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Nodal prices Oscar Volij Iowa State University Nodal prices – p. 1/44
Nodal pricesOscar Volij
Iowa State University
Nodal prices – p. 1/44
MotivationThere are two regions: Oblivia and Rodrigombia. Each of thetwo regions has generators and consumers.
Nodal prices – p. 2/44
The main actors
Consumers GeneratorOblivia
Consumers GeneratorRodrigombia
Figure 1: The main actors
Nodal prices – p. 3/44
ObliviaThe generator in Oblivia is characterized by the following costfunction:
CO(q) =16 q
5+
q2
15.
The consumers in Oblivia have the following utility function:
UO(m,x) = m + (112− x) x
Nodal prices – p. 4/44
RodrigombiaThe generator in Rodrigombia is characterized by the followingcost function:
CR(q) =204 q
25+
14 q2
25.
The consumers in Rodrigombia have the following utilityfunction:
UR(m,x) = m + 2 (54− x) x
Nodal prices – p. 5/44
The Generators ProblemGenerators take the price of power p as given, and generate theamount of power that brings their profits to the maximum.Formally, node i’s generator solves
maxq≥0
pq − Ci(q).
The solution to this problem is the quantity that solves thefirst order condition
p =∂Ci
∂q(q∗).
The solution to this equation is a function of the price, and isknown as the generator’s supply function.
Nodal prices – p. 6/44
Oblivia’s Supply FunctionIn the case of Oblivia, the profit maximizing output solves
−
(
16
5
)
+ p−2 q
15= 0
Consequently, Oblivia’s generator’s supply function is
SO(p) =3 (−16 + 5 p)
2.
Nodal prices – p. 7/44
Rodrigombia’s Supply FunctionIn the case of Rodrigombia, the profit maximizing output solves
−
(
204
25
)
+ p−28 q
25= 0
Consequently, Rodrigombia’s generator’s supply function is
SR(p) =−204 + 25 p
28.
Nodal prices – p. 8/44
The Consumers’ ProblemConsumers take the price of power p and their income I asgiven, and buy the amount of power that brings their utility tothe maximum. Formally, node i’s consumer solves
maxm,q≥0
m + ui(m, q)
s.t. m + pq = I.
The solution to this problem is the quantity that solves thefirst order condition
p =∂ui
∂q(q∗).
The solution to this equation is a function of the price, and isknown as the consumer’s demand function.
Nodal prices – p. 9/44
Oblivia’s Demand FunctionIn the case of Oblivia, the utility maximizing quantity solves
112− p− 2x = 0
Consequently, Oblivia’s consumers’ demand function is
DO(p) =112− p
2.
Nodal prices – p. 10/44
Rodrigombia’s Demand FunctionIn the case of Rodrigombia, the utility maximizing quantitysolves
−p + 2 (54− x)− 2x = 0
Consequently, Rodrigombia’s consumers’ demand function is
DR(p) =108− p
4.
Nodal prices – p. 11/44
Autarkic equilibriumAssume that there is no line connecting Oblivia andRodrigombia. Then the equilibrium condition of the Oblivianmarket is
DO(p) = SO(p)
or112− p
2=
3 (−16 + 5 p)
2.
The equilibrium price is given by
pAO = 10
and the corresponding equilibrium quantity is
qAO = 51.
Nodal prices – p. 12/44
Oblivia in Autarky
45 50 55 60quantity
10
20
30
price Oblivia
OBLIVIAN MARKET
Nodal prices – p. 13/44
Autarkic equilibriumSimilarly, the equilibrium condition of the Rodrigombianmarket is
DR(p) = SR(p)
or108− p
4=−204 + 25 p
28.
The equilibrium price is given by
pAR = 30
and the corresponding equilibrium quantity is
qAR = 39/2.
Nodal prices – p. 14/44
Rodrigombia in Autarky
15 20 25 30quantity
20
40
60
price Rodrigombia
RODRIGOMBIAN MARKET
Nodal prices – p. 15/44
Unconstrained equilibriumAssume now that there is a line that connects Oblivia withRodrigombia and that an unlimited amount of power can betransmitted along the line.
Nodal prices – p. 16/44
The grid
Consumers GeneratorOblivia
Consumers GeneratorRodrigombia
Figure 2: The main actors
What is the socially optimal generation and production levelsat each node?
Nodal prices – p. 17/44
Social optimumIn order to find the social optimum we need to solve thefollowing problem:
max UO(xO) + UR(xR)− CO(qO)− CR(qR)
s.t. xO + xR = qO + qR
In our case, the problem is
max (112− xO) xO + 2 (54− xR) xR
−16 qO
5− qO
2
15− 204 qR
25− 14 qR
2
25
s.t. xO + xR = qO + qR
Nodal prices – p. 18/44
Social optimumThe Lagrangian is
L = (112− xO) xO + 2 (54− xR) xR
−16 qO
5−
qO2
15−
204 qR
25−
14 qR2
25− λ(xO + xR − qO − qR)
Nodal prices – p. 19/44
Social OptimumThe first order conditions are:
112− 2 xO − λ = 0
2 (54− xR)− 2 xR − λ = 0
−16/5− 2 qO/15 + λ = 0
−204/25− 28 qR/25 + λ = 0
qO + qR − xO − xR = 0
The solution to this system of equations is
{xO →199
4, xR →
191
8, qO →
279
4, qR →
31
8, λ →
25
2}
Nodal prices – p. 20/44
Competitive equilibriumThe competitive equilibrium obtains when the market-clearingconditions is satisfied: aggregate demand equals aggregatesupply.Aggregate demand:
D(p) = DO(p) + DR(p)
=108− p
4+
112− p
2
= 83−3 p
4.
Nodal prices – p. 21/44
Competitive EquilibriumAggregate Supply:
S(p) = SO(p) + SR(p)
=3 (−16 + 5 p)
2+−204 + 25 p
28
=−876 + 235 p
28.
Nodal prices – p. 22/44
Competitive equilibriumThe market clearing condition is
D(p) = S(p)
83−3 p
4=
−876 + 235 p
28.
Nodal prices – p. 23/44
Competitive equilibrium
50 60 70 80 90quantity
-10
10
20
30
price Free Trade
UNCONSTRAINED MARKET
Nodal prices – p. 24/44
Competitive equilibriumThe market clearing price is
p∗ =25
2
and the quantities produced and consumed in each one of thenodes are:
{xO →199
4, xR →
191
8, qO →
279
4, qR →
31
8}
Nodal prices – p. 25/44
Competitive EquilibriumNote that in this equilibrium Oblivia exports 20 units toRodrigombia.Note that the competitive equilibrium allocation is sociallyoptimal.
Nodal prices – p. 26/44
Competitive equilibrium
199������������
4279������������
4
quantity
25���������2
price Oblivia
OBLIVIA
Nodal prices – p. 27/44
Competitive equilibrium
191������������
831���������8
quantity25���������2
price Rodrigombia
RODRIGOMBIA
Nodal prices – p. 28/44
Competitive equilibrium
191������������
831���������8
199������������
4279������������
4
quantity25���������2
price Oblivia
THE WHOLE MARKET
Nodal prices – p. 29/44
Social SurplusThe change in the social surplus in Oblivia is
∫
10
25
2
DO(p) dp +
∫25
2
10
SO(p) dp = 25
The change in the social surplus in Rodrigombia is
∫ 30
25
2
DR(p) dp +
∫25
2
30
SR(p) dp = 175
Nodal prices – p. 30/44
Constrained equilibriumAssume that the maximum amount of power that can flowthrough the line is 16 units.
Therefore, the above competitive equilibrium cannot beimplemented because according to it, there are 20 units ofpower flowing along the Oblivia-Rodrigombia line.
What is the socially optimal allocation of resources?
What is the consumption and generation levels in eachregion that maximize the social surplus?
Nodal prices – p. 31/44
Social optimumIn order to find the social optimum we need to solve thefollowing problem:
max UO(xO) + UR(xR)− CO(qO)− CR(qR)
s.t.
{
xO + xR = qO + qR
qO − xO ≤ 16
In our case, the problem is
max (112− xO) xO + 2 (54− xR) xR
−16 qO
5− qO
2
15− 204 qR
25− 14 qR
2
25
s.t.
{
xO + xR = qO + qR
qO − xO ≤ 16
Nodal prices – p. 32/44
Social optimumThe Lagrangian is
L = (112− xO) xO + 2 (54− xR) xR
−16 qO
5−
qO2
15−
204 qR
25−
14 qR2
25−λ(xO + xR − qO − qR)− µ(qO − xO − 16)
Nodal prices – p. 33/44
Social OptimumThe first order conditions are:
112− 2 xO − λ + µ = 0
2 (54− xR)− 2 xR − λ = 0
−16/5− 2 qO/15 + λ− µ = 0
−204/25− 28 qR/25 + λ = 0
qO + qR − xO − xR = 0
qO − xO − 16 ≤ 0
(qO − xO − 16)µ = 0
The solution to this system of equations is
{xO → 50, xR → 23, qO → 66, qR → 7, λ → 16, µ → 4}
Nodal prices – p. 34/44
Competitive equilibriumQuestion: Can the above outcome be obtained as a result ofdecentralized trade?Answer: Yes!Question: How?Answer: As follows.
Nodal prices – p. 35/44
Competitive equilibriumWhat would happen in Rodrigombia if the local price ofpower was 16?
To answer this question we need to look at theRodrigombian market
237quantity
12
price Rodrigombia
Nodal prices – p. 36/44
Competitive equilibriumWhat would happen in Rodrigombia if the local price ofpower was 16?
To answer this question we need to look at theRodrigombian market
50 66quantity
12
price Oblivia
Nodal prices – p. 37/44
Competitive equilibrium
237 50 66quantity
1216
price Oblivia
THE WHOLE MARKET
Nodal prices – p. 38/44
Competitive equilibriumWe see that if the price in Oblivia is 12 and the price inRodrigombia is 16, the quantities demanded and supplied ineach of the regions coincide with the socially optimal quantities.
Nodal prices – p. 39/44
Competitive equilibriumIn this equilibrium:
Oblivian generators sell 66 units at $12/MW
Oblivian consumers buy 50 units at $12/MW
Rodrigombian generators sell 23 units at $16/MW
Rodrigombian consumers buy 7 units at $16/MW
Therefore
Oblivian generators export 16 units to Rodrigombianconsumers
Oblivian generators get $16× 12
Rodrigombian consumers pay $16× 16
Where does the difference go?
Nodal prices – p. 40/44
Transmission rentThe difference goes to the transmission owners.
7 16 20quantity
12
16
price Transmission Rents
Nodal prices – p. 41/44
Transmission rentThe transmission owners charge $4 for each unit that transitsalong the line.
7 16 20quantity
12
16
price Transmission Rents
Nodal prices – p. 42/44
Competitive equilibrium
Consumers�
Oblivia
Consumers
50
Rodrigombia
�
y
)
23
66
7
?
16
pO = 12
pR = 16
Figure 3: Equilibrium dispatch Nodal prices – p. 43/44
Competitive equilibriumHow can we characterize our equilibrium?The competitive equilibrium consists of
A price of $12/MW in Oblivia
A price of $16/MW in Rodrigombia
A transmission charge of $4/MW
such that
SO(12)−DO(12) = DO(16)− SO(16)
The power transmitted does not exceed the capacity of theline. In fact it equals the capacity of the line given that thetransmission charge is positive
One cannot make money by buying power from Obliviangenerators, transmitting it through the line and selling it toRodrigombian consumers.
Nodal prices – p. 44/44
In Reality . . .In reality there is no benevolent dictator that knows thedetails of the economy and imposes on society the socialoptimum.
There is no fully competitive market either.
There is a independent system operator that coordinatesthe market.
Nodal prices – p. 45/44
The System Operator. . .Asks the consumers and generators for their bids (demandand supply functions).
Consumers and generators at each node report (hopefullytruthfully) their respective bids.
In our case Oblivians report DO(p) and SO(p) and Rodrigombians report
DR(p) and SR(p).
The ISO calculates the social optimum given the reportedbids.
The ISO announces the corresponding economic dispatchand the nodal prices.
{xO → 50, xR → 23, qO → 66, qR → 7, pR → 16, pO → 12}
Contracts are carried out.
The ISO pays the transmission owners the transmissionrents.
Nodal prices – p. 46/44
