Stepwise Investment and Capacity Sizing under Uncertainty

Transcription

1 OR Spectrum manuscript No. (wi be inserted by the editor Stepwise Investment and Capacity Sizing under Uncertainty Michai Chronopouos Verena Hagspie Stein Erik Feten Received: date / Accepted: date Abstract The reationship between uncertainty and manageria fexibiity is particuary crucia in addressing capita proects. We consider a firm that can invest in a proect in either a singe (umpy investment or mutipe stages (stepwise investment under price uncertainty and has discretion over not ony the time of investment but aso the size of the proect. We confirm that, if the capacity of a proect is fixed and the investment premium associated with stepwise investment is positive, then umpy investment becomes more vauabe than a stepwise investment strategy under high price uncertainty. By contrast, if a firm has discretion over capacity, then we show that the stepwise investment strategy aways dominates that of umpy investment. In addition, we show that the tota amount of instaed capacity under a stepwise investment strategy is aways greater than that under umpy investment. Keywords investment anaysis capacity sizing fexibiity rea options Acknowedgements The authors woud ike to express their gratitude to Peter Kort for his vauabe comments that heped improve the paper. Michai Chronopouos University of Brighton, Schoo of Computing Engineering and Mathematics, Brighton, United Kingdom Department of Business and Management Science, Norwegian Schoo of Economics, Bergen, Norway Te.: E-mai: Michai.Chronopouos@nhh.no Verena Hagspie Norwegian University of Science and Technoogy, Department of Industria Economics, Trondheim, Norway Stein Erik Feten Norwegian University of Science and Technoogy, Department of Industria Economics, Trondheim, Norway

2 2 Michai Chronopouos et a. 1 Introduction In irreversibe investment, firms tend to spit the proects in distinct phases. We expain this behaviour in a setup where there is uncertainty, discretion over timing, and over the choice of proect scae. According to standard economic iterature (Arrow and Fisher [5]; Henry [21], investment decisions are infuenced by three main factors, namey, uncertainty, irreversibiity, and fexibiity. The first refers to the uncertainty in the cash fows that a proect generates, the second to the inabiity to recover the investment cost after investment has taken pace, and the third to the discretion over the timing of investment. The atter aows for uncertainty in underying parameters to resove before making an irreversibe investment decision. Consequenty, the abiity to optimise the investment timing raises the expected vaue of the investment opportunity, which, in turn, impies that investment is deayed reative to the traditiona net present vaue (NPV approach due to the opportunity cost of kiing the timing option. In fact, this hesitation is proonged as uncertainty increases, since the vaue of waiting increases. Interestingy, however, not ony is the negative reationship between uncertainty and investment subect to mode specifications (Lund [27], but, aso, it does not extend anaogousy to other types of fexibiities. Thus, in spite of the extensive iterature that chaenges the traditiona views of how uncertainty and irreversibiity expain investment behaviour (Avarez and Stenbacka [3]; Abe and Ebery [1], the interaction between uncertainty and different types of fexibiities has not been thoroughy examined yet. For exampe, modes for sequentia capacity expansion sedom present a comparison of umpy and stepwise investment (Gahungu and Smeers [15], whie the ones that do, either ignore discretion over capacity (Goier et a. [17] or restrict the concusions of this comparison to the now or never investment case (Pindyck [33]. By deveoping an anaytica framework for investment under uncertainty, we expore how discretion over proect scae impacts a firm s incentive to invest in stages. Indeed, one crucia type of manageria discretion is the fexibiity to either invest in an entire proect at a singe point in time (umpy investment or divide it into smaer, moduar proects and then invest in each one at distinct points in time (stepwise investment. Within the context of sequentia capacity expansion, it has been shown that moduarity can have crucia impications for the vaue of a proect. For exampe, Goier et a. [17] compare a sequence of sma nucear power pants with a singe nucear power pant of arge capacity under eectricity price uncertainty. Assuming that the capacity of each proect is fixed, they find that the option vaue of moduarity may trigger investment in the initia modue at a eve beow the now or never NPV. By contrast, Kort et a. [24] show how uncertainty reduces the vaue of moduarity. More specificay, they show that, in the absence of an investment premium, the stepwise investment strategy dominates umpy investment. However, if the moduarisation of a proect is costy and the size of each modue is fixed, then,... higher uncertainty makes the singe-stage investment more attractive reative to the more fexibe stepwise investment strategy. This happens because high uncertainty owers a firm s incentive to make costy switches

3 Stepwise Investment and Capacity Sizing under Uncertainty 3 between stages, thereby promoting a umpy investment strategy. A imitation of these modes is that they study a particuar type of fexibiity in isoation, when, in reaity, firms can typicay combine different types of manageria fexibiities. Consequenty, how various types of manageria discretion, e.g., discretion over capacity, option to abandon, etc., interact to affect the vaue of moduarity under increasing uncertainty remains an open question. Indeed, athough Kort et a. [24] has motivated other research in the area of stepwise investment under price uncertainty that considers aso the impact of information change between subsequent investment stages (Adkins and Paxson [2], considerations regarding the impact of other types of manageria discretion have not been formuated anayticay. We address this disconnect by anaysing how the fexibiity to choose between a umpy and a stepwise investment strategy interacts with discretion over capacity under price uncertainty. This situation is reevant to various industries, e.g., renewabe energy (RE power pants. Indeed, in the case of both on and off shore wind farms an area can, and often is, deveoped in stages. Additionay, for capita intensive proects, discretion over capacity is particuary crucia, since the instaation of a arge proect increases a firm s exposure to downside risk in the case of a potentia downturn, whereas the instaation of a sma proect imits a firm s upside potentia if market conditions suddeny become favourabe. Thus, we contribute to the existing iterature by deveoping an anaytica framework in order to expore how discretion over capacity interacts with the fexibiity to choose between umpy and stepwise investment under price uncertainty. Additionay, we derive anaytica resuts regarding the impact of uncertainty on the optima investment threshod, the optima capacity, and the choice of investment strategy. The anaytica tractabiity of our mode faciitates a detaied exporation of how market parameters infuence investment and capacity sizing decisions, thus enabing further insights on how different types of manageria discretion interact to affect the choice of investment strategy. Hence, the contribution of this work is that it provides a framework for anaysing both the dynamics underying the interaction between different types of manageria discretion and the impications of this interaction for the optima investment poicy. We proceed in Section 2 by discussing some reated iterature and introduce assumptions and notation in Section 3. In Section 4, we formuate the probem and derive anaytica expressions for the vaue of the option to invest, the optima investment threshod, and the corresponding optima capacity under umpy (Section 4.1 and stepwise investment (Section 4.2. In addition, we present anaytica resuts regarding the impact of uncertainty on the choice of investment strategy. We present numerica resuts in Section 5 and concude in Section 6.

4 4 Michai Chronopouos et a. 2 Reated Work The semina work of Mad and Pindyck [30] and Dixit and Pindyck [13] has spawned a substantia iterature in the area of sequentia investment. The former show how traditiona vauation methods understate the vaue of a proect by ignoring the fexibiity embedded in the time to buid, whie the atter deveop a sequentia investment framework with infinite investment options assuming that the proect vaue depreciates exponentiay. Extensions in the same ine of work incude Pawina and Kort [32], who impement strategic interactions in capita budgeting under market uncertainty and anayse the optima repacement timing of a production faciity. The vaue of moduarity is emphasised in Machow Møer and Thorsen [31], who iustrate how the investment poicy resembes the simpe NPV rue when investing sequentiay in subsequent upgrades of a technoogy. More specificay, they find that the expected vaue of subsequent upgrades reduces the vaue of waiting to invest in the current version, whie the investment rue is ess sensitive to changes in uncertainty. Taking the perspective of hydro pump storage pant, Doege et a. [14] show how operationa fexibiity can be used for hedging against adverse movements in a portfoio. Within the context of investment in distributed generation capacity, Siddiqui and Maribu [36] anayse how sequentia investment may reduce the exposure of a microgrid to natura gas price voatiity, and find that a direct (sequentia investment strategy is more preferabe for ow (high eves of voatiity. By contrast, Kort et a. [24] show that, whie the vaue of umpy investment is aways smaer than that of a stepwise investment strategy when the atter entais no investment premium, if the investment premium is positive, then the umpy investment strategy may dominate provided that uncertainty is sufficienty high. This happens because, high price uncertainty reduces the incentive to make costy switches between stages. Siddiqui and Takashima [35] combine strategic interactions with sequentia capacity expansion in order to expore how sequentia decision making offsets the effect of competition. They find that the oss in the vaue of a firm due to competition is reduced when the firm invests in stages and specify the conditions under which sequentia capacity expansion is more vauabe for a duopoist firm than for a monopoist. From a more empirica standpoint, Rodrigues and Armada [34] present a rea options approach to the vauation of moduar proects, and show that moduarisation can increase the vaue of a proect depending on the reative vaues, costs, and risk of each moduar configuration. Gamba and Fusari [16] deveop a rea options vauation approach in order to address the issues that a moduarisation process poses in terms of financia vauation for capita budgeting. They first create a stochastic optima contro framework for the six moduar operators proposed by Badwin and Cark [7] and then adopt the east squares Monte Caro method of Longstaff and Schwartz [26] in order to cope with the dynamic programming feature of the vauation probems. Whie the aforementioned iterature offers a thorough anaytica and empirica treatment of the vaue of moduarity and sequentia investment under uncertainty, it ignores the potentia impications

5 Stepwise Investment and Capacity Sizing under Uncertainty 5 of aowing for other types of manageria discretion that firms typicay consider when designing an optima investment poicy, e.g., discretion over capacity, suspension and resumption options, etc. Indeed, apart from discretion over the investment strategy, e.g., umpy versus stepwise, firms typicay aso have discretion over the size of a proect, in the form of instaed capacity. Dang [12] addresses the probem of a firm that invests in a proect with continuousy scaabe capacity under demand uncertainty, and shows that, even when demand is high, ow uncertainty makes waiting for further information the optima strategy. A simiar approach is adopted by Bøckman et a. [9] for vauing sma hydropower proects under eectricity price uncertainty, however, unike Dang [12], they assume a cost function that is convex in capacity, and, therefore, their mode is more pertinent to the energy sector. Huisman and Kort [22] introduce game theoretic considerations and show how, in a duopoistic competition, a eader can use discretion over capacity strategicay in order to deter a foower s entry temporariy. A poicy oriented mode for investment and capacity sizing is presented by Boomsma et a. [8], who anayse the impact of uncertainty stemming from different types of poicy mechanisms on investment and capacity sizing decisions. The impact of risk aversion on such decisions when a firm has operationa fexibiity is addressed in Chronopouos et a. [11], who find that higher risk aversion faciitates investment by decreasing the optima capacity of a proect. Athough discretion over capacity has crucia impications for investment and operationa decisions, anaytica modes that study the impications of this fexibiity within the context of umpy versus stepwise capacity expansion are imited. Apart from anaysing the vaue of discretion over capacity in isoation, a strand of iterature combines it with various types of operationa fexibiities. For exampe, He and Pindyck [20] aow for demand uncertainty and examine the technoogy and capacity choice probem of a firm that can insta either output specific or fexibe capita, which may be used to produce different outputs. They formuate the capacity choice probem as a stochastic contro probem, and show that the vaue of the firm equas the vaue of its instaed capita pus the expected vaue of its options to add capacity in the future. Considering the choice between two types of technoogies, Takashima et a. [37] find that price uncertainty induces investors to maximise expected profits by buiding arger pants, whie the consideration of mutuay excusive proects increases the option vaue of the entire investment opportunity. Hagspie et a. [19] compare a fexibe scenario, in which a firm can adust production over time with the capacity eve as the upper bound, to the infexibe scenario, in which a firm fixes production at capacity eve from the moment of investment onward. Among other resuts, they find that the fexibe firm invests in higher capacity than the infexibe firm and that the capacity difference increases with uncertainty. We extend the existing iterature by deveoping and anaytica framework that combines two important types of manageria discretion, i.e., the fexibiity to invest in either a singe or mutipe stages with discretion over capacity. Athough increasing uncertainty favours a umpy over a more fexibe, yet more costy, stepwise

6 6 Michai Chronopouos et a. investment strategy when the capacity of a proect is fixed, the impications from aowing for discretion over capacity are not thoroughy examined yet. For this reason, we assume that the capacity of the proect is continuousy scaabe, and, in ine with Dang [12], the firm has the option to fix the capacity of the proect at investment. We first confirm the resuts of Kort et a. [24] and then show that, athough the reative vaue of the two strategies decreases with greater uncertainty, the stepwise investment strategy aways dominates that of umpy investment. This seemingy counter intuitive resut happens because the firm can optimise the size of the proect in response to an increase in the cost of the stepwise investment strategy reative to that of umpy investment. Intuitivey, the extra fexibiity to optimise the size of the proect mitigates the oss in proect vaue due to the higher cost associated with the fexibiity to proceed in stages, thereby offsetting the benefit of a ower investment cost via umpy investment. 3 Assumptions and Notation We consider a price taking firm that hods an option to invest in a proect of infinite ifetime that may be competed in either a singe or a sequence of i < discrete stages with i N. Aso, the firm can either exercise an investment option immediatey or deay investment in the ight of price uncertainty. We assume that there is no variabe production cost and that the output price at time t, P t, where t 0 is continuous and denotes time, foows a geometric Brownian motion (GBM that is described in (1 dp t = µp t dt + σp t dz t, P 0 P > 0 (1 where µ is the annua growth rate, σ is the annua voatiity, and dz t is the increment of the standard Brownian motion. Aso, ρ > µ is the subective discount rate. The capacity of the proect is denoted by K when the firm has discretion over investment timing and by K when the firm invests immediatey, thus exercising a now or never investment opportunity. Additionay, F ( is the expected vaue of the now or never investment opportunity, where {, s i } (denoting umpy and staged investment respectivey, whie K is the corresponding optima capacity. For exampe, F ( denotes the expected vaue of the now or never investment opportunity under umpy investment and K is the corresponding optima capacity. If the option to defer investment is avaiabe, then F ( denotes the maximised vaue of the option to invest in stage excusive of subsequent stages, whie τ, P, and K denote the time of investment, the optima investment threshod, and the corresponding optima capacity, respectivey. The investment cost, I ( K, is indicated in (2 I ( K = a K + bk γ, a, b, and γ > 1 (2

7 Stepwise Investment and Capacity Sizing under Uncertainty 7 where γ > 1 impies that this mode is more suitabe for describing proects that exhibit diseconomies of scae. In the energy sector, this is the case with RE power pants, whie more genera exampes where the use of a convex cost function can be reaistic incude a monopsonistic environment in which a firm contempates investment facing increasing prices due to increasing demand. As it becomes cear in Section 4, the assumption γ > 1 shoud be considered as an impication of the mode itsef, and, therefore, is not restricting the anaytica resuts. For the purpose of comparing a umpy investment to a strategy that entais a series of moduar investments, we assume that each individua stage of the stepwise investment strategy is ess costy than the entire proect. However, in ine with Kort et a. [24], we assume that the fexibiity to proceed in stages is costy, and, thus, requires the firm to incur a premium. Thus, the tota investment cost under a stepwise investment strategy is greater than that under umpy investment, as indicated in (3. I (K si < I ( K, i N and i I (K si > I ( K (3 This assumption is required in order to distinguish a umpy from a moduar proect based on their cost, since the capacity of the proect is scaabe. Athough condition (3 may hod for an arbitrary capacity eve, we express it in terms of the optima capacity, K, and not the state variabe, K. This enabes a direct association with the investment cost parameters and faciitates a better intuition of the endogenous nature of the optima capacity. Note that condition (3 is equivaent to a si < a, i N and i a s i > a. In order to compare our resuts to Kort et a. (2010, we aso consider the benchmark case in which the capacity of the proect is fixed and such that K s1 + K s2 = K, whie the investment cost satisfies the condition I < I s1 + I s2. This impies that, under stepwise investment, the sum of the capacities of the moduar proects equas that of the umpy proect, yet, due to the investment premium, the investment cost of the stepwise investment strategy is greater than that of the umpy investment. Note that in the case of Kort et a. [24] the capacities are set exogenousy, and, therefore, it is possibe to repicate the singe stage investment by setting the capacity of first stage equa to that of the umpy investment. This is not possibe if a firm has discretion over proect scae, because the optima capacity is determined endogenousy and depends on the optima investment threshod, which, in turn, depends on the investment cost by assumption 3. The atter, is different for umpy and stepwise investment. Finay, note that, athough a firm may have the fexibiity to respond to ow prices by producing at a eve beow the instaed capacity, in this paper, we assume that a firm does not have production fexibiity. This is often referred to as the cearance assumption and is widey used in the iterature (Chod and Rudi [10]; Anand and Girotra [4]. For exampe, in the energy sector this assumption is reevant to baseoad and RE power pants. Additionay, fixed costs, e.g., commitments to suppiers and production ramp up, make it too costy to produce beow the capacity eve

8 8 Michai Chronopouos et a. (Goya and Netessine [18]. In the car industry, firms often prefer to reduce prices in order to maintain production at fu capacity, instead of producing beow capacity (Mackintosh [29]. 4 Mode The firm s optimisation obective under each investment strategy, i.e., umpy and stepwise investment, is summarised in (4. The outer maximisation corresponds to the genera decision on whether to invest immediatey or deay investment. If the firm decides to wait for an infinitesima time interva dt, then, according to the Beman principe, the vaue that the firm hods is the discounted expected vaue of the capita appreciation of the option to invest. This is represented by the first argument of the maximisation on the right hand side of (4. By contrast, the second argument of the outer maximisation represents the vaue that the firm receives if it decides to exercise a now or never investment opportunity. More specificay, the inner maximisation indicates that when the firm decides to invest it wi choose the capacity of the proect in such a way that maximises its expected NPV. { [ F (P = max (1 ρdte P F (P + dp ] } [ ( ], max F P, K, =, s i and i = 1, 2 (4 K Initiay, we assume that investment takes pace immediatey, i.e., at P 0 P. This impies that the output price at investment is known and enabes the cacuation of the corresponding optima capacity by maximizing the vaue of the now-or-never investment opportunity. In turn, this yieds the expression reating the initia output price to the corresponding optima capacity, i.e., K (P. We then aow for the option to defer investment and maximize the vaue of the investment opportunity by determining the optima investment threshod taking into account the inner extremum of optima capacity choice at investment. The soution to this optimization probem is obtained by equating the margina benefit of deaying investment, MB, to the margina cost, MC. Thus, we obtain the expression reating the optima investment threshod to the optima capacity, i.e., P P ( K. Inserting this expression into the condition of optima capacity choice at τ, we obtain the optima capacity of the proect, i.e.,, i.e., K K ( P. Finay, using K, we can determine the corresponding optima investment threshod price P. 4.1 Lumpy Investment We begin by assuming that the firm adopts a umpy investment strategy. Foowing the approach of Chronopouos et a. [11], the firm can deay investment unti τ, at which point it must fix the capacity of the entire proect, K, and incur the investment cost, I(K, thereby receiving a perpetua stream of stochastic cash fows, as shown in Figure 1. Consequenty, K is a function of the output price, P, at time τ.

9 Stepwise Investment and Capacity Sizing under Uncertainty 9 0 Fig. 1 Lumpy investment P, K τ e ρt P t K dt I (K t We first address the inner maximisation in (4. Hence, we assume that the firm ignores the option to wait for more information and invests in the proect immediatey. The expected vaue of the now or never investment opportunity is indicated in (5. F ( P, K = P K ρ µ I ( K (5 Consequenty, at investment, the output price, P, is known, and, therefore, the firm needs to determine ony the corresponding optima capacity, K, by maximising the vaue of the now or never investment opportunity, as indicated in (6. [ ( ] 1 ( 1 P max F P, K K (P = K bγ ρ µ a (6 We proceed by considering the outer maximisation in (4. If the firm can defer investment, then the vaue of the option to invest is described in (7, where S denotes the set of stopping times of the fitration generated by the price process and E P is the expectation operator, which is conditiona on the initia vaue, P, of the price process. F (P = sup τ S [ E P e ρt P t K dt I ( K ] (7 τ Using the aw of iterated expectations and the strong Markov property of the GBM, which states that price vaues after time τ are independent of the vaues before τ and depend ony on the vaue of the process [ ] ( β at τ, we can rewrite (7 as in (8. The stochastic discount factor E P e ρτ = P P (Dixit and Pindyck [13], p.315, where β > 1 is the positive root of 1 2 σ2 x(x 1 + µx ρ = 0. Notice that, at investment, the optima capacity, K, is reated to the optima investment threshod via (6, i.e., K K ( P. Thus, the now-or-never NPV rue serves as an intermediate step for determining the endogenous reationship between the optima capacity of the proect and the optima investment threshod. F (P = sup τ S where K K (P = [ [ ] E P e ρτ EP e ρt P t K dt I ( K 0 [ ( 1 P (K bγ ρ µ a ] 1 ] ( β [ P P K = max P P P ρ µ I ( K ] (8

10 10 Michai Chronopouos et a. Soving the unconstrained maximisation probem (8, we can express the maximised option vaue, F (P, as in (9. The endogenous constant, A, the optima investment threshod, P, and the corresponding optima capacity, K, can be determined equivaenty via vaue matching and smooth pasting conditions between the two branches of (9 together with the condition for optima capacity choice at investment (6 and are indicated in (A 7, (A 8, and (A 9, respectivey for = (a proofs can be found in the appendix. A P β, for P < P F (P = P K ρ µ I ( K, for P P (9 4.2 Stepwise Investment Next, we assume that the firm adopts a stepwise investment strategy, that comprises of two stages, i.e., i 2. Whie it is possibe that i > 2, the number of stages shoud, nevertheess, be finite in order to ensure that the size of the entire proect does not diverge. As indicated in Figure 2, the firm has the option to deay investment in the first stage unti τ s1, at which point it must fix the corresponding capacity, K s1, and incur the investment cost, I ( K s1. The firm receives the revenues of the first stage unti τs2, at which point it fixes the capacity of the second stage, K s2 and incurs the investment cost I ( K s2. After the firm invests in the second stage, it receives the revenues from both stages. P s1, K s1 P s2, K s2 τ s 2 2 e ρt P t K s1 dt I (K s1 e ρt P t K si dt I (K s2 τ s1 0 τ s1 Fig. 2 Stepwise investment τ s2 τ s2 i=1 t The optima capacity at each stage of the proect when the firm invests immediatey is obtained by maximising the vaue of the now or never investment opportunity. Foowing the same approach as in the case of umpy investment, the optima capacity for each stage is indicated in (10. ( [ ( ] 1 max F si P, K si K 1 P (P = s K i si bγ ρ µ a s i (10 Notice that the vaue of the now or never investment opportunity is the sum of the maximised NPVs from each stage, i.e., F s (P = ( i F P, K. s i s i

11 Stepwise Investment and Capacity Sizing under Uncertainty 11 If the option to deay investment is avaiabe, then the optimisation obective is described in (11 F s (P = sup τ s1 S E P [ sup τ s2 τ s1 E P [ τs 2 τ s1 e ρt P t K dt I (K s1 s 1 + e ρt P t τ s2 2 i=1 ( ]] K dt I K (11 s s i 2 where K s i K si ( P s i, i = 1, 2. Notice that by competing the first stage, the firm receives the option to proceed to the second. As a resut, the option to invest in the first stage may be seen as a compound option. In fact, since the cash fows accrue over disoint time intervas the vaue of the option to invest is separabe. By decomposing the first integra on the right hand side of (11, we can express the origina probem as two separate optima stopping time probems, as in (12 [ ( ] F s (P = sup E P e ρt P t K dt I K s τ s1 S τ 1 s 1 s1 [ + sup τ s2 τ s1 E P τ s2 ( ] e ρt P t K dt I K (12 s s 2 2 and, foowing the same steps as in (8, we obtain (13. ( β1 [ P F s (P = max E Ps1 e ρt P t K dt I (K ] s1 s P s1 P P s max P s2 P s1 ( [ ( ( ] 1 P Notice that, ike in (8, Ks K si P 1 s K s i s = i i i bγ ρ µ a si ( [ β1 P ( ] E Ps2 e ρt P t K dt I K (13 s P s2 0 2 s 2. The soution of each of the two optima stopping time probems on the right hand side of (13 is expressed in (14, where A si, P s i, and K s i are indicated in (A 7, (A 8, and (A 9, respectivey. Notice that the vaue of the option to invest is the sum of the respective option vaues of each stage, i.e., F s (P = i F s i (P. A si P β F si (P = P K ( s i ρ µ I K s i, for P < P s i, for P P s i (14 Proposition 1 The optima investment threshod and the corresponding optima capacity under umpy and stepwise investment are: P ( ( I K K β(ρ µ = K β 1 and K = [ a b ] 1 1, γ(β 1 β > 0 (15 γ(β 1 β

12 12 Michai Chronopouos et a. Assuming that τ s1 < τ s2, Proposition 2 indicates that the decision to invest in the first stage is independent of the presence of the second. Notice first that the structure of the cost function in (2 and Proposition 1 impy that P s 1 < P s 2 if a s1 < a s2. In turn, the atter condition impies that the two investments within a stepwise investment strategy are not undertaken at the same time. Consequenty, the assumption τ s1 < τ s2 is satisfied. Intuitivey, Proposition 2 is a consequence of the optimaity of myopic behaviour based on which a firm disregards subsequent investment decisions when evauating the current one. Within the context of capacity expansion, this property impies that an investment in new capacity is evauated assuming that it is the ast one in the horizon. The optimaity of myopic behaviour is not generay true but hods in the benchmark cases of monopoy (Pindyck [33] and perfect competition (Leahy [25]; Badursson and Karatzas [6]. Optimaity of myopia aso hods within a context of strategic interactions provided that the profit is additivey separabe if more that one technoogy is considered (Badursson and Karatzas [6]. Proposition 2 P is independent of P. s s 1 2 In ine with the standard rea options intuition, Proposition 3 indicates that greater uncertainty raises both the optima capacity of the proect and the optima investment threshod. This happens because greater uncertainty increases the opportunity cost of an irreversibe investment decision, thereby raising the vaue of waiting. Furthermore, from (6 we know that the optima capacity of the proect is a monotonic function of the output price. Consequenty, an increase in the optima investment threshod resuts in the instaation of a bigger proect. Proposition 3 K P > 0 and > Lumpy versus Stepwise Investment In this section, we anayse the impications of discretion over capacity for the optima investment strategy. Interestingy, as Proposition 4 indicates, if the firm has discretion over capacity, then the vaue of the option to proceed in stages is aways greater than that under umpy investment. This is in contrast to Kort et a. [24] who show that, under reativey arge uncertainty, the singe stage investment is more attractive reative to a more fexibe, yet more costy, stepwise investment strategy. This seemingy counter intuitive resut is based on the endogenous reationship between the price at investment and the capacity of the proect. Notice that, if a firm has discretion over capacity, then, according to (15, the optima capacity, K, is non negative if γ(β 1 β > 0. However, whie greater uncertainty owers the reative vaue of the two strategies, it aso decreases β. According to Proposition 4, the reative vaue of the two strategies does not decrease beow one for non negative vaues of K. Intuitivey, athough the vaue of the stepwise investment strategy is reduced due to the cost that a firm incurs for the fexibiity to proceed in stages, the extra fexibiity to scae the capacity of the proect aows the firm to offset the reduction in the vaue of the stepwise investment strategy

13 Stepwise Investment and Capacity Sizing under Uncertainty 13 competey. Indeed, if the capacity of the proect was fixed, then greater uncertainty woud deay investment but the amount of instaed capacity woud remain unaffected. By contrast, discretion over capacity aows a firm to respond to an increase in the investment cost by optimising the endogenous reationship between the size of the proect and the time of investment. Hence, contrary to Kort et a. [24], there exists no investment premium for which the firm is indifferent between the two strategies. This woud ony happen if the cost of any given modue was equa to that of the umpy proect. However, this woud vioate assumption (3 since then there woud be no distinction between the umpy and the moduar proect. Proposition 4 If a firm has discretion over capacity, then F s (P > F (P. From (15 we see that the existence of an optima soution to the investment probem under each strategy requires that the cost function is stricty convex, i.e., γ(β 1 β > 0 γ > β β 1 > 1. Therefore, the convexity of the cost function is not an assumption, as indicated in (2, but rather a property impied by the anaytica framework itsef. More specificay, convexity ensures that the optima stopping time probem is finite, since, otherwise, it is aways optima to deay investment. Indeed, if γ > whereas if γ β β 1, then K β β 1, then 0 < K <,. Consequenty, the resut of Proposition 4 is in ine with the more genera intuition that a firm is typicay induced to adust its capita stock more sowy due to diseconomies of scae associated with rapid changes in the investment cost. Hence, a convex investment cost impies that it is more expensive to perform adustments, e.g., expand capacity, at a greater than at a ower rate (Jørgensen and Kort [23]. Additionay, note that γ > β β 1 is a consequence of the exogenous price, whie the resut of Proposition 4 depends upon the endogenous reationship between the price at investment and the capacity of the proect, as this is described in (6. Aowing for the price to depend on the amount of quantity produced via an inverse demand function wi resut in a concave cost function (Dang [12], yet the quaitative (positive reationship between the price and capacity wi remain the same. Indeed, Dang [12] iustrates how the optima capacity increases monotonicay with the output price under economies of scae, i.e., γ < 1. Since the endogenous reationship between the price at investment and the capacity of the proect remains unaffected, the quaitative resut of Proposition 4 shoud hod under both diseconomies and economies of scae. Nevertheess, the rigorous derivation of this resut for γ < 1 is eft for future work. Another consequence of the endogenous reationship between the output price at investment and the size of the proect, is that, if condition (3 hods, then the amount of instaed capacity under umpy investment is ower than the tota amount of capacity instaed under a stepwise investment strategy, as shown in Proposition 5. Indeed, as the investment cost associated with the stepwise investment strategy increases, it raises both the optima investment threshod and the amount of instaed capacity. Consequenty, the firm compensates for the extra cost it incurs for the fexibiity to proceed in stages by adusting the size of the proect so that it offsets the reduction in the vaue of the investment opportunity. As a resut, the stepwise investment strategy eads to the instaation of a bigger proect than that under umpy investment. This is in

14 14 Michai Chronopouos et a. contrast to Kort et a. [24], where a firm may deay investment due to an increase in the investment cost, yet it can insta a fixed amount of capacity. Notice aso that, if assumption (3 is extended to refect a proect with more than two stages, then intuition suggests that the resut of Proposition 5 can be extended to aow for an arbitrary number of stages. Proposition 5 K < n i=1 K a 1 s i < n i=1 a 1. s i In order to obtain a deeper intuition of Proposition 4 and the underying dynamics that determine the optima investment poicy, we anayse the impact of uncertainty on the margina benefit (MB and the margina cost (MC of deaying investment under each investment strategy assuming that the capacity of the proect is either fixed or scaabe. Therefore, we first express the firm s maximised option vaue as in (16 ( β [ P P K F (P = max P P P ρ µ I ( ] K (16 and then describe the first order necessary condition for the optimisation probem (16 by equating the MB of deaying investment to the MC, as in (17. ( βi K P + K ρ µ = βk ρ µ (17 The first term on the eft hand side of (17 is positive and represents the incrementa proect vaue created by a margina increase in the output price. Notice that this term is a decreasing function of the output price, since waiting onger enabes the proect to start at a higher initia price, yet the rate at which this benefit accrues diminishes due to the effect of discounting. The second term is aso positive and represents the reduction in the MC of waiting to invest due to saved investment cost. Together, these two terms constitute the MB of deaying investment. The right hand side of (17 represents the MC of deaying investment. This term is positive and refects the opportunity cost of forgone cash fows. As shown in Coroary 1, when the output price is ow it is worthwhie to postpone investment since the MB is greater than the MC. Coroary 1 The MB is steeper than the MC. As Proposition 6 indicates, if the capacity of the proect is fixed, then greater uncertainty decreases both the MB and the MC of deaying investment, however, the impact of uncertainty on the MC is more pronounced than that on the MB. By contrast, the opposite is true if the firm has discretion over capacity. In fact, athough in both cases greater uncertainty postpones investment, the incentive to deay investment is greater when the firm has the fexibiity to scae the capacity of the proect. Proposition 6 If K is fixed, then MB < 0, scaabe, then MB > 0, MC > 0, and MB > MC. MC < 0, and MB < MC, whereas if K is

15 Stepwise Investment and Capacity Sizing under Uncertainty 15 Indeed, if the capacity of the proect is fixed, i.e., K MC of deaying investment decrease with greater uncertainty, since β K, then from (17 we see that both the MB and < 0. In addition, from (A 13 we have I(K P < K ρ µ, and, therefore, greater uncertainty owers the MC by more than the MB. As a resut, the margina vaue of deaying investment increases, thereby raising the incentive to postpone investment. Intuitivey, athough the extra benefit from aowing the proect to start at a higher output price is fixed, the extra benefit from saving on the investment cost and the extra cost of the forgone cash fows decrease due to the effect of discounting. In fact, the atter becomes more pronounced as both the output price and the voatiity increase. By contrast, if the capacity of the proect is scaabe, then the increase in the optima capacity of the proect with greater uncertainty presents an opposing force, which mitigates the reduction in the vaue of β. As Proposition 6 indicates, in the atter case both the MB and MC of deaying investment increase with greater uncertainty, and, unike the case of fixed capacity, the MB increases by more than the MC, thus increasing the incentive to deay investment. Consequenty, discretion over capacity aows the firm to manage price uncertainty more efficienty by adusting the size of the proect in response to an increase in the investment cost. 5 Numerica Exampes For the numerica exampes we assume that µ = 0.01, ρ = 0.1, and σ [0, 0.4]. Aso, the cost parameters are a = 30, a s1 = 15, a s2 = 25, b = 0.5, and γ = 3. In order to compare our resuts with the case of fixed capacity, we assume that the investment cost in the atter case is I = 1000, I s1 = 500, 510, 520, and I s2 = 900 for umpy and stepwise investment respectivey, whie the corresponding capacity eves are K = 10, K s1 = 3.4, and K s2 = 6.6. Notice that if the capacity of the proect is not scaabe, then, in ine with Kort et a. [24], stepwise is more costy than umpy investment, i.e., I = 1000 < I s1 + I s2, yet K s1 + K s2 = 10 = K. Figure 3 iustrates the option and proect vaue as we as the maximised NPV in the case umpy investment for σ = 0.2, 0.3. Notice that greater price uncertainty increases the opportunity cost of investment and raises the vaue of the investment opportunity. In turn, this postpones investment and increases both the optima investment threshod and corresponding optima capacity. Figure 4 iustrates the impact of uncertainty on the optima investment threshod under scaabe capacity, as we as on the optima capacity of the proect. According to the eft pane, P s 1 the numerica assumptions satisfy the condition I (K s1 ( < K s 1 I K s 2 < P s 2, and, therefore, K s 2. Additionay, as the right pane iustrates, with the fexibiity to scae the size of the proect the tota capacity when proceeding in stages exceeds that of the umpy investment, as shown in Proposition 5. In fact, the wedge between K s 1 + K s 2 and K refects the extra vaue that the firm has due its discretion over capacity. Notice that, since K s 1 + K s 2 > K and a s1 + a s2 > a, the condition that stepwise investment is more costy than umpy investment, as indicated in

16 16 Michai Chronopouos et a. Option Vaue, Proect Vaue K = 5.48 P = 6.75 K = 8.36 P = σ = 0.3 Option Vaue, F (P -200 σ = 0.2 Proect Vaue, F (P,K ( -400 Maximised NPV, F P,K (P Optima Investment Threshod Output Price, P t Fig. 3 Option vaue, proect vaue, and maximised NPV under umpy investment for σ = 0.2, 0.3 (3, is aso satisfied. Consequenty, apart from discretion over capacity, the remaining assumptions are the same as the ones underying the mode of Kort et a. [24]. Optima Invesment Threshod P P s 1 P s 2 Optima Capacity K K s 1 K s 2 K s 1 +K s Voatiity, σ Voatiity, σ Fig. 4 Optima investment threshod (eft and optima capacity (right versus σ Figure 5 iustrates the impact of uncertainty on the reative vaue of the two strategies, i.e., F s (P F (P, under fixed (eft pane and scaabe capacity (right pane. The eft pane confirms the resuts of Kort et a. [24] for the case in which stepwise investment requires a positive investment premium. Indeed, Kort et a. (2010 show that greater price uncertainty owers the critica eve of the investment cost premium for which the umpy

17 Stepwise Investment and Capacity Sizing under Uncertainty 17 and stepwise investment strategies are equay good. Equivaenty, this impies that a greater investment cost premium owers the critica eve of price uncertainty for which the firm is indifferent between the two strategies. The critica eve of price uncertainty for each eve of investment cost premium is iustrated in the eft pane of Figure 5 and is ocated at the intersection between the curves and the horizonta ine. As the eft pane iustrates, there exists a eve of uncertainty for which the firm woud be indifferent between a umpy and a stepwise investment strategy. Indeed, if the capacity of the proect is fixed, then the reative vaue of the two strategies is greater than one for ow eves of uncertainty, yet drops beow one as uncertainty increases (eft pane. This resut is more pronounced as the investment premium, i I I s, of the stepwise i investment strategy increases. Hence, with greater uncertainty, umpy investment becomes more attractive than stepwise investment when the atter entais a positive investment premium. This happens because greater uncertainty increases inertia and raises the incentive to avoid costy switches between stages, thereby promoting a umpy investment strategy. By contrast, if a firm has discretion over capacity, then the stepwise investment strategy aways dominates that of umpy investment, as shown in Proposition 4. Intuitivey, the fexibiity to scae the capacity of the proect offsets the reduction in the vaue of the stepwise investment strategy due to the cost that a firm must incur for the fexibiity to proceed in stages. Additionay, as the right pane iustrates, the reative vaue of the two strategies is not ony stricty greater than one, but shifts upwards as the investment cost becomes more convex, i.e., as γ increases. This impies that a more pronounced increase in the margina cost of investment raises the incentive to adopt a stepwise investment strategy. Reative Option Vaue premium = 400 premium = 410 premium = 420 Reative Option Vaue γ = 3 γ = 3.5 γ = Voatiity, σ Voatiity, σ Fig. 5 Reative vaue of the two investment strategies, i.e., umpy and stepwise, versus σ under fixed capacity (eft and scaabe capacity (right

18 18 Michai Chronopouos et a. Figure 6 iustrates the impact of uncertainty on the excess capacity, i K K, and the investment s i premium, ( i I K I ( K s, under stepwise investment for different vaues of a. As the eft pane i iustrates, greater uncertainty increases the wedge between the tota capacity instaed via stepwise and umpy investment, whie this resut is more pronounced as the cost of the stepwise investment strategy increases. This happens because an increase in the investment premium raises the firm s incentive to increase the amount of instaed capacity, and, thus, compensate for the extra cost associated with stepwise investment. As the right pane iustrates, the investment premium that is required in order to proceed in stages aso increases with greater uncertainty. This is in contrast to Kort et a. [24], who show that if stepwise investment is associated with an investment premium, then greater uncertainty owers a firm s wiingness to proceed in stages by decreasing the investment premium for which a firm is indifferent between the two strategies. Since, in our mode, the stepwise investment strategy aways dominates, a direct comparison with Kort et a. [24] is not possibe. Nevertheess, the right pane indicates that a firm is wiing to incur an extra cost in order to proceed in stages so ong as the cost of each moduar proect is ess than that of umpy investment, i.e., provided that condition (3 is satisfied. 14 a s1 = a s1 = 15 Excess Capacity a s1 = 20 a s1 = 25 Investment Premium a s1 = 20 a s1 = Voatiity, σ Voatiity, σ Fig. 6 Impact of σ on the excess capacity (eft and the investment premium (right The eft pane in Figure 7 iustrates the MB and MC of deaying investment under scaabe capacity for each stage of the proect. Notice that for ow price eves the MB exceeds the MC, and, as a resut, the firm has an incentive to postpone investment. Furthermore, the MB decreases as the output price increases due to the effect of discounting, whie the MC is constant. The right pane iustrates the impact of uncertainty on the tota MB and MC of deaying investment in each stage of the proect under fixed and scaabe capacity. In the former case, the MB and MC decrease with greater uncertainty, whie, in the atter case, the MB and MC increase, as shown in Proposition 6. Intuitivey, the incentive to deay investment is greater when

19 Stepwise Investment and Capacity Sizing under Uncertainty 19 the capacity of the proect is scaabe because a moduar investment enabes fexibiity, thereby making it possibe to adapt to uncertain market conditions. Margina Benefit, Margina Cost MB: Stage 1 MC: Stage 1 MB: Stage 2 MC: Stage 2 Optima Investment Threshod Margina Benefit, Margina Cost MB MC Fixed Capacity Scaabe Capacdity Output Price, P t Voatiity, σ Fig. 7 MB and MC of deaying investment for stages i = 1, 2 and σ = 0.2 under scaabe capacity (eft and tota MB and MC under stepwise investment (right 6 Concuding Remarks Manageria fexibiity is crucia for addressing the vauation and tradeoffs invoved in capita proects, that are typicay more compex than simpe now or never investments. In this paper, we extend the resuts of Kort et a. [24] by assuming that a firm does not ony have the fexibiity to choose the investment strategy, in terms of umpy versus stepwise investment, but aso has discretion over both the investment timing and the size of the proect. Thus, we determine not ony the optima investment threshod and the corresponding optima capacity under umpy and stepwise investment, but aso the impact of price uncertainty on the reative vaue of the two investment strategies. Therefore, the contribution of this work is in deineating the interaction between various forms of manageria fexibiity when a firm faces externa pressures, e.g., market voatiity. Whie Kort et a. [24] show that, in the presence of an investment premium, the fexibiity to proceed in stages becomes ess vauabe than umpy investment with greater uncertainty, which is in contrast to the traditiona rea options intuition that emphasises the positive reationship between fexibiity and uncertainty, impications from incuding different types of manageria fexibiities have not been examined thoroughy yet. We confirm the resuts of Kort et a. [24], however, in addition we show that, if a firm has discretion over capacity, then the stepwise investment strategy aways dominates that of umpy investment. This resut emphasises that the reationship between fexibiity and uncertainty requires further investigation. Indeed,

20 20 Michai Chronopouos et a. not ony is the positive reationship between the vaue of fexibiity and uncertainty case specific, but, more importanty, the impact of uncertainty on an isoated type of manageria discretion may be competey mitigated if the atter is combined with another type of fexibiity. In this paper, we show that, athough the fexibiity to proceed in stages becomes ess vauabe than umpy investment with greater uncertainty when a proect has a fixed capacity, aowing for discretion over capacity mitigates this effect competey. More specificay, the reduction in the vaue of the stepwise investment strategy due to the cost that a firm incurs in order to have the fexibiity to proceed in stages is competey offset by the extra vaue from the fexibiity to scae the capacity of the proect. Additionay, we show that the amount of instaed capacity under stepwise investment is aways greater than that under umpy investment. A imitation of this work is the exogenous price process, which impies that investment decisions do not affect future prices. This assumption can be reaxed by inking the output price with the amount of instaed capacity via an inverse demand function. However, considering the resuts of Dang [12], this in not expected to infuence the main resut of the paper. Nevertheess, it woud sti be interesting to investigate any quantitative difference due to the impications of instaing a very arge proect. In order to obtain further insights on the robustness of the resuts regarding the reationship between uncertainty and various combinations of different types of fexibiities, we may aso aow for production fexibiity in the context of Hagspie et a. [19], operationa fexibiity in the form of options to suspend and resume operations, or an aternative stochastic process, e.g., arithmetic Brownian motion or mean reverting process. Additionay, the impications of irreversibiity may be further anaysed by introdusing agency conficts as in Löffer [28]. Finay, in ine with Siddiqui and Takashima [35], this setup aows for exporation of game theoretic considerations, e.g., how the presence of a riva impacts the decision to invest and the reative vaue of the two investment strategies under duopoistic competition. APPENDIX Proposition 1: The optima investment threshod and the corresponding optima capacity under umpy and stepwise investment are: P ( ( I K K β(ρ µ = K β 1 and K = [ a b ] 1 1, γ(β 1 β > 0 (A 1 γ(β 1 β Proof: By maximising the vaue of the now or never investment opportunity, we obtain the expression for the optima capacity, K, corresponding to the current output price P, as indicated in (A 2 for =, s i. [ ( ] 1 ( 1 P max F P, K K (P = K bγ ρ µ a (A 2