1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

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

9. 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)

10. 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.

11. 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.)

12. 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

13. 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)

14. EPOC Winter Workshop, September 7, 2007 Calculus of variations

15. 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 )

16. 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

17. 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.

18. 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

19. 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

20. 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).

21. 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

22. 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.

23. 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

24. 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)

25. 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 )

26. 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

27. 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

28. EPOC Winter Workshop, September 7, 2007 The End