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

* We now provide a model of strategic behavior in this market that incorporates the uncertainty faced by each firm when making their bidding decisions. To do this, we follow the uniform price share auction setup of Wilson (1979). For ease of exposition, assume that firms sign forward contracts and then bid all electricity through the auction, that is, assume that there is no day-ahead scheduling. This allows us to simplify notation. After deriving the model, we explain how we map this model into a market with a day-ahead schedule followed by a balancing auction.

We index the costs of generation (at time t) of the N firms in this market by {[C.sub.it](q), i = 1,..., N}. We take total demand [[??].sub.t](P) = [D.sub.t](p) [[epsilon].sub.t] to be the sum of a deterministic price-elastic component and a stochastic constant term. (1) Prior to the auction, each firm has signed contracts to deliver certain quantities of power each hour, given by [QC.sub.it], at a fixed price [PC.sub.it]. We assume that these contracts have been written a long enough time ago that they are taken as "sunk" decisions from the perspective of a bidder making real-time decisions on the balancing market. (12)

In each time period t, each firm simultaneously submits a supply schedule, [S.sub.it](p, [QC.sub.it]), which we restrict to be continuously differentiable, with bounded derivatives. Given the supply schedules of each firm, the auctioneer computes the market clearing price, [p.sup.c.sub.t], which satisfies the market clearing condition

[N.summation over (i=1)] [S.sub.it]([p.sup.c.sub.t], [QC.sub.it]) = [[??].sub.t]([p.sub.c.sub.t]), (1)

Each firm gets paid [S.sub.it]([p.sup.c.sub.t], [QC.sub.it])[p.sup.c.sub.t] due to the uniform pricing rule. Hence, firm i's ex post profit, upon the realization of market clearing price [p.sup.c.sub.t], is

[[pi].sub.it] = [S.sub.it] ([p.sup.c.sub.t], [QC.sub.it])[p.sup.c.sub.t] - [C.sub.it]([S.sub.it]([p.sup.c.sub.t])) - ([p.sup.c.sub.t] - [PC.sub.it])[QC.sub.it].

The firm's payoff from its contract position is -([p.sup.c.sub.t] - [PC.sub.it])[QC.sub.it] because it has to refund its customers any differential between the contract and market prices for the contracted sales. Wolak (2003a) has shown this is identical to a contract for differences.

The most important source of uncertainty in the profit equation above is [p.sup.c.sub.t], the market clearing price at time t. In a strategic equilibrium, the uncertainty in [p.sup.c.sub.t], from the perspective of firm i, is due to two factors: the uncertainty in market demand, [[??].sub.t], and the unobserved components of i's competitors' profit maximization problems, that is, the contract positions and prices of rival firms, {([QC.sub.jt], [PC.sub.jt]), j [member of] -i }.

Following the discussion in Section 2, each firm bidding into the market is assumed to know its rivals' total cost functions. This knowledge would be sufficient to calculate equilibrium bids by rival firms, except the firm does not know its rivals' contract positions. Thus, following Wilson (1979), we look for a Bayesian-Nash equilibrium characterization of the game, in which firms' strategies are of the form [S.sub.it](p, [QC.sub.it]).

To characterize a Bayesian-Nash equilibrium, we define a probability measure over the realizations of the market clearing price, from the perspective of firm i, conditional on firm i's private information contract quantity, [QC.sub.it], and the fact that firm i submits the supply schedule, [[??].sub.it](p), while his competitors are playing their equilibrium bidding strategies, {[S.sub.jt](p, [QC.sub.jt]), j [member of] - i},

[H.sub.it](p, [[??].sub.it](p); [QC.sub.it]) [equivalent to] Pr ([p.sup.c.sub.t] [less than or equal to] p | [QC.sub.it], [[??].sub.it](p)),

Utilizing the definition of the market clearing price in equation (1), we can rewrite this probability distribution as,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the first line follows from the fact that the event "[p.sup.c.sub.t] [less than or equal to] p" is equivalent to there being excess supply at price p. In the second line, 1 {.} is the indicator function for the enclosed event, and F([QC.sub.-it], [[epsilon].sub.t] | [QC.sub.it]) denotes the joint distribution of the vector of contract quantities, {[QC.sub.jt, j [member of] - i}, and the demand noise, [[epsilon].sub.t], conditional on the contract position of bidder i, [QC.sub.it]. (13)

We now rewrite the bidder's expected utility maximization problem, where U([pi]) is the utility enjoyed by the bidder from making [pi] dollars of profit. This general utility formulation allows for both risk-averse and risk-neutral firms.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the expectation is taken over all possible realizations of the market clearing price, weighted by the probability density, d [H.sub.it](p, [[??].sub.it](p); [QC.sub.it]). The Euler-Lagrange necessary condition for the (pointwise) optimality of the supply schedule [S.sup.*.sub.it] (p) is given by (14)

p - [C'.sub.it]([S.sup.*.sub.it](p)) = ([S.sup.*.sub.it](p) - [QC.sub.it] [H.sub.S](p, [S.sup.*.sub.it](p); [QC.sub.it)/[H.sub.p](p, [S.sup.*.sub.it](p); [QC.sub.it) (2)

where

[H.sub.p](p, [S.sup.*.sub.it](p); [QC.sub.it) = [partial derivative]/[partial derivative]p Pr ([p.sup.c.sub.t] [less than or equal to] p | [QC.sub.it], [S.sup.*.sub.it](p))

[H.sub.S](p, [S.sup.*.sub.it](p); [QC.sub.it) = [partial derivative]/[partial derivative]S Pr ([p.sup.c.sub.t] [less than or equal to] p | [QC.sub.it], [S.sup.*.sub.it](p)),

[H.sub.p](p, [S.sup.*.sub.it] (p);[QC.sub.it]) is the "density" of market clearing price when firm i bids [S.sup.*.sub.it] (p). [H.sub.S](p, [S.sup.*.sub.it] (p);[QC.sub.it]) can be interpreted as the "shift" in the probability distribution of the market clearing price, due to a change in [S.sup.*.sub.it] (p); that is, this is the term that captures the "market power" of firm i. Notice that this derivative is always nonnegative, because an increase in supply weakly lowers the market clearing price, which weakly increases the probability that the market clearing price is lower than a given price p.

The above derivation relied on the assumption that supply schedules are continuously differentiable. In reality, however, firms are allowed to bid 40 price-quantity points, restricting supply schedules to (nondifferentiable) step functions. A characterization of bidding behavior in uniform price auctions when the strategy space is restricted to a discrete number of steps is pursued in McAdams (2006) and Kastl (2006a, 2006b). Kastl (2006b) obtains the result that as the number of steps available to bidders grows without bound, necessary conditions characterizing bidding behavior in the discrete strategy game converge to the necessary conditions for the game with differentiable supply schedules. (15)

First-order condition (2) can be seen as a "markup" expression, where the markup in price above the marginal cost depends on how much market power firm i can exercise by shifting the distribution of the market clearing price through its own supply function [S.sup.*.sub.it](p). As an intuitive consequence, observe that if [H.sub.S] [right arrow] 0, that is, there is no market power, price equals marginal cost. Also, note that where [S.sup.*.sub.it] (p) < [QC.sub.it], the firm is a net buyer and bids below marginal cost.

Finally, observe that where [S.sup.*.sub.it] (p) - [QC.sub.it] = 0, p = [C'.sub.it]([S.sup.*.sub.it] (p)). This allows us to infer the unobserved contract positions of the bidders.

Proposition 1. If [C'.sub.it]([S.sup.*.sub.it] (p)) is observed, one can calculate the contract position [QC.sub.it], by finding the quantity where the supply function of the firm intersects its marginal cost function.

The empirical implementation of (2) requires the estimation of [H.sub.it](p, [S.sup.*.sub.it] (p);[QC.sub.it]) (and its partial derivatives), for each bidder i, in every period t. [H.sub.it](p, [S.sup.*.sub.it] (p);[QC.sub.it]) is the equilibrium belief of bidder i regarding the distribution of the market clearing price in auction t, conditional on his bidding strategy, [S.sup.*.sub.it] (p). Obtaining econometric estimates of this probability distribution might require strong parametric assumptions regarding the specification of bidder-specific beliefs, and especially the role played by economic unobservables entering into bidders' beliefs across different time periods. (16) The latter concern is potentially the most troublesome: if bidders condition their beliefs on factors unobservable to the economist, estimating [H.sub.it](p, [S.sup.*.sub.it] (p);[QC.sub.it]) by pooling data from a series of auctions without taking these unobservable factors into account may lead to incorrect estimates.

Observe also that the set of first-order conditions (2) for each firm, when written as a system of equations, characterizes equilibrium strategies, [S.sub.it] (p, [QC.sub.it]), for given primitives of the game. (17) The computation of equilibrium strategies is not a trivial task, however, because [H.sub.it](p, [S.sup.*.sub.it](p);[QC.sub.it]) is determined endogenously through the market clearing condition (1) and depends on the joint distribution of contract positions and the distribution of demand noise.