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EPOC Winter Workshop, September 7, 2007 Andy Philpott The 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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EPOC Winter Workshop, September 7, 2007 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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EPOC Winter Workshop, September 7, 2007 Uniform price auction (single node) price quantity price quantity combined offer stack demand p price quantity T 1 (q) T 2 (q) p

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EPOC Winter Workshop, September 7, 2007 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) Residual demand curve = D(p) – S(p) p q Optimal dispatch point to maximize profit

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EPOC Winter Workshop, September 7, 2007 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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EPOC Winter Workshop, September 7, 2007 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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EPOC Winter Workshop, September 7, 2007 The market distribution function [Anderson & P, 2002] p q quantity price 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 generators D(p) = demand function = random demand shock F = cdf of random shock

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EPOC Winter Workshop, September 7, 2007 Symmetric SFE with D(p)=0 [Rudkevich et al, 1998, Anderson & P, 2002]

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EPOC Winter Workshop, September 7, 2007 Example: n generators, ~U[0,1], p max =2 n=2n=3n=4n=5 p Assume cq = q, q max =(1/n)

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EPOC Winter Workshop, September 7, 2007 Example: 2 generators, ~U[0,1], p max =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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EPOC Winter Workshop, September 7, 2007 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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EPOC Winter Workshop, September 7, 2007 Pay-as-bid price auction (single node) price quantity price quantity combined offer stack demand p price quantity T 1 (q) T 2 (q) p

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EPOC Winter Workshop, September 7, 2007 Modelling a pay-as-bid auction Probability that the quantity between q and q + q is dispatched is Increase in profit if the quantity between q and q + q is dispatched is Expected profit from offer curve is quantity price Offer curve p(q)

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EPOC Winter Workshop, September 7, 2007 Calculus of variations

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EPOC Winter Workshop, September 7, 2007 Necessary optimality conditions (I) Z(q,p)<0 q p Z(q,p)>0 qBqB qAqA x x ( the derivative of profit with respect to offer price p of segment (q A,q B ) = 0 )

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EPOC Winter Workshop, September 7, 2007 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 generator D(p) = demand function = random demand shock Z(q,p)>0

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EPOC Winter Workshop, September 7, 2007 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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EPOC Winter Workshop, September 7, 2007 Prices: PAB vs uniform Source: Holmberg (2006) Uniform bid = price PAB average price PAB marginal bid Demand shock Price

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EPOC Winter Workshop, September 7, 2007 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 generator D(p) = demand function = random demand shock Z(q,p)>0

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EPOC Winter Workshop, September 7, 2007 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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EPOC Winter Workshop, September 7, 2007 Example Two players A and B each with capacity q max. Regulator sets a price cap of p max. D(p)=0, can exceed q max but not 2q max. Suppose player B offers q max at a fixed price p with distribution G(p). Market distribution function for A is Suppose player A offers q max 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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EPOC Winter Workshop, September 7, 2007 Determining p max from K Can now find p max for any K, by solving G ( p max )=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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EPOC Winter Workshop, September 7, 2007 Example (cont.) Suppose c ( q )= cq and (q max,p) is offered with density Each generator will offer at a price p no less than p min >c, where

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EPOC Winter Workshop, September 7, 2007 Example Suppose c=1, p max = 2, q max = 1/2. Then p min = 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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EPOC Winter Workshop, September 7, 2007 Expected consumer payment Suppose c=1, p max =2. g(p) = 0.5(p-1) -2 If < 1/2, then clearing price = min {p 1, p 2 }. If > 1/2, then clearing price = max {p 1, p 2 }. Generator 1 offers 1/2 at p 1 with density g(p 1 ). Generator 2 offers 1/2 at p 2 with density g(p 2 ). Demand ~ U[0,1]. E[Consumer payment] = (1/2) E[ | < 1/2] E[min {p 1, p 2 }] +(1/2) E[ | > 1/2] E[max {p 1, p 2 }] = (1/4) + (7/32) ln (3) ( = 0.49 )

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EPOC Winter Workshop, September 7, 2007 Welfare Suppose c=1, p max =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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EPOC Winter Workshop, September 7, 2007 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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EPOC Winter Workshop, September 7, 2007 The End

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