Trade 1

ECON 4925 Autumn 2010 Hydropower economicsLecture 13

Lecturer:

Finn R. Førsund

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

Trade and transmission 18

Combining the bathtub diagram and the thermal diagram for two periods

θ2

Period 1 Period 2

c' c'

Import Export

p1Th=p1

H= p2Th=p2

H=

ImportExport

Hydro ThermalThermal

A' A M' M B' B