Product Innovation and Optimal Capital Investment under Uncertainty. by Chia-Yu Liao Advisor Ching-Tang Wu

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1 Produc Innovaion and Opimal Capial Invesmen under Uncerainy by Chia-Yu Liao Advisor Ching-Tang Wu Insiue of Saisics, Naional Universiy of Kaohsiung Kaohsiung, Taiwan 8 R.O.C. July 2006

2 Conens Z`Š zz`š iii iv Chaper. Inroducion Chaper 2. The Basic Model and Parameers 3 Chaper 3. One-Period Model Opimizaion problem a ime Opimizaion problem a ime Example 2 Chaper 4. Two-Period Model Opimizaion problem a ime Opimizaion problem a ime Opimizaion problem a ime Example 25 Chaper 5. Coninuous-Time Model Convex cos funcions Linear cos funcions Invesmen wihou consrains Invesmen wih consrains 36 Chaper 6. Conclusion 39 Appendix A. Differen Forms of V in All Condiions 4 Appendix B. Proof of Exisences of I0 and R0 45 i

3 ii CONTENTS Appendix C. Derivaion of Equaion (5.6) 49 Bibliography 53

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5 Produc Innovaion and Opimal Capial Invesmen under Uncerainy Advisor: Dr. Ching-Tang Wu Deparmen of Applied Mahemaics Naional Chiao Tung Universiy Suden: Chia-Yu Liao Insiue of Saisics Naional Universiy of Kaohsiung ABSTRACT This hesis analyzes he opimal invesmen sraegy of a compeiive firm in an uncerain environmen. In conras o he previous conribuion, he curren sudy akes ino accoun he impacs on price of R&D invesmen in he framework of opimal invesmen under uncerainy. This formulaion capures an imporan observaion of he business world, ha is, firms consanly face rade-off in R&D aciviies which can raise he firm value by delivering new echnology. We characerize he firm s opimal invesmen policy explicily and derive soluions for firm value in some differen siuaions by considering discree-ime and coninuous-ime models. We show ha demand uncerainy has a posiive effec on he opimal invesmens and he presen value of a firm. Keywords: opimal invesmen, uncerainy, R&D iv

6 CHAPTER Inroducion The issue of invesmen policy under uncerainy has been exensively discussed in recen years. Wih regard o his issue, Avinash Dixi and Rober Pindyck s influenial work Invesmen under uncerainy (994) has ever provided subsequen researchers wih significan perspecives. However, here are wo opposie conclusions abou he effec of uncerainy on he opimal invesmen in he sudies. Some auhors, such as Pindyck (982) and Novy-Marx (2004), argue ha here will exis an opion value o drive he decision maker o delay an invesmen projec unil new informaion abou he marke condiions arrives. In conras, Richard Harman (972) and Andrew Abel (983) brough up a disinc viewpoin. They showed ha increased oupu price uncerainy will raise he marginal revenue producs of capial and hence lead o an increase in he opimal invesmen. Abel (983) presened a model in which a compeiive firm faces oupu price wih uncerainy and underakes gross invesmen by incurring an increasing convex cos. In his paper, we use a simple generalizaion of Abel s framework o analyze he invesmen policy under uncerainy and he operaing cash flows of a firm. One of he differences beween he model we use and Abel s model is he source of uncerainy. Abel considered uncerainy in oupu price, hough we go a sep furher o hink ha demand of oupu could flucuae due o some exernal facors in he marke and hen proceed o influence he price of oupu. Therefore we se demand of oupu as he uncerainy erm. In addiion, an exra facor invesmen on research is added in our model. Generally speaking, many facors could affec he operaion of a compeiive firm, and a firm migh underake mulifarious

7 2. INTRODUCTION producion coss. Besides he paymen of hiring labor and invesmen in capial, expenses for research should be an essenial iem of a firm s operaing coss. This supplemenary variable will increase he degree of complexiy in he soluion for he opimal invesmen and he operaing cash flows of he firm. The purpose of our sudy is o find he invesmen policy which can make he presen value of a firm achieve opimizaion and o analyze he effec of demand uncerainy. The res of he hesis is organized as follows. Chaper 2 inroduces he parameers and he basic model. Then we sar he opimizaion problem from an analysis of he simples discree ime in Chaper 3. In Chaper 4, we expand he one-period ime o wo periods and calculae some examples. In Chaper 5, we consider a coninuous-ime model wih convex and linear cos funcions and analyze he characerisics of he soluions. Chaper 6 concludes.

8 CHAPTER 2 The Basic Model and Parameers Consider a filered probabiliy space (Ω, F, (F ), P) wih F 0 =, Ω}. We invesigae in his hesis a model of a compeiive firm ha uses research invesmen, A, labor invesmen, L, and capial invesmen, K o produce oupu, Y, wih he Cobb-Douglas producion funcion Y = K α L β, where 0 < α, β <, and α + β =. The price of oupu a ime is given by P = X Y ε A wih he elasiciy of demand ε > 0 and he elasiciy of research > 0. Compared wih research, demand is a more sensiive facor of influencing he prices of producs. Therefore i is generally assumed ε >. X is an exogenous demand shifer which is a sochasic process following he sochasic differenial equaion dx X = σ dz, where (Z ) is a sandard Brownian moion. In oher words, X has a log-normal disribuion of he form X = X 0 exp σz } 2 σ2, where X 0 = x is he iniial value, and σ is he consan volailiy rae. Assume ha R and C denoe he firm s revenue and cos a ime, respecively. We would like o find he opimal value of a firm V by maximizing he presen value of is cash flow which equals revenues minus coss a ime. Then he problem we discuss is V := max(r C ), 3

9 4 2. THE BASIC MODEL AND PARAMETERS where R = P Y, and he form of C depends on siuaions. Moreover, he firm akes I and R o inves in capial and research a ime. A and K depreciae a consan raes λ and δ, respecively. In he discree-ime model, hey can be wrien as general formulas: A + = R + λa K + = I + δk, for all = 0,, 2,. In he coninuous-ime model, dk = (I δk ) d, da = (R λa ) d, which will be discussed in Chaper 5. Since he coninuous ime model is usually no easy o solve, we invesigae firs wo ypes of discree ime model in he sequel and discuss some of is properies. We will deliberae abou some differen saes in he following chapers.

10 CHAPTER 3 One-Period Model In his chaper, we consider he siuaion only for one period of he operaing ime of a firm. Since he case we discuss here is for one period, he operaing ime of a firm is considered as wo specified inervals, ime 0 and ime. We use backward inducion o invesigae he opimizaion problem. We firs discuss he problem V := max(r C ), and ge is opimal soluions. Afer ha, we plug he resul in he case of ime 0 o ge he soluion o he opimizaion problem V 0 := max E R 0 C 0 + V ]. 3.. Opimizaion problem a ime Using backward inducion, we firs invesigae he opimizaion problem in he second inerval. The oal cos of he firm C is nohing bu paymen for labor s wages a ime. The firm employs labor a a wage rae ρ > 0 so ha is he operaing profi a ime is R C = P Y ρ L which equals cash flow of he firm. Le V (A, K, X ) denoe he firm s value a ime : V (A, K, X ) = max L P Y ρ L } } = max X K α( ε) L β( ε) A ρ L, L 5

11 6 3. ONE-PERIOD MODEL where X = X 0 exp σz 2 σ2}, X 0 = x is he iniial value which is a consan, and Z is disribued as Normal(0, ). Maximizing V wih respec o L yields Then we have V (A, K, X ) = X L = ] β( ε)x K α( ε) A. ρ α( ε) K A β( ε) Thus, we ge he opimal value of he firm a ime. ρ ] β( ε) β( ε)] Opimizaion problem a ime 0 Now ha he opimal value of he firm a ime was obained, we consider he opimizaion problem in he firs period, ime 0. We deermine he value of he firm by maximizing he expeced presen value of is cash flow. When i is a ime 0, he fundamenal value of he firm is no longer V. Besides he discoun rae r, r 0, ], he firm s value a ime urns ino he condiional expecaion of V wih respec o pas informaion F 0, and EV F 0 ] = EV ] = K = K α( ε) A α( ε) A β( ε) ρ β( ε) ρ } σ 2 β( ε) exp. 2 β( ε)] 2 ] β( ε) β( ε)] EX ] ] β( ε) β( ε)] x The firm akes invesmen I 0 by incurring cos c 2 (I 0 ) = ω 0 I 0 a ime 0, ω 0 > 0, and depreciaes a a consan rae δ (0, ]. K 0 denoes he original capial of he firm. Therefore, capial a ime is K = I 0 + δk 0.

12 3.2. OPTIMIZATION PROBLEM AT TIME 0 7 Before regularly operaing, he firm invess A 0 in researching on he procedure for producion. The cos of research a ime 0 is given by c 3 (R 0 ) = θ 0 R 0 where θ 0 > 0. Research fund depreciaes a a consan rae λ (0, ]. Thus, research a ime is given by A = R 0 + λa 0. Furhermore, c (L 0 ) = ρ 0 L 0 denoes he cos of employing labor a ime 0 where ρ 0 > 0. Then he oal cos in ime 0 is C 0 = c (L 0 ) + c 2 (I 0 ) + c 3 (R 0 ). The revenue funcion is R 0 = P 0 Y 0. The value of he firm a ime 0 will be he sum of he maximized profi and he discouned expecaion of he firm in he previous period: V 0 (A 0, K 0, X 0 ) = max I 0,L 0,R 0 R + = max I 0,L 0,R 0 R + P 0 Y 0 c (L 0 ) + c 2 (I 0 ) + c 3 (R 0 ) + EV } ] + r X 0 K α( ε) 0 L β( ε) 0 A 0 ρ 0 L 0 ω 0 I 0 θ 0 R r (I 0 + δk 0 ) α( ε) (R0 + λa 0 ) β( ε) ρ ] β( ε) } } β( ε)] x σ 2 β( ε) exp. (3.) 2 β( ε)] 2 In order o solve above opimizaion problem wih consrains I 0, L 0, R 0 R +, we may consider he problem wihou consrain, i.e., we consider he problem max P 0 Y 0 ρ 0 L 0 ω 0 I 0 θ 0 R 0 + EV } ]. (3.2) I 0,L 0,R 0 + r

13 8 3. ONE-PERIOD MODEL We can ge opimizaion soluion Ī 0, L 0 and R 0 of (3.2) of he form L 0 = ] β( ε)x 0 K α( ε) 0 A 0, ρ 0 Ī0 = ( + r) ε exp R 0 = ( + r) ε exp ω 0 α( ε) ] α( ε) +ε ε ( ) ] β( ε) θ0 ε ρ ε x ε β( ε) } σ 2 β( ε) δk 0, (3.3) 2 β( ε)](ε ) ] α( ε) ( ) ε ω ] β( ε) ε 0 θ0 ε ρ ε x ε α( ε) β( ε) σ 2 β( ε) 2 β( ε)](ε ) } λa 0. (3.4) However, i is possible for Ī 0 and R 0 o be negaive, which depends on he values of A 0 and K 0 given in he beginning. Here we assume ha he firm does no sell is capial or achievemen of research, herefore invesmen and research mus be nonnegaive. I can be divided ino four possible cases: Case : The iniial values saisfy and K 0 δ ( + r) ε exp A 0 λ ω 0 α( ε) σ 2 β( ε) 2 β( ε)](ε ) ( + r) ε exp ω 0 α( ε) ] α( ε) +ε ε }, σ 2 β( ε) 2 β( ε)](ε ) ] α( ε) ε }, ( ) ] β( ε) θ0 ε ρ ε x ε β( ε) ( ) ε ] β( ε) θ0 ε ρ ε x ε β( ε) ha is, Ī 0 0 and R 0 0. If so, we ake posiive values of I 0 in (3.3) and R 0 in (3.4) o maximize V 0, i.e., he opimal soluion o (3.) is given by (L 0, I 0, R 0) = ( L 0, Ī 0, R 0). Subsiuing (3.3) and (3.4) ino (3.), we can obain ha he value of

14 3.2. OPTIMIZATION PROBLEM AT TIME 0 9 he firm a ime 0 is V 0 (A 0, K 0, X 0 ) = X α( ε) 0 K0 A0 + ( + r) ε exp ω 0 α( ε) β( ε) ] α( ε) ε σ 2 β( ε) 2 β( ε)](ε ) ρ 0 ( θ0 ] β( ε) β( ε)] ) ε β( ε) ρ ] β( ε) ε } (ε ) + ω 0 δk 0 + θ 0 λa 0. x ε Case 2: The iniial values saisfy K 0 > δ ( + r) ε exp ω 0 α( ε) σ 2 β( ε) 2 β( ε)](ε ) ] α( ε) +ε ε }, ( ) ] β( ε) θ0 ε ρ ε x ε β( ε) and A 0 λ ( + r) ε exp ω 0 α( ε) σ 2 β( ε) 2 β( ε)](ε ) ] α( ε) ε }. ( ) ε ] β( ε) θ0 ε ρ ε x ε β( ε) The firm invesed so much capial such as machinery insrumens of facory before operaing ha i does no need o pu in any more capial o srenghen is equipmen a ime 0. Hence, we ake L 0 = L 0, I0 = 0 and ge he opimal R0 for (3.) in he following form ( ) α( ε) θ0 R0 = ( + r) +β( ε) β( ε) +β( ε) (δk0 ) ρ } σ 2 β( ε) exp λa 0. 2 β( ε)] β( ε) ] ] β( ε) x

15 0 3. ONE-PERIOD MODEL Subsiue (L 0, I 0, R 0) ino (3.), we ge ha he value of he firm a ime 0 is given by V 02 (A 0, K 0, X 0 ) = X α( ε) 0 K0 A0 β( ε) ρ 0 ( ) α( ε) θ0 ] β( ε) β( ε)] ] β( ε) + ( + r) +β( ε) β( ε) +β( ε) (δk0 ) ρ } σ 2 β( ε) x exp β( ε) ] 2 β( ε)] β( ε) ] + θ 0 λa 0. Case 3: The iniial values saisfy K 0 δ ( + r) ε exp ω 0 α( ε) σ 2 β( ε) 2 β( ε)](ε ) ] α( ε) +ε ε }, ( ) ] β( ε) θ0 ε ρ ε x ε β( ε) and A 0 > λ ( + r) ε exp ω 0 α( ε) σ 2 β( ε) 2 β( ε)](ε ) ] α( ε) ε }. ( ) ε ] β( ε) θ0 ε ρ ε x ε β( ε) I means ha he firm had already made a horough sudy of research and developmen for is producs in advance. As a resul, he firm migh inend o suspend invesmen of research a ime 0. So we ake L 0 = L 0, R 0 = 0 and ge I0 = ( + r) ε (λa 0 ) ε α( ε) ω 0 } σ 2 β( ε) exp δk 0. 2 β( ε)]ε ] ε β( ε) ρ ] β( ε) ε x ε

16 Subsiuing L 0, I 0 and R 0 ino (3.) yields V 03 (A 0, K 0, X 0 ) = X 3.2. OPTIMIZATION PROBLEM AT TIME 0 α( ε) 0 K0 A0 + ε( + r) β( ε) ε (λa 0 ) ε β( ε) ρ 0 α( ε) ω 0 } σ 2 β( ε) exp + ω 0 δk 0. 2 β( ε)]ε ] β( ε) β( ε)] ] α( ε) ε β( ε) ρ ] β( ε) ε x ε Case 4: The iniial values saisfy K 0 > δ ( + r) ε exp ω 0 α( ε) σ 2 β( ε) 2 β( ε)](ε ) ] α( ε) +ε ε }, ( ) ] β( ε) θ0 ε ρ ε x ε β( ε) and A 0 > λ ( + r) ε exp ω 0 α( ε) σ 2 β( ε) 2 β( ε)](ε ) ] α( ε) ε }. ( ) ε ] β( ε) θ0 ε ρ ε x ε β( ε) In his case, jus like reasons in he wo preceding cases, expending large sums of money in equipmen and research previously keeps he firm from invesing a he nex sage. Consequenly, we allow he occurrence of L 0 = L 0, I 0 = 0, and R 0 = 0, and he value of he firm a ime 0 is V 04 (A 0, K 0, X 0 ) = X α( ε) 0 K0 A0 β( ε) ρ 0 β( ε) + (δk 0 ) α( ε) (λa0 ) + r } σ 2 β( ε) exp. 2 β( ε)] 2 ] β( ε) β( ε)] β( ε) ρ ] β( ε) x According o hese four cases, we obain he following proposiion.

17 2 3. ONE-PERIOD MODEL Proposiion 3.. Soluions of he opimal invesmen policy, L 0, I 0, and R 0, exis in one-period model Example The siuaion is no quie complicaed in his chaper while he procedure we consider is a one-period model. As long as he given values of A 0 and R 0 are known, we can recognize in which case i should be. And hen we are able o find he opimizaion of he firm. The following example gives us an overview how o compue he opimal soluions in he one-period model. Example 3.. We fix α = β = 2, ε = 2 3, = 3 and ω = ρ = θ = for = 0,, 2. In order o make i as simple as possible, we do no ake he effec of he ineres rae ino consideraion and assume ha equipmen of he facory and he accomplishmen in research of he firm will no depreciae as ime goes on. Therefore, We know ha he criical values of A 0 and K 0 decide he cases. In his example, i is of Case if A 0 < and K 0 < 0.025; i is of Case 2 if A 0 < and K 0 > 0.025; i is of Case 3 if A 0 > and K 0 < 0.025; i is of Case 4 if A 0 > and K 0 > We selec wo numbers for boh A 0 and K 0 : 0.0, smaller hen 0.025, and 0.08, exceeding We find ou he opimal invesmen, I 0 and R 0, and he maximized value of he firm a ime 0 respecively in hese four cases. We consider wo siuaions for σ 2 : σ 2 = and σ 2 = 4. The resuls are as abled below. Any change in he value of each parameer migh lead anoher invesmen sraegy. Wih he fixed parameers in his example, he fourh invesmen policy would be he bes choice for he firm; i brings he firm he highes presen value. And we observe ha he firm s presen value wih larger σ 2 is higher. We may wonder if demand uncerainy has a posiive effec on he presen value of a firm. This quesion will be resolved in following chapers.

18 3.3. EXAMPLE 3 σ 2 = Case Case 2 Case 3 Case 4 A K I R V Table 3.. σ 2 = σ 2 = 4 Case Case 2 Case 3 Case 4 A K I R V Table 3.2. σ 2 = 4 In he nex chaper, we will discuss a more complex siuaion which is for wo periods.

19 CHAPTER 4 Two-Period Model In he las chaper, we obain he resul of one period. Now we exend he period for observaion from an one-period ime o a wo-period ime. The noaion and resricion of parameers is similar o ha in he las chaper. Now we consider hree operaing ime poins of he firm, which are ime 0, ime and ime 2. The same as he previous chaper, we use he backward inducion o find he opimal invesmens in his model. Tha is, a ime 2, we consider V 2 := max(r 2 C 2 ), where R 2 and C 2 are he firm s revenue and cos a ime 2, respecively, and he maximum is aken over all possible variables which he firm can conrol a ime 2. A ime, we consider V := max(r C + EV 2 F ]), and a ime 0, he opimizaion problem is given by V 0 := max(r 0 C 0 + EV ]). (4.) 4.. Opimizaion problem a ime 2 Consider he hird ime poin, ime 2. The oal cos of he firm C 2 = ρ 2 L 2 ; i is spen hiring labor. The operaing profi which equals revenue minus wages is 5

20 6 4. TWO-PERIOD MODEL P 2 Y 2 ρ 2 L 2. Thus he value of he firm a ime 2 is where X 2 V 2 (A 2, K 2, X 2 ) = max L 2 P 2 Y 2 ρ 2 L 2 } } = max X 2 K α( ε) 2 L β( ε) 2 A 2 ρ 2 L 2, L 2 = X 0 exp σz 2 } 2 σ2 2, X 0 = x is a consan, and Z 2 follows a sandard normal disribuion. We can find he L 2 ha maximizes he firm s value a ime 2, and L 2 = ] β( ε)x 2 K α( ε) 2 A 2. ρ 2 Then he value of he firm a ime 2 would be V 2 (A 2, K 2, X 2 ) = X α( ε) 2 K2 A2 β( ε) ρ 2 ] β( ε) β( ε)] Opimizaion problem a ime As ime goes o he second period, he oal cos will increase iems which include cos of invesmen, cos of wages and cos of research, ha is, C = c (L ) + c 2 (I ) + c 3 (R ). The value of he firm a ime is V (A, K, X ) = max P Y c (L ) c 2 (I ) c 3 (R ) + EV } 2 F ], I,L,R R + + r in which EV 2 F ] denoes he expeced value of V 2 a ime. Since (Z ) is a Brownian moion, Z is F -measurable, and Z 2 Z is independen of F such ha EX 2 F ] = E x exp σz 2 σ 2} F ] = x expσ(z σ)} E exp σ(z 2 Z )} ] ( = x exp σ Z σ )}, 2

21 4.2. OPTIMIZATION PROBLEM AT TIME 7 where Z 2 Z has a sandard normal disribuion and is independen of Z. Noe ha EV 2 F ] is no a value bu a random variable. We rewrie V as V (A, K, X ) = max X K α( ε) I,L,R R + L β( ε) A ρ L ω I θ R + α( ε) + r K β( ε) 2 A2 ρ 2 2σ(Z σ) β( ε)] + σ 2 exp 2 β( ε)] 2 ] β( ε) β( ε)] x }}, (4.2) where X = x exp σz 2 } σ2. We know ha K 2 = I +δk and A 2 = R +λa. Using a similar argumen as in Secion 3.2, we consider he opimizaion problem max X K α( ε) L β( ε) A ρ L ω I θ R + EV } 2 F ], I,L,R R + r whose soluions Ī, L, and R are given by L = ] β( ε)x K α( ε) A, ρ Ī = α( ε)θ C x 2σ(Z σ) β( ε)] + σ 2 ε exp δk, (4.3) ω 2 β( ε)](ε ) } R = C x 2σ(Z σ) β( ε)] + σ 2 ε exp λa, (4.4) 2 β( ε)](ε ) where C = ( + r) ε ( ) ε ] α( ε) θ ε ω ε α( ε) ρ 2 β( ε) } ] β( ε) ε. As we menioned in he previous chaper, we equally reques ha he opimal invesmens L, I, and R o (4.2) o be nonnegaive here. Clearly, we can ge L = L. In order o ge I and R, we need o look he properies of Ī and R. We noice ha Ī and R are random variables and heir sign will depend on he value of Z. According o he range of Z, here would be four siuaions abou I and R :

22 8 4. TWO-PERIOD MODEL Siuaion : If Z > ε σ called d, and Z > ε σ log(λa ) + β( ε) σ log( + r) + α( ε) σ ( ε)β( ε) + σ β( ε)] log ρ 2 β( ε) σ log x + σ σ 2 β( ε)], + σ log θ log(δk ) + β( ε) σ log( + r) + log α( ε) + ε σ ω α( ε) + ε σ log θ log ω α( ε) + ( ε)β( ε) σ β( ε)] log ρ 2 β( ε) σ log x + σ σ 2 β( ε)], called d 2, boh R and Ī are non-negaive, which implies ha (I, R ) = (Ī, R ). Then we subsiue (4.3) and (4.4) ino (4.2) and obain he firm s value a ime in his case is given by V (A, K, X ) := X α( ε) K A ( θ β( ε) ρ ] β( ε) β( ε)] ) ] α( ε) ] β( ε) + ( + r) ε ω ε ε ρ ε 2 x ε α( ε) β( ε) } 2σ(Z σ) β( ε)] + σ 2 exp (ε ) + ω δk + θ λa. 2 β( ε)](ε ) Siuaion 2: If Z > maxd, d 3 } and Z d 2, where d 3 = β( ε) log(λa ) σ β( ε) + log θ β( ε) + σ σ α( ε) σ log log(δk ) + β( ε) σ log( + r) ρ 2 β( ε) σ log x + σ σ 2 β( ε)], he cumulaive capial invesmen K is large enough ha he firm manufacures oupu by using is original equipmen K wihou puing in exra invesmen. Thus, no addiional invesmen on he capial is needed. Replacing I wih zero in (4.2), we have R = ( + r) +β( ε) (δk ) ( α( ε) θ } 2σ(Z σ) β( ε)] + σ 2 exp 2 β( ε)] β( ε) ] ) +β( ε) ρ 2 β( ε) ] β( ε) +β( ε) x λa, (4.5)

23 4.2. OPTIMIZATION PROBLEM AT TIME 9 which is posiive, since Z > d 3. The value of he firm can be calculaed wih I 0 = 0 and R in (4.5) V 2 (A, K, X ) := X α( ε) K A β( ε) ρ ( α( ε) θ ] β( ε) β( ε)] ) +β( ε) ρ 2 ] β( ε) +β( ε) + ( + r) +β( ε) (δk ) β( ε) } 2σ(Z σ) β( ε)] + σ 2 x exp β( ε) ] 2 β( ε)] β( ε) ] + θ λa. Siuaion 3: If Z d and Z > maxd 2, d 4 }, where d 4 = ε σ log(δk ) σ log(λa ) + β( ε) σ log( + r) + β( ε) ρ 2 + log σ β( ε) σ log x + σ σ 2 β( ε)], β( ε) σ log ω α( ε) i means ha he previous research A is enough o work. The firm does no need o develop new echnology ye. Therefore, we ake R = 0 and ge he opimal I given by I = ( + r) ε (λa ) ε α( ε) ω ] ε } 2σ(Z σ) β( ε)] + σ 2 exp δk, 2 β( ε)] ε β( ε) ρ 2 ] β( ε) ε which is posiive due o Z > d 4. And hen we can find he value of he firm in his case is given by V 3 (A, K, X ) = X α( ε) K A +ε( + r) ε (λa ) ε β( ε) ρ α( ε) ω x ε ] β( ε) β( ε)] ] α( ε) ε β( ε) } 2σ(Z σ) β( ε)] + σ 2 exp + ω δk. 2 β( ε)]ε ρ 2 ] β( ε) ε x ε

24 20 4. TWO-PERIOD MODEL Siuaion 4: The final siuaion is ha Z d, Z d 2 }, d 2 > Z > d, Z < d 3 }, and d > Z > d 2, Z < d 4 }. The firm does no inves anyhing in equipmen of he facory or research and developmen. I coninues using primary equipmen and echnology o operae. Hence, we ake I = R = 0 and have V 4 (A, K, X ) = X α( ε) K A β( ε) + + r β( ε) ρ (δk ) α( ε) (λa ) ] β( ε) β( ε)] 2σ(Z σ) β( ε)] + σ 2 exp 2 β( ε)] 2 }. β( ε) ρ 2 ] β( ε) x In conclusion, we can ge he following lemma. Lemma 4.. Given fixed cumulaive research and capial invesmens A and K, hen he opimal invesmen sraegies a ime, I and R, exis. Moreover, he firm s opimal wealh is given by V = V I Z >d,z >d 2 } + V 2 I d2 >Z >d,z >d 3 } + V 3 I d >Z >d 2,Z >d 4 } +V 4 I d2 >Z >d,z <d 3 } + V 4 I d >Z >d 2,Z <d 4 } + V 4 I Z <d,z <d 2 } Opimizaion problem a ime 0 Now we consider he firs ime poin, ime 0. The oal coss in his period C 0 = c (L 0 ) + c 2 (I 0 ) + c 3 (R 0 ). Due o (4.) he presen value of he firm a ime 0 is given by V 0 (A 0, K 0, X 0 ) = max P 0 Y 0 c (L 0 ) c 2 (I 0 ) c 3 (R 0 ) + EV } F 0 ] L 0,I 0,R 0 R + + r = max X 0 K α( ε) L 0,I 0,R 0 R + 0 L β( ε) 0 A 0 ρ 0 L 0 ω 0 I 0 θ 0 R 0 + EV } ]. (4.6) + r

25 4.3. OPTIMIZATION PROBLEM AT TIME 0 2 We would like o find L 0, I 0, and R 0 which make V 0 reaches is maximum. Since EV ] is independen of L 0, we may rewrie (4.6) as V 0 (A 0, K 0, X 0 ) } = max X 0 K α( ε) L 0 R + 0 L β( ε) 0 A 0 ρ 0 L 0 + max I 0,R 0 R + I is easy o find ha L 0 = ] β( ε)x 0 K α( ε) 0 A 0, ρ 0 } EV ] + r ω 0I 0 θ 0 R 0. (4.7) which is obviously posiive. I remains o deal wih he second opimizaion problem } EV ] max I 0,R 0 R + + r ω 0I 0 θ 0 R 0. (4.8) Our goal is o find nonnegaive opimal soluions I 0 and R 0 o (4.8). Thus we differeniae he funcion U(I 0, R 0 ) := EV ] ( + r)ω 0 I 0 ( + r)θ 0 R 0 wih respec o I 0 and R 0, respecively, such ha EV ] = ω 0 ( + r) and EV ] = θ 0 ( + r). However, we do no know he acual value of V so far because i will be affeced by he range of he random variable Z. Therefore, he order of d, d 2, d 3 and d 4 is wha we should figure ou firs. We compare d, d 2, d 3 and d 4 wih one anoher and discover ha wo proporions deermine he relaion beween hem. (See Appendix A.) There are oally six possible cases for V. We will discuss he cases one by one. Case : There are wo condiions conforming wih his case: () ρ 2 > β( ε), α( ε)θ λa > ω δk and d 4 < d 2 < d < d 3. (2) ρ 2 < β( ε) and α( ε)θ λa > ω δk. Under hese wo condiions, V has he same form as V = V I Z >d } + V 3 I d >Z >d 2 } + V 4 I Z <d 2 }.

26 22 4. TWO-PERIOD MODEL Le where EV ] = d V f(z ) dz + G I 0 = EV ] ω 0 ( + r), (4.9) d d 2 V 3 f(z ) dz + where f is he probabiliy densiy funcion of Z. Then d2 V 4 f(z ) dz, G I 0 = d V f(z ) dz + d d 2 V 3 f(z ) dz + d2 V 4 f(z ) dz V +V 4 (d 2 ) V 3 (d 2 )]f(d 2 ) d 2 ω 0 ( + r)., V 3 and V 4 have similar forms such as ai I 0+c b where a, b and c are consans 0 o I 0 and b <. In addiion, lim V 4(d 2 ) V 3 (d 2 )] < 0. Then i is clear ha I 0 Hence, I 0 exiss such ha G I 0 = 0. Similarly, le lim I 0 GI 0 < 0. G R 0 = EV ] θ 0 ( + r) (4.0) = V f(z ) dz + d d V 3 f(z ) dz + d 2 d2 +V 3 (d ) V (d )]f(d ) d θ 0 ( + r). V 4 f(z ) dz Also, we obain lim R 0 GR 0 < 0. Thus, R0 exiss such ha G R 0 = 0. (See he horough proof in Appendix B.) We can use he same echnique o verify ha I 0 and R 0 exis in he oher five cases, so he deails of compuaion will be omied in he following cases. Case 2: ρ 2 > β( ε), α( ε)θ λa > ω δk, and d 2 < d 4 < d < d 3. Then V = V I Z >d } + V 3 I d >Z >d 4 } + V 4 I Z <d 4 }.

27 4.3. OPTIMIZATION PROBLEM AT TIME 0 23 By he similar argumen as in Case, we ge ha d lim I 0 GI 0 = lim I 0 d V f(z ) dz + d 4 V 3 f(z ) dz + +V 4 (d 4 ) V 3 (d 4 )]f(d 4 ) d } 4 ω 0 ( + r) V d lim R 0 GR 0 V 3 = lim f(z ) dz + f(z ) dz + R 0 d d 4 < 0. Then I 0 and R 0 exis. d4 < 0. d4 V 4 f(z ) dz V 4 f(z ) dz +V 4 (d 4 ) V 3 (d 4 )]f(d 4 ) d 4 + V 3 (d ) V (d )]f(d ) d ω 0 ( + r) } Case 3: Two condiions conform wih his case: () ρ 2 > β( ε), α( ε)θ λa > ω δk, and d 2 < d < d 4 < d 3. (2) ρ 2 = β( ε), and α( ε)θ λa > ω δk. Then V = V I Z >d } + V 4 I Z <d }. The relaive derivaives (4.9) and (4.0) in his case are given by G I 0 = G R 0 = V d f(z ) dz + V f(z ) dz + d θ 0 ( + r). d d V 4 f(z ) dz ω 0 ( + r), V 4 f(z ) dz + V 4 (d ) V (d )]f(d ) d I can be verified ha lim I 0 GI 0 < 0 and lim I 0 GR 0 < 0. Then I 0 and R 0 exis. Case 4: Two condiions conform wih his case: () ρ 2 > β( ε), α( ε)θ λa < ω δk, and d 3 < d < d 2 < d 4.

28 24 4. TWO-PERIOD MODEL (2) ρ 2 < β( ε), and α( ε)θ λa < ω δk. Then V = V I Z >d 2 } + V 2 I d2 >Z >d } + V 4 I Z <d }. The relaive derivaives (4.9) and (4.0) in his case are given by G I 0 = G R 0 = d 2 V f(z ) dz + d2 d V 2 f(z ) dz + +V 2 (d 2 ) V (d 2 )]f(d 2 ) d 2 ω 0 ( + r), V f(z ) dz + d 2 d2 V 2 f(z ) dz + d d d V 4 f(z ) dz V 4 f(z ) dz +V 4 (d ) V 2 (d )]f(d ) d θ 0 ( + r). I can be verified ha lim I 0 GI 0 < 0 and lim I 0 GR 0 < 0. Then I 0 and R 0 exis. Case 5: Two condiions conform wih his case: () ρ 2 > β( ε), α( ε)θ λa < ω δk, and d < d 3 < d 2 < d 4. (2) ρ 2 = β( ε), and α( ε)θ λa < ω δk. Then V = V I Z >d 2 } + V 2 I d2 >Z >d 3 } + V 4 I Z <d 3 }. The relaive derivaives (4.9) and (4.0) in his case are given by G I 0 = G R 0 = d 2 V f(z ) dz + d2 d 3 V 2 f(z ) dz + d3 V 4 f(z ) dz +V 2 (d 2 ) V (d 2 )]f(d 2 ) d 2 + V 4 (d 3 ) V 2 (d 3 )]f(d 3 ) d 3 ω 0 ( + r), V f(z ) dz + d 2 d2 V 2 f(z ) dz + d 3 d3 V 4 f(z ) dz +V 4 (d 3 ) V 2 (d 3 )]f(d 3 ) d 3 θ 0 ( + r). I can be verified ha lim I 0 GI 0 < 0 and lim I 0 GR 0 < 0. Then I 0 and R 0 exis.

29 4.4. EXAMPLE 25 Case 6: ρ 2 > β( ε), α( ε)θ λa < ω δk, and d < d 2 < d 3 < d 4. Then V = V I Z >d 2 } + V 4 I Z <d 2 }. The relaive derivaives (4.9) and (4.0) in his case are given by G I 0 = d 2 V f(z ) dz + d2 V 4 f(z ) dz G R 0 = +V 4 (d 2 ) V (d 2 )]f(d 2 ) d 2 ω 0 ( + r), V f(z ) dz + d 2 d2 V 4 f(z ) dz θ 0 ( + r). I can be verified ha lim I 0 GI 0 < 0 and lim I 0 GR 0 < 0. Then I 0 and R 0 exis. I remains a condiion which belongs o a special case. When α( ε)θ λa = ω δk, i is suiable for boh Case 3 and Case 6. The condiion here will induce ha d equals o d 2, and V = V I Z >d} + V 4 I Z <d}, where d = d = d 2. There would be no difference beween Case 3 and Case 6. In hese six cases, we confirm he exisence of opimal invesmen policy, I0 and R0. Then we are able o formulae he following proposiion. Proposiion 4.. The opimal capial invesmen policy in wo-period ime exiss. In summary, I and R exis a ime and ime Example I is no easy o find ou he definie forms of I 0 and R 0 here. We can only prove ha he soluion of he opimal invesmen exiss. Nex, we will ry o perform an

30 26 4. TWO-PERIOD MODEL example by subsiuing numerical values of parameers ino he model and o find he soluion of he opimal invesmen. Example 4.. We choose he hird case of hose six ones o perform and fix α = β = 2, ε = 2 3, = 3 and ω = ρ = θ = for = 0,, 2. Here we ignore no only he depreciaion in capial asses of he firm bu also he effec of he ineres rae, hus we have δ = λ = and r = 0. A ime 2, he value of he firm is V 2 (A 2, K 2, X 2 ) = } max X 2 K 6 2 L 6 2 A 3 2 L 2 L 2 = 5 ( ) 5 6 X 5 2 K A 2 5 2, where X 2 = X 0 expz 2 }, and we se X 0 =. Then he expeced value of V 2 a ime is EV 2 F ] = 5 ( ) } 5 K 5 2 A exp 5 Z. A ime, he value of he firm is } V (A, K, X ) = max X K 6 L 6 A 3 L I R + EV 2 F ] = max I,L,R I,L,R X K 6 L 6 A 3 L I R where X = X 0 expz 2 }. We can find L, Ī and R ha make V maximized: ( )6 L 5 6 = X 5 K 5 6 A 2 5, ( ) 3 Ī = 2 exp 3Z 5 } K, 6 6 ( ) 3 R = 4 exp 3Z 5 } A. 6 6 ( ) } } 5 (I + K ) 5 (R + A ) exp 6 5 Z, We only consider he siuaion ha Ī and R are nonnegaive. As we have discussed formerly, i depends on he range of Z, and here are four possible siuaions abou he range of Z.

31 4.4. EXAMPLE 27 Siuaion : When boh of Ī and R are posiive, we ake (L, I, R)=( L, Ī, R ), and we have wo following inequaliies abou he range of Z : Z > 3 log A +.32 = d, Z > 3 log K +.76 = d 2. Then he value of he firm is V (A, K, X ) = 5 6 Siuaion 2: ( ) 5 6 X 5 K 5 6 A ( ) 3 exp 3Z 6 } + K + A. 3 5 R is posiive, bu Ī is so small ha we ake I = 0. We can find he opimal R which is posiive and ge wo ranges of Z. Z > d, Z > 2 log A 6 log K +.64 = d 3. In his siuaion, he value of he firm is V 2 (A, K, X ) = 5 6 Siuaion 3: ( ) 5 6 X 5 K 5 6 A ( )4 3 3 exp 2Z 4 } + A. 6 5 Ī is posiive, bu R is so small ha we ake R = 0. We have I 0 > 0 and ge he following wo ranges of Z. Z > d 2, Z > 2 3 log K 3 log A = d 4. Then he value of he firm is V 3 (A, K, X ) = 5 6 ( ) 5 6 X 5 K 5 6 A ( ) 2 3 exp 6 2 Z 3 } A K.

32 28 4. TWO-PERIOD MODEL Siuaion 4: Z < d and Z < d 2, hen ake I = R = 0. The value of he firm is V 4 (A, K, X ) = 5 6 ( ) 5 6 X 5 K 5 6 A ( ) 5 K 5 A exp 5 Z 2 }. 25 In his case, we know ha d 2 < d < d 4 < d 3 and V = V I Z >d } + V 4 I Z <d }. Then we can find I0 and R0 from he maximizaion a ime 0: } V 0 (A 0, K 0, X 0 ) = max X 0 K 6 0 L 6 0 A 3 0 L 0 I 0 R 0 + EV ], I 0,L 0,R 0 where EV ] = d V f(z ) dz + d V 4 f(z ) dz. Noe ha A 0 and K 0 are given in he beginning. I is obviously ha L 0 exiss and L 0 = X K 5 0 A However, I 0 and R 0 need o be found from he simulaneous equaions: V d f(z ) dz + d V f(z ) dz + d d V 4 f(z ) dz =, V 4 f(z ) dz + V 4 (d ) V (d )]f(d ) d =. Though i is difficul o find he explici soluion, we can obain he numerical roos, which are A , K We know ha A = R 0 + A 0. If A 0, he invesmen in research of he firm before regular operaing, is oo much (ending o ), R 0 will be chosen as zero. Tha means he firm will suspend is invesmen in research and developmen a ime 0. I is similar o K 0 and I 0. If he equipmen of facory is qui enough, he firm will no end o pu capial in purchasing machinery equipmen or expanding he facory s scale.

33 4.4. EXAMPLE 29 We ry o subsiue some differen values of β and σ 2 ino he example and see wha changes will happen in A, K and V 0. Choose A 0 and K 0 o be fixed a 0.02 and ake β = 3, 2, 3 4 ; σ2 =, 4. Table (4.) shows he resuls. β /3 /2 3/4 σ K A V Table 4. We observe ha higher σ 2 leads exremely higher value of V 0. This resul will be verified once again in Chaper 5. In he nex chaper, we will ry o exend he observaion period from discree ime o coninuous ime and see if here is any disinc propery abou he opimal soluion. On he occasion, we will also clarify he effec caused by uncerainy σ 2.

34 CHAPTER 5 Coninuous-Time Model Now we consider a coninuous-ime model wih a demand shifer X a ime. The sochasic process (X ) follows a geomeric Brownian moion wih drif σ, i.e., dx X = σ dz and X 0 = x, (5.) where (Z ) is a sandard Brownian moion. The firm akes I o expand facory faciliies and invess R in developing new echnology a ime. The equipmen of he facory and accomplishmens in research of he firm depreciae a consan raes δ and λ, respecively. The laws of moion for capial and research are dk = (I δk ) d, (5.2) da = (R λa ) d. (5.3) The fundamenal value of he firm a ime is given by maximizing he expeced presen value of is cash flow wih he discoun rae r; explicily, V is given by ] V (A, K, X ) = max E P s Y s c (L s ) c 2 (I s ) c 3 (R s )]e r(s ) ds I,L,R F,(5.4) which is subjec o he consrains in (5.2) and (5.3). Here he cos funcions of capial and research are considered convex such ha c 2 (I ) = ω I u and c 3 (R ) = θ R v where u and v are consans. In his chaper, we will separae i ino wo cases: he cos funcion is sricly convex and is linear. 3

35 32 5. CONTINUOUS-TIME MODEL 5.. Convex cos funcions In his secion, we consider c 2 (I ) and c 3 (R ) o be convex. In oher words, we ake u, v >. V (A, K, X ) saisfies he following Bellman equaion: rv (A, K, X ) = max P Y c (L ) c 2 (I ) c 3 (R ) + EdV F } ]. (5.5) I,L,R d The value of he firm is a funcion concerned wih variables A, K and X, hen apply Iô formula and Equaions (5.), (5.2) and (5.3) o obain ha dv (A, K, X ) = σx dz + V A (R λa )d + V K (I δk )d + 2 X2 σ 2 V XX d. Then Equaion (5.5) can be rewrien as rv (A, K, X ) = (5.6) P Y ρ L ω I u θ R v + V A (R λa ) + V K (I δk ) + 2 } X2 σ 2 V XX. max I,L,R The exhausive deriving process from (5.4) o (5.6) is shown in Appendix C. Differeniaing he righ-hand side of he above equaion wih respec o I, L and R, respecively, yields V K = ω ui u, (5.7) V A = θ vr v, (5.8) ] L β( ε)x K α( ε) A =. ρ Subsiue hese hree equaions back ino (5.6), hen we have rv (A, K, X ) = hx K α( ε) A + (u )ω I u + (v )θ R v δk V K λa V A + 2 X2 σ 2 V XX, (5.9)

36 ] β( ε) β( ε) where h = β( ε)]. ρ 5.. CONVEX COST FUNCTIONS 33 We can find a paricular soluion of V (A, K, X ), and i is wrien as V (A, K, X ) = h r + δα( ε)+λ σ2 β( ε) 2] 2 X K α( ε) A + ω (u ) r From Equaion (5.7), we have where E = ( ω u K = E X ε A ε And from Equaion (5.2), we know ha I u + θ (v ) R v. r I ( u)] ε, (5.0) hα( ε) r β( ε)] + δα( ε) + λ σ2 β( ε) 2] K = 0 )] ε. e δ(s ) I s ds + e δ K 0. (5.) Combine (5.0) and (5.) and hen use Iô formula. I saisfies he following equaion I d = δe X ε A ε + E ε X ε + E X ε A ε + E 2ε I ( u)] ε d + E ε X A ε I ( u)] ε ( u) β( ε)] ε ) X ε 2 A ε ε ( ε ε A ε I ( u)] ε (R λa ) d I ( u)] From mahemaical poin of view, we should wrie (5.9) as rv (A, K, X ) = hx ( v θ K ) α( ε) A + I ( u)] ε ( u σ 2 d. ω ) d u ( u)v v v ( v)v v A λa V A + 2 X2 σ 2 V XX. u u K σdz δk V K However, we have no much informaion abou his nonlinear parial differenial equaion (exisence, uniqueness,...). Hence, we follow here he mehod inroduced by Abel (983).

37 34 5. CONTINUOUS-TIME MODEL (5.8): Similarly, we can find R saisfying he following equaion by Equaion (5.3) and R d = λe 2 X where E 2 = E 2 K α( ε) + β( ε) X + E 2 α( ε) β( ε) X + E 2 X + E 2X K 2 ( θ v ( v)] R α( ε) α( ε) K K 2 β( ε) ] K d α( ε) ( v)] R α( ε) ( v)] R ( v) β( ε)] β( ε) ( v)] R h r β( ε)] + δα( ε) + λ σ2 β( ε) 2] R σdz (I δk ) d ( v)] d ( ) β( ε) σ 2 d, )]. I is clear ha E and E 2 are increasing funcions of σ 2. An increase in σ 2 will also make he demand shifer X increase. However, for a given level of he curren demand shock X, increased σ 2 will lead o an increase in he opimal invesmen; uncerainy has a posiive effec on invesmen for capial I and research R. Moreover, we can observe ha if raising he value of σ 2, he presen value of he firm V (X, A, K ) will also increase for a given level of X. This resul is he same as he performance of calculaion on Table (4.) in Secion Linear cos funcions In he las secion we have go he opimal invesmen when cos funcions of capial and research are convex such as c 2 (I ) = ω I u and c 3 (R ) = θ R v, ha is u, v >. From he srucures of I and R, we discover ha i could be an ineresing siuaion when u = v =, i.e., he cos funcions of I and R are linear. We wonder wha will happen if he cos funcions are linear o I and R. Now we consider a

38 5.2. LINEAR COST FUNCTIONS 35 special case for u = v =. The value of he firm a ime would be ] V (X, A, K ) = max E (P s Y s ρ s L s ω s I s θ s R s )e r(s ) ds L,I,R R F. + The maximizaion is also subjec o he consrains in (5.2) and (5.3). Wih he similar procedure, he value funcion will be derived o be he following opimaliy condiion rv (A, K, X ) = max P Y ρ L ω I θ R + V A (R λa ) + V K (I δk ) + 2 } X2 σ 2 V XX L,I,R R + = max L R + P Y ρ L } + max I R + (V K ω )I } + max R R + (V A θ )R } λa V A δk V K + 2 X2 σ 2 V XX. (5.2) I is usual o assume ha he invesmens I and R are bounded. However, i is ineresing o consider he case where I and R are unbounded Invesmen wihou consrains. I is easy o ge ha L β( ε)x K α( ε) A = ρ ]. Now we firs discuss he case where he maximum value V in (5.2) is finie, hen he boh opimizaion values max I R + (V K ω )I } and max R R + (V A θ )R } are finie, which implies ha V K ω and V A θ. (5.3) This means ha he uni cos of capial and research invesmens is larger han heir marginal profi, respecively. I is obviously ha if V K < ω, he opimal I = 0, and if V K = ω, he opimal I is no unique, bu in boh case, max (V K ω )I } = 0. I R +

39 36 5. CONTINUOUS-TIME MODEL Similarly, we can ge max (V A θ )R } = 0. R R + The opimaliy condiion (5.2) hen becomes rv (A, K, X ) = hx K α( ε) A δk V K λa V A + 2 X2 σ 2 V XX. (5.4) This differenial equaion is much simpler han ha wih convex cos funcions and we can find ha he soluion o (5.4) is given by V (A, K, X ) = E 3 X α( ε) K A, (5.5) where E 3 = 2 β( ε)] 3 2r β( ε)] 2 + 2αδ( ε) + λ] β( ε)] β( ε)σ 2 β( ε) Of course, we sill have o check if he condiions (5.3) hold or no. Observing (5.3) and V given by (5.5), we know ha if he uni cos of he capial, research and labor invesmen is oo expensive, or alernaively, he iniial invesmens of capial and research, K 0 and A 0, are oo small, he firm will no inves much in capial and research. The only opporuniy for he firm o increase is invesmen in A and K is when he uni price of capial, research or labor invesmen sinks such ha V K = ω or V A = θ. As for he general case where V is no necessary o be finie, we have o consider he siuaion V K > ω or V A > θ. I is easy o see ha under hese siuaions, he firm should inves as much as possible in his iem, and hen i resuls in V =. However, his case does no make much sense, since in pracice all of he invesmen are limied. Thus, i is reasonable o consider he invesmen wih consrains. ρ ] β( ε) Invesmen wih consrains. An ineresing and more pracical case is o consider he invesmen wih consrains. More explicily, we consider he case, where 0 I Ī and 0 R R for some posiive consans Ī and R. Similarly

40 5.2. LINEAR COST FUNCTIONS 37 o he above argumen, we ge he firm s opimal sraegies are given by 0, if V I K < ω, 0, if V = and R A < θ, = Ī, if V K ω, R, if V A θ. This implies ha he firm eiher invess nohing or invess so much as he can. Thus, he cumulaive invesmens are given by K = e δ (K 0 + Ī A = e λ (A 0 + R 0 and he opimal condiion (5.2) can be wrien as rv (A, K, X ) = hx K α( ε) 0 ) e δs I VK ω s} ds, ) e λs I VA θ s} ds, A + Ī(V K ω )I VK ω } + R(V A θ )I VA θ } λa V A δk V K + 2 X2 σ 2 V XX. Figure 5. shows one possible sample pah of I which behaves like a sep funcion. Figure 5.2 shows he corresponding capial invesmen of he sample pah given in Figure 5.. We can see ha in he inerval where I = 0, he corresponding K is convex decreasing; in he inerval where I = Ī, he capial invesmen K is eiher increasing or decreasing wih a smooher decay (which depends on he value of Ī, K 0, and δ). I is clear o see he posiive effec of uncerainy on V (A, K, X ). We can also find he opimal invesmen and analyze he soluion wih respec o he parameers, jus like he analysis for he general case. So he auology is omied here.

41 38 5. CONTINUOUS-TIME MODEL I _ 2 3 Figure 5.. One sample pah of I K Figure 5.2. The corresponding K of Figure 5.

42 CHAPTER 6 Conclusion In our analysis for discree ime, we sysemized oal possible condiions o six cases. Even a sligh change of a single parameer will drive he case o swich o anoher one, hen he form of he opimal invesmen and he presen value of he firm will be compleely differen. The jump is fierce. The model we use does no differ from Abel s model in 983 oo much. The soluion of a firm s fundamenal value and he effec caused by uncerainy are similar as well. The added facor research complicaes he soluion, bu he basic resul does no change. The effec of uncerainy on invesmen for capial and research is posiive. The firm will inves more on capial and research if demand uncerainy increases, hen he firm value will be raised. This is possible in realiy, for he firm migh inves more capial and make more effor on research o deal wih he unexpeced flucuaion of oupu demand in he marke. Any sudden inciden could induce a violen aack on he operaion of each firm. Besides, we conclude ha given he curren demand shifer, increased uncerainy will also make he presen value of a firm raise. This resul can be observed no only from he numerical appearance of examples bu also from he feaure of he general formula. However, he opimal invesmen has a high degree of complexiy so ha he explici form of he soluion is no easy o find ou. And his siuaion makes he analysis for effecs of some parameers on he opimal invesmen policy and he presen value of a firm become much more difficul. I is a obsacle remaining o overcome in he fuure. In addiion, recen sudies on his opic have highlighed he imporance of irreversibiliy for invesmen. I would be a feasible direcion for our sudy. 39

43 APPENDIX A Differen Forms of V in All Condiions Firs of all, we would like o find ou he order of d, d 2, d 3 and d 4. We have d = ε σ d 2 = ε σ d 3 = log(λa ) + β( ε) σ log( + r) + α( ε) σ ( ε)β( ε) + σ β( ε)] log ρ 2 β( ε) σ log x + σ σ 2 β( ε)]. + σ log θ log(δk ) + β( ε) σ log( + r) + log α( ε) + ε σ ω α( ε) + ε σ log θ log ω α( ε) + ( ε)β( ε) σ β( ε)] log ρ 2 β( ε) σ log x + σ σ 2 β( ε)]. β( ε) log(λa ) σ β( ε) + log θ β( ε) + σ σ d 4 = ε σ log(δk ) σ log(λa ) + α( ε) σ log β( ε) σ log(δk ) + β( ε) σ log( + r) ρ 2 β( ε) σ log x + σ σ 2 β( ε)]. log( + r) + β( ε) ρ 2 + log σ β( ε) σ log x + σ σ 2 β( ε)]. β( ε) σ log ω α( ε) Compare d, d 2, d 3 and d 4 wih one anoher by subracion: d d 2 = ε σ log α( ε)θ (λa ) ω δk. α( ε) d d 3 = log α( ε)θ (λa ) β( ε) ε + β( ε)] ρ 2 + log σ ω δk σ β( ε)] β( ε). d d 4 = ε σ log α( ε)θ (λa ) ω δk + β( ε) ε + β( ε)] σ β( ε)] 4 log ρ 2 β( ε).

44 42 A. DIFFERENT FORMS OF V IN ALL CONDITIONS d 2 d 3 = + β( ε) σ log α( ε)θ (λa ) ω δk β( ε) ε + β( ε)] ρ 2 + log σ β( ε)] β( ε). d 2 d 4 = σ log α( ε)θ (λa ) ω δk d 3 d 4 = β( ε) σ + log α( ε)θ (λa ) ω δk. β( ε) ε + β( ε)] σ β( ε)] We noice ha wo proporions decide he order,which are All possible cases are as following:. ρ 2 > β( ε), α( ε)θ λa > ω δk and assume d 4 < d 2. We have d 4 < d 2 < d < d 3, hen V = V I Z >d } + V 3 I d >Z >d 2 } + V 4 I Z <d 2 }. 2. ρ 2 > β( ε), α( ε)θ λa > ω δk and assume d 2 < d 4 < d. We have d 2 < d 4 < d < d 3, hen V = V I Z >d } + V 3 I d >Z >d 4 } + V 4 I Z <d 4 }. 3. ρ 2 > β( ε), α( ε)θ λa > ω δk and assume d 4 > d. We have d 2 < d < d 4 < d 3, hen V = V I Z >d } + V 4 I Z <d }. 4. ρ 2 > β( ε), α( ε)θ λa < ω δk and assume d 3 < d. We have d 3 < d < d 2 < d 4, hen V = V I Z >d 2 } + V 2 I d2 >Z >d } + V 4 I Z <d }. 5. ρ 2 > β( ε), α( ε)θ λa < ω δk and assume d < d 3 < d 2. We have d < d 3 < d 2 < d 4, hen V = V I Z >d 2 } + V 2 I d2 >Z >d 3 } + V 4 I Z <d 3 }. 6. ρ 2 > β( ε), α( ε)θ λa < ω δk and assume d 3 > d 2. log ρ 2 β( ε). ρ 2 β( ε) and α( ε)θ (λa ). ω δk

45 A. DIFFERENT FORMS OF V IN ALL CONDITIONS 43 We have d < d 2 < d 3 < d 4, hen V = V I Z >d 2 } + V 4 I Z <d 2 }. 7. ρ 2 > β( ε) and α( ε)θ λa = ω δk. We have d 4 = d 3 > d 2 = d, hen V = V I Z >d =d 2 } + V 4 I Z <d =d 2 }. I is he same as 3. or ρ 2 < β( ε) and α( ε)θ λa > ω δk. We have d > d 2 > d 4 and d 3 > d 4, hen V = V I Z >d } + V 3 I d >Z >d 2 } + V 4 I Z <d 2 }. I is he same as. 9. ρ 2 < β( ε) and α( ε)θ λa < ω δk. We have d 2 > d > d 3 and d 4 > d 3, hen V = V I Z >d 2 } + V 2 I d2 >Z >d } + V 4 I Z <d }. I is he same as ρ 2 < β( ε) and α( ε)θ λa = ω δk. We have d 3 = d 4 < d = d 2, hen V = V I Z >d =d 2 } + V 4 I Z <d =d 2 }. I is he same as 3. or 6.. ρ 2 = β( ε) and α( ε)θ λa > ω δk. We have d 2 < d < d 3 < d 4, hen V = V I Z >d } + V 4 I Z <d }.

46 44 A. DIFFERENT FORMS OF V IN ALL CONDITIONS I is he same as ρ 2 = β( ε) and α( ε)θ λa < ω δk. We have d < d 3 < d 2 < d 4, hen V = V I Z >d 2 } + V 2 I d2 >Z >d 3 } + V 4 I Z <d 3 }. I is he same as ρ 2 = β( ε) and α( ε)θ λa = ω δk. We have d = d 2 = d 3 = d 4 = d, hen V = V I Z >d} + V 4 I Z <d}. I is he same as 3. or 6. Afer caegorizing, V (A, K, X ) has six possible forms in oal.

47 APPENDIX B Proof of Exisences of I 0 and R 0 Le G I 0 = d V f(z )dz + d d 2 V 3 f(z )dz + d2 V 4 f(z )dz +V 4 (d 2 ) V 3 (d 2 )]f(d 2 ) d 2 ω 0 ( + r). The way we use o prove he exisence of he opimal invesmen I 0 is o verify ha he value of G I 0 V = V 3 and V 4 is negaive as I 0 ends o infiniy. We have = α( ε) X (I 0 + δk 0 ) = α( ε) X (I 0 + δk 0 ) ε A ε A δα( ε) ε + δ(i 0 + δk 0 )] (λa ) + r } 2σ(z σ) β( ε)] + σ 2 exp. 2 β( ε)] 2 Obviously, V lim f(z )dz + I 0 d d d 2 V 3 f(z )dz + where he value of c is finie. Furhermore, we have d2 V 4 (d 2 ) = β( ε)] δ(i 0 + δk 0 )] (λa0 ) ( ) ] ( ε )β( ε) θ ρ 2, β( ε) 45 β( ε) ρ β( ε) ρ 2 β( ε) ρ ] β( ε) ] β( ε) + ω δ, ] β( ε) x } V 4 f(z )dz = c, ω α( ε) ]

48 46 B. PROOF OF EXISTENCES OF I 0 AND R 0 and V 3 (d 2 ) = ε δ(i 0 + δk 0 )] ε (λa0 ) ε +ω δ(i 0 + δk 0 ). We observe ha ω α( ε) ] ε ε ( ) ] ( ε )β( ε) θ ε ρ ε 2 β( ε) lim V 4(d 2 ) V 3 (d 2 )} =. I 0 Then he proof of lim I 0 GI 0 < 0 is done. where V The par of proving he exisence of R 0 is similar. G R 0 = = X V = X and V 4 = X V f(z )dz + d d V 3 f(z )dz + d 2 +V 3 (d ) V (d )]f(d ) d θ 0 ( + r), α( ε) K α( ε) K d2 λ(r 0 + λa 0 )] β( ε) λ(r 0 + λa 0 )] β( ε) +λ ( + r) β( ε) ε λ(r 0 + λa 0 )] α( ε) ε } 2σ(Z σ) β( ε)] + σ 2 exp 2 β( ε)]ε α( ε) K ω ρ ρ ] α( ε) ε λ(r 0 + λa 0 )] β( ε) ρ V 4 f(z )dz ] β( ε) + θ λ, ] β( ε) β( ε) ρ 2 ] β( ε) ] β( ε) ε + λ ] β( ε) α( ε) + r K λ(r 0 + λa 0 )] β( ε) x ρ 2 } 2σ(Z σ) β( ε)] + σ 2 exp. 2 β( ε)] 2 Also, we know ha V lim f(z )dz + I 0 d d V 3 f(z )dz + d 2 d2 } V 4 f(z )dz = c 2, x ε

49 where he value of c 2 is finie. V 3 (d ) = λ(r 0 + λa 0 )] V (d ) = λ(r 0 + λa 0 )] B. PROOF OF EXISTENCES OF I 0 AND R 0 47 ( θ ( θ +λθ (R 0 + λa 0 ). ) ρ 2 β( ε) ) ρ 2 β( ε) ] β( ε) ε +β( ε)] ε] ] β( ε) ε +β( ε)] (ε )] I is verified ha lim R0 V 3 (d ) V (d )} =. Therefore, The soluion of he opimal invesmen exiss. ε + ω δk, (ε ) + ω δk lim R 0 GR 0 < 0.

50 APPENDIX C Derivaion of Equaion (5.6) The maximizaion subjec o he consrains in (5.2) and (5.3) is ] V (A, K, X ) = max E (R s C s )e r(s ) ds I,L,R F, where R s = P s Y and C s = c (L s )+c 2 (I s )+c 3 (R s ) = ρ s L s +ω s I u s +θ s R v s. Muliplying e r o boh sides of he above equaion yields ] e r V (A, K, X ) = max E (R s C s )e rs ds I,L,R F. (C.) In order o make he following equaions look simpler and clearer, we will abbreviae Equaion (C.) as where e r V = max E I,L,R ] f(s)ds F, f(s) = (R s C s )e rs. Since e r V = max E I,L,R = max I,L,R E = max I,L,R E + + ] f(s)ds F f(s)ds + + ] f(s)ds F f(s)ds + e r(+ ) V + F ], we have + ] max E f(s)ds + e r(+ ) V + e r V F = 0. (C.2) I,L,R 49

51 50 C. DERIVATION OF EQUATION (5.6) Divide boh sides of (C.2) by, we have + f(s)ds max E + e r(+ ) V + e r V I,L,R ] F = 0, which implies Ee r(+ ) V + e r V F ] = max I,L,R E + f(s)ds ] F. (C.3) Noe ha max I,L,R E + f(s)ds ] F By using Iô formula o obain max f() = max (R C )e r}. 0 I,L,R I,L,R dv = V A da + V K dk + V X dx + 2 V AA(dA ) V KK(dK ) V XX(dX ) 2 + V AK (da )(dk ) + V KX (dk )(dx ) + V XA (dx )(da ) = V A (R λa ) d + V K (I δk ) d + V X σx dz + 2 X2 σ 2 V XX d, which is compued according o he rules (d) 2 = (d)(dz ) = 0 and (dz )(dz ) = d. By Equaion (C.3), we obain and max (R C )e r} E re r V + e r V F ], I,L,R Ee r(+ ) V + e r V F ] E re r V + e r F ]. Divide boh sides of he above equaion by e r, hen max R C } I,L,R E rv + V F ] E rv + V A (R λa ) + V K (I δk ) + 2 ] X2 σ 2 V XX F = rv + V A (R λa ) + V K (I δk ) + 2 X2 σ 2 V XX,

52 C. DERIVATION OF EQUATION (5.6) 5 where EdZ F ] = 0 since Z is a maringale. Then we obain he opimaliy condiion rv = max I,L,R P Y ρ L ω I u θ R v + V A (R λa ) + V K (I δk ) + 2 X2 σ 2 V XX }, which is equivalen o Equaion (5.6) in he main ex. Remark C.. Equaion (5.6) is named he Bellman equaion.

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