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

Several characteristics of bid schedules appear to drive the foregone ex post profits. We examined many sets of actual and ex post optimal bid functions similar to Figure 2. The key characteristic of underperforming bidders is that the actual bid functions tend to be "too steep" relative to optimal bids. By bidding too high during INC hours and too low during DEC hours, firms sell less than under ex post optimal bidding and forego individually profitable sales. As shown in columns 6 and 7 of Table 1, firms sell less than the expost optimal quantity on average. Interestingly, the cause of foregone profits is not that firms are bidding more competitively than individually optimal, but rather that bid prices are too high.

It is important to keep in mind that these profitability measures do not necessarily represent generators' overall performance in trading electricity. Many generators may focus their strategic efforts in the bilateral market where the vast majority of transactions occur. If firms focus strategic efforts on the bilateral markets, the overall performance is substantially higher. The upper bound on total profitability, shown in column 8 of Table 1, ranges from 41% to 98% with a mean of 80%. These metrics are more likely to reflect the firms' overall performance.

Additional tests of profit maximization. It is important to note that our calculations of foregone profits rely on restrictions that we place on the economic environment. In particular, we assume that bidders' equilibrium perceptions of the uncertainty results in "parallel shifts" rather than "pivots" in residual demand. If this assumption is violated, it is possible that the set of ex post optimal price-quantity points is a "cloud" of points that cannot be connected by an increasing supply function. If this were the case, a profit-maximizing firm's solution to the (ex ante) expected profit maximization problem may not be the set of expost optimal price-quantity points. That is, testing whether firms maximize expost profits as we did would not be informative about whether firms maximize ex ante profits. Because ex ante profitability is a more accurate measure of bidder performance, we employ several testing strategies that do not impose restrictions on the relationship between uncertainty and residual demand. All of these tests suggest that the "additively separable in private information" restriction does not drive our findings of foregone profits.

Best-response to previous rival bids. First, we test whether firms could significantly increase profits by using a simple bidding rule that utilizes only information available to traders at the time of bidding. The bidding rule we employ is a naive best response to recent rival bidding.

As discussed in Section 2, traders have access to the aggregate bid function with a two day lag. Using their own bids from the past, traders can calculate the aggregate bids by rivals in the recent past. (28) To construct the naive best-response of firm i for upcoming day t, we

(i) use aggregate bids and own bids for day t-3 and calculate aggregate rival bids on day t3;

(ii) assume rivals use the t-3 bid schedule on upcoming day t,

(iii) calculate ex post optimal bid function for various realizations of day t total balancing demand.

This algorithm uses only information available to firms when bids are submitted. We view this bidding rule as fairly unsophisticated--it uses only a small fraction of the information available to traders and it would be simple to program as an add-in to the trading interface used by the generators. We calculate producer surplus under "naive best-response" bidding and compare to producer surplus under actual bidding. If uncertainty causes the ex ante expected profit-maximizing bid to differ from expost optimal bids, the actual profitability should be much closer to the naive best-response benchmark.

Results are shown in columns 2 and 4 of Table 1. Across all firms, naive best response profits are substantially higher than actual profits and very close to ex post optimal profits. The performance of actual bidding is significantly below the naive best-response benchmark for all firms except Reliant. In fact, most bidders' measure of Percent Achieved rises very little when compared against the naive best-response benchmark. Naive best-response profits are very close to expost optimal profits--the former average $1193 and the latter average $1204. This suggests that our findings of foregone profits do not arise from the additive separability restriction. Moreover, this is consistent with the additive separability restriction. The results in columns 4 and 5 reveal that a firm's conditioning on [RD.sub.t-3] instead of [RD.sub.t] leads to negligible profit losses. Thus, under a profit metric, the shifts in RD are purely parallel.

We also test the restriction of how uncertainty affects residual demand using a Generalized Method of Moments (GMM)-based approach inspired by Wolak (2003a). we allow for a much more general relationship between uncertainty and residual demand, and derive first-order conditions for the choice of each bid point. These conditions yield moment conditions that are zero in expectation under the null of expected profit maximization. The results indicate that, except for Reliant, firms in this market violate the first-order optimality conditions that need to hold for expected profit maximization. Details are in Hortacsu and Puller (2005).

Testing additive separability. Because we can estimate bidders' (private information) contract positions, we also investigate whether the additive separability restriction holds in the data. According to this restriction, assuming that the firm's and its competitors' marginal costs are unaffected, exogenous shifts in a bidder's contract position (QC) should affect the intercept of the bid function, but not the slope.

To operationalize this test, we first fit a linear function to each day's bid function and calculate a slope term. The linear specification yielded excellent fit especially in the price range between $0 and $30. We also used a linear specification to calculate the slopes of bidders' daily marginal cost functions (intercepts change very little during the sample period).

Optimal bids can depend on (the parameters of) competitors' marginal cost schedules in complex ways. Unfortunately, incorporating every competitor's marginal cost parameters in a regression specification would exhaust degrees of freedom rapidly. Therefore, instead of using competitors' marginal cost slopes and intercepts as control variables, we use the slope of the (realized) residual demand curve seen by each bidder (we do not use the intercept, as this is the uncertain part of residual demand not seen by the bidder). Note that, under additive separability, this residual demand derivative can be seen as a "sufficient statistic" encoding changes in competitors' costs.

The first regression in Table 2 reports the panel regression of the bid function slope on the estimated contract quantity, controlling for residual demand and (own) marginal cost variation, along with a linear time trend and bidder fixed effects. Although the coefficient on contract quantity is statistically significant at the 5% level and positive, the economic significance of this correlation is not large, as the amount of variation in bid function slope that is explained by changes in contract quantity is small. Specifically, the average standard deviation of daily contract quantities (across firms) is 338, and multiplying this by the coefficient estimate 0.001 leads to only 4.2% of the average bid function slope in the sample (which is 8.0).

In the second column of Table 2, we repeat the regression in the first column, but also control for auction fixed effects, which can be interpreted as factors (common across bidders) that bidders take into account when formulating their bids, but not explicitly taken into account by our simplified econometric specification. This specification yields a weaker (both economically and statistically) relationship between contract quantities and bid function slope. Note that accounting for the auction fixed effect leads to a dramatic increase in the coefficient on residual demand slope, which, in theory, should be an important determinant of bid functions. This latter fact is suggestive of an omitted variables problem in the first regression, which is ameliorated by the fixed effect specification.

In Table 3, we report the results of the first regression in Table 2 estimated at the bidder level. We display the results for those bidders who submit their own bid schedules, and do not use an intermediary "qualified scheduling entity" (QSE). (29) Reliant, the most successful bidder (in terms of ex post and ex ante profit maximization), appears to conform to the additive separability restriction--changes in contract quantity do not have a statistically significant effect on the slope of Reliant's bid function. The next most successful bidder in terms of ex post profit maximization, City of Bryan, also satisfies this restriction. Although TXU violated additive separability during its first month of bidding, we fail to reject additive separability in its subsequent bidding patterns. In contrast, additive separability appears to break down for Calpine and City of Austin; however, the amount of variation in bid function slope explainable by shifts in contract position is less than 20% in both of these cases. The violation of additive separability is strongest for LCRA; variation in contract quantities can explain 50% of the variation in bid function slope.