Term paper 2007 1

ECON 4930 Term paper

Finn R. Førsund

Term paper 2007 2

1a. Define the situation of overflow of the reservoir The water accumulation equation

Strict inequality means that the amount of water at the end of period t is less than the sum of what was received from period t-1 plus inflow during period t subtracted the release during period t → overflow

Overflow implies that , the reservoir capacity

1t t t tR R w r

tR R

Term paper 2007 3

1b. What is the unit of measurement of the fabrication coefficient a? Explain the calibration of the coefficient Unit for a:

Calibration: The height of the fall from the reservoir to the

turbine, called head. Gravity gives the energy of water

The efficiency losses due to friction in pipes, turbine not perfect, totalling 10-15% loss

3

( )tHt

r ma

e kWh

Term paper 2007 4

1c. Converting variables measured in water to variables measured in kWh Inserting the production function into the

water accumulation equation:

1 1

1

Ht t t t t t t

Ht t tt

R R w r R w ae

R R we

a a a

Term paper 2007 5

2. The social planning problem

1 0

1

max ( )

subject to

, 0

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

HteT

tt z

Ht t t t

t

H Ht

Ht t

Ht o T

p z dz

R R w e

R R

e e

R e

T w R R e R t T

Term paper 2007 6

2a. Discuss the objective function for the planning problem The objective function is the area under the inverse demand curve (NB! Choke price finite) Demand function can be linked to utility function The model is partial because there are no links to

other activities, goods, etc. in the economy A typical general objective function is the consumer

plus producer surplus. Because variable production costs are zero we are left with the area under the demand curve.

Discounting is neglected due to short total time period

Term paper 2007 7

2b. Why is the planning problem formulated as a dynamic problem? Having a reservoir means that water used

today can alternatively be used tomorrow, water has an opportunity cost

Term paper 2007 8

2c. Discuss reasons for a constraint on production to be realistic Production measured in kWh can have an

upper limit for a period due to technical reasons The flow of water through the pipe hitting the

turbines is constrained by the diameter The conversion to electricity is constrained by

installed turbine capacity The production of electricity may be constrained

by the size of the generator

Term paper 2007 9

2d. Kuhn – Tucker conditions

The Lagrangian function

1 0

11

1

1

( )

( )

( )

( )

HteT

tt z

TH

t t t t tt

T

t tt

TH H

t tt

L p z dz

R R w e

R R

e e

Term paper 2007 10

2d., cont.

The Kuhn – Tucker conditions

1

1

( ) 0 ( 0 for 0)

0 ( 0 for 0)

0( 0 for )

0( 0 for )

0( 0 for ) , 1,..,

H Ht t t t tH

t

t t t tt

Ht t t t t

t t

H Ht t

Lp e e

e

LR

R

R R w e

R R

e e t T

Term paper 2007 11

2d., cont. Interpretation of shadow prices:

Change in the optimised objective function of a marginal change in the constraint, found by partial differentiation of the optimised Lagrangian

Shadow price on the water accumulation constraint Change in the objective function of a marginal change in

the constraint (i.e., change in Rt-1,wt)

1 0

1 1

( ( ) )*

HteT

tt z

tt t

p z dzL

R R

Term paper 2007 12

2d., cont.

Shadow price on the reservoir capacity constraint

Shadow price on the production constraint

1 0

( ( ) )*

HteT

tt z

t

p z dzL

R R

1 0

( ( ) )*

HteT

tt z

tH H

p z dzL

e e

Term paper 2007 13

2e. Circumstances that may lead to a binding constraint for production. Concept of locking in of water and manoeuvrability of the reservoir Constraining production

Satisfying consumption in a high-demand period Producing in order to prevent overflow

Locking in of water Impossible to prevent overflow physically

Manoeuvrability The rate of maximal production relative to

reservoir size

Term paper 2007 14

2f. Kuhn – Tucker conditions for period T

Realistic assumptions

No satiation of demand: price positive Binding production constraint in period T

Not realistic unless T is a high-demand period, prevention of overflow is not realistic in the last period

( ) 0 ( 0 for 0)

0 ( 0 for 0)

H HT T T T TH

T

T T TT

Lp e e

e

LR

R

0, 0, 0 0HT T T Te p

Term paper 2007 15

2g. A bathtub diagram illustration

pT-1

λT-1

M DCBA

Period TPeriod T-1

λT

Total available water

pT

Term paper 2007 16

2h. Events that may lead to social price and shadow price changes Threat of overflow t

Emptying the reservoir t

Binding production constraint t

1

1 1 1 1

0 ( 0), 0( 0)

, ,t t t t t

t t t t t t t t t

R

p p p p

1

1 1

0 ( 0), 0( )

,t t t t t t

t t t t

R R R

p p

( ) 0 ( 0), 0( )

( ) ( )

H Ht t t t t t

H Ht t t t t t t

p e e

p e p e

Term paper 2007 17

2i. Shadow prices on stored water for period u+2, u+1 and u Production constraint binding for period u+1,

but not for period u, and u+2 to T Reservoir in between full and empty from T-1

to u

2 1

2 1 1 1 1

,

,u u u T

u T u u u u T

u u T

p p p p

p

Term paper 2007 18

2i. Illustration: two-period bathtub diagram for periods u and u+1pu

pu=λu=pT

DCA

Period u+1Period u

ρu+1

Pu+1

pT=λu+1

Pu+1=λu+1+ρu+1

BB’’