1
Supply-function equilibrium with price-taking
suppliers
Keith Ruddell, under supervision of Andy Philpott and Tony Downward
Department of Engineering Science, University of Auckland
Abstract—In classical Supply-Function Equilibrium firms compete in supply curves and maximise their revenue over the
distribution of residual demand. These firms will have the power
to set prices through tacit collusion.
We can model a finite number of price-taking firms by
making the bounded-rationality assumption that firms cannot
influence the distribution of market prices. When all firms supply
curves are maximal against the distribution of prices induced by
those same supply curves then we have a rational-expectations
equilibrium.
This price-taking model is simpler to solve in network settings
than price-making SFE.
I will present some important mathematical results about
these price-taking equilibria over constrained networks and their
relationship to the strategic supply-function equilibrium.
These models can help measure the benefits arising from
increased competition as the NZ electricity transmission grid is
expanded.
I. I NTRODUCTION
Supply-function equilibrium (SFE) [3] is a market model
where a finite number of producers compete in supply functions to satisfy a stochastic demand. In SFE, producers maximise their profit as if they were monopolists facing a residual
demand curve.
We can represent price-taking behaviour by altering the apparent control problem that producers face. As the producer’s
optimisation problem is equivalent to a SFE optimisation problem, many of the known results about SFE can be immediately
applied to our partial-price-taking equilibrium.
A key parameter in our models will be the degree to which
players behave as price-takers, β. There are several possible
interpretations of β. It can represent producers awareness or
ignorance of their true market power. It can be used to model
situations in which producers are only in partial competition
with each other; it greatly simplifies existing (and probably
intractable) models of transport-constrained markets. Finally,
it allows us to continously transform from fully-price-taking
equilibria to oligopolistic SFE; thus opening up qualitative
and numerical methods for the analysis of more difficult SFE
models.
When speaking to market traders in the NZ wholesale electricity market, they often claim to be price-takers even if they
posses significant market share. Because of this ideological
frame, analsyes of changes to the NZEM rules that make a
price-taking assumption have an important role in justifying
regulatory policy to market participants.
After defining the market clearing mechanism and pricing
rules, we will derive the first-order equilibrium conditions,
which are a system of ordinary differential equations (ODE)
whose solutions are candidate equilibria. We then translate
some important properties of PPTE from SFE. This allows us
to solve some simple pool-market examples.
Finally, we explore price-taking behaviour in a
transmission-constrained market.
II. D ERIVATION
We have a market in which a welfare-maximising system
operator collects supply-function bids from producers and
decreasing demand function D(p) from consumers, and then
observes a random demand shock. There may be transport or
other constraints. When the demand shock is realised, prices
are chosen so as to satisfy demand at minimum system cost.
When producers are fully aware of their influence over price
outcomes in the market we have Supply-Function Equilibrium
(SFE). In this case they are strategic oligopolists. When
producers are totally ignorant of their influence over price
outcomes, it is as if they observe a distribution of prices and
optimise for each possible price level. We shall analyse here a
convex combination of these two extreme awareness scenarios.
A. Optimisation of expected profit
A given producer has a profit function that gives the net
profit when they are dispatched a given quantity at a given
price. For example, if their cost function is C (q) and the
market operates under uniform pricing, a producer’s profit
function will be
R (q, p) = pq − C(q).
(1)
The producer also observes their apparent market distribution function Ψ(q, p), which is their (subjective) probability of
an offer of quantity q at price p being rejected by the system
operator. Assuming all dispatched prices occur the interval
[p, p], the producer chooses a monotone supply function q(p)
and their expected payoff is given by the integral
Z p
dΨ
R (q, p)
dp.
dp
p
The first-order condition to maximise such a functional
(ignoring the monotonicity constraint for now) is that
∂Ψ ∂R ∂Ψ ∂R
Z (q, p) =
−
=0
(2)
∂p ∂q
∂q ∂p
at every point along the supply function. The second-order
condition for a local maximum is that ∂Z
∂q ≤ 0 everywhere
along the solution curve.
2
By a theorem we prove in [4], sufficient conditions for
optimality are that q 0 > 0 and that, for all p, Z(x, p) > 0
for all x < q(p) and Z(x, p) ≥ 0 for all x > q(p).
As in classical SFE, there may be a continuum of valid
equilibria. We shall focus our attention on the most competitive
equilibrium, which is determined by the boundary condition
that the highest demand levels are satisfied by offers at
marginal cost.
III. R ELATION TO SFE
A classical SFE is just a rational expectations equilibrium
in which all producers are perfectly aware of their market
power. The results in this section assume a strong equilibrium
and symmetric cost functions between n firms.
Theorem 1. Like Klemperer and Meyers’ SFE, the partial price-taking equilibrium lies between a Cournot and a
Bertrand schedule, defined respectively by
B. Rational Expectations Equilibrium
p − C 0 (S B ) = 0
The market distribution function is the apparent distribution
of residual demand. When the system operator receives all the
offer curves, the true slope of the aggregate supply curve is
the sum of the slopes of all the individual supply curves.
An equilibrium is reached when the apparent market distribution function along the offer curve has identical value to
the true market distribution function (that with β = 0).
In a pool market with a one-dimensional demand shock ε
having distribution F (ε), we arrive at the following definition
for the apparent market distribution function


X
Ψ (q, p) = F 
qj (p) + βxi (p) + (1 − β) q − D(p)
−D0 (p)
SC
=
.
p − C 0 (S C )
1−β
It also has zero markup at zero output, i.e.
p = C 0 (q(p)) when q(p) = 0.
Consequently, if zero is in the support of the demand shock,
then the only curves that satisfy the first-order optimality
condition and the monotonicity constraint are symmetric (all
producers offer the same supply function).
Proof. The argument is exactly as in [3].
Theorem 2. If all players are perfectly price-taking and the
(3) auction rule is uniform-pricing, then all output will be offered
where xi (p) = q(p), but producer i does not consider this to at marginal cost. Moreover, as β % 1, the equilibrium supply
be under their control. For any value of β, this is clearly the in- functions converge to marginal cost (the Bertrand schedule).
duced joint distribution on a producers offer quantity
Pand price
=
the
ODE
becomes
1,
(q, p), given the market clearing condition ε = q + j6=i qj (p) Proof. 0When 0 β
0
0
(p
−
C
(q))
q
+
βq
−
D
(p)
=
0.
This
can
only be
−i
(demand equals supply at the clearing price). When β = 1,
satisfied
by
a
curve
that
follows
the
marginal
cost
function
the apparent market distribution function is constant in the
0
p
=
C
(Q).
quantity direction. This models a producer acting only on the
For the convergence, we first consider the case where
system operator’s declared distribution of prices, unaware of
0
−D
(p) > 0. Using the two-policemen theorem and Theorem
any market power it may possess.
1,
it
suffices to show that S C → S B as β % 1. By the
The actual payoff is independent of the awareness β, since
definition
of the Cournot schedule

!
X
X
dΨi
SC
−D0 (p)
=f
qi − D(p) 
qj0 + βx0i + (1 − β) qi0 − D0 (p)
lim =
=
lim
=
= ∞.
dp
β%1
p − C 0 (S C ) β%1
1−β
i
j6=i
i6=j
= f (Q)Q0 ,
P
where Q = i qi (p) − D(p) and f is the probability density
of the demand shock.
Substituting the revenue function (1) and the apparent
market distribution function (3) into (2) gives the first-order
optimality condition
0
(p − C 0 (q)) q−i
+ βq 0 − D0 (p) − (1 − β) q = 0,
(4)
where we have divided through by the density f (Q) to
obtain conditions that are independent of the demand-shock
distribution. Thus the equilibrium will be a strong equilibrium
in the sense of [1]; it is ex-post optimal as well as ex-ante
optimal.
To show the second-order condition is satisfied for strong
equilibrium, define Ẑ = Z/f and observe that
∂ Ẑ
0
+ βq 0 − (1 − β) ,
= −C 00 (q) q−i
∂q
which is non-positive as long as C 00 ≥ 0 and β ≤ 1.
This is only possible if either S C → ∞, which is impossible
as S C ≤ S B < ∞, or p − C 0 (S C ) → 0, i.e. S C → S B .
When −D(p) = 0, the result follows since the ODE (4)
is continuous in −D0 and we have just shown that for even
the smallest positive demand elasticity the equilibrium curve
approaches marginal cost.
Corollary 3. The prices obtained in a PTE are lower than
for the equivalent SFE, for all demand outcomes.
Proof. Take two different values of the parameter β and
suppose that β1 > β2 . Then
q10 − D0 =
1 − β1
q1
,
n − 1 − β1 p − C 0 (q1 )
2 n−1+β1
and similarly for q20 − D0 . Since 1−β
1−β1 , n−1+β2 > 1, it follows
that q10 < q2 at every point (q, p). Therefore, given the same
boundary conditions, the symmetric equilibrium with higher
β will be closer to the marginal cost curve, i.e. have lower
prices.
3
Theorem 4. If in a pool market there are n symmetric firms
but every firm bids as if there were m symmetric firms, then
this is equivalent to a partial-price-taking equilibrium with
parameter
n
β =1− .
m
Proof. With perfect awareness and m symmetric firms, the
Klemperer-Meyer equation is
B. Asymmetric costs
Suppose there are two producers with cost functions
C1 (q) = q 2
1
C2 (q) = q 2 .
2
The system of first-order equilibrium equations is
(p − 2q1 ) (βq10 + q20 ) = (1 − β) q1
(p − q2 ) (q10 + βq20 ) = (1 − β) q2 .
(p − C 0 (q)) ((m − 1) q 0 − D0 (p)) = q.
With m symmetric firms and parameter β, the K-M equation
is
(p − C 0 (q)) ((n − 1 + β) q 0 − D0 (p)) = (1 − β) q.
These are equivalent when
n−1+β
=m−1
1−β
n
β =1− .
m
It is plausible that a firm not have a definite count of its
competitors. Think of a market whose products are sold at
specific locations and where transport is costly, like pints
of beer in a pub. As a publican, it is quite clear that one
is competing directly with the pub across the road, but the
location-preferences and travel costs of drinkers mean that one
is competing with a pub on the other side of town to a much
lesser degree. Under this interpretation, β is a one-dimensional
parameter that, combined with the market size, measures the
competitiveness of a market, like the Herfindahl-Hirschman
Index but different.
When producers underestimate the number of competitors
they have, we can have β < 0. The KM equation makes sense
as long as m−1+β > 0, since otherwise we obtain a negative
slope for positive output and markup.
IV. P OOL EXAMPLES
To avoid cluttered equations, we shall assume throughout
these examples that the price-dependent component of demand
is identically zero.
To solve this system numerically, we first make the change
of variable ui = qi /p, which gives the semi-autonomous nonlinear system
u2
β
u1
1
−
− u1
pu01 =
1 + β 1 − u2
1 + β 1 − u1
1
u1
β
u2
pu02 =
−
− u2 .
1 + β 1 − 2u1
1 + β 1 − u2
Figure 2 shows the direction field for the semi-autonomous
system after the change of variables, as well as a solution of
the original system.
C. Asymmetric beliefs
Suppose that two producers have identical cost function
C(Q) = 21 q 2 , but different β parameters, β1 = 45 and β2 = 25 .
This asymmetry could be due to different beliefs about market
power or to preferential market access for producer 1.
The resulting system of ODE is
4 0
1
(p − q1 )
q1 + q20 = 1 −
q1
5
5
2
3
(p − q2 ) q10 + q20 = 1 −
q2 .
5
5
We can solve this by the same method as the asymmetric cost
model above.
V. T RANSMISSION - CONSTRAINED EQUILIBRIA
In [2], [5], is derived the equilibrium conditions for SFE
in an electricity market with transmission constraints. In
these models the KM equation is augmented by probability
coefficients P̂ on the supply slopes, which are functions of
locational price and output levels. With partial-price-taking
behaviour, the first-order optimality condition in a transmission
constrained market becomes


X
(p − Ci0 (qi )) βqi0 +
P̂ij (q, p) qj0  = (1 − β) qi .
i6=j
A. Symmetric example
If we make the ansatz that the equilibrium curves will also
be symmetric, we obtain the single first-order ODE
(p − C 0 (q)) (n − 1 + β) q 0 = (1 − β) q.
(5)
This admits a closed-form solution (see the appendix), but is
also stable when evaluated numerically in the decreasing price
direction.
Figure 1 shows the equilibrium supply functions for this
example.
A. Network Example
If there are two nodes with two suppliers in each node, then
the matrix of probabilities looks like


0 1 P P
 1 0 P P 

P̂ = 
 P P 0 1 ,
P P 1 0
With P (q, p) the probability that the line connecting the two
nodes is not congested.
Such a model can also be solved by our numerical methods.
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Price
Price
4
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
−0.2
0
0.2
0.4
Quantity
0.6
0.8
0
1
0
0.2
0.4
0.6
0.8
1
Quantity
(a) 2 producers
(b) 5 producers
Fig. 1. PPTE with β = −1/2, 0, 1/2, 1 from top to bottom.
1.4
45
1.2
40
1
30
0.6
25
Price
u2
35
0.8
0.4
20
15
0.2
10
0
−0.2
−0.1
5
0
0.1
0.2
0.3
u1
0.4
0.5
0.6
0
0.7
(a) Diretion field of u0i
0
5
10
15
20
25
Quantity
30
35
40
45
(b) Equilibrium offer curves
Fig. 2. PPTE for 2 asymmetric producers with β = 1/2.
VI. C ONCLUSION
Partial-price-taking equilibrium is an extension of supplyfunction equilibrium which allows us to treat producers that
have false beliefs about their market position, or make only
partial contribution to others’ residual demand. As the parameter beta does not vary with price and output levels, the
resulting systems of ODE are much more tractable than, for
instance the general network SFE models of [2], [5].
In this brief paper we have not considered the natural
extension to discriminatory pricing. In such settings it could
prove to be of even more value, given the result, shown in [4],
that for a perfectly price-taking equilibrium with discriminatory pricing the second-order optimality condition is always
satisfied, which is not the case with strategic SFE.
A PPENDIX
A NALYTIC SOLUTION OF SYMMETRIC EQUILIBRIUM
With k =
1−β
n−1+β ,
the equation (5) is
kQ0 =
Q
.
p − γQ
By making the substitution
w=
which renders the ODE separable, we obtain
dw
w
=
−w
dp
k (1 − γw)
dw
dp
=
w
p
k(1−γw) − w
Z
Z
k (1 − γw)
dp
=
dw
p
(1 − k) w + kγw2
kγw − (k − 1) 1
+ log |kγw − (k − 1)| + c
log |p| =
log k−1
w
Q
1
(k − 1) kγ − (k − 1) + c,
log |p| =
log kγ − p
+
log
k−1
Q
p
p
where c is a constant of integration. It is not possible to solve
this for Q as a function of p or vice versa in elementary
functions.
ACKNOWLEDGMENT
The authors would like to thank an anonymous referee
for the journal Operations Research who first suggested we
investigate price-taking behaviour in a network SFE setting.
R EFERENCES
Q
,
p
[1] E. J. Anderson and X. Hu. Finding supply function equilibria with
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5
[2] P. Holmberg and A.B. Philpott. Supply function equilibria in transportation networks. Working paper, Electric Power Optimization Centre,
www.epoc.org.nz, 2015.
[3] Paul D. Klemperer and Margaret A. Meyer. Supply function equilibria in
oligopoly under uncertainty. Econometrica, 57(6):1243–1277, Nov. 1989.
[4] K. Ruddell, T. Downward, and A. Philpott. Supply-function equilibrium
with taxed benefits. EPOC working paper, EPOC.org.nz, 2015.
[5] R. Wilson. Supply function equilibrium in a constrained transmission
system. Operations research, 56(2):369–382, 2008.