Valuation of Irreversible Investments and Agency Problems Jril Mland February 24, 1999 Preliminary draft Abstract This article examines dynamic invest

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1 Valuation of Irreverible Invetment and Agency Problem Jril Mland February Preliminary draft Abtract Thi article examine dynamic invetment deciion when there i an agency problem. A principal delegate the deciion of an invetment trategy of a project to an agent. The agent ha private information about the invetment cot wherea the principal only know the probability ditribution of the cot. The principal' problem i how to compenate the agent inorder to optimize the value of the principal' invetment opportunity. Owing to aymmetric information about the invetment cot it may be optimal for the principal to leave the agent ome \information rent". Optimal compenation function dependent on the obervable outcome from the invetment are found. 1 Introduction In the literature on real option the option value reulting from the interaction of uncertainty exibility and (partly) irreveribility i recognized. The uncertainty taken into account i mainly \ymmetric" uncertainty i.e. the uncertainty in future income i common knowledge. However in many ituation there i alo aymmetric information. An example of uch a ituation i when a manager (an agent) of an invetment project ha better information than the owner of the invetment poibility (the principal) about the invetment cot and the manager alo ha diverging interet from thoe of the owner. The ituation i known from the principal-agent and the regulatory literature. I ketch amodel where a principal delegate the invetment trategy of a project to an agent. The agent ha private information about the exact invetment cot wherea the principal only know the probability ditribution of the cot. One reaon for an owner of an invetment poibility to delegate the management of a project to an agent may be that the management require expertie that the principal doe not poe or that i too cotly for him to obtain. In other cae it may be impoible for the principal to make the deciion himelf but it may be poible for him to commit to a delegation contract. The information aymmetry create a ituation where advere election may occur. The agent i compenated according to a contract. The principal oberve the outcome from the invetment project and the contracted compenation i a 1

2 function of thi variable. The project may generate prot above capital cot. Both the principal and the agent aim to maximize the value of the project. Owing to aymmetric information about the invetment cot it may be optimal for the principal to leave the agent ome \information rent". The model applie to ituation where the production from the project i old in perfect market wherea there are imperfection due to the cot of project. An application of the model conidered in thi article i the cae where a government own ome natural reource. Production of natural reource involve large and (partly) irreverible invetment and uncertainty due to future output price. A feature of production of natural reource i that uncertainty in output price uually i common knowledge wherea invetment and production cot may be private information for thoe inveting in and operating uch project. To exploit the reource the government delegate the production of the reource to companie. The companie may have incentive to ignal higher cot than the true cot in order to obtain a larger prot within the companie. The model preented in thi paper give the government a method of how to nd the mot ecient contract between the government and the companie to which itgive the right toinvet in production of natural reource. The contract can be in the form where the companie are paid a compenation for the management of the reource or it can be in the form of a taxation ytem. Shareholder veru corporate management i another example where the model may apply. The problem i then how to compenate the management given their private information of the cot of the invetment trategy. A in the example above the management maywant to ignal higher cot than the true one. An alternativeinterpretation i that the companie may have incentive to maximize lack in the organization thereby increaing the realized invetment cot compared to the neceary cot. A tandard approach for olving advere election problem i to ue the revelation principle ee e.g. Baron and Myeron [] and Laont and Tirole [9]. Under a revelation mechanim the agent report hi private information to the principal and the deciion in quetion i then made according to a deciion rule to which the principal ha committed himelf. Looely peaking the revelation principle make ue of the fact that for every contract between the principal and the agent that lead the agent to lie there i another contract with the ame outcome but with no incentive for lying. Thi reduce the principal' optimization problem to optimizing over the et of truthful mechanim. In the model the invetment deciion i delegated to the agent. Conequently the revelation principle doe not apply directly here: there i no deciion to be made by the principal and therefore the agent doe not have to report hi private information. However Melumad and Reicheltein [10] have found that under certain condition the performance of an optimal revelation mechanim can be replicated by a delegation cheme which doe not involve communication. We know that in ituation where the revelation principle i valid (i.e. where we have cotle communication and unlimited information) the performance under delegated mechanim will never dominate a centralized mechanim. But a Melumad and Reicheltein [10] and [11] point out the elf-election contraint may be o retrictive that no admiible contract dominate the optimal

3 no-communication contract even if communication i cotle. They how that when both the principal and the agent are rik-neutral and the agent either ha perfect information or the uncertainty can be panned communication ha no value. If we in addition aume that the delegated deciion i obervable delegated cheme perform equally well a communication-baed centralization cheme. Thi ituation i valid in the preented model. Melumad and Reicheltein [10] alo point out that delegation cheme may even gain a comparative advantage to centralized cheme in ituation where communication i cotly. Bjerkund and Stenland [3] have formulated an advere election model omewhat imilar to the model decribed in thi paper where an owner of ome reource may exploit the reource in two way: (i) Sell the reource in a competitive pot market at a contant price or (ii) hip the reource to an agent for proceing and ell the proceed reource in a competitive market where the price of the proceed reource i tochatic. Bjerkund and Stenland aume that the proceing may be witched on and o at no cot (i.e. they formulate a \witching option" imilar to Brennan and Schwartz [5]). In alternative(ii) the owner of the reource (\the regulator") mut compenate the agent for the cot of proceing the reource. The cot of proceing i perfect private information to the agent wherea the regulator know the probability ditribution of the cot. The tochatic income proce ued in Bjerkund and Stenland [3] i more general than the diuion proce preented in the model in thi article. The interaction between option and diverging incentive between a principal and an agent i alo analyzed in Antle Bogetoft and Stark [1]. They how how timing and incentive eect interact to aect invetment trategie in a twoperiod model. At each of the two point in time where invetment i poible the manager (the agent) know the invetment cot wherea the owner (the principal) doe not. Before the time of an invetment poibility neither the owner nor the manager know the invetment cot. However they both agree on the ditribution of future cot. Antle et al. nd that incentive eect a timing eect lower the target cot. Incentive problem alo have the eect of puhing invetment toward period of lower uncertainty i.e. the target cot at time zero (today) may be increaed by incentive eect o much that the overall probability of invetment can increae with incentive problem. The article i organized a follow: In ection the problem i formulated and model aumption are given. In ection 3 future cah ow in the model are evaluated uing the market-baed valuation approach (auming dynamically complete market) and the claical theory of diuion. Section 4 preent the principal' and the agent' optimization problem and the revelation principle. The optimal invetment trategie are given in ection 5 and 6 for the cae where the information about the invetment cot i ymmetric and aymmetric repectively. In ection 7 the optimal compenation function i found. The reult are illutrated in ection 8 uing the uniform ditribution for the invetment cot and the geometric Brownian motion for the income proce. Section 9 conclude the article. Model aumption An invetor (a principal) ha an opportunity to invet in a project. The invetment deciion of the project i undertaken by an agent and the principal 3

4 compenate the agent baed on the output from the project. The output i obervable by both partie wherea the agent ha private information about the invetment cot. In order to keep a larger part of the prot from the project the agent ha incentive to bae hi invetment trategy on ignaling a higher invetment cot than the true cot. Thu the problem for the principal i how to compenate the agent to maximize the value of the principal' invetment opportunity. The project may generate monopoly rent. The principal aim to obtain the prot from the project and only compenate the agent with the neceary invetment cot. However becaue of the agent' private information about the cot it may be optimal for the principal to leave the agent ome \information rent". The agent ha perfect knowledge of the true invetment cot of the project wherea the principal know only the probability denity f( ~ ) of an aeed tochatic cot ~. The cumulative ditribution i denoted by F ( ~ ) and upper and lower level of the invetment cot are and repectively. It i aumed that the option to invet i perpetual and that the value of the income tream from the project follow a tochatic proce where the uncertainty i common knowledge. The value of the income tream at time t i denoted S t. The tochatic proce i dened by a complete ltered probability pace (; ff t g t0 ; F;P) where the ltration atie the uual condition (ee e.g. Borodin and Salminen [4] ch. I.3). Under the equivalent martingale meaure Q (ee e.g. Due [7] ch. 6.H) the tochatic income proce i given by ds t =(rs t (S t ))dt + (S t )dw t ; S 0 ; (1) where r i a contant rik free rate (S t ) reduce the drift in the tochatic proce becaue of the convenience yield and w t i a tandard Brownian motion with repect to the equivalent martingale meaure. It i aumed that (S t ) and (S t ) are continuou and that 0 i an unattainable lower boundary for S t. The expectation operator E t [] denote the expectation conditioned on the time t information with repect to the equivalent martingale meaure Q. The tranfer function from the principal to the agentmut be baed on ome obervable variable. In the model it i aumed that the value of the income i obervable. Alo recall that the information with repect to the proce S t i aumed to be ymmetric. To avoid the agent from behaving opportunitically the value of the compenation mut not be paid before the time of invetment. The principal' time zero value of the project i n h io W (; G(S K )) = up G(S K ) E ~ E 0 e rk (S K G(S K )) + = up G(S K ) R E 0 h e rk (S K G(S K )) +i f( ~ )d ~ : The expectation with repect to the cot level i denoted E ~. It i aumed that the uncertainty in the invetment cot i the ame under the P and the Q meaure. K i a topping time with repect to the ltration F t. The topping time i a function of K where K i the "cot" upon which the agent bae hi invetment trategy. The ignaled cot K K( ~ ) i higher than or equal to ince the agent prot on ignaling a higher cot than the true one. The exercie value of income i denoted S K and G(S K ) i the agent' compenation tranferred at the invetment time. 4

5 The agent' value function may be formulated a V (; K; ) = up E 0 he rk (G(S K ) ) +i : K The pricing of the principal' and the agent' value function are baed on the market valuation approach i.e. it i aumed that the partie are well diveried in the capital market. 3 Valuation of future cah ow Becaue of the trong Markov property and the time homogeneity of the problem we know that the optimal topping time K will be of the form K = infft 0jS t ^S(K)g: The \trigger value of income" ^S(K) i independent of time. Thu we can rewrite the principal' and the agent' value function a repectively and W (; G(S K )) = Z up G(S K ) E 0 [e rk ] ^S(K) G(S K ) + f( ~ )d ~ ; V (; K; ) = up S K E 0 [e rk ](G(S K ) ) + ; where the expected value of the dicount factor i written independently of the value of the income tream and the compenation function. Thi independence implie the problem of nding the optimal invetment trategy ince we will be able to optimize with repect to a \determinitic" trigger level ^S(K) intead of the tochatic trigger S K. Uing reult from the claical theory of diuion the expected value of the dicount factor can be formulated a a function of the trigger level ^S(K) and the time 0 value of the income (Borodin and Salminen [4] ch. II.10 and Ito and McKean [8] ect. 4.6) E 0 [e rk ]= ( if < ^S(K) ( ^S(K)) () 1 if ^S(K): Dening u = E 0 [e rk ] the function () i the trictly poitive and increaing unique olution to the ordinary dierential equation 1 () u +(r )u ru =0; (3) with boundary lim " ^S(K) u =1. Thu the principal' and the agent' value function can be reformulated to (where K K( ~ )) W (; G( ^S(K))) ( R = up ( ^S(K) G( ^S(K)) f( )d ~ ~ if < ^S(K) (4) ^S(K)) G() G if ^S(K); 5

6 and V (; K; ) = up ^S(K) G( ^S(K)) ( if < ^S(K) ^S(K)) (5) G if ^S(K); ( repectively. Note that the value function now are function of the "determinitic" trigger level ^S() and the time zero value of the income proce only. 4 The optimization problem The principal' problem i to optimize the value function given by equation (4). Retriction on the optimization problem are the agent' participation contraint V (; K; ) 0 8K; [; ]; (6) and the agent' incentive compatibility =0 8K; [; ]: (7) The participation contraint enure that the agent doe not reject the contract by letting the agent earn at leat hi reervation utility. Here it i aumed that the reervation utility equal 0 and therefore the participation contract mut be non-negative for all. Thu the binding contraint i the cae where the agent ha the highet cot level. The contract between the principal and the agent i incentive compatible when the agent ha no incentive to ignal a higher cot than hi true cot. The contraint 7 enure that thi condition i atied. The incentive compatibility contraint correpond to the agent' optimization problem which i evaluated in the ubection below. 4.1 The agent' optimization problem The agent optimize hi value of the invetment opportunity given by equation (5) with repect to invetment trategy ^S(K) and the ignaled cot K(). The rt-order condition with repect to the invetment trategy ^S(K) = G ^S( ^S(K)) ^S ( ^S(K)) ( ^S(K)) 8K; [; ]; G( ^S(K)) =0 where G ^S( ^S(K)) and ^S( ^S(K)) denote the rt-order partial derivative of G and repectively with repect to ^S(K). The agent' rt-order condition with repect to the ignaled cot K n (;K;) = ^S0 (K) ^S( ^S(K)) ^S ( G( ^S(K)) ^S(K)) ( =0 ^S(K)) (9) 8K; [; ]; (8) 6

7 i equivalent to the rt-order condition with repect to the invetment trategy ^S(K) in equation 8. The econd-order condition mut be V (; K; = ( ^S(K)) ^S ( ^S(K)) ( ^S) ^S ^S ( ^S(K)) ^S ( ^S(K)) ( ( ^S) ^S0 (K)) (G( ^S(K) ) ^S00 (K)(G( ^S(K)) )+(^S(K)) G ( ^S ^S(K)) ( ( ^S(K)) ^S0 (K)) (G( ^S(K) )+G^S ^S( ^S(K))( ^S0 (K)) o +G ^S( ^S(K)) ^S00 (K) 0 8K; [; ]: To implify the incentive compatibility contraint the revelation principle i ued. The application of the revelation principle i hown in the next ubection. 4. The revelation principle The incentive compatibility condition in equation (9) require that the agent' rt-order condition i atied for all K [; ]. The revelation principle i ued to reduce the principal' optimization problem to the et of truthful mechanim. Therefore implementation of the revelation principle require that the agent' rt-order condition in equation (7) i atied at K() =. Uing the envelope theorem the rt-order condition for optimization 1 i dv (;K();) = = 8K(); [; ]: ( ^S()) (10) K()= Incentive compatibility implie V (; ; ) =V (; K; ). In order to implify the notation I dene V (; ) V (; ; ). The econd-order condition for K mut be atied at K() = i.e. the function V (; ) mut be more convex than V (K(); ) (; K(); (; ) : Dierentiating the rt-order condition in equation (9) when K() = with repect to V (; = ( ^S()) S( ^S()) ( ^S) ^S ^S ( ^S()) ^S ( ^S()) ( ( ^S) ^S0 ()) (G( ^S() ) ^S00 ()(G( ^S()) )+(^S()) G ( ^S ^S()) ^S0 () ( ( ^S()) ^S0 ()) (G( ^S() )+G^S ^S( ^S())( ^S0 ()) o +G ^S( ^S()) ^S00 () 0: 1 dv (;K();) (;K();) d = d The agent optimize K() given hi cot level. The rt term on the right-hand ide i zero when K() i optimal. (1) 7

8 Thi lead to the econd-order condition (uing the retriction in (; K(); K()= = ( ^S()) ^S( ^S()) ( ^S()) ^S0 () 0: Integrating the condition in (10) give an equivalent condition on the reward function (when < ^S()): V (; ) = Z ( ^S(u)) du + V (; ): (13) Equation 13 give the agent' value of accepting the contract. The rt term on the right-hand ide of equation (13) i the agent' value of private information. The lower the agent' true cot level the higher the agent' value of information. The lat term on the right-hand ide V (; ) i the value of the reervation utility. From the participation contraint (6) we know that the agent at leat mut earn hi reervation utility in order to accept the contract. Alo in the cae where the agent' true cot i at the highet poible cot level the agent mut earn hi reervation utility. In thi model the reervation utility i aumed to be zero i.e. V (; ) = 0. Thu equation (13) repreent the agent' value of accepting the contract that the principal oer. 5 Benchmark: Symmetric information A a benchmark we rt tudy the cae where the information about the invetment cot i ymmetric. When the agent ha no private information there i no need for the principal to compenate the agent with more than hi true cot. Thu the agent i compenated for hi capital cot only i.e. 0 if < ^S() G = (14) if ^S(): Inerting G( ^S()) = into the agent' value function in equation (5) we nd V ym (; ) = 0 where the ubcript ym indicate that thi i the value under ymmetric information. The agent ha no private information and therefore the term R =( ^S(u))du of equation (13) i zero. Determinitic and ubtitution of G( ^S()) with into the principal' value function in equation (4) lead to ^S() W ym (; ) = up ^S() ( if < ^S() ( ^S()) (15) if ^S(): Equation (15) how that when we have no aymmetric information we have an optimization problem imilar to the \tandard" real option problem of exerciing an innite (American) option with exercie price and ^S() athe critical level of exerciing the option. 8

9 The optimal trigger value of income i given by the rt-order ym (; ^S() =1 ^S( ^S()) ( ^S()) ^S() =0: (16) For the trigger value in equation (16) to be optimal the econd-order condition ha to be W ym (; ^S() = ( ^S()) ^S ^S( ^S()) ( ^S()) ^S()) 0; The rt-order condition (16) can be written a S ym() = (S ym()) S ym (S ym()) ; (17) where Sym() i the optimal critical value for invetment. The lat term on the right-hand ide can be interpreted a the opportunity cot of exerciing the option with payo Sym(). The fraction capture the wedge between the critical value Sym and the invetment cot. By (15) and (17) the value of the invetment opportunity i ( (S S W ym (; ) = ym ()) ym () if <Sym() (18) if Sym(): 6 Aymmetric information: The optimal exercie trategy In thi ection we olve the principal' problem of nding the optimal invetment trategy given the agent' private information. In order to implify the problem of nding an optimal trategywe ubtitute the unknown function G() in the principal' value function in equation (4) with an expreion of known function of ^S(). Uing equation (5) and (13) the value of the compenation function may be written a the um of the value of the true invetment cot and the value of the agent' private information G( ^S()) ( = ^S()) ( = ^S()) + V (; ) + R du: (19) ( ^S()) ( ^S(u)) The right-hand ide of the equation give an repreentation of the value of the compenation which contain known function and only. Subtituting the expreion for G( ^S()) ( in equation (19) into the principal' optimization problem in equation (4) lead to ^S()) Z ( Z ) W (; ) = up ^S() du f()d: (0) ^S() ( ^S()) ( ^S(u)) From equation (0) we ee that the ubtitution of G( ^S()) implie that the principal' problem i reduced to nding an optimal trigger income S (). 9

10 A further implication of the optimization problem can be done by partial integration of the term R R =( ^S(u))duf()d. Integration lead to Z Z Z duf()d = ( ^S(u)) ( ^S()) F ()d: (1) Inerting the right-hand ide of (1) into the objective function (0) we nd Z W (; ) = up ^S() ( ^S()) ^S() F () f()d: () f() From the lat term in equation () we ee that the principal' optimization problem i now imilar to the problem of optimally exerciing an American call option with optimal exercie price + F ()=f(). The term F ()=f() can be interpreted a the ineciency due to the agent' private information. Pointwie dierentiation give the rt- and econd-order condition for the optimal \exercie value" S =1 ^S( ^S()) ^S() F () =0; ^S() ( ^S()) f() The condition for the trigger value are atied a long a the econd-order W (; G()) = ^S ^S() ^S()) F () f() ^S() ( ^S()) ( ^S()) f() hold. Thu the optimal trigger value for the principal i given by S () F () f() = (S ()) S (S ()) : (4) Given the compenation function (to be evaluated in the next ection) the trigger value in equation (4) i alo the optimal exercie trategy for the agent. Equation (4) how that the trigger value i baed on the principal' total cot of exerciing the invetment option i.e. it i baed on + F ()=f(). A in equation 17 the right-hand ide repreent the opportunity cot of exerciing the option. Compared to the optimal invetment trategy under ymmetric information equation (17) the critical value for invetment ha increaed due to the aymmetric information. Thi ineciency lead to underinvetment becaue of the longer "waiting time" of invetment. Z! Z du ( ^S(u)) f()d = " Z # Z duf () () ( ^S(u)) F ()d: ( ^S(u)) By inerting the bound and in the rt term on the right-hand ide we ee that thi term i zero: ubtituting with yield R (; ^S(u))du = 0 and ubtituting with yield F () =0. Thu we are left with the right-hand ide term of equation (1). 10

11 7 Implementation of the optimal compenation function We are now left with the problem of nding an implementable compenation function that lead to the optimal invetment trategy. Conidering equation (19) and (4) the time zero value of the optimal compenation function when <S () igiven by (S ()) G(S ()) = = (S + R ()) h (S + u ()) (S du (u)) i (S (u)) R u S (S (u)) ((S (u))) S udu: (5) The rt right-hand ide equality in (5) tate that the compenation function mut cover the agent' true cot (the rt term) and the agent' value of private information (the lat term). Notice that the compenation function in equation (5) i not written in a contractable form a it i a function of the unobervable variable a well. The right-hand ide of the equation mut therefore be found a a function of obervable variable only. From Melumad and Reicheltein [10] we know that a compenation function G(; S ()) under a communication-baed centralized contract (by the revelation principle) i compatible with the compenation function G(S ()) under a direct delegation contract if for all [; ] G(; S ()) = G(S ()). Thi retriction i atied when the function S () i one-to-one. Auming that thi i valid for S () 3 we denote #(S ()). Thi lead to (S ()) G(S ()) = = (S R ()) #(S (u)) R S () (S ()) S () #(S (u)) S (S (u)) ((S S (u))) udu S (S (u)) ((S ds (u) (u))) = #(S ()) (S + R S () ()) S () #0 (S (u)) (S (u)) ds (u) (6) Thu from equation (6) and the aumption that only the outcome of the invetment i obervable we nd that the contracted optimal compenation function i given by 8 0 if <S () >< G = #+ R S () # 0 (S (u)) (S (u)) ds (u) if S () <S () >: if S () (7) The above expreion repreent an implementable compenation function dependent upon the obervable variable and S () only. When S () 3 S () i a one-to-one function a long a it i continuou and trictly increaing in the interval S () [S ();S ()]. 11

12 the compenation i zero a the invetment ha not taken place in thi range of the value of. A long a <S () the agent will wait with exerciing the option until the point in time where the time zero value of the income tream reache S (). When S () < S () the compenation i dependent on only. The compenation i increaing in. However note that the compenation never can be higher than. The reaon i that the principal know that the invetment cot i not higher than the upper level. The value of the compenation function in equation (6) and (7) inerted into equation (5) implie that the agent' value from the invetment project can be written a 8 R S () S () #0 (S (u)) (S (u)) ds (u) if <S () >< # V (; ) = + R S () # 0 (S (u)) (S (u)) ds (u) >: if S () <S () if S () (8) In ection 5 it wa hown that the agent' value from the invetment i zero under ymmetric information about the invetment cot. Equation (8) tate that the agent' value from the invetment when he ha private information about the cot i poitive a long a hi invetment cot i below. The agent' hare of the total value of the invetment i larger the larger i. However the agent' value from the project will never exceed. The principal' value of the invetment option i repreented by 8 (S ()) (S () ) R S () S >< () #0 (S (u)) (S (u)) ds (u) W (; ) = # >: R S () # 0 (S (u)) (S (u)) ds (u) if <S () if S () <S () if S (): (9) A i to be expected the principal' time zero value i lower under aymmetric information than under the ymmetric information cae (compare (9) and (18)). The reaon i that the invetment occur at a later time and at a higher cot (a the compenation i higher than the true invetment cot) thereby lowering the value of the invetment. The principal' lo will however never be higher than the lo in the interval >S () i.e. it will not exceed W ym (; ) W (; ) =. Though the principal will have a lo under aymmetric information for all (30) how that the total dead-weight lo L(; ) W ym (; )+V ym (; ) 1

13 (W (; )+V (; )) will only be poitive when >S (): 8 (S S ym ()) ym () (S >< ()) (S () )) if <Sym() L(; ) = >: (S ()) (S () ) if S ym() <S () 0 if S (): (30) The total dead-weight lo i 0 when S () becaue in thi range the agent' invetment trategy lead to the ame deciion a in the full information cae and the contracted compenation function only give a haring rule between the principal and the agent. The agent' gain exactly equal the principal' lo becaue of the aymmetric information. 8 Illutration of the reult The preceding ection ued a general diuion (equation (1)) for the income proce S t and an unpecied probability denity f( ) ~ for the aeed invetment cot. ~ To illutrate the reult the imple uniform ditribution and the geometric Brownian motion are aumed for the invetment cot and the income proce repectively. A uniform ditribution implie that F ()=f() =. The geometric Brownian motion proce of the value of the income i repreented by ds t =(r )S t dt + S t dw t ; S 0 = ; (31) under the equivalent martingale meaure Q. The trictly poitive and increaing olution () to the ordinary dierential equation (compare equation () and (3)) 1 u ()+(r )u () ru() =0 i then found to equal () = where 3 = (r )+ (r ) 1 +r 5 > 1: Hence the olution to the expectation E 0 [e rk ] i (uing equation ()) ( E 0 [e rk ]= (S = ()) S () if <S () 1 if S (): (3) For the benchmark ymmetric information cae the right-hand ide of equation (17) become Sym = and hence the optimal critical value for invetment i Sym() ==( 1) >a>1. From equation (18) the correponding value of the invetment opportunityiw ym (; ) =(=Sym) Sym() = 13

14 =( 1)(=Sym) for <Sym(). Recall that the agent obtain no prot under ymmetric information i.e. V ym (; ) =0. For the aymmetric information cae however the optimal \trigger income" i found by equation (4) to be S () =( ) 1 ; (33) which (when > ) i higher than the trigger under ymmetric information Sym() ==( 1). The fraction =( 1) > 1 caue a wedge between the critical value for exerciing the invetment opportunity and the principal' cot of the invetment even in the cae of ymmetric information. The dierence ( )=( 1) i the increae in the trigger income caued by aymmetric information. The variable #(S ()) equal by equation (33) #(S ()) = 1 S () 1 +. In order to nd the expreion for the compenation function G we rt inert the above variable into the integration in the econd equality in (7). Thi lead to Z S () 1 1 S (u) " ds (u) = 1 S () # S (). In addition oberve that # in (7) equal # =1=(( 1)= + ). Thi give 8 0 if <S () 1 >< [ + G = S >: S () ( ) if S () <S (34) () () if S (); Further we nd that the time zero value of the agent' and the principal' value function (equation (8) and (9)) are 8 >< V (; ) = 1 S ()) S () S () [S () ( ) S () ( ) 1 [ ( ) S () ( ) S () if <S () if S () <S () (35) >: if S (); 14

15 and 8 >< W (; ) = 1 S () + S () S () S () [S () S () ( ) 1 [ + S () ( ) if <S () if S () <S () (36) >: if S (); repectively. Oberve that the total combined value for the principal and the agent i ( W (; )+V (; ) = S () (S () ) if S () (37) if >S () in the cae of aymmetric information. Similar expreion held for the ymmetric information cae a well but with S () replaced by Sym < S (). Thee relation are conitent with (30) which in the cae the aumption of a geometric Brownian motion and a uniform denity equal 8 S S ym () ym () S () >< (S () )) if <Sym() L(; ) = (38) (S () ) if Sym() <S () >: S () 0 if S (): The reult are illutrated graphically. In the bae cae the invetment cot i et to 1 the lower level cot = 0:5 and the upper level cot =. For the parameter in the income proce we et the rik-free rate r = 0:04 the convenience yield = 0:03 and the volatility = 0:1. With a uniformly ditributed invetment cot and an income proce that follow a geometric Brownian motion thee parameter lead to =:37 Sym() =1:73 S () = :59 and S () =6:05. In gure 1 the compenation i plotted a a function of. The compenation i zero when i lower than the critical value of invetment S () =:59 a the compenation i not paid prior to the invetment time. Therefore at S () the function jump to the amount paid when S () and it i increaing from thi point until = S () =6:05. For 6:05 the compenation i contant at it maximum level =. Both within regulation and corporate nance we often nd that compenation function are linear in the obervable output from a project. In the numerical example given here the compenation function i concave. The reaon i that the upper level for the cot ha a ignicant eect. If the upper level 15

16 G Figure 1: The compenation G a a function of. 6 W + V 5 W W ym V Figure : W ym W + V W and V a function of. 16

17 the cot had been very high the compenation function would have approached a linear function of. In gure the principal' and the agent' value function are hown a function of. The principal' value function under ymmetric information i convex when <S () =:59 and it i linear in the interval where the optimal deciion i to invet immediately. Thi correpond to the value of a "tandard" real option a a function of the output price. Under aymmetric information it i alo the cae that the principal and the agent have convex value function in the interval where it i ex ante protable to potpone the invetment. Thi i for the ame reaon a under ymmetric information: avolatility higher than zero implie a poibility of higher protability in the future. In the interval S () <S () the agent' value i concave for the ame reaon a for the concavity in the compenation function: the upide potential for future prot i limited. For S () the principal alone benet from higher and the agent' value of the contract i contant at =1. Since the agent' value of information leave le prot to the principal and the agent' value function i concave in the interval [S ();S ()) the principal' value i convex in the ame interval. When S () the principal' value under aymmetric information increae linearly a the agent' value of information i zero in thi interval. Figure alo how the um of the principal' and the agent' value function under aymmetric information W (; )+V (; ). A long a i higher than or equal to S () = :59 thi curve i identical with the principal' value under ymmetric information W ym (; ). The reaon i that in thi interval the contract between the principal and the agent give a haring rule without having any eect on the invetment trategy compared to the ituation of full information. In the interval (0 S()) W (; )+V (; ) ilower than W ym (; ) due to an inecient invetment trategy. Thi fact i alo illutrated in gure 3 where the relative dead-weight lo a a function of i plotted in the lower curve. The relative dead-weight lo i dened a (W ym W V )=W ym ). 4 The gure how that dead-weight lo i poitive when <S () =:59. In gure 3 the principal' relative lo (W ym W )=W ym i plotted in the upper curve. Both the principal' relative lo and the relative dead-weight lo i contant a long a the bet deciion under both aymmetric and ymmetric information i to potpone the invetment i.e. when <Sym() =1:73. The loe are decreaing in the interval [Sym();S ()) ince the inef- ciency in the econd-bet invetment trategy i decreaing a approache S () =:59. For all higher than thi point the invetment trategy i the ame for the ymmetric and the aymmetric information cae i.e. there i no dead-weight lo. In the interval [S ()S ()) the principal' relative lo rt increae and then decreae. The reaon i that two eect pull in oppoite direction: higher lead to higher dierence between the principal' value under ymmetric and aymmetric information which increae the relative lo wherea an upper limit for the invetment cot tend to decreae the agent' value of information a get cloer to S (). Figure 4 plot the partie' function of when = 3. In the "tandard" real option problem of valuing an invetment poibility correponding to the value 4 In the gure the notation V ym i not included a V ym =0. 17

18 WymW Wym L Wym Figure 3: Principal' lo and dead-weight lo a function of..5 W ym W V Figure 4: W ym W and V a function of =1. 18

19 L Wym WymW Wym Figure 5: Principal' lo and dead-weight lo a a function of = W ym 1 W V Figure 6: W ym W and V a function of =1. 19

20 WymW Wym 0. L Wym Figure 7: Principal' lo and dead-weight lo a a function of =1. of W ym (; ) the value i increaing with repect to in the interval where the bet deciion i to potpone the invetment. The reaon i that a long a the option i not exercied higher volatility increae the poibility of a higher future prot. The principal' value function under aymmetric information depend on alo in the interval where the optimal deciion i to invet immediately i.e. the interval S () correponding to 0:14. The reaon i connected to the agent' value of information: a increae the agent' value of information decreae and therefore the hare of the prot left to the principal i increaing. The agent' value i decreaing in becaue of the upper limit on the agent' compenation. For <S () correponding to > 0:14 there i an additional eect on the principal' value under aymmetric information which tend to depre the principal' value: the lo in value becaue of an inecient invetment trategy. Thi eect i dominating when i between 0.14 and The ame eect are reected in gure 5. At the correponding to S () = 3 the relative dead-weight lo get poitive becaue then it reache the interval <S () in which we know that the lo i poitive. Both the relative deadweight lo and the relative principal' lo increae in thi interval a long a the eect of a econd-bet invetment trategy dominate the eect that the agent' value of information decreae with an increaing volatility. The principal' lo when S () decreae becaue of the agent' decreaing value of information a increae. Figure 6 plot the principal' and the agent' value a function of the 0

21 invetment cot. Both the principal' and the agent' value function are nonincreaing with repect to a a higher cot lower the value of the invetment for both. For 1:1 correponding to S () the principal' value i independent of the agent' invetment cot. The reaon i that the compenation paid to the agent cannot be a function of the unobervable variable. Figure 7 how that the relative dead-weight lo i increaing in. Thi i becaue higher cot lead to higher critical value for exerciing the option and thereby larger ineciency in the invetment deciion. The principal' relative lo i decreaing in for lower than or equal to 1.1 correponding to S (). Once again the reaon i connected to the fact that when S () W i independent of and therefore an increae in reult in a correponding increae in the principal' lo. For correponding to <S () the dominating eect i the ame a in the dead-weight lo a long a i lower than 1.6. For higher than 1.6 the dominating eect i the agent' value of information getting lower the cloer to the upper level cot the true invetment cot i. Thi tend to decreae the lo. At = the principal' lo and the dead-weight lo coincide a the value of the agent' information i zero at thi point. 9 Concluion In thi article we tudy eect of aymmetric information on dynamic invetment deciion. A principal own an invetment opportunity and delegate the invetment trategy of the project to an agent. The agent ha private information about the invetment cot wherea the tochatic output i common knowledge. Thi etting applie to a number of ituation both within regulation (the principal i a regulator and the agent i a company) and corporate nance (hareholder repreent the principal and manager repreent the agent). The agent' private information about the cot implie that it i optimal for the principal to compenate the agent according to hi value of private information. Thu the compenation will be higher than the true invetment cot in mot cae thereby increaing the principal' cot of hi invetment opportunity. A higher cot lead to a higher critical value for invetment. Thu it i found that the agent' private information about the invetment cot may lead to underinvetment. The agent' value of private information will however not alway lead to an inecient invetment trategy. Inecient deciion will occur only in the interval where the critical value of invetment given aymmetric information i higher than the time zero value of the output from the invetment. If the time zero value of the output i higher than the critical value of invetment the compenation function only give a rule for haring the prot between the principal and the agent without having any ineciency eect. In the ame way a aymmetric information about invetment may depre activity an agent' private information about the cot of hutting down an activity may lead to higher activity than when there i no private information. More generally in an model where one can witch between option private information about witching cot lead to higher cot and therefore fewer 1

22 witche. For intance in Dixit [6] entry and exit deciion of production are dicued. In thi model Dixit nd that entering and exiting an activity lead to a "hyterei band" due to the uncertainty of future outcome and to the irreverible entry and exit cot. If an agent ha private information about the cot of witching between activity and no activity the hyterei band will be even larger than in Dixit' model. Thu the cot of witching between the two option may lead to both too much and too little activity. Thu on a macroeconomic level even though the level of activity when there i private information hould happen to be not far from the aggregate level when we have no private information the activity may not necearily take place in the activitie where the prot i highet. A witching option model can alo be applied for nancial invetment. An example i the holder of a fund who delegate the trading trategy of the nancial portfolio to an agent and where there are ome tranaction cot. If the agent ha private information about ome xed tranaction cot the invetor can ue a variant of the method decribed in thi article to deign the compenation to the agent in uch a way a to optimize the agent' rik management. Acknowledgement I thank Steinar Ekern for many helpful comment. I am alo grateful to Petter Bjerkund an anonymou referee participant at the Sympoium on Real Option held in September 1997 at the Copenhagen Buine School and participant at a PhD-eminar on Dynamic Aet Pricing held by Darrell Due in May 1998 at the Norwegian School of Economic and Buine Adminitration. Reference [1] Rick Antle Peter Bogetoft and Andrew W. Stark. Incentive problem and the timing of invetment [] David P. Baron and Roger B. Myeron. Regulating a monopolit with unknown cot. Econometrica 50(4):911{930 July 198. [3] Petter Bjerkund and Gunnar Stenland. A elf-enforced dynamic contract for proceing of natural reource [4] Andrei N. Borodin and Paavo Salminen. Handbook of Brownian Motion - Fact and Formulae. Birkhauer Verlag Bael Boton Berlin [5] M.J. Brennan and E.S. Schwartz. Evaluating natural reource invetment. Journal of Buine (58):135{157 January [6] Avinah Dixit. Entry and exit deciion under uncertainty. Journal of Political Economy (97):60{638 June [7] Darrell Due. Dynamic aet pricing theory. Princeton Univerity Pre Princeton New Jerey nd edition [8] Kiyoi It^o and Henry P. Jr. McKean. Diuion procee and their ample path. Springer 1t edition [9] Jean-Jacque Laont and Jean Tirole. A Theory of Incentive in Procurement and Regulation. MIT Pre Cambridge Maachuette 1993.

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