Uniform-price auctions versus pay-as-bid auctions

1. 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)

2. 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. price T1(q) quantity demand Uniform price auction (single node) price T2(q) p quantity price combined offer stack p quantity

4. 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

5. 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

6. 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

7. Define: y(q,p) = Pr [D(p)+ e – 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)] price p q quantity The market distribution function[Anderson & P, 2002] • S(p) = supply curve from • other generators • D(p) = demand function • = random demand shock F = cdf of random shock

8. Symmetric SFE with D(p)=0[Rudkevich et al, 1998, Anderson & P, 2002]

9. Example: n generators, e~U[0,1], pmax=2 n=5 n=4 n=3 n=2 p Assume cq = q, qmax=(1/n)

10. Example: 2 generators, e~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.

11. 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. price T1(q) quantity demand Pay-as-bid price auction (single node) price T2(q) p quantity price combined offer stack p quantity

13. Offer curve p(q) price quantity 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

14. Calculus of variations

15. Necessary optimality conditions (I) Z(q,p)<0 p Z(q,p)>0 ( the derivative of profit with respect to offer price p of segment (qA,qB) = 0 ) q x x qB qA

16. Example: S(p)=p, D(p)=0, e~U[0,1] • S(p) = supply curve from • other generator • D(p) = demand function • = random demand shock q+p=1 Z(q,p)<0 Optimal offer (for c=0) Z(q,p)>0

17. Finding a symmetric equilibrium[Holmberg, 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) > 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. Prices: PAB vs uniform Price Uniform bid = price PAB marginal bid PAB average price Demand shock Source: Holmberg (2006)

19. Example: S(p)=p, D(p)=0, e~U[0,1] • S(p) = supply curve from • other generator • D(p) = demand function • = random demand shock q+p=1 Z(q,p)<0 Optimal offer (for c=0) Z(q,p)>0

20. 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. B undercuts A A undercuts B Example • Two players A and B each with capacityqmax. • 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

22. 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.

23. Example (cont.) Suppose c(q)=cq Each generator will offer at a price p no less than pmin>c, where and (qmax,p) is offered with density

24. g(p) = 0.5(p-1)-2 Example Suppose c=1, pmax= 2, qmax= 1/2. Then pmin= 4/3, and K = 1/8 Average price = 1 + (1/2) ln (3) > 1.5 (the UPA average)

25. g(p) = 0.5(p-1)-2 Expected consumer payment Suppose c=1, pmax=2. Generator 1 offers 1/2 at p1 with density g(p1). Generator 2 offers 1/2 at p2 with density g(p2). Demand e ~ U[0,1]. If e < 1/2, thenclearing price =min {p1, p2}. If e > 1/2, thenclearing price =max{p1, p2}. E[Consumer payment] = (1/2) E[e|e < 1/2] E[min {p1, p2}] +(1/2) E[e|e > 1/2] E[max {p1, p2}] = (1/4) + (7/32) ln (3) ( = 0.49 )

26. g(p) = 0.5(p-1)-2 Welfare Suppose c=1, pmax=2. E[Profit] = 2*(1/8)=1/4. < E[Profit] = 1/3 for UPA E[Consumer surplus] = A E[e] – E[Consumer payment] = (1/2)A– E[Consumer payment] = (1/2)A – 0.49 > (1/2)A – 5/6 for UPA E[Welfare] = (1/2)A – 0.24 > (1/2)A – 0.5 for UPA

27. 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. The End