...the properties of the derived SFE are valid for real-time markets in general. Firms' capacity constraints bind at different prices (i). Still, firms with non-binding capacity constraints have smooth residual demand (ii). Approximating an asymmetric real-time market with symmetric one, tends to overestimate mark-ups for small positive imbalances and underestimate mark-ups for large positive imbalances (iii).

1. INTRODUCTION

Electric energy is expensive to store compared to its production cost. As a result, stored energy is negligible in most power systems. Thus power consumption and production have to be roughly in balance at all times. Most of the electric power is sold long before delivery. However, because neither consumption nor production is fully predictable, adjustments need to be made in real-time to maintain balance. The real-time market, also called a balancing market, is an important component in this process. The market functions as an auction--often a uniform-price auction--in which power producers can offer additional power relative to their contracted position. Both increments, balancing up/up-regulation, and decrements, balancing down/down-regulation, can be offered. The paper focuses on balancing-up bids, but corresponding results can be readily derived for balancing-down bids. A bid consists of a non-decreasing supply function and is submitted before the start of the delivery period. The demand in the real-time market is given by the system imbalance, and it is not known when bids are submitted, as unexpected demand shocks as well as generator and transmission outages may occur during the delivery period. The period is typically an hour, as in California, Pennsylvania-New Jersey-Maryland (PJM), and the Nordic countries, or half an hour as in Britain.

The Supply Function Equilibrium (SFE) with uncertain demand was introduced by Klemperer and Meyer (1989). Later on Green and Newbery (1992) and Bolle (1992) observed that the set-up of the model is similar to the organization of most electricity markets, and SFE is now an established model of bidding behavior in electricity auctions. In the non-cooperative Nash equilibrium of the static game, each producer commits to the bid that maximizes his expected profit given the bids of competitors. Klemperer and Meyer (1989) show that all smooth supply function equilibria are characterized by a differential equation, which in this paper is called the KM first-order condition.

The assumption of symmetric producers is convenient as it allows straightforward calculation of SFE for general cost functions, as shown by Rudkevich et al. (1998), Anderson and Philpott (2002), and Holmberg (2004). However, firms in electric power markets are typically asymmetric. In order to assess efficient antitrust policy and merger control, models that can analyze asymmetric markets are important. To get analytic results, the models of asymmetric markets tend to be greatly simplified, but are anyway useful as they improve the qualitative understanding of bidding in electricity auctions.

The case of linear SFE for asymmetric firms with linear marginal costs was analyzed by Green (1996). Baldick et al. (2004) extended this concept to piece-wise linear SFE that can be used to analyze asymmetric firms with asymmetric intercepts. However, both linear and piece-wise linear SFE are problematic in the presence of capacity constraints, an important feature of electricity markets.

For asymmetric capacities, SFE have only been analytically derived for firms with identical constant marginal costs. For this case, Newbery (1991) and Genc and Reynolds (2004) derive piece-wise symmetric and symmetric SFE (1), respectively. This paper extends their asymmetric duopoly models to multiple asymmetric producers. For two firms the unique equilibrium is similar to the piece-wise symmetric equilibrium presented by Newbery, but as in the analysis by Genc and Reynolds (2004), Newbery's linear demand has been replaced by perfectly inelastic demand and a price cap. This paper makes contributions beyond the work by Newbery (1991) in that it is thoroughly proven that the outlined equilibrium exists and that there are no other possible equilibria. Uniqueness occurs when maximum demand is so high that the up-regulation capacities bind for all firms, except possibly the largest. Similarly, minimum demand should be so low that the down-regulation capacities bind for all firms, except possibly the largest. As argued by Holmberg (2004), these assumptions are in theory satisfied by all real-time markets. Still it is too early to rule out other "equilibria". In practice, the probability of getting power shortages during a delivery period is very small, of the order [10.sup.-4]-[10.sup.-6], and even lower during off-peak periods. This may result in a very long learning period before the market finds the unique SFE. In addition, the risk may be so small that it is not considered at all by the bidders, i.e. they are not perfectly rational. Further, a regulator's implicit threat of reregulation might restrain the bids more than the price cap. These are all empirical questions, which are not further addressed here.

The unique SFE has the following properties for balancing-up bids. The supply functions of any two producers are identical and satisfy the KM first-order condition until the capacity constraint of the smaller firm binds. To compensate the smaller firm's kink at this price, all firms with non-binding capacity constraints have kinks in their supply functions. This ensures that firms with non-binding capacity constraints face a smooth residual demand. The capacity constraint of the second largest firm binds when the price reaches the price cap. Thereafter, the largest firm offers its remaining up-regulation capacity at a price equal to the price cap.

The piece-wise symmetric nature of the equilibrium is a specific result of piece-wise symmetric costs, and this result is only applicable to the few electricity markets that are dominated by one production technology. With this exception, the derived properties of the equilibrium: that firm's capacities bind at different prices, that the remaining firms (with non-binding capacity constraints) become more elastic at these prices, and that one firm sells its remaining capacity as a monopolist, is believed to be a general result if maximum demand is sufficiently high and if minimum demand is sufficiently low, and if an equilibrium exists under those circumstances. Further, it is likely that one can generally rule out perfectly elastic supply segments (except at the price cap) (2). Moreover, it is unlikely that there are SFE where firms with non-binding capacity constraints have perfectly inelastic supply segments also when marginal costs are non-constant. In Holmberg (2005) the conjectured properties are used to numerically calculate SFE for markets where firms have both asymmetric costs and asymmetric capacities. This is a task that has been problematic in the past, see e.g. Baldick and Hogan (2002). Lastly, it is believed that when comparing an asymmetric market with a symmetric market that has the same average competition, e.g. with the same Herfindahl-Hirschman index, the asymmetric market is more competitive for small demand outcomes--when most asymmetric firms have non-binding capacity constraints--and less competitive for large demand outcomes, when most asymmetric firms have binding capacity constraints.

The structure of the paper is as follows. Notation and assumptions are introduced in Section 2 and the unique SFE is derived in Section 3. In Section 4, the unique SFE is numerically illustrated for the case of three asymmetric producers. In Section 5, the unique SFE is calculated for 153 firms in the Norwegian real-time market and Section 6 concludes.

2. NOTATION AND ASSUMPTIONS

The analysis in this paper is similar to that in Holmberg (2004). However, symmetric producers with strictly convex cost functions are replaced by producers with identical constant marginal costs c and asymmetric capacities. In the derivation of the equilibrium, the analysis is confined to real-time and balancing markets with positive imbalances but corresponding results can be readily derived for negative imbalances as illustrated in Section 5. The contract/forward positions are assumed to be exogenously given.

There are N[greater than or equal to]2 producers, all of whom have different production capacities. The bid of a producer is assumed to be valid for one delivery period only. In the original work by Klemperer and Meyer (1989), analysis was confined to twice continuously differentiable supply functions. Here, as in Holmberg (2004), the set of admissible bids is extended to allow for kinks as well as perfectly inelastic and perfectly elastic segments. Perfectly elastic segments imply that there are several possible supplies at some prices. Thus in the general case the bid is not a supply function, but a supply correspondence. However, the correspondence can still be represented by a piece-wise smooth supply function, which is assumed to be left continuous in this paper. (3) Let [S.sub.i](p) be the supply function of an arbitrary producer i. It is then understood that this firm is prepared to sell any supply in the range [[S.sub.i](p), [S.sub.i](p+)] at the price p. Aggregate supply of his competitors is denoted [S.sub.-i](p) and total aggregate supply is denoted S(p). All supply functions are assumed to be non-decreasing, as this is required by most electricity auctions.

Let [[??].sub.i] be the up-regulation capacity of producer i. It is given by the firm's production capacity that can be regulated on short notice and that has not been contracted in advance. Without loss of generality, firms can be ordered according to their up-regulation capacity, i.e. [[??].sub.1] < [[??].sub.2] < ... < [[??].sub.N]. Total up-regulation capacity is designated by [??], i.e. [??] = [N.summation over (i=1) [[??].sub.i]. Denote consumers additional demand (relative to their contracts) by [epsilon] and the additional demand's probability density function by f([epsilon]). The density function is continuously differentiable and has a convex support set that...

NOTE: All illustrations and photos have been removed from this article.



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