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Integrating Contracting and Spot Procurement with Capacity Options

D.J. Wu Department of Management Bennett S. LeBow College of Business Drexel University Philadelphia, PA 19104, U.S.A. Tel: 215 895 2121 Email: [email protected] Paul R. Kleindorfer Department of Operations and Information Management The Wharton School University of Pennsylvania Philadelphia, PA 19104, U.S.A. Tel: 215 898 5830 Email: [email protected] Jin E. Zhang Department of Finance Hong Kong University of Science and Technology Clear Water Bay Kowloon, Hong Kong Tel: 852 2788 7315 Email: [email protected] First Version: June 1999 This Version: May 2001 Keywords: Capacity Options; Bidding; Contracting

Integrating Contracting and Spot Procurement with Capacity Options

Abstract

This paper develops a framework for integrating contract markets with spot markets using tradable options for capacity. Sellers and Buyers may either contract for delivery in advance the contracting" option or they may sell and buy some or all of their output input in a spot market. Contract pricing involves both a reservation fee per unit of capacity and an execution fee per unit of output if capacity is called. The key question addressed is the structure of the optimal portfolios of contracting and spot market transactions for these Sellers and Buyers, and the pricing thereof in market equilibrium. We show that when Sellers properly anticipate demands to their bids, then bidding a contract execution fee equal to variable cost dominates all other bidding strategies yielding the same contract output. The optimal capacity reservation fees are determined by Sellers to trade o the risk of underutilized capacity against unit capacity costs. Buyers' optimal portfolios are shown to follow a merit order or greedy shopping rule, under which contracts are signed following an index that is an increasing function of the Sellers' reservation cost and execution cost. Existence and structure of market equilibria are characterized for the associated competitive game between Sellers, under the assumption that they know Buyer demand functions.

1

1 Introduction

This paper studies the use of contracting mechanisms that rely on capacity options. One or more Sellers compete to supply a set of Buyers in a market in which, in the short run, Seller capacities are xed. Buyers can reserve capacity through options obtained from any Seller. Output on the day can be either obtained through executing such options or in a spot market. Examples of such contract-spot markets abound, and include electric power, natural gas, various commodity chemicals and transportation services. These markets can be expected to become more prominent under e-Commerce. In this paper, we extend the single-Seller results in Wu, Kleindorfer and Zhang 2001a, to multiple Sellers to investigate the impact of competition among Sellers and to characterize the necessary and su cient conditions for a market equilibrium to exist in the contract market. We indicate as well the nature of long-term equilibria, in which Sellers can adjust capacity, anticipating the payo s they will receive from the xed capacity game that results. This long-term game follows the basic structure of the classic Kreps and Scheinkman 1983 paper and illustrate also the nature of e cient technology mixes likely to survive in long-run equilibrium when rms with heterogeneous cost structures compete. This paper may be best viewed as a contribution to the integration of nancial risk management instruments, in this case capacity options, with the classic problem of coordinating supply and demand through markets. We focus here on capacity and production decisions, and not on inventory decisions. Inventory-related issues have been dealt with in recent work on supply chains e.g., Barnes et al. 1998, Brown and Lee 1997, Erhun, Keskinocak and Tayur 2000, Serel, Dada and Moskowitz 2001, Tayur, Ganeshan and Magazine 1999, but these papers neglect competition, nancial and market integration, which are the focus here. The non-storability assumption applies directly to cases such as electric power or dated services like car rentals and airline seats, but also to contexts in which JIT delivery is essential and Sellers and Buyers intend to have continuous deliveries with near-zero inventories in the pipeline. A primary application area for this analysis is the design of electronic exchanges, in which the Internet is used to coordinate supply and demand of Sellers and Buyers, with purchases possible either through contracting markets or through auction-based spot markets. Examples of such exchanges are www.ChemNet.com, 2

Commerxplasticsnet.com, and www.apx.com for electric power, and others available through the New York Mercantile Exchange in the U.S. and similar exchanges throughout Europe, Asia and Latin America. The information being generated from these long-term and short-term contract exchanges is further giving rise to a new range of risk hedging instruments, such as the weather derivatives recently introduced by Enron. Electric power is especially interesting. In the newly restructured electricity market e.g., Chao and Huntington 1998, Kleindorfer, Wu, and Fernando 2001, producing Sellers Generators and Buyers Load Serving Entities and Distribution Companies can sign long-term bilateral contracts to cover Buyer needs, which are in turn derived from the demands of the Buyers' customers. Alternatively, Sellers and Buyers can interact on the day" in a spot market. How much of their respective capacity and demand Sellers and Buyers should or will contract for in the bilateral contracting market, and how much they will leave open for spot transactions, is a fundamental question. Similar options markets are developing in a number of other areas, including semi-conductors see e.g., Brown and Lee 1997 and chemicals and plastics, as noted above. Interestingly, the underlying theory for addressing the associated pricing and contracting questions has not been fully developed, though pieces of the puzzle have been addressed as we now brie y discuss. The problem of commodity pricing, including futures contracts, options and other derivative instruments, has been the subject of study for some time in the nance and economics literature: e.g., Abel et al. 1996, Chambers and Bailey 1996, Deaton and Lazoque 1996, Dunn and Spatt 1999, Kawai 1983, Newbery and Stiglitz 1981, and Schwartz 1997. But these contributions are all incomplete in terms of modeling the interaction between long-run capacity decisions and short-run contracting and output decisions. In the area of contracting and options theory, previous contributions cited above neglect the interaction of xed capacities in the short run and contract and forward options, and nearly all of them neglect the heterogeneity of costs so-called diverse technology" across competing Sellers, which is rather fundamental to most real problems. In the area of diverse technologies, the works of Clark 1961, Crew and Kleindorfer 1976, Hiebert 1989 and Stigler 1939, characterize the cost conditions for production decisions and e cient 3

technology mixes under competition and price uncertainty, but these authors do not consider contract and spot market interactions. This issue has been considered by Allaz 1992, and Allaz and Vila 1993, but with no consideration of capacity constraints or heterogeneous production costs. The framework developed here clari es the well-known ine ciency results of Allaz and Vila 1993 and shows that these are, in fact, partly the result of the structure of forward contracts assumed by Allaz and Vila 1993, and these ine ciencies are reduced or disappear altogether when properly structured options contracts are used in place of xed forward contracts. A natural interpretation of the contracts here is in the real options context, in which Sellers sell a call option on their capacity to Buyers which gives the right but not the obligation to these Buyers to use the capacity, for an additional execution fee, should they exercise the option during the agreed contract period. There is a rich literature on real options, accessible through both texts such as Dixit and Pindyck 1994, and Trigeorgis 1997 as well as through growing literatures in Management Science and related disciplines see, for example, Birge 2000, Huchzermeier and Loch 2001. The central feature of this previous literature is that it treats either single projects or single technologies. Its primary concern is the decision-theoretic analysis of the bene ts from adaptive or contingent responses to uncertainty in project implementation or production decisions. The central focus of the present paper is the integration of contracting and spot markets, with heterogeneous options available to Buyers. In the spirit of Operations Research, our primary focus is on computable pricing strategies for rational market participants. The rest of this paper is organized as follows. Section 2 provides necessary preliminaries on notation and the model. Section 3 generalizes our earlier results from Wu, Kleindorfer and Zhang 2001a to the Multi-Seller, Single-Buyer case, solving this case completely. Section 4 provides a number of numerical examples to highlight key insights from the analysis. Section 5 discusses some interesting generalizations of this framework, including the e ciency of alternative options and forward contract types, initial results on the continuous-time version of the model, straightforward extensions to the Multi-Seller, Multi-Buyer case, and a sketch of the long-run capacity game between multiple Sellers and Buyers. Section 6 concludes with a number of suggestions for future research. 4

2 Preliminaries

Following Wu, Kleindorfer and Zhang 2001a, we use the following notation. = f1; : : : I g: Sellers, with number of Sellers = I J : Number of Buyers P : spot market price, with cumulative distribution function F P and density function f P , where is the mean of the spot market price : Seller i's unit capacity cost per period. Let = ; : : : ; b : Seller i's short-run marginal cost of providing a unit. Let b = b ; : : : ; b K : Seller i's total capacity. Let K = K ; : : : ; K L : Seller i's posted capacity to the contract market. Let L = L ; : : : ; L s : Seller i's reservation cost per unit of capacity if the contract is signed. Let s = s ; : : : ; s g : Seller i's execution cost per unit of output actually used from the contract. Let g = g ; : : : ; g Q : Buyer j 's contract market demand for Seller i's output. Let Q = P Q . Let Q = Q ; : : : ; Q q : Buyer j 's actual contract market consumption of Seller i's output, where q Q ; 8i; 8j . Let q = P q . Let q = q ; : : : ; q x : Buyer j 's actual spot market purchase U z: Buyer j 's aggregate Willingness-To-Pay for output level z D : Buyer j 's total consumption De ne p = s + Gg , with the e ective price function Gv de ned as Z Gv = 1 , F ydy = EfminP ; vg

s s s i

1

I

i

1

I

i

1

I

i

1

I

i

1

I

i

1

I

J

ij

i

j =1

ij

1

I

ij

ij

ij

J

i

j =1

ij

1

I

j

j

Tj

i

i

i

v

0

s

^ where G, is the inverse function of G and p is the short-term equilibrium price of set M , which ^ is a subset of D v: Buyer j 's normal demand function when there exists only the spot market, i.e., D v = argmax fU D , vDg, so that D v = U 0, v. We note below assumptions on U that assure that D is well de ned.

1

sj sj D

0

1

j

sj

j

j

sj

5

De ne D v = D G, v De ne D v = P D v De ne U 0 z = D, z for z 0 m P : The probability that Seller i can nd a last-minute buyer on the spot market when the realized spot price is P . We will typically suppress the dependence of m on P in what follows De ne c = s + Gb , in which s = Efm P P , b g is Seller i's unit opportunity cost on the spot market if the buyer chooses to exercise his contract De ne X M = P 2 K as the total capacity of all Sellers in set M . For every k 2 M , we de ne the following sets depending on the Sellers' bids s; g

j sj J

1

s

j =1

sj

1

s

i

s

s

i

s

i

i

i

i

i

s

s

i

+

i

M

i

M s; g = fi 2 js + Gg s + Gg g

1

k i i k k

M s; g = fi 2 js + Gg s + Gg g

2

k i i k k

M s; g = fi 2 js + Gg = s + Gg g

3

k i i k k

M s; g = fi 2 js + Gg s + Gg g

4

k i i k k

we will usually suppress the dependence of the sets M on s; g whenever s; g is xed and clear from the context.

j k

We make the following assumptions, which we assume to hold throughout the paper Further discussion of these assumptions and conditions is in Wu, Kleindorfer and Zhang 2001a. . A1: In keeping with decreasing marginal utility of consumption i.e., assuming the normal demand curve D is downward sloping, we will assume that the Buyer j 's WTP U z is strictly concave and increasing so that

sj j

U 0 z 0; U 00 z 0; for z 0:

j j

1

Concerning A1, these are standard assumptions on the Willingness-to-Pay function Varian 1992. From A1, it follows that D p is monotonously descending. A2: zD00 z + 2D0 z 0; 8z 0 The intuition of this assumption is the following. A2 is equivalent to R0 0 and R 1 where R = ,U 00 zz=U 0 z is the Arrow-Pratt measure of relative risk aversion on the U function

sj s s

6

however, this does not imply any risk attitude on the Buyers or Sellers in our model below, where both the Buyers and Sellers are risk neutral . This is standard in the nancial economics literature e.g., Rothschild and Stiglitz 1971. A3 No Excess Capacity Condition: Let Sellers' o ers be indexed so that g g : : : g . Then the No Excess Capacity Condition is said to hold if and only if

1 2

I

Q D g ,

i s i

X

i l=1

Q 0;

l

i = 1; : : : ; I:

This No Excess Capacity Condition implies that Buyers will not contract for more than what they are sure they will use if they buy under contract on the day, i.e., if Q 0 then D g P Q . Put another way, this condition says that if Q 0 then the sum of all contracted capacity with execution fees less than or equal to g must not exceed D g . By way of justi cation for this condition, we note that, on the day, the optimal execution order of contracts is to use all contracts available in order of increasing g up to the point at which the next such execution fee exceeds P , satisfying all additional demand from the spot market. Thus, if the No Excess Capacity Condition were violated, it would mean that the Buyer had contracted for some capacity that would guaranteed never be used on the day. This follows since the Buyer would have to be willing to pay at least the execution fee g per unit of output in order to make use of contract capacity. This non-optimal behavior is ruled out by the condition stated. A4 Bid Tie Allocation Mechanism: When there is a bid-tie among a set of Sellers M , then the Buyers' demand for Seller i's output is proportionally allocated to the Sellers according to their bid capacity, thus Q = D^ p . For example, suppose two Sellers have the same bid, and their bid capacities are 40 and 60. If the Buyer demands a total of 50 units from both Sellers, then this assumption indicates that the demand for Seller 1 will be 20 and the demand for Seller 2 will be 30. This would be the expected outcome of a process in which random selection of Sellers with the same bid were undertaken, with visibility of capacity proportional to bid capacity. This type of allocation mechanism makes sense when there are many Buyers the focus of our intended application below and is rather typical in the literature on price competition e.g., Kreps and Scheinkman 1983, Friedman 1988. The results given on the demand side below are not very sensitive to the manner in which bid

i s i i l=1 l i i s i i s i i X M Ki

7

ties are broken, but Seller equilibrium conditions can be a ected by varying assumptions in this regard. The important thing, of course, is that the assumption made matches the actual market allocation mechanism for the speci c application of interest. The von Stackelberg Game: Seller i bids the following information e.g., via an electronic bulletin board: s ; g ; L ; i = 1; : : : ; I . Buyer j decides how much Q to reserve from each Seller i. Given a common knowledge distribution F P of the spot price P , Sellers and Buyers adjust their bids and o ers until equilibrium is reached. This is e ectively a two-period game: Sellers and Buyers sign a contract in advance period 1 and then on the day" period 2 when the spot market price is revealed, decide how much to deliver exercise from the contract and how much to sell purchase on the spot market. Below we show that when Sellers properly anticipate demands to their bids, s ; g ; L ; i = 1; : : : ; I , then the contract execution price should be set at the Sellers' variable cost, i.e., g = b , with reservation fees s determined by Sellers to trade o the risk of underutilized capacity against unit capacity costs. Buyers' optimal portfolios are shown to follow a merit order or greedy strategy shopping rule, under which contracts are signed following the index s + Gb that is an increasing function of both the Sellers' reservation fee s and execution fee b .

i i i ij s s i i i i i i i i i i

3 Multi-Seller, Single-Buyer

This section considers the Multi-Seller-Single-Buyer case. Assume there are I Sellers and one Buyer. We use the subscript i to denote Seller i's costs b ; and Capacity K . This type of cost structure has been studied in the public economics literature, e.g., Williamson 1966, further advanced by Bailey 1972, Bailey and White 1974, Crew and Kleindorfer 1976, and Crew and Kleindorfer 1986. Seller i bids the following information on an electronic bulletin board, s ; g ; L ; i = 1; : : : ; I , while the Buyer posts its demand information on the bulletin board, D P ; s ; g ; Q ; Q ; q : As noted, we assume that the demand function of the Buyer is common knowledge among Sellers. The problem confronting the Buyer is to choose an optimal portfolio of contracts from those available on the bulletin board. Some of the contract o ers carry high subscription rates s but

i i i i i i T s i i i i i i

8

low execution fees g . These must be compared to other o ers carrying lower subscription fees but higher execution fees. We will show below that the Buyer's optimal solution is basically to rank contracts in increasing order of the index s + Gg . This index re ects the cost of reserving capacity at price s plus the cost of maintaining the option to execute the contract at price g rather than utilizing the spot market. This solution enriches the literature on the diverse technology problem in economics Crew and Kleindorfer 1986 to the integrated two-goods" framework as studied here. Imitating the single-Seller case in Wu, Kleindorfer and Zhang 2001a, de ne the Buyer's utility as: Note that this does not imply anything about risk preferences. The Buyer is, in fact, risk neutral in our model since V below is linear in money. The consequences of risk aversion by the Buyer would be to encourage more contracting to avoid exposure to spot market volatility. The assumed decreasing marginal utility in consumption U 00 z 0 simply means that the Buyer's normal demand curve D is downward sloping, since as noted earlier D = U 0, . X X V D ; q; x; = U D , s Q , g q , P x; 2

i i i i i s s

1

I

I

T

T

i

i

i

i

s

i=1

i=1

where = P ; Q is the spot price and the vector of contract capacities, q is the vector of purchases under contract from the Sellers, x is the amount purchased in the spot market, and D is the total consumption of the Buyer, so that X D =x+ q:

s T I T i

Lemma 1: Let = P ; Q be given. W.o.l.g., assume that Sellers' o ers are indexed so that g g : : : g . Let the Buyer's optimal total demand be given by the solution D to the

s

i=1

1

2

I

T

following problem: MaximizeD ; q; x V D ; q; x; ;

T T

3

subject to:

D =x+

T

X

I i=1

q;

i

x 0;

q 0;

X

k l=1

D 0:

T

4 5

Then under A3 the solution D to 3- 4 is

T

D = D P , maxf P , g

T s s k s k

Q j 1 k I g ;

+

l

9

and optimal purchases under contract from Seller i = 1; : : : ; I and from the spot market are given by

q = Q P , g ;

i i s i

x = D ,

T

X

I i=1

q :

i

6

where is the indicator function which takes the value of 1 if its argument is positive and 0 else. Alternatively D may be expressed as D = max D P ; P Q , where Seller k = k provides the last unit of contract output to the Buyer. De ning g = 0 and g = 1, Seller k is determined as the rst Seller in the indicated order of increasing g satisfying g P g . If k = 0, no contract capacity whatsoever is used.

k T T s s l=1 l

0

I +1

i

k

s

k +1

Proof: See Appendix. 2

The optimal quantities 6 purchased under contract from Sellers follow the normal merit order" indicated by the execution fee rank order in the Lemma. Contracts are executed in order up to the point at which the spot price dominates any further available contracts. This implies also, as seen in 6, that either all units of a contract or none are executed on the day, depending on the spot price. We note from Lemma 1 that when facing dual sources spot market and contract market for procurement, the Buyer's demand curve is kinked, as captured in 5. Given the de nition of G, we have the following Theorem 1.

Theorem 1 Buyer's Optimal Contract Portfolio: Let s; g; L be posted bids by the

Sellers. W.o.l.g. assume that Seller bids are ranked in order of the index s + Gg , so that s + Gg s + Gg : : : s + Gg . If GU 0 0 s + Gg , then the Buyer's solution will be to set Q = 0; 8i, i.e., no contracting is optimal. Otherwise, Greedy Contracting in order of the given index is optimal for the Buyer, i.e., the optimal Portfolio of contracts has the form: 8i 2 M ; Q = L ; 8i 2 M ; Q s; g; L = 0; and for i 2 M

i i

1

1

2

2

I

I

1

1

i

1

h

i

i

4

h

i

3

h

X Q s; g; L = P L 3 L D G, s + Gg , L 2 2 1

i

1

i

s

i

i

j

l

M

h

l

7

j

M

h

where h 2 is any Seller there may be more than one in the case of tied bids with the largest 10

value of the index s + Gg satisfying

i i

s + Gg GU 0

h h

X

1 i2M

h

L :

i

8

Proof: See Appendix. 2

The structure of the optimal portfolio captured in Theorem 1 is relatively simple. It calls for the Buyer to rank all o ers in terms of a single index s + Gg and then to pull o as much capacity as allowed by Seller i, proceeding in rank order of the contract index until the marginal WTP is exceeded by the contract index. Since G and therefore G, is strictly increasing G0x = 1 , F x 0, and since by concavity see A1 D is strictly decreasing, and therefore Q is decreasing in the contract index for all i 2 . Before WTP is exceeded, the Buyer takes all capacity o ered by Sellers from whom it contracts. Of course, WTP may be exceeded with the rst Seller and the Buyer may, in fact, sign no contracts whatsoever if GU 0 0 s + Gg . Returning to some of the examples mentioned in the introduction, it is interesting to note the characteristics of the type of contracting captured in Theorem 1. In consolidators for airline tickets or in hotel convention planning, a block of seats or rooms is reserved by a travel planner. The cost of this is the reservation cost s , which gives the travel planner a considerably lower execution cost g than would be obtained in the spot market showing up on the day and requesting a seat or a room. When the travel planner is relatively certain of being able to re-sell the seats or rooms, the bene ts are clear. It is also clear that providing the travel planner with a contracted number of seats or rooms with no subscription fee would put all the risk on the Seller, an unacceptable outcome for the Seller. Bilateral contracts in electricity planning also have this two-part tari structure to assure that both the opportunity cost of capacity reservation and the cost of execution are covered. Notwithstanding these standard practices, to the best of our knowledge, this is the rst characterization of the structure of the optimal portfolio of contracts for a Buyer. We now turn to the Sellers' optimal bidding strategies for s; g; L. We assume throughout that Sellers are risk neutral and maximize expected pro ts from both contracting and spot markets. If Sellers were risk-averse, they would naturally be more active in the contract market, using it as a hedge against low prices in the spot market. We follow standard nance theory in assuming

i i

1

s

i

1

1

i

i

11

that Sellers, except for issues of nancial distress and bankruptcy, are risk neutral. We assume throughout that Sellers face stringent penalties for non-performance under contract so that they will, in fact, post no more than L K as available capacity and that they will set prices s ; g so that contracted amounts will not exceed L .

i i i i i

Lemma 2 Optimal Sellers' Bidding Capacity: L = K ; 8i.

i i

Proof: The pro t of Seller i is given by

s ; g ; L ; P = s Q + g q , K + b q + P , b m P L , q

i i i i s i i i i i i i i s i

+

i

s

i

i

= s Q + g , b q , P , b m P q , K + P , b m P L ;

i i i i i s i

+

i

s

i

i

i

s

i

+

i

s

i

where, from Lemma 1, q = Q P , g . Expected pro t is therefore given by Z1 E s ; g ; L = s Q + g , b 1 , F g Q + L , Q P , b m P dF P Z +Q g , b P , b m P dF P , K : 9

i i s i i i i i i i i i i i i i s i i s s bi gi i i i s i i s s i i bi

Seller i's problem is to maximize its overall pro t from both the contract market and the spot market by deciding how much to bid L on the contract bulletin board, i.e. Maximize i E s ; g ; L . Since @ [email protected] 0, bidding a larger capacity into the contract market relaxes the constraint region since Q L is required and therefore can not decrease Seller i's pro t. 2

i

L

i

i

i

i

i

i

i

i

Lemma 3 Optimal Execution Fee Bids by Sellers: g = b ; 8i.

i i

Proof: From Theorem 1, we know that the Buyer's optimal contract Q s; g will remain uni

changed as long as s + Gg remains constant. To prove the Lemma, we therefore show that for any xed value of s + Gg , the optimal solution for g is b . To see this, note that along any iso-quant of s + Gg = c, we must have: ds = , dGg = ,1 , F g : 10 dg dg Thus, assuming a constant level of contract capacity Q s; g along the iso- quant s + Gg = c, we compute from 10 and 9 that dE = @ E , 1 , F g @ E = ,g , b f g Q ; g b; ,1 , m g g , b f g Q ; g b : dg @g @s

i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i

12

where f is the probability distribution density of spot market price. Therefore, the FOC implies g =b. 2

i i

Lemma 3 establishes that E is a strongly unimodal function along any bid iso-quant of s + Gg = c for which Q 0 except in the degenerate cases where m g = 1 or f g = 0, with a global maximum at g = b unique except in the noted degenerate cases. Thus, we will refer to g = b as a g-dominant bidding strategy for Sellers, since it is optimal no matter what total e ective bid is made, i.e., no matter what the value of s + Gg is. Note that this strategy is not dominant" in the usual game-theoretic sense since the unquali ed term dominant strategy" means that it is optimal for a player no matter what s is and no matter what the strategies are played by other players in the game. As is rather clear, and will be made explicit below, the optimal value of s , and therefore also of s + Gg , bid by a particular Seller will depend on what bids are made by other Sellers. However, the term g-dominant" seems appropriate here given the strong optimality properties of the strategy g = b . The rationale for Lemma 3 is that there is a tradeo for Sellers between charging higher s and higher g , depending on their market power relative to competitors. Charging higher g erodes the options value recall the contract index s + Gg of the bene t Buyers see from contracting more quickly than the marginal bene ts associated with increases in s . The lowest level for g , namely b , is therefore the result.

i i i i i i i i i i i i i i i i i i i i i i i i i i i

Lemma 4 Conditions for both parties to trade: Let g = b , 8i. Then the conditions for

i i

both the Seller and the Buyer to pro t from trade are the following:

GU 0 0 , Gb = s s s = Efm P P , b g; 8i:

def def i i i i i s s i

+

11

Proof: The rst inequality holds since for any Seller i who wants to secure any positive contract

with the Buyer, from Theorem 1, its index has to be less than Buyer's maximum WTP, i.e. s + Gg GU 0 0 ; 8i. Thus, the upper bound for any feasible reservation fee including the optimal is s GU 0 0 , Gb = s . Now we show the second inequality holds. Since g = b , the expected pro t of Seller i is given

i i i i i i i

13

from 9 as E s ; b ; K = s , Efm P P , b gQ + Efm P P , b g , K :

i i i i i i s s i

+

i

i

s

s

i

+

i

i

12

We see from 12 that no contracting Q = 0 would be preferable to any positive contract unless expected contract revenues are superior per unit of capacity committed to the expected revenues available on the spot market, i.e., unless s Efm P P , b g. The same logic establishes assuming g = b that Seller i will earn at least as much from each unit of capacity subscribed by a Buyer in the contract market as i would earn by o ering the same unit in the spot market, as long as s s . 2 The rationale for Lemma 4 is that the Seller does not want to make any less per unit of capacity than it could by committing to sell its capacity into the spot market although access to the spot market is imperfect and occurs on the day only with probability m P . Equating the two possibilities leads to the lower bound of s in Lemma 4. From Lemma 4, we know that when there is a long-term contract market, Seller i could still be used even if Efm P P , b g, but Seller i may or may not break even if is su ciently large, though participation in the contract market, when it is appropriate, will lead in any case to increased pro tability. Recall that we are dealing with the short-run problem here in which capacities are not variable, so that negative pro ts are indeed possible if investments are sunk and capital costs su ciently high relative to what the contract and spot market will bear. Note also from Lemma 4 that if m P is with very high probability near unity, then the Seller will also face diminished incentives to contract, since the spot market then provides a viable alternative to contracting. The reason the Seller nds participating in the contract market attractive at any s s is that the contractor has nothing to lose at such a reservation fee. If any units of the contract are purchased at s ; g s ; b , then the Seller earns at least as much from such units as in the spot market. Whatever portion of the bid capacity is not purchased in the contract market can always be sold on the spot market, incurring precisely the same risk as if the capacity had not been o ered in the contract market. The reason is that we assume Sellers can always sell unused capacity in the spot market subject to market access risk m P , which is not a ected by the amount they bid in the contract market. We will refer in the sequel to s as the minimum

i i i s s i

+

i

i

i

i

i

s

i

i

i

s

s

i

+

i

i

s

i

i

i

i

i

i

i

s

i

14

contract-feasible reservation price strategy for Seller i. Figures 1 and 2 explain the intuition of Theorem 1 and Lemmas 3 & 4 for the single Seller case as well as for the two-Seller case for corresponding numerical examples, see Example 1 and 2 in the next section. Based on Theorem 1, Lemmas 3 & 4, Figure 1 provides a partition of the contract strategy space s; Gg into the following regions: I Both parties want to trade or contract; II No Seller want to contract; III No Seller and no Buyer want to contract; IV No Buyer want to contract. Note also that the left most region when g = 0 or Gg = 0 illustrate a pure forward i.e., must produce must exercise market, while the right most region illustrate the pure spot market i.e., g is very high such that g GU 00 no Buyer want to contract but to rely on the spot market. The optimal contracting is a mixed two-part option contract Point D in the Figure 1. Figure 2 shows the two-Seller case when there is competition in the contract market. Clearly, the Buyer bene ts from such competition and becomes more active in the contract market the feasible trade spread enlarges due to the adding of another Seller. The optimal contract strategy again Point D is to reserve all capacity for the Seller i who provides the lowest index bid s + Gg , and ful ll residual contract demand from the second Seller, i.e., to be greedy. We give several numerical examples in the next section.

i i

Insert Figures 1 and 2 about here We are now in a position to derive market equilibrium prices in the short-term and long-term contract market. Several de nitions of market equilibrium might be used. The approach we take is to assume that Sellers all know the total demand function of the Buyer, consistent with sophisticated Sellers and well-developed markets. Sellers only know their own costs and capacities and bid these into the market via an electronic bulletin board. Sellers adjust their bids until they achieve a Nash equilibrium in this market. We refer to this equilibrium as a von Stackelberg Leader-Follower Equilibrium vSLFE, Fudenberg and Tirole 1991 to account for the assumption that Sellers anticipate Buyer responses to their actions, given what they observe other Sellers to be bidding on the electronic bulletin board. The normal form Fudenberg and Tirole, 1991 game of interest is straightforward. The players are the Sellers. The strategies are the triples s; g; L and the utility functions of the players are the expected pro t functions 12. The von Stackelberg 15

assumption is evident in this game through our substitution into the expected pro t functions of the anticipated Buyer demand functions Q s; g; L as derived from Theorem 1. From 12, we have the following pro t functions in the short run, i.e., neglecting adjustments in capacity:

i

E s; b; K = s , s Q s; b; K + s , K :

i i i i i i i

We are interested for the moment only in short-term pricing strategies satisfying Lemmas 1-4. From Theorem 1, for such strategies, the only interesting parameter is the price" index. From Lemma 4, we know that p c with c = s + Gb . Using this notation, we can express the pro t functions for Sellers given in general by 12 in the following form:

i i i i i

E p = p , c Q p + c , , Gb K ;

i i i i i i i i

where Q p = Q s; b; K is given by our Theorem 1. We will characterize contract equilibria of interest in p-vector space. For notational simplicity, de ne Dz = D G, z; 8z 0 as noted previously. From Theorem 1, we have, 8 K; Dp X M ; Q p = K =X M Dp , X M ; X M Dp X M ; : 0; Dp X M :

i i s

1

k

k

2

k

k

3

k

k

1

1

k

k

k

k

2

k

1

k

k

De nition: For any potential equilibrium set M and any k 2 M , de ne f p = p , c Dp , X M , where M = M n fkg.

k k k k k

0

0

k

k

Lemma 5: If there exists an equilibrium, then it must be symmetric for all Sellers providing

positive capacity in the contract market. That is, every equilibrium p must be of the form ^ p = p ; 8i; 8j 2 M , where Q ^ 0; i 2 M and Q ^ = 0; i 2 n M . p p

i j i i

Proof: Take any supposed equilibrium p and let M be the subset of Sellers at this equilibrium

supplying positive capacity in the contract market. Suppose that the equilibrium p is not symmetric, so that p = Minfp j i 2 M g Maxfp j i 2 M g = p , and such that Q 0 and Q 0. Then, as an equilibrium strategy for player 1, p must be a best response, and moreover Q = K .

1

i i

2

1

2

1

1

1

16

But consider the new bid for player 1: p0 = p + p =2 which is now strictly greater than p and strictly less than p . Now Q = K still obtains, but clearly pro ts are strictly higher for player 1 at p0 than at p , keeping all other players' strategies xed at their equilibrium values. Thus, the original p-vector cannot have been an equilibrium. This contradiction obtains as long as the lowest bid is lower than the highest bid for Sellers in M . 2

1 1 2 1 2 1 1 1 1

^ Given Lemma 5, we denote by p the equilibrium price, such that for those Sellers i 2 M ^ ^ where M is the short-term equilibrium set having positive contract capacity at equilibrium ^ ^ p = p; 8 i 2 M . We rst state a Theorem covering the case in which M is non-singleton, reserving ^ ^ for Corollary 1 the case in which M is singleton.

i

Theorem 2 Non-singleton Short-Term von Stackelberg Leader-Follower Equilibrium: ^ Let K; p; M be any short-term equilibrium, where M K is the equilibrium set of all Sellers ^ ^ ^ ^ having positive capacity contracts, i.e., Q ^ 0; i 2 M and Q ^ = 0; k 2 n M . If Minfc j i 2 p p ^ g GU 0 0, then no Seller will participate in the contract market. Assume M is non-singleton ^ such that jM j 1 and Minfc j i 2 g GU 0 0. Then the necessary and su cient conditions

i k i i

for an equilibrium p to exist are ^ ^ C1: D^ = P 2 K = X M p C2: @f p [email protected] 0 if p p ^ ^ ^ C3: 8k 2 n M; p c .

i

^ M

i

k

k

k

k

k

Proof: See Appendix. 2

Condition C1, noted as the symmetric condition" in Lemma 5, says that in the short-term ^ equilibrium, for any Seller in the money", i.e., for any k 2 M , its entire capacity will be contracted in the contract market. Condition C2 might seem a bit technical to begin with, but it is a standard regularity assumption on the demand function D to assure quasi-concavity of the pro t function in prices. See, for example, Friedman 1988 for a discussion of this and stronger regularity conditions typically imposed in price-based equilibrium analysis. Ours is a special case of the Friedman 1988 assumption for the behavior of the pro t function. In Friedman 1988, it is 17

assumed to be strictly concave, here we only require the function to be non-increasing to the right of the equilibrium price p, where p argmaxf p . Figure 3 illustrates the cases when C2 is ^ ^ satis ed and Figure 4 when C2 is not satis ed. Condition C3 implies that any Seller out of the money" does not have any incentive to join in the short-term contract market equilibrium, as doing so results a net loss in its pro t.

k k

Insert Figures 3 and 4 about here

Corollary 1 Singleton Short-Term vSLFE: When jM j = 1, the only Seller providing positive contract output which we denote as Seller 1 satis es c = Minfc ji 2 g GU 00.

1

i

The necessary and su cient conditions for a single-Seller equilibrium p to exist are i p = ^ ^ Maxfp ; x g c , where p = argmaxf p = argmaxp , c Dp , and x = D, K , and ii p Minfc ji 2 n f1gg. ^

H H

1

H

1

1

1

1

1

H

1

1

i

Corollary 2 Identical Cost Sellers: In the case where all Sellers have the same cost, i.e., ^ 8i; j 2 ; c = c , then if an equilibrium exists see Theorem 2 for conditions it will entail M = .

i j

^ ^ Theorem 3 Uniqueness of set M : If there exists any equilibrium group M , it must be unique.

^ ^ if there are two separate equilibrium groups M and M , then one must be the strict subset ^ ^ of the other, assume M = f1; : : : ; lg M = f1; : : : ; l; l + 1; : : : ; mg, denote p i = 1; 2 as the ^ ^ equilibrium price of M i = 1; 2, then we must have p p . Consider any Seller k 2 fl +1; : : : ; mg ^ ^ ^ ^ ^ which belongs to M but not M , since it does not belong to M , we have p c . On the other ^ ^ ^ hand, M is a larger set than M , we have p p , thus we obtain p c , this suggests Seller k ^ ^ ^ ^ ^ is price out of M , which is a contradiction to the fact that Seller k is indeed a member in M . 2

1 2 1 2

i i

Proof: First we notice that, for any equilibrium group, it must accept group members along the ^ ^ index line, i.e., if Seller k is in the equilibrium group M , then, 8i k must be in M . Therefore,

1

2

2

1

1

1

k

2

1

2

1

2

k

2

2

^ ^ Theorem 4 Computation of set M : M can be computed in the following way. i We index Sellers in order of c , i.e., c c : : : c , for convenience, de ne c = 1, then

i

1

2

N

N +1

18

M = f1; : : : ; h , 1g, where h is the smallest index that satis es, GU 0

k k k h

,1 X

i=1

K c = s + Gb :

i h h h k

^ ii For any k 2 M , if @f p [email protected] 0, when p p, then M is the unique equilibrium group M , ^ otherwise, there does not exist any equilibrium subsets of .

U 0, G, ^ = P 2 K , therefore p = GU 0 P , K . From iii of Theorem 2, we know that p ^ 8k 2 n M; p c , since Seller h does not belong to M , we have p c = s + Gb . Hence we ^ ^ obtain GU 0 P , K c = s + Gb , and identify set M . If every member in this set passes the test of condition ii of Theorem 2, then it satis es all necessary conditions of theorem 2; thus ^ it is the unique equilibrium group M . 2

1 1

h i M i i=1

Proof: From i of Theorem 2 we have D^ = P 2 K , which is equivalent to D G, ^ = p p

i M i s

1

1

i

k

h

h

h

h

i=1

1

i

h

h

h

4 Numerical Examples

Suppose the risk factor m = 0:5; 8i 2 and the spot market price follows an exponential distribution, f y = e, , so the mean of the spot market price is = 30. Then the e ective price function is Gv = ,30e, , 1, where 0 v 1, and thus we have G, v = 30ln , , where 0 v 30. Suppose the WTP function is U z = 30zln + 1, where 0 z 30, it is obvious that this function satis es A1 as follows: U 0 z = 30ln 0 and U 00 z = , 0; thus we have D v = U 0, v = 30e, , where 0 v 1. So the contract market demand function is Dv = D G, v = U 0, G, v = 30 , v, where v 2 0; 30. Numerical Example 1. Assume a single Seller 1 with parameters Gb = 6, = 10, and K = 4:2. From this c is computed via Lemma 4 as c = s + Gb = m , m Gb + Gb = 18. Then Seller 1 would like to bid a price p = 25:8 and her overall expected pro t is ^ E = 25:8 , 184:2 + 18 , 10 , 64:2 = 41:2. This case is illustrated in Figure 5.

i

1 30

y=30

v=30

1

30 30 v

30

z

30

z

30

z

s

1

v=30

s

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Insert Figure 5 about here

Numerical Example 2. Assume another Seller 2 with parameters Gb = 10,

2 2 2 2

= 8, and K = 2:6, so that from Lemma 4, c can be computed to be c = 20. So Seller 1 and 2 can

2

19

form an equilibrium in the contract market at p = 23:2, and Seller 1's equilibrium expected ^ pro t is E = 23:2 , 184:2 + 18 , 10 , 64:2 = 30:2 while Seller 2's expected pro t is E = 23:2 , 202:6 + 20 , 8 , 102:6 = 13:5. As a result of the competition, Seller 1's total pro ts decrease from 41.2 the monopoly case in Example 1 to 30.2, while Seller 2 pro ts from both markets even though it has an inferior technology compared with Seller 1. The sum of the total pro ts for both Sellers is 30:2 + 13:5 = 43:7, which is bigger than 41:2 the one Seller case, showing an increase of the social welfare due to competition. Figure 6 illustrates this case.

1 2

Insert Figure 6 about here

Numerical Example 3. However, still in Example 2, if Gb = 24 so that c = 27, Seller 1 will

2 2

bid the price p = 25:8 as in Example 1 and Seller 2 will no longer being able to participate in the ^ contract market but only take part in the spot market, as shown in Figure 7. Insert Figure 7 about here

Numerical Example 4. In Example 2, suppose Seller 2's e ective cost Gb = 18 so that c =

2 2

24. Then if Seller 1 and 2 can form an equilibrium", then the price would be p = 23:2 c = 24, which leads to Seller 2's withdrawal from the contract market. However, after Seller 2 drops out, Seller 1 will increase the bid the price to p = 25:8 and this will motivate Seller 2 to join in again. So in this case, the contract market will exhibit an oscillating situation with Seller 2's entering and exiting. In particular, no equilibrium exists in the contract market. This oscillating behavior is depicted in Figures 8a and 8b.

2 1

Insert Figures 8a and 8b about here

Numerical Example 5. Still consider Example 2 when there is an equilibrium formed by Seller

1 and 2. If the two plants, b ; ; K and b ; ; K , are owned by a single Seller, either by Seller 1 or by Seller 2, then the single owner will bid prices for the two plants as p = 25:8 and ^ p = 1 i.e., Plant 2 only takes part in the spot market, while Plant 1 participates fully in ^ the contract market. This is because Plant 2 has a higher operating cost and lower capacity cost than Plant 1 and if a symmetric price with Plant 2 is bid, Plant 2's opportunity cost will

1 1 1 2 2 2 1 2

20

be 11 1 2 + 12 2 2 = 18:8, which is dominated by bidding asymmetric prices. Under the noted asymmetric prices, Plant 1 always has larger marginal pro t than Plant 2, with expected pro ts from Plant 1 being 25:8 , 184:2 + 18 , 10 , 64:2 = 41:2 and expected pro ts from Plant 2 being 20 , 8 , 104:2 = 8:4, and overall expected pro ts from both plants of 41:2 + 8:4 = 49:6. Thus, as expected, when a single Seller owns both plants, the resulting monopoly pro t 49.6 is higher than the total pro t 43.7 when these two plants are owned separately by two Sellers in competition.

c K c K K

+K

K

+K

5 Some Extensions

5.1 E ciency of the Two-Part Options Contract

It is well known e.g., Allaz and Vila 1993 that forward markets are generally ine cient under Cournot competition among sellers though these results typically ignore capacity constraints. Our results can be seen to extend this nding to the case of Bertrand-Nash competition with capacity constraints in the short run arguably the natural form of competition for electronic markets. That this is so follows from the following observation. Our two part options s ; g are equivalent to forward contracts when g = 0; if g = 0, then clearly the Buyers will always exercise the contracts on the day since U 0z 0; 8z, and Sellers will therefore be forced to deliver the full amount of any option committed with g = 0. Such a contract is therefore a must-produce, must-take" contract, i.e. a forward. As we note in Lemma 3, however, this contract is strictly dominated by an appropriately designed options contract from the Seller's perspective without changing the Buyer's utility. Thus, any such forward contract is Pareto dominated by some options contract when both contract and spot markets are active. Naturally, if custom features of a product make spot markets infeasible, then contract contracts can still be e cient, especially if they allow better, cheaper planning of production through advance reservation. In a similar fashion, one should note the more intuitively obvious fact that contracts that precommit capacity without a reservation fee, of the form s; g = 0; g are also Pareto dominated. A referee has asked the question whether generalizations to allow for state-dependent options contracts would perform better than the simpler contracts studied here. Such a contract would 21

take the form s; g!; Q! where g and Q both may depends explicitly on the state of the world !. By retracing our arguments in Lemma 2, such contracts are easily shown to be dominated by the two-part options contract studied here. But such contracts might be of interest if either Sellers' costs b depend on ! or if Buyers' demands depend on !, for example, if the strength of Buyer demands depends on the weather". However, the characterization of the optimal Buyer's choices are considerably more complicated when costs or demands are state-dependent, as worked out in detail in Spinler, Huchzermeier and Kleindorfer 2000 generalizing the single-Seller results of Wu, Kleindorfer and Zhang 2001a to the state-dependent case. In particular, we do not yet know whether an appropriately generalized form of Theorem 1 continues to hold when costs or demands are state-contingent; if it does, the logic of the above argument would go through and our two-part options contract would still be optimal. Thus, the resolution of the question raised by the Referee on the more general issue of state-dependent contracts awaits future research.

5.2 Continuous Time

We de ne f P as the probability density function of the spot market price at time T , given the information available until time t T . As in the above analysis, we assume the density function f P is common knowledge. We impose some structure on the evolution of this density function as t progresses. As in standard options theory, we assume that the logarithm of the commodity spot market price X = ln P is normally distributed and follows the Ornstein-Uhlenbeck meanreverting process Dixit and Pindyck 1994 Schwartz 1997 proposes that the commodity spot price P follows a stochastic process given by

t T t T t t t

dP = , ln P P dt + P dz :

t t t t t

Applying Ito's lemma to this process, the logarithm of the price, X = ln P , follows the process 13. :

t t

dX = , X dt + dz

t t

t

13

where is the speed of reversion 0 measuring the degree of mean reversion to the long-run mean log price, ; the second term in equation 13 characterizes the volatility of the process, 22

with dz being an increment to a standard Brownian motion. The following single-Seller result is generalized in Wu et al. 2001 from the results of this paper.

t

Theorem 5 Buyers' Optimal Contracting Policy at time t; Wu et al. 2001: Assume lim ! + U 0 z 1. If st + G gt G U 0 0, then Qst; gt; t = 0 is optimal; otherwise,

z

0

t

t

if the Buyer purchases a capacity contract at time t, then the optimal Qst; gt; t will satisfy the following identity:

st = G U 0 Qt , G gt = E P , gt , E P , U 0 Qt :

t t T ;t

+

T ;t

+

14

The Seller's optimal contract at time t will be of the form st; b where st solves the pro t maximization problem of 12 with the spot price distribution given by f P .

t T

The bidding price st in equation 14 has two terms, in which the rst term E P , gt is essentially the Black-Scholes-Merton BSM, Black and Scholes 1973, Merton 1973 option value and the second term E P , U 0 Qt is related to the willingness-to-pay of the Buyer. The second term is the new feature of the real option we are studying. It shows that the price of the real option here is always smaller than the BSM option price E P , gt . Note that this is the case notwithstanding the fact that in the classical BSM model the contract market is competitive, whereas here the Seller has some market power. The reason that the reservation fee here is bounded above by the BSM option price is directly related to the fact that Buyers can also purchase from the spot market and this constrains the reservation price that can be supported in the market. See also Wu, Kleindorfer, and Sun 2001 that characterizes the longterm capacity choices of competing sellers in the presence of capacity options. We show there that competition among sellers, as expected, further constrains the supportable reservation price level. For bounding results on options, see also the recent work of Bertsimas et al. 2000. It is straightforward to show that, as t ! T , the di erence between the BSM and the option value here st gets smaller and smaller, and eventually disappears at option maturity at t = T . Remark Disincentive for Selling Short; Wu, Kleindorfer and Zhang 2001: Neither the Sellers nor the Buyers have incentives to sell short in the contract market and cover their

T ;t

+

T ;t

+

T ;t

+

23

positions in the spot market. This is true since for any rm either the Seller or the Buyer to engage in such an activity, the net unit expected pro t would be Efs + g fP

i i s

g g , P g = s + g 1 , F g ,

i s i i i

= s + Gg , ,

i i

Z

gi

0

P f P dP

s s

s

0:

From Lemma 4, we know no Buyer j would be willing to sign any contract if the index exceeds the expected spot market price. Thus, s + Gg GU 0 0 must be satis ed if Buyer j is to have an incentive to participate in the contract market. Hence the net unit expected pro t from short sales is strictly negative. Thus, no rm would be interested in such an activity. A key companion result to the above Theorem 5 is that the standard non-arbitrage condition holds for Buyers at the equilibrium contract st; b so that Buyers do not have any incentive to wait to purchase options. This continuous-time result extends the above Remark that, in the context of this paper, neither the Buyers have an incentive to buy excess options and sell these on the spot market on the day, nor do Sellers have an incentive to sell short and cover their positions on the day through spot purchases. These results show that in our framework, at least for the Single Seller, the dynamic or continuous time case can be reduced to the static two period case. If this is true for the multiple Sellers case, then the signi cance of the results reported in this paper would pave the way for a general solution to the capacity options pricing problem for non-storable goods.

i i j

5.3 Multiple Buyers

Now we consider multiple Sellers and multiple Buyers and derive the most general results in this paper. These results will be seen to follow very closely the structure of previous results, so we will spare the reader the full development of the arguments. From the point of view of a particular Buyer, the existence of other Buyers does not a ect its own demand. Thus, on the day, each Buyer's problem remains the same as in the Multi-SellerSingle-Buyer case. Hence the following Lemma 6 is a direct consequence of Lemma 1.

Lemma 6: Let

j

= P ; Q i = 1; : : : ; I , j 2 f1; : : : ; J g be given, where Q is contracted

s ij ij

24

capacity by Buyer j with Seller i. De ne Buyer j 's demand function D as the solution to MaximizefV D ; q ; x ; jD 0; q 0; x 0g, where q and x are purchases by Buyer j under contract and from the spot market, respectively, and where X X V D ; q ; x ; = U D , s Q , g q , P x : 15

j j j j j j j j j j ij j I I j j j j j j j i ij i ij s j i=1 i=1

Then D , q and x are given by Lemma 1 with obvious adjustments for the subscripts j .

j j ij j

Although the presence of other Buyers will not a ect the structure of demand, a key issue when multiple Buyers are present is who will have precedence for the more preferable Sellers those for which the shopping index s + Gg is lowest. As we note below, this will not be an issue in equilibrium, since just as in Theorem 2, so too here the shopping index of all Sellers in the money will be set equal to the market index". But clearly the operation of the market can depend on how competing Buyers are allocated among Sellers. We follow the standard economic assumption that whenever two or more Buyers compete for the same Seller contract, the Buyer with the highest WTP will be the Buyer awarded the Seller's capacity, at least up to the point at which some other Buyer does not have a higher WTP. For a discussion of WTP and other rationing mechanisms under conditions of excess demand, see Crew and Kleindorfer 1986. Under this allocation procedure, Seller i o ering capacity at s ; g will sell just as much as he would to a single Buyer with aggregate demand equal to the sum of the demands of all the Buyers. Thus, the consequence of this standard economic allocation procedure is the following Corollary.

i i i i

Corollary 3 Buyer's Optimal Consumption Portfolio: Given the demand functions D p; p 0; j = 1; : : : ; J , de ne the aggregate Retail demand as D p = P D p and aggregate marginal WTP as U 0z = D, z; z 0. Greedy Contracting is optimal for every

J sj s j =1 sj

1

s

Buyer j , as speci ed in Theorem 1, in the sense that Buyers will pick o contracts from Sellers o ered to them in the order of the Shopping Index s + Gg . In a similar fashion, Lemmas 2-4 in the previous sections hold since the Sellers' problems all remain the same, where now Q is to be understood as aggregate supply to all Buyers by Seller i. In particular, from Lemma 2 and Lemma 3, L = K and g = b continue to be, respectively, dominating and g-dominating strategies for the Sellers. Most importantly, Corollary 3 implies

i i i i i i i

25

that the game among Sellers, and its equilibria following the von Stackelberg model assumed here, is given by Theorem 2, with aggregate Retail demand, as speci ed in Corollary 3, with the inverse aggregate marginal WTP speci ed by U , z = D z. The remaining logic of Theorem 2 and its proof are then identical. The result is the following.

1

s

Corollary 4 Multi-Buyer, Multi-Seller vSLFE: Consider the Multi-Buyer, Multi-Seller

Case in which competing demands by Buyers for a particular Seller are allocated in order of Buyer WTP. For this case, the conditions for the existence of a vSLFE to the Multi-Buyer, Multi-Seller Case fs; g; L j i = 1; : : : ; I g are identical to those speci ed in Theorem 2, with Dp = D G, p, where D p is given in Corollary 3.

i i i s

1

s

Thus, under the assumption that competing Buyer demands for the same Seller contract are rationed in order of Buyer WTP, the solution to the general network case is solved. This general solution results from using the demand and equilibrium contracting building blocks of Buyer contract demand and of the market equilibrium results for the Multi-Seller, Single-Buyer case. The equilibrium characterized in both Theorem 2 and Corollary 4 has important prescriptive properties that should be mentioned in concluding our discussion. The g-strategy is g-dominant for the contract capacity provided, the capacity strategy is a dominant strategy, and the s-strategy is the transparent outcome of the pivot Seller h who is the price setter in the contract market. Whether this equilibrium is stable or has other important properties behaviorally than the dominance properties noted remains to be seen through further theoretical and experimental work. The key matter to note here is that once the game is understood to be completely speci ed in terms of the shopping index s + Gb , considerable degrees of freedom are removed from the problem. Most importantly, the above results establish this index as the critical focal point for rational strategies for a particular Seller, whatever strategies are played by other Sellers.

i i

5.4 Long Run Equilibrium

The above results are short-run in the sense that capacities K are assumed xed. It is of some interest to understand the consequences for optimal capacity adjustment in the long run of having an integrated contracting and spot market of the type examined here. We sketch here the basic 26

results for the long run, where we assume, as in Kreps and Scheinckman 1983, that the long-run capacity setting game is played in the anticipation that thereafter the short-term options-pricing game studied above will materialize. We seek a sub-game perfect equilibrium Fudenberg and Tirole, 1991 for the long-run capacity game. Given this structure, the pro t function for any Seller k in the long-run capacity game will be the anticipated outcome of the short-term options game, i.e. rearranging 12: E ^; K = ^ , c Q + c , p p

k k k k k k

, Gb K

k

k

Seller k's problem is to choose an optimal capacity K to maximize k's long-run expected pro t, i.e.,

K = argmaxE ^K ; K : p

k k

The basic results for this long-run game can be summarized as follows for detailed results, see Wu, Kleindorfer and Sun 2001. Lemma 7 Wu, Kleindorfer and Sun 2001: Let pK be the short-run equilibrium price ^ ^ ^ and let M K be the set of Sellers in the contract-market equilibrium. Assume M K is non^ ^ singleton so that jM K j 1. The case of singleton when jM j = 1 is somewhat di erent, as was specially dealt with in the short-term case previously; for details, see Wu, Kleindorfer and Sun 2001. Then, in the long-run capacity game, the best response capacity strategy for each Seller ^ k 2 M K is ^ , K = maxf p , p , cGb X; 0g: 16 ^ Corollary 5 Wu, Kleindorfer and Sun 2001: De ne

- a modi ed Tobin's marginal q for Seller k as: @ p , c Dp [email protected]

= @ + Gb , c K [email protected] 17 Let K ; p; M be a long-run equilibrium solution. Then, whether M is singleton or not, for any Seller k 2 M ,

= 1. When j M j 1, then p must satisfy in addition to being a short-run equilibrium price corresponding to K X + Gb , c =j M j ,1: 18 p , c 2

k k k k k k Kk X k k k k k k k k i i i i M i

27

Theorem 6 Wu, Kleindorfer and Sun, 2001: The long-term equilibrium set M ,

which may be empty, is characterized by the following algorithm. Index Sellers in the order of c , i.e., c c : : : c and set M = . i p = argmaxp , , Gb Dp. If c + Gb then exit else if p c , then M = f1g exit. Else M = f1g and i = 2. ii Loop While p + Gb and + Gb c begin M = M Sfig: compute p M via 18. if i I then i = i + 1 else exit. end. iii If p c and c + Gb then M = . iv If @f p [email protected] 0 and p p then M = . An immediate insight of Theorem 6 is that the structure of the long-run equilibrium, if it exists, entails a segmentation of Sellers into three non-overlapping subsets, ordered by the index c : Segment 1 consists of those Sellers with the lowest index values c and consists of Sellers that participate in both the contract market and the spot market; Segment 2 consists of Sellers with the next lowest index values c , which participate only in the spot market; Segment 3 are Sellers who are unable to compete in either market and exit in the long run. The characterizing condition for long-run equilibrium is that Tobin's marginal q be equal to unity for all Sellers, with the de nition of Tobin's marginal q Abel et al. 1996, Tobin 1969 modi ed to re ect the presence of two markets the contract and the spot for procurement. This brings new insights in putting together the traditional conditions for e cient equilibrium based on Tobin's marginal q with options markets. See Dixit and Pindyck 1994, pp. 4-7 for a discussion. A further issue of some interest in the present context is the characterization of e cient technology mixes in long-run equilibrium. This has been discussed by Allaz 1992, Allaz and Vila 1993, and Crew and Kleindorfer 1976. The conditions characterizing the e cient mix are extended here to account for the integration of the two markets of interest. The usual conditions

i

1

2

I

1

1

1

1

1

2

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

28

are only cost-based where the e cient technology mix embodies tradeo s between unit capacity costs and unit variable costs across di erent technologies. These conditions are generalized in the present context to account for the interaction of each technology with the characteristics especially the volatility of the spot market and access conditions m P .

i s

6 Summary

This paper has provided a general solution to the Multi-Seller, Multi-Buyer procurement problem in an integrated contract-spot framework with capacity options. The key question addressed is the structure of the optimal portfolios of contracting and spot market transactions for these Sellers and Buyers, and the pricing thereof in market equilibrium. We show that when Sellers properly anticipate demands to their bids, then a g-dominant strategy for the contract execution price is to truthfully reveal the Sellers' variable cost, with reservation fees determined by Sellers to trade o the risk of underutilized capacity against unit capacity costs. Buyers' optimal portfolios are shown to follow a merit order or greedy shopping rule, under which contracts are signed following an index that combines the Sellers' reservation cost and execution cost. These results are extensions of our earlier theoretical results for the Single- Seller case to the Multi-Seller Multi-Source case. This solution has important applications in a number of industrial and service sector contexts where capital intensity and non-storability are essential operations characteristics. Non-storability including cases of JIT delivery is important to our analysis as is the implicit assumption throughout of capital intensity of the production process. Absent either of these and inventory or rapid scale-up of production would be suitable substitutes for long-term contracting or spot market purchases. A number of future research topics are evident. It would be interesting to explore the opportunities for both market research and Seller Buyer contracting support systems e.g., the design of electronic exchanges in e-Commerce based on these results in sectors of interest. As noted, on the theoretical side, several areas are potentially interesting. Concerning the present model, extensions to allow the WTP function to depend on the state of the world ! would be useful. On the supply side, it would be useful to consider the case in which bids into the spot market required 29

an additional preparation cost think of this as a xed charge or set-up cost to maintain a plant in ready condition", incurred before the state of the world ! is known. Such costs are typical in many capital intensive industries and could be expected to make contracting more desirable since contracting would allow plants to recover such xed preparation costs in contract fees. It would also be important to extend this framework to the case of limited storability of goods, as well as to the case where contracting or re-contracting could take place along a temporal continuum for the multiple sellers case. In the latter case, options pricing results can be expected to re ect the changes in information about the value of the spot price as time progresses to the day of physical delivery. We would expect, based on the results of this paper, that not only will the stochastic evolution of the spot market price be important to valuing such options, but also the evolution of predicted access to the market by various Sellers the m P of our framework. Since one might expect that spot market price and m P are likely to be correlated depending on congestion and on search intensity of Buyers, the resulting framework could be quite interesting. In particular, it could link to the e ects of Internet access and shopping since such access is likely to increase the probability of nding customers at the last minute. These belong to ongoing research projects.

i s i s

7 Acknowledgement

Corresponding author is Dr. D.J. Wu, Drexel University, 101 North 33rd Street, 324 Academic Building, Philadelphia, PA 19104. Tel.: +1-215-8952121; fax: +1-215-8952891. Email address:[email protected] D.J. Wu. D.J. Wu thanks Yanjun Sun for his excellent graduate assistance as well as nancial support by a mini-Summer research grant and a research fellowship by the Safeguard Scienti cs Center for Electronic Commerce Management, LeBow College of Business at Drexel University. Partial support of this research by Project ADVENTURES and the German Ministry for Education and Research BMBF is also gratefully acknowledged. The work of Jin E. Zhang has been supported by City University of Hong Kong. Partial results in this paper have been presented at the seminars of City University of Hong Kong Hong Kong, China, CMU, LeBow College of Business, Drexel University, INSEAD France, London Business School UK, Naval Postgraduate School, PECO Energy, the Smeal College of Business, Pennsylvania 30

State University, UCLA, WHU Germany, as well at the 34th Annual Hawaii International Conference on System Sciences Wu, Kleindorfer and Zhang 2001b. We thank the participants for their comments. Helpful discussions with Arnd Huchzermeier, Christopher Loch, Tim Mount, Shmuel Oren, and insightful comments from the anonymous referee, the AEs, and participants at the above mentioned seminars or conferences are also acknowledged.

8 Appendix

Proof of Lemma 1 Take any i 2 and any xed total demand D . Contract purchases will be in order of increasing

T

execution fees g as long as g P . We can therefore assume that, for any l i, all capacity under contract l will be used prior to taking any output under contract i. Now let k 0 be the Seller providing the last unit of contract output, so that k satis es g P g . Under A3, the Buyer will never contract for more than needed, so that for every k 2 , Q = 0 unless P Q D g or equivalently, since D = U 0, , g U 0 P Q . Thus, on the day, depending on the spot market price, the Buyer faces only the following two cases under which q 0 and q = 0 for i k:

i i s k s k +1 k k i=1 i s k s

1

k

k

i=1

i

k

i

1. g P U 0 P Q ;

k k s i=1 i

2. g U 0 P Q P :

k k i=1 i s

Using the fact that Seller k provides the last unit of contract capacity, we have q = Q for i k, q = 0 for i k and q Q . Thus, noting that P maxfg ; : : : ; g g, we obtain the following expression for Buyer's utility V after substituting the derived expressions for optimal q and x = D , P q as functions of D :

i i i k k s

0

k

i

T

i

T

V D ; qD ; xD ;

T T T

= U D ,

T T

X

I i=1 s

sQ ,

i i I T

X

I i=1 i

g q , P D ,

i i s T k

X

I i=1 i

q

i k

= U D , P D ,

X

i=1

sQ +

i

,1 X

i=1

P , g Q + P , g min D ,

s i s k T

,1 X

l=1

Q ;Q :

l k

19

31

To determine D we need to maximize this concave function V D ; q; x; over D 0. We do this in two steps. In step 1 which corresponds to case 1 above, we solve this problem where the min is the second quantity in the in 19. This leads directly to D = D P . In step 2 which corresponds to case 2 above, we solve this problem where the min is the rst quantity in the in 19, i.e., subject to D P Q . Since in any case g U 0 P Q , this leads to D = P Q . Combining these two cases, shows the validity of the alternative characterization of D in the statement of Lemma 1. The characterization 5 is easily seen to be equivalent to this, so that Lemma 1 follows. 2

T T T T s s k T l=1 k k k i=1 i k T i=1 i T

Proof of Theorem 1

The reader should keep in mind in this proof that the order assumed in this Theorem, implied by non-decreasing values of the index s + Gg , may be a di erent order than the order assumed in Lemma 1 in terms of non- decreasing values of g . Assume s + Gg GU 0 0 since otherwise the Buyer is not willing to pay the e ective price of even the lowest cost Seller i.e., otherwise the Buyer will have no incentive to accept any contracts. We rst note that since s + Gg is nondecreasing in i by assumption and since GU 0 P , L is nonincreasing in i by the properties of G and U 0 , then if Seller h has the largest value of the index s + Gg satisfying 8, then every i h will also satisfy the following condition:

i i i

1

1

i

i

i

l=1

1

l

i

i

s + Gg s + Gg

i i i+1 i+1

X GU 0 L ;

i l l=1

20

with the left inequality following from the assumed ordering in terms of the shopping index. For suppose the right inequality in 20 did not hold at some i h. Then again by the ordering of the index and the monotonicity of U 0 and G, we would have:

s + Gg s + Gg GU 0 L GU 0

i h h i+1 i+1 l l=1

X

h

,1 X

l=1

L ;

l

contradicting the de nition of h in 8. We note, in particular, from 20 and the monotony of G that the optimal Q ; i 2 , given in the Theorem statement satisfy the No Excess Capacity Condition.

i

32

Now for any = P ; s ; g ; Q, we can substitute the optimal contract and spot purchases q and x from Lemma 1 to obtain the following simpli ed form of the Buyer's utility 2 as a function of X X V = U D , P D , s Q + P , g Q ; 21

s i i i I I T s T i i s i

+

i

i=1

i=1

where D is given by 5. De ne EV Q as the expected value w.r.t. P of V . Clearly there exists an optimal solution to the problem of maximizing the EV Q since V is continuous in its arguments and we can, w.o.l.g., restrict attention to the compact set of non-negative contract levels Q for which the No Excess Capacity Condition is satis ed. To characterize the optimal contract amounts, we derive FOCs by considering the derivatives of V for a xed realization of P and then take the expected values of these derivatives w.r.t. P . From 21 we obtain for any i 2 that

T s i s s

@V = U 0 D , P @D , s + P , g ; 22 @Q @Q where D is given by 5. We wish to evaluate @D [email protected] . ^ Let k be the index of the Seller with the highest execution fee for which contract capacity is non-zero, i.e. g = maxfg j 1 i I ; Q 0g. Then the following relation holds for any k such that Q 0:

T T s i s i

+

i

i

T

T

i

^ k

i

i

k

g g U 0 Q U 0 Q :

I k k

^ k

X

l=1

X

l=1

l

l

23

The rst inequality follows since g is the maximum execution fee for non-zero callable capacity. ^ ^ The middle inequality is essentially equivalent to A3 since, by de nition of k, Q = 0; 8i k, so that P Q = P Q . The nal inequality in 23 follows since U 0 = D, is monotone decreasing. From Lemma 1 we know that D = maxD P ; P Q , where k is the last contract executed given . Thus, D depends on Q if and only if, for some k i, P Q D P and g P g . Using the Non Excess Capacity Condition, we further note that if Q 0, then D g P Q . We conclude from this and 23 that @D = 1 i 9 k with Q 0 and g g U 0 P Q P ; 0 else: @Q

^ k

i

^ k

l=1

I

l

l=1

l

1

s

k

T

s

s

l=1

l

k

T

i

l=1

l

s

s

k

s

k +1 k

k

s

i

l=1

l

T

k

k

i

k

l=1

l

s

i

33

Since Gv = E fmin v; P g = E fP , P , v g, the rst two terms in this expression reduce to GU 0 P Q , E fP g and E fP g , Gg , respectively, yielding @ EV Q = GU 0X Q , s + Gg : 24 @Q

s s s

We also note from 23 that precisely when @D [email protected] = 1, we have D = P Q , since if P U 0 P Q for any k i with Q 0, then P U 0 P Q . Thus, substituting D = P Q and interchanging expectation and di erentiation, we nally obtain from 22 Z1 X @ EV Q = Z 1 U 0 Q , P dF P + P , g dF P , s : P 0 @Q =1

I T i T l=1 l k s l=1 I l k s l=1 l I T l=1 l I i U

^ But, given the de nition of k and 23, this is equivalent to the following: ^ @D = 1 i i k and U 0 P Q P ; 0 else: @Q

T I l=1 l s i

I l

Ql

l

s

s

s

i

s

i

l=1

gi

+

I

l=1

l

s

s

i

I

l

i

i

i

l=1

Now note that since the rst term is independent of i in 24, @[email protected] is monotonic decreasing in s + Gg . Thus, if for some k 2 g, @[email protected] 0, then @[email protected] 0 for any i such that ^ s + Gg s + Gg . Finally, note that for the contract with the maximum such index, say h, with Q 0, we must have @[email protected] 0 with inequality unless Q = L . It follows that every ^ contract with a value of the index s + Gg no greater than h will have @[email protected] 0 so that ^ for such contracts Q = L . From this, we see that, in fact, h = h as de ned in 8. The Theorem follows. 2

i i i k i i i k k

^ h

^ h

^ h

^ h

i

i

i

i

i

We rst note that if the assumption Minfc ji 2 g GU 00 is not satis ed, then there would be no positive contract equilibrium since, as noted in Theorem 1, in this case no Seller would be able o er a pro table contract that would attract non-zero Buyer demand. Su ciency. First we show that if there is a set M that satis es i and ii, then 8k 2 M , the optimal contract price for k is to charge p = p as computed via i. Furthermore, denote p, as ^ the price vector of other players in set M . Given p, = p, k has no incentive to either decrease or ^ increase its price p from p. ^ Case a: Choose any k 2 M , and suppose, p p. We see immediately that all k's capacity ^ will be used, i.e., Q p ; p, = K , that gives him a pro t of E p ; p, = p , c K .

i k k k k k k k k k k k k k k k

Proof of Theorem 2

34

By de nition, we have E ^ = ^ , c D^ p p p E ^ = ^ , c K . Thus we obtain p p

k k k k k

X M

Kk

, by i, we get D^ = X M , therefore, p 25

E p ; p, = p , c K

k k k k k

k

^ , c K = E ^; p p

k k k

which shows that Seller k has no incentive to decrease its price given that other players are charging p. ^ ^ ^ p Case b: Choose any k 2 M , and suppose, p p. Since p p and from i we have D^ = X M , since the demand function D is strictly decreasing, we get Dp ; p, D^ = X M , p therefore, Seller k's capacity cannot be fully used, i.e., Q p K : We see,

k k k k k k k

lim ! E p ; p, = ^ , c Q ^ = ^ , c K = E ^: p p p p

pk p ^ k k k k k k k k

If we take limits on both sides of equation 25, we get lim ! E p ; p, = ^ , c K = E ^: p p

pk p ^ k k k k k k

Hence, E p ; p, is continuous at p. Since p ^

k k k k k k k k k k k k

k

p, we have ^

k k k k k

E p ; p, = f p ; p, = p , c Q p ; p, = p , c Dp ; p, , X M :

k

0

k

26

Take derivatives of w.r.t. p in the above equation, we have

k

@f p ; p, = Dp ; p , X K + p , c @Dp ; p, = Q p ; p + p , c @Q p ; p, : , , @p @p @p 2 0

k k k k k k k k k k i k k k k k k k k i M

k

k

k

From ii, we have @f p [email protected]

k k

k

0 if p

k k k

k

p, we obtain ^

k k k k k

@ p ; p, = @f p ; p, 0: @p @p

We conclude that, E p ; p, = f p ; p, decreases if p p. Hence Seller k has no incentive ^ to increase its price given other players' charge p, = p. ^ ^ Condition C3 insures that no one outside group M would have any incentive to charge a price p, since doing so will lead to negative pro ts by Lemma 4. ^ ^ Necessity. Assume there exists an equilibrium set, M , of positive contract Sellers. By Lemma 5, ^ ^ we can further assume all Sellers in M charge an identical price p. Consider any Seller k 2 M , ^

k k k k k k k k

35

^ since M is an equilibrium, k has no incentive to deviate from the equilibrium price p. That is to ^ say, if we x other Sellers' strategy at p, k does not want to decrease or increase his contract price ^ p . We discuss these two cases accordingly. ^ Case a: Seller k does not want to decrease its price by charging p p. If he decreases his price, then he would either extract all the demand from the Buyer by being the rst and last unit ^ provider Q p ; p, = Dp ; p, , leaving nothing for other Sellers in M , or still be able to sell all its capacity Q p ; p, = K . Clearly he has incentive to do so in the former case, since lim ! E p ; p, = ^ , c D^; p, ^ , c D^; p, XKM = E ^; p, : p p p p p The fact that he does not decrease his price indicates that the latter is true, i.e., Q p ; p, = K , and E p ; p, E ^; p, . Therefore p E p ; p, = p , c K E ^; p, = ^ , c D^; p, K ^ : p p p X M or equivaIf we let p ! p on both sides of the above equation, we obtain, K D^; p, ^ p ^ lently, D^ X M . p Case b: Seller k does not want to increase price by charging p p. If he increases his ^ price, then he would either be able to sell all its capacity Q p ; p, = K or ful ll the residual ^ ^ demand from other Sellers in group M at Q p ; p, = Dp ; p, , X M 0. Clearly he has incentive to do so in the former case since he enjoys more pro t by increasing his price and selling the same output, the fact that he does not increase price indicates that the latter is true, ^ i.e., Q p ; p, = Dp ; p, , X M , and E p ; p, E ^; p, . Therefore p ^ E p ; p, = p , c Dp ; p, , X M E ^; p, = ^ , c D^; p, K ^ : p p p X M ^ If we let p ! p in both side of the above equation, we obtain, D^; p, ,X M D^; p, ^ p p 0 ^ ^ or equivalently, D^; p, p X M , or D^ X M . p If we combine the results of case a and case b, we obtain C1 in the Theorem 2, i.e., ^ D^ = X M . Furthermore, the above reasoning shows that E p ; p, is continuous at p, p ^ and decreases if p p, so that it is straightforward to obtain C2. Condition C3 holds since if ^ ^ ^ ^ 9j 2 n M; p c , the Seller j has an incentive to join in M , which contradicts our assumption ^ that M is an equilibrium group. 2

k k k k k k k k k k k k pk p ^ k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k

^ X M

Kk

k

k

k

k

k

k

k

k

k

k

0

k

k

k

k

k

k

0

k

k

k

k

k

k

k

k

k

k

k

k

k

0

k

k

k

k

k

k

k

k

0

k

k

^ X M

Kk

X M

k

^ k ^ X M

0

k

k

k

k

k

j

36

References

1 Abel, A., A. Dixit, J. Eberly AND R. Pindyck. 1996. Options, The Value of Capital, and Investment. The Quarterly Journal of Economics August, 753-777. 2 Allaz, B. 1992. Oligopoly, Uncertainty and Strategic Forward Transactions. International Journal of Industrial Organization 10, 297-308. 3 Allaz, B., AND J.-L. Vila. 1993. Cournot Competition, Forward Markets and E ciency. Journal of Economic Theory 59, 1-16. 4 Bailey, E. 1972. Peak Load Pricing with Regulatory Constraint. Journal of Political Economy 80, 662-679. 5 Bailey, E., AND L. White. 1974. Reversals in Peak and O peak Prices. The Bell Journal of Economics and Management Science 51, 75- 92. 6 Barnes-Schuster, D., Y. Bassok AND R. Anupundi. 1998. Coordination and Flexibility in Supply Contracts with Options. Working paper, J.L. Kellogg Graduate School of Management, Northwestern University, Evanston, IL. 7 Bertsimas, D., I. Popescu and J. Sethuraman, 2000, Moment Problems and Semide nite Optimization" Working Paper, Sloan School of Management, MIT, April. 8 Birge, J. 2000. Option Methods for Incorporating Risk into Linear Capacity Planning Models Manufacturing & Service Operations Management 2:1. 9 Black, F., AND M. Scholes. 1973. The Pricing of Options and Corporate Liabilities. Journal of Political Economics 81, 637-654. 10 Brown, A. O., AND H. Lee. 1997. Optimal 'Pay-to-Delay' Capacity Reservation with Application to the Semiconductor Industry. Working paper, Department of Industrial Engineering and Engineering Management, Stanford University, Stanford, CA. 11 Chambers, M.J., AND R.J. Bailey. 1996. A Theory of Commodity Price Fluctuations. Journal of Political Economy 1045, 924-957. 37

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25 Kleindorfer, P., D.J. Wu AND C. Fernando. 2001. Strategic Gaming in Electric Power Marktes. European Journal of Operational Research 1301, 156-168. 26 Kreps, D., AND J. Scheinkman. 1983. Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes. Bell Journal of Economics 14, 326-337. 27 Merton, R. 1973. The Theory of Rational Option Pricing. Bell Journal of Economics and Management Science 4, 141-183. 28 Newbery, D., AND J. Stiglitz. 1981. The Theory of Commodity Price Stabilization: A Study in the Economics of Risk. Oxford University Press, New York. 29 Rothschild, M. AND J. Stiglitz. 1971. Increasing Risk: II. Its Economics Consequences. Journal of Economic Theory, 3, 66-84. 30 Schwartz, E.S. 1997. The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging. Journal of Finance L II3, July, 923-973. 31 Serel, D., M. Dada AND H. Moskowitz. 2001. Sourcing Decisions with Capacity Reservation Contract. European Journal of Operational Research 1313, 635-648. 32 Spinler, S., A. Huchzermeier AND P. R. Kleindorfer. 2000. Capacity Allocation and Pricing of Capital Goods Production. Working Paper, Wharton Risk Management and Decision Processes Center, The University of Pennsylvania, November. 33 Stigler, G. 1939. Production and Distribution in the Short-Run. Journal of Political Economy 305-327. 34 Tayur, S., R. Ganeshan AND M. Magazine. 1999. Quantitative Models for Supply Chain Management. Kluwer Academic Publishers, Boston. 35 Tobin, J. 1969. A General Equilibrium Approach to Monetary Theory. Journal of Money, Credit and Banking February, 15-29. 36 Trigeorgis, L. 1997. Real Options. MIT Press. 39

37 Varian, H. 1992. Microeconomic Analysis 3rd edition. W. W. Norton & Company, Inc. 38 Williamson, O. 1966. Peak Load Pricing and Optimal Capacity under Indivisibility Constraints. The American Economic Review 564, 816-827. 39 Wu, D.J., P. Kleindorfer AND Y. Sun. 2001. Optimal Capacity Expansion in the Presence of Capacity Options. Working Paper, Department of Operations and Information Management, The Wharton School, University of Pennsylvania. 40 Wu, D.J., P. Kleindorfer, Y. Sun AND J. E. Zhang. 2001. The Price of a Real Option on Capacity. Working Paper, Department of Operations and Information Management, The Wharton School, University of Pennsylvania. 41 Wu, D.J., P. Kleindorfer AND J. E. Zhang. Forthcoming 2001a. Optimal Bidding and Contracting Strategies for Capital-Intensive Goods. European Journal of Operational Research. 42 Wu, D.J., P. Kleindorfer AND J. E. Zhang. 2001b. Optimal Bidding and Contracting Strategies in the Deregulated Electric Power Market: Part II. In Proceedings of the Thirty-fourth Annual Hawaii International Conference on System Sciences CD ROM, R.H. Sprague, Jr. IEEE Computer Society Press, Los Alamitos, California.

40

s

I IV D s II 0 G(b) Pure Forward Market G(U'(Q)) µ III G(g) Pure Spot s+G(g)=µ s+G(g)=G(U (Q))

'

Figure 1: Illustration of Optimal Bidding and Contracting Strategies One Seller and One Buyer. I: Feasible Trade (Contract) for the Buyer and the Seller; D: Optimal contracting for the Buyer and for the Seller; II: No Seller wants to contract; III: No Seller and no Buyer wants to contract; IV: No Buyer wants to contract.

s s+G(g)=µ s+G(g)=G(U'(K2)) s+G(g)=G(U'(K1)) s+G(g)=G(U'(K1+K2)) s G(U'(K1+K2)) µ G(g) G(b2)-G(b1) Figure 2: Illustration of Optimal Bidding and Contracting Strategies Two Sellers and One Buyer. The Buyer benefits from such competition and becomes more active in the contract market (the feasible trade spread enlarges due to the adding of another Seller). The optimal contract strategy (again Point D) is to be "greedy". 0 41

D

Ek

f k ( pk )

0

ck

pkH

^ p (non-singleton) xH (singleton)

pk

Figure 3: An Illustration when Condition 2 in Theorem 2 is satisfied.

Ek

^ xH (singleton) or p (non-singleton) is the market clearing price, i.e. would be the equilibrium price if existed.

f k ( pk )

0

ck

xH ^ p

pkH

pk

Figure 4: An Illustration (Counter-Example) when Condition 2 is violated.

42

18 c1

25.8 p^

Figure 5: Illustration of Numerical Example 1 (One Seller). Technology parameters: G(b1) = 6, 1 = 10, K1 = 4.2. The optimal price is 25.8.

18 c1

20 c2

23.2 p^

Figure 6: Illustration of Numerical Example 2 (Two Sellers). Technology parameters: G(b1) = 6, 1 = 10, G(b2) = 10, 2 = 8; K1 = 4.2, K2 = 2.6. Two sellers form an equilibrium group on the contract market and set the optimal bid to be 23.2, which is determined by their total capacity, K1 + K2.

18 c1

25.8 p^

27 c2

Figure 7: Illustration of Numerical Example 3 (Two Sellers). Technology parameters: G(b1) = 6, 1 = 10, G(b2) = 24, 2 = 8; K1 = 4.2, K2 = 2.6. Seller 1 sets the contract bid price to be 25.8 < c2. Seller 2 only participates on the spot market.

43

18 c1

23.2 p1

24 c3

25.8 p2

Figure 8a: Illustration of Numerical Example 4 (No equilibrium with Two Sellers). Technology parameters: G(b1) = 6, 1 = 10, G(b2) = 18, 2 = 8; K1 = 4.2, K2 = 2.6. p1 and p2 represent respectively the optimal prices with Seller 2 and without Seller 2.

26

25.5

Contract Market Price

25

24.5

24

23.5

23 1 2 3 4 5 6 7 8 9 10

Time

Figure 8b: The oscillation bidding behavior when there are two Sellers competing with each other.

44

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