# Oxford University Press 1999 Oxford Eonomi Papers 51 (1999), 559±573 559 Irreversibility and restoration in natural resoure development By Jinhua Zhao* and David Zilberman{ * Department of Eonomis, Heady Hall, Iowa State University, Ames, IA 50011; email: jzhao@iastate.edu { University of California at Berkeley We extend Real Option Theory to evaluate natural resoure development projets that may bring negative net bene ts and require ostly restoration. Based on a new onept, irreversibility ost, we show that the degree of irreversibility beomes an endogenous hoie, rather than an exogenously given eonomi onstraint. Fixed osts of restoration have ontinuous impats, over and above the widely reognized xed effets, on development and restoration levels (and the marginal q). The projet's value may not neessarily be onvex in the underlying random variable, and disounting may in fat enourage the pattern of developing now and restoring later. 1. Introdution The standard tool of projet evaluation, the Net Present Value riterion, seems appliable to natural resoure projets, suh as a hydroeletri dam or lear-utting land for agriulture. But as Arrow and Fisher (1974) argue, resoure projet evaluation has to aount for restoration dif ulties in ase of projet failures. This feature has generated a rih literature on investment deisions under unertainty and irreversibility, or Real Option Theory (ROT). See Arrow and Fisher (1974), Henry (1974), Dixit and Pindyk (1994), Epstein (1980), Freixas and Laffont (1984), Ulph and Ulph (1997), and Kolstad (1996). The major argument of ROT is that faing irreversibility and unertainty, it may pay to delay the investment and wait for information to avoid the downside risk. Thus, onventional projet evaluation leads to too early or too muh investment. The assumption of absolute irreversibility in ROT is in most ases too strong, given resoure restoration efforts urrently going on around the world. 1 The sh population redued by a dam may be restored by dereasing the apaity of or even destroying the dam. 2 Clear utting forest may endanger valuable speies, and restoration involves reforestation to preserve these speies.... 1 There are efforts to restore the Everglades in Florida, the San Franiso Bay Delta in California, the RhoÃne River in Frane, the Pearl River in China, and the Hula Lake in Israel, among others. 2 Destroying the Elwha dam in the state of Washington has been studied seriously as a way of restoring the eosystem and sheries in the Elwha River area.
560 irreversibility and restoration Most of the ROT literature, inluding Arrow and Fisher (1974), Henry (1974), Capozza and Li (1994), MDonald and Siegel (1986), Majd and Pindyk (1987), Dixit and Pindyk (1994), and a series of papers by Pindyk, fouses on the timing of investment and does not allow variable projet size. We all this the 0±1 approah sine the deision for the urrent period is either full (the 1) or no development (the 0). The major onlusion of the 0±1 model is that inreased unertainty tends to delay the investment. Pindyk (1988) onsiders variable projet size but does not allow ostly restoration. He onludes that inreased unertainty redues the level of investment. This paper aims to extend ROT to allow for restoration of natural resoure projets that may result in negative environmental bene ts. Irreversibility is not assumed; instead eonomi optimization determines the extent of restoration and whether zero restoration (absolute irreversibility) is optimal. We go beyond the 0±1 approah and assume that the size of the projet is endogenously hosen in all periods. Thus, resoure management is modeled as a losed-loop stohasti dynami optimization problem that determines the extent of both investment and restoration, and the probability and degree of irreversibility. We introdue a new onept, alled irreversibility ost (IC), whih measures the expeted loss due to ostly restoration and irreversibility. IC is monotonially related to the Arrow±Fisher±Henry option value (OV) in problems of xed projet size and absolute irreversibility, but is appliable to more general problems of variable projet size and ostly restoration. IC highlights the impliations of the negative net bene t of development, and the adjustment in the optimal projet sale it entails. The onept of restoration emphasized here is related to the disinvestment onept in Abel and Eberly (1994). In their ase, disinvestment is undertaken when selling the equipment is more pro table than ontinuing the operation. Our model is different beause in resoure development, the reason for restoration is negative environmental impat. We obtain some ounterintuitive results, showing that the value of resoure projets may deline with future unertainty, and that inreased disounted rates may enourage investment. For natural resoure projets, there is an important distintion between tehnial and eonomi irreversibility. The former happens when there does not exist a tehnology that an mitigate the negative impats of the development. Extintion of endangered speies due to resoure development falls into this ategory. The original work of Arrow and Fisher (1974) and some of its appliations (suh as Fisher et al., 1972) assumes tehnial irreversibility. Eonomi irreversibility arises when it is not optimal to restore the development even though tehnologies exist for doing so. For example, Caballero (1991) onsidered irreversibility as orresponding to in nite marginal ost for negative adjustments of the projet size. Our framework also applies to tehnial irreversibility although we fous on eonomi irreversibility. The paper is organized as follows. In Setion 2, we assume absolute irreversibility to illustrate an analytial framework based on a new onept, irreversibility ost.
jinhua zhao and david zilberman 561 We ompare irreversibility ost with the option value de ned in Arrow and Fisher (1974) and Henry (1974). We repliate some of the standard results of ROT using the new framework, and show that in our model a higher disount rate promotes development. Setion 3 onsiders ostly restoration and shows that the degree of irreversibility is an endogenous hoie. It also illustrates that the xed ost of restoration has ontinuous impats on levels of development and restoration. Setion 4 onludes the paper and disusses impliations of the results and possible future researh. The appendies inlude model details and proof of the theorems. 2. Basi model and irreversibility ost We onsider a risk neutral deision maker faing a net bene t funtion B t K t ; w t, where K t ˆ Pt sˆ1 I s is the total apaity of the projet, I t is the level of development (or restoration if I t is negative), and w t is the random variable in period t, for t ˆ 1; 2. 3 We assume that (A1) B t K t ; w t is onave in K t, (A2) B t K t w t > 0, (A3) B t K t w t w t 4 0, and (A4) B t K t, B t w t, and B t K t w t are bounded for all w t, where subsripts denote partial derivatives. (A2) and (A3) mean that higher w t raises the marginal bene t of development but at a dereasing rate. (A2) an be hanged without affeting the results of the paper. (A3) provides a suf ient, but not neessary, ondition for Theorem 1 whih states that inreased unertainty redues development. This assumption is widely used inanalyzing the effets of unertainty, as in Dixit and Pindyk (1994) and Pindyk (1988) where B t K t w t w t ˆ 0 is assumed. 4 (A4) gives some regularity onditions. Consider two period sequential resoure development with omplete resolution of unertainty (that is, the value of the random variable is observed at the beginning of eah period). The deision maker's optimization problem is max B 1 I 1 E w2 max B 2 I 1 I 2 ; I 1 I 2 s:t: I t 5 0; t ˆ 1; 2; ˆ 2 " 2 ; " 2 0; 2 ; 2 ; Š 1 where 2 represents the level of unertainty, may be 1 and may be 1. For simpliity, we ignored w 1 in the B 1 funtion.... 3 Choosing total apaity, instead of total apaity and urrent period investment level, as an argument for B t is for the sake of simpliity. In priniple, the net bene t funtion should have three arguments: B t K t ; I t ; w t ˆR t K t ; w t C t I t ; K t 1, where R t and C t are revenue and ost funtions respetively. A simpli ed speial ase of the ost funtion is C t T t K t 1. The impliit assumption of this simpli ation is that the ost of building K t 1 is divided over all periods, so that even if I t ˆ 0, there is still ost of onstruting K t 1 appropriated to period t. This ost funtion C K t leads to the spei ation of B K t ; w t. This bene t funtion makes I 1 and I 2 perfet substitutes in ase of perfet reversibility, whih helps highlight the role of ostly restoration and makes the analysis muh easier without hanging the nature of the problem. Visusi (1985) made a similar assumption. 4 In a stati framework, B t K t w t w t <0 means that inreased unertainty in w t redues expeted optimal K t. We will show that, as in ROT, dynami optimization further enhanes this effet of unertainty. (A3) guarantees that the stati and dynami effets of unertainty work in the same diretion.
562 irreversibility and restoration Fig. 1. Optimal I 2 depending on " 2 and I 1. Appendix 1 derives the optimal seond period development, ^I 2, depending on I 1 and, illustrated in Fig. 1. The dashed urves indiate the optimal levels of I 2 when the investment is perfetly reversible: I 1 is so high and is so low that the optimal strategy is to restore. However, due to irreversibility ^I 2 ˆ 0 in this ase. For any I 1 (e.g. I 1 ˆ 2 in Fig. 1), there is a ritial level of denoted as w2 I 1 e.g. b 2 in Fig. 1, below whih the irreversibility onstraint is binding (formal de nition in Appendix 1). When it is binding, there is a loss in the seond period ompared with the senario of perfet reversibility (e.g. optimization without the onstraint I 2 5 0). We de ne irreversibility ost to be the expeted loss in the seond period assoiated with I 1 IC I 1 ˆ B 2 I 1 I r 2 I 1 ; x ; x B 2 I 1 ; x f x dx 2
jinhua zhao and david zilberman 563 where f is the density funtion of, and I2 I r 1 ; is the optimal seond period development without the irreversibility onstraint, that is I2 I r 1 ; ˆarg max B 2 I 1 I 2 ; 3 I 2 Theorem 4 in Appendix 1 shows that IC I 1 is inreasing and onvex in I 1. The maximization problem in (1) an be transformed to inlude diretly the irreversibility ost in the objetive funtion. The maximization is equivalent to a situation where the deision maker pays the irreversibility ost in return for a perfetly reversible investment environment. Let M r ˆ E w2 max B 2 I 1 I 2 ; 4 I 2 be the expeted seond period bene t with perfet reversibility. Then the maximization problem in (1) (see Theorem 5 of Appendix 1) an be rewritten as max I 1 5 0 B1 I 1 M r IC I 1 5 Note that M r is independent of I 1, beause for any I 1, the optimal seond period apaity K2 is always ahieved by adjusting the level of I 2. This is due to the perfet substitutability between I 1 and I 2 in the net bene t funtion. Thus the irreversibility ost fully aptures I 1 's impat on the seond period deision. The presentation in (5) enables us to evaluate the impat of unertainty on investment through its impat on the irreversibility ost: Theorem 1 Let F i, i ˆ a; b be two umulative distribution funtions of, with F a seond order stohastially dominating F b. The marginal irreversibility ost assoiated with F a is smaller than that with F b. That is, the rst period investment under F a is higher than under F b. A diret impliation of Theorem 1 is that inreased future unertainty redues optimal I 1, whih is similar to Pindyk (1988). In his interpretation, higher unertainty raises the `option value' of delaying partial investment, leading to a smaller projet size. Here, inreased unertainty raises the probability of the irreversibility onstraint being binding (regretting the original investment), prompting more autious investment deisions. The effets of unertainty on the value of the projet an be analyzed using (4) and (5). It depends on two effets, through IC and through M r :As 2 inreases, the irreversibility ost IC inreases sine higher unertainty raises the probability that the projet is over-sized. The unonstrained seond period bene t M r may or may not inrease depending on the urvature of the bene t funtion B 2 K 2 ;. We an show that when B 2 K 2 ; is linear in, the inrease in M r dominates that in IC, and the projet value is inreasing in unertainty, whih is onsistent with Pindyk (1988). 5 However, in resoure development projets, the most signi ant... 5 Pindyk studied prie unertainty where the per period payoff funtion is linear in the random variable.
564 irreversibility and restoration unertainty is of environmental impats. In most ases, the loss assoiated with serious environmental damage is more than proportional to the gain assoiated with environmental improvement. The bene t funtion may be onave in, espeially where the irreversibility onstraint is binding. In this ase, the inrease in IC due to higher unertainty may dominate the inrease in M r, and the projet value may be onave in the level of unertainty. It beomes lear then that the risk-neutral deision maker may demonstrate risk averse behavior: inreased unertainty may lead to both less development and less payoff. In this ontext, the irreversibility ost may be onsidered as an insurane premium: the deision maker pays IC I 1 in exhange for the insurane of perfet reversibility. The potential future loss from urrent investment hanges the diretion of the impat of disount rate on resoure alloation. Thus far, we have not onsidered disounting the seond period expliitly, even though we allow B 1 and B 2 to be different. Assuming B 2 ˆ B 2 0 k 2 ; = 1 i where i is the disount rate and B 2 0 is the temporal bene t of the projet, one an show that higher i leads to higher I 1 (Theorem 6 in Appendix 1). That is, a higher disount rate leads to more, instead of less, initial development. Higher i redues the seond period bene t, but also redues the irreversibility ost IC and MIC. Beause of irreversibility, investment in rst period has a muh bigger impat on future ost than future bene t. Disounting thus redues the net ost inurred in the future due to urrent development, leading to more urrent development. This result is ontrary to the onventional wisdom that disounting disourages investment. The inter-temporal pattern of ost and bene t in the onventional investment model is that ost ours rst and bene ts later. But many natural resoure projets have the opposite pattern: bene ts are initially enjoyed at the ost of the future. One the timing sequene of ost and bene t is reversed, the impat of disount rate on investment is reversed. Then with a higher disount rate, it is more likely that a big projet is built now and restoration is onduted later. 6 Irreversibility ost is monotonially related to the muh used onept in ROT, the Arrow±Fisher±Henry option value (OV). In fat, OV provides the lower bound for IC and equals IC for small projets. 7 To see this, we modify our model following the investment problem in Arrow and Fisher (1974), where the deision of exeuting a projet of size K is made in either the urrent or the next period. Absolute irreversibility means that if the projet is exeuted in period 1, the apaity remains at K in both periods. Perfet reversibility means that the projet an be reversed in the seond period so that the apaity an be zero, depending on the state of nature.... 6 A nulear power station may be an example. 7 Sine OV is the value of information onditional on the projet being delayed (see Hanemann, 1989), our statement about the relationship between IC and OV also applies to that between IC and the (onditional) value of information.
jinhua zhao and david zilberman 565 Theorem 2 The relation between IC and OV an be expressed as ( 2 ) w2 OV K ˆIC K max 0; B 2 0; df B 2 K; df 6 where the seond term on the right hand side is the expeted gain of delaying the projet alulated using the open-loop approah. Note that OV K ˆIC K when K is small, sine the differene between the integrals in (6) is negative when K is small and positive when K is large. To see the intuition of (6), note that in losed-loop optimization, if the projet is exeuted, the deision maker gains by the extra bene ts provided by the projet in period one, B 1 K B 1 0, and loses by the possibility that the projet is too big (i.e. IC(K)). Thus, the net bene t of not delaying is B 1 K B 1 0 IC K, and IC K measures the bene t of delaying the projet in the losed-loop approah. Similarly, the differene between the integrals in (6) measures the bene t of delaying in the open-loop approah. Sine option value is in fat a orretion fator imposed on the open-loop objetive funtion, (6) follows from the fat that the bene t of exeuting the projet in period one is the same B 1 K B 1 0 in both open and losed-loop approahes. 3. Restoration and endogenous degree of irreversibility Now suppose that restoration is possible in the seond period at a ertain ost. Let I R be the amount of restoration (i.e. it is I 2 when I 2 is negative), and 0 and I R the xed and variable osts. Restoration efforts usually inur physial osts and transation osts that are independent of restoration level. We assume that is onvex with 0 ˆ0. The new deision problem is max I 1 B 1 I 1 2 w 2 I 1 B 2 I 1 I r 2 I 1 ; ; f d max I R B 2 I 1 I R ; I R 0 s I R Š f d where s I R ˆ > 0 ifi R ˆ > 0. The seond omponent in (7) is the expeted seond period bene t when further development is needed (so I r 2 is used). The third omponent is the expeted bene t when restoration is needed and its amount has to be optimally hosen. The optimal restoration deisions are illustrated in Fig. 2. At ˆ w 2 I 1, there is no inentive for restoration. As dereases, restoration is not feasible at rst, sine the variable bene t of restoration is small and annot ompensate for the xed ost that would have to be inurred if restoration is undertaken. However, as further delines, the variable bene t of restoration inreases, and eventually will surpass the xed ost when falls below a ritial level, denoted as w r 2 I 1 ; 0, and restoration ours. We denote this optimal restoration level as I R I 1 ;. It is 7
566 irreversibility and restoration straightforward to show (see Appendix 2) that the ritial level w2 I r 1 ; 0 is inreasing in I 1 and dereasing in 0. Using the irreversibility ost approah of the last setion, (7) an be rewritten as max B 1 I 1 ; w 1 M r IC R I 1 ; 0 8 I 1 where M r, de ned in (4), is the seond period bene t with free restoration (or perfet reversibility), and IC R I 1 ; 0 (de ned in Appendix 2) is the irreversibility ost with restoration, an extension of irreversibility ost IR. IC R is inreasing, although not neessarily onvex, in I 1. But the onavity of B 1 and B 2 and onvexity of ensure that a unique solution exists in (8). More importantly: Theorem 3 The marginal irreversibility ost with restoration, MIC R I 1 ; 0, is inreasing in 0. Spei ally, MIC R I 1 ; 0 ˆ Fig. 2. Restoration given I 1. w2 r I 1; 0 r 2 I 1 ; 0 @B2 I 1 ; @I 1 f d 0 I R I 1 ; f d 9
jinhua zhao and david zilberman 567 From (9), we see that MIC R is inreasing in the marginal restoration ost 0. Fixed ost 0 affets MIC R I 1 ; 0 through w2 I r 1 ; 0, or by affeting the range of where restoration is eonomially feasible. Sine higher 0 redues optimal rst period development ^I 1, and as Appendix 2 shows, ^I R I 1 ; is inreasing in ^I 1, we know @^I R ^I 0 ; @^I 4 0; E R ^I 1 0 ; @ w2 4 0 10 0 @ 0 Higher xed ost of restoration redues the expeted restoration and the restoration effort for eah state of nature. The intuition is that as the xed restoration ost inreases, the deision maker would prefer a lower probability of restoration, leading to less development in the rst period. Thus less restoration is optimal for eah state of nature. We an then onlude that: Remark 1 In a dynami and stohasti framework, the xed ost of restoration affets not only whether or not there is restoration but also the level of restoration. In fat, we an use our framework to show that the marginal value of investment, i.e. the marginal q in the neolassial investment theory (Abel and Eberly (1994)), depends ontinuously on 0. For example, by (8) we an dedut that the value funtion of seond period investment is V 2 I 1 ; 0 ˆM r IC R I 1 ; 0. Then learly the marginal value of a size I 1 projet (whih is negative in our speial ase) is dereasing in 0. The literature has not expliitly modeled the role of xed osts in affeting marginal q. The disussion so far indiates that in the seond period restoration may be needed. The probability and magnitude of restoration depend on the downside support of the random variable, the xed ost 0, and initial development I 1. For a given I 1, absolute irreversibility orresponds to a situation where the xed ost of restoration is so high that the probability of restoration is zero, or w2 I r 1 ; 0 ˆ. It does not depend on the marginal ost of restoration, sine a higher marginal ost would only lead to a smaller amount of restoration (but does not lead to no restoration), unless the marginal ost is in nite. Caballero (1991) identi ed absolute irreversibility with in nite marginal restoration ost. Then absolute irreversibility is a rare event sine in nite marginal ost rarely happens. However, we show here that absolute irreversibility happens as long as the xed ost of negative adjustment is reasonably high. This senario is muh more likely to happen. This interpretation means that irreversibility is a dynami onept and evolves with hanges in parameters. For example, hanges in tehnologies that redue 0 and hanges in tastes that inrease the damage of development may make irreversible projets in the past beome restorable now. The degree of irreversibility is endogenously determined when I 1 is hosen. Absolute irreversibility is less likely for large projets than for small projets. ROT assumes away this endogeneity by assuming absolute irreversibility. Therefore, allowing variable projet size in ROT on its with the absolute irreversibility assumption.
568 irreversibility and restoration The ost of assuming absolute irreversibility (and ignoring restoration possibilities) depends on the nature of the problem. Ceteris paribus, it is small if both the xed and variable osts of restoration are high. If resoure development leads to tehnial irreversibility, suh as extintion of speies, then the restoration ost is in nite and the assumption of absolute (eonomi) irreversibility is justi ed. 8 In some ases suh as urban sprawl when agriultural or forest lands are onverted to urban use, the ost of restoration is high and the assumption of irreversibility is realisti. However, in other ases suh as restoring the sheries that are damaged by water development projets, the restoration ost is suf iently low so that the assumption of absolute irreversibility is unrealisti. In fat, we have witnessed signi ant restoration effets in sheries. 9 4. Disussion and onlusion This paper expands ROT to situations where a natural resoure development projet may make the deision maker worse-off and ostly restoration is feasible. We have introdued a notion of irreversibility ost whih is the expeted ost due to the irreversibility onstraint and restoration ost. This onept is related to, but more general than the Arrow±Fisher±Henry option value. Even without relaxing the assumption of absolute irreversibility, the impliations of ROT may be different for resoure projets than those for onventional investment projets. We showed that the value of a projet may not be onvex in the underlying random variable, due to the possibility of bene t loss of an over-sized projet. A risk neutral deision maker may thus demonstrate risk averse behavior: both the sale of development and the assoiated payoff go down as the level of unertainty rises. This result highlights the importane of information gathering and unertainty resolution in projet formulation. We also showed that for resoure projets, a higher disount rate may in fat lead to more development. It enourages the mentality of `building now and restoring later'. This result has important impliations for the hoie of disount rate in resoure projet evaluation. Low disount rates have been blamed for exessive resoure development. We argue that this is mainly due to the pratie of negleting the possibility of ostly restoration in formulating and evaluating... 8 Even in this ase, we should be areful when asserting absolute irreversibility. If a speies exists in different parts of the world, then extintion in one area is not irreversible: with some ost, the same speies an be reintrodued in this area. In another ase, if there are lose substitutes to a speies (either in terms of their gene struture or in terms of onsumption substitution), its extintion is likely to inur eonomi losses (i.e. we wish to restore it) only in very extreme situations. In this ase, irreversibility an be safely ignored: there may be absolute irreversibility, but essentially it does not affet the optimal resoure development. 9 One example is the Central Valley Projet Improvement At (CVPIA) in California whih alloates water to help restore a salmon shery in San Franiso Bay Delta that was negatively impated by water development.
jinhua zhao and david zilberman 569 resoure projets. One ostly restoration is onsidered, the reverse may be true: low disount rate may enourage more autious development deisions. When ostly restoration is allowed, the degree of irreversibility beomes an endogenous hoie. Absolute irreversibility happens only when the xed restoration ost is suf iently high and when the random variable has a suf iently high downside support (i.e. a truly bad senario annot happen). ROT, by assuming absolute irreversibility, assumes away an endogenous hoie. Further, ontrary to the popular argument that large projets tend to be irreversible, we nd that absolute irreversibility is more likely for small projets. We showed that understanding the struture of the restoration ost funtion is important: the xed ost of restoration affets not only whether or not restoration should be undertaken, but also the level of development (and restoration) deisions. More generally, the marginal value of investment, i.e. the marginal q in neolassial investment theory, ontinuously depends on the xed ost. Although we have foused on resoure development projets, the framework an be used to analyze more onventional investment deisions where ostly orretion of mistakes is allowed. In essene, restoration is a way of orreting the mistake of too muh initial investment. Similar senarios may arise in other settings, suh as an investment projet leading to the possibility of environmental liability. For example, with the federal Superfund leanup program in the United States, a urrently pro table investment may bring forth future environmental liabilities that outweigh the earned pro ts (Aton, 1989). Aknowledgements We would like to thank Anthony Fisher, Mihael Hanemann, Hayne Leland, Brian Wright, and two anonymous reviewers for their helpful omments. The usual dislaimer applies. Referenes Abel, A. B. and Eberly, J. C. (1994). `A Uni ed Model of Investment Under Unertainty', Amerian Eonomi Review, 84, 1369±84. Aton, J. P. (1989). `Understanding Superfund: a progress Report', Tehnial Report RAND/ R-3838-ICJ, RAND Corporation, CA. Arrow, K. J. and Fisher, A. C. (1974). `Environmental Preservation, Unertainty, and Irreversibility', Quarterly Journal of Eonomis, 88, 312±19. Caballero, R. J. (1991). `On the Sign of the Investment±Unertainty Relationship', Amerian Eonomi Review, 81, 279±88. Capozza, D. and Li, Y. (1994). `The Intensity and Timing of Investment: the Case of Land', Amerian Eonomi Review, 84, 889±904. Dixit, A. K. and Pindyk, R. S. (1994). Investment Under Unertainty, Prineton University Press, Prineton, NJ. Epstein, L. C. (1980). `Deision Making and The Temporal Resolution of Unertainty', International Eonomi Review, 21, 269±83.
570 irreversibility and restoration Fisher, A. C., Krutilla, J. V. and Cihetti, C. J. (1972). `The Eonomis of Environmental Preservation: a Theoretial and Empirial Analysis', Amerian Eonomi Review, 62, 605±19. Freixas, X. and Laffont, J. (1984). `On the Irreversibility Effet', in M. Boyer and R. Kihlstrom (eds), Bayesian Models in Eonomi Theory, North Holland, Ch. 7, 105±14. Hanemann, W. M. (1989). `Information and the Conept of Option Value', Journal of Environmental Eonomis and Management, 16, 23±37. Henry, C. (1974). `Investment Deisions Under Unertainty: the Irreversibility Effet', Amerian Eonomi Review, 64, 1006±12. Kolstad, C. (1996). `Fundamental Irreversibilities in Stok Externalities', Journal of Publi Eonomis, 60, 221±33. Majd, S. and Pindyk, R. (1987). `Time to Build, Option Value and Investment Deisions', Journal of Finanial Eonomis, 18, 7±27. MDonald, R. and Siegel, D. (1986). `The Value of Waiting to Invest', Quarterly Journal of Eonomis, 101, 707±27. Pindyk, R. S. (1988). `Irreversible Investment, Capaity Choie, and the Value of the Firm', Amerian Eonomi Review, 79, 969±85. Ulph, A. and Ulph, D. (1997). `Global Warming, Irreversibility and Learning', Eonomi Journal, 107, 636±50. Visusi, W. K. (1985). `Environmental Poliy Choie with an Unertain Chane of Irreversibility', Journal of Environmental Eonomis and Management, 28±44. Appendix 1 Details of the basi model We denote the seond period maximization problem in (1) as J I 1 ; ˆ max I2 5 0B 2 I 1 I 2 ; and its solution as ^I 2 I 1 ;. For ^I 2 I 1 ; > 0, it is easy to verify that @^I 2 I 1 ; =@ > 0 and @^I 2 I 1 ; =@I 1 < 0. This result leads to Fig. 1. The formal de nition of w 2 I 1 is: De nition 1 For a given I 1, the assoiated ritial is de ned as w2 I 1 ˆsup f 2 ; Š : ^I 2 I 1 ; ˆ0g [ inf f 2 ; Š : ^I 2 I 1 ; > 0g Theorem 4 The irreversibility ost is inreasing and onvex in I 1. Proof. From the de nition of threshold level, we know I2 I r 1 ; w2 I 1 ˆ 0. The rst order ondtion of (3) gives @B 2 I 1 I2; r =@I 2 ˆ 0. We then have MIC ˆ @IC ˆ B 2 I @I 1 I2 I r 1 ; w2 I 1 ; w2 I 1 B 2 I 1 ; w2 I 1 f w2 I 1 @ I 1 1 @I 1 @B 2 I 1 I r 2; x 1 @Ir 2 @B2 I 1 ; x f x dx @I 2 @I 1 @I 1 ˆ 0 0 1 @I r 2 @B2 I 1 ; x f x dx @I 1 @I 1 ˆ 11 @B 2 I 1 ; x f x dx 12 @I 1
For any < w2 I 1, @B 2 I 1 ; =@I 1 <@B 2 I 1 I2; r =@I 1 sine I2 I r 1 ; < 0 and B 2 is onave in K. But @B 2 I 1 I2; r =@I 1 ˆ @B 2 I 1 I2; r =@I 2 ˆ 0. Therefore, for any < w2 I 1, @B 2 I 1 ; =@I 1 < 0. Substituting this result into (12), we get MIC > 0. Similarly, differentiating Equation 12 with respet to I 1, we have @MIC I 1 ; w 1 ˆ 0 @ 2 B @I 1 @I1 2 f x dx Sine B 2 is onave in I 1, @MIC I 1 ; w 1 =@I 1 > 0. Theorem 5 For any I 1, E w2 J I 1 ; ˆM r IC I 1. Proof E w2 J I 1 ; ˆ 2 jinhua zhao and david zilberman 571 max I 2 5 0 B2 I 1 I 2 ; x f x dx & ˆ B 2 I 1 ; x f x dx 2 w 2 I 1 B 2 I 1 ^I 2 ; x f x dx 13 M r ˆ 2 max w I 2 2 B 2 I 1 I 2 ; x f x dx ˆ 2 B 2 I 1 I r 2; x f x dx ˆ B 2 I 1 I2; r x f x dx B 2 I 1 I2; r x f x dx: w2 I 1 Subtrating (13) from (14), and noting that for 5 w2 I 1, I2 r ˆ ^I 2, we get M r E w2 J I 1 ; ˆ ˆ IC I 1 2 B 2 I 1 I r 2; x B 2 I 1 ; x f x dx Theorem 6 Optimal I 1 inreases as disount rate i inreases. Proof. Note that IC I 1 ˆIC 0 I 1 = 1 i where IC 0 I 1 is given in (2) with B 2 by B 2 0. Then the theorem follows immediately from (5). 14 15 replaed & Appendix 2 Details of the model with endogenous irreversibility We rst deal with the third omponent of (7), that is, the restoration problem given I 1 and < w2 I 1. Denote the restoration payoff assoiated with a ertain I 1 and as V I 1 ; ; I R ˆB 2 I 1 I R ; I R 0 s I R 16 and let ^I R I 1 ; ˆarg max IR V I R ; ; I R. To understand the behavior of ^I R I 1 ;, let IR I 1 ; be the optimal restoration without the xed ost, i.e. it solves @V=@I R ˆ 0. Then there is positive amount of restoration with ^I R I 1 ; ˆIR I 1 ; only if the variable bene t of restoration surpasses the xed ost. This result happens only when is below a ritial level, denoted as w2 I r 1 ; 0.
572 irreversibility and restoration De nition 2 The ritial level of for restoration, w2 I r 1 ; 0, is de ned as w2 I r 1 ; 0 ˆsup f 2 ; w2 I 1 Š : ^I R I 1 ; > 0g [ inf f 2 ; w2 I 1 Š : ^I R I 1 ; ˆ0g 17 It is obvious that ˆ IR I ^I 1 ; if < w2 I r 1 ; 0 R I 1 ; ˆ 0 if 5 w2 I r 18 1 ; 0 It is straightforward to show that @w2 I r 1 ; 0 @w 5 0; 2 I r 1 ; 0 @^I 4 0; R I R I 1 ; @^I 5 0; R I 1 ; 4 0 19 @I 1 @ 0 @I 1 @ Higher initial development or lower xed restoration ost inreases w2 I r 1 ; 0 and the probability that restoration is optimal. The amount of restoration inreases with the initial investment and delines with. To inorporate the restoration onsideration in the optimal hoie of I 1, substitute the solution ^I R I 1 ; into (7). Then the maximization problem beomes 2 max B 1 I 1 ; w 1 I 1 w2 I 1 adding apaity z } { B 2 I 1 I2 I r 1 ; ; f d r 2 I 1 ; 0 no ation z } { B 2 I 1 ; f d w2 r I 1; 0 B 2 I 1 IR I 1 ; ; IR I 1 ; 0 f d {z } restoration Using the de nition of M r, (20) an be rewritten as (8), and the expeted loss in period two due to ostly restoration is IC R I 1 ; 0 ˆ w r 2 I 1; 0 loss due to absolute irreversibility z } { B 2 I 1 I2 I r 1 ; ; B 2 I 1 ; Š f d r 2 I 1 ; 0 B 2 I 1 I2 I r 2 ; ; B 2 I 1 IR I 1 ; ; IR I 1 ; 0 Š f d {z } loss due to ostly restoration 21 The rst term on the right hand side of (21) is the ost of absolute irreversibility (when restoration is not optimal), and the seond term is the sum of the ost of restoration and irreversibility (when the restoration is partial). Note that if 0 is in nity, w2 I r 1 ; 0 ˆ and we are left with only the rst term, while if 0 is zero, w2 I r 1 ; 0 ˆw2 I 1 and we have only the seond term. 20 Appendix 3 Proof of theorems Theorem 1 Let h ˆ @B 2 I 1 ; =@I 1. We know h > 0 for < w 2 I 1 and h w 2 I 1 ˆ 0. Further, from assumptions (A2)±(A4), we know h 0 < 0, h 00 5 0, and h and h 0 are bounded for all. Using these onditions and integrating by parts, we an show that
MIC a MIC b ˆ h 0 w 2 I 1 x F a y F b y dy F a y F b y dyh 00 x dx < 0: & Theorem 2 With the new formulation, the ritial level below whih the irreversibility onstraint is binding beomes w2 K and the irreversibility ost is 2 K IC K ˆ B 2 0; B 2 K; df : 22 The AFH option value is de ned by OV K ˆ ^V 0 ^V 1 Š ~V 0 ~V 1 Š, where 2 K ^V 0 ˆB 1 0 B 2 0; df B 2 K; df w2 K is the expeted payoff of delaying the projet in a losed-loop optimization ^V 1 ˆB 1 K B 2 K; df is the expeted payoff of exeuting the projet in a losed-loop optimization 2 2 ~V 0 ˆB 1 0 max B 2 K; df ; B 2 0; df is the payoff of delaying exeution in the open-loop optimization, and 2 2 ~V 1 ˆB 1 K B 2 K; df is the payoff of not delaying in the open-loop optimization. (6) then follows immediately. & Theorem 3 Similar to proving Theorem 4, by applying (17), @B 2 I 1 I2; r =@I 1 ˆ 0, and B 2 1 I 1 IR; 0 IR ˆ0, we an show MIC R I 1 ; 0 ˆ w r 2 I 1; 0 B 2 1 I 1 ; f d jinhua zhao and david zilberman 573 r 2 I 1 ; 0 2 B 2 1 I 1 I R I 1 ; ; f d whih is equal to (9). Taking derivatives on (23), we an show @MIC R I 1 ; 0 @ 0 ˆ B 2 1 I 1 ; w r 2 I 1 ; 0 B 2 1 I 1 I R I 1 ; w r 2 I 1 ; 0 ; w r 2 I 1 ; 0 23 f w r 2 I 1 ; 0 @wr 2 I 1 ; 0 @ 0 > 0 24 &