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

However, our results also suggest some amount of caution when analyzing and predicting the behavior of smaller players in newly restructured markets. Smaller firms submit bids that differ substantially from the benchmarks we construct for optimal bidding. This finding is not inconsistent with rational economic behavior by these bidders, however. As argued in Section 5, "participation" in the balancing market may have nontrivial costs, and the behavioral pattern across firms appears to confirm this hypothesis.

Our third result is that small firms' deviations from optimal bidding is economically important. In Section 6, we calculated that small firms' bidding patterns led to the major portion of losses in productive efficiency. This suggests interesting new avenues for market design that explicitly take into account the strategic complexity, hence the participation costs, imposed by proposed market mechanisms. Such a consideration may favor dominant strategy implementable mechanisms, such as the Vickrey-Clarke-Groves (VCG) mechanism, over others. However, as pointed out by Milgrom (2004), the VCG mechanism suffers from serious pitfalls of its own. Nevertheless, we view theoretical research in this area to prove extremely fruitful for real-world applications.

Appendix A

* Proof of proposition 2. Given the additively separable form of the bidding strategies [S.sub.i](p, [QC.sub.i]) = [[alpha].sub.i](p) + [[beta].sub.i]([QC.sub.i]), use the market cleating condition (1) above to represent the event {[p.sup.c.sub.t] [less than or equal to] p | [QC.sub.i], [S.sub.i]}, that is, there is excess supply at p, conditional on firm i bidding [S.sub.i] at this price,

[summation over (j[not equal to]i)][[beta].sub.j]([QC.sub.j])-[epsilon][greater than or equal to] D(p)-[S.sub.i]- [summation over (j[not equal to]i)][[alpha].sub.j](p).

The left-hand side of this inequality can be labeled as a (bidder-specific) random variable, [[theta].sub.i], that does not depend on p, and the right-hand-side is a deterministic function of price. Let [[GAMMA].sub.i](.) denote the cdf of [[theta].sub.i] and [[gamma].sub.i] (.) denote the pdf (both conditional on the bidder's contract quantity, [QC.sub.i]). Given these,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Evaluating the derivatives gives [H.sub.p](p,[S.sub.i];Q[C.sub.i])/[H.sub.S](p,[S.sub.i];Q[C.sub.i]) = -[D'(p) - [[summation].sub.j[not equal to]i][alpha]'j(p)]. Now, observe that with the above restrictions, the residual demand function faced by firm i (for a given realization of the random variables {[epsilon], Q[C.sub.-i], i = 1, ..., N}), is given by

R[D.sub.i](p, [epsilon], Q[C.sub.-i]) = D(p) + [epsilon] - [summation over (j[not equal to]i)] [[alpha].sub.j](p) - [summation over (j[not equal to]i)] [[beta].sub.j](Q[C.sub.j]) (4)

with derivative

R[D'.sub.i](p) = D'(p)- [summation over (j[not equal to]i)][[alpha]'.sub.j] (p),

which yields the result in the proposition.

Note that in one other ease where we can collapse the multiple stochastic terms into a scalar random variable, we do not obtain the ex post optimality result. This is the case when [S.sub.i](p, Q[C.sub.i]) = [[alpha].sub.i](p) + [[beta].sub.i] pQ[C.sub.i] and [??](p) = D(p) + p[epsilon], that is, private information and aggregate uncertainty leads to pure rotations of the residual demand curve. In this case, the market clearing condition becomes [[theta].sub.i] = [summation over (j[not equal to]i)] [[beta].sub.j]Q[C.sub.j] - [epsilon] [greater than or equal to] 1/p [D(p) - [S.sub.i] - [summation over (j[not equal to]i)] [[alpha].sub.j](p)],but [H.sub.s](p,[S.sub.i])/[H.sub.p](p,[S.sub.i] [not equal to] 1/R[D'.sub.i](p,[epsilon]Q[C.sub.-i]). Q.E.D.

Appendix B

* Data description. Hourly bid schedules by each bidder, or qualified scheduling entity (QSE), are from ERCOT. QSEs occasionally bid for more than one firm. For example, in the South zone in 2001, the QSE named Reliant bid for both Reliant and the City of San Antonio. We match the bid functions to all units for which the QSE bids. So for all units owned by both Reliant and the City of San Antonio in the South in 2001, we match the bid function to the generation data. However, interpretation of the results becomes problematic when an observed bid function represents the bids by more than one firm. Because the results are some combination of two firms' behavior, we will not interpret results in such situations. We only interpret our results as firm-level behavior when at least 90% of all electricity generated by owners using that QSE can be attributed to a single owner. We make one exception to this 90% rule--TXU generation, which comprises 87 % of the generation for TXU the QSE in North 2002.

We measure the variable costs of output using data on each unit's fuel costs and the rate at which the unit converts the fuel to electricity. For each 15-minute interval, we have data from ERCOT on whether a generating unit is operating, its day-ahead scheduled generation, and its hourly available generating capacity. We measure the marginal cost of units that bum natural gas and coal. For each unit, we have data on the fuel efficiency (i.e., average heat rate). Each unit is assumed to have constant marginal cost up to its hourly operating capacity, an assumption that is common in the literature. The ERCOT system is largely separated from other electricity grids in the country so there are virtually no imports.

Daily gas spot prices measure the opportunity cost of fuel for natural gas units. We use prices at the Agua Dulce, Katy, Waha, and Carthage hubs for units in the South, Houston, West, and North zones, respectively. We assume a gas distribution charge of $0.10/mmBtu. Coal prices are monthly weighted average spot price of purchases of bituminous, subbituminous, and lignite in Texas, reported in Form FERC-423. Coal-fired plants in Texas are required to possess federal emission permits for each ton of S[O.sub.2] emissions. In order to measure average emission rates, we merge hourly net metered generation data from ERCOT with hourly emission data from EPA's CEMS to calculate each unit's average pounds of S[O.sub.2] emissions per net MW of electricity output. The emissions each hour are priced at the monthly average EPA permit price reported on the EPA website.

In order to deal with complications posed by transmission congestion, we restrict our sample to daily intervals 6:00-6:15 pm during which there is no interzonal transmission congestion during the 6-7 pm bidding hour. We believe intrazonal (or local) congestion is likely to be rare during these intervals.

We do not incorporate the possibility that some of the available capacity to INC in our data may be sold as reserves. However, the amount of operating reserves procured are small as a fraction of total demand.

We measure the marginal cost of INCing or DECing from the day-ahead schedule of output. We account for the fact that units cannot DEC down to zero output without incurring costs of startup and facing constraints on minimum downtime. It is unlikely that revenue from the balancing market would be sufficiently lucrative to compensate a unit for shutting down. Therefore, we assume that each operating unit cannot DEC to a level below 20% of its maximum generating capacity.

We thank seminar participants at various universities and conferences. We are grateful for assistance with data and institutional knowledge from Parviz Adib, Tony Grasso, and Danielle Jaussaud at the Public Utility Commission of Texas. The editor Igal Hendel, two anonymous referees, Severin Borenstein, Jim Bushnell, Stephen Holland, Marc Ivaldi, Julie Holland Mortimer, Shmuel Oren, Peter Reiss, Steve Wiggins, Joaquim Winter, and Frank Wolak provided very helpful comments. Hailing Zang, Anirban Sengupta, Jeremy Shapiro, and Joseph Wood provided capable research assistance. Hortacsu was a visitor at Harvard University and the Northwestern University Center for the Study of Industrial Organization during the course of this research, and gratefully acknowledges both institutions' hospitality and financial support. Puller was a visitor at the University of California Energy Institute's Center for the Study of Energy Markets, for whose hospitality he is grateful. Hortacsu acknowledges financial support from the National Science Foundation (SES-0449625) and the Alfred P. Sloan Foundation, and Puller acknowledges support from the Texas Advanced Research Program (010366-0202).

References

ATHEY, S.C. AND HAILE, P.A. "Nonparametric Approaches to Auctions." In J.J. Heckman and E. Leamer, eds., Handbook of Econometrics, Vol. 6. New York: Elsevier, forthcoming.

AUSUBEL, L.M. AND CRAMTON, P. "Demand Reduction and Inefficiency in Multi-Unit Auctions." Mimeo, Department of Economics, University of Maryland, 2002.

BAJARI, P. AND HORTACSU, A. "Are Structural Estimates of Auction Models Reasonable? Evidence from Experimental Data." Journal of Political Economy, Vol. 113 (2005), pp. 703-741.

BALDICK, R. AND NIU, H. "Lessons Learned: The Texas Experience." In J. Griffin and S.L. Puller, eds., Electricity Deregulation: Choices and Challenges. Chicago: University of Chicago Press, 2005.

--, GRANT, R., AND KHaN, E. "Theory and Application of Linear Supply Function Equilibrium in Electricity Markets." Journal of Regulatory Economics, Vol. 25 (2004), pp. 143-167.

BORENSTEIN, S., BUSHNELL, J.B., AND WOLAK, F.A. "Measuring Market Inefficiencies in California's Restructured Wholesale Electricity Market." American Economic Review, Vol. 92 (2002), pp. 1376-1405.