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Andy PhilpottThe University of Auckland

www.esc.auckland.ac.nz/epoc

(joint work with Eddie Anderson, UNSW)

Uniform-price auctions versus pay-as-bid auctions

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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

quantity

price

quantity

combined offer stack

demand

p

price

quantity

T1(q) T2(q)

p

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Residual demand curve for a generator

S(p) = total supply curve from other generatorsD(p) = demand function

c(q) = cost of generating q R(q,p) = profit = qp – c(q)

Residual demand curve = D(p) – S(p)

p

q

Optimal dispatch point to maximize profit

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A distribution of residual demand curves

(Residual demand shifted by random demand shock )

D(p) – S(p) +

p

q

Optimal dispatch point to maximize profit

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p

q

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

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The market distribution function[Anderson & P, 2002]

p

q quantity

price

)p,q(

Define: (q,p) = Pr [D(p) + – S(p) < q]= F(q + S(p) – D(p)) = Pr [an offer of (q,p) is not fully dispatched]= Pr [residual demand curve passes below (q,p)]

S(p) = supply curve from other generatorsD(p) = demand function = random demand shockF = 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, ~U[0,1], pmax=2

n=2n=3n=4n=5

p

Assume cq = q, qmax=(1/n)

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Example: 2 generators, ~U[0,1], pmax=2

• T(q) = 1+2q in a uniform-price SFE • Price p is uniformly distributed on [1,2].• Let VOLL = A.• E[Consumer Surplus] = E[ (A-p)2q ] = E[ (A-p)(p-1) ] = A/2 – 5/6.• E[Generator Profit] = 2E[qp-q] = 2E[ (p-1)(p-1)/2 ] = 1/3.• E[Welfare] = (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

quantity

price

quantity

combined offer stack

demand

p

price

quantity

T1(q) T2(q)

p

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Modelling a pay-as-bid auction

• Probability that the quantity between q and q

+ q is dispatched is 1 ( , ( ))q p q

• Increase in profit if the quantity between q and q + q is dispatched is ( ( ) '( ))p q c q q

• Expected profit from offer curve is

max

0[ ( ) '( )][1 ( , ( ))]

qp q c q q p q dq

quantity

price Offer curve p(q)( , ( ))q p q

maxq

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Calculus of variations

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Necessary optimality conditions (I)

Z(q,p)<0

q

p

Z(q,p)>0

qBqAxx

( the derivative of profit with respect to offer price p of segment (qA,qB) = 0 )

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Example: S(p)=p, D(p)=0, ~U[0,1]

q+p=1

Z(q,p)<0

Optimal offer (for c=0)

S(p) = supply curve from other generatorD(p) = demand function = random demand shock

Z(q,p)>0

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Finding a symmetric equilibrium[Holmberg, 2006]

• Suppose demand is D(p)+ where 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) > 0

then there exists a symmetric equilibrium.• If –f(x)2 – (1 - F(x))f’(x) < 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

0 2000 4000 6000 8000 100000

200

400

600

800

1000

Source: Holmberg (2006)

Uniform bid = price

PAB average price

PAB marginal bid

Demand shock

Price

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Example: S(p)=p, D(p)=0, ~U[0,1]

q+p=1

Z(q,p)<0

Optimal offer (for c=0)

S(p) = supply curve from other generatorD(p) = demand function = random demand shock

Z(q,p)>0

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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, 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

B undercuts A A undercuts B

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Determining pmax from K

Can now find pmax for any K, by solving G(pmax)=1.

Proposition: [A&P, 2007] Suppose demand is inelastic, random and less than market capacity. For every K>0 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

and (qmax,p) is offered with density

Each generator will offer at a price p no less than pmin>c, where

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ExampleSuppose c=1, pmax= 2, qmax= 1/2. Then pmin= 4/3, and K = 1/8

g(p) = 0.5(p-1)-2

Average price = 1 + (1/2) ln (3) > 1.5 (the UPA average)

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Expected consumer paymentSuppose c=1, pmax=2.

g(p) = 0.5(p-1)-2

If < 1/2, then clearing price = min {p1, p2}.If > 1/2, then clearing price = max {p1, p2}.

Generator 1 offers 1/2 at p1 with density g(p1).

Generator 2 offers 1/2 at p2 with density g(p2).

Demand ~ U[0,1].

E[Consumer payment] = (1/2) E[| < 1/2] E[min {p1, p2}] +(1/2) E[| > 1/2] E[max {p1, p2}]

= (1/4) + (7/32) ln (3) ( = 0.49 )

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WelfareSuppose c=1, pmax=2.

E[Profit] = 2*(1/8)=1/4.

g(p) = 0.5(p-1)-2

E[Consumer surplus] = A E[] – E[Consumer payment]

= (1/2)A – E[Consumer payment]

= (1/2)A – 0.49

E[Welfare] = (1/2)A – 0.24 > (1/2)A – 0.5 for UPA

< E[Profit] = 1/3 for UPA

> (1/2)A – 5/6 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