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