Spot Market Competition
in the UK Electricity Industry∗
Nils-Henrik M. von der Fehr
University of Oslo
David Harbord
Market Analysis Ltd
12 February 1992
Abstract
With particular reference to the structure of the UK industry, price competition in a
deregulated wholesale market for electricity is modelled as a sealed-bid, multiple-unit auction
with a random number of units. It is argued that under the existing regulatory rules one must
expect volatile prices, above marginal-cost pricing and inefficient despatching. Evidence from
the pricing performance of the UK industry is presented and shown to be compatible with the
model predictions. We also discuss alternative regulatory rules and show that offering to
supply at marginal cost can be induced as a dominant strategy for all firms, thereby securing
efficient despatching.
JEL Classification Numbers: D44. L94, L10, L5
Keywords: Multi-unit auctions, electricity spot markets, regulatory reform
∗
The first author has benefited considerably from contact with the ESRC project "The Regulation of Firms with
Market Power” and wishes to thank its participants, and in particular John Vickers, for helpful discussions. The
second author's work was partially supported by London Economics Ltd. We are grateful to Friedel Bolle, Robin
Cohen, Finn Førsund, John Kay, Kai-Uwe Kühn, Meg Meyer, Jorge Padilla, Tony Curzon Price, Lars Sørgaard,
workshop participants at Nuffield College and two anonymous referees for valuable comments and suggestions
on earlier versions. Financial support from NORAS (Norges råd for anvendt samfunnsforskning) is gratefully
acknowledged (NHF).
1. Introduction
At the core of the recently deregulated and privatized UK electricity industry is the wholesale
spot market.1 Before each period that the market is open, the generating companies (the
generators) submit minimum prices at which they are willing to supply power. On the basis of
these “offer prices”, the National Grid Company, which plays a central role as coordinator
and which is responsible for running the transmission grid, draws up a least-cost plan of
generating units for despatching in the next period. This rank order, together with demand,
determines which units will actually be despatched.2 Payments to supplying units, or ‘sets’,
are based on a ‘system marginal price’, determined as the offer price of the marginal
operating unit in every period.
The particular organization of the electricity spot market makes standard oligopoly models
inadequate as analytical tools. We propose instead to model this market as a sealed-bid,
multiple-unit auction. In the first stage of the model, firms simultaneously submit offer prices
at which they are willing to supply their (given) capacities. As in the UK industry, firms
(generators) can submit different offer prices for each individual plant or generating set, i.e.
firms offer step supply functions. Sets are then ranked according to their offer prices (i.e. a
supply function is constructed). In the final stage, demand is realized and the system
marginal price is determined by the intersection of demand and supply; that is, by the offer
price of the marginal operating unit.
It turns out that pure-strategy equilibria do not always exist in such a model. The reason is
basically the same as that in standard oligopoly models of capacity-constrained price
competition (Kreps and Scheinkman, 1986). Since, when demand is sufficiently large, a firm
is unable to serve the whole market at the competitive price, there is an incentive to raise bids
above marginal cost, and thus the competitive outcome cannot be an equilibrium. It can then
be shown that for a range of demand distributions no other pure-strategy combinations
constitute an equilibrium either. We believe that this result does not necessarily reflect an
inadequacy of our modelling approach, but rather suggests that there is an inherent price
instability in the present regulatory set up. Indeed, our empirical evidence would seem to
confirm that experimentation and abrupt changes in pricing strategies is a feature of the new
industry.
This particular result (the non-existence of equilibrium in pure strategies) also casts some
doubt on the relevance of the model analyzed by Richard Green and David Newbery (1991)
(see also Bolle,1990, and Newbery, 1991). These authors argue that the “step-length”, i.e. the
size of individual generating sets, is small enough to justify approximating the step-supply
schedules by smooth (i.e. continuously differentiable) functions, thus applying the supplyfunction framework developed by Klemperer and Meyer (1989). As we demonstrate however,
the particular types of equilibria they derive do not generalize to a model where sets are of
positive size. Although theirs is a seemingly very useful contribution, it remains “an open
question whether the bidding strategies of the firms will differ significantly if they are forced
1
For details on the UK electricity industry, new and old, see Vickers and Yarrow (1991), Green
(1991a) and James Capel & Co.(1990).
2
Allowances for particular circumstances such as transmission constraints and system stability
considerations may lead to deviations in despatching from that which follows from the rank order.
Further details are provided in appendix B.
1
to provide a step function, or whether they are allowed to provide a smooth schedule” (Green
and Newbery, 1991, footnote 2, page 5).3
Nevertheless the most important result, inefficient pricing, turns out to be robust to alternative
forms of modelling. Indeed, we find an even stronger tendency than Green and Newbery
towards above marginal-cost pricing. Thus the conjecture that the Bertrand outcome is
unlikely in the present institutional set-up of the UK electricity industry, even if there is no
collusive behaviour, seems to be strongly supported. In addition, our model suggests that
high-cost sets may be bid in at lower offer prices than lower-cost sets and thus be despatched
before these more efficient units. Hence despatching may be inefficient in the sense that
overall economic generation costs are not minimized.
An important advantage of our framework is that it makes it possible to model explicitly the
role of the grid company (the auctioneer), and then use insights from the auction literature to
study the effects of different pricing rules, i.e. the rules determining the prices paid to
different supplying units. With a pricing rule like the one used in the present UK electricity
industry, the game corresponds to a first-price sealed-bid auction. However, by letting the
system marginal price be determined by the offer price of one of the marginal non-operating
sets, the game corresponds to a (generalized) second-price sealed-bid auction, and in this
case offering to supply at marginal cost can be shown to be a dominant strategy for each
firm. This result is in accord with what is typically found in the literature on optimal
auctions where it is well known that second-price (or Vickery) auctions lead to higher
revenues for the auctioneer than do first-price auctions (Myerson (1981) or Maskin and
Riley (1989)).
The remainder of the paper is organized as follows. Our auction model of the U.K. electricity
spot market is presented in section 2 and then analyzed in section 3. Section 4 then relates the
model to empirical evidence from the UK electricity industry. In section 5 we consider an
alternative regulatory rule. Section 6 contains a short summary and conclusions.
2. The Model
Before the actual opening of the market, N independent firms simultaneously submit offer
prices at which they are willing to supply electricity from each of their generating units, or
sets. On the basis of these bids, the market organizer (or ‘auctioneer’) draws up a ranking of
units, i.e. a market supply curve is constructed. When the market opens, demand is
determined as a random variable independent of price, and the auctioneer, by calling suppliers
into operation, equates demand and supply. Operating units, i.e. units actually supplying
output, are paid the system marginal price which is equal to the offer price of the marginal
operating unit.
It is assumed that generators have constant marginal costs, cn ≥ 0 , n = 1,2,...,N, at production
levels below capacity, while production above capacity is impossible. We let the index n rank
firms according to their marginal costs, i.e. cn ≤ cn +1 . The total capacity of firm n is given by
kn , n = 1,2,...,N. The capacity of firm n consists of mn sets where kni is the capacity of the ith
set, i = 1,2,..., mn , and
∑
i
kni =kn . We let M denote the total number of sets, i.e. M = ∑ n mn .
3
Green and Newbery also assume downward sloping demand curves, whereas completely inelastic
demand would seem to be more appropriate for the UK industry. Bolle (1990) proves that in the latter
case, no equilibrium exists in the supply-function model.
2
Firms can submit different bids for each of their sets. If two or more sets (of any firm) are
offered at the same price, they are equally likely to be called into operation.
We consider a game G with N+1 players: N suppliers and Nature. The move order is as
follows:
pni ≤ p ,
Stage 1: The suppliers simultaneously submit offer prices
n = 1,...., N , cN ≤ p < ∞ .
i = 1, 2,....., mn ,
Stage 2: Sets are ranked according to their bids such that if the bid of the set with rank r is p r
and that of the set with rank r ′ is p r ′ and p r < p r ′ then r < r ′ . If m sets are offered at the
same price p, then these sets are designated numbers r, r+1,...,r+m-1, with (marginal)
probabilities 1/m, for some r ∈ {1, 2,..., M − m + 1} .
Stage 3: Nature chooses a number d ∈ ⎡⎣ d , d ⎤⎦ ⊆ [ 0, K ] , K = ∑ n =1 kn , according to some
N
probability distribution G ( d ) .4
Let K 0 = 0 and K r = ∑ j =1 k j , r = 1, 2,...., M , where k r is the capacity of the set with rank r.
r
Let ρ = max {r | K r −1 < d } . Then all sets with rank r = 1,2,...,ρ-1 get paid p ρ k r while set ρ
gets p ρ ⎡⎣ d − K ρ −1 ⎤⎦ .
Let
sn
be
the
actual
supply
of
firm
n,
i.e. sn = ∑ r =1 δ n ( r ) k r + δ n ( ρ ) ⎡⎣ d − K ρ −1 ⎤⎦ , where δ n ( r ) is 1 if the set with rank r belongs to
ρ −1
firm n and zero otherwise. The payoff to firm n is then ⎡⎣ p ρ − cn ⎤⎦ sn , n = 1, 2,...., N . All
players are assumed to be risk neutral and hence aim to maximize their expected payoff in the
game. All aspects of the game, as well as the players’ marginal costs and capacities and the
probability distribution G ( d ) , are assumed to be common knowledge.
Note that firms’ offer prices are constrained to be below some threshold level p < ∞ , since
otherwise, in cases when there is a positive probability that all sets will be called into
operation, expected payoffs could be made infinitely large. A natural interpretation of p is
that it is a (regulated) maximum price, either officially, or as perceived by the generators (i.e.
firms believe that the regulation authorities will effectuate price regulation if the price rises
above p ). An alternative interpretation is that p is a reservation price, below which demand is
completely inelastic.
Our model may be considered as a special case of an auction, in fact, as a first-price, sealedbid, multiple-unit auction with a random number of units, where all units are sold
simultaneously (McAfee and McMillan, 1987, Hausch, 1986). It is ‘sealed-bid’ because of the
simultaneous move structure and ‘first-price’ in the sense that the market price is determined
by the marginal successful supplier. In Section 5, we show that this interpretation is
particularly convenient for analyzing alternative pricing rules.
4
Note that to assume d ∼ [ 0, K ] is without loss of generality since supply is limited to K and, thus,
demand will have to be rationed if it increases beyond K. In particular, G ( d ) will typically have an
atom at d = K, reflecting the fact that rationing may occur with positive probability. In addition, the
present UK electricity-supply industry is characterized by significant excess capacity, and this is likely
to remain true for the foreseeable future. Hence d < K would appear to be the relevant case.
3
3. Analysis
In this section we characterise the Nash-equilibria of the model presented in Section 2. Most
of the discussion will centre on the duopoly case, for which we are able to derive explicit
results. Apart from being the relevant case for the UK electricity industry (see the discussion
in Section 4), explicit formulae for optimal strategies are difficult to derive in the more
general oligopoly case. Hence our discussion of oligopoly in this section is in most cases
limited to pointing out where and how the duopoly results generalize.
We start by presenting a result on the types of pure-strategy equilibria that can occur.5
Proposition 1: In a pure-strategy equilibrium, (generically) at most one firm will determine
the system marginal price with positive probability.
Remark: By genericity is here meant that firms have different marginal costs, i.e. cn ≠ cnˆ for
all n ≠ nˆ . If firms have identical marginal costs there may exist Bertrand-type equilibria
where firms submit offer prices equal to the marginal costs of each set, and in which more
than one firm owns sets which with positive probability may become the marginal operating
unit.
The intuition underlying the result is the following. A player who owns a set which has a
positive probability of becoming the marginal operating unit, will always want to increase the
offer price of that set by some small amount towards the next higher bid, since that does not
affect the ranking, but increases the firm’s payoff in the event that this is the marginal set. On
the other hand, it cannot be optimal to submit an offer price equal to or just above that of a set
of another player, since as long as the offer price is above marginal cost (which it will be for
at least one firm), profits can be increased by undercutting the rival slightly, thereby
increasing the probability of being called into operation, without significantly reducing the
price received in any state. These two opposing forces destroy any candidate for a purestrategy equilibrium in which two or more firms both have sets which with positive
probability will become the marginal operating unit.
Proposition 1 implies that the types of pure-strategy equilibria that may exist are very
restricted, and, furthermore, it rules out the existence of pure-strategy equilibria for a wide
range of demand distributions (we return to this latter point in later sections). Significantly, it
follows that the types of equilibria found by Green and Newbery (1991) in their model, do not
generalize to the case where individual generating sets are of positive size. The reason that
such equilibria exist in their supply-function framework is that when individual sets are of
size zero (the cost function is continuously differentiable everywhere), the gain from
undercutting any individual rival set is negligible, and thus the second part of the above
argument does not apply.
In the following section we consider circumstances under which pure-strategy equilibria will
exist, as well as presenting examples of mixed-strategy equilibria when pure strategy
equilibria are non-existent. The existence, multiplicity and the type of equilibria will be seen
to depend crucially on the support of the demand distribution. We will therefore distinguish
between three generic cases, which, for N = 2, are defined as follows:
Case 1 (Low demand): Supp G ( ⋅) = [ d , d ] ⊂ [0, min{k1 , k2 }] , i.e. a single firm can supply the
whole of demand.
5
All proofs have been relegated to the appendices.
4
Case 2 (High demand): Supp G ( ⋅) = [d , d ] ⊂ [max{k1 , k2 }, K ] , i.e. both firms will be
producing with probability one.
Case 3 (Variable demand): d − d > max{k1 , k2 } , i.e. there is positive probability for both the
event that a single firm can supply the whole of demand and the event that both firms will
have to be called into operation, irrespective of their bids.
3.1
Low-demand periods
We begin by considering the case where, with probability 1, demand is less than the capacity
of the smallest firm. This case turns out to correspond to the standard Bertrand model of
oligopoly and thus there is a unique equilibrium outcome in which both firms offer to supply
at a price equal to the marginal cost of the least efficient firm:
Proposition 2: If supp G ( ⋅) ⊂ [0, min{k1 , k2 }] then there exist pure-strategy equilibria, in all of
which the system marginal price equals the marginal cost of the least efficient firm, c2, and
only firm 1 produces.6
Remark: When k1 > k2 , such equilibria continue to exist for all G ( ⋅) such that Pr(d ≤ k1 ) = 1
and E d ≤ k2 [ p − c1 ] [ p − c2 ] ,7 but other equilibria may exist also (see the next section).
The argument in the proof is identical to that of the standard Bertrand model. Since, with
probability 1, demand can be covered by one firm, there will be competition to become the
single operating firm. In particular, a firm will always undercut its rival so long as its rival’s
bids are above its own marginal costs. Thus any equilibrium must have the most efficient firm
(firm 1) submitting offer prices for a capacity sufficient to cover demand, at or below the
marginal cost of the least efficient firm. Since in this range, firm 1’s profit is increasing in its
own offer price, these bids must equal c2. We conclude that in low demand periods, the
system marginal price is bounded above by the marginal costs of the less efficient firm.
A similar result can be shown to hold in the oligopoly model. If, with probability 1, demand is
less than the total capacity of the n most efficient firms (n < N), then in any equilibrium
system marginal price cannot exceed the marginal cost of the n+1st most efficient firm.
However, as we show in section 3.3, pure-strategy equilibria will generally not exist in this
model.
3.2
High-demand periods
In this sub-section we assume that with probability 1 both firms will be called into operation.
Since the high-pricing firm will be operating for sure, and in equilibrium firms never submit
equal bids (see Proposition 1), its profit will be increasing in its own offer price. Thus, the
extreme opposite to the result of the previous section holds; whereas in low-demand periods
the system marginal price equals the marginal cost of the least efficient firm, in high-demand
periods it always equals the highest admissible price.
6
To avoid non-existence, we impose the tie-breaking rule that firm 1 is called into operation with
probability 1 whenever the firms' offer prices tie at c2 . This captures the idea that the most efficient
firm marginally underbids its rival, while simplifying the description of the equilibrium.
7
E is the expectations operator.
5
Proposition 3: If supp G ( ⋅) ⊂ [max{k1 , k2 }, K ] all pure-strategy equilibria are given by offer-
price pairs ( p1 , p2 ) satisfying either p1 = p and p2 ≤ b2 or p2 = pˆ and p1 ≤ b1 for some
bi < p , i = 1,2.
Remark: If k1 > k2 ( k1 < k2 ), then all ( p1 , p2 ) such that p1 = p and p2 ≤ b2 ( p2 = p and
p1 ≤ b1 ) continue to be equilibria for all G ( ⋅) such that Pr(d > k2 ) = 1 ( Pr(d > k1 ) = 1 ). That
is, a sufficient condition for the existence of this type of equilibrium is that, with probability
1, demand is greater than the capacity of the smaller firm.
As already noted, the intuition for the result is straightforward. The high-bidding firm will
always determine the system marginal price. Therefore its payoff is increasing in its own offer
prices and profit maximization requires bidding at the highest admissible price. The lowbidding firm is indifferent between any offer price lower than that of the high-bidding firm.
However, to ensure that the high-bidding firm does not deviate, the low-bidding firm has to
bid low enough so that the high-bidding firm’s payoff from undercutting is less than the
payoff earned in equilibrium. Thus the upper bound on the low-bidding firm’s offer price.
In all of the equilibria characterized by Proposition 3, the system marginal price equals the
highest admissible price. The low-bidding firm is despatched with full capacity while the
high-bidding firm supplies the residual demand. It follows that both firms prefer equilibria
where they act as the low-bidding firm, since the received price is the same while a firm’s
output is greater in the equilibrium in which it is ranked first.
Note that some of these equilibria involve inefficient despatching: the high-cost firm may be
the firm with the lowest bid and thus will be despatched with its total capacity, while the low
cost firm is only despatched with part of its capacity. In such equilibria, generation costs are
not minimized.
It is easy to see that in the oligopoly case we get a corresponding result: Whenever demand is
such that the highest-bidding firm determines the system marginal price with probability 1,
any vector of offer prices such that the highest-pricing firm submits p while the rest bid
sufficiently below this, will be an equilibrium.
3.3. Variable-demand periods
We turn now to the intermediate case in which there is a positive probability of either firm
becoming the marginal firm, whatever their offer prices (i.e. rank). It is clear that offer-price
pairs like those in Proposition 3 cannot constitute equilibria in this case since the low-bidding
firm will now always wish to increase its offer price; in doing so it thereby increases the
system marginal price in the event that it becomes the marginal operating firm. In fact, we
have the following result:
Proposition 4: If d − d > max{k1 , k2 } , where [d , d ] is the support of the demand distribution
G ( ⋅) , then there does not exist an equilibrium in pure strategies.
This result follows directly from Proposition 1. Since the range of possible demand
distributions exceeds the capacity of the largest firm, it follows that for any strategy
combination there is a positive probability that sets of either firm will be the marginal
operating unit. We can then apply the result of Proposition 1; there cannot exist pure strategy
6
equilibria for which more than one firm has a positive probability of determining the system
marginal price.8
In the remainder of this section we consider an example where for all n, mn = 1, i.e. each firm
owns only one set, or can submit only one offer price for the whole of its capacity, and we
characterize mixed-strategy equilibria for both the duopoly and the oligopoly cases.9 In the
duopoly case we are able to show that there exists a unique mixed-strategy equilibrium, and
we derive the explicit form of the two players’ strategies. In particular, we find that the lowest
price in support of the players’ strategies is strictly greater than the marginal cost of the least
efficient firm, and that this lowest price is an increasing function both of the highest possible
price p , the probability that both firms will be operating (i.e. demand), and the marginal cost
of the least efficient firm. We conclude this section by characterizing the (unique) equilibrium
of the symmetric oligopoly model and demonstrate that the (expected) system marginal price
is a decreasing function of the number of firms in the industry.
The analysis is considerably simplified by restricting attention to the following special case:
All firms are assumed to have equal capacities normalized to 1, and demand is discrete and
distributed on {1,2,..,N} with probabilities π n = Pr(d = n) , n = 1,2,...,N, with Pr ( d = n ) ≥ 0
and ∑ π n = 1 . Since the main results carry over to the more general model, for the rest of this
n
section we concentrate on this special case.
Assume N = 2 and define π ≡ π 1 . Without loss of generality, normalize c1 to zero and let
c ≡ c2 . The assumption on the support of the demand distribution in proposition 4 now
corresponds to the case where 0 < n < 1, i.e. the events that one and two firms are called into
operation both occur with positive probability. Define
⎧ ⎛ p−c ⎞
⎪ln ⎜ e
⎟
⎪ ⎝ p −c ⎠
F1 ( p ) = ⎨
1− 2π
⎪ π −1 ⎡ p − c ⎤ π
π
+
⎪
⎢
⎥
2π − 1
⎩ 2π − 1 ⎣ p − c ⎦
when π =
1
2
(1)
when π ≠
1
2
8
In the oligopoly case this result is only "generically" true. By "generic" we mean here that two
assumptions must be fulfilled. First, at least two firms must have positive probability of becoming the
marginal firm irrespective of the rank order. Second, if m is the minimum and M the maximum
number of firms that can be called into operation, then the M+1st most efficient firm must have higher
marginal cost than the m’th most efficient firm. If these conditions are not fulfilled, then there are
(multiple) pure strategy Bertrand-type equilibria where the system marginal price equals the marginal
cost of the M+1st most efficient unit, and only (a subset of) the M most efficient units supply (see von
der Fehr (1990)).
9
Analyses of mixed strategies in models with a similar structure to the model presented here can be
found in Shilony (1977), Varian (1980), and Padilla (1990). See Brandenburger (1990) for a
discussion of the interpretation of mixed-strategy equilibria as equilibria in beliefs.
7
⎧ ⎛
⎞
p
⎪ln ⎜ e
⎟
⎪ ⎝ p + [e − 1]c ⎠
F2 ( p ) = ⎨
1− 2π
⎪ π −1 ⎡
⎤ π
π
p
+
⎪
⎢
⎥
2π − 1
⎩ 2π − 1 ⎣ p + [α π − 1]c ⎦
when p < p , and π =
1
2
(2)
1
when p < p , and π ≠
2
F 2 ( p ) = 1,
π
where α π
⎡ π ⎤ 2 π −1
≡ ⎢
⎣ 1 − π ⎥⎦
and
pm
⎧ p−c
⎪ e +c
⎪
= ⎨
π
⎪ [ p − c ] ⎡ π ⎤ 1− 2 π + c
⎢⎣ 1 − π ⎥⎦
⎪
⎩
w hen π =
1
2
(3)
w hen
1
π ≠
2
where ln(e) ≡ 1 . We can then prove the following result:
Proposition 5: When N = 2 there exists a unique mixed-strategy Nash equilibrium in which
player i‘s strategy is given by: Play p ∈ [ p m , p ] according to the probability distribution
Fi ( p ) , where Fi ( p ) , i = 1, 2 are given by (1) and (2), and p m is given by (3).
From (1) and (2) it follows that when the probability that both firms will be operating is large
(i.e. π is small), more probability mass is placed on higher prices (and vice versa). The
intuition for this is straightforward: the incentive to raise the offer price is checked by the
likelihood of ending up as the higher pricing firm and not being called into operation. When
π is small however, it is very likely that both firms will be despatched, i.e. there is a
substantial probability that a firm will be operating even if it offers to supply only at a very
high price. Thus, for small π both firms will tend to submit high bids. In particular, the
following is easily demonstrated:
limπ →1 p m = c , and
(4)
limπ →0 p m = p
(5)
(Note that the limit in (5) corresponds to the case discussed in section 3.1, while the limit in
(6) corresponds to that discussed in section 3.2.)
From (2) one sees that the higher-cost firm’s mixed strategy distribution has a mass point at
the highest admissible price, p . Furthermore, from (1) and (2) it follows that ∀p ∈ ⎡⎣ p m , p ⎤⎦ ,
F2 ( p ) ≤ F1 ( p) , i.e. the high-cost firm’s strategy profile first-order stochastically dominates
the strategy profile of the low-cost firm. Thus, the high-cost firm will generally (i.e. in
8
expected terms) submit higher bids than the low-cost firm. We have not been able to find an
algebraic expression for the probability that the high-cost firm submits a bid below that of
the low-cost firm, but a lower bound can be established by considering the probability
that p2 < p1 − c . When π =1/2 , this reduces to
1
e 2
Pr( p2 < p1 − c) = [1 − ln(1 +
)]
α −1
2
(6)
where α ≡ p / c . If α = 5 (10) , i.e. p is 5 (10) times the marginal cost of the high-cost firm,
this probability equals 12% (27%). Thus, although the typical outcome is that the high-cost
firm prices above the low-cost firm, there is a potentially significant positive probability that
the high-cost firm submits the lowest offer price and thus becomes the only operating firm.
Therefore we may conclude that, as in the case discussed in section 3.2, the regulatory rule, as
it is modelled here, is not ex-post efficient.
We now extend the analysis of the duopoly case to the oligopoly model under the assumption
that firms have equal marginal costs which, without loss of generality, are normalized to zero.
We consider a symmetric model since this is the only case in which it is possible to
characterize equilibria in any detail. We assume that there is a positive probability that all
units will be called into operation, i.e. π N > 0 , since otherwise, given the symmetry
assumption, only the perfectly competitive outcome would be possible (see section 3.3 and
footnote 8). We obtain the following result:
Proposition 6: Assume cn = 0 , n = 1,...., N . Then there exists a unique symmetric mixed-
strategy equilibrium for the game G in which each player plays prices p ∈ [ p m , p ] according
to the probability distribution F ( p) where F ( p) is the solution to
F ´( p) =
α ( F ( p))
≡ Ω( p, F ( p ))
p β ( F ( p))
(7)
and
N
α (q ) ≡ ∑ π i b(i − 1; N − 1, q)
i =1
N −1
β (q) ≡ ∑ π i Bq (i; N − 1, q) .
i =1
b(i; N , q) is the density function of the binomial probability distribution with parameters N
and q, B(i; N , q ) is one minus the corresponding cumulative binomial probability distribution,
and Bq ≡ ∂B / ∂q . Furthermore, F ( p m ) = 0 , F ( p ) = 1 , and p m > 0 .
From the uniqueness of the solution to (7), it follows that p m is decreasing in p . Note that
Bq (i; N − 1, q) is always decreasing in i for sufficiently small q. For larger q, Bq (i; N − 1, q) is
increasing (decreasing) in i for small (large) i. b(i − 1; N − 1, q) , as a function of i, is shaped
as an inverse V. Thus, reducing π i for small i and increasing π i for larger i, typically
increases Ω( p, F ( p)) for given p. Therefore one would expect that p m is larger the more
probability weight there is on π i for large i’s. One has the following limiting results:
9
lim p m = 0
(8)
lim p m = p
(9)
π1 →1
π N →1
One question of particular interest is how the number of suppliers in the market will affect the
price structure. There are in general two different ways of analyzing this. We could either
think of a situation where, for a given level of demand, additional firms are introduced into
the market, i.e. total capacity is increased, or a situation in which existing firms are split up
into smaller units, i.e. a given total capacity is divided between a larger number of firms. If
the question of primary interest is the organization of the deregulated structure of an existing
industry, the latter approach seems the most natural and this is what will be considered here.
We analyze a particular example, where π i = 1/N, by comparing the outcome for different
N’s. By substituting for π i and solving (7), we have the following:
Result: When ∀n : cn = 0 , and ∀i : π i = 1/ N ,
F ( p) =
⎛
1
p⎞
ln ⎜ e N −1 ⎟
N −1 ⎝
p⎠
p m = pe1− N , and
Εp =
p
⎡⎣1 − e1− N ⎤⎦
N −1
(10)
(11)
(12)
Thus, both p m and Εp are decreasing in N. That is, prices will tend to be lower on average in
a more fragmented industry. The intuition for this may be explained as follows: By increasing
its offer price a firm reduces the probability that it will receive a positive payoff. On the other
hand, submitting a high offer price increases, in expected terms, the system marginal price.
The system marginal price effect, however, benefits the firm only when it happens to be the
marginal firm, an event which is less likely the more firms there are in the industry.
This intuition also suggests that in the general model with multi-unit firms, prices will tend to
be higher than in the model in which these same units act independently. As indicated above,
raising the offer price of one unit will have an external effect on other units in that it increases
the expected system marginal price. An owner who controls many units will internalize part
of this externality and will thus have an additional incentive to increase their offer prices. In
particular, this “coordination incentive” is stronger the more units an owner controls. It
therefore seems reasonable to conclude that for a given number of generating sets in the
industry, the system marginal price will be a decreasing function of the number of owners,
or generators controlling the sets, i.e. the industry concentration ratio.
4.
The UK Electricity Industry
In this section we present empirical evidence on the bidding behaviour of the major
generators in the U.K. electricity industry10. Since our model of price competition in the
10
See Vickers and Yarrow (1991), Green (1991 a) and James Capel & Co. (1990) and Holmes and
Plaskett (1991) for descriptions of the new UK electricity industry. Vickers and Yarrow (1991) in
10
electricity spot market is obviously too abstract and simplified to be tested directly against the
empirical evidence, our purpose here is rather to demonstrate that the types of strategic
behaviour we have identified in our model are at least consistent with actual historical bidding
behaviour, and that our most important conclusion for policy purposes, viz. that bids will tend
to be well above generation costs, is supported by the evidence.11
4.1. Structure of the U.K. industry
There are three main generating companies in the system for England and Wales: National
Power with approx. 52% of the generating capacity of the old Central Electricity Generating
Board, PowerGen with 33% and Nuclear Electric with 15%. In addition there are some
suppliers in Scotland and on the continent connected to the system in England and Wales.
These, however, are for the time being of minor importance. National Power and PowerGen
are private companies (with the government as a minority shareholder), while Nuclear
Electric is publicly owned. Nuclear Electric’s production is completely based upon nuclear
power. It therefore functions entirely as a baseload producer and its capacity is bid in at
(virtually) zero. Thus there are in reality only two significant strategic players in the
electricity spot market.
The electricity spot market has been designed very much like a multi-unit auction. Each day
the generating companies submit ‘bids’ to the National Grid Company which give the
minimum prices at which they are willing to supply electricity from each generating unit (or
‘genset’).12 A ‘merit order’ is then constructed from the bids, with sets ranked in ascending
order, and a despatch schedule determined to match supply and predicted demand for each
half-hour of the following day (this is called the ‘unconstrained schedule’). System marginal
price (SMP) – the major component of the price paid to each despatched genset – is
determined by the bid price of the marginal despatched set. In most half-hour periods (Table
A periods), each despatched genset is paid, in addition to system marginal price, a ‘capacity
element’, intended to reflect the probability of loss of load, i.e. a power shortage. They may
also receive ‘uplift payments’. In Table B periods, when there is expected to be an excess of
running, partly-loaded capacity, capacity payments are not made, and only the ‘incremental
bid prices’ are used to determine system marginal price. There are further additional
complexities to the system. These have been described elsewhere (c.f. NGC (1991), Green
(1991a), Green (1991b), James Capel & Co. (1990)), and further details are provided in
appendix B.
At vesting, on March 31 1990, ‘contracts for differences’ were placed between the two major
generators and the regional electricity supply companies covering approximately 85% of the
generators’ capacities.13 These are option contracts under which the difference between the
spot, or ‘pool’ price of electricity and the contract strike price is paid to the purchaser (i.e. the
regional electricity company) on a specified number of units. These option contracts have
significantly reduced the incentives of the generators to bid pool prices above the level of
particular provide a discussion of a broad range of issues relevant for the evaluation of the
deregulation of the UK electricity industry.
11
Green (1991b) claims to have found bids to be at or near generation costs on most of the two
major generators' generating units. However the recent report by the regulator (OFFER (1991)) cites
bids well above estimates of 'avoidable generation costs' as a cause of concern in its attempt to
evaluate how well competition in the new system is working.
12
Amongst a great deal of other information - see appendix B and NGC (1991) for further details.
13
Details may be found in the share offer prospectus, Kleinwort Benson Ltd. (1991); Powell (1991)
also contains a discussion of the importance of contracts the new British electricity industry.
11
contract strike prices, since any difference between the pool price and contract strike prices is
paid back to the regional supply companies in the form of a difference payment on the amount
of capacity contracted for. One would therefore not expect to see the type of ‘noncompetitive’
bidding behaviour predicted by the theoretical model mirrored in the historical bidding data.14
By March 31 1991 however, a proportion of these contracts had expired (approx. 15%), and
the rest are due to expire by March 1993. With contract coverage lowered to about 70% of the
generators’ capacities, ‘strategic’ or ‘noncompetitive’ bidding behaviour becomes more
likely, and so one expects to see in the first year of operation of the new system, bids
reflecting generation costs – since contract strike prices were chosen to represent expected
marginal generation costs – and after February/March 1991 a possible change in ‘regime’ to
more aggressive, noncompetitive bidding. It is precisely this kind of ‘change in regime’ that
we see reflected in the data to April 31 1991, and which is described here.
In appendix B we describe in some detail the operation of electricity pool, as operated by the
National Grid Company. We also provide information concerning our data sources and our
analysis of the bid data. The interested reader is referred there for this information and we turn
here directly to a discussion of the results of our empirical investigations.
4.2.
Generator bidding behaviour
In the figures below, two different ways of describing the bidding behaviour of National
Power and PowerGen for the period from July 1990 to April 1991 are depicted. Figures 1 and
2 show the ‘actual’ bids15 of the two major generators for each level of output on particular
weekdays of the year, i.e. the generators’ ‘supply curves’. Figure 3 on the other hand
represents the average weekly bids of each generator for generating sets of a particular size
and fuel type. The former are thus ‘snapshots’ of generator bids at particular points in time,
while the latter gives a longer-term picture of generator bidding behaviour over the period.
Figure 1a shows the supply curves of both generators on July 2 1990. Since PowerGen’s
capacity is approximately 64% of National Power’s, its supply curve becomes vertical much
earlier, at approximately 12,000 MGWhs. Figures 1b and 1c compare the generators’
estimated cost curves to their supply curves. Since the generators’ cost curves were
constructed by summing over all of their capacity, while on a given day some capacity will be
declared unavailable (due to maintenance, etc.), it is to be expected that the supply curves
become approximately vertical before the cost curves do. Apart from this however, the
generators seem to have been bidding a very close approximation to their cost curves.16 Thus
during the first 8 to 10 months of the operation of the new system, the evidence seems to suggest
that the two major suppliers were bidding ‘competitively’, i.e. at cost.
The figures for 22 February 1991 begin to indicate a different pattern. PowerGen’s supply
curve on this day lies uniformly above National Power’s (fig. 2a), with marked differences in
bids in 2,000 to 10,000 MGWh range. In figures 2b and 2c we see that PowerGen’s supply
curve also lies uniformly above its cost curve, while National Power’s supply curve is below
its cost curve from 0 to approximately 10,000 MGWhs, the range of output covered by its
14
von der Fehr and Harbord (1992) model the strategic incentives of the generators with simple oneway and two-way difference contracts.
15
See appendix B for a discussion of how these were constructed.
16
Figures B1a, Bib and B1c in appendix B show similar comparisons for an early winter day,
November 22 1990. Again the generators' bids closely reflect their costs at all levels of output.
12
large, coal-burning sets (once adjustments for availability have been made), and thereafter is
above it.17
While these figures do not provide enough data to allow us to reach any firm conclusions,
they do seem to indicate a change in the pattern of bidding behaviour. In particular the figures
for February indicate more sophisticated patterns of bidding behaviour than simply bidding in
at cost. This is confirmed by an examination of the weekly averages of bids on gensets of a
particular size and fuel type over the entire period from May 1 1990 to April 30 1991.18 In
figure 3 average weekly bids for National Power’s and PowerGen’s large coal sets are
shown.19 It is apparent that at around week 40 (early December 1990) both generators
altered their bidding behaviour significantly, and in opposite directions. National Power’s
bids on its large coal sets drop dramatically from an average of approximately £14/MGWh
to well below £10/MGWh in almost all weeks, and its largest coal sets were occasionally
bid in at below £2 /MGWh. PowerGen, on the other hand, increased its bids on its large coal
sets by an average amount of approximately £1/MGWh. This pattern of bids remained
stable from December 1990 to the end of April 1991.20
As noted above, a possible explanation for these results is that since in the first part of this
period contract coverage for each generator was approximately 85% of their capacities,
contract strike prices put a downward pressure on spot prices, while in the latter part, when
contract coverage was less, this pressure was eased. However, contracts only started expiring
right towards the end of the period (March 31 1991), while the change of pattern occurs in
December 1990.
An alternative explanation is given by our model. The first period more or less coincides with
the warm season, and based on the model predictions we expect to observe prices closer to
costs when demand is low. Thus for demand levels below approximately 27,500 MGWhs,
which is typical during warm seasons, large coal sets are almost exclusively the ‘marginal
sets’ which determine system marginal price.
Furthermore, since in the warm season demand may fall very low at night and in early
morning, there will be strong competition to be despatched (see the discussion in sections 3.1
and 3.2). In the colder season, however, demand is always so high that sets with low rank will
never become marginal. Therefore, from December 1990 onwards, PowerGen’s large coal
sets were determining system marginal price over a large number of periods, while National
Power’s large coal sets were being bid in low enough so that they were certain to be
17
The figures for April 18 1991 in appendix B tell a similar story. In figure B2a PowerGen's supply curve lies
significantly above National Power's up to approximately 8000 MGWhs of output, and is uniformly above its
cost curve (fig. B2b). National Power's supply curve for April 18 shows a similar pattern to that of February
22. The first 8000 MGWhs of output have been bid in below cost, and the remainder at offer prices above
generation costs
18
In figure 5, week 5 is the week beginning May 1 1990.
19
C5 indicates a coal set of greater than 500 MGWhs and C6 a coal set of greater than 600 MGWhs.
We concentrate on coal sets since for other sets the picture is blurred by the frequent changes in input
prices, i.e. prices of oil and gas.
20
In appendix B figures B3 through B6 give some further evidence. Figures B3 and B4 compare
bids on large coal sets to the generators' average costs. As is evident, National Power's bids fell
significantly below costs from December 1990, while PowerGen's bids remained above average
costs and increased after November 1990. Finally, figures B5 and B6 show the average weekly bids
of National Power and PowerGen respectively over all coal sets and all sets, for the entire period. It
is clear that the picture for other sets is similar to that for the coal sets, although the former is
somewhat blurred by the frequent changes in oil and gas prices.
13
despatched. This type of bidding behaviour, then, has the flavour of the equilibria described in
section 3.3.
5.
An Alternative Payoff Rule
As we have shown in section 3, firms will in general choose bids greater than their marginal
costs, and thus the system marginal price will tend to exceed the marginal costs of each of the
operating units. Furthermore, since less efficient sets may submit lower offer prices than more
efficient sets, inefficient despatching may result. It is therefore an interesting question
whether the regulatory rule can be modified so as to induce truthful revelation of costs and, as
a result, efficient despatching. In this section we show that by extending an insight on optimal
auctions due to Vickrey (1961), such a modification is indeed possible.
The game G may be interpreted as a first-price, sealed-bid, multiple-unit auction with a
random number of units. In particular, the system marginal price is determined by the offer
price of the marginal operating set, and thus a firm’s bids will determine the price received in
the event that one of its sets is the marginal operating unit. The fundamental insight of
Vickrey (1961) was that by making the price received by a firm independent of its own
offer price, marginal cost pricing can be induced as a dominant strategy for all firms. The
reason for this is that in such a set-up a firm can only influence its own payoff to the extent
that it affects the probability of being called into operation. A firm will prefer to be
operating for all realizations of demand such that its payoff is positive, and will prefer not to
operate whenever its payoff is negative. Therefore, offering to supply at a price equal to
marginal cost becomes a dominant strategy because it maximizes the probability of being
called into operation whenever the firm’s payoff is expected to be non-negative.
In a standard Vickrey auction, price is determined by the marginal unsuccessful, i.e. nonoperating, player. To generalize this result, we must construct a mechanism which is both
incentive compatible and individually rational. This can be done by letting the price paid to
firm n be determined by the intersection of demand with the residual (i.e. net of the capacity
of firm n) supply curve. Consider therefore a slight variation of the game G where the only
change involves the payoff rule: The intersection of the demand and the supply curves
determines which units will be called into operation. All operating units are paid firm-specific
prices determined by the intersection of the demand and the respective residual supply curves
if such an intersection exists, and set equal to pˆ ≥ max{cn } otherwise. Call this game Ĝ .
Then the following result holds:
Proposition 7: The game Ĝ has a unique dominant strategy equilibrium in which pni = cn ,
n = 1, 2,..., N
Remark: Other Nash equilibria typically exist. However, since offering to supply at marginal
cost (weakly) dominates all other strategies, we consider this the natural ‘focal’ point, and
thus base our discussion solely on this equilibrium21.
In Ĝ , as opposed to G, despatching is efficient since firms are always despatched in order of
increasing marginal cost. Thus, our alternative regulatory rule leads to minimization of real
generation costs.22
21
In addition, since all other strategies are weakly dominated, no other equilibria are ‘strategically
stable’ in the sense of Kohlberg and Mertens (1986) (while it is easy to check that the equilibrium in
question is stable); this is another reason for not considering them.
14
In addition to technical efficiency, one might ask how total (expected) payments to the
generators compare in the two auctions. Denote by ΕC total expected payments in G,
and by ΕĈ total expected payments in Ĝ , respectively. It is easy to verify that revenue
equivalence holds when valuations are drawn from the same distribution (if we let
p̂ = p ), as we would except from the Revenue-Equivalence Theorem (Vickrey (1961),
McAfee and McMillan (1987)). For example, in the symmetric oligopoly model
considered in section 3.3, ΕC = N π n p (since players receive the same payoff whichever
of the prices in the support of the strategies they play, and, in particular, the profit from
playing p is π N p ). On the other hand, ΕCˆ = N π N pˆ (since payments are zero when some
firms do not operate and p̂ to each of the N firms otherwise). It turns out to be difficult to
establish the sign of ΕC − ΕCˆ in the general model. However, in the duopoly case one can
show that ΕC is never smaller than ΕĈ . This is obvious in the cases discussed in sections
3.1 and 3.2. In the case analyzed in section 3.3, ΕC can be found by considering ϕ 2 ( p ) and
ϕ1 ( p) as p → p , where ϕi ( p ) is the profit of firm i from playing p, i = 1, 2 , from which it
follows that ΕC = π Kc + [1 − π ]2 p , where
⎧
α ⎞
1
⎛
⎪α ln ⎜1 + α + e − 1 ⎟ , when π = 2
⎝
⎠
⎪⎪
1− 2π
K =⎨
⎧
⎫
π
⎡
⎤
π
−
1
α
1
⎪
⎪
⎪α
1− ⎢
, when π ≠
⎨
⎬
⎥
⎪ 2π − 1
2
⎪ ⎣α + απ − 1 ⎦
⎪
⎩
⎭
⎩⎪
(13)
π
p
⎡ π ⎤ 2π −1
α = , and απ ≡ ⎢
c
⎣1 − π ⎥⎦
and ΕCˆ = π c + [1 − π ]2 p . Now, K ≥ 1 , and K = 1 when α = 1 and is increasing in α . For a
given α , K is maximized at π = 1/ 2 and K ≤ e − 1 . We summarize the duopoly result in the
following proposition:
Proposition 8: When N = 2 , ΕĈ is a lower bound for ΕC .
Such an improved pricing performance echoes the result in the optimal-auction literature that
second-price sealed-bid auctions yield higher payoffs to the auctioneer than do first-price
sealed-bid auctions (McAfee and McMillan (1987), Myerson (1981), and Maskin and Riley
(1989)). Thus, some of the first-price/second-price comparison results found in the auction
literature extend to this setting as well.
We conclude that (disregarding collusion and long-term contracts) an institutional setup
which induces firms to make offer prices equal to marginal costs is perfectly possible even
22
Efficiency considerations in electricity supply industries are considerably complicated by the
network characteristics of such industries, and being able to rank generating units according to
production costs is only a necessary condition for short run efficiency. For a treatment of efficiency
and optimal pricing in electrical networks, see Bohn, Caramanis, and Schweppe (1984).
15
when firms are capacity-constrained, something which seems not to have been appreciated in
the literature. As such, it also shows that applying results from standard oligopoly models
(such as those found in for example Kreps and Scheinkman (1983) and Tirole (1988), ch. 5)
can be misleading as a description of the outcome of competition in the U.K. electricity
market.
6.
Conclusion
In this paper price competition in the deregulated wholesale market for electricity for England
and Wales has been analyzed as a first-price, sealed-bid, multiple unit auction. In doing so,
we have demonstrated that under the existing institutional set-up there is likely to be both
inefficient despatching and above marginal cost pricing, even in the absence of collusion and
long term contracts. While these points have been argued elsewhere (see for instance Vickers
and Yarrow (1991) or Green (1991a)), the arguments have been largely informal and usually
based upon standard models of oligopoly pricing, and hence somewhat inconclusive. A
major purpose of the present paper has been to address these issues in a formal model
specifically designed to capture the essential elements of the new electricity pricing system
in the United Kingdom.
To our knowledge Newbery and Green (1991) (see also Newbery (1991)) is the only other
model specifically designed to study the bidding behaviour of the generators under the new
U.K. system.23 While our conclusions echo theirs in many respects, our results have also cast
some doubt upon the type of equilibrium analysis they have employed, i.e. Klemperer and
Meyer’s (1989) ‘supply function equilibrium’ model. This is because the equilibria found
under the assumption that firms submit smooth, i.e. continuously differentiable, supply
functions do not appear to generalize to the case where supply functions must be discrete
‘step functions’, even when the ‘step-length’ can be made very small. Indeed, we have found
that for a wide range of demand distributions pure-strategy (i.e. supply function) equilibria
will not exist in this case. It is therefore reassuring to find that Green and Newbery’s most
significant conclusion for policy purposes, viz. above marginal cost pricing, is also a property
of the model analysed here, and hence does not depend upon the particular assumptions they
impose.
In section 4 we have presented empirical evidence on the bidding behaviour of the two major
generators in the U.K. industry which has tended to support the conclusions of our theoretical
model. While not claiming to have ‘tested’ the model in any sense, we have been able to
demonstrate that at the very least the model is not contradicted by the empirical evidence, and
that the bidding strategies of the generators may be viewed, at least in part, as conforming to
the types of strategies described by the theory. While our empirical conclusions, in particular
bids greater than generation costs, do not agree with those of Green (1991b), they have been
confirmed elsewhere, significantly in the recent report of the regulator OFFER (1991). There
thus now exists serious evidence, both theoretical and empirical, that competition in the new
electricity supply industry for England and Wales may not be achieving the purposes for
which it was originally designed, i.e. the efficient generation of electricity, sold at competitive
prices to consumers.
23
While the model of Bolle (1990) is very close that of Green and Newbery (1991) in many respects,
its purpose is somewhat more general.
16
While the analysis presented here would appear to be useful in providing a framework for
studying pricing behaviour in the deregulated U.K. electricity industry, the importance of our
conclusions is limited by the extent to which they do not take into account opportunities for
collusive behaviour between the generators, or the effects of long-term contracts between
suppliers and purchasers (or third parties). These problems call for further research.24
24
In a companion paper - von der Fehr and Harbord (1992) - the model of the present paper is
extended to account for the presence of long-term contracts.
17
18
19
20
21
22
23
24
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26
Appendix A: Proofs
Proof of proposition 1: Assume by way of contradiction that more than one firm has sets
which with positive probability will become the marginal operating unit, and, thus determine
the system marginal price. Then, since the support of the demand distribution is an interval,
there must exist two sets with rank r and r + 1 , for some r ∈{1, 2,..., M − 1} , belonging to two
different firms, both of which will become the marginal operating unit with positive
probability. Call the firm that owns the set ranked r firm n and the other firm n̂ . Note that
since firms can secure non-negative profits by bidding at marginal cost, one must have
p r ≥ cn and p r +1 ≥ cnˆ . p r < p r +1 cannot be part of an equilibrium since by increasing the bid
of set r towards p r +1 , firm n will increase its profit. Increasing the bid in this a way does not
affect the ranking but does increase the system marginal price in the event that the r’th set
becomes the marginal operating unit. p r = p r +1 cannot be an equilibrium either, since if
cn ≠ cnˆ , at least one firm can increase its profit by undercutting. For example, if cn < cnˆ ,
p r = p r +1 > cn by the argument above, and thus firm n can increase its profit by undercutting
firm n̂ by an arbitrarily small amount, thereby strictly increasing its chance of being called
into operation without affecting the (expected) system marginal price. QED.
Proof of proposition 2: In any equilibrium firm 1 will determine the system marginal price
with probability one. Assume otherwise, i.e. that firm 1 has bid in so many of its sets at high
prices that some of the sets of firm 2 have a positive probability of becoming the marginal
operating unit. Since firm 2 can secure non-negative profits by bidding at marginal cost, firm
2 will not bid in these sets below c2 . But then firm 1 can increase its profit by undercutting
such firm 2 sets by some arbitrary amount, since this increases the (expected) amount
supplied by firm 1 without affecting the system marginal price in any event.
Next we show that that the system marginal price will not exceed c2 . Assume otherwise.
Then since firm 1 always determines the system marginal price, it must have bid in sets at a
price greater than c2 . But then firm 2 can increase its profit by undercutting firm 1.
Lastly, since firm 1’s profit is increasing in bids on the sets that may become the marginal
operating unit, the only candidate for equilibrium involves these being bid in at c2 . (This
will be an equilibrium if one imposes the tie-breaking rule that if firms tie at c2 , firm 1 is
despatched with probability 1.) QED.
Proof of proposition 3: From proposition 1 it follows that only sets of one firm will
determine the system marginal price. Without loss of generality, assume that these belong to
firm 2. Then since the profits of firm 2 are increasing in its own bids, all these sets will be
bid in at p . Furthermore, firm 1 must bid in all its sets at offer prices strictly less than p .
Now, for this to be an equilibrium, firm 1 must bid low enough so that firm 2 cannot
increase its profit by undercutting. Firm 1 bidding at or below firm 2’s marginal cost is
sufficient to guarantee this. (Since firm 1 never becomes the marginal firm, it will be
willing to do so even if c2 < c1 .) QED.
Before considering the proofs of propositions 5 and 6, we derive some properties of the
general oligopoly model’s mixed-strategy equilibria when firms have equal capacities and
27
demand is distributed discretely as assumed in the main text and under the assumption that
any number of units will be called into operation with positive probability. We start by
proving that no firm submits offer prices below its marginal cost. We then show that p is
always part of some player’s strategy. Lastly, we prove the following two results: There can
be no mass point at any price in a player’s mixed strategy, with the possible exception of p ,
i.e. no price less than p is played with positive probability. And if p m is the smallest price in
the support of any player’s strategy, then all prices p ∈ [ p m , p ) are in the support of at least
two players’ strategies.25
Lemma 1:
(Lower bounds for the offer prices) In any equilibrium and for all
n = 1, 2,..., N , pn ≥ max{cn , c2 } , (remember that c2 is the marginal cost of the
second-most efficient firm).
The result is rather obvious and follows from the observation that a player, by offering a
price below his marginal cost, has a positive probability of obtaining a non-positive payoff
which could be avoided by choosing a higher offer price. From this, and the fact that no
player will choose an offer price below and bounded away from the lowest price ever
chosen by anyone else, one concludes that the system marginal price will not fall below the
marginal cost of the second-most efficient firm.
Lemma 2:
(Upper support) If π N > 0 , at least one player will have p as part of his equilibrium
strategy.
Proof: Let p̂ be the highest price in the support of any player’s strategy and assume that
p̂ < p . Let n be one of the players who has p̂ as part of his strategy (in particular, if one
player plays p̂ with positive probability, let n be him). Playing p̂ yields him an expected
payoff of pˆπ N while playing p yields pπ N > pˆπ N . QED.
Lemma 3:
(No interior mass points) In equilibrium no offer price p < p will be played
with positive probability by any player. Furthermore, if p is played with
positive probability by some player, no other player will play p with positive
probability.
Proof: We start by showing that if p̂ > c2 is (believed to be) played with positive
probability by some player, p̂ is not played with positive probability by any other player.
Let pˆ > max{c2 , cnˆ } be an offer price which is played with positive probability by a player n̂
and assume that only n̂ plays p̂ with probability greater than zero (the argument below
extends in a straightforward manner to the case where more than one player plays p̂ with
positive probability). Then if a player n, for which cn < pˆ (by lemma 1 such a player exists),
plays p̂ , an element in his expected payoff is
25
Note that for any equilibrium (mixed) strategy profile there exist infinitely many (generically)
equivalent strategy profiles which differ only on sets of measure zero. We do not make any distinction
between such strategies.
28
N −1
πi
i =1
2
Pr( pnˆ = pˆ )∑ Pr([ pni+1 < pˆ ] ∩ [ pni+2 > pˆ ] | pnˆ = pˆ )[ pˆ − cn ]{
+ π i +1}
(A.1)
where
Pr( pn0 < pˆ | pnˆ = pˆ ) ≡ Pr( pnN +1 > pˆ | pnˆ = pˆ ) ≡ 1
This is the expected payoff in the event that there is a tie at p̂ . Given a tie, player n is ranked
below n̂ with probability ½ and gets a payoff of p̂ whenever he or n̂ is the marginal firm.
With probability ½, n is ranked above n̂ and receives p̂ only when he is the marginal firm.
If n plays p̂ − ε for some ε > 0 , then in the limit, as ε → 0 , the corresponding element in his
expected payoff is
N −1
Pr( pnˆ = pˆ )∑ Pr([ pni+1 < pˆ ] ∩ [ pni+2 > pˆ ] | pnˆ = pˆ )[ pˆ − cn ]{π i + π i +1}
(A.2)
i =1
The difference between (A.1) and (A.2) is that the latter corresponds to the case where n is
always ranked below n̂ whenever n̂ plays p̂ since ∀ε > 0 , pn = pˆ − ε < pˆ . All other
elements in the sum which constitute player n’s expected payoff from playing p̂ and p̂ − ε ,
respectively, can be made arbitrarily close by choosing ε small enough. Thus, there
exists ε > 0 , such that playing p̂ − ε yields a strictly higher payoff than playing p̂ and,
therefore, playing p̂ with positive probability cannot be part of an equilibrium strategy for
player n .
We next show that if p̂ < p is played with positive probability by some player n̂ , any offer
price exceeding p̂ by any other player will be bounded away from p̂ . Consider the payoff to
player n who plays p̂ + ε for some ε > 0 . Then in the limit, as ε → 0 , the element in his
expected payoff corresponding to (A.1) is
N −1
Pr( pnˆ = pˆ )∑ Pr([ pni+1 < pˆ ] ∩ [ pni+2 > pˆ ] | pnˆ = pˆ )[ pˆ − cn ]π i +1
(A.3)
i =1
The difference between this and (A.1) is that n is always ranked above n̂ whenever n̂ plays
p̂ , since for ∀ε > 0 , pn = pˆ + ε > pˆ , i.e. there is never a tie. All other corresponding
elements in the sum which constitutes player n ’s expected payoffs from playing p̂ and
p̂ + ε , respectively, can be made arbitrarily close by choosing ε small enough. Thus there
exists ε > 0 such that for all ε ∈ (0, ε ) , playing p̂ yields a strictly higher payoff than
playing p̂ + ε , and therefore any offer price which forms part of n ’s strategy and is not less
than p̂ , must exceed p̂ + ε .
Since the above result must hold for all n ≠ nˆ , it follows that player n̂ would gain by playing
p̂ + ε instead of p̂ , for some 0 < ε < ε . This contradicts the assumption that p̂ is part of his
equilibrium strategy and completes the proof. QED.
Lemma 4:
(No ‘holes’) If p is part of any equilibrium strategy, then for any interval
S ⊂ ( p, p ] , ∀p ∈ S are part of at least two players’ equilibrium strategies.
29
Proof: We first show that there cannot exist any interval S ⊂ ( p, p ] such that no player has
elements in S as part of his strategy. Assume, for a contradiction, that such an interval S
exists, and let pinf = inf{ p | p ∈ S } and psup = sup{ p | p ∈ S} . Then, for the player n̂ with
pinf − ε , for some ε > 0 , as part of his strategy, playing p̂ = pinf + εˆ ∈ S instead of pinf − ε
yields in the limit as ε → 0 an increase in expected payoff of
N
∑π
i =1
i
Pr([ pni+1 < pˆ ] ∩ [ pni+1 > pˆ ] | pnˆ = pˆ )εˆ
N
= ∑ π i Pr([ pni+1 < pˆ ] ∩ [ pni+1 > pˆ ] | pnˆ ∈ [ pinf , psup ])εˆ > 0
(A.4)
i =1
where
Pr([ pn0 < pˆ ] | pnˆ ∈ [ pinf , psup ])εˆ = Pr([ pni+1 > pˆ ] | pnˆ ∈ [ pinf , psup ]) ≡ 1
A contradiction.
By applying a similar argument we can show that it is not possible that only one player has
elements in S as part of his equilibrium strategy. If n̂ is the only player with p̂ = psup − εˆ ∈ S
as part of his strategy, then by playing psup + ε , ε > 0 , instead of p̂ , this yields in the limit
as ε → 0 an increase in his expected payoff equal to that in (A.4). QED.
Proof of proposition 5: From lemmas 1-4 we know that there is at most one player who plays
p with positive probability (this will be player 2, a result which follows from the observation
that the argument below leads to a contradiction if one makes the assumption that firm 1 plays
p with positive probability), that no player plays any price p < p with positive probability,
and that if p m is the smallest price played by any player, both players’ mixed strategies have
full support on ⎡⎣ p m , p ⎤⎦ . Then, 2’s expected payoff from playing p is
Φ 2 ( p ) = [1 − π ][ p − c ]
(A.5)
Prices below c cannot be in the support of his equilibrium strategy since
∀p ≤ c : Φ 2 ( p ) ≤ Φ 2 (c) = π *0 + [1 − π ][ Ep1 − c ] < [1 − π ][ p − c ]
(A.6)
Thus, the smallest offer price which is in the support of 2’s strategy must be strictly greater
than his marginal cost. Furthermore, it is clear that if p m is the smallest price in the support of
2’s strategy, then 1 never offers anything less than p m , and vice versa.
Let F1 ( p ) ≡ Pr( p1 ≤ p ) and f1 ( p ) ≡ F1´( p) . Then the expected payoff to player 2 of playing
p ∈ ⎡⎣ p m , p ⎤⎦ is
p
Φ 2 ( p) = π [1 − F1 ( p) ][ p − c ] + [1 − π ] F1 ( p) [ p − c ] + [1 − π ] ∫ [ ρ − c ] f1 ( ρ )d ρ
p
30
(A.7)
The first two elements in the sum are the expected payoff when firm 2 is the marginal firm,
i.e. determines the market price, and one and two firms are active respectively. The third
element is the expected payoff given that both firms are called into operation and firm 2 has
the lower price.
From (A.7)
Φ´2 ( p) = π {1 − f1 ( p ) [ p − c ]} + [1 − 2π ] F1 ( p)
(A.8)
Using the fact that in equilibrium ∀p ∈ ⎡⎣ p m , p ) , Φ 2 ´( p ) = 0 , one gets
f1 ( p) −
1 − 2π F1 ( p )
1
=
π p−c p−c
(A.9)
This and the fact that F1 ( p) = 1 , imply the following unique solution to (A.9):
⎧ ⎛ p−c ⎞
⎪ln ⎜ e
⎟
⎪ ⎝ p −c ⎠
F1 ( p ) = ⎨
1− 2π
⎪ π −1 ⎡ p − c ⎤ π
π
+
⎪
⎢
⎥
2π − 1
⎩ 2π − 1 ⎣ p − c ⎦
⎧ p−c
⎪ e +c
⎪
pm = ⎨
π
1
−
2π
π
⎡
⎤
⎪ p−c
+c
[
]
⎢
⎥
⎪
⎣1 − π ⎦
⎩
when π =
1
2
when π ≠
1
2
when π =
1
2
when π ≠
1
2
(A.10)
(A.11)
where ln(e) ≡ 1 . A similar reasoning gives
⎧ ⎛
⎞
p
⎪ ln ⎜⎜ e
⎟⎟
⎪⎪ ⎝ p [ e − 1] c ⎠
F2 ( p ) = ⎨
1− 2 π
⎪ π −1 ⎡
⎤ π
π
p
+
⎪
⎢
⎥
2π − 1
⎪⎩ 2π − 1 ⎣ p + [α π − 1] c ⎦
F2 ( p ) = 1 ,
π
⎡ π ⎤ 2π −1
where απ ≡ ⎢
⎣1 − π ⎥⎦
QED.
31
when p < p and π =
1
2
when p < p and π ≠
1
2
Proof of proposition 6: Note that it follows from lemma 2 that no symmetric mixed-strategy
equilibrium can contain mass points at any offer price. Furthermore, from lemmas 3 and 4, it
follows that in any symmetric mixed strategy equilibrium, if p m is the smallest offer price, all
p ∈ [ p m , p ] are in the support of the players’ strategies.
Let Φ n ( p ) be the expected payoff to firm n of choosing offer price p. Then one has the
following
N
p
i =1
p
Φ n ( p ) = ∑ π i {Pr([ pni+1 < p ] ∩ [ pni+1 > p ] | pn = p ) p + ∫ ρ dFni ( ρ )} ,
(A.13)
where
⎛ N − 1⎞ j
N −1− j
, i = 1,..., N
⎟q [1 − q ]
j ⎠
j =i −1 ⎝
N −1
∑⎜
(A.14)
⎛ N − 1⎞ i −1
N −1
Pr ⎡⎣ pni−1 < p ⎤⎦ ∩ ⎡⎣ pni+1 > p | pn = p ⎤⎦ = ⎜
⎟ q [1 − q ] , i = 1,..., N
⎝ i −1 ⎠
)
(A.15)
q ≡ F ( p) ≡ Pr( pk ≤ p) , k ≠ n
(A.16)
Fni ( p) = Pr( pni ≤ p | pn ≤ p) =
(
The first part of each element of the sum in (A.13) represents the payoff in the event that firm
n is the marginal supplier, while the second part is the payoff given that firm n supplies but is
not the marginal operating unit. The sum is over all possible demand realizations.
In a mixed strategy equilibrium, it (generically) must be the case that for any two points p̂
and p in the support of the players’ strategies, Φ n ( pˆ ) = Φ n ( p ) . Thus,
0 = Φ n ´( p ) = α (q) − pq´β (q)
(A.17)
where q´≡ f ( p ) ≡ F´( p ) , and
N
α (q ) ≡ ∑ π i b(i − 1; N − 1, q)
(A.18)
i =1
N −1
β (q) ≡ ∑ π i Bq (i; N − 1, q)
(A.19)
⎛ N − 1⎞ i −1
N −i
b(i − 1; N − 1, q) ≡ ⎜
⎟ q [1 − q ] , and
⎝ i −1 ⎠
(A.20)
N −1 N − 1
⎛
⎞ j
N −1− j
.
B (i; N − 1, q) ≡ ∑ ⎜
⎟q [1 − q ]
j ⎠
j =i ⎝
(A.21)
i =1
b(i; N , q) is the density function of the binomial probability distribution with parameters N
and q, while B(i; N , q ) is one minus the corresponding cumulative binomial probability
32
distribution. Note that for N ≥ 2 and q ∈ [ 0,1) , b(1 − 1; N − 1, q ) ≥ 0 for all i, with strict
inequality for at least one i, and Bq (i; N − 1, q) > 0 . Thus, ∀q ∈ [ 0,1) ,
α (q) > 0 , and
(A.22)
β (q) > 0 .
(A.23)
Consider the differential equation
q' =
α (q)
≡ Ω( p , q ) > 0 .
pβ (q )
(A.24)
Define
K 1 = max {α ( q) | q ∈ [ 0,1]} > 0 ,
(A.25)
K1 = min {α (q ) | q ∈ [ 0,1]} > 0 ,
(A.26)
K 2 = max { β (q) | q ∈ [ 0,1]} > 0 , and
(A.27)
K 2 = min { β (q ) | q ∈ [ 0,1]} > 0 .
(A.28)
Since Ω( p, q ) and Ω q ( p, q) are continuous for p > 0 and q ≥ 0 , and
∀q ∈ [ 0,1) : Ω( p, q ) ≤ K p
(A.29)
where K = K 1 / K 2 , for every p m ≥ 0 (A.24) has a unique solution F ( p ) where F ( p m ) = 0
(see Sydsæter (1984), p. 25). Since the solution F ( p ) is continuous in p m and f ( p) = q′ > 0 ,
there exists a p m < p such that F ( p ) = 1 .
Next, one has that
∀q ∈ [ 0,1] : Ω( p, q ) ≥ K log( p / p m ) ,
(A.30)
where K = K1 / K 2 < ∞ . Therefore,
p
1 = F ( p) ≥
1
∫ K p dp = K log( p / p
p
m
)
(A.31)
m
from which it follows that pm > 0. QED.
Proof of proposition 7: Observe that the payoff to a particular operating firm is independent
of its own offer price. Fix the strategies of firms m ≠ n , and consider the best response of firm
n. Let one of firm n’s offer prices be pni = p . First, for realizations of demand for which set i
of firm n is operating and gets positive payoff, i.e. p < P and P > cn , the firm would have
been equally well off offering pni = cn . Second, for realizations of demand for which set i is
33
not operating and the system marginal price exceeds its marginal cost, i.e. p ≥ P > cn , firm n
would have been strictly better off offering pni = cn . For realizations of demand such that the
system marginal price is below its marginal costs, firm n is at least as well off by offering
pni = cn as any other price. Thus, pni = cn is a (weakly) dominant strategy for firm n. QED.
Reference
Sydsæter, K. (1984) “Matematisk analyse”, vol. 2, Universitetsforlaget, Oslo.
34
Appendix B: The U.K. Electricity industry
In this appendix we describe in more detail than in the text the daily operation of the pool by
the National Grid Company, and our analysis of the generators’ bid and cost data. We also
provide additional figures presenting evidence on the bidding behaviour of the generators.
B.1 Operation of the Pool and Generator Bids
Much of the detail provided here may be found in the NGC publication, An Introduction to
the Initial Pool Rules, March 1991, available from NGC Settlements Ltd.. Other sources of
information are Schedule 9 to the Initial Settlement Agreement (as amended), which contains
the text of the Initial Pool Rules as of July 1 1990 (copies are available from NGC
Settlements Ltd.), the generators’ share offer prospectus (Kleinwort Benson Ltd. (1991)),
Holmes and Plaskett (1991), James Capel & Co. (1990) and London Economics Ltd. (1990).
On March 31 1990 a new electricity market was established in England and Wales for the
trading of electricity between generators and suppliers (the regional electricity companies).
Under the new trading arrangements electricity is sold by the generators and purchased by the
regional electricity companies and large (industrial) consumers, through ‘the pool’ at prices
determined by the rules which govern the market’s operation (the Initial Pool Rules). At the
same time the National Grid Company (NGC), as operator of the transmission system (the
Grid Operator), seeks to schedule and despatch generating units, subject to certain technical
restrictions, to meet demand (including a margin for reserve). This is done principally on the
basis of a merit order constructed from the generators’ offer prices and availability
declarations for their generating units (gensets). In the absence of technical constraints,
gensets with the lowest offer prices are despatched first. It is important to understand that the
pool is not a place, but a mechanism to allow the orderly trading of electricity; in particular it
does not either purchase or sell electricity on its own account. A computerised system (the
Settlement System, which includes the computer program Settlement GOAL) is used to
calculate prices and to calculate payments due under the pool trading arrangements. The
Settlement System Administrator is NGC Settlements Ltd., a subsidiary of the NGC.
All electricity generating units of greater than 100 MW (megawatts) are despatched through
the pool by the grid operator. Smaller gensets may opt to be centrally despatched, or may
operate independently of the Grid Operator. Practically all electricity consumed in England
and Wales is supplied from centrally despatched gensets.
The market for electricity created by the pool trading arrangements establishes prices for the
sale and purchase of electricity for each half-hour period (a Settlement Period) of each day.
The principal component of generators’ revenues from the pool is the Pool Purchase Price
(PPP), while the price paid by suppliers is the Pool Selling Price (PSP). PSP is derived from
PPP, plus any other payments made to generators (known as Uplift Payments), such that on
any given day pool revenues exactly balance.
The generators submit their bids for any given day by 10:00 A.M. on the previous day. From
their price offers and capacity availability declarations, a notional generation, i.e. despatch
schedule (the Unconstrained Schedule), is calculated using forecasts of the following days’
demands. Gensets are scheduled to meet forecasted demand plus a margin of reserve, and PPP
for each half-hour is calculated. The pool purchase price – PPP – consists of two components:
35
System Marginal Price (SMP) and a Capacity Element. SMP for any Settlement Period (halfhour of the day) is determined by the offer price of the most expensive genset to be
despatched in the Unconstrained Schedule. This genset is known as the Marginal Genset.
Subject to certain constraints, cheaper gensets are always despatched before more expensive
gensets. SMP is determined by the costs of the Marginal Genset over its period of running in
the Unconstrained Schedule.
The capacity element of PPP is a payment which is intended to provide an incentive for
capacity to be made available to the system, particularly at times of high demand. It is
determined using the probability of loss of load (LOLP), i.e. the probability that demand will
exceed the available generating capacity of the system, as calculated by Settlement GOAL,
and by the Value of Lost Load (VLL), a parameter set in advance.26 PPP is then calculated as:
PPP = SMP + LOLP(VLL − SMP)
(B.1)
VLL is taken as an upper bound for SMP: that is, SMP is the cost of the Marginal Genset
unless these costs exceed VLL in which case SMP = VLL.
In the Unconstrained Schedule each day is divided into two types of periods. In Table B
periods (normally periods of low demand), when there is an excess supply of scheduled
capacity (because it is uneconomic or technically infeasible to turn off and on certain
generating units within a short period of time), despatched gensets are paid only SMP, i.e. no
capacity element is paid; in all other periods (known as Table A periods), despatched gensets
receive the full pool purchase price (PPP). The vast majority of Settlement Periods during a
day are Table A periods.27
By 4:00 P.M. the day ahead, the value of PPP for each half-hour, as determined in the
unconstrained schedule, is made public. This permits large (industrial) consumers of
electricity to adjust their demands, and thus provides for, at present, a small amount of
demand price-responsiveness. There are at present approximately 1000 such customers in
England and Wales. The operational (despatch) schedule is calculated by the Grid Operator
for actual operation on the day, taking into account various system constraints, and the initial
running conditions of gensets, using a computer program identical to Settlement GOAL. The
operational despatch schedule will in general call into operation gensets which were not ‘in
merit’, but which for reasons of system stability need to be run. These gensets are paid their
bid price rather than SMP, as well as the capacity element. These additional payments to
generators, over and above PPP, form the larger part of Uplift payments, which are then
included in the pool selling price.28
Generator offer data consist of price offer data and declarations of available capacity, as well
as a set of ‘operating characteristics’, for each genset. The price offer data include, for each
genset, two fixed components, Start Up Price and the No Load Heat Rate, which are paid
independently of the level of generation, plus three incremental prices, or Incremental Heat
Rates (in addition there is also a Maxgen Price for generation above normal availability). The
incremental prices are separated by two ‘elbow points’. The availability declarations state the
MWh quantity which a genset is willing to supply, and a time from which it applies. In
addition gensets are able to place conditions on their running (i.e. whether, and how often
26
VLL for 1990/91 was set at £2000/MWh (MWh = a megawatt hour)
OFFER (1991) contains a discussion of the relative frequency of Table A vs. Table B periods.
28
OFFER (1991) describes the controversy which now surrounds this practice, because of the
potential for its abuse by the generators.
27
36
they are able to shut down) and the output levels at which they are run, as well as certain other
restrictions on their operation (not detailed here). Each genset’s price offer is fixed by 10:00
A.M. the previous day; however, capacity declarations may be revised (‘redeclared’) any time
prior to the running. A Revised Unconstrained (despatch) Schedule is determined on the basis
of the generator’s re-declared availabilities. However, SMP is calculated only once (for each
half-hour) on the basis of the original offer data.29
Gensets declared (or ‘redeclared’) available, but not scheduled for generation (in any halfhour) in the Revised Unconstrained Schedule, are paid the Capacity Element of SMP,
modified to take account of their bid price if this is greater than SMP. The bid price for a
genset is the average price at which that genset would provide electricity when operating at its
Declared Availability, including the price of one start-up. (Gensets scheduled for reserve in
the Revised Unconstrained Schedule are paid PPP less the genset’s incremental cost to reflect
the ‘cost saving’ brought about by operating at a lower level of output.)
Genset Price (the genset’s ‘bid price’) is the price used in the determination of SMP. In Table
B periods it is simply (one of) the incremental price(s) of each genset (depending upon its
level of scheduled operation); in Table A periods the fixed components of the genset’s price
offer are also included. Start Up Price and No Load Heat Rates are allocated across all Table
A periods for that period of continuous running in which the start-up occurred. These are
added to the incremental price to determine a £/MWh genset price. To take an example, a
genset with a Start Up Price of £1000, a No Load Price of £50/h, an Incremental Price of
£15/MWh and an offered availability of 200 MW, which is scheduled for 1.5 hrs. of Table A
running (we ignore Table B periods), has a ‘bid price’ of: (£1000 + ( 1.5h ⋅ 50£ / h ))/300MWh
+ £15/MWh = £18.58/MWh = Genset Price.
The Genset Price of the marginal genset in any half-hour is then taken as System Marginal
Price (SMP). Only ‘flexible’ gensets, i.e. gensets which may be turned off and on, and which
satisfy certain other conditions, are eligible to determine SMP.
B.2 Analysis of Bid Data
Bid data for the generators are made available by the NGC. In examining bid data it is not
possible to replicate the operation of the Settlement GOAL program to arrive at a Genset
Price, since for that the scheduled operation of each genset must be known.30 It is therefore
necessary to use an estimate of the normal running hours of each type and size of genset to
arrive at an approximation of Genset Price. We have done this using the following table
(Table 1), which contains rough estimates of average running times.
29
There is serious potential for abuse in this system. Available, but non-despatched, gensets receive a
capacity payment (see below). By initially declaring capacity unavailable, a generator can increase
SMP and the Capacity Element (by increasing LOLP); by later redeclaring the capacity available the
redeclared gensets will receive (at least) the capacity payment, and if their original bid prices were low
enough, SMP as well. In addition all of the generator’s other (available or despatched) gensets receive
the higher payments as well. OFFER (1991) has documented the recent manipulations of this
mechanism by PowerGen.
30
In our analysis of bid data we assume that all periods are Table A periods.
37
Table 1: Average running times.
Coal Sets:
Oil Sets:
6 hrs.
< 100 MW:
4 hrs.
Coal/Oil Sets:
4 hrs.
< 200 MW:
6 hrs.
Coal/Gas Sets:
4 hrs.
< 300 MW:
8 hrs.
Gas Turbine Sets:
< 400 MW:
10 hrs.
< 500 MW:
12 hrs.
> 500 MW:
16 hrs.
0.5 hrs.
Using the above figures a Genset Price (‘bid’) for each genset may then be calculated as
follows:
Genset Price (£/MWh) =
{Start Up Price}/{(hours of normal running) ⋅ (available capacity)}
+ {No Load Heat Rate}/{available capacity}
+ {(Capacity Weighted Average) Incremental Heat Rate}
This is the method we have used to calculate the bids depicted in figures 1 to 5 in the text.
Green (1991b) uses a slightly cruder method to calculate bid prices; OFFER (1991) also
includes calculations of the generators’ bid prices, but the methodology used is not described
in detail.
B.3 Generator cost data
Since privatisation there has been no publicly available generator cost information. OFFER
(1991) notes the difficulty, informational problems aside, of arriving at a unique correct
estimate of ‘avoidable generation’ costs for the two major generators, given the nature of the
fuel supply contracts they have had in place since vesting, which typically include a ‘musttake’ element. In particular the generators’ contracts with British Coal, which supplies almost
all of the fuel for their coal burning sets31, are ‘must take’ contracts, which makes the
‘avoidable (i.e. opportunity) costs’ of many of their coal burning stations virtually zero in
many periods.
These issues aside, generation costs can in principle be calculated from knowledge of station
thermal efficiencies and their fuel input costs, with an appropriate adjustment for
transportation costs. (This is done in Green (1991b); it is also the method used by OFFER
(1991)). Thermal efficiencies were published by the old Central Electricity Generating Board.
Apart from adjusting for stations which have since been decommissioned, these remain
31
See the share offer prospectus for details; also Holmes and Plaskett (1991).
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reasonably accurate. Fuel input costs are most appropriately taken to correspond to world
market prices (plus transport costs). This is true despite the generators’ contracts with British
Coal, which fixes a coal price well above its world market value, because the generators’
contracts for differences with the regional electricity companies include a British Coal
payment which effectively reimburses them for any difference between the two prices of
coal.32 Accordingly this is the method we have followed here.
Thermal efficiencies (in percentages) have been taken from the Central Electricity Generating
Board’s statistical yearbooks for 1987/88. Fuel input costs (in £’s per gigajoule) have been
taken from various sources. (These data are available on request from the authors.) Given
these, generation costs may be calculated as follows. One gigajoule equals approx. 0.2777778
megawatt hours of energy. Thus a price expressed in £/GJ can be converted to £/MWh by
dividing by .2777778. Multiplying the fuel input cost (£/MWh) by one over the thermal
efficiency (times 100), yields generation costs in £/MWh.
32
See the share offer prospectus, Holmes and Plaskett (1991), or James Capel & Co. (1990) for further
details.
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