Understanding strategic bidding in multi-unit auctions: a case study of the Texas electricity spot market.

(13) Observe that we have not imposed independence on this joint distribution; contract quantities and demand noise can be correlated. However, as will be evident from the bidder's objective function below, this is not a common value environment, as other bidders' contract quantities do not enter into the bidder's ex post utility.

(14) The proof can be found in Hortacsu and Puller (2005).

(15) Although this is a comforting result from the perspective of assuming differentiable bid schedules, Kastl (2006a) finds that if bidding extra points is a costly activity, the constrained optimal bidding behavior may depart significantly from equation (2). We discuss some evidence regarding this possibility in Section 5.

(16) Some of these assumptions and possible estimation strategies based on such assumptions are discussed in the discriminatory (pay-as-bid) share auction context by Hortacsu (2002a).

(17) The primitives of this game are the set of firms that are participating, N, their cost curves [C.sub.it](q), i = 1, ..., N, the joint distribution of contract quantities, and the distribution of the uncertain demand component.

(18) The additive separability restriction appears to be crucial. Note that without this separability restriction, we cannot, in general, collapse the stochastic terms (from the perspective of bidder i) into a single scalar random variable. See Appendix A for further discussion of the case where private information leads to rotations in residual demand, as opposed to shifts.

(19) In particular, for certain specifications of marginal costs, a bidder's best-response to additively separable bidding strategies by her opponents may not be additively separable.

(20) This could be extended to show that given [QC.sub.i], one can compute the entire marginal cost curve rationalizing a supply curve [S.sub.i](p) observed in the data using a single realization of the residual demand curve.

(21) Notice that the lumping of the day-ahead quantity and the balancing bids does not affect the strategic nature of the game because the bidders are not provided any information about each others' actions until the market clears.

(22) One could be concerned that there is a small set of distinct contract quantities; however, the empirical distribution we recover using Proposition 1 suggests that our assumption of a continuous differentiable distribution of contract quantities is very reasonable.

(23) Let {([p.sub.1], [q.sub.1]), ..., ([p.sub.K], [q.sub.K]) represent the price and incremental quantities that form the residual demand curve seen in the data. The smoothed version of this function is RD(p) = [[sigma].sup.K.sub.k=1] [q.sub.k][kappa] (p - [p.sub.k]/h), where [kappa](.) is a kernel function. With this representation, the derivative of residual demand is RD'(p) = [[sigma].sup.K.sub.k=1] [q.sub.k] 1/h[kappa]' (p - [p.sub.k]/h). We used a normal kernel and the smoothing parameter, h = 10 MW throughout our analysis.

(24) As an illustration, Reliant's expost profitability is 79% under "smoothed" residual demand and 70% under the grid search.

(25) Some hydroelectric units may be able to respond to balancing calls; however, these units represent less than 1% of total capacity and are primarily owned by the Lower Colorado River Authority.

(26) We cannot use a measure such as the fraction of possible profits achieved because some firms tend to be short on their contract positions entering the balancing market, and we would have to make an assumption about the contract price. The measure we construct differences out the contract price and avoids this complication.

(27) Note that two firms, Extex Laporte and Air Liquide, earn lower profits under actual bidding than bidding to "avoid the market." Both firms, which are infrequent participants in the balancing market, have positive contract positions and relatively high-cost units available. Both bid so they are called to INC despite the fact that it would be more profitable to not participate and buy its contract position from the market at a price lower than marginal cost.

(28) Note that individual firm bid data are only available (to firms and analysts) with a six-month lag, so firms are unable to use Proposition 1 in real time to estimate rival firms' contract quantities and resolve some of their uncertainty. Some of rivals' uncertainty stems from variation in [QC.sub.it]. We find that balancing contract positions by a firm varies across time; for example, the standard deviation of [QC.sub.it] is 449, 161, and 7 for Reliant, Calpine, and Guadalupe, respectively.

(29) We discuss this sample further in Section 5.

(30) An annual expenditure of $1 million corresponds to $114 per hour for one year of operations (365 days, 24 hours), without correcting for the fact that the hour we are analyzing is a peak hour.

(31) We also used a similar approach to get a lower bound on the implied cost of using an additional bid point. To do this, we calculated TXU's NBR profits using 12 versus 13 bid points (we kept the 12 equally spaced points constant, and varied the 13th point to maximize TXU's profit). TXU's incremental profit gain from adding the 13th bid point was $1.59.

(32) It is difficult to make a direct comparison between "rounds" in the laboratory and "days" in electricity markets. However, if we are willing to equate rounds with days, bidding behavior appears to converge (in percentage profit terms) to theoretical predictions quicker in many laboratory experiments. See Kagel (1995) for examples from auction experiments.

(33) Because total demand is perfectly inelastic, prices higher or lower than competitive levels do not cause suboptimal levels of consumption. All inefficiencies are productive rather than allocative. For a more general discussion of the efficiency properties of uniform price multi-unit auctions, see Ausubei and Cramton (2002).

(34) Borenstein, Bushnell, and Wolak (2002) find the actual production costs to be 14% higher than competitive levels in the California market in 2000. Mansur (forthcoming) refines the methodology and finds that production costs in the PJM market exceed competitive costs by 3%-8%. Note that our analysis only calculates productive inefficiencies in the balancing market for units that have already started and submitted day-ahead schedules.

(35) Note that the "reverse" of this counterfactual, where we set nonstrategic bids equal to marginal cost and do not allow strategic bidders to respond, would be less realistic, because our results show that the strategic bidders do respond to changes in the residual demand they are facing. Counterfactuals involving the equilibrium response of strategic bidders are complicated by the fact that multi-unit auctions can have multiple equilibria.

Ali Hortacsu *

and

Steven L. Puller **

* University of Chicago and NBER; hortacsu@uchicago.edu.

** Texas A&M University; puller@econmail.tamu.edu. TABLE 1 Outcomes under Actual, Ex post Optimal, and Naive Best-Response Bidding

Percent Achieved

Relative to

XP Optimal Naive BR Firm (1) (2) Reliant 79% 80% Brownsville PUB 50% 50% City of Bryan 45% 45% Tenaska Gateway Partners 41% 41% TXU 39% 41% Calpine Corp 37% 38% Denton Municipal Electric 35% 35% Ingleside Cogeneration 31% 31% City of Austin 30% 31% Rio Nogales LP 28% 28% Lower Colorado River Auth 25% 25% City of San Antonio 23% 24% Gregory Power Partners 20% 20% Midlothian Energy 17% 17% Cogen Lyondell Inc 16% 16% Tractebel Power Inc 16% 16% Brazos Electric Power Coop 15% 15% Lamar Power Partners 15% 15% Mirant Wichita Falls 14% 14% BP Energy 14% 14% City of Garland 13% 13% Hays Energy 8% 8% West Texas Utilities 8% 8% Central Power and Light 8% 8% Guadalupe Power Partners 6% 6% Tenaska Frontier Partners 5% 5% South Texas Electric Coop 3% 3% Sweeney Cogeneration 2% 2% Brazos Valley Energy LP 0% 0% AES Deepwater 0% 0% Frontera General LP 0% 0% TGC 0% 0% South Houston Green Power 0% 0% Air Liquide America -8% -8% Extex Lanorte LP -81% -81%

Producer Surplus

($/hour)