Journal of Regulatory Economics; 22:2 161±183, 2002 # 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.

Multi-Dimensional Procurement Auctions for Power Reserves: Robust Incentive-Compatible Scoring and Settlement Rules* HUNG-PO CHAO Stanford University

Electric Power Research Institute PO Box 10412, Palo Alto, CA 94303 USA E-mail: [email protected]

ROBERT WILSON

Stanford University

Stanford, CA 94305-5015 USA E-mail: [email protected]

Abstract System operators in the electricity industry purchase reserve capacity in a procurement auction in which suppliers offer two-part bids, one part for making capacity available and another part for supplying incremental energy when called. Key ingredients of an ef®cient auction design are the scoring rule for comparing bids, and the settlement rule for paying accepted bids. We use the principle of incentive compatibility to establish that very simple rules suf®ce. In particular, the scoring rule uses only the capacity part of the bid, and energy supplies are paid the spot price. This is the design adopted in California.

1. Introduction Procurement auctions in which bidders submit two-part bids are used in situations where each winning supplier provides two ingredients or two dimensions of service. Typically these auctions rely on a scoring rule to compare bids and a settlement rule to determine payments. Two-part bids are common in the electricity industry due to the distinction between an initial commitment of capacity availability and a subsequent determination of the amount of energy supplied from that capacity, contingent on later events. Some notorious procurement auctions of this kind were California's 1993 series of * The authors are grateful for research support from the Electric Power Research Institute and from National Science Foundation grant SBR9511209, although the views expressed are solely our own, and to Sam Lovick who ®rst proposed re-design of California's 1997 auction along the lines described here.

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biennial resource planning update (BRPU) auctions. Each auction resembled a Vickrey (1961) auction in that the bidder with the lowest score won the right to negotiate a detailed supply contract on terms comparable to the bid with the second-lowest score. This feature was used to justify predictions that the auction outcome would be ef®cient, as in an ordinary Vickrey auction with single-part bids and standard technical assumptions. In fact, however, the scoring and settlement rules encouraged bidding strategies that resulted in winning bids proposing huge upfront capacity payments and negative energy payments (the BRPU auctions were eventually voided by the Federal Energy Commission on other grounds). Bushnell and Oren (1994) and Gribik (1995) show that anomalous results of this kind stem from naive reliance on a scoring rule that uses a linear function of the two parts of each bid. The relation between Bushnell and Oren's proposed solution and ours is described in section 5. The design of procurement auctions with two-part bids is a central problem in current efforts to restructure wholesale electricity markets. In particular, this problem arises in the auctions conducted by system operators (SOs) to purchase suf®cient reserve capacity to meet contingencies. Thus, the two ingredients supplied by a winning bidder are capacity availability, and then depending on events, energy generated from that capacity when called by the SO during real-time operations. A typical example is (incremental) spinning reserve, in which a supplier operates a generator below its maximum rate so that it can be ramped up to higher rates when called to meet load surges. Besides the maximum operating rate of the generator, reliability standards restrict the amount of spinning reserve that a generator can provide to the amount that it can ramp to within a speci®ed time, such as 10 min or 30 min. A typical ramp rate for a thermal generator is 1% of its maximum operating rate per minute, so at most 10% of capacity can be available within 10 min. Reliability standards are also the main determinant of the amount of spinning reserves purchased by the SO. In California, the procurement auction to obtain spinning reserves is conducted roughly as follows. For spinning reserves to be available in a speci®ed hour of the next day, an auction is held after the close of the day-ahead forward markets for energy and transmission, which establish the amount of spinning reserves required by the SO to maintain reliability of the transmission grid. On Tuesday, for instance, 24 simultaneous auctions are conducted independently for reserves to be provided in each of the 24 h on Wednesday. Each supplier who proposes to provide spinning reserve in a given hour submits a two-part bid for a quantity that its ramp rate and maximum operating rate allow. Each bid speci®es an offered price ($/MW) for capacity availability and an offered price ($/MWh) for delivered energy. The SO then evaluates these bids using a scoring rule, and accepts enough bids to meet its reliability requirement. Those suppliers whose bids are accepted are obligated to maintain spinning reserves during the speci®ed hour on Wednesday, enforced by occasional spot checks and penalties for noncompliance. During an hour of real-time operations, the SO calls the spinning units as needed to meet load surges. The spinning units are called in order of increasing energy costs, called the merit order, as speci®ed in the energy portion of their bids. The settlement rule speci®es the payments made to the suppliers whose bids are accepted. These payments include a capacity payment derived from the initial evaluation, and an energy payment for the energy actually called and supplied.

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This sketchy description of the market for spinning reserves omits various features that we ignore later. For instance, each bid is actually a schedule of quantities offered at different prices. Transmission constraints can alter the merit order. And there are other categories of reserves acquired in parallel auctions. Here, we use the procurement auction for 10-min spinning reserves as the context for our analysis. We will, however, consider both incremental and decremental reserves. As in the description above, incremental reserves are provided by generators able to ramp up to higher production rates; similarly, decremental reserves are provided by generators able to ramp down to lower production rates in order to meet load decreases. Separate auctions are held for incremental and decremental reserves. The reasons for this important distinction will be explained in section 4. (The energy part of a bid is relatively unimportant for the reserve category called regulation, in which both increments and decrements are provided by automatic generation control (AGC) to follow the load continually, since such a unit is required to return to its set point every few minutes and therefore often has little or no net generation.)

1.1. Scoring and Settlement Rules To indicate the role of scoring and settlement rules, we describe a version of the rules included in California's ``Phase II'' ®ling to the Federal Energy Regulatory Commission, for the case of incremental spinning reserve (California Trust for Power Industry Restructuring, March 1997). The bid format requires each supplier to specify for each available mega-Watt (MW) an offered capacity price, say R measured in $/MW, and an offered energy price, say P measured in $/MW-hour. If the SO accepts the supplier's bid then the settlement rule speci®es that the supplier is paid R ‡ hP in total for that MW, where h is the actual fraction of the hour that its MW of energy is provided when called by the SO. On the other hand, the scoring rule is based on the SO's estimate, say H, of the fraction of the hour in which generation from spinning reserve will be needed during realtime operations. Thus, the SO assigns to each bid …R; P† the score I…R; P†:R ‡ H6P: Bids are accepted in increasing order of these scores until the SO's requirement for spinning reserves is satis®ed. This scoring rule (the linear function I, depending on the parameter H) was justi®ed by the objective of minimizing the SO's expected total cost of ful®lling its reserve requirement. Problems with these scoring and settlement rules are evident from the previous experience with the 1993 BRPU auctions. Recall that during real-time operations the spinning-reserve units are to be called in merit order; i.e., in increasing order of their energy bids. Each supplier can anticipate, therefore, that the duration h in which it is called will differ from the SO's overall estimate HÐa scalar parameter undifferentiated by bidder, unit, or MW. Consequently, there are ample incentives for a supplier to `game the system' by selecting its bid …R; P† to maximize its expected pro®t. For instance, based on an anticipated functional relationship h…P† between its energy bid and the expected

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duration h that its unit is called, the supplier might choose its bid …R; P† to maximize the expectation of its realized pro®t R C ‡ h…P†‰P cŠ based on its true ®xed cost C and marginal cost c, taking account that the probability its bid is accepted depends only on its total score R ‡ HP. For any speci®ed score, this maximization could result in either a very high or very low energy bid P that is unrelated to its actual marginal cost c. This kind of gaming is similar to what occurred in the BRPU auctions (Gribik, 1995), as anticipated by Bushnell and Oren (1994). Gaming of this sort distorts the energy portion P of a bid away from the true marginal cost c. Consequently, the SO's reliance on the merit order to call the spinning reserves, required for an ef®cient outcome, con¯icts with the incentives of bidders. There is no assurance that the called energy is provided by those suppliers whose true marginal costs of generation are lowest. This is a classic problem of incentives. What is needed are scoring and settlement rules that encourage each supplier to offer as the energy portion of its bid its actual marginal cost, and at the same time, enable the SO to procure the reserves it needs at the minimum expected total cost. Problems of this kind are addressed by the theory of mechanism design, in which procedural rules are constrained by ``incentive compatibility.'' In the present context, this constraint requires truthful revelation, in the sense that if the procedural rules imply an optimal strategy of offering the energy price P…c† when one's energy cost is c, then we require that P…c† ˆ c. This constraint ensures that the merit order re¯ects suppliers' costs accurately. In sections 2±4 we examine the design problem in terms of standard results from mechanism design theory, and then derive scoring and settlement rules that implement an ef®cient design. Our method reverses the forward-looking approach of the 1997 design and instead works backward from the real-time market to the day-ahead procurement auction. Thus, our starting point is the scheduling of real-time calls in merit order: incentive compatibility constraints dictate the settlement rule for energy, which then enables a simple scoring rule for the procurement auction. In section 5 we compare our design with the alternative design proposed by Bushnell and Oren (1994). The key feature of our design is uniform pricing; that is, our settlement rule speci®es that each MW of reserved capacity is paid the same price and each MWh of called energy is paid the same price. The Bushnell and Oren design relies on a settlement rule that uses non-uniform pricing; in particular, each MWh of called energy is paid the energy price offered by its supplier in the previous procurement auction, and therefore payments for both called energy and capacity availability differ among suppliers. Both settlement rules are feasible and the two corresponding auction designs accomplish the same results in principle. However, we argue in section 5 that uniform pricing is preferable because it implies a rule for capacity availability payments that is more robust, in the sense that it does not require the SO and the bidders to agree on the probability distribution of energy calls. In sections 2±4 we simply assume uniform pricing and defer to section 5 our justi®cation for adopting a settlement rule of this form. The exposition in sections 2 and 3 emphasizes the intuitive rationale for our design and the practical aspects of its implementation. The Appendix provides concise statements and proofs of the two main theorems.

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2. Design of the Real-Time Market In California as in other jurisdictions, the economic role of reserves is to moderate the volatility of the real-time spot price. The SO ordinarily uses regulation and energy bids in the real-time market for following the load and for balancing the transmission grid, but when these bids are exhausted or unduly expensive it calls ®rst on spinning reserves. When reserves are used, the energy bid of the last unit in the merit order among those called sets the spot price. To keep matters simple, initially we consider only incremental reserves, so it is the most expensive among those units called that sets the spot price. In this case, the probability distribution of the spot price can be represented as follows. Let F…Q† represent the probability that the quantity of generation called from incremental reserves will not exceed Q. Represent the merit order by the aggregate supply function S…p†, where S is a nondecreasing function of the price p. That is, S…p† is the quantity of generation from those reserve units whose energy bids are less than the spot price p. Then the probability that the spot price will be less than p is G…p† ˆ F…S…p††: For instance, there is a high probability that the spot price will be less than p if there is a high probability that the supply available at price p exceeds the quantity called. Thus, G…p† is the probability that the price required to meet load surges from the reserved capacity is less than p. A basic assumption that we maintain throughout is that the distribution G of the spot price has a positive density. One part of this assumption is that the distribution F of quantities called has a positive density, so there is no lumpiness in the distribution of the SO's calls for energy. This does not assume that there is no chance of a large call, due perhaps to failure of a generator or a transmission line; rather, it assumes only that there is enough stochastic variability that each speci®c called quantity has negligible probability. This ensures that no supplier can design its energy bid to exploit some particular contingency. This depends on the time frame, of course, but since the procurement auction is held a day ahead this assumption is reasonable. Another part of this assumption is that the supply function S is smooth. This conveys the requirement that no supplier has a signi®cant role in the aggregate supply. It is a strong assumption that incorporates the major feature of a competitive market, namely that each supplier's offered capacity is small relative to the aggregate. This is more realistic when one takes account that each unit can offer only the quantity that it can ramp in 10 min. This assumption already provides an indication of how the settlement rule can be constructed to meet the requirement for truthful revelation of each supplier's marginal cost via its energy bid. We present the construction ®rst for a perfectly competitive market, and then provide a reinterpretation in terms of a Vickrey auction. 2.1. Revelation in a Perfectly Competitive Spot Market In a competitive market, each supplier has a negligible chance of affecting the spot price. Therefore, if the settlement rule merely pays the spot price for called energy, then a

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supplier's optimal bidÐconditional on winning in the day-ahead auctionÐis simply to name its marginal cost as its energy bid. To see this, re-interpret the energy bid P as the supplier's reserve price, namely the least spot price at which it wants to be called. Then the expected pro®t of a supplier whose marginal cost is c is P…P; c† ˆ ‰1

G…P†Š6E‰ p

c j p  PŠ:

This formulation re¯ects the fact that the supplier's unit is called whenever the spot price p exceeds its reserve price P, so the probability of being called is 1 G…P†, and when it is called its expected pro®t is the conditional expectation of the pro®t margin p c. When G has a positive density at p ˆ c, as assumed, the unique maximizer P…c† of this expected pro®t is P…c† ˆ c, as required. This settlement rule implies that the expected pro®t from called energy for a supplier with marginal cost c is P…c; c†, which is a convex and decreasing function of c. 2.2. Revelation in a Vickrey Auction An analogous result is obtained from a Vickrey auction. In the simplest case where each supplier provides a single MW, the settlement rule speci®es that called energy is paid the lowest rejected energy bid in the merit order. Now the probability distribution G has the analogous interpretation that G…p† is the probability that the lowest rejected bid is no more than p. As before, the expected pro®t is P…P; c† ˆ ‰1

G…P†Š6E‰ p

c j p > PŠ;

and again the optimal bid P…c† ˆ c reveals the supplier's marginal cost accurately. More generally, a Vickrey auction requires that the qth MW provided by a supplier is paid the qth from the lowest energy bid (one for each MW) among those rejected that are not bids from this supplier. This settlement rule is more complicated, and in particular some large suppliers may be paid more than the nominal spot price, but it has the advantage once again that, if suppliers' costs are statistically independent, then an optimal strategy is to specify one's marginal cost for each MW as the reserve price, namely P…c† ˆ c. The settlement rule for a Vickrey auction ensures truthful revelation of marginal costs even if the energy market is imperfectly competitive. In the sequel, however, we concentrate on the perfectly competitive case in order to clarify the derivation of the scoring rule used in the initial procurement auction. Thus, we use P…c; c† as the formula for a supplier's expected pro®t in the energy market even though this is not exactly accurate for a Vickrey auction when competition is imperfect. 2.3. Interpretation For both the competitive spot market and the Vickrey auction, the above derivations are incomplete because in each case the analysis is conditioned on the supplier having won a place in today's merit order when it bid in the reserve auction conducted yesterday. The result says only that, for a settlement rule that pays the spot price for energy, after winning spinning reserve status in the day-ahead auction, a supplier's optimal reserve price for

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energy is its true marginal cost, namely P…c† ˆ c. This leaves open the question of whether the supplier might prefer to distort its day-ahead energy bid to improve its chances of winning reserve status. The resolution of this question depends on the scoring rule, which we examine next.

3. Design of the Procurement Auction A brief summary of section 2 is a sequence of implications. Productive ef®ciency of the real-time energy market requires calling the reserved units in the merit order of their marginal costs. A settlement rule that pays the spot price for called energy suf®ces to encourage each supplier to offer its actual marginal cost as its energy bid, interpreted as a reserve price below which it prefers not to be calledÐprovided this does not alter the chances of being selected to provide spinning reserve. Therefore, a procurement auction that ignores suppliers' energy bids in selecting those awarded reserve status solves the problem of productive ef®ciency in the next day's energy market. One such auction is a Vickrey auction in which those suppliers offering the lowest capacity prices are accepted. That is, the scoring rule is simply I…R; P† ˆ R: Again, the settlement rule of a Vickrey auction pays for the qth MW of spinning capacity accepted from a supplier the qth lowest rejected bid (one for each MW) among those from other suppliers. To simplify, however, we again invoke the assumption of perfect competition. In this case every accepted MW is paid the same price, say R*, which is the capacity price offered by the lowest rejected bid. In sum, the settlement rule in the perfectly competitive case is that each MW accepted in the procurement auction is paid R* for making spinning capacity available, plus the spot price p for energy that is actually called and supplied. Our aim in the remainder of this section and the next is to establish that such an auction in the day-ahead market for reserved capacity is ef®cient for a standard speci®cation of the economics of supply for reserve capacity. In this speci®cation, the costs incurred by a supplier providing spinning reserve are independent of its energy bid P. For example, a supplier might incur a direct cost maintaining a unit in spinning condition, and further, an opportunity cost representing the expected pro®t that could otherwise have been earned if the reserved capacity were instead committed to producing energy for sale in the dayahead and/or real-time energy markets. The latter may be problematic if the supplier's real-time sales are large, but here we assume that, as in competitive spot markets, a supplier's energy bid P has a negligible chance of affecting the spot price p in the real-time market. Thus, we assume that a supplier with marginal cost c has a reserve price in the procurement auction that is the difference R…P; c† ˆ V…c†

P…P; c†;

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between its foregone expected pro®t V…c† and its expected pro®t P…P; c† from called energy paid the spot price. For instance, if the opportunity cost is the pro®t that could have been earned in the dayahead energy market where the clearing price is p, then V…c† ˆ maxf0; p cg independently of the energy bid P offered in the auction of spinning reserve. As in this example, we assume that V is a nonincreasing function of the marginal cost c.

3.1. The Scoring Rule To examine the effect of the scoring rule, it suf®ces to illustrate the case that it has the additive form I…R; P† ˆ R ‡ H…P†; where (as in the 1997 ®ling) H is a smooth nondecreasing function of the energy part of the bid; and, the settlement rule pays each accepted unit the capacity price R* …P† ˆ I *

H…P†;

where I * is the smallest score among those rejected. Thus, if the bid …R; P† is accepted then that supplier is paid R* …P† for reserving capacity, plus the subsequent spot price p for energy called when p > P. We claim that in this case the incentive compatibility constraint requires that H…P† does not depend on the energy bid P; that is, it is necessary for productive ef®ciency and the validity of the merit order that bids are evaluated solely on the basis of the capacity part of each bid. To show this requires several steps. 1. Observe ®rst that a supplier's bid …R; P† will be accepted if and only if I…R; P† < I * , where I * does not depend on its bid, and if accepted its expected pro®t will be R* …P† R…P; c† net of any opportunity and ®xed operating costs. For any ®xed choice of P, therefore, the bidder's incentive is to offer the capacity price R ˆ R…P; c†, since that maximizes the probability its bid is accepted without altering its expected pro®t if accepted. 2. The next step observes that the expected pro®t from an accepted bid is R* …P†

R…P; c† ˆ I *

H…P†

V…c† ‡ P…P; c†:

The incentive compatibility constraint that the optimal energy bid must be P ˆ c therefore requires that H 0 …c†

P1 …c; c† ˆ 0:

Note further that this constraint reinforces the optimality of the choice R ˆ R…P; c†

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for the capacity bid, since R…P; c† ˆ V…c† P…P; c† is minimized by choosing P ˆ c as we saw in section 2. 3. The third step uses the fact that P1 …c; c† ˆ 0 to infer that H 0 …c† ˆ 0, which implies that H must be a constant. From this derivation, we conclude that the SO's optimal auction design can be simpli®ed by setting H ˆ 0. Then the score value I * equals the capacity payment R* …c† made to the winning bidders. A similar construction applies to decremental reserves. In this case a winning bidder's expected pro®t from real-time calls from the SO is P…P; c† ˆ G…P†6E‰c

p j p < PŠ:

This formula re¯ects the fact that for a decrement the supplier pays the spot price to the SO to purchase energy that replaces its previous delivery commitments contracted in the dayahead energy market, for which it would otherwise incur the marginal cost c. The opportunity cost is typically nil, V…c† ˆ 0, in view of the supplier's previous opportunity to sell less in the day-ahead energy market. Example: Suppose that the spot price p in the real-time market is distributed uniformly on the interval ‰p D; p ‡ DŠ. Then for incremental spinning reserve, P…P; c† ˆ

1 ‰…p ‡ D 4D

c†2

…P

c†2 Š;

provided P  p ‡ D. The optimal bid offers the energy price P ˆ c and the capacity price R ˆ R…c; c† ˆ V…c†

P…c; c†:

In particular, if p represents the clearing price in the day-ahead energy market and V…c† ˆ maxf0; p cg, then  Rˆ

…1=4D†…p D …1=4D†…p ‡ D

2

c† 2 c†

if if

c  p; c  p:

Notice that the capacity bid R is negative in this example, re¯ecting the fact that reserve status is essentially a call option exercised whenever the spot price exceeds marginal cost. Indeed, in actual operation, procurement auctions for spinning reserve sometimes clear at zero prices, or if allowed (as in California), at negative prices. The clearing price is usually positive, however, because maintenance of spinning reserve incurs ®xed costs, and for units that provide only spinning reserve there are also start-up costs, possibly augmented by unpro®table sales of energy from generation at the minimum feasible rate.

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4. Overall Productive Ef®ciency Lastly, we derive conditions under which this auction design provides an ef®cient schedule of overall capacity and energy purchases. Our derivation follows closely the traditional peak-load pricing analysis as in Chao (1983); see Crew et al. (1995) for a recent survey. This analysis involves three key ingredients: 1. the load-duration curve, whose role is played here by the probability distribution of the amount of energy called in real-time operations, 2. the ®xed capacity cost, whose role is played here by the ®xed cost of reserving capacity in advance, and 3. the marginal cost, which includes the outage cost or the value of unserved energy in the event load is curtailed. Each ingredient of the traditional analysis has an analog in an auction market for reserves. The load-duration curve is usually constructed by recording for each load Q, the number or fraction H…Q† of hours in the year that the load is at least Q; or in reverse, by recording the load Q…H† of the Hth hour when the hours are ranked in order of their loads. The corresponding representation in the present context is the probability 1 F…Q† that the real-time call for energy exceeds Q, where this probability is interpreted as the expected fraction of the hour of real-time operations. The ®xed capacity cost, say k, is represented here by the opportunity cost V…c† of reserving capacity. The marginal cost of each technology is represented by the incremental cost c of generation and the outage cost v. Assume that bids are received from n generators with capacities Xi , where i ˆ 1; . . . ; n. These bids are ranked in merit order according to their marginal costs c1  c2 


 cn < v. In general, their corresponding capacity costs ( per unit of capacity) ki need not be ranked in any order, but in our case ki ˆ V…ci † is nonincreasing in i. As in traditional analyzes, a production technology i with ®xed cost ki and marginal cost ci is ef®cient if there is a duration h for which its total cost ki ‡ hci is less than the total cost of any alternative. While the ef®cient technology locus consists of points …ki ; ci † that form a declining curve, the property that ki is nonincreasing in ci is insuf®cient in itself to assure that every pair …ki ; ci † is on the ef®cient locus. A more rigorous analysis is provided below.

4.1. The Ef®cient Production Schedule Denote by D…o† the actual load demanded in the event o, which is stochastic, and let Z…Yi †:E‰minfD…o†; Yi gŠ be the expected quantity of energy generated from units deployed in the merit order up to i with a total capacity of Yi . An important requirement for ef®ciency in the management of reserve capacity is that the expected total cost in each hour is minimized subject to the constraints on available capacities. This minimization problem can be stated as:

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MinimizefXi ;Yi g

n X iˆ1

fki Xi ‡ ci ‰Z…Yi †

Z…Yi

1 †Šg

‡ vE‰D…o†

Y n Š‡

subject to Yi ˆ

i X jˆi

Xj

…1†

Xi  Xi  0;

i ˆ 1; . . . ; n:

…2†

De®ne F…Y†:1 F…Y† ˆ PrfD…o† > Yg, which can be interpreted as a probabilistic version of the load duration curve. The derivative of Z…Y†, i.e., Z 0 …Y† ˆ F…Y†, can be interpreted similarly as the load duration curve. Denote by yi and mi the shadow prices associated with the upper and lower bounds on Xi in the above formulation. Then, the optimality condition for the above problem is: ki ‡

n X jˆi

cj F…Yj †

n X jˆi‡1

cj F…Yj

1†

ˆ vF…Yn †

yi ‡ m i

…3†

for each increment of reserved capacity from bidder i ˆ 1; . . . ; n. The above condition has a simple cost-bene®t interpretation. The left side of this equation represents the expected cost of scheduling an additional unit of capacity available for reserve duty. The optimality condition requires that this cost of supply must not exceed the expected value of avoiding an additional unit of lost load (the ®rst term on the right side) for a unit that is selected to be in the reserve …mi ˆ 0; yi  0†. For the marginal unit, the cost of supply equals the bene®t of meeting the demand …mi ˆ 0; yi ˆ 0†, whereas units with higher costs of supply should be rejected …mi > 0; yi ˆ 0†. Our auction design adopts the marginal cost pricing rule, which requires that the spot price is p…o† ˆ ci , when Yi 1 < D…o† < Yi , and p…o† ˆ v, when Yn < D…o†. Therefore, we can write Prfp…o† ˆ ci g ˆ F…Yi 1 † F…Yi †, and the optimality condition (3) can be rewritten as ki

 E ‰ p…o†

ci Š

‡

ˆ

yi ‡ mi ; for

i ˆ 1; . . . ; n

…4†

This form of the optimality condition can be interpreted as the result of arbitrage among pro®t-seeking ®rms. As we have shown earlier, the expression on the left side represents the opportunity cost of reserving capacity, which equals the marginal capacity cost minus the expected pro®t from selling power at the spot price. An ef®cient operation reserves units according to the increasing order of this opportunity cost. As a result, all units with negative opportunity costs …mi ˆ 0; yi > 0† should be selected (i.e., Xi > 0), and all units with positive opportunity cost …mi > 0; yi ˆ 0† should be rejected (i.e., Xi ˆ 0). Therefore, the scoring rule in the above auction design is operationally ef®cient for ranking the reserve capacity. In actual implementation, the system operator may be required to meet a given reserve capacity requirement, i.e., Yn  Y. This requirement essentially replaces the role of v, the

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value of unserved demand, in the above formulation by the shadow price on this constraint. 4.2. The Tie-Breaking Rule If k is the same for each marginal cost in an interval c < c < c* then only the pair …k; c † can be on the locus of ef®cient technologies. Thus, it is necessary for productive ef®ciency to exclude those technologies with the same ®xed cost and higher marginal costs. This consideration implies that a particular tie-breaking rule should be used in the procurement auction: if two bids offer the same capacity price then the one with the lower reserve price for called energy has priority for acceptance. In practice this tie-breaking rule may have few effects because each bid for 10-min spinning reserve offers a small quantity, so even if one with lower marginal cost is accepted ®rst it is still necessary to accept the second too in order to meet the SO's demand for reserves. However, for the particular formula V…c† ˆ maxf0; p cg that is commonly used in examples to illustrate markets for incremental reserve, the tie-breaking rule might play a more signi®cant role. All suppliers with costs in the upper range c > p of marginal costs exceeding the day-ahead price p have no opportunities to sell in the day-ahead energy market and therefore all have the same opportunity cost V…c† ˆ 0, and therefore each offers the capacity bid P…c; c† depending on its marginal cost c. In this case, a lower marginal cost c is associated with a higher pro®t P…c; c† from called energy, and so indeed the capacity bid R…c; c† ˆ P…c; c† is lower and therefore is accepted ®rst. More generally, however, such suppliers might differ in other cost ingredients, such as start-up or no-load costs required to make capacity available, so the tie-breaking rule can play a role in selecting between two or more with the same capacity bids and differing energy bids by giving priority to the one with the smaller energy bid. The tie-breaking rule is especially important in those systems that constrain bids to be nonnegative, because then there are often several capacity bids that are zero. These same considerations apply to decremental reserve, since again the opportunity cost is often supposed to be nil, so the tie-breaking rule can be a major factor in selecting among identical capacity bids by assigning priority to those offering the highest energy bids for called decrements. It is important to realize, however, that constraining capacity bids to be nonnegative should be avoided to prevent ``gaming the system.'' The chief risk is that an inef®cient generator (one for which k ‡ hc is not minimal for any duration h within the hour) could bid a positive capacity price that is always accepted by the SO and an energy price so high that it is never called for generation: in this case the supplier collects capacity payments without providing any useful service. This gaming strategy is precluded if the capacity payment can be negative (i.e., the supplier pays the SO for reserve status) or less than the start-up and no-load cost of making the unit availableÐprovided of course that, as in California, the SO occasionally checks that reserved capacity is in fact operable and responsive to calls for incremental generation. 4.3. Inter-Market Ef®ciency Essentially, we have applied the traditional peakload pricing analysis to the procurement auction for reserves. The load-duration curve is replaced by the probability

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173

1 F…Q† and the set of ef®cient technologies is replaced by the pairs …V…c†; c† of suppliers' ®xed costs of reserving capacity and marginal costs of generation, where V…c† is a decreasing function of c. The merit order remains the same, since paying the spot price for energy encourages each bidder to offer P…c† ˆ c as its energy bid. A main conclusion is that the lowest-cost technology, say …k* ; c* †, is selected to serve the peak load of short duration H * ; thus k* is typically small and c* is typically large, which represents an ef®cient tradeoff because the capacity is idle most of the time. A technology with a higher capacity cost but lower marginal cost, say …k; c† that is most ef®cient for the longer duration H > H* is called whenever the load exceeds Q…H†. This corresponds to the familiar operating rule that calls the ef®cient technologies in the merit order of their marginal costs as the load rises over an interval of real-time operations. The technology …k; c† recovers its total cost from the spot price that is the marginal cost of the last unit called in merit order. In particular, even though its capacity cost k is higher than k* , its lower marginal cost c < c* enables a baseload unit to recover its full cost from longer pro®table runs at energy prices exceeding its marginal cost. In the ideal world of the mathematical model this cost recovery is exact when each technology has constant returns to scale: compared to a peaking unit, the baseload unit's greater pro®ts from energy sales exactly compensate for its higher capacity cost, so k P…c; c† ˆ k* P…c* ; c* †. With limited capacity available from each technology, however, there may be additional scarcity rents that can be earned so these differences can be negative, as in the example above. There is one very signi®cant difference, however, when we take into account the interaction between the energy and the reserve markets. To describe and analyze this difference we rely on the standard example in which there are no ®xed costs and the opportunity costs are V…c† ˆ maxf0; p cg for incremental reserve and V…c† ˆ 0 for decremental reserve, where p is the clearing price in the day-ahead energy market. To distinguish between these two reserve categories, let RI …c† ˆ V…c† P…c; c† be the predicted capacity bid in the auction for incremental capacity, and similarly RD …c† is the predicted capacity bid for decremental capacity. The relevant fact is that for incremental reserve the predicted capacity bid RI …c† is not an increasing function of the marginal cost c as required for full ef®ciency. Indeed, RI …c† decreases with slope G…c† for c < p and increases with slope 1 G…c† for c > p. Consequently, an auction in which suppliers with costs on both sides of p participate will typically accept some from each side. This non-monotonicity undermines the ef®ciency of the outcome that would otherwise be predicted by the theory of mechanism design. But apart from this mathematical problem it is already evident that acceptance of bids from suppliers with costs below p violates the intent of the example's formulation. That is, attracting low-cost suppliers away from the day-ahead energy market is inef®cient because energy sales in that market are sure whereas energy sales from reserved capacity are uncertain. Ef®ciency therefore requires that the auction design not attract away from the energy market those suppliers who can successfully sell there at the clearing price p. To resolve this problem we examine the role of decremental reserves. A simple calculation shows that RD …c† decreases with slope G…c† in the range c < p, just as RI …c† does. Therefore, either these two functions are the same in this range or one dominates the other. For ef®ciency, we require that RI …c†  RD …c† so that those suppliers with low costs

174

HUNG-PO CHAO AND ROBERT WILSON

c < p prefer to sell in the day-ahead energy market at the price p and then offer bids for decremental reserve (rather than incremental reserve). There is, in fact, a simple and realistic assumption that ensures that this requirement is satis®ed: E‰ pŠ ˆ p: That is, if the day-ahead price p is an unbiased predictor of the real-time spot price p then RI …c† ˆ RD …c† for every marginal cost c  p. This assumption is just the no-arbitrage condition that one can realistically suppose to characterize a sequence of markets. For example, if E‰ pŠ > p then suppliers would prefer to defer sales from the day-ahead market to the real-time market. A touch of realism strengthens this result. Decremental reserve requires no ®xed costs, whereas incremental reserve incurs no-load costs and possibly start-up costs, so typically the no-arbitrage condition E‰ pŠ ˆ p actually implies the strong inequality RI …c† > RD …c† when c < p. We interpret this resolution of the problem as a basic explanation for the separate auctions of incremental and decremental reserves conducted by system operators. It conforms, moreover, to the conventional wisdom that all supply units that can sell in the day-ahead market do so and offer bids only for decremental reserve, whereas it is only those whose costs are too high to sell in the day-ahead market who offer bids for incremental reserve.

5. The Argument for Uniform Pricing The key assumption in our construction is a settlement rule that relies on uniform pricing of called energy, which in turn enables uniform pricing of capacity availability. In contrast, Bushnell and Oren (1994, 1995) rely on non-uniform pricing of called energy, which requires non-uniform pricing of capacity availability; a closely related analysis is by Che (1993). One operational difference is that our design interprets an energy bid as a reserve price used only to construct the merit order for real-time calls, but this reserve price is not used for settlements as it is in the Bushnell and Oren design. The key difference, however, is that uniform pricing enables a payment rule for capacity availability that does not depend on any data speci®ed by the SO; that is, each accepted MW is paid the lowest rejected capacity bid. In contrast, the Bushnell and Oren design depends sensitively on the SO's speci®cation of the probability distribution of real-time calls. Thus, using the probability 1 G…p† that a MWh offered at price p will be called, they derive from the requirements of ef®ciency and incentive compatibility that the score for a bid …R; P† must be Z I…R; P† ˆ R ‡

P 0

and the capacity payment for this bid must be

‰1

G…p†Š dp

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MULTI-DIMENSIONAL PROCUREMENT AUCTIONS

R* ‡

Z

P



P

‰1

G…p†Š dp;

where …R* ; P* † is the rejected bid with the lowest score. The two integrals in these formulas (which are absent in our design) stem from their assumption that called energy is paid the bid price P rather than the spot price. In our design a winning bidder obtains the ``extra'' capacity payment represented by the integral from energy payments at spot prices above its reservation price. For instance, if G…P* † ˆ 1 then the capacity payment in the Bushnell and Oren design can be written as R* ‡

Z P

P

‰p

PŠ dG… p†;

where the integral is precisely the bidder's expected net revenue from sales at spot prices p that are above its reservation price P. These two formulas reveal the dif®culties that an actual implementation encounters. An obvious practical problem is that the SO must specify the distribution function G continually, one for each hour of the next day; further, the databases and software required for settlements are elaborate and expensive. The main problem, however, is a lack of robustness. The hidden assumption in the derivation of these two formulas is that the distribution function G is an objective fact; that is, it is common knowledge among the SO and the bidders. This assumption is rarely (if ever) satis®ed in practice because ordinarily the SO and the bidders have differing information and opinions about the prospects of energy calls. We prefer a formulation that depends only on each bidder's subjective assessment of its prospects of being called, and that does not require any ``objective'' data from the SO. The SO can provide bidders with a forecast but to make the scoring and settlement rules depend on this forecast makes the procurement auction vulnerable to gaming by bidders with different predictions, much like the California BRPU auctions and the 1997 Phase II ®ling.

6. Concluding Remarks We have shown that elementary considerations suf®ce to derive an ef®cient design for a procurement auction in which the system operator purchases reserve capacity that it can later call for energy generation. The key to our approach is to work backward from the requirement that, for productive ef®ciency, the spinning reserve units must be called in merit order based on marginal costs that are accurately revealed by supplier's bids in the initial procurement auction. The distinguishing features of such auctions are two-part bids, one part offering a price for capacity availability and another offering a reserve price for energy called in real-time. Because ef®cient generation requires the merit order to re¯ect accurately the suppliers' marginal costs, we impose the incentive compatibility condition that each supplier's optimal energy bid must accurately reveal its privately known marginal cost. One way to

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ensure this condition is to use a variant of the Vickrey auction in the spot market. In the perfectly competitive case, each energy bid is interpreted as a reserve price to construct the merit order, and then all energy is paid the spot price obtained as the lowest unused energy bid. To ensure that incentive compatibility is not distorted by the scoring rule used to compare bids in the initial reserve auction, it is necessary that only the capacity prices are used in the scoring rule for comparing bids, except for a tie-breaking rule. This ensures that the system operator obtains the required reserves at least cost, since those bids accepted are the ones for which the difference between the ®xed and opportunity costs of reserving capacity and the expected pro®t from spot market sales is smallest. AVickrey auction can be used at the initial stage, or in the competitive case, all winning bidders are paid the lowest rejected bid offered for reserving capacity. This payment corresponds exactly to the demand charge familiar from traditional analyzes of ef®cient capacity planning, since it represents the ®xed and opportunity costs of the last ( peaking) unit in the merit order among those accepted for reserve status, net of pro®ts from called energy paid the spot price. Overall productive ef®ciency depends crucially on two additional features. One is that parallel auctions for incremental and decremental reserves are conducted, and the second is that arbitrage between the day-ahead and real-time markets is suf®cient to ensure that the day-ahead energy price is an unbiased predictor of the spot price, namely E‰ pŠ ˆ p. These features ensure that low-cost suppliers prefer to sell energy in the day-ahead market, rather than providing incremental reserves, and offer bids only in the auction for decremental reserves.

Appendix In this appendix, we summarize our main theoretical contributions in two theorems. Essentially, the main conclusion of our paper is extremely simple: (1) The Vickrey-ClarkeGroves (VCG) auction1 provides an ef®cient form of multi-dimensional auctions for power reserves, and (2) a reserve market design based on sequential Vickrey auctions for the capacity and energy price bids is ef®cient. A more subtle insight is that since truthful revelation is a dominant strategy in this design, it is more robust than a mechanism that satis®es the standard incentive compatible condition.

Basic Assumptions The reserve capacity requirement, Q, is ®xed. For y [ ‰0; QŠ; f …y† represents the probability that y units of reserved capacity will be called. There are n generators or bidders who own the capacity, represented by qi, for i ˆ 1; . . . ; n. The expected cost of reserve is k^i xi ‡ Ey ‰^ ci gi …y†Š, where xi is the capacity of unit i, k^i and c^i are the expected unit capacity 1 For recent applications of the VCG auction to power markets, see Hobbs et al. (2000). An alternative ef®cient design is presented in Bushnell and Oren (1994).

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MULTI-DIMENSIONAL PROCUREMENT AUCTIONS

and energy costs, and gi …y† is the actual amount of energy generated by the unit when the demand for reserve energy is y. We assume that the energy and capacity costs are random variables with known probability distributions. For clarity, we adopt the following information structure for the cost components: k~i ˆ k^i ‡ ai ˆ ki ‡ ai ‡ b; c~i ˆ c^i ‡ gi ˆ ci ‡ gi ‡ d;

Efai g ˆ Efbg ˆ 0; Efgi g ˆ Efdg ˆ 0;

for

i ˆ 1; . . . ; n:

In the above expression, the actual costs are represented by random variables k~i and c~i . The random variables ai and gi represent private information. In other words, the probability distributions of k^i ˆ E‰k~i jai Š and c^i ˆ E‰~ ci jgi Š are privately known to bidder i. We assume that ai and gi are stochastically independent of a i ; g i ; b, and d.2 The remaining two random variables b and d represent common factors, such as spot electricity prices and fuel costs, that affect capacity and energy costs in a uniform fashion among all bidders. We assume that the values of b and d are not available before the auction, but the probability distributions of these random variables are publicly known. Finally, we assume that each generator will submit the maximum available capacity, qi , to the auction. This assumption is not restrictive for two reasons. First, as we will show, there is no ®nancial advantage for the generators to withhold capacity. Second, should withholding be pro®table, a generator can always do it by submitting a suf®ciently high capacity price bid.

Simultaneous Vickrey-Clarke-Groves Auction We ®rst consider a simultaneous VCG auction. Generators submit bids f…ki ; ci ; qi †ji ˆ 1; . . . ; ng, which are entered into a clearing model described below. Q n n X X X ki xi ‡ ci gi …y†f …y†; J…k; c†: min g… ? †;x iˆ1 iˆ1 yˆ0 n X gi …y† ˆ y; iˆ1

n X iˆ1

xi ˆ Q;

gi …y†  xi ; x i  qi ; gi …y†  0; xi  0;

2 In this paper, we adopt the convention that

for

i:f1; 2; . . . ; i

i ˆ 1; . . . ; n:

1; i ‡ 1; . . . ; ng:

…1†

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HUNG-PO CHAO AND ROBERT WILSON

The objective of the clearing model is to minimize the total cost based on the data submitted. The ®rst two constraints represent the reserve demand for the energy and capacity components. The next two constraints represent the reserve supply conditions for the energy and capacity components. Lemma 1: J…k; c† is concave. Proof: The result follows immediately from the fact that J…k; c† is the minimum of a family of linear functionals on a convex set. & For later reference in the paper, we present here the optimality conditions for market clearing in a simultaneous auction. Let f …y†p…y†; l; f …y†mi …y† and Zi denote the shadow prices associated with the constraints in (1). Note that these shadow prices are always nonnegative. From the duality theorem of linear program, we obtain, p…y† l‡

mi …y†  ci ;

Q X yˆ0

‰p…y† " l‡

f …y†mi …y† mi …y†

Q X yˆ0

for

Zi  ki ;

ci Šgi …y† ˆ 0; #

f …y†mi …y†

Zi

ki xi ˆ 0;

i ˆ 1; . . . ; n:

…2†

The clearing model determines the allocation xi and gi …y† for bidders i ˆ 1; . . . ; n. Then, in the settlement process, each bidder (say i) is provided a non-negative payment: J

i …k

i; c

J…k; c† ‡ ki xi ‡ ci

i†

Q X yˆ0

gi …y†f …y†:

…3†

After accounting for the actual capacity and energy costs, the net payoff for the bidder can be written, Vi …ki ; ci †: J

i …k

‡ …ki

i; c

i†

J…ki ; k

k^i †xi ‡ …ci

c^i †

i ; ci ; c Q X yˆ0

i†

gi …y†f …y†:

…4†

Lemma 2: Vi …k^i ; c^i † is always non-negative. Proof: By de®nition, we have Vi …k^i ; c^i †: J i …k i ; c i † J…k^i ; k i ; c^i ; c i †, which is always non-negative because J i …k i ; c i † can be equivalently obtained by solving (1)

179

MULTI-DIMENSIONAL PROCUREMENT AUCTIONS

with virtually the same data as for J…k^i ; k i ; c^i ; c i † but with the additional restriction, qi ˆ 0. The result follows from the fact that the value of J…k; c† is always non-increasing in qi . & Therefore, truthful revelation always satis®es the individual rationality condition independent of the other bidders' strategies. In the following, we provide a new mathematical proof of the properties of the VCG scheme for multi-dimensional auctions of power reserves. Theorem 1: In a simultaneous VCG auction, the bids …k^i ; c^i †, for i ˆ 1; . . . ; n, represent a set of dominant strategies that implement an ef®cient allocation. PQ Proof: From the optimization theory, we obtain …xi ; y ˆ 0 gi …y†f …y†† [ qJ…k; c†, where qJ denotes the subgradient of J…k; c†. In other if J…k; c† is differentiable, we can P words,  write qJ…k; c†=qki ˆ xi and qJ…k; c†=qci ˆ Q g …y†f …y†. Since J…k; c† is concave, we yˆ0 i have J…k^i ; k

^i ; c i; c

i †  J…ki ; k

i ; ci ; c

^ i † ‡ …k i

ki †xi ‡ …^ ci

ci †

Q X yˆ0

gi …y†f …y†:

Then, we can compare the payoffs for alternative bidding strategies, Vi …ki ; ci †

Vi …k^i ; c^i †;

ˆ J…k^i ; k

^i ; c i; c

i†

J…ki ; k

i ; ci ; c

i † ‡ …ki

k^i †xi ‡ …ci

c^i †

Q X yˆ0

gi …y†f …y†

 0: This shows that independent of the bids of the other generators, …k i ; c i †, bidder i gains no ®nancial advantage by bidding anything different from …k^i ; c^i †. This conclusion holds for every bidder. Therefore, truthful revelation represents a dominant strategy for every bidder. As a consequence, the clearing model attains a truly ef®cient allocation. & Remark: The payoff function Vi is non-decreasing in qi . Therefore, there should be no reason for the bidder to withhold capacity.

Sequential Vickrey Auctions We now consider a sequential auction design applying simple Vickrey auctions to capacity and energy components separately. As before, generators submit bids f…ki ; ci ; qi †ji ˆ 1; . . . ; ng. However, there are two main differences. First, the capacity

180

HUNG-PO CHAO AND ROBERT WILSON

price bids are a set of non-decreasing price functions, fki … ? †ji ˆ 1; . . . ; ng, which need not be linear.3 Second, the settlement is now conducted in two stages separately for the capacity and energy components. In the ®rst stage, the reserve capacity is selected according to the following clearing model. L…k†: min x

n X iˆ1

n Z X

xi

ki …z†dz;

iˆ1

0

for

i ˆ 1; . . . ; n:

xi ˆ Q; x i  qi ; xi  0;

…5†

The following result follows from the same line of reasoning as in Lemma 1. Lemma 3: L…x†: L…k ‡ xh† is concave in x. Let the symbols l and Zi represent the shadow prices for the constraints in (5). We can express the optimality condition for the clearing model as follows. l Zi  ki …xi †; i h l Zi ki …xi † xi ˆ 0:

…6†

Each selected bidder will receive a capacity payment according to the following settlement rule: Z L

i …k

i†

L…k† ‡

xi 0

ki …z†dz:

…7†

Therefore, the net payoff for bidder i in the ®rst stage can be written, Z L

i …k

i†

L…k† ‡

xi 0

ki …z†dz

k^i xi :

…8†

In the second stage, the reserves are dispatched, as needed, according to the following cost minimization model. 3 See Wilson (1979) for an introduction to nonlinear auctions.

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MULTI-DIMENSIONAL PROCUREMENT AUCTIONS

G…c; x; y†: min g

n X iˆ1

n X iˆ1

c i gi ;

gi ˆ y; gi  x i ; gi  0;

for

i ˆ 1; . . . ; n:

…9†

The following result follows the same line of reasoning as in Lemma 1. Lemma 4: G…c; x; y† is concave in c. Let the symbols p…y† and mi …y† represent the shadow prices for the demand and supply constraints in (9). The optimality condition can be written as follows. p…y† mi …y†  ci ; ‰p…y† mi …y† ci Šgi …y† ˆ 0; for

i ˆ 1; . . . ; n:

…10†

After the real-time dispatch, each reserved unit will receive the following energy payment:4 G

i …c

i; x

i ; y†

G…c; x; y† ‡ ci gi :

…11†

The expected net payoff after taking into account of the energy cost is given by, Pi …ci ; xi † ˆ Ey ‰G

i …c

i; x

i ; y†

G…c; x; y† ‡ …ci

c^i †gi Š:

…12†

To simply the analysis, we assume5 that G is differentiable. From (12), we obtain, qPi …^ ci ; xi † ˆ qxi

qEy ‰G…^ ci ; c i ; x; y†Š ˆ Ey ‰p…y†Š qxi

c^i :

Summing (8) and (12) yields the net payoff for bidder i as follows: 4 When xi ˆ 1, for i ˆ 1; . . . ; n, and y assumes only integer values, it is straightforward to show that the settlement rule is equivalent to the second-price, or the highest-rejected-price, rule in Vickrey auction. 5 This can be assured by requiring all the capacity bids k… ? † to be differentiable.

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HUNG-PO CHAO AND ROBERT WILSON

Z Ui …ki ; ci †: L

i …k

i†

L…k† ‡

xi 0

ki …z†dz

k^i xi ‡ Pi …ci ; xi †:

…13†

Using the ®rst variation of GaÃteaux differential, we obtain a necessary optimality condition for bidder i's capacity bid as follows: k^i …xi † ˆ k^i

Ey ‰p…y†Š ‡ ci :

…14†

In the following, we show that (14) is indeed suf®cient and globally optimal. Theorem 2: In a sequential Vickrey auction, the bids …k^i ; c^i †, for i ˆ 1; . . . ; n, represent a set of dominant strategies that implement an ef®cient allocation. Proof: The proof proceeds backward starting with the second stage. Since G…c; x; y† is concave in c and gi …y† [ qci G…c; x; y†, we have G…^ ci ; c

i ; x; y†

 G…ci ; c

i ; x; y†

c^i †gi …y†:

‡ …ci

Therefore, we obtain, h Pi …^ ci ; xi † ˆ Ey G…^ ci ; c

Pi …ci ; xi †

i; x

i ; y†

G…c; x; y† ‡ …ci

i c^i †gi  0:

This proves that bidding the true value of energy cost, c^i , is a dominant strategy in the second stage of auction. Next, we consider the bidding strategy in the ®rst stage, assuming that bidder i adopts the optimal energy price bid, c^i . Suppose that bidder i's capacity bid is ki … ? †. We de®ne hi ˆ k^i ki ; h i ˆ 0 and L…x†: L…k ‡ xh†, where k^i is de®ned in (14). Not that L…0† : L…ki ; k i † and L…1†: L…k^i ; k i †. Since L…x† is concave in x, we have L…k^i ; k

qL…k ‡ xh† ˆ L…ki ; k i †  L…ki ; k i † ‡ qx xˆ0

Z i†

‡

xi 0

k^i …z†

ki …z†dz:

We then compare i-th bidder's net payoff under the two bidding strategies, …ki ; c^i † and …k^i ; c^i †: Ui …k^i † ˆ L…k^i ; k

Ui …ki † Z 

0

xi

k^i …z†

Z i†

L…k† ‡

Z ki …z†dz ‡

0

xi

ki …z†dz

xi 0

ki …z†dz

k^i xi ‡ Pi …^ ci ; xi †

k^i xi ‡ Pi …^ ci ; xi † ˆ 0:

MULTI-DIMENSIONAL PROCUREMENT AUCTIONS

183

Therefore, the bidding strategy, …k^i ; c^i †, is a globally dominant strategy. Finally, combining (6), (10) and (14) yields the following optimality conditions. p…y†

mi …y†  ci ;

l ‡ Ey ‰mi …y†Š

Zi  k^i ;

ci Šgi …y† ˆ 0; ‰l ‡ Ey ‰mi …y†Š Zi k^i Šxi ˆ 0; for i ˆ 1; . . . ; n:

‰p…y†

mi …y†

…15†

Since (15) implements the optimality conditions in (2), we conclude that sequential Vickrey auction achieves an ef®cient allocation. &

References Bushnell, J., and S. Oren. 1994. ``Bidder Cost Revelation in Electric Power Auctions.'' Journal of Regulatory Economics 6: 5±26. Bushnell, J., and S. Oren. 1995. ``Internal Auctions for the Ef®cient Sourcing of Intermediate Products.'' Journal of Operations Management 12: 311±320. California Trust for Power Industry Restructuring 1997. ``Phase II Filing.'' Submitted to the Federal Energy Regulatory Commission. Che, Y. 1993. ``Design Competition Through Multidimensional Auctions.'' Rand Journal of Economics 24: 668±681. Chao, H. 1983. ``Peak-Load Pricing and Capacity Planning with Demand and Supply Uncertainty.'' Bell Journal of Economics 14: 170±190. Crew, M., C. Fernando, and P. Kleindorfer. 1995. ``The Theory of Peak-Load Pricing: A Survey.'' Journal of Regulatory Economics 8: 215±248. Gribik, P. 1995. ``Learning from California's QF Auction.'' Public Utilities Fortnightly, April 15. Hobbs, B., M. Rothkopf, L. Hyde, and R. O'Neill, ``Evaluation of a Truthful Revelation Auction in the Context of Energy Markets with Nonconcave Bene®ts.'' Journal of Regulatory Economics 18: 5±32. Vickrey, W. 1961. ``Counterspeculation, Auctions, and Competitive Sealed Tenders.'' Journal of Finance 16: 8± 37.