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

However, we now show that the characterization of equilibrium strategies (2) is greatly simplified when the functional form of the supply function strategies, [S.sub.i](p, [QC.sub.i]), is restricted to a class of strategies that are additively separable in the private information possessed by bidders. In the subsequent discussion, we drop the time subscript.

Proposition 2. If supply function strategies [S.sub.i](p, [QC.sub.i]) are restricted to the class of strategies [S.sub.i] (p, [QC.sub.i]) = [[alpha].sub.i] (p) + [[beta].sub.i]([QC.sub.i]), the markup relation (2) is given by the familiar "inverse-elasticity" markup rule p - [C'.sub.i]([S.sub.i](p, [QC.sub.i])) = [S.sub.i](p, [QC.sub.i]) - [QC.sub.i]/-[RD'.sub.i] (p), where [RD'.sub.i](p) is the price derivative of the ex post realization of the residual demand curve faced by bidder i.

Proof. See Appendix A.

The intuition underlying this result is straightforward. The additive separability restriction implies that, in equilibrium, the residual demand function faced by bidders is also additively separable in its random component--that is, all uncertainty (from the perspective of bidder i) shifts the residual demand curve but does not rotate it. Given this, it is easily seen that (subject to the concavity of the profit function), the bid function [S.sub.i](p) provides a pointwise best-response to every possible realization of the residual demand curve. This also means that the class of additively separable equilibrium strategies, when they exist, are expost optimal, in the sense that seeing other bidders' supply functions would not change bidder i's choice of supply function. (18)

As an important caveat, however, note that the additive separability restriction is an a priori restriction on bidding strategies. It is not necessarily true that every specification of marginal cost functions, [C'.sub.i](q), and joint distribution of contract quantities, [QC.sub.i], will lead to equilibrium strategies of this form. (19) Hortacsu and Puller (2005) work through an example in which firms possess linear marginal cost curves (these can be asymmetric across firms), under which equilibrium strategies are analytically characterizable, and satisfy the additive separability restriction. Based on our inspection of actual marginal cost curves, linearity appears to be a reasonable approximation. Finally, we note that the additive separability restriction is testable and we provide several tests in Section 4.

This result has an immediate practical application.

Proposition 3. Suppose supply function strategies [S.sub.i](p, [QC.sub.i]) are restricted to the additively separable class of strategies, [S.sub.i](p, [QC.sub.i]) = [[alpha].sub.i](p) + [[beta].sub.i]([QC.sub.i]). Then, given data on the marginal cost function, one can compute the ex post optimal supply curve [S.sup.xpo.sub.i](p), which is the ex post best-response to the observed realization of the residual demand curve.

Observe that under the above restriction, a single realization of the residual demand curve, [RD.sub.i](p, [epsilon], [QC.sub.-i]), is enough to compute d/dp [RD.sub.i](p, [epsilon], [QC.sub.-i]) = [RD'.sub.i](p) for all realizations. Then, for a range of prices, p [epsilon] [[p.bar], [bar.p]], one can solve the equation for S, in terms of p and [QC.sub.i],

p - [C'.sub.i](S) = S - [QC.sub.i]/-[RD'.sub.i](p) (3)

to trace out [S.sup.xpo](p, [QC.sub.i]), which constitutes an ex post best response to all possible realizations of residual demand. (20)

Thus, although additive separability is a restrictive assumption, its imposition aids us greatly in solving for optimal supply schedules. Perhaps more importantly, ex post optimality aids us in dealing with economic unobservables. Observe once again that the ex post optimal supply schedule corresponding to a given marginal cost schedule is derived using a single, ex post observation of residual demand. The proposed empirical procedure does not pool data across auctions and thus avoids the problem of making strong assumptions as to how to model the role of unobservables.

4. Analysis of observed bid schedules.

* In our empirical application, we implement Proposition 3. We use data on bidders' marginal cost functions to calculate the ex post optimal supply curve, which is also an equilibrium bid function under the restrictions imposed in Proposition 2. Then we compare ex post optimal bid schedules to actual bids.

Now we explain how we map the model of Section 3 into the actual market. As described in Section 2, firms in ERCOT do not bid all electricity through the auction. Rather, the firms submit a fixed-quantity day-ahead schedule, and then compete in the balancing auction to increase or decrease supply from that day-ahead quantity. In order to implement Proposition 3 to model bidding in the balancing auction, we modify our marginal cost function and contract quantity. First, we account for the fact that the day-ahead schedule implies that some of the firm's generating capacity is already committed to produce the day-ahead quantity. Thus, we shift the total marginal cost function to the left by the day-ahead quantity. This balancing marginal cost function represents the marginal cost of increasing output and the marginal savings of reducing output relative to the day-ahead schedule. Second, the day-ahead quantity is used to satisfy some of the firm's forward contract positions. Any remaining contract position, the balancing contract quantity, is the [OC.sub.it] that affects bidding into the balancing market. (21) Therefore, in the context of our notation in Section 3, [S.sub.it](.) represents supply bids into the balancing market, [C.sub.it](q) represents the costs savings of increasing/reducing output relative to the day-ahead schedule, and [QC.sub.it] represents the quantity that the firm is long or short on its contracted sales after the day-ahead schedule and upon entering the balancing market.

Our empirical strategy is illustrated in Figure 1. In order to measure the contract position, [QC.sub.it], we use Proposition 1. [QC.sub.it] is measured as the quantity at which the actual (balancing) bid schedule, [S.sup.0.sub.i](p, [QC.sub.i]), intersects the (balancing) marginal cost function (point A in Figure 1). (22) Note that Proposition 1 allows us to identify the contract position under certain forms of suboptimal or nonequilibrium bidding. This identification approach is valid as long as the firm is sufficiently sophisticated to bid above (below) marginal cost when it is a net seller (buyer), even if it errs in the size of the markup. More formally, we can identify [QC.sub.it] in equation (2) even if firms have incorrectly calculated [H.sub.S](p,[S.sup.*.sub.it](p);[QC.sub.it]/ [H.sub.p](p,[S.sup.*.sub.it](p);[QC.sub.it]. Our conversations with various market participants lead us to believe that traders clearly recognize the rationale for marking up bids for quantities greater than the contract position (and vice versa), but traders have different heuristics for choosing the size of the markup.

Each firm's residual demand [RD.sub.i](p) is the realized total demand minus the bids by all rival firms. Suppose [RD.sub.1] in Figure 1 is the actual realization of residual demand for firm i. We calculate [RD'.sub.1](p) and find the ex post optimal (price, quantity) bid to be point B, where the marginal revenue curve corresponding to [RD.sub.1](p), [MR.sub.1] intersects the marginal cost curve MC. Also, we can calculate the ex post optimal bid under other possible realizations of uncertainty (that is, other realizations of total balancing demand, [epsilon], and rivals' private information, [QC.sub.-i]). Because each form of uncertainty acts to shift residual demand in a parallel fashion, we can consider another possible realization of residual demand as [RD.sub.2]--the realized residual demand shifted parallel to the left. Under this realization of uncertainty, the optimal bidpoint is given by point C, where [MR.sub.2] = [MC.sub.i](q). We repeat this operation by adding parallel shifts to the actual realization of the residual demand curve to find the set of ex post optimal points for various realizations of uncertainty. The ex post optimal bid function is traced out by the set of ex post optimal bid points to generate [S.sup.xpo.sub.i](p, [QC.sub.i]). Note that our assumption that the slope of residual demand is independent of uncertainty is necessary for the set of ex post optimal points to be on a monotonic bid function.

[FIGURE 1 OMITTED]

We should note that the residual demand function is a step function whose derivatives are either zero or infinity, which renders the literal evaluation of equation 3 impossible. We follow two methods to address this: first, we follow Wolak (2003a, 2003b) to obtain a "smoothed" version of the residual demand function to calculate the marginal revenue curve. (23) Second, we perform a grid search on the "unsmoothed" residual demand function and find the ex post profit-maximizing point for each parallel shift in residual demand. In practice, we found that the ex post profitability benchmark that is generated by the grid search departs negligibly from the ex post profitability benchmark generated by the "smoothed" residual demand. (24)