Andy Philpott

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Uniform-price auctions versus pay-as-bid auctions

• Andy Philpott
• The University of Auckland
• www.esc.auckland.ac.nz/epoc
• (joint work with Eddie Anderson, UNSW)

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Summary

• Uniform price auctions
• Market distribution functions
• Supply-function equilibria for uniform-price case
• Pay-as-bid auctions
• Optimization in pay-as-bid markets
• Supply-function equilibria for pay-as-bid markets

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Uniform price auction (single node)
price
T2(q)
p
quantity
price
combined offer stack
p
quantity
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Residual demand curve for a generator
S(p) total supply curve from other
generators D(p) demand function c(q) cost of
generating q R(q,p) profit qp c(q)
p
Residual demand curve D(p) S(p)
Optimal dispatch point to maximize profit
q
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A distribution of residual demand curves
p
e
D(p) S(p) e
(Residual demand shifted by random demand shock e
)
Optimal dispatch point to maximize profit
q
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One supply curve optimizes for all demand
realizations
The offer curve is a wait-and-see solution. It
is independent of the probability distribution
of e
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The market distribution functionAnderson P,
2002

• S(p) supply curve from
• other generators
• D(p) demand function
• random demand
• shock
• F cdf of random shock

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Symmetric SFE with D(p)0 Rudkevich et al,
1998, Anderson P, 2002
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Example n generators, eU0,1, pmax2
n2
n3
n4
n5
p
Assume cq q, qmax(1/n)
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Example 2 generators, eU0,1, pmax2

• T(q) 12q in a uniform-price SFE
• Price p is uniformly distributed on 1,2.
• Let VOLL A.
• EConsumer Surplus E (A-p)2q
• E
  (A-p)(p-1)
• A/2 5/6.
• EGenerator Profit 2Eqp-q
• 2E
  (p-1)(p-1)/2
• 1/3.
• EWelfare (A-1)/2.

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Pay-as-bid pool markets

• We now model an arrangement in which generators
  are paid what they bid a PAB auction.
• England and Wales switched to NETA in 2001.
• Is it more/less competitive?
• (Wolfram, Kahn, Rassenti,Smith Reynolds versus
  Wang Zender, Holmberg etc.)

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Pay-as-bid price auction (single node)
price
T2(q)
p
quantity
price
combined offer stack
p
quantity
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Modelling a pay-as-bid auction

• Probability that the quantity between q and q
  dq is dispatched is

• Increase in profit if the quantity between q and
  q dq is dispatched is

• Expected profit from offer curve is

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Calculus of variations
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Necessary optimality conditions (I)
Z(q,p)lt0
p
Z(q,p)gt0
( the derivative of profit with respect to offer
price p of segment (qA,qB) 0 )
q
x
x
qB
qA
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Example S(p)p, D(p)0, eU0,1

• S(p) supply curve from
• other generator
• D(p) demand function
• random demand
• shock

qp1
Z(q,p)lt0
Optimal offer (for c0)
Z(q,p)gt0
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Finding a symmetric equilibriumHolmberg, 2006

• Suppose demand is D(p)e where e has distribution
  function F, and density f.
• There are restrictive conditions on F to get an
  upward sloping offer curve S(p) with Z negative
  above it.
• If f(x)2 (1 - F(x))f(x) gt 0
• then there exists a symmetric equilibrium.
• If f(x)2 (1 - F(x))f(x) lt 0 and costs
  are close to linear then there is no symmetric
  equilibrium.
• Density of f must decrease faster than an
  exponential.

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Prices PAB vs uniform
Price
Uniform bid price
PAB marginal bid
PAB average price
Demand shock
Source Holmberg (2006)
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Example S(p)p, D(p)0, eU0,1

• S(p) supply curve from
• other generator
• D(p) demand function
• random demand
• shock

qp1
Z(q,p)lt0
Optimal offer (for c0)
Z(q,p)gt0
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Consider fixed-price offers

• If the Euler curve is downward sloping then
  horizontal (fixed price) offers are better.
• There can be no pure strategy equilibria with
  horizontal offers due to an undercutting
  effect
• .. unless marginal costs are constant when
  Bertrand equilibrium results.
• Try a mixed-strategy equilibrium in which both
  players offer all their power at a random price.
• Suppose this offer price has a distribution
  function G(p).

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Example

• Two players A and B each with capacity qmax.
• Regulator sets a price cap of pmax.
• D(p)0, e can exceed qmax but not 2qmax.
• Suppose player B offers qmax at a fixed price p
  with distribution G(p). Market distribution
  function for A is
• Suppose player A offers qmax at price p
• For a mixed strategy the expected profit of A is
  a constant

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Determining pmax from K
Can now find pmax for any K, by solving
G(pmax)1. Proposition AP, 2007 Suppose
demand is inelastic, random and less than market
capacity. For every Kgt0 there is a price cap in a
PAB symmetric duopoly that admits a
mixed-strategy equilibrium with expected profit K
for each player.
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Example (cont.)
Suppose c(q)cq
Each generator will offer at a price p no less
than pmingtc, where
and (qmax,p) is offered with density
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Example
Suppose c1, pmax 2, qmax 1/2.
Then pmin 4/3, and K 1/8
Average price 1 (1/2) ln (3) gt 1.5 (the
UPA average)
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Expected consumer payment
Suppose c1, pmax2.
Generator 1 offers 1/2 at p1 with density g(p1).
Generator 2 offers 1/2 at p2 with density g(p2).
Demand e U0,1.
If e lt 1/2, then clearing price min p1,
p2. If e gt 1/2, then clearing price max p1,
p2.
EConsumer payment (1/2) Eee lt 1/2 Emin
p1, p2
(1/2) Eee gt 1/2 Emax p1, p2
(1/4) (7/32) ln (3)
( 0.49 )

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Welfare
Suppose c1, pmax2.
EProfit 2(1/8)1/4.
lt EProfit 1/3 for UPA
EConsumer surplus A Ee EConsumer
payment
(1/2)A EConsumer payment
(1/2)A 0.49
gt (1/2)A 5/6 for UPA
EWelfare (1/2)A 0.24 gt (1/2)A 0.5
for UPA
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Conclusions

• Pay-as-bid markets give different outcomes from
  uniform-price markets.
• Which gives better outcomes will depend on the
  setting.
• Mixed strategies give a useful modelling tool for
  studying pay-as-bid markets.
• Future work
• N symmetric generators
• Asymmetric generators (computational comparison
  with UPA)
• The effect of hedge contracts on equilibria
• Demand-side bidding

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The End