max ( ) ( )

subject to

, , 0 , 1,..,

, , , , given, free

TTh

t t t tt

H Tht t t

Ht t t t

t

Th Tht

H Tht t t

Tht o T

p x x c e

x e e

R R w eR R

e e

R e e t T

T w R R e R

• The Lagrangian

Market power 17

1

1

11

1

[ ( )( ) ( )]

( )

( )

( )

TH Th H Th Th

t t t t t tt

TTh Th

t ttT

Ht t t t t

tT

t tt

L p e e e e c e

e e

R R w e

R R

• Necessary first‐order conditions

Market power 18

1

1

( )( ) ( ) 0

( 0 for 0)

( )( ) ( ) '( ) 0

( 0 for 0)

0 ( 0 for 0)

0 ( 0 for )

0 ( 0 for

H Th H Th H Tht t t t t t t t tH

tHt

H Th H Th H Th Tht t t t t t t t t tTh

tTht

t t t tt

Ht t t tTh

t t

L p e e e e p e ee

eL p e e e e p e e c e

e

eL RR

R R w e

e

)0 (0 for )

Th

t t

eR R

• Using both hydro and thermal in the same period the marginal revenue substitutes for the marginal willingness to pay in the social optimal solution

Market power 19

( )(1 ) ( )Tht t t t t tp x c e

Hydro and thermal plants

Market power 20

Period 2Period 1

p2

λ

p1

Hydro energy

λ

a                                      A                                     B           M                              C     D                                  d

c' c'

1 1( )p x2 2(1 )p

2 2( )p x

1 1(1 )p

Thermal

Dominant hydro with a thermal competitive fringe

• Optimisation problem

Market power 21

1

1

max ( )

subject to

( ) ( )

, , 0, 1,..,, given, free

TH

t t tt

H Tht t t

Ht t t t

tTh

t t tH Th

t t t

T

p x e

x e e

R R w eR R

p x c e

x e e t TT R R

• The fringe response to change in hydro production– Differentiation of the fringe 1. order condition

– The relationship between the fringe output and hydro output

Market power 22

( ) ( )( 1,.., )H Th Tht t t tp e e c e t T

( )( ) ( )

( ) 0 ( 1,.., )( ) ( )

H Th H Th Th Tht t t t t t t

Th H Tht t t tH H Th Tht t t t t

p e e de de c e de

de p e e t Tde p e e c e

( ), 0 ( 1,.., )Th Ht t t te f e f t T

• The Lagrangian

Market power 23

1

11

1

( ( ))

( )

( )

TH H H

t t t t tt

TH

t t t t ttT

t tt

L p e f e e

R R w e

R R

1

1

( ) ( ) (1 ) 0

( 0 for 0)

0 ( 0 for 0)

0 (0 for )0 (0 for ) , 1,..,

ThH Th H Th H t

t t t t t t t tH Ht t

Ht

t t t tt

Ht t t t t

t t

L dep e e p e e ee de

eL RR

R R w eR R t T

• First‐order order conditions

• Inserting the fringe output reaction

Market power 24

( ) ( )1 1 0( ) ( ) ( ) ( )

Th H Th Tht t t t tH H Th Th H Th Tht t t t t t t t t

de p e e c ede p e e c e p e e c e

• The conditional marginal revenue function

Market power 25

(1 ) , 1,..,t

H ThHt t

t t t t tp c H Th Ht t t

e deMR p p e t Te e de

Competitive thermal fringe

Market power 26

Period 2Period 1

λ

p1

Hydro energy

λ

a                                   A                                       B                    M              C             D                                                  d

c'

c'p2

1 1( )p x 2 2( )p x

11 p cMR

Thermal fringe

22 p cMR

Monopoly and trade• Optimisation problem

1

1

max ( ( ) )

subject to

,

, , 0

, , , given, free, 1,..,

TXI XI

t t t t tt

H XIt t t

XI XI XIt

Ht t t t

t

Ht t t

XI XI XIt t

p x x p e

x e e

e e e

R R w eR R

x e R

T R e p e t T

27Market power

• The Lagrangian

1

11

1

1

1

( ) ( )

( )

( )

( )

( )

TH XI H XI XI XI

t t t t t t tt

TH

t t t t tt

T

t tt

TXI XI

t tt

TXI XI

t tt

L p e e e e p e

R R w e

R R

e e

e e

28Market power

• The first‐order conditions

1

1

( )( ) ( ) 0

( 0 for 0)

( )( ) ( ) 0

0 ( 0 for 0)

0 ( 0 for )0 ( 0 for )

0 ( 0 for

H XI H XI H XIt t t t t t t t tH

tHt

H XI H XI H XI XIt t t t t t t t t t tXI

t

t t t tt

Ht t t t t

t t

t t

L p e e e e p e ee

eL p e e e e p e e p

eL RR

R R w eR R

e

)

0 ( 0 for )

XI XI

XI XIt t

e

e e

29Market power

Monopoly and trade, figure

1He

1 1( )p x

2 2(1 )p

1He

1XIe

2He

XIe

R

2 2( )p x

1 1(1 )p

1 1XIp

2XIp

A B C D

Period 2

Period 1

λ2

Waterperiod 1

Importcapacity

Inflowperiod 2

γ1

α2

p2M

p1M

30Market power

1XIp