trade1 econ 4925 autumn 2010 hydropower economics lecture 13 lecturer: finn r. førsund
TRANSCRIPT
Trade 2
Unconstrained trade
Consider two countries (regions) linked by an interconnector, home country and abroad
Loss on the interconnector is disregarded Trade during a period is the net flow, i.e., either
import or export The price of electricity abroad is exogenous Money is a new good in our partial model Trade income (expenditure) is just added
(subtracted from) to the social value of electricity consumption
Trade 3
Opening up for free trade
The social planning problem
New symbols
1 0
1
max [ ( ) ]
subject to
, , 0, unrestricted
, , , , given , 1,..,
txTXI XI
t t tt z
H XIt t t
Ht t t t
t
H XIt t t t
XIt t o
p z dz p e
x e e
R R w e
R R
x e R e
T w p R R t T
export/import price
export (positive)/
import (negative)
consumption (home)
energybalance
XIt
XIt
t
H XIt t t
p
e
x
x e e
Trade 4
The Lagrangian
Inserting the energy balance because it has to hold as an equality
1 0
11
1
( ( ) )
( )
( )
H XIt te eT
XI XIt t t
t z
TH
t t t t tt
T
t tt
L p z dz p e
R R w e
R R
Trade 5
The Kuhn – Tucker conditions
The second equation holds with equality because trade variable is unrestricted in sign
1
1
( ) 0 ( 0 for 0)
( ) 0
0 ( 0 for 0)
0 ( 0 for )
0 ( 0 for ) , 1,..,
H XI Ht t t t tH
t
H XI XIt t t tXI
t
t t t tt
Ht t t t t
t t
Lp e e e
e
Lp e e p
e
LR
R
R R w e
R R t T
Trade 6
Trade and the bathtub diagram for two periods
γ1
p1Period 2Period 1
Total available water
p2
Import
A x1A' B AAU B' C C' x2
D
p1AU p2
AU
p2XI = 2
p1XI = 1
Trade 7
The social planning problem with trade constraint
1 0
1
max [ ( ) ]
subject to
, , 0 , unconstrained in sign
, , , , , given, free, 1,..,
txTXI XI
t t tt z
H XIt t t
Ht t t t
t
XI XI XIt
H XIt t t t
XI XIt o t T
p z dz p e
x e e
R R w e
R R
e e e
x e R e
T w R R p e R t T
Trade 8
The Lagrangian function
1 0
11
1
1
1
( ( ) )
( )
( )
( )
( )
H XIt te eT
XI XIt t t
t z
TH
t t t t tt
T
t tt
TXI XI
t tt
TXI XI
t tt
L p z dz p e
R R w e
R R
e e
e e
Trade 9
The Kuhn – Tucker conditions
1
1
( ) 0 ( 0 for 0)
( ) 0
0 ( 0 for 0)
0 ( 0 for )
0 ( 0 for )
0 ( 0 for ) ( 0)
0 ( 0 for ) ( 0) , 1,
H XI Ht t t t tH
t
H XI XIt t t t t tXI
t
t t t tt
Ht t t t t
t t
XI XI XIt t t
XI XI XIt t t
Lp e e e
e
Lp e e p
e
LR
R
R R w e
R R
e e e
e e e t
..,T
Trade 10
The bathtub diagram for two periods
p1
Prod.1 Cons.2Import Export
Total available water
p2
α2
β1
γ1pAU
Period 1 Period 2
x1A B C Dx2
p1=1
p1XI
p2XI
p2=2
Trade 11
Trade between countries Hydro and Thermal The social planning problem in Thermal
Unlimited trade to exogenous prices
1 0
max [ ( ) ( )]
subject to
, 0, unrestricted
, , given , 1,..,
txTXI XI Th
t t t tt z
Th XIt t t
Th Tht
Th XIt t t
XI Tht
p z dz p e c e
x e e
e e
x e e
T p e t T
Trade 12
The Lagrangian function
Inserting the energy balance
1 0
1
[ ( ) ( )]
( )
Th XIt te eT
XI XI Tht t t t
t z
TTh Th
t tt
L p z dz p e c e
e e
Trade 13
The Kuhn – Tucker conditions
The second equation holds with equality because the trade variable is unrestricted in sign
( ) '( ) 0 ( 0 for 0)
( ) 0
0 ( 0 for ), 1,..,
Th XI Th Tht t t t t tTh
t
Th XI XIt t t tXI
t
Th Tht t
Lp e e c e e
e
Lp e e p
e
e e t T
Trade 14
Illustrating two periods
Period 2Period 1
ExportImport
c'
c'
θ2
c' (0)
x2x1
x1
A B CO D
p1AU
p1XI=p1
p2AU
p2XI=p2
Trade and transmission 15
Trade between Hydro and Thermal The cooperative social planning problem
1 0 0
, ,
, ,
1
, ,
max [ ( ) ( ) ( )]
subject to
, , , 0
, , given , 1,..,
H Tht tx xT
H Th Tht t t
t z z
H H XI XIt t Th t H t
Th Th XI XIt t Th t H t
THt
t
Th Tht
H H XI XIt t Th t H t
Th
p z dz p z dz c e
x e e e
x e e e
e W
e e
x e e e
T W e t T
Trade and transmission 16
The Lagrangian function
Inserting the energy balances Export for one country is import for the other
, , , ,
1 0 0
1
1
[ ( ) ( ) ( )]
( )
( )
H XI XI Th XI XIt Th t H t t Th t H te e e e e eT
H Th Tht t t
t z z
TTh Th
t tt
THt
t
L p z dz p z dz c e
e e
e W
Trade and transmission 17
The Kuhn – Tucker conditions
,,
,,
1
( ) 0 ( 0 for 0)
( ) ( ) 0 ( 0 for 0)
( ) '( ) 0 ( 0 for 0)
( ) ( ) 0 ( 0 for 0)
0 ( 0 for )
0 ( 0 fo
H H Ht t tH
t
H H Th Th XIt t t t H tXI
H t
Th Th Th Tht t t t tTh
t
H H Th Th XIt t t t Th tXI
Th t
THt
t
t
Lp x e
e
Lp x p x e
e
Lp x c e e
e
Lp x p x e
e
e W
r )Th Th
te e