Supply function equilibrium with taxed bene…ts
Anthony Downward, Andy Philpott and Keith Ruddell
Electric Power Optimization Centre
University of Auckland
March 25, 2014
Abstract
Supply function equilibrium models are used to study electricity market auctions. We
study the e¤ects on supply function equilibrium of a system tax on the observed bene…ts of
suppliers. Such a tax provides an incentive for agents to submit more competitive o¤ers. The
model is extended to a setting in which the agents are taxed on the bene…ts accruing to them
from a transmission line expansion (in order to help fund the line).
1
Introduction
The supply function equilibrium (SFE) is a natural concept to use in studying electricity pool
markets. Here generators submit supply schedules in the form of increasing o¤er curves to an
auction that dispatches generation to those suppliers o¤ering at the lowest prices. The …rst use of
SFE in this context was by Green and Newbery [6] in a study of the England and Wales electricity
pool. The paper [6] was based on the theory of supply function equilibrium laid out by Klemperer
and Meyer [11]. This theory has been extended to inelastic demand and pay-as-bid auctions
by several authors, notably [1],[2],[3],[4],[5],[7],[12],[14]. As well as [6], a number of authors have
applied SFE in a practical setting. For example [10] and [13] use the SFE model to study generator
behaviour in the Texas electricity pool. The recent survey paper by Holmberg and Newbery [9]
gives a good overview of the state of the art in SFE models.
In this paper we examine a supply-function equilibrium model in circumstances where suppliers
are taxed on the bene…ts they earn. At …rst sight, paying a (non-progressive) tax that is a …xed
proportion of pro…ts will not alter agent behaviour. Each supplier will still want to maximize
after-tax pro…t, which will be achieved by maximizing pro…t before tax. The pro…t in any market
outcome can be estimated by computing the di¤erence between clearing price and marginal cost
at each level of production and integrating over all production levels below the cleared level.
Taxation can be used as a mechanism to redistribute wealth, or to extract payment for assets (like
transmission lines) that are shared between market participants.
In some settings, like electricity markets, suppliers o¤er supply curves to the market that in
equilibrium are marked up over marginal cost. Here the o¤er curves are revealed to the market
auctioneer, but the true marginal costs are private information. In such circumstances the auctioneer can estimate supplier pro…ts based on the di¤erence between clearing price and o¤ered price,
This paper is a revision of a previous manuscript, "Taxation and supply function equilibrium" by A.B. Philpott.
1
and tax this observed pro…t at a …xed proportion. Since the o¤er curve is a choice of the supplier,
it can a¤ect the observed pro…t by its o¤er, without having too much e¤ect on its actual pro…t. In
the simplest case where demand is certain, a supplier might increase the price on infra-marginal
units, i.e. units that have o¤ered below the clearing price, and make observed pro…t very small
without a¤ecting the actual pro…t.
When demand is uncertain, increasing o¤er prices on inframarginal units must be done carefully
since higher o¤er prices might decrease the probability of being dispatched. In this setting one
can use the theory of market distribution functions to derive a SFE that illustrates the incentive
to increase prices on inframarginal o¤ers. This analysis draws heavily on the market distribution
theory of both uniform-price auctions [3] and pay-as-bid auctions [2].
Our study of taxation has been motivated by a proposed cost-recovery scheme in which the
bene…ciaries of an electricity transmission line capacity expansion contribute to its cost in proportion to the additional bene…ts (pro…t) accrued by each agent. This is slightly di¤erent from
the model in which all pro…ts are taxed, in the sense that pro…ts that would have accrued in a
counterfactual model (with the original line capacity) are exempt from the tax.
The paper is laid out as follows. In the next section we review supply function equilibrium
through the lens of market distribution functions. This is used to derive optimality conditions for
suppliers who are taxed a …xed proportion of their observed pro…t. We show that pure-strategy
SFE can be computed as long as the tax is not too high. In the extreme case where the tax is 100%,
it is easy to see that the payment mechanism becomes a pay-as-bid scheme. It is well known [2]
that pure-strategy SFE occur very rarely in these auctions and mixed strategies prevail. Section 3
deals with the conditions for supply function equilibrium in symmetric duopoly when demand is
inelastic and additive demand shocks have a uniform distribution. In section 4 we study an SFE
model of a line capacity expansion with inelastic demand. The last section concludes the paper.
2
Supply function equilibrium
As shown in [3] the optimal o¤er curve p(q) for a generator with cost C(q) facing a market
distribution function (q; p) will maximize
Z
= (qp C(q))d (q; p):
The market distribution function (q; p) de…nes the probability that a supplier is not fully dispatched if they o¤er the quantity q at price p. It can be interpreted as the measure of residual
demand curves that pass below and to the left of the point (q; p). Suppose we treat p as function
of q. Then
Z qm
@ (q; p)
@ (q; p) 0
p (q) +
dq
=
(qp(q) C(q))
@p
@q
0
The Euler-Lagrange equation that p(q) must satisfy to minimize a general functional
Z qm
H(q; p; p0 )dq
0
is
Z(q; p) =
d
Hp0
dq
2
Hp = 0:
In the case where the functional is
we obtain
Hp0 = (qp(q)
Hp = q
and
C(q))
@ (q; p) 0
@ (q; p)
p (q) +
@p
@q
d
Hp0 = (p + qp0 (q)
dq
C 0 (q))
@ (q; p)
@p
+ (qp(q)
@ (q; p)
+ (qp(q)
@p
C(q))(
C(q))(
pp p
0
(q) +
pp p
0
qp );
(q) +
qp ).
This gives
@ (q; p)
@ (q; p)
d
Hp0 Hp = (p C 0 (q))
q
dq
@p
@q
which can be identi…ed with the standard Z function of [3].
Suppose that some fraction 2 (0; 1) of the pro…t earned by a generator is paid as tax. When
the market clears at quantity q for a generator at price then the generator receives
Z q
(
p(t))dt
q
C(q)
0
Z q
Z q
p(t)dt C(q)):
p(t))dt = (1
)(q
C(q)) + (
= q
C(q)
q +
0
0
This is a convex combination of uniform and pay-as-bid pricing with multiplier . Thus the total
payo¤ will be
Z
Z
= (1
) (qp C(q))d (q; p) +
(p C 0 (q)) (1
(q; p))dq.
We can write down the optimality conditions for the problem faced by a generator maximizing
d
. These use the scalar …eld de…ned by Z(q; p) = dq
Hp0 Hp . Thus
Z(q; p) = (1
= (p
)((p C 0 (q))
C 0 (q)) p (1
p
q
)q
q)
q
(1
(1
(q; p)
(q; p))
C 0 q))
(p
p
The …rst-order conditions are given by Z(q; p) = 0 and global optimality is guaranteed for a
@
Z(q; p) 0.
monotonic solution to Z(q; p) = 0 if @q
3
Symmetric duopoly for inelastic demand
We use the optimality conditions to look for an equilibrium in symmetric duopoly. Suppose
the other player o¤ers a smooth supply function S(p), and demand has cumulative probability
distribution function F . Then
(q; p) = Pr[h < q + S(p)]
= F (q + S(p))
and
Z(q; p) = (p
= (p
C 0 (q)) p (1
)q q
0
0
C (q))S (p)f (q + S(p))
(1
(1
3
(q; p))
)qf (q + S(p))
(1
F (q + S(p)))
Thus Z = 0 is equivalent to
(p
C 0 (q))S 0 (p) = (1
(1
)q +
F (q + S(p))
:
f (q + S(p))
(1)
The global optimality conditions (see [2]) are
(p
(p
(p
C 0 (q))S 0 (p)
C 0 (q))S 0 (p)
C 0 (q))S 0 (p)
These can be guaranteed by
(1 F (q+S(p))
f (q+S(p))
(1 F (q+S(p))
f (q+S(p))
(1 F (q+S(p))
f (q+S(p))
)q
)q
)q
@
Z(q; p)
@q
C 00 (q)S 0 (p)
3.1
(1
(1
(1
0; q < S(p)
= 0; q = S(p)
0; q > S(p)
0 which amounts to
(1
(1
)
F (q + S(p))
f (q + S(p))
0:
q
Examples
1
.
2
Assume a symmetric duopoly where each
1 guarantees that
Suppose F (t) = t is uniform on [0; 1] and
generator has capacity 21 . Then C 00 (q)S 0 (p) 2
C 00 (q)S 0 (p)
(1
(1
)
F (q + S(p))
f (q + S(p))
C 00 (q)S 0 (p)
=
(1
)+
q
0
The …rst order condition is then enough to give a supply-function equilibrium. Replacing q by
S(p) in (1) yields
pS 0 (p)
(1
)S(p)
(1
2S(p)) = 0
This di¤erential equation can be solved using an integrating factor, whereby
pS 0 (p) + (3
p3
1
S 0 (p) + (3
p3
1
S(p) = ap1
S(p) =
3
1)S(p) =
1)p3
2
S(p) = p3
0
S(p) = p3
3
+ p1
1
+ ap
2
3
3
2
1
p3
1
1 3
A unique equilibrium can be found by imposing a price cap P at which both generators o¤er their
capacity [7].
4
y
4
3
2
1
0
0.0
0.1
0.2
0.3
0.4
0.5
x
Figure 1: Plot of supply-function equilibrium solutions in a duopoly when inelastic demand is
uniformly distributed on [0,1] and each player has capacity 0.5. The red curve is an equilibrium
when each supplier must pay 25% of his surplus (above the red curve) as a tax. The green curve
gives the equilibrium o¤er for a 33% tax.
Example 1: Suppose
=
1
2
and P = 4, and each generator has capacity 12 . Then
S(p) =
3
1
3
2
a
q
+ ap1
1
=p
To pass through ( 12 ; 4) we choose a = 1.
Example 2: Suppose
through ( 12 ; 4) is
=
1
4
and P = 4, and each generator has capacity 12 . Then the solution
3 1
1 + p p4
2 2
64
p = (1 + q)4
81
q=
These solutions are shown in Figure 1.
3.2
Welfare calculations.
We can compute the changes in welfare of each agent from the change in equilibrium. Consider
…rst the case where = 0, and there is a price cap at 4. In perfect competition each generator
would o¤er at price zero and earn no pro…t. The welfare of the consumers in demand realization
h is 4h, and so the expected competitive total welfare is
Z 1
W =
4hdh = 2
0
5
In a supply function equilibrium, both players o¤er linear supply functions q = p8 . Thus the total
supply is S(p) = p4 . When demand is h the clearing price is 4h, so the total generator pro…t
assuming zero marginal cost is 4h2 . The expected total generator pro…t is then
Z 1
4h2 dh
G =
0
4
=
3
The consumer welfare (assuming a maximum valuation of 4), is
Z 1
h(4 4h)dh
C =
0
2
=
3
Expected total welfare of 2 is divided 32 to the demand and 23 to each generator. The observed
pro…t of each generator in demand realization h is de…ned by the area above their curve, namely
the integral of the clearing price at their dispatch h2 minus the o¤ered price at quantity q from
q = 0 to h2 .
G(h) =
Z
h
2
8
0
2
h
2
8q dq
= h
R1
The expected observed pro…t for both suppliers is then 2 0 h2 dh = 32 , or half their actual pro…t.
Suppose we now impose a tax by choosing = 14 . Then in equilibrium, each generator o¤ers
q=
3 1
1 + p p4
2 2
(2)
or
64
(1 + q)4 :
(3)
81
The (before-tax) observed pro…t in demand realization h of each generator if they o¤er this curve
is de…ned by the area above their curve, namely the integral of the clearing price at their dispatch
h
minus the o¤ered price at quantity q from q = 0 to h2 . This gives
2
p=
G(h) =
Z
0
=
h
2
64
h
(1 + )4
81
2
64
(1 + q)4 dq
81
4 2
h 2h3 + 15h2 + 40h + 40
405
The (before-tax) expected observed pro…t for both suppliers is then
Z 1
128
2
4 2
2
h 2h3 + 15h2 + 40h + 40 dh =
= 0:527 <
243
3
0 405
which is the total observed pro…t under a linear supply curve o¤er. The new o¤er is arranged to
32
reduce the tax while maintaining a healthy pro…t. The total tax paid is then 128
= 243
< 16 , the
243
tax collected when linear curves are o¤ered.
6
Since their costs are zero, the before-tax actual pro…t in demand realization h of each generator
(1 + h2 )4 . The total expected before-tax actual pro…t for both
if they o¤er the optimal curve is h2 64
81
suppliers is then
Z 1
h 64
h
1586
2
(1 + )4 dh =
2
1215
0 2 81
The total expected after-tax actual pro…t for both suppliers is
T =
Recall that
1586
1215
32
1426
=
:
243
1215
(4)
8
q + S(p) 0
< 0;
q + S(p); 0 < q + S(p) < 1
(q; p) =
:
1;
q + S(p) 1
so
(q; p(q)) = 2q
and
@ (q; p) @ (q; p) dp(q)
+
@q
@p
dq
0
0
= (1 + S (p)p (q)) dq
= 2dq
d (q; p) =
dq
and so the after-tax pro…t is
= (1
3
=
4
Z
(qp
(qp
C(q))d (q; p) +
1
C(q))d (q; p) +
4
Z
1
2
Z
(p
p(q)(1
C 0 (q)) (1
(q; p))dq
(q; p))dq
0
Z 1
64
1 2 64
4
(q (1 + q) )d (q; p) +
(1 + q)4 (1
(q; p))dq
81
4 0 81
Z 1
Z 1
3 2
64
1 2 64
4
=
(2q (1 + q) )dq +
(1 + q)4 (1 2q) dq
4 0
81
4 0
81
713
=
1215
3
=
4
Z
)
Z
which is half the …gure T we computed in (4) as expected.
The welfare of consumers is slightly improved by the tax. Without the tax, generators o¤er
linear supply functions, and the consumer welfare is 23 . When a tax is imposed, the generators
64
change their o¤ers, and the price under demand realization h is 81
(1 + h2 )4 . We can then compute
the expected total welfare for consumers as
Z 1
64
h
C =
h(4
(1 + )4 )dh
81
2
0
844
2
=
>
1215
3
7
The total welfare is then the sum of consumer welfare, generator pro…t, and tax giving
1426
32
844
+
+
= 2:
1215 1215 243
In summary if each generator o¤ers a linear supply curve (as they would in an untaxed equi7
1
librium) then they each earn 23 before tax and 12
after tax, after paying 12
in tax on observed
pro…t of 31 . If they instead o¤er the curve (2) then each generator will appear to earn a pro…t
793
16
64
but in fact will earn 1215
. They will then pay less tax of 243
and each retain a pro…t of
of 243
793
16
713
7
=
>
.
The
total
welfare
is
2,
and
so
consumers’
welfare
increases from 23 to
1215
243
1215
12
844
2 (2) = 1215
. The total welfare is consumer welfare plus generator welfare plus tax, giving
844
713
16
+ (2)
+ (2)
=2
1215
1215
243
So the reaction of the suppliers after the imposition of the tax is to o¤er to improve their welfare
and minimize the tax. The e¤ect of this is to transfer some wealth to consumers.
4
Line capacity expansion
We now consider a model for allocating the costs of a transmission line expansion, by taxing
the bene…ts accruing to generators and demand. The model is a simple two-node network, with
symmetric players at one node, and an inelastic demand shock " at the end of a line that originally
has capacity J and expands its capacity to K. Output capacities of …rms are denoted q and the
price cap is p.
Network example. Line is expanded from Kc to K.
Suppose player 2 o¤ers a supply function S2 (p). If player 1 o¤ers quantity q at price p then the
market distribution function is just the probability that either the total quantity o¤ered q + S2 (p)
exceeds the link capacity K or that the demand shock " exceeds the combined o¤ers of the two
…rms q + S2 (p) at price p;
(q; p) = Pr(q + S2 (p) > min(K; "):
If the demand shock is uniformly distributed on [0; "], then we obtain the following piecewise
de…nition for .
(
q+S2 (p)
if q K S2 (p)
"
(q; p) =
(5)
1
if q > K S2 (p):
The partial derivatives are
otherwise.
q
=
1
"
and
p
=
S20
"
8
= S20
q when
q
K
S2 (p) and both are zero
4.1
Payo¤s
We now look at the payo¤s. The actual pre-tax producer surplus if a generator is dispatched q at
price is
(
c)q:
The system operator observes a di¤erent surplus, assuming that p(q) is marginal cost. The observed
surplus is then
Z
q
P =
(
p(t))dt.
0
There is some ambiguity in the computation of the producer surplus if the o¤ers of two agents
are the same and are vertical. We assume that when the o¤er curves are both vertical the market
clears at the smallest price where S1 ( ) + S2 ( ) = ".
Suppose that producer 1 submits an o¤er of q at price p. We consider two cases.
Case 1: S2 ( ) + q < J: The perceived surplus as computed from the o¤er curve p(q) is the
same with the line at J as with the line at K. There is no tax imposed.
Case 2: S2 ( ) + q > J: The perceived surplus as computed from the o¤er curve p(q) is
Z q
P =
(
p(t))dt.
0
The perceived surplus as computed from the o¤er curve p(q) if the line has capacity J is
Z
0
P =
q0
(p(q 0 )
p(t))dt
0
where q 0 satis…es
S2 (p(q 0 )) + q 0 = Kc
and the tax imposed is (P P 0 ).
The generator then constructs an o¤er curve to maximize tax-adjusted pro…t which is
R(q; ) =
(
(
c) q;
c)q
q < q0
P 0 ); q > q 0
(P
Now
(
c)q
(P
0
P) = (
Z
c)q
(
0
= (
c)q
q
p(t))dt +
Z
)(
= (1
) q+
q0
(p(q 0 )
p(t))dt
!
Z q0
q 0 p(q 0 ) +
p(t)dt
0
q
p(t)dt
0
= (1
Z
q
Z
0
q
q 0 p(q 0 ) +
p(t)dt cq
q0
Z q
0
0
q p(q ) +
p(t)dt
cq
c)q +
q0
Thus
R(q; ) =
(
q
(1
cq;
) q+
q 0 p(q 0 ) +
9
Rq
q0
q < q0
p(t)dt
cq; q > q 0
:
Writing p for , the objective is
Z
=
q0
(p
0
+
Z
c) qd (q; p)
Z
Q
(1
0
q
p(t)dt
cq d (q; p)
q p(q ) +
q0
Z q
0
0
p(t)dt
cq (1
(Q; p(Q)))
q p(q ) +
)pq +
q0
+ (1
0
)pq +
q0
Q
Observe that
Z
Q
f (q; p)d (q; p) + f (Q; p(Q))(1
(Q; p(Q)))
q0
= f (Q; p(Q)) (Q; p(Q)) f (q 0 ; p(q 0 )) (q 0 ; p(q 0 ))
Z Q
d
(f (q; p)) (q; p)dq + f (Q; p(Q)) f (Q; p(Q)) (Q; p(Q))
q 0 dq
Z Q
d
0
0
0
0
= f (Q; p(Q)) f (q ; p(q )) (q ; p(q ))
(f (q; p)) (q; p)dq:
q 0 dq
Setting
Z
f=
q
p(t)dt
q0
gives
Z
Z
Q
p(q)dq
q0
Q
p(q) (q; p)dq =
q0
Z
Q
p(q)(1
(q; p))dq:
q0
Setting
f = q 0 p(q 0 )
gives
q 0 p(q 0 ) (1
(q 0 ; p(q 0 ))) :
The pro…t to be optimized by the generator is therefore
=
Z
0
+
q0
[pq
Z
cq] d (q; p)
Q
[(1
)pq
cq] d (q; p)
q0
+
Z
Q
p(1
(q; p))dq
q0
+ q 0 p(q 0 ) (1
(q 0 ; p(q 0 )))
+ [(1
)p(Q)Q cQ] (1
The pro…t to be optimized can be viewed as follows:
10
(Q; p(Q)))
Z
=
Q
pqd (q; p) + p(Q)Q(1
(Q; p(Q)))
Z Q
Z Q
pqd (q; p)
p(1
(q; p))dq
0
q0
q0
p(Q)Q(1
(Q; p(Q)))
0
0
+ q p(q ) (1
(q 0 ; p(q 0 )))
Z Q
cqd (q; p) cQ(1
(Q; p(Q)))
0
Where
Z
Q
pqd (q; p) + p(Q)Q(1
(Q; p(Q)))
0
is total expected generator revenue, and
Z Q
pqd (q; p)
q0
+Qp(Q)(1
q 0 p(q 0 ) (1
Z
Q
p(1
(q; p))dq
q0
(Q; p(Q)))
(q 0 ; p(q 0 )))
is total expected perceived bene…t of the extra line capacity, and
Z Q
cqd (q; p) + cQ(1
(Q; p(Q)))
0
is expected variable cost. Figure 2 illustrates these terms.
Figure 2: Taxation of net bene…ts
Total expected generator revenue is area A+B+C+D+E+F integrated over the range from
0 to Q, as well as the revenue that accrues for all realizations where q = Q. Total expected
perceived bene…t of the extra line capacity is the integral of area A+B integrated over the range
from q = q 0 to q = Q. We compute this by integrating A+B+C+D+E+F (pq) over the range
from q 0 to Q with density de…ned by and subtracting the expectation of area F+C (which is
RQ
p(1
(q; p))dq). We need to lower the total by the expectation of D+E (q 0 p(q 0 )) whenever
q0
q q 0 (which happens with probability (1
(q 0 ; p(q 0 )))). When q < q 0 there is no tax counted.
11
We also need to account for some extra contribution to expected bene…ts of the line when q = Q
(with probability (1
(Q; p(Q)))).
We now consider the case where there is a tax on the bene…ts of the line. We denote by the
market distribution function with the expanded line. The payo¤ for the o¤er curve p(q) is then
Z
(p (q)) =
F (q; p; p0 ) dq
Z q0
@
@ 0
p (q) +
dq
(qp (q) C (q))
=
@p
@q
0
Z Q
@ 0
@
((1
)qp (q) C (q))
+
p (q) +
dq
@p
@q
q0
Z Q
p (q) (1
(q; p (q))) dq :
+
q0
The Z function will have two ranges. For q < q 0 , we have
d
dq
@F
@p0
= (p + qp0
@F
=q
@p
@
+ (qp
@p
C 0)
@ 0 @
p +
@p
@q
Z (q; p) =
+ (qp
d
dq
@2 0
@2
p
+
@p2
@p@q
C)
@F
@p0
= (p
C 0 (q))
) (p + qp0 ) C 0 )
@
+ ((1
@p
@2
@2
+ 2 p0
@p@q
@p
C)
@F
@p
@
@p
q
@
@q
as we would expect from [3].
For q > q 0 , we have
d
dq
@F
@p0
@F
@p
= ((1
= (1
+
)q
1
@ 0 @
p +
+ ((1
@p
@q
@
(q; p) p
:
@p
12
)qp
)qp
C)
C)
@2
@2
+ 2 p0
@p@q
@p
@2 0
@2
p
+
@p2
@p@q
Thus
@F
@p0
d
dq
Z (q; p) =
@F
@p
) (p + qp0 ) C 0 )
= ((1
@
@p
@ 0 @
p +
@p
@q
@
1
(q; p) p
@p
@
= ((1
)p C 0 )
(1
@p
@
1
(q; p) p
@p
@
@
(1
)q
= (p C 0 )
@p
@q
(1
)q
)q
@
@q
(1
(q; p)) :
Observe that this is the same as the Z function for an o¤er to an auction with a convex combination
between uniform and pay-as-bid pricing, as in section 2.
4.2
Example
We now consider an example. The base levels of parameters are as follows:
g=
c=
K=
J=
=
r=
maximum shock
marginal cost
enlarged line capacity
restricted line capacity
tax rate
price cap (i.e p)
10
1
8
2
1
4
6
The …rst order condition is
(p
@
@p
(1
)q
(q; p) =
(
q+S2 (p)
g
C 0)
Here
so
@
@q
(1
(q; p)) = 0:
if q K
if q > K
1
@
S 0 (p)
= 2 ;
@p
g
S2 (p)
S2 (p):
@
1
=
@q
g
Replacing S2 (p) and q by S(p) in (1) yields
1
(p
g
c)S 0 (p)
1
(1
g
(p
c)S 0 (p)
(1
)S(p)
(1
2S(p)
)=0
g
or
)S(p)
13
(g
2S(p)) = 0
can be solved using an integrating factor, whereby
c)S 0 (p) + (3
(p
(p
c)3
1
S 0 (p) + (3
1)S(p) = g
c)3
1)(p
(p
c)3
1
S(p) = k(p
c)1
3
S(p) = k(p
c)1
3
2
S(p) = g(p
0
c)3
S(p) = g(p
c)1
+ (p
3
g
1 3
2
2
g
3
c)3
1
(p
c)3
1
(r
c)3
1
To pass through the price cap we require
k(r
c)1
S(p) =
3
(
g
1 3
K
2
+
K
2
K
g
k =
+
2
1 3
=
g
1 3
p
r
Jp
2 t
c 1 3
c
c
c
g
1 3
if p t
if p < t
where setting p = t makes this function continuous at q = J2 .
Observe that the exponent of pr cc vanishes when = 13 . Then the di¤erential equation
(p
c)S 0 (p) + (3
1)S(p) = g
becomes
S 0 (p) =
g
(p
so
S(p) = g log
The section of the supply curve below q =
J
2
S(t) =
p
r
c)
c K
+ .
c
2
is found by solving S(t) = J2 , for t, and setting
Jp
2t
c
;
c
p < t.
The supply-function equilibria for di¤erent choices of
14
are plotted below.
p
6
4
2
0
0
1
2
3
4
5
q
Figure 3: Plot of untaxed equilibrium o¤er (red) and taxed equilibrium o¤er (blue) when maximum
demand is 10. Equilibrium o¤er when demand is 20 is shown in green.
p
6
4
2
0
0
1
2
3
4
5
q
Plot of untaxed equilibrium o¤er (red) and taxed
equilibrium o¤ers when maximum demand is 10 and
= 14 (blue), = 31 (green), and = 12 (magenta).
The degree to which the taxed equilibrium is marked up above the untaxed equilibrium depends
on the range of the demand shock. If the range of the demand shock is large, then there is a high
probability that the expanded line will be congested, which even so provides signi…cant beme…ts
compared with the unexpanded line. This means that the equilibrium o¤ers try to avoid taxation
of these by ‡attening the o¤er curve. This can be observed in Figure 3.
We …nish this example by computing the equilibrium for = 14 , g = 10, and di¤erent values of
J. These are shown in Figure 4. Observe that for small increases in line capacity (from J = 6 to
J = 8) the magenta and red curves almost coincide, so there is minimal change in o¤er strategy
to avoid the tax.
15
p
6
4
2
0
0
1
2
3
4
5
q
Figure 4: Plot of untaxed equilibrium o¤er (red) and taxed equilibrium o¤er when J = 2 (blue),
J = 4 (green), J = 6 (magenta).
5
Conclusions
The supply-function equilibrium models outlined in this paper show that taxes imposed on electricity suppliers do not necessarily lead to less competitive outcomes. It is interesting to speculate
whether these results remain true for increases in the number of players, asymmetry in suppliers,
and contracting. Since they lead to lower overall welfare for suppliers, one might conjecture that
some recovery of these losses will be achieved possibly by some out-of-market mechanism. However
that is outside the scope of this work.
References
[1] Anderson, E. J. and X. Hu. 2008. Finding supply function equilibria with ssymmetric …rms.
Operations Research 56(3), 697–711.
[2] Anderson, E. J., P. Holmberg, A. B. Philpott. 2013. Mixed strategies in discriminatory
divisible-good auctions. RAND Journal of Economics, 44(1), 1756-2171.
[3] Anderson, E. J., A. B. Philpott. 2002. Optimal o¤er construction in electricity markets. Mathematics of Operations Research 27, 82–100.
[4] Anderson, E.J., A. B. Philpott. 2002. Using supply functions for o¤ering generation into an
electricity market. Operations Research 50 (3), 477–489.
[5] Genc, T. and Reynolds S. 2004. Supply function equilibria with pivotal electricity suppliers.
Eller College Working Paper No.1001-04, University of Arizona.
[6] Green, R. J., D. M. Newbery. 1992. Competition in the British electricity spot market, Journal
of Political Economy 100, 929–953.
[7] Holmberg, P. 2008. Unique supply function equilibrium with capacity constraints, Energy
Economics 30, 148–172.
16
[8] Holmberg, P. andPhilpott A.B. 2012. Supply function equilibria in networks with transport
constraints, EPOC working paper, downloadable from www.epoc.org.nz..
[9] Holmberg, P., D. Newbery. 2010. The supply function equilibrium and its policy implications
for wholesale electricity auctions, Utilities Policy 18(4), pp. 209–226.
[10] Hortacsu, A., S. Puller. 2008. Understanding Strategic Bidding in Multi-Unit Auctions: A
Case Study of the Texas Electricity Spot Market. Rand Journal of Economics 39 (1), 86–114.
[11] Klemperer, P. D., M. A. Meyer. 1989. Supply function equilibria in oligopoly under uncertainty. Econometrica 57, 1243–1277.
[12] Rudkevich, A., M. Duckworth and R. Rosen. 1998. Modelling electricity pricing in a deregulated generation industry: The potential for oligopoly pricing in poolco. The Energy Journal
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5.1
Appendix: Plot of Z for example
In this appendix we verify the equilibrium candidate of section 4 by plotting the Z function. Given
(
p c 1 3
g
K
+ 1 g3
if p t
2
r c
1 3
S(p) =
Jp c
if p < t
2 t c
we get
0
S (p) =
(
7
3
1 4
10(
5)
38 416
73 205
@
@p
@
@q
1
p
5
if p
t
if p < t
S 0 (p)
g
1
=
g
=
We can now plot contours of
Z(q; p) = (p
= (p
c)
@
@p
(1
)q
7
c)
10g
1
p
5
1
5
3
4
@
@q
(1
(1
(q; p))
1
)q
g
1
q+
K
2
+
p c 1 3
r c
g
1 3
g
g
1 3
!
which are presented in Figure 5. These show that Z(q; p) is a decreasing function of q for any …xed
p, which demonstrates that the blue curve is an optimal supply function response to itself.
17
p
7
6
5
4
3
2
1
0
1
2
3
4
5
q
Figure 5: The blue curve is Z = 0. Green curves show where Z > 0 and red curves where Z < 0.
18