Center for Economic Research. No INVESTMENT UNDER UNCERTAINTY AND POLICY CHANGE. By Grzegorz Pawlina and Peter M. Kort.

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1 Center for Economic Researc No INVESTMENT UNDER UNCERTAINTY AND POLICY CHANGE By Grzegorz Pawlina and Peter M. Kort January 2001 ISSN

2 Investment under Uncertainty and Policy Cange Grzegorz Pawlina y and Peter M. Kort z January 15, 2001 Abstract Existingrealoptions literature provides relativelylittle insigt into te impact of structural canges of te economic environment on te investment decision of te rm. We propose a metod to model te impact of a policy cange on investment beavior in wic, contrary to te earlier models based on Poisson processes, uncertainty concerning te moment of te cange can be explicitly accounted for. Moreover, probabilities of te cange depend on te state of te dynamic system, wat ensures te consistency ofte action of te policy maker. We model te policy cange as an upward jump in te (net) investment cost, wic is, for instance, caused by a reduction in te investment tax credit. Te rm as an incomplete information concerning te trigger value of te process for wic te jump occurs. We derive te optimal investment rule maximizing te value of te rm. It is sown tat te impact of trigger value uncertainty is non-monotonic: te investment tresold decreases wit te trigger value uncertainty for low levels of uncertainty, wile te reverse is true for ig uncertainty levels. Finally, we present policy implications for te autority tat result from te rm s value-maximizing beavior. Keywords: investment under uncertainty, policy cange JEL classi cation: C61, D81, G31 Tis researc was undertaken wit support from te European Union s Pare ACE Programme Te content of te publication is te sole responsibility of te autors and in no way represents te views of te Commission or its services. Te autors would like to tank Kuno Huisman and Uli Hege for elpful comments and suggestions. All remaining errors are ours. y Corresponding autor, Department of Econometrics and Operations Researc, and CentER, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, te Neterlands, g.pawlina@kub.nl, pone: , fax: z Department of Econometrics and Operations Researc, and CentER, Tilburg University, te Neterlands. 1

3 1 Introduction Corporate investment opportunities may be represented as a set of (real) options to acquire productive assets. In te literature it is widely assumed tat te present values of cas ows generated by tese assets are uncertain and teir evolution can be described by a stocastic process. Consequently, an appropriate identi cation of te optimal exercise strategies for te real options plays a crucial role in capital budgeting and in te maximization of a rm s value (cf. Lander and Pinces [13]). So far, existing real options literature provides relatively little insigt into te impact of structural canges of te economic environment on te investment decision of te rm. We propose a metod to model suc canges. Te modernteory of investmentunder uncertainty (DixitandPindyck [6]) provides tools for evaluating te rm s investment opportunities and determining te value-maximizing investment rules. Witin tis teory, mainly use is made of two stocastic processes. Te rst process is calledbrownianmotion, or Wiener process, wic can be applied to cases were an economic variable is continuous over time. Often, owever, it is more realistic to model an economic variable as a process tat makes infrequent but discrete jumps. Ten use is made of Poisson (jump) processes. Under Poissonprocesses jumps always occur wita constant probability. Altoug itis convenient to model external socks using a Poisson process, tis tecnique as a major drawback. It does not allow for modeling situations, suc as a policy cange, were te sock occurs under speci c circumstances. We propose a new metodology tat allows for modeling a structural cange occurring after passing a certain trigger by an underlying variable. As an example consider a reduction of an investment tax credit wic was previously analyzed by Hassett and Metcalf [11] (see also Dixit and Pindyck [6], C. 9) wit te use of a Poissonprocess. Suc a reduction is typically imposedby te autority wen te economy is booming and an active investment policy is no longer needed or desired. 1 Hence, te moment of te reduction sould depend on te state of te economy so tat its probability sould not be constant like under a Poisson process. 2 Anoter example, were it is realistic to assume tat te occurrence of te sock depends on te state of te economy, is a foreign direct investment (FDI) decision of a rm tat is aiming at te purcase of a local company from te government of a developing country (see Smets [18] andcerianandperotti [3] for a discussion of te e ects of strategic interactions and political risk). Te government observes te performance of te local company and may decide to 1 For example, after a period of fast economic growt, in 1999 Ireland announced an end to its special 10% rate for new foreign manufacturing and nancial investors as one of te means to avoid overeating of te economy. 2 Hassett and Metcalf [11] try to correct tis by letting te arrival rate depend on te output price. But still it is ten possible tat an investment subsidy is reduced for low output prices, wile te subsidy was maintained under ig output prices. Tis kind of inconsistency in te autority s beavior is no longer possible under our approac. 2

4 increase te o ering price for tis company after it obtains iger pro ts. 3 In te paper we introduce a possibility of an upward jump in te (net) investment cost. Tis jump will, for instance, be caused by a reduction of an investment tax credit. Rater tan letting te probability of te jump to be constant, we propose to let te jump occur at te moment tat an underlying variable reaces a certain trigger. Here, te underlying variable is te value of te investment project. Te rm is not aware of te exact value of te trigger but it knows te probability distribution of te trigger instead. In te investment credit example, tis corresponds to te situation were te rm as incomplete information about te autority s tax strategy. Taking into account consistent government beavior, te rm knows tat a jump will not occur as long as te current value of te variable remains below te maximum tat tis variable as attained in te past. Wen te underlying variable reaces a new maximum and te jump does still not occur, te rm updates its conjecture about te value of te barrier. Consequently, our objective is to determine te optimal timing of an irreversible investment wen te investment cost is subject to cange and te rm as incomplete information about te moment of te cange. It is clear tat te value of te project drops to zero at te moment tat te investment cost jumps to in nity. However, we mainly consider scenarios were te cost of investment is still nite after te upward jump occurred. In tis way tis work generalizes Lambrect and Perraudin [12], Scwartz and Moon [17], and Berrada [2], were te value of te project drops to zero at te unknown point of time. Our main results are te following. We derive an equation tat implicitly determines te value of te project at wic te rm is indi erent between investing and refraining from te investment. Tis value is te optimal investment tresold and it is sown tat tis tresold is decreasing in te azard rate of te cost-increase trigger. For te mostfrequently used density functions it olds tat, for a given value of te project, te azard rate rst increases and ten decreases wit trigger value uncertainty. Tis leads to te conclusion tat te investment tresold decreases wit te trigger value uncertainty wen te uncertainty is low, wile it increases wit uncertainty for ig uncertainty levels. Hence, for a policy maker interested in accelerating investment, an optimal level of uncertainty can be identi ed wic is te level corresponding to 3 Te same idea can also be applied witin te topic of tecnology adoption. In Farzin et al. [7] te arrival of a more e cient tecnology satis es a Poisson process (tis assumption is also adopted by Baudry [1] were te new tecnology as te advantage of being less polluting, and by Mauer and Ott [14] were maintenance and operation cost are lowered after te tecnological breaktroug). Tis way of modeling is satisfactory only wen te rm as no insigt at all in te innovation process of new tecnologies. If, instead, te rm could observe progress (but as no perfect information), a way to model it is to introduce a variable tat stands for te state of tecnological progress. Te rm is able to observe perfectly te realizations of tis variable. As soon as te state of tecnological progress its a certain barrier, wic is ex ante unknown to te rm, te new tecnology is invented. Tis approac is similar to te one in Grenadier and Weiss [8] but tere it was assumed tat te value of te barrier is known beforeand. 3

5 te minimal investment tresold. 2 Value Maximizing Investment Rule We apply te value-maximizationcriterion to determine te optimal investment rule of te rm. First, we consider te basic model of investment under uncertainty wit no cange in te investment cost. Subsequently, we develop a model wicallows for an increase inte investment cost after te value of te project reaces a certain trigger. 2.1 Basic Model We start by considering te basic model of investment under uncertainty. Te model was rst developed by McDonald and Siegel [15] and is extensively analyzed in Dixit and Pindyck [6], C. 5. Te general problem is to nd te optimal timing of an irreversible investment, I, given tat te value of investment project, V, follows a geometric Brownian motion: dv t = V t dt + ¾V t dw t : (1) Te parameter denotes te deterministic drift parameter, ¾ is an instantaneous standard deviation and dw t is an increment of a Wiener process. Witin tis setting a rm s investment opportunity is a perpetual American option wit an exercise price equal to I and were V is te value of te underlying asset of V. 4 In our case I denotes a lump sum payment needed to undertake te project. Te rm is assumed to be risk-neutral. 5 After making te payment, te rm owns te project, wic generates a present value of cas ows V. Te rm maximizes te expected present value of cas ow by coosing te optimal V at wic te project is undertaken. A well-described procedure (see Dixit and Pindyck [6]), involving te use of Ito s lemma and solving a di erential equation under te corresponding value-matcing and smoot-pasting conditions, yields te value of te optimal investment tresold, V m : were V m = = ¾ s µ ¾ I; (2) r > 1; (3) ¾2 and r is an instantaneous interest rate. 6 For te value of te investment opportunity, W(V ); it olds tat Z 1 µ V W(V ) = E V t (r )e rt dt Ie rtm = (V m I) ; (4) T m V m 4 To simplify notation, we skip te time subscripts wenever it does not yield ambiguity. 5 Alternatively, we can apply te replicating portfolio argument. 6 Te problem as a nite solution for <r. 4

6 were T m is te rst passage time corresponding to te tresold V m. 7 By following Dixit [4] and substituting variables, we obtain µ 1 V E [T m ] = 1 ln : (5) 2¾2 V m Tere are two factors determining te value of te investment opportunity. Te rst factor, V m I, corresponds to te net payo realized at te time of te optimal exercise. Te second one, often referred to as a probabilityweigted discount factor, ³ V V m, allows for translating te future payo from te investment opportunity into its present value. Te value ofte optimal investment tresoldis positively relatedto te volatility of te project s value (te iger it is, te iger V must be reaced for te project to be undertaken) and negatively related to its growt rate. W(V ) increases in te volatility of te value of te project ( is a decreasing function of ¾ and W is decreasing in ) wat results from te convex payo of te investment opportunity. W is increasing in te growt rate,, since te e ective discount rate of future cas ows decreases linearly in. 2.2 Model wit Switcing Cost We introduce an upward cange in te investment cost tat occurs if te project becomes more valuable. 8 Tis cange may be attributed to an action of te autority suc as reducing te investment tax credit or increasing te o ering price for a privatized enterprise. Moreover, it can be a result of an arrival of a competitive rm o ering a iger bid for a particular project (as soon as its value is su ciently ig). Te investment cost jumps upwards at te moment tat te project value becomes su ciently large. In case of te investment tax credit it is no longer needed to stimulate investment policy, in case of FDI te government simply requires a iger price for an asset tat is wort more, and in case of a competitive rm it olds tat often potential buyers are attracted wen te project value increases. Let us denote by V suc a realization of te process for wic te (net) investment cost canges wit probability 1 from I l to I, were I > I l. At tis stage we assume tat I is deterministic. Later we consider a straigtforward extension to te stocastic case and discuss its implications. We assume tat te rm knows only te distribution, F(V ), of te cost-increase trigger. 9 Te 7 Te expected timing of reacing te tresold can only be calculated under te condition 2 >¾ 2 (cf. Harrison [9]). 8 Te obtained results also old if, instead, a downward jump in V is considered. 9 Tis assumption is consistent wit Lambrect and Perraudin [12] wo analyze a duopoly game under incomplete information. In teir model te investment opportunity deteriorates completely for some realization of te process and te rms know only te density function generating te arrival. In te Lambrect and Perraudin framework te deterioration of te investment opportunity results from a competitive entry. Our model allows for a partial deterioration of te value of te investment opportunity wat is captured by settingi <1. 5

7 key assumption ere is tat te distribution function governing te cange is stationary over time. If by time bt te investment cost as not increased for V V, were V is te igest realization of te process so far, te cost will not increase at any t > bt for wic V (s) V _ for all s t: Hence, te probability of te jump in investment cost is a function of V alone. In order to restrict our analysis to te most relevant case, we impose te following assumptions on te values of te variables used in te model: 8 >< >: V 0 < I l ; (i) I l < V ; (ii) V < 1 I l; (iii) ³ V (V I ) V < V I l ; (iv) were V 0 denotes te initial value of te project and V is te unconditional optimal investment tresold corresponding to te cost I. Te assumption (i) means tat te initial value of te project is su ciently low to exclude an immediate investment. Violating eiter (ii) or (iii) would imply tat te investment would be undertaken eiter at a ig cost or at a low cost, respectively, wit probability one. 10 Finally, (iv) ensures tat ex post it is never optimal to wait wit investing until te upward cange in cost occurs Value of Investment Opportunity Since te cost-increase trigger is not known beforeand, two scenarios are possible. In te rst scenario, te investment occurs before te cange in te price of te asset, and, in te second scenario, te investment takes place after te upward cange. Consequently, te value of te investment opportunity re ects te structure of te expected payo s: Z W s (V; V _ ji = I l ) = p s ( V _ 1 )E V t (r )e rt dt I l e rt s + T s ³ + 1 p s ( V _ ) E Z 1 V t (r )e rt dt I e rt T (6) ; (7) were p s ( _ V ) is a conditional (on te igest realization of V, _ V ) probability tat te investment cost will increase after te investment is made optimally, and T s and T denote rst passage time corresponding to te optimal investment tresold at te low and at te ig cost, respectively. Teir expected values can be calculated in a similar fasion as (5). After rearranging and including tese expected values, we obtain te following maximization problem allowing 10 Tis would degenerate te problem to te basic McDonald and Siegel [15] model. 6

8 for nding te optimal investment tresold: W s (V; _ V ji µ V = I l ) = max(v s I l ) V s µ V +(V I ) V V s à 1 F(V s ) 1 F( V _ ) +! 1 1 F(V s) _ : (8) 1 F( V ) V s is te optimal investment tresold in case te investment takes place before te cange in cost, V _ is te igest realization of te process so far and F(:) denotes te cumulative density function of te cost-increase trigger. Hence, 1 F(V _ s) is te probability tat te jump in te investment cost will not occur 1 F( V ) _ by te moment V is equal to V s, given tat te jump as not occurred for V smaller tan V _. Equation (8) is interpreted as follows: te value of te investment opportunity is equal to te weigted average of te values of two investment opportunities. Tey correspond to te investment cost I l and I, respectively, given tat investment is made optimally (at V s if te price is still equal to I l and at V if te upward cange as already occurred). 11 Te value of te investment opportunity depends on te igest realization of te process, V. A iger V _ (tus a one closer to V s ) implies a lower probability of te trigger falling into te interval ( V _ ; V s ) and, as a consequence, a iger probability of making te investment at te lower cost, I l. In order to calculate te value of te investment opportunity, rst we need to establis te value of V s by solving te maximization problem Optimal Timing of Investment Te optimal investment tresold, V s, is determined by maximizing te value of te investment opportunity or te RHS of te Equation (8). Proposition 1 Under te su cient condition tat 0 (V s )V s + (V s ) 0; (11) 11 It is wort pointing out tat for I!1 te value of te investment opportunity boils down to: _ µ V W s (V; VjI=I 1 )=max(v s I 1 ) V s V s 1 F(V s) _ ; (9) 1 F( V) wat directly corresponds to te result of Lambrect and Perraudin [12]. In te oter limiting case, i.e. fori!i l, te value of investment opportunity converges to _ µ V W s(v; VjI =I l )=(V l I l ) (10) wic is te formula obtained by McDonald and Siegel [15]. V l 7

9 te investment is made optimally at V s wic is a solution to te following equation: (V s )Vs 2 ( 1) 1 Vs +1 + ( 1)V s (V s (V s ) + )I l (V s ) I 1 = 0; (12) were (x) = F 0 (x) 1 F(x) denotes te azard rate.12 Proof. See Appendix. A su cient condition for (11) to old is tat te azard rate as to be non-decreasing. 13 Tis condition (11) is satis ed for most of te common density functions as, e.g., exponential, uniform and Pareto Solution Caracteristics In tis section we analyze te sensitivity of te optimal tresold wit respect to te canges in te parameters caracterizing te dynamics of te projectvalue. Moreover, we determine te direction of te impactofte canges in te investment costs in bot scenarios. Subsequently, we examine ow te uncertainty concerning te moment of imposing te cange in uences te rm s optimal investment rule. 3.1 Canging Parameters of Investment Opportunity We are interested in ow potential canges in te caracteristics of te investment opportunity in uence te optimal investment rule. Terefore we establis te sign of relationsips between te value of V s and te parameters capturing suc features of te project (via ) as its growt rate,, volatility, ¾, and te interest rate, r. Moreover, we analyze te impact of te canges in a current investment cost, I l, as well as of a sift in te expectations about its future value, I. For tis purpose we formulate te following proposition. Proposition 2 Te e ects on te investment tresold level of te canges in 12 In our case, te azard rate as te following interpretation. Te probability of te upward cange in te investment cost during te nearest increment of te value of te project, dv, (given tat te cost-increase as not occurred by now) is equal to te appropriate azard rate multiplied by te size of te value increment, i.e. to (x; )dv. 13 More precisely, te relevant azard rate as to be not too fastly decreasing so tat te component wit a negative derivative is not greater (in absolute terms) tan te value of te function. 14 In fact, te azard rate based on te Pareto function is decreasing at an order of1=x and te property (11) is still met. 8

10 te di erent parameters are as l > 0; < 0; < 0; (15) 8I l ;I satisfying 0 < I l < I ; 8 2 (1;r= ) if > 0 and 8 2 (1;1) if 0: Proof. See Appendix. Consequently, te optimal tresold (ceteris paribus) increases in te initial investmentcostanddecreases inte size ofte potential cost-increase as well as in te parameter. Te latter implies tat te tresold increases wit uncertainty of te value of te project and decreases wit te wedge between interest rate and te project s growt rate. All tese results are intuitively plausible. 3.2 Impact of Policy Cange Te optimal investment rule depends not only on te caracteristics of te projectitself but also onte rm s conjecture about te probability distribution underlying te expected policy cange. Te parameters of tis distribution can be in uencedby actions ofte autority. Forinstance, aninformationcampaign about te expected canges in investment tax credit will lead to a reduction of te variance of te distribution underlying te value triggering te cange. Terefore, it is important to know ow te canges in te uncertainty related to te project value triggering te jump in te investment cost in uence te rm s optimal investment rule. Knowing tat te rms are going to act optimally, te autority can implement a desired policy, wicis, for instance, accelerating te investment expenditure by canging te level of te rms uncertainty about te tax strategy Hazard Rate Te azard rate of te arrival of te cost-increase trigger is one of te basic inputs for calculating te optimal investment tresold. Altoug it is exogenous to te rm, it may well be controlled by anoter party suc as te autority. Here, we determine te impact of its cange on te rm s investment rule. Later, we discuss some of te policy implications of te obtained result. After applying te envelope teorem (see Appendix) to te LHSof (12), we can formulate te following proposition. Proposition 3 Te optimal investment tresold is decreasing in te corresponding azard rate, i.e. te following inequality < 0; (16) 9

11 8 2 (0;1): Tis result implies tat an increasing risk of te switc leads to an earlier optimal exercise. Te intuition is quite simple: an increasing risk of a partial deterioration of te investment opportunity after a small appreciation in te project value decreases te value of waiting. Consequently, (16) implies tat for any parameter of te density function underlying te jump, denoted by a, te following condition olds: = : (17) Using (17) we can establis ow te azard rate is a ected by canges in te parameters of te distribution underlying te occurrence of te jump. It is easy to sow tat, in te relevant interval, te azard rate is monotonic in V wic denotes te mean of te corresponding density function Trigger Value Uncertainty We analyze te impact of te uncertainty related to te value of te costincrease trigger on te optimal investment tresold. For tis purpose, te concept of a mean-preserving spread (see Rotscild and Stiglitz [16]) is applied. Following Proposition 3, we know tat te optimal investment tresold is monotonic in te azard rate corresponding to te trigger. Hence, wat is left is to establis te sign of te relationsip between te azard rate and te uncertainty related to te value of te trigger. If te cost-increase trigger is known wit certainty, te investment is made optimally at an in nitesimal instant before V is reaced. At tis point, te azard rate is zero (tere is no risk tat te cost increases before te optimal tresold is reaced). As te uncertainty marginally increases, te azard rate is a ected by: 1) an increase in te value of te density function, f(v ), underlying te trigger, and 2) a cange in te value of te survival function, 1 F(V ). It is easy to verify tat, for te most frequently used density functions, suc as normal, uniform, exponential and Pareto, te value of te azard rate, for any V 2 [V 0 ;V ), rst increases and ten decreases in te mean-preserving spread. An example for te normal density function is sown on Figure Tis property olds for te most frequently used density functions, suc as normal, uniform, exponential and Pareto. 16 Altoug te concepts of te mean-preserving spread and increased standard deviation are, in general, not equivalent, tey may be treated as suc for te types of density functions referred to in tis paper. 10

12 H azard rat e V = 140 V = 120 V = ω Figure 1 Figure 1. Te relationsip between te azard rate and standard deviation of a normal density function wit a mean equal to 150. Hazard rates are plotted for V = 100, 120 and 140. Moreover, for eac degree of te trigger value uncertainty, tere exists suc a value of V < V, say ev, tat for V 2 [V 0 ; ev ) te azard rate increases, and for V 2 (ev ;V ) decreases, in tis uncertainty. Tis form of te relationsip between te azard rate and te uncertainty implies (via Proposition 3) tat V s decreases in te uncertainty if it falls into te interval [V 0 ; ev ) and increases oterwise, as depicted in Figure 2. V s decreases V s increases in uncertainty ev in uncertainty Figure 2 Figure 2. Te relationsip between trigger value uncertainty and te optimal investment tresold. Consequently, in orderto determine te signofte e ectofuncertainty on V s, we need to establis te relative position of V s wit respect to e V. Let us denote te standard deviation of a density function underlying te cost-increase trigger as!. Since te expression for V s is already known (see (12)), all we ave to calculate is e V as a function of!, suc tat, for eac pair (V;!), te following condition olds: = 0: 17 Altoug ev(!) cannot be written explicitly in a general form, its values corresponding to a given density function may be easily found numerically. 11

13 For most frequently used density functions it can be sown tat ev decreases in uncertainty. For a relatively low degree of uncertainty, it olds tat V s < ev. Since for V < ev te azard rate increases in!, V s is moving to te left wen te uncertainty rises. After te uncertainty reaces a critical level, say! e, at wic V s = ev ; te azard rate at V s is decreasing in! and te optimal tresold begins to increase. Tis implies tat optimal investment tresold attains te minimum for! =! e. Now, we are able to formulate te following proposition. Proposition 4 For density functions suc tat lim f(v; ) = 0; 8V and!!1 f(v; ) is unimodal, (19) tere exists a non-monotonic relationsip between te optimal investment tresold and te trigger value uncertainty. At a low degree of uncertainty, te marginal increase in uncertainty leads to an earlier optimal investment. Te reverse is true for a ig degree of uncertainty. Tere exists a unique point! e, suc tat V s (! e ) = ev (! e ); wic separates te areas of low and ig uncertainty levels. Figures 3 and 4 sow te relationsip between te uncertainty,!, and te optimal investment tresold. V s, V V s HI =120;ωL V s HI =150;ωL V s HI =200;ωL V HωL ω Figure 3 Figure 3. Te relationsip between te uncertainty,!, and te optimal investment tresold, V s, for di erent sizes of te ig investment cost (I = 120;150 and 200). Te values are calculated for a normal density function wit mean 150. An intersection of V s and e V corresponds to te minimal investment tresold, V s (! e ). Te parameters of te underlying process are: = 0; r = 0:025 and ¾ = 0:1: Tis set of parameters is used in Dixit [5], and Lambrect and Perraudin [12] (we rescale te investment cost wit te factor100). 12

14 In Figure 3 it can be seen tat te optimal investment tresold is rst decreasing and ten increasing in te uncertainty concerning te value of te trigger. Te minimum is always reaced wen V s (!) intersects ev (!). Te azard rate increases in! in te area located to te sout-westfrom ev (!) and decreases in te nort-eastern region. Te opposite olds for V s. Moreover, te optimal tresold is iger if te expected cange in te investment cost is smaller (cf. Proposition 2) ω = 15 ω = ω = 25 ω bac V Figure 4 Figure 4. Te relationsip between V and te derivative of te azard rate wit respect to te trigger value uncertainty. Te optimal investment tresolds for I = 150 and di erent uncertainty levels are sown on te orizontal axis (Point a corresponds to V s (15), b to V s (! e = 19:26) and c to V s (25): Te values are calculated for a normal density function wit mean 150. Te parameters of te underlying process are: = 0; r = 0:025 and ¾ = 0:1: In Figure 4 it can be noticed tat te point, ev, at wic te derivative of te azard rate is equal to zero moves to te left wen te trigger uncertainty increases. As long as V s < ev, te optimal tresold also moves to te left (cf. te location of V s (15)). Wen te standard deviation is equal to 19:26, V s equals ev. After a furter increase in te uncertainty, ev continues moving to te left and V s starts moving to te rigt (cf. V s (25)). For a su ciently ig degree of uncertainty, V s exceeds V and for te uncertainty tending to in nity, V s converges to te unconditional tresold V l : 19 Tis fact as implications for te optimal investment tax credit policy, discussed as an example inte subsequent section. 19 Te necessary and su cient condition for lim Vs=Vs is lim (Vs; )=0:!!1!!1 13

15 4 Implications for te Investment Credit Tax Policy Inn our setting, o a policy implemented by te autority may be expressed as a triple I I l ;V ;!. In te example we assume tat te ratio I I l is predetermined by te current amount of te tax credit (and is a priori a common knowledge). Te variables V and! are te autority s decision variables. As we already know, a decrease in V results in a lower optimal tresold. Consequently, in case of a single rm a reduction in te trigger value is going to accelerate investment. However, incase of multiple eterogenous rms, lowering te trigger as two opposite e ects. First, as in te single- rm case, it triggers an early investment for tose rms for wic Assumption iv (6) is still satis ed. On te oter and, it results in te oter rms waiting longer and investing at a ig cost (if Assumption iv (6)) no longer olds). Terefore, if te rms are su ciently eterogenous, reducing V does not yield a desired e ect. Terefore, te autority may prefer to resort to anoter instrument, suc as!. An appropriately designed treat of abandoning te investment tax credit cantrigger an early investment (see Dixit and Pindyck [6], C. 9). Since te rm s optimal investment tresold reaces a minimum for a certain degree of uncertainty,! e, te objective of te autority interested in accelerating te investment sould be to set te standard deviation of te density function underlying te cost-increase trigger equal to! e. Suc a policy allows for Assumption iv (6) to be satis ed for a larger fraction of rms tat in case of reducing V. Altoug a relatively small deviation from te optimal policy would result in a small delay in te aggregate investment, a su ciently large deviation would ave muc more severe adverse e ects. Tere exists a critical level of!, say b!, above wic te optimal tresold is greater tan V. In suc a situation, te cange in te cost is occurs before te investment is made and te project of a bencmark rm is undertaken at V. Terefore, increasing! beyond b! leads to a discontinuous cange in te investment tresold wat results in a considerable delay of te investment. 20 According to Proposition 1, b! satis es te following equation 21 0 = (V ; b!; )(V ) 2 + ( 1)V (V (V ; b!; ) + )I l (20) (V ( 1) 1 ; b!; ) (V ) +1 I 1 If te uncertainty related to te timing of imposing te trigger for a given rm is iger tan b!, ten te optimal tresold, V s, is larger tan V : 20 Te expected delay, T,can be calculated from a direct application of te rst passage time. In tis case T = 1 1 ln V 2 ¾2 Vs : 21 Equation (20) is also satis ed for!=0, since te optimal tresold in te deterministic case is equal tov. : 14

16 Consequently, te impact of uncertainty associated wit te timing of te cange may be presented in te following way: 8 < [0; b!)n! e : feasible policy,! 2! e : most e ective policy, : [b!;1) : policy resulting in te investment delay. Te treat of te increased investment cost is used as an investment stimulus most e ectively wen tere exists a positive degree of informational noise concerning te timing of imposing te measure. Te level of noise corresponds to! e. Perfect information allows investors to wait until V is reaced. Excessive noise (above b!) results in te treat of imposing te trigger being lowered too muc to trigger an early investment. As an e ect, te cange in te cost occurs before te investment is made. Te optimal degree of uncertainty results in te optimal investment tresold being lower tan V and, at te same time, does not excessively dilute te treat. In suc a case te preemption is most signi cant. 5 Extension: Stocastic Jump Size Now we introduce a stocastic size of te upward cange in te investment cost. Similarly to te previous case, te value of te investment opportunity, W s, re ects te structure of te expected payo s maximized wit respect to te optimal investment tresold, V s. Allowing for a stocastic I, distributed according to te cumulative density function G(I ), te value of investment opportunity becomes (cf. (8)): W s (V; _ V ji µ V = I l ) = max(v s I l ) V s Z I µ V + (V I ) I V 1 F(V s ) 1 F( V _ ) + V s à 1 1 F(V s) 1 F( _ V )! dg(i ): (21) I and I denote te lower and te iger bound of I, respectively. Equation (21)is interpretedanalogously to (8), andte second component is te expected value of te option to wait after te upward switc. We prove in te Appendix tat te following proposition olds. Proposition 5 In case te size of te jump is stocastic, te optimal investment rule can be determined by replacing I by v I = u 1 t Z I dg(i ) (22) I I 1 in te expression for te optimal tresold (12). 15

17 Formula (22) can be interpreted as a certainty equivalent of te ig investment cost. 22. Tis allows for a relatively simple analysis of te impact on te optimal investment timing of te uncertainty of te size of te jump. Te impact of te uncertainty concerning te size of te jump can be analyzed by directly comparing (12) and (14). By Jensen s inequality it olds tat Z I I I 1 dg(i ) > Ã Z I I I dg(i )! 1 ; (23) since te function f(x) = x a ;a < 0; is convex for all x > 0. Te RHS of Equation (23) is aninversely monotonic transformationof te expected value of I. Since, < 0; te tresold increases in I 1 : Consequently, te tresold is iger for LHS tan for RHS. In oter words, te uncertainty in te size of te jump of te investment cost leads to te iger optimal investment tresold. Tis result may be explained in te following way. Te optimal timing is a convex function of te new investment cost, I. Terefore, te gains from below average realizations of te jump are assigned a larger weigt by te rm tan te symmetric losses from above-te-average realizations. Consequently, te rm is going to wait longer if te realizations are random tan in te case wen all of tem are equal to te average. Compared to te basic model were investment cost is constant, te treat of an upward cange in te investment cost reduces te optimal investment tresold. Now, we can see tat te uncertainty in te size of te jump mitigates tis reduction of te tresold value. Again, it olds tat te increased uncertainty raises te option value of waiting. Apart from te overall di erence between te uncertain and deterministic outcome, we are interested in a marginal impact of uncertainty on te optimal investment strategy. In oter words, we aim at establising ow te investment tresold beaves for te di erent degrees of uncertainty concerning te size of te jump. Terefore, we compare te investment triggers corresponding to a relatively small and a ig degree of uncertainty. For tis purpose, we use te concept ofmean preserving spread (Rotscild andstiglitz [16]). In tis setting, te e ect of increasing uncertainty is examined by replacing te original random variable I ( low uncertainty case) by a new random variable I +» ( ig uncertainty case), were E[»] = 0 and ¾» 2 (0;1): By applying Jensen s inequality it can be proven tat te expected value of a convex function (in our case f(i ) = I 1 ) increases as its argument undergoes a mean preserving spread (cf. Hartman [10]). Consequently, an increase in te uncertainty leads to te iger expected value of I 1 wat corresponds to te lower I, and a iger (or less distant from te basic case) investment tresold. 22 Using te term certainty equivalent is a simpli cation since te rm is assumed to be riskneutral. In our sense, (22) corresponds to suc a value of a certain investment cost (witin te ig regime) tat yields an identical optimal investment rule as wen uncertain costs are distributed according to G(I H ): 16

18 Te impact on te optimal investment rule of uncertainty related to te magnitude of te cange in te cost is monotonic. Terefore, in te investment credit example, increasing tis type of uncertainty as te same e ect on te investmentas te reductionof te magnitude ofte cange. Furtermore, (13) implies tat a lower potential increase in te investment cost is associated wita iger optimal investment tresold. Terefore, a iger degree of uncertainty associated wit te magnitude of te potential cost-increase results in a later investment. An e ective policy triggering an early investment sould, terefore, be associated wit minimizing te investors uncertainty about te size of te expected cange. 6 Conclusions In te paper we consider an investment opportunity of a rm. Te investment cost is irreversible and subject to an increase resulting from a policy cange. Te value of te cost-increase trigger is unknown to te rm and te rm knows te underlying density function instead. Tis corresponds to te situation were te rm as some information concerning te autority s future policy and tis information is incomplete. Moreover, it is taken into account tat policy canges are more likely to occur under certain economic conditions. Recent tax debates across Europe introduce a signi cant source of uncertainty for potential investors. Altoug some of te canges result from te need to unify te EU tax systems, in many cases te policy cange can be attributed to te pace of economic growt. Te booming Iris economy will face an increase in corporate tax from a special 10% rate for new foreign manufacturing and nancial investors to 12.5%. 23 Oter proposals include abandoning corporate tax exemptions in Germany and witdrawing approximately seventy tax reliefs used by EU governments use to draw investment. 24 We sow tat te treat of a policy cange resulting in a iger (net) investment cost leads to a reduction in te option value to wait. Consequently, te rm invests earlier tan in te case of te constant investment cost. Te optimal investment tresold decreases in te magnitude of te cange in investment cost and increases in te market volatility (te latter result also old for te Dixit and Pindyck [6] framework). Te impact of te trigger value uncertainty on te optimal investment tresold is non-monotonic. If te uncertainty is su ciently low, ten te investment tresold is negatively related to te trigger value uncertainty. However, a rise in te uncertainty beyond a certain critical point reverses tis relationsip and leads to an increase of te optimal investment tresold. We use our ndings to determine te optimal design of a policy cange 23 Ireland: Burning Too Brigt; Can Ireland control its rapid growt?, Businessweek, 10 Apr., Te tax reliefs subject to cange range from Belgian exemptions on multinational eadquarters to incentives made by Spain for investors in te Basque region. For te details, see Hey, Let s All Get Togeter and Raise Taxes!, Businessweek, 25 Nov.,

19 tat accelerates investment expenditures. Tere exists a certain (strictly positive) level of te uncertainty concerning te policy cange tat triggers te earliest investment. Hence, a policy maker interestedinaccelerating investment sould aim at acieving te level of uncertainty, corresponding to tis minimal investment tresold. Finally, we extend te analysis by considering a case wen te size of te cange is stocastic. Te uncertainty in te magnitude of te cange appears to mitigate te degree of preemption so it leads to te outcome wic is closer to te unconditional optimal level. 7 Appendix Proof of Proposition 1. Te implicit solution for te optimal investment tresold is found by calculating te rst order condition of (8). Consequently, by di erentiating (8) wit respect to V s ; and equalizing to zero, we obtain: 0 = V Vs +1 (V s V s + I l ) 1 F(V µ Ã! s) V 1 F( V _ ) + (V f (V s ) s I l ) V s 1 F( V _ ) µ V f (V s ) (V I ) V 1 F( V _ ) ; (24) were f Furter simpli cation yields: 0 = 1 V +1 s µ 1 (V s V s + I l )(1 F(V s )) (V s I l ) µ 1 +(V I ) V f (V s ); V s f (V s ) tus ( 1) V s I l + (V s )(V s I l )V s (V I )V +1 s µ 1 V (V s ) = 0: Since V = 1 I (after te jump te McDonald-Siegel problem is left), tis is equal to (V s )Vs 2 + ( 1)V s ((V s )V s + )I l Vs +1 I µ 1 (V s ) 1 I = 0; wat in a straigtforward way leads to (12). In order to prove tat (12) is te expression for te maximal value of te project, we calculate te second order condition, wic is equal to te following 18

20 derivative: (V s )V 2 ( 1) 1 s ( 1)V s + (V s (V s ) + )I l (V s s! Vs +1 I 1 : After di erentiating, we obtainexpression for te secondorder condition of (8): 2 W 2 = ( 0 (V s )V s + (V s )) V s I l V 1 s µ 1 V s I! 1 µ 1 V s (V s )V s Ã1 ( 1): (25) I Te sign of te second component can be proven to be negative by observing tat: 1 1 µ 1 V s µ 1 V 1 > 1 = 0: (26) I I Te sign of te rst component can be determined by notifying tat te lower bound of V s ; denoted by V s, is a solution to te following equation: V s I l = (V I ) µ Vs V : (27) For V s = V s te second factor in te rst component of (25) is equal to zero and for V s > V s it is positive. Terefore te wole expression is surely negative if (11) olds. Proof of Proposition 2. Let us de ne te LHS of (12) as a function: H(V s ;I l ;I ; ) (28) = (V s )Vs 2 ( 1) 1 + ( 1)V s (V s (V s ) + )I l (V s ) Di erentiating (28) wit respect to I l ; I and, respectively, = (V s (V s ) + ) < @ Vs +1 I 1 : ( 1) 1 Vs +1 = ( 1)(V s ) > 0; (29) I H µ 1 ( 1) 1 Vs +1 = V s I l (V s ) I 1 ln V s > 0; I 19

21 8I l ;I satisfying 0 < I l < I ; 8 2 (1;r= ) if > 0 and 8 2 (1;1) if 0: Furtermore, di erentiating (28) wit respect to V s gives: Ã = ( 0 Vs (V s )V s + (V s )) V s I l (V I ) + s V! 1 µ 1 V s +(V s )V s Ã1 + ( 1): I From te proof of Proposition 1 it is known tat under s is positive. Finally, by (30), we know tat 8z ; (31) wat completes te proof. Proof of Proposition 3. By di erentiating (28) wit respect to te azard rate, wile taking into account tat V = 1 I, we obtain: = V Vs s V s I l (V I ) > 0: (32) Te inequality olds since te bot factors are positive (cf. (27) and te proof of s is also positive, from te envelope teorem we directly obtain te sign of (16). Proof of Proposition 5. Equation (22) requires te optimal investment tresold wit a deterministic size of te jump be equal to te tresold wit a jump wit a stocastic size distributed according to G(I ): Since te maximization problem wit a stocastic size of te jump can be expressed as follows: µ W s (V; V _ V 1 F(V s ) ji = I l ) = max(v s I l ) V s V s 1 F( V _ ) + (33) Ã! F(V s) ( 1) 1 V 1 F( V _ R ; ) I I 1 I dg(i ) te expression for te optimal investment tresold is a sligt modi cation of (12): 0 = (V s )Vs 2 + ( 1)V s (V s (V s ) + )I l (34) Vs +1 ( 1) 1 (V s ) Z I V I I 1 dg(i ): Comparing (34) wit (12) allows for observing tat te tresold values are equal if: ( 1) 1 V +1 Z I s ( 1) 1 Vs +1 (V s ) I 1 = (V s ) I I 1 dg(i ): (35) 20

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