Option exercise with temptation

Transcription

1 Economic Theory 2008) 34: DOI /s RESEARCH ARTICLE Jinjun Mio Option exercise with tempttion Received: 25 Jnury 2006 / Revised: 5 December 2006 / Published online: 10 Jnury 2007 Springer-Verlg 2006 Abstrct This pper dopts the Gul nd Pensendorfer self-control utility model to nlyze n gent s option exercise decision under uncertinty over n infinite horizon. The gent decides whether nd when to do n irreversible ctivity. He is tempted by immedite grtifiction nd suffers from self-control problems. The cost of self-control lowers the benefit from continution or stopping nd my erode the option vlue of witing. When pplied to the investment nd exit problems, the model cn generte the behvior of procrstintion nd prepropertion. In ddition, unlike the hyperbolic discounting model, the model here provides unique prediction. Keywords Time in)consistency Self-control Tempttion Procrstintion Prepropertion Rel options JEL Clssifiction Numbers D81 D91 G31 I thnk Drew Fudenberg, Brt Lipmn, Neng Wng, nd prticipnts in the BU theory lunch workshop nd in the Cndin Economic Theory Conference 2005 for helpful discussions. I m especilly grteful for vluble suggestions from n nonymous referee nd coeditor Steve Willimson. The BU ISP Seed Grnt hs supported this reserch. J. Mio B) Deprtment of Economics, Boston University, 270 By Stte Rod, Boston, MA 02215, USA E-mil: mioj@bu.edu J. Mio Deprtment of Finnce, Hong Kong University of Science nd Technology, Hong Kong, Hong Kong J. Mio CEMA of Centrl University of Finnce nd Economics, Beijing, Chin

2 474 J. Mio 1 Introduction Suppose you hve referee report to write tody. You feel writing the referee report is unplesnt nd prefer to put off nd do it tomorrow. But when tomorrow comes, you tend to dely gin. This behvior is often referred to s procrstintion wit when you should do it. Suppose you hve coupon to see one movie over the next severl weeks, nd your llownce does not permit you to py for movie. You tend to see movie in n erlier week even though there my be better movie in lter week. This behvior is often referred to s prepropertion do it when you should wit. 1 The procrstintion nd prepropertion behvior is prevlent in mny choice situtions. Motivted by this behvior, this pper studies generl environment where n gentwith time-consistent preferences mkes irreversible binry choices under uncertinty over n infinite horizon. I dopt the Gul nd Pesendorfer 2001, 2004) self-control utility model nd interpret tht behvior s n gent s struggling with tempttions. 2 In this model, preferences re defined over domin of sets of lterntives or decision problems. Utility depends on the decision problem from which current consumption is chosen. The interprettion is tht tempttion hs to do with not just wht the gent hs consumed, but lso wht he could hve consumed. The gent lso seeks immedite grtifiction becuse n immedite benefit constitutes tempttion to the gent, but not becuse it hs higher reltive weight. The gent my either succumb to tempttions or exercise costly self-control to resist tempttions. The Gul-Pesendorfer model is time consistent becuse utility stisfies recursivity under the domin of decision problems. Thus, the stndrd recursive methods such s bckwrd induction nd dynmic progrmming cn be pplied. Importntly, the Gul-Pesendorfer model cn explin time inconsistent behvior observed in some experiments s illustrted in Gul nd Pesendorfer 2001, 2004). In ddition to its trctbility, the Gul-Pesendorfer model hs cler welfre implictions becuse it is bsed on the stndrd reveled preference principle. This is in contrst to the hyperbolic discounting model in which there is no generlly greed welfre criterion. In the hyperbolic discounting model, the gent t different dtes is treted s seprte self. An lterntive or policy my be preferred by some selves, while it my mke other selves worse off. The Preto efficiency criterion nd the long-run ex nte utility criterion re often dopted. In Sect. 2, I model n gent s irreversible binry choice problem under uncertinty s n option exercise problem, or more techniclly, n optiml stopping problem. Irreversibility nd uncertinty re importnt in mny binry choice problems such s entry, exit, defult, liquidtion, project investment, nd job serch. According to the stndrd theory see Dixit nd Pindyck 1994), ll these problems cn be viewed s problem where gents decide when to exercise n option nlogous to finncil cll option it hs the right but not the obligtion to buy 1 These exmples nd the term prepropertion re borrowed from O Donoghue nd Rbin 1999). As there, the comprison is bsed on the stndrd time consistent preferences benchmrk. 2 The Gul-Pesendorfer model hs been pplied to study txtion Krusell et l. 2001), sset pricing Krusell et l. 2002), DeJong nd Ripoll 2006), nd nonliner pricing Estebn et l. 2006).

3 Option exercise with tempttion 475 n sset t some future time of its choosing. This rel options pproch emphsizes the positive option vlue of witing. Unlike the stndrd theory, I mke the distinction ccording to whether rewrds nd costs re immedite or delyed, s in O Donoghue nd Rbin 1999) who first nlyze procrstintion nd prepropertion using hyperbolic discounting model in finite horizon setting. 3 This distinction is importnt to explin procrstintion nd prepropertion in the hyperbolic discounting model since it mkes present bis criticl. This distinction is lso importnt in the present model since it mkes immedite tempttion criticl. After stting the model setup nd ssumptions, I present the self-control utility model developed by Gul nd Pesendorfer 2001, 2004) nd compre it with the hyperbolic discounting model. I then present propositions to chrcterize the optiml stopping rules for the generl infinite-horizon model when the gent hs self-control preferences. I describe the optiml stopping rules s trigger policy whereby the gent stops the first time the stte process hits threshold vlue. I lso explin the impct of tempttion nd self-control on the optiml stopping rules. In prticulr, I show tht the cost of self-control my lower the benefit from both stopping nd continution nd erode option vlue of witing. Moreover, it my outweigh this option vlue if the level of self-control is sufficiently low. In Sect. 3, I pply the results in Sect. 2 to investment nd exit problems when the decision mker hs self-control preferences. I lso conduct welfre nlysis. The investment nd exit problems represent two different clsses of option exercise problems. The project investment decision is n exmple in which n gent decides whether or not to exercise n option to pursue upside potentil. Entry nd job serch re similr problems. I show the following: When the investment cost is immedite, the investor is tempted to dely investment. Thus, he procrstintes nd the welfre loss is the forgone project vlue, which is equl to the cost of selfcontrol. When the project vlue is immedite, the investor is tempted to invest erly. Thus, he prepropertes nd the welfre loss is the forgone option vlue of witing. If his level of self-control is sufficiently low, the investor my invest in negtive net present vlue NPV) projects. This reflects the trde-off between investing now but incurring finncil losses nd witing but incurring self-control costs. When both the project vlue nd investment cost re immedite, the investor lso prepropertes nd the welfre loss is the forgone option vlue of witing. In this cse, he never invests in negtive NPV projects. If his level of self-control is sufficiently low, he invests ccording to the myopic rule which compres the current period benefit nd cost only. After nlyzing the investment problem, I turn to the exit problem, in which n owner/mnger with self-control preferences decides when nd if to terminte project. This problem represents n exmple in which n gent decides whether or not to exercise n option to void downside losses. Other exmples include defult nd liquidtion decisions. I show the following: When the profits re immedite, 3 Strotz 1956) first studies time-inconsistent preferences in economics. Akerlof 1991) nlyzes procrstintion, but frmes his discussion very differently. The O Donoghue nd Rbin model hs been generlized by number of ppers, e.g., O Donoghue nd Rbin 1999b, 2001), Brocs nd Crrillo 2001, 2005). The hyperbolic discounting model hs been pplied to study consumption-sving Libson 1994, 1997), job serch DellVign nd Psermn 2005), socil security Imrohoroglu et l. 2003), retirement Dimond nd Koszegi 2003), investment Grendier nd Wng 2006), nd generl equilibrium Herings nd Rohde 2006).

4 476 J. Mio the owner is tempted to continue the project even when he should terminte. Thus, he procrstintes. The welfre loss is equl to the cost of self-control. By contrst, when the fixed cost of continuing the project is immedite, the owner is tempted to void this cost nd prepropertes to terminte, even though he my mke positive net profits. The welfre loss is the forgone current nd future profit opportunities. When both the cost nd profit re immedite, the owner lso prepropertes, but never termintes the project t time when he mkes negtive net profits. If the owner s level of self-control is sufficiently low, the owner termintes the project ccording to the myopic rule. O Donoghue nd Rbin 2001) nlyze similr infinite horizon deterministic tsk choice problem using the hyperbolic discounting model. They show tht their model typiclly hs multiple equilibri using the perception-perfect strtegy solution concept. They lso show tht some equilibri re cyclic, with some fixed intervls of length between ction dtes. I show tht cyclic equilibri lso rise in the problem under uncertinty nlyzed here. These cyclic equilibri re counterintuitive nd unppeling. By contrst, the Gul-Pesendorfer model dmits unique prediction. The importnce of uniqueness is emphsized by Fudenberg nd Levine 2006). Fudenberg nd Levine provide dul-self model which is lso motivted by time inconsistency issues. They show tht their reduced-form model is similr to the Gul-Pesendorfer model. They independently nlyze n optiml stopping problem which is specil cse of my generl setup. As here, they lso chrcterize the optiml stopping rule by trigger policy nd derive unique solution. Their model differs from mine in tht they ssume the cost is stochstic nd the rewrd is constnt. Moreover, they consider only the cse of immedite costs nd future benefits. I conclude the pper in Sect. 4 nd relegte technicl detils to n ppendix. 2 The model I model n gent s option exercise decisions s n optiml stopping problem. Specificlly, consider discrete time nd infinite horizon environment. In ech period, the gent decides whether to stop process nd tke termintion pyoff, or to continue for one more period nd mke the sme decision in the future. The decision is irreversible in the sense tht if the gent chooses to stop, he mkes no further choices. Formlly, time is denoted by t = 1, 2,..., nd uncertinty is generted by stte process x t ) t 1. For simplicity, I ssume tht x t is drwn identiclly nd independently from distribution F on, A], where A > > 0. Continution t dte t genertes pyoff π x t ) nd incurs cost c c, while stopping t dte t yields pyoff x t ) nd incurs cost c s, where π nd re continuous nd incresing functions. I will provide pplictions in Sect. 3 to show tht this simple model covers wide rge of economic problems. As in O Donoghue nd Rbin 1999), I mke n importnt distinction ccording to whether costs nd rewrds re obtined immeditely or delyed. The term of immedite costs is used to refer to the sitution where the cost is incurred immeditely while the rewrd is delyed. The term of immedite rewrds is used to refer to the sitution where the rewrd is incurred immeditely while the cost is delyed. For simplicity, I consider the cse of one period dely only. In ddition, I lso consider the cse where both costs nd rewrds re immedite. This cse is not

5 Option exercise with tempttion 477 explicitly nlyzed by O Donoghue nd Rbin 1999). O Donoghue nd Rbin 1999) give mny exmples to illustrte tht the preceding distinction is meningful in relity. Moreover, this distinction is importnt to generte procrstintion nd prepropertion. Unlike O Donoghue nd Rbin 1999), I consider uncertinty nd infinite horizon. Uncertinty is prevlent in intertemporl choices nd infinite horizon is necessry to nlyze long-run sttionry decision problems. These two elements re building blocks in mny economic models, especilly in mcroeconomics nd finnce. Incorporting them llows me to study some interesting pplictions in mcroeconomics nd finnce, s illustrted in Sect Self-control preferences O Donoghue nd Rbin 1999) explin procrstintion nd prepropertion by dopting the time-inconsistent hyperbolic discounting model proposed by Phelps nd Pollk 1968). This model cn be described s follows. Let U t c t,..., c T ) represent n gent s intertemporl preferences from consumption strem c t,..., c T ) in period t. T could be finite or infinite. The hyperbolic discounting preferences re represented by T t ] U t c t,..., c T ) = u t c t ) + β E δ k u t+k c t+k ), t 1, where 0 < β,δ 1ndu t+k ) represents period t + k utility function, k = 0,...,T t. In ddition, δ represents long-run, time-consistent discounting nd β represents bis for the present. The gent t ech point in time is regrded s seprte self who is choosing his current behvior to mximize current preferences, while his future selves will control his future behvior. In this model, n gent must form expecttion bout his future selves preferences. Two extreme ssumptions re often mde. In one extreme, the gent is nive nd believes his future selves preferences will be identicl to her current self s, not relizing chnging tstes. In the other extreme, the gent is sophisticted nd knows exctly wht his future selves preferences will be. The solution concept of subgme perfect Nsh equilibrium is often dopted. As typicl in dynmic gmes, multiple equilibri my rise see Proposition 6 below, Fudenberg nd Levine 2006; Krusell nd Smith 2003, nd O Donoghue nd Rbin 2001). The hyperbolic discounting model provides n intuitive explntion for procrstintion nd prepropertion. The key intuition relies on the following feture of the hyperbolic discounting model. When β<1, the gent gives more reltive weight to period t when he mkes choice in period t thn he does when he mkes the choice in ny period prior to period t. Tht is, the gent hs time-inconsistent tste for immedite grtifiction. There seems to be mple experimentl evidence on the time-inconsistent behvior. 4 In typicl experiment, subjects choose between smller period t rewrd nd lrger period t + 1 rewrd. If the choice is mde in period t then the smller erlier rewrd is chosen. If the choice is mde erlier, then the lrger lter rewrd is chosen. k=1 4 See, for exmple, Thler 1981), Ainslie nd Hslm 1992), Kirby nd Herrnstein 1995).

6 478 J. Mio Gul nd Pesendorfer 2001, 2004) propose n lterntive interprettion of this behvior bsed on time-consistent preferences. Their key insight is tht the gent finds immedite rewrds tempting. When the gent mkes the choice in period t, the period t rewrd constitutes tempttion to the gent. So he my choose smller period t rewrd rther thn lrger period t + 1 rewrd. However, if he mkes choice prior to period t, neither period t rewrd nor period t +1 rewrd cn be consumed immeditely nd hence his decisions re unffected by tempttions. To cpture this intuition, Gul nd Pesendorfer 2001, 2004) develop model of self-control bsed on choice theoretic xiomtic foundtion. 5 They define self-control preferences over sets of lterntive consumption levels or decision problems domin different from the usul one. The interprettion is tht tempttion hs to do with not just wht the gent chooses, but wht he could hve chosen. Specificlly, let B t be the gent s period t decision problem nd W t represent his intertemporl utility in period t. Then the self-control preferences re represented by W t B t ) = mx c t B t ut c t ) + v t c t ) + δe W t+1 B t+1 ) ]} mx c t B t v t c t ), t 1. If T is finite, since there is no continution problem in period T, W T B T ) = mx u T c T ) + v T c T )} mx v T c T ). 2) c T B T c T B T Here u t +δw t+1 represents the commitment utility in period t nd v t is the tempttion utility in period t. The expression u t c t )+v t c t )+δe W t+1 B t+1 ) ] reflects the compromise between commitment nd tempttion. The gent s optiml choice in period t mximizes this expression. When this choice is identicl to the tempttion choice in the second mximum in 1) or 2), the gent succumbs to the tempttion nd there is no self-control cost. However, when the two choices do not coincide, the gent exercises costly self-control nd v t c t ) mx ct B t v t c t ) represents the cost of self-control. If T =, I consider sttionry model nd drop time subscripts, W B) = mx c B 1) u c) + v c) + δe W B )]} mx v c). 3) c B Here B denotes the choice problem in the next period nd E ] denotes the expecttion opertor. An importnt feture of the Gul Pesendorfer model is tht it is time consistent since utility in 1) 3) is defined recursively. Thus, the usul recursive method such s bckwrd induction nd dynmic progrmming cn be pplied. Importntly, in ddition to this trctbility, this model hs cler welfre implictions. Tht is, this model follows the reveled preference trdition of stndrd economic models: if the gent chooses one lterntive over nother, then he is better off with tht choice. By contrst, time inconsistent models do not hve universlly greed welfre criterion. Some reserchers such s Libson 1994, 1997) dopt Preto efficiency criterion, requiring ll period selves wekly prefer one strtegy to nother. Other 5 See Gul nd Pesendorfer 2001, 2004) for detiled xioms. The key xiom is set betweenness. Their model is more generl thn the one presented in this pper.

7 Option exercise with tempttion 479 reserchers such s O Donoghue nd Rbin 1999) dopt n ex nte long-run utility criterion. The problem of the welfre nlysis of the time inconsistent models is tht the connection between choice nd welfre is broken. In the present pper, I dopt the Gul Pesendorfer model to nlyze the option exercise problem. In this problem, the set B consists of two elements representing the current period pyoffs from stopping nd continution since the choice problems re binry. If the gent chooses to stop, then there is no continution problem so tht B = nd W B ) = 0. If the gent chooses to continue, then he fces the sme decision problem in the next period so tht B consists of two elements representing the pyoffs from stopping nd continution in the next period. To simplify exposition, I ssume risk neutrlity throughout. Tht is, u c) = c nd v c) = λc, λ>0. Here λ is the self-control prmeter. An increses in λ rises the weight on the tempttion utility nd leds to decrese in the gent s instntneous) selfcontrol. 6 When λ = 0, the model reduces to the stndrd time-dditive expected utility model with exponentil discounting. 2.2 Optiml stopping rules I now dopt the Gul Pesendorfer utility model 3) to solve the gent s option exercise problem by dynmic progrmming. 7 The key is to formulte Bellmn equtions. These Bellmn equtions re different for the cses of immedite costs, immedite rewrds, nd immedite costs nd rewrds. They re described s follows: 1. Immedite costs W x) = mx δ x) 1+λ) c s,δπx) 1+λ) c c +δ W x ) df x )} λ mx c c, c s }. 4) 2. Immedite rewrds W x) = mx 1+λ) x) δc s, 1+λ) π x) δc c +δ W x ) df x )} λ mx π x), x)}. 5) 3. Immedite costs nd rewrds W x) = mx 1+λ) x) c s ), 1+λ)π x) c c )+δ W x ) df x )} λ mx π x) c c, x) c s }. 6) 6 See Gul nd Pesendorfer 2004) for the definition nd chrcteriztion of mesures of selfcontrol. To distinguish between differences in imptience nd differences in self-control, one should fix intertemporl choices nd consider instntneous self-control only. 7 See Stokey nd Lucs 1989) nd Dixit nd Pindyck 1994) for the theory of dynmic progrmming. The existence of bounded nd continuous vlue function is gurnteed by the contrction mpping theorem.

8 480 J. Mio I explin 4) in some detil. The interprettions for the other two equtions re similr. Suppose costs re immedite. In the current period, the gent fces the decision problem of whether to continue or to stop fter observing the current stte tkes the vlue x. Stopping incurs n immedite cost c s nd yields the pyoff x). However, the gent obtins this pyoff in the next period so tht he gets discounted vlue δ x). After stopping, the gent hs no further choice, nd hence the continution vlue is zero. Becuse of the compromise between the tempttion nd the commitment utilities, the totl pyoff of stopping is δ x) 1 + λ) c s, which is the first term in the first curly brcket in 4). Similrly, continution incurs n immedite cost c c nd gets discounted pyoff from the next period δπ x). The gent hs to mke the sme choice of whether to stop or to continue in the next period, nd hence gets continution vlue δ W x ) df x ). Thus, we hve the second term in the first curly brcket in 4). Finlly, the gent is tempted by whether to stop now nd void the cost of continution c c or to continue nd void the cost of stopping c s. Thus, the tempttion choice is described by λ mx c c, c s }, which is the lst term in 4). Note tht the rewrds x) nd π x) do not enter the tempttion utility since they re obtined with one period dely nd hence do not tempt the gent. Clerly, continution is optiml for those vlues of x for which the mximum in the first line of 4) is ttined t the second expression in the curly brcket. Immedite termintion is optiml when the mximum is ttined t the first expression. Cll the corresponding divisions of the rnge of x the continution region nd the stopping region, respectively. A similr nlysis pplies to 5) nd 6). In generl, for rbitrry pyoffs π x) nd x), the continution nd stopping regions could be rbitrry. In most pplictions, these regions cn be esily chrcterized. In prticulr, there is threshold vlue such tht it prtitions the stte spce into continution region nd stopping region. Consequently, the optiml stopping rule is chrcterized by trigger policy. Tht is, the gent stops the first time the process x t ) t 1 hits the threshold vlue. Importntly, depending on the pyoff structure, the stopping region could be bove the threshold vlue or below it. The former cse describes the problems of pursing upside potentil such s investment nd job serch. The ltter cse describes the problems of voiding downside losses such s exit nd defult. In the pplictions in Sect. 3, I will impose explicit ssumptions nd provide more complete nd trnsprent nlysis of these problems. Here I do not provide generl conditions for the structure of the continution nd stopping regions. 8 Insted, I provide explicit chrcteriztions of the threshold vlue for the cse where the gent pursues upside potentil only. Tht is, I consider tht the continution region is below the threshold vlue. I chrcterize the optiml stopping rule in the following proposition: Proposition 1 Consider ech problem in 4) 6).If x) π x) is strictly incresing in x, then the optiml stopping rule is described by one of the following cses: ) The gent never stops. b) The gent stops immeditely. c) There is unique threshold vlue, A] such tht the gent stops the first time the process x t ) t 1 hits from below. In wht follows, I will focus on the third cse since it is the most interesting cse. To compre with the stndrd model, I denote by x the threshold vlue for n 8 See Dixit nd Pindyck 1994) pp ], for discussions for stndrd preferences.

9 Option exercise with tempttion 481 gent hving stndrd preferences with λ = 0. Since the men vlue of the option exercise time increses with the threshold vlue, comprtive sttic nlysis for the threshold vlue revels properties of the verge option exercise time. Proposition 2 Let the ssumption in Proposition 1 hold. Suppose costs re immedite. i) The threshold vlue stisfies the eqution 1 δ) δ ) c s ] + λ 1 δ) cs mx c c, c s }] = δπ ) c c + δ + δ δ π x ) π )] df x ) δ x ) ) ] df x ) + λ c c mx c c, c s }]. 7) ii) If c s c c, then x. If c s < c c, then < x. The interprettion of 7) is s follows. The expression on the left side of Eq. 7) describes the normlized per period benefit from stopping, while the expression on the right side describes the benefit from continution or the opportunity cost of stopping. The gent optimlly stops t the threshold vlue such tht he is indifferent between stopping nd continution. Note tht the benefit from continution consists of not only the current period vlue but lso n option vlue of witing when the gent wits for one more period nd gets better drw x > or worse drw x <. The option vlue is represented by the integrtion terms on the right side of 7). Importntly, ll terms contining λ represent the cost of self-control. Specificlly, the term on the first line of 7) represents the normlize per period cost of self-control if the gent chooses to stop. The term on the fourth line of 7) represents the cost of self-control if the gent chooses to continue. When λ = 0, the model reduces to the one with stndrd preferences. For prt ii), if the cost of stopping is higher thn the cost of continution, i.e., c s c c, then the gent is tempted to continue. Thus, if the gent chooses to continue, there is no self-control cost so tht the term on the third line of 7) vnishes. By contrst, if the gent chooses to stop, then he hs to exercise self-control nd incurs cost given in the second term in the first line of 7 ). Consequently, compred with the stndrd model, the benefit of stopping is lowered nd the gent procrstintes to exercise the option. The interprettion of the other cse c s < c c ) is similr. Proposition 3 Let the ssumption in Proposition 1 hold. Suppose rewrds re immedite.

10 482 J. Mio i) The threshold vlue stisfies the eqution 1 δ) ) δc s ] + λ 1 δ) ) mx π ), )}] = π ) δc c + δ + δ x ) ) ] df x ) π x ) π )] df x ) + λ π ) mx π ), )}] + λδ π x ) mx π x ), x )} π ) mx π ), )})] df x ) + λδ x ) mx π x ), x )} ) mx π ), )})] df x ). 8) ii) If x) π x) for ll x, then x. The interprettion of 8) is similr to tht of 7). Unlike in the cse of immedite costs, the gent is tempted by stochstic rewrds in ech period. Thus, self-control incurs not only current period cost but lso future period cost. The former is represented by the fourth line of 8). The ltter is represented by the lst two lines of Eq. 8). In prticulr, in the next period the gent is tempted to stop or continue depending on whether the stte in the next period is better thn x > ) or worse thn x < ). To resist this tempttion, the gent must incur future self-control cost. Consider prt ii). When the rewrds from stopping re lwys higher thn the rewrds from continution, x) π x), the gent is tempted to stop. Stopping t mens the gent succumbs to the tempttion nd hence there is no cost of self-control. Thus, the second term in the first line of 8) vnishes. If the gent decides to continue t, he hs to resist the tempttion to stop nd hence incurs cost of self-control represented by the third line of 8). Consider next the future cost of self-control. In the next period, if x >, the gent stops nd succumbs to the tempttion. There is no cost of self-control nd hence the term in the lst line of 8) vnishes. If x <, the gent should continue nd incur cost of self-control in the next period. This implies tht the benefit from continution is lowered, compred with the stndrd model. Consequently, the gent prepropertes to exercise the option. Proposition 4 Let the ssumption in Proposition 1 hold. Suppose both costs nd rewrds re immedite.

11 Option exercise with tempttion 483 i) The threshold vlue stisfies the eqution 1 δ) ) c s ] + 1 δ) λ ) c s mx π ) c c, ) c s }] = π ) ) x c c + δ + δ x ) ) ] df x ) π x ) π )] df x ) + λ π ) c c mx π ) c c, ) c s }) + λδ π x ) c c mx π x ) c c, x) c s } π ) c c mx π ) c c, ) c s })] df x ) + λδ x ) c s mx π x ) c c, x ) c s } ) c s mx π ) c c, ) c s })] df x ). 9) ii) If x) c s π x) c c for ll x, then x. The interprettion of this proposition is similr to tht of Proposition 3. So I omit it. Finlly, when the continution region is bove the threshold vlue, the gent tries to void downside losses. This hppens in the exit problem s described in the next section. One cn provide chrcteriztions for the threshold vlue similr to Propositions Applictions This section pplies the setup nd results in Sect. 2 to study investment nd exit problems. 3.1 Investment An importnt type of option exercise problems is the irreversible investment problem. 9 Consider tht risk-neutrl investor decides whether nd when to invest in project with stochstic vlues x t in period t. Investment incurs lump sum cost I t 9 The stndrd rel options pproch ssumes tht investment pyoffs cn be spnned by trded securities so tht preferences do not mtter for investment timing Dixit nd Pindyck 1994). Here we ssume tht these securities re not vilble to the decision mker.

12 484 J. Mio the time of the investment. This investment problem cn be cst into the frmework lid out in Sect. 2 by setting 10 x) = x, c s = I, π x) = c c = 0. In stndrd investment problems, costs nd benefits come t the sme time. In relity, there re mny instnces where costs nd benefits do not rrive t the sme time. For exmple, n importnt feture of rel investment is time to build. It is often the cse tht it tkes time to complete fctory or develop new product. This is n instnce of immedite costs nd delyed rewrds. As different exmple, some firms strt investing in project finnced by borrowing. Debts my be grdully repid fter the firms ern profits. This is n instnce of immedite rewrds nd delyed costs. I now nlyze these different cses by rewriting the Bellmn equtions 4) 6) s follows: 1. Immedite costs W x) = mx δx 1 + λ) I,δ 2. Immedite rewrds W x) = mx 1 + λ) x δi,δ W x ) df x )} λ mx 0, I }. 10) W x ) df x )} λ mx x, 0}. 11) 3. Immedite costs nd rewrds W x) = mx 1 + λ)x I ),δ W x ) df x )} λ mx x I, 0}. 12) From the bove equtions, the effect of self-control is trnsprent. When rewrds re immedite, the investor is tempted to invest now. He my either succumb to tempttion or exercise costly self-control. Self-control cts s if the benefit of witing is lowered by λx in utility vlue. Thus, the investor hs n incentive to preproperte. By contrst, when costs re immedite, the investor is tempted to wit. Self-control cts s if the cost of investment is incresed by n mount of λi. This cuses the investor to procrstinte. The interesting cse is when both costs nd rewrds re immedite. When x > I, the investor is tempted to invest erlier. But when x < I, the investor is tempted to wit. Thus, the result seems to be mbiguous. Using Propositions 1 4, I formlize the preceding intuition nd chrcterize the optiml investment rule for ech cse in the following: Proposition 5 Under the conditions given in the ppendix, there is unique threshold vlue, A] x, A]) such tht the investor with self-control preferences stndrd preferences) invests the first time the process x t ) t 1 reches this vlue. 10 The stopping problem nlyzed in Fudenberg nd Levine 2006) is specil cse of my generl frmework by setting π x) = x, c c = c s = 0, nd x) = 1 δ δ v for some constnt v>0. They nlyze the cse with immedite costs nd future rewrds only.

13 Option exercise with tempttion 485 i) ii) If costs re immedite, then stisfies δ I ) 1 δ) λi 1 δ) = δ δ x x ) df x), 13) nd x is the solution for λ = 0. Moreover, > x > I/δ nd increses with λ. If rewrds re immedite, then stisfies δi ) 1 δ) = δ x x ) df x) + λδ x ) df x) λ, iii) 14) nd x is the solution for λ = 0. Moreover, < x, x >δi nd decreses with λ. If both costs nd rewrds re immedite then stisfies I ) 1 δ) = δ x x ) df x) λ I ) + λδ I ) mx 0, x I ) ] df x), 15) nd x is the solution for λ = 0. Moreover, I < x nd decreses with λ. The left nd right sides of Eq. 13) 15) describe the utility benefits from investment nd witing, respectively. At the investment threshold, the investor is indifferent between investing nd witing. Before nlyzing the impct of selfcontrol, I first discuss briefly the solution for the stndrd model corresponding to λ = 0. As is well known, becuse of irreversibility nd uncertinty, witing hs positive option vlue. The option vlue in ech cse is represented by the first term on the right side of the corresponding equtions 13) 15). Due to this option vlue, the investor with stndrd preferences invests t the time when the threshold vlue is higher thn the cost e.g., in prt i) of Proposition 5, δ x > I ). Thus, the stndrd net present vlue NPV) rule leds to non-optiml erly investment time. This result is well known in the finnce literture e.g., Dixit nd Pindyck 1994). I next turn to the cse with self-control. Consider prt i) of Proposition 5. If costs re immedite, the investor is tempted to wit. To resist this tempttion, investing now must incur self-control cost λi 1 δ), this lowers the benefit from investment s reveled on the left side of 13). Thus, the investor chooses to procrstinte in the sense tht he invests t time lter thn tht when he hs stndrd preferences. Since increses with the self-control prmeter λ, the gent delys further s the self-control prmeter becomes lrger. 11 When λ is 11 The men vlue of the witing time is given by 1 F )) 1. It is incresing in the threshold vlue.

14 486 J. Mio sufficiently lrge, my exceed the upper bound A so tht the investor never undertkes the investment project. By contrst, if rewrds re immedite, then the investor is tempted to invest now. This cse is nlyzed in prt ii) of Proposition 5. Witing incurs direct current period self-control cost λ, which lowers the option vlue of witing. Importntly, self-control dds positive option vlue of witing to invest, λδ x) df x). This is becuse, when the investor wits for one more period to invest t nd gets worse drw x <, the cost of self control is less thn the project vlue mesured by the tempttion utility. 12 One cn show tht this positive vlue is dominted by the current self-control cost. Thus, compred with the stndrd model with λ = 0, the benefit from witing is lowered nd the investor chooses to preproperte. Prt ii) of Proposition 5 lso shows tht decreses with the self-control prmeter λ. Thus, s λ gets lrger nd lrger, the investor invests sooner nd sooner. When λ is sufficiently lrge, the investor invests t threshold vlue lower thn tht prescribed by the NPV rule. Under this rule, the threshold vlue is δi. 13 This result implies tht the investor my obtin negtive NPV t the time of investment. This result seems counterintuitive. In fct, tempted by investing now, the investor my reson, If I invest now, I get rewrd nd incur cost in the future. If I do not invest now, I hve to exercise costly self-control. The cost of self-control my outweigh the option vlue of witing. Thus, I prefer to invest now even though I get negtive NPV. In relity, we do observe the phenomen tht investors rush to embrk on investments with negtive NPV. For exmple, Rook 1987) finds empiricl evidence tht the presence of credit opportunities results in present-oriented, unplnned, nd impulse buying. I now consider prt iii) of Proposition 5 where both costs nd rewrds re immedite. It is importnt to observe tht the investor would never invest t project vlue less thn the cost; tht is, cnnot be less thn I. This is becuse when < I, the investor hs no tempttion to invest nd cn choose costlessly not to invest, thereby obtining the outside vlue zero. Given I, t the threshold vlue the investor is tempted to invest. Thus, there is no self-control cost of investing t. Consider now the self-control cost of witing. Witing incurs current period self-control cost λ I ). Witing lso hs n option vlue mesured by the tempttion utility) represented by the lst term in 15) when the investor gets worse drw x <. One cn show tht the current self-control cost domintes so tht the benefit from witing is lowered. Thus, compred with the stndrd model, the investor prepropertes. Note tht s in the cse of immedite rewrds, decreses with the self-control prmeter λ. As λ is sufficiently lrge, the threshold vlue converges to the vlue I under the myopic rule. Since O Donoghue nd Rbin 1999) seminl work, the behvior of procrstintion nd prepropertion hs been often nlyzed using the hyperbolic discounting 12 When he gets better drw x >, the investor succumbs to the tempttion of investing so tht there is no self-control cost. 13 Note tht the risk-neutrl gent discounts future csh flows ccording to the long-run discount fctor δ.

15 Option exercise with tempttion 487 model. To compre with this model, I consider only procrstintion when costs re immedite. 14 Proposition 6 Suppose costs re immedite nd the gent hs the sophisticted hyperbolic discounting preferences. i) ii) If βδ δe x]) / 1 βδ) < I <βδa, then there is sttionry equilibrium where ech self invests when x t is bigger thn some threshold, A). There is n open set of prmeter vlues for which there re other equilibri. In prticulr, there is 2-cycle equilibrium where the odd-numbered selves never invest nd even ones lwys invest. This proposition demonstrtes tht the hyperbolic discounting model hs multiple equilibri. One equilibrium hs similr feture to Proposition 5. However, there is nother equilibrium hving cycles. Similr results re obtined by Fudenberg nd Levine 2006) nd O Donoghue nd Rbin 2001) for the tsk choice problem different from the one nlyzed here. As in Fudenberg nd Levine 2006), I view tht the cyclic equilibrium is counterintuitive nd the multiplicity of equilibrium is unppeling. I now turn to welfre implictions. I sk the question: How severely does the self-control problem hurt person? I compute the utility loss from investment for n investor with self-control preferences, compred with n investor with stndrd preferences. Let V x) be the vlue function for the investor with stndrd preferences corresponding to λ = 0. The utility loss from self-control problems could be mesured s V x) W x). One cn interpret V x) s the commitment preference s in Gul nd Pesendorfer 2001, 2004). Then V x) W x) mesures the utility loss if the gent cnnot precommit nd suffers from self-control problems. I evlute this mesure t the time when the gent with self-control preferences invests. Tht is, this vlue is given by V ) W ). The following proposition gives the utility loss. Proposition 7 Let nd x be given in Proposition 5. When costs re immedite, the utility loss from investment is given by λi. When rewrds re immedite or both costs nd rewrds re immedite, the utility loss from investment is given by x. By Proposition 5, when costs re immedite, the investor with self-control preferences procrstintes he wits when he should invest if he hd stndrd preferences. The utility loss is the forgone project vlue. This loss is incresing in the self-control prmeter λ. However, it does not increse with λ without bound. This is becuse s λ pproches the vlue δ A/I 1, the investment threshold pproches the upper bound of the project vlue A. When λ is incresed further, no investment is ever mde nd the gent gets zero. Thus, the upper bound of the utility loss is δ A I. When rewrds re immedite or both costs nd rewrd re immedite, the investor with self-control preferences prepropertes he invests 14 I lso consider sophisticted gent s behvior only. A complete chrcteriztion for the nive or prtilly sophisticted gent is not my focus nd is beyond the scope of the present pper.

16 488 J. Mio Tble 1 This tble presents solutions for the investment threshold vlues, the men witing time until investment nd the utility loss. The utility loss is mesured s V ) W )) /V ) λ Threshold Witing time Utility loss Immedite costs Immedite rewrds Immedite costs nd rewrds when he should wit if he hd stndrd preferences. The utility loss is then the forgone option vlue of witing. The following exmple illustrtes Propositions 5 nd 7 numericlly. Exmple 1 Let = 0, A = 1, δ = 0.9, I = 0.5, nd F x) = x. Tble 1 reports the solution. It revels tht even for smll self-control problems, i.e. smll λ, the utility loss could be quite lrge. For exmple, when costs re immedite nd λ = 0.6, the investor procrstintes bout 70 periods to invest. The utility loss ccounts for 77% of the project vlue. When rewrds re immedite nd λ = 0.6, the investor prepropertes to invest in negtive NPV projects since the investment threshold 0.32 is less thn δi = 0.45 ccording to the NPV rule. The utility loss ccounts for 144% of the option vlue. When both costs nd rewrds re immedite, the investor lso prepropertes. But the utility loss is less thn tht in the cse of immedite rewrds. 3.2 Exit Some reserchers hve found experimentl evidence tht people procrstinte to terminte projects see, for exmple, Stw 1976; Stw nd McClne 1984, nd Sttmn nd Cldwell 1987). While severl explntions re vilble in the literture, the interprettion suggested here is simple. If owners/mngers perceive tht the rewrds of the projects re immedite, but the costs of continution come with dely, then they re tempted by the immedite benefits. To resist this tempttion, they must suffer from self-control costs. These self-control costs lower the benefit from termintion. Thus the owners/mngers prefer to dely termintion. While my model cn explin this procrstintion behvior, it cn lso generte the behvior of prepropertion if the costs of continution come erlier thn the benefits from the projects. I now pply the generl setup lid out in Sect. 2 to the project termintion problem or exit problem. 15 I interpret the process x t ) t 1 s the stochstic profit 15 The problem cn be reinterpreted s one in which n owner/mnger decides when to shut down firm.

17 Option exercise with tempttion 489 flows from project. It incurs fixed cost c f > 0 to continue the project. Normlize the scrpping vlue of the project to zero. A risk-neutrl owner/mnger with self-control preferences decides when nd if to terminte the project. This problem fits into our frmework by setting x) = 0, c s = 0, π x) = x, c c = c f. As in the investment problem described in the preceding subsection, there re mny instnces tht profits nd costs my not come t the sme time. Thus, I consider three cses nd rewrite the Bellmn equtions 4) 6) s follows: 1. Immedite costs W x)=mx 0,δx 1+λ) c f +δ 2. Immedite rewrds W x)=mx 0, 1 + λ) x δc f + δ 3. Immedite costs nd rewrds W x)=mx 0, 1+λ) ) x c f +δ The following proposition chrcterizes the solution. W x ) df x )} λ mx ) 0, c f. 16) W x ) df x )} λ mx x, 0). 17) W x ) df x )} λ mx x c f, 0 }. 18) Proposition 8 Under the conditions given in the ppendix, there is unique threshold vlue, A] x, A]) such tht the owner with self-control preferences stndrd preferences) termintes the project the first time the process x t ) t 1 flls below this vlue. i) If costs re immedite, then stisfies 0 = δ c f + δ δ x x ) df x) λc f, 19) nd x is the solution for λ = 0. Moreover, > x, x < c f /δ nd increses with λ. ii) If rewrds re immedite, then stisfies λ 1 δ) λδ x ) df x) = δc f +δ x x ) df x), 20) nd x is the solution for λ = 0. Moreover, < x <δc f nd decreses with λ.

18 490 J. Mio iii) If both costs nd rewrds re immedite, then stisfies 0 = c f + δ x x ) df x) + λ A ) c f + λδ x mx x c f, 0 }) df x), 21) nd x is the solution for λ = 0. Moreover, c f > x nd increses with λ. One cn interpret equtions 19) 21) s follows. Their left nd right sides represent the utility benefits from termintion nd continution of the project, respectively. At the threshold vlue, the owner is indifferent between termintion nd continution.in the stndrd model with λ = 0, becuse of irreversibility nd uncertinty, there is positive option vlue of witing in the hope of getting better shocks. The owner will not terminte the project s soon s he incurs losses since keeping it live hs n option vlue. The option vlue of witing for ech cse is represented by the third term on the corresponding right side of equtions 19) 21). Only when the loss is lrge enough, will the owner terminte the project. I next turn to the cse where the owner hs self-control preferences. When costs re immedite, the owner is tempted to terminte the project. Exercising self-control is costly. The cost of self-control is represented by the lst term in 19). It lowers the benefit from continution of the project by eroding the option vlue of witing. Thus, he prepropertes to terminte erly. Note tht the termintion threshold is incresing in the self-control prmeter λ. 16 When λ is lrge enough, pproches the upper bound of profits A. In this cse, the owner succumbs to tempttion nd termintes the project immeditely even if the project cn still mke positive net profits. When rewrds re immedite, the owner is tempted to continue the project even though he my suffer from losses. To resist this tempttion, he incurs current period self-control cost represented by the first term on the left side of eqution 20). He lso incurs future self-control cost represented by the second term on the left side of 20). The ltter cost rises when the vlue of future profits is less then. These two components of self-control cost lower the benefit from termintion. Thus, the owner procrstintes to terminte the project. In prticulr, the termintion threshold is lower thn the vlue x when the owner hs stndrd preferences. When λ is lrge enough, pproches zero nd the owner will lwys keep the project live even though he mkes no profits. Consider the cse where both costs nd rewrds re immedite. Since profits re stochstic, the owner is tempted to continue the project if its profits re higher thn the fixed cost nd is tempted to terminte the project if its profits re lower thn the fixed cost. It seems tht there is no unmbiguous conclusion. However, it is importnt to note tht the owner will never terminte the project t the profit level higher thn the fixed cost. Otherwise, t tht profit level the owner hs no tempttion to exit. Thus, sty for one more period incurs no self-control cost nd 16 The men vlue of the exit time is given by F ) 1, which decreses with the threshold vlue.

19 Option exercise with tempttion 491 the owner cn still mke positive profits. Becuse of this fct, t the termintion threshold, the owner cnnot mke positive profits nd hs tempttion to terminte the project. Exercising self-control is costly, which lowers the benefit from continution of the project. The cost of self-control is represented by the lst two terms in eqution 21). Thus, the owner prepropertes to terminte the project t time erlier thn tht in the model with stndrd preferences. Prt iii) of Proposition 8 lso implies tht the termintion threshold increses with λ. In prticulr, when λ is sufficiently lrge, pproches the fixed cost c f so tht the project is terminted ccording to the myopic rule. In this cse, the cost of self-control erodes completely wy the option vlue of witing. I finlly nlyze welfre implictions. Similrly to Proposition 7, the following proposition gives the utility loss due to self-control problems. Proposition 9 Let nd x be given in Proposition 7. When costs re immedite, the utility loss is given by δ δx. When rewrds re immedite, the utility loss is given by λ. When both rewrds nd costs re immedite, the utility loss is given by x. By Proposition 8, when costs re immedite or both rewrds nd costs re immedite, the owner prepropertes he termintes the project when he should continue if he hd stndrd preferences. The utility loss is the forgone profit opportunities. When rewrds re immedite, the owner procrstintes he continues the project when he should terminte it if he hd stndrd preferences. The utility loss is the cost of self-control incurred from resisting the tempttion to sty. As in the investment model, this cost does not increses with λ without bound. When λ is sufficiently lrge, the termintion threshold pproches zero nd the owner never termintes the project. The mximl utility loss from keeping the project live is δ c f Ex ) / 1 δ), which is the bsolute vlue of the NPV of profits nd is positive by the ssumption in the ppendix. The following exmple illustrtes Propositions 8 9 numericlly. Exmple 2 Let = 0, A = 1,δ= 0.9, c f = 0.6, nd F x) = x. Tble 2 reports the solution. It revels the following: When costs re immedite, the owner termi- Tble 2 This tble presents solutions for the exit threshold vlues, the men witing time until exit nd the utility loss. The utility loss is mesured s the frction of the profits t exit λ Threshold Witing time Utility loss Immedite costs Immedite rewrds Immedite costs nd rewrds

20 492 J. Mio ntes the project too erly even if he suffers from very smll self-control problems, i.e., λ = 0.2. The project is terminted t the profit level 0.78, which is bigger thn the fixed cost 0.6. If the owner hd stndrd preferences, he should terminte the project t the vlue 0.59 less thn the fixed cost becuse of the option vlue of witing. The utility loss ccounts for 24% of the profits. When rewrds re immedite, the owner procrstintes. He suffers from lrger loss if he hs lower level of self-control since the profit level t termintion becomes smller. The utility loss is proportionl to λ since it is equl to λ / = λ. When both costs nd rewrds re immedite, the owner prepropertes. However, the utility loss is less thn tht when costs re immedite. 4Conclusion This pper dopts the Gul Pesendorfer self-control utility model to nlyze n option exercise problem under uncertinty over n infinite horizon for n gent who is tempted by immedite grtifiction nd suffers from self-control problems. Unlike the time-inconsistency pproch which depends on the expecttions bout future selves preferences, there is no multiplicity of predictions. When pplied to the investment nd exit problems, the present model hs number of testble implictions. For exmple, the present model implies tht overinvestment, excess entry, procrstintion to terminte project or shut down firm my be the rtionl choices of those investors/mngers/entrepreneurs hving self-control preferences, who re tempted by immedite profit opportunities. On the other hnd, the opposite phenomen cn be cused by such decision mkers who re tempted to void immedite costs. Further, when both costs nd rewrds re immedite, the myopic option exercise rule my be optiml for such decision mkers, who hve sufficiently low levels of self-control. Appendix: Proofs Proof of Proposition 1 I provide the proof for the problem in 4) only. The proof for other cses is similr. Subtrct δπ x) from the two sides in Eq. 4) to obtin W x) δπ x) = mx δ x) δπ x) 1 + λ) c s, 1 + λ) c c + δ W x ) df x )} λ mx c c, c s }. A.1) Note tht the gent s optiml choice is determined by the first mx opertor in A.1). Since the second term in this mx opertor is constnt independent of x nd since x) π x) is strictly incresing in x, A], the following three cses my rise. ) The first term is lrger thn the second term for ll x, A]. In this cse the gent stops immeditely. b) The first term is less thn the second term for ll x, A]. In this cse, the gent never stops. c) There is unique threshold vlue, A] such tht the first term is equl to the second term. In this cse, the gent stops when x. Tht is, he stops the first time the process x) t 1 hits from below.