ECON 4930 Term paper
Finn R. Førsund
Term paper 2007
1
1a. Define the situation of overflow of the
reservoir

The water accumulation equation
Rt  Rt 1  wt  rt


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 Rt  R , the reservoir
capacity
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1b. What is the unit of measurement of the
fabrication coefficient a? Explain the
calibration of the coefficient

Unit for a:
rt m3
a H (
)
et kWh

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
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1c. Converting variables measured in water
to variables measured in kWh

Inserting the production function into the
water accumulation equation:
Rt  Rt 1  wt  rt  Rt 1  wt  aetH 
Rt Rt 1 wt

  etH
a
a
a
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2. The social planning problem
T
max
etH

t 1 z  0
pt ( z ) dz
subject to
Rt  Rt 1  wt  etH
Rt  R
etH  e H
Rt , etH  0
T , wt , Ro , R , e H given, RT free, t  1,.., T
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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
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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
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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
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2d. Kuhn – Tucker conditions

The Lagrangian function
T
L
etH

t 1 z  0
pt ( z ) dz
T
 t ( Rt  Rt 1  wt  etH )
t 1
T
  t ( Rt  R )
t 1
T
 t (etH  e H )
t 1
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2d., cont.

The Kuhn – Tucker conditions
L
H
H

p
(
e
)





0
(

0
for
e
t
t
t
t
t  0)
H
et
L
 t  t 1   t  0 ( 0 for Rt  0)
Rt
t  0( 0 for Rt  Rt 1  wt  etH )
 t  0( 0 for Rt  R )
t  0( 0 for etH  e H ) ,
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t  1,.., T
10
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)
T
 (
etH

t 1 z  0
pt ( z ) dz )
Rt 1
L *

 t
Rt 1
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2d., cont.

Shadow price on the reservoir capacity constraint
T
 (
etH

t 1 z  0
pt ( z ) dz )
R


L *
 t
R
Shadow price on the production constraint
T
 (
etH

t 1 z  0
pt ( z ) dz )
e H

L *
 t
H
e
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2e. Circumstances that may lead to a
binding constraint for production.
Concept of locking in of water and
manoeuvrability of the reservoir

Constraining production



Locking in of water


Satisfying consumption in a high-demand period
Producing in order to prevent overflow
Impossible to prevent overflow physically
Manoeuvrability

The rate of maximal production relative to
reservoir size
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2f. Kuhn – Tucker conditions for period T
L
H
H

p
(
e
)





0
(

0
for
e
T
T
T
T
T  0)
H
eT
L
 T   T  0 ( 0 for RT  0)
RT



Realistic assumptions
eTH  0, pT  0,  T  0  T  0
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
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2g. A bathtub diagram illustration
pT-1
Period T-1
pT
Period T
λT-1
A
λT
B
M
C
D
Total available water
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2h. Events that may lead to social price and
shadow price changes

Threat of overflow
t  t 1   t  0 ( Rt  0),  t  0( 0) 
t  t 1   t , pt  t , pt 1  t 1

Emptying the reservoir
t  t 1   t  0 ( Rt  0),  t  0( Rt  R ) 
t  t 1 , pt  pt 1

Binding production constraint
pt (etH )  t  t  0 ( etH  0), t  0() 
pt (etH )  t  t  pt (etH )  t
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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
u  2  u 1  u  T ,
pu  2  T , pu 1  u 1  u 1  pu 1  pT
pu  u  T
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2i. Illustration: two-period bathtub diagram
for periods u and u+1
pu
Period u
Pu+1
Period u+1
Pu+1=λu+1+ρu+1
ρu+1
pT=λu+1
pu=λu=pT
A
B
B’’
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C
D
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