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A model for optimal infrastructure investment in boom towns Poklitar, Joanne Carol 1980

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A MODEL FOR OPTIMAL INFRASTRUCTURE INVESTMENT IN BOOM TOWNS by JOANNE CAROL POKLITAR ( B . A . , Carleton University, 1975) A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in! THE FACULTY OF GRADUATE STUDIES Institute of Applied Mathematics and Statistics Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1980 (c) Joanne Carol Poklitar, 1980 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Joanne Carol Poklitar Department of Mathematics The University of British Columbia 2075 Wesbrook Place Vancouver, V6T 1W5 Canada Date April 25, 1980 A B S T R A C T A Model for Optimal In f rastructure Investment in Boomtowns A linear model to determine the optimal policy for investment in social in f ras t ruc ture is formulated and its solution is obtained us ing the Maximum Pr inc ip le . T h e unique solution is character ized by a-b a n g - b a n g c o n t r o l , with only one interval of investment in social cap i ta l , • and the endpoints of this interval can be numerically determined, g iven values for the parameters of the model. A general ization of the model which allows instantaneous jumps in the level of social capital is also ana lyzed , and the solution to the modified problem is shown to be a uniquely determined impulse c o n t r o l . T h e final extension of the model allows us to determine an upper bound for the optimal time h o r i z o n . iii T A B L E OF CONTENTS Page A B S T R A C T • ii LIST OF TABLES iv LIST OF FIGURES v ACKNOWLEDGEMENTS vi i Section 1. INTRODUCTION 1 2. THE BASIC MODEL * 3. SOLUTION TO THE BASIC PROBLEM 11 4. QUALITATIVE BEHAVIOUR OF THE SOLUTION 31 5. EXTENSION TO UNBOUNDED CONTROL SET 43 6. OPTIMIZATION OF THE TIME HORIZON 56 7. CONCLUSION 69 BIBLIOGRAPHY 70 APPENDIX A - THE ABNORMAL FORM OF THE CONTROL PROBLEM 71 APPENDIX B - SELECTING AN INITIAL SET OF PARAMETER VALUES 73 APPENDIX C - METHOD OF NUMERICAL SOLUTION FOR THE BASIC MODEL , . 7 8 APPENDIX D - PROOF OF LEMMA 1 81 iv L IST O F T A B L E S Table Page I. Behaviour of Solution With Ghanges in Al ternate Parameters 34 II. Behaviour of Solution as I., Increases 35 M III. Sensi t iv i ty of Solution to Changes in T and m. With Negative t (y) 37 V LIST O F F I G U R E S F igure Page 1. T h e Adjoint Funct ion X ( t;y) for Al ternate S igns of X (T;y) 17 2. T h e Adjoint Funct ion X ( t;y) for Increasing Values of y 19 — (5 t 3. C u r v e s e and A.(t;y) for V a r y i n g y 19 m (St 4. C u r v e s X ( t;y), X ( t;y), and e , for a G iven y . . 25 5. Optimal C u r v e X m ( t ; y m ) is T r a p p e d Between X m ( t ; u i 1 ) and X m ( t ; y 2 ) 27 6. Adjoint C u r v e X ( t ; y ^ for Increasing n 48 * 7. Switching Times for T = A 2 60 8. Cost C u r v e s f and g with Al ternate Benefit C u r v e s B B_ , and B 66 VI Figure Page 9. G r a p h of K 83 10. Tra jectory K(t) Determines p: (a) p = 0; (b) p > 0 86 11. Const ruct ion of K when K has more than ct One Jump 90 vi i A C K N O W L E D G E M E N T I would like to give special thanks to my a d v i s o r , Professor Col in W. C l a r k , for b r i n g i n g this problem to my at tent ion, and for his guidance and suppor t throughout the writ ing of my thes is . I would also like to thank the other members of my committee. Professors F r e d Wan and Don L u d w i g , for their helpful comments and s u g g e s t i o n s , and Professors Frank C larke and Ulr ich Haussmann for their advice on certain technical detai ls . For their f inancial suppor t d u r i n g my graduate s tud ies , I am indebted to the National Science and Engineer ing Research Counci l of C a n a d a , and the Department of Mathematics at U . B . C . T o M r s . Maryse E l l i s , I am grateful for her eff icient work in t yp ing this thes is . Las t , but far from least, I wish to thank my f iance, David Moloney, whose constant encouragement and expert ise in economics have been invaluable to me in completing my thes is . 1 1. I N T R O D U C T I O N A n isolated town or community close to the site of a p r o -posed mining or power-generat ing faci l i ty rarely has the social in f ras t ruc ture needed to suppor t the large influx of labour requi red d u r i n g the construct ion phase of the fac i l i ty . T h e r e f o r e , this "boom" p e r i o d , which is character ized by large increases in populat ion, must be accompanied or preceded by some growth in the level of social capi ta l . Health c a r e , educat ion , sani tat ion, serv iced l and , e tc . must all be suppl ied in order to suppor t the town's populat ion. Once construct ion of the facil i ty has been completed, however , the town typical ly exper iences a drast ic population dec l ine , due to the fact that the labour force requi red for operation of the facil i ty is usual ly much smaller than that requi red d u r i n g cons t ruc t ion . T h u s the need for provis ion of the h igher level of serv ices is restr ic ted to the construct ion phase of the project . Determining the optimal investment in social in f ras t ruc ture in boomtowns provides an excellent problem to which control theory can be app l ied . What makes the in f ras t ruc ture investment decisions interest ing is the fact that this investment i s , for the most par t , i r reve rs ib le . Once in p lace, schools , hospi ta ls , roads and sewage systems a r e , in the main, not easily moveable, so that capital 2 invested in them is often not recoverable through r e s a l e . 1 T h e r e f o r e , the decisions which face the boomtown community are important ones . Over - inves tment in social capital d u r i n g the relat ively short boom period can result in substantial idle stocks when the population settles down to its long run post -boom. Such a misallocation of cap i ta l , which is observed in many boomtowns i s , of c o u r s e , socially undes i rab le . Th is problem is of some interest , then , as a pract ical matter of publ ic po l icy , and as a suitable application of control techn iques . T h e problem of determining the optimal investment strategy has been prev ious ly addressed by Cummings and Schulze [5]. T h e i r mathematical formulation of the problem, however , appears to be too complex to permit a complete analytical solut ion; consequent ly , we need to f ind a simpler form of model for which more complete analytical results can be obta ined. Simplif ication of a model, achieved by imposing addit ional assumptions or const ra in ts , usual ly resu l ts , of c o u r s e , in a greater depar ture from the real wor ld . However , there is some justif ication for s tudy ing simpler models for which the solution can be ful ly spec i f i ed , in that subsequent analysis can sometimes p r o -vide some insight for possible solutions of more complex models. T h e " i r revers ib i l i ty" aspect of investment is not unique to social cap i ta l . T h i s problem is also encountered to some degree in the investment of capital assets to exploit certain natural resource deposi ts . For example, harvest ing a stock of f ish requires a decision on the level of investment in a f ish ing f leet. For a general treatment of this problem, see C l a r k , C l a r k e , and Munro [ 3 ] . 3 In this paper , t h e n , we s tudy a simplif ied vers ion of the model formulated by Cummings and S c h u l z e . T h e problem of determining the optimal investment strategy is posed as a linear optimal control problem, and a unique solution is obtained us ing the Maximum Pr inc ip le . O u r simplif ied model permits an easy character izat ion of the optimal time horizon for the construct ion project . T h i s question was not considered by Cummings and S c h u l z e . T h e organization of the paper is as follows: in Section 2 we br ie f ly d iscuss the Cummings and Schulze model and show how our model is der ived from it; Section 3 contains the solution to the new optimal control problem; a qualitat ive analysis of the solut ion, i l lustrated by numerical resu l ts , is presented in Section 4; in Section 5, we solve a more general vers ion of our model which extends the set of admissible controls to impulse investment policies-, f ina l ly , in Section 6, we d iscuss the problem of extending the model to determine the optimal time h o r i z o n . 2. T H E B A S I C M O D E L T h e problem posed by Cummings and Schulze [5] centers a round the construct ion of a large energy extract ion /convers ion facil i ty near a small town. Const ruct ion of the facil i ty requires high levels of labour , which must be "imported" from ne ighbour ing communities, and the inf lux of labourers and their families necessitates investment in social i n f r a s t r u c t u r e . Determination of the cor rec t level of in f ras t ruc ture inves t -ment revolves a round a simple " t rade-of f" . On one h a n d , it is desirable to maintain a h igh level of per capita in f ras t ruc ture (or , equ iva lent ly , a h igh in f ras t ruc tu re / l abour ratio) throughout construct ion of the fac i l i ty , because this will tend to decrease the h igh wage rate that incoming labour will otherwise demand. On the other h a n d , in f ras t ruc ture is c o s t l y , and the labour force requi red to operate the faci l i ty after its completion is usual ly much smaller than the level requ i red to complete the construct ion on time, so that , d u r i n g the operational phase , a smaller amount of social capital is suff ic ient to maintain a reasonable per capita in f ras t ruc ture level . S ince inves t -ment in social capital is i r reve rs ib le , large amounts of social i n f r a -s t ructure put in place d u r i n g the boom period may be redundant when construct ion has been completed. T h e problem, t h e n , is one of choosing how much labour to u s e , and how much investment in in f ras t ruc ture to make d u r i n g the 5 construction phase so that the total costs of the project (wages plus social capital costs) will be minimized, in accordance with certain physical constraints. The earliest point in time at which construction of the facility may begin is taken as time t = 0, and the facility must be operational by a fixed time T . Construction of the facility is modelled by the equation c = f (L ) , o ( t a . . . . . ( 2 .D where C(t) is the' level of construction completed at time t, L(t) is the amount of labour used in construction at time t, and f(L) is a concave production function. Of course, the amount of labour used at any time is non-negative, so L ( t ) > 0 Y 0 ( t s< T . . . . .(2.2) Initially, no construction has been done on the facility, so C(0) = 0 . . . .(2.3) and if Cj denotes the specified level of construction for the completed facility, then C ( T ) = C T . . . . .(2.4) Increases in the stock of social capital, K(t ) , are determined by the equation K = I , 0 i \ i J , . . . .(2.5) 6 where I (t) is the rate of investment in social capital at time t. T h e i r revers ib i l i ty of investment in in f ras t ruc ture is enforced by the constra int It is assumed that before the project is s ta r ted , there is some n o n -negative level of in f ras t ruc ture K(0) at the site of the proposed faci l i ty . to minimize c o s t s , subject to the constra ints in (2.1) to (2 .6 ) . Most of the costs expl ic i t ly stated in the model are costs which are incur red d u r i n g the construct ion phase; these include wage payments, W ( K , L ) , for labour used in construct ion of the fac i l i ty , investment 1, in social i n f r a s t r u c t u r e , and maintenance c o s t s , M ( K ) , of ex is t ing in f ras t ruc ture In add i t ion , there may be costs F ( K ( T ) ) which are incur red d u r i n g the ^operational phase of the fac i l i ty , but which are dependent only on the amount of in f ras t ruc ture in place at the end of the const ruct ion phase . (For example, F could represent maintenance costs for in f ras t ruc ture over the operational life of the fac i l i ty . ) Assuming that the decision maker will minimize the present value of these c o s t s , his objective functional will be Kt) > 0 0 i t i T . . (2.6) T h e decision maker responsible for p lanning the project wants T minimize 0 . ( 2 . 7 ) where 6 is the posit ive rate of d iscount . 7 T h e model posed by Cummings and Schulze [5] is the 2 general model descr ibed above , in (2.1) to (2.7), with specif ic functional forms assumed for f, W, M, and F. T h e product ion funct ion f. is def ined as HC) = \J s for some constant p £ ( o j ) . . . .(2.8) T h e wage funct ion is also non- l inear in both K and L: W ( K , L ) - 3 • L , . . . .(2.9) ( K / L ) 1 where 3 and n are posit ive constants . T h e wage rate B / f K / L ) 1 1 ref lects the t rade-of f which is assumed to exist between wages and the i n f r a -s t r u c t u r e / l a b o u r rat io. The model abstracts from depreciat ion of the stock of social capital by assuming that with a rate of unit maintenance costs m per p e r i o d , capital stocks do not deter iorate. T h u s mainten-ance of ex ist ing in f ras t ruc ture is given by M ( K ) = m K . . . . . . . . (2.10) Fina l ly , the terminal funct ion F is def ined to include total wages paid to labour d u r i n g the operational phase of the fac i l i ty , plus maintenance costs for social cap i ta l . T h e rate of labour requi red to operate the 2 In their model, Cummings and Schulze expl ic i t ly allow for " f r o n t - e n d " investments, which increase the level of social capital from K(0~) to K(0 +) at t = 0. T h i s instantaneous jump in the level of infra s t ruc ture is simply the result of an impulse control (or 6 - funct ion) at t = 0, which may or may not be al lowed, depending on the control set for investment speci f ied in the model. facil i ty is assumed to be f ixed at L = L , so that cO F (K (T)) = j e 8 t { w ( K ( T ) , L) t mK(T)}dt W | H ( T ) L ) t m K ( T ) J ( 2 . i i ) T -ST _L e 5 Subst i tut ion of Equations (2.8) to (2.11) into the general model results in a nonlinear optimal control problem, with two state var iab les , C and K, and two control var iab les , L and I. Such problems are usual ly di f f icul t to solve analyt ical ly and this one appears to be no except ion . Cummings and Schulze [5] der ive some necessary c o n -dit ions for a solut ion, but these are by no means a complete spec i f ica -tion of the solut ion. T h e model solved in this thesis is der ived from the model in Equations (2.1) to (2 .11) , by means of two additional assumpt ions: (i ) T h e product ion funct ion f (L ) is l inear . ( I . e . , p = 1.) (ii ) T h e in f ras t ruc tu re / l abour ratio is constant over the time interval [ 0 , T ] . I . e . , L = aK, for some constant a, and by scal ing the labour and construct ion var iab les , we may assume that a = 1. Note that this results in f ix ing the wage rate 3 / ( K / L ) n = 3 . With these assumpt ions, the state equation for c o n s t r u c t i o n , (2 .1 ) , becomes £ = |^  . . . . ( 2 . 1 2 ) 9 From (2.9) and (2 .10) , the sum of wages and maintenance costs d u r i n g construct ion reduces to W(K/L) + M(K) = + mK - YK . . . . (2.13) with Y = 3 +m. Since the wage rate for construct ion labour is now f i x e d , it is somewhat inconsistent to have the wage rate for operation of the completed facil i ty dependent on the level of in f ras t ruc ture avai lable. T h e r e f o r e , we assume that the wage rate for operation of the facil i ty is f i x e d , and with labour for the operational phase f ixed at L = L , total wage payments made after time T are a speci f ied constant and can be eliminated from the cost funct ional . T h e terminal funct ion in Equation (2.11) is replaced by F(K(T)) = jn e S t K(T) , . . . . (2.14) S ' which is just the cost of maintaining ( forever) the level of in f ras t ruc ture which exists at the end of the construct ion phase . Equations (2.12) to (2.14) show that in the simplif ied model, L has been el iminated, so there remains only one control var iab le , I. For this model, we specify the control set for I to be the interval [ 0 , l . , ] , where I,, is a constant . (The case in which 1., = + °° which makes impulse controls admissible , is d iscussed in Section 5). F ina l ly , we shall replace the initial condit ion in Equation (2.3) with C(0) = Cn . . . . (2.15) 10 where is a non-negat ive constant . T h i s generalization simply includes the case in which construct ion of the facil i ty has been started before the time from which optimization begins ( i . e . , t = 0). Incorporat ing the changes in Equation (2.12) to (2.15) into the or iginal model thus yields the vers ion which will be analyzed in this thes is . The essence of the problem can be viewed as follows. T h e level of in f ras t ruc ture at any time specif ies how much labour is h i r e d , which in turn determines how quick ly the facil i ty is being c o n s t r u c t e d . T h e r e f o r e , extra units of in f ras t ruc ture allow the c o n s t r u c -tion per iod to be compressed into a shorter length of time so that construct ion costs can be de layed , thus reduc ing the present value of those c o s t s . On the other h a n d , an increased level of in f ras t ruc ture will result in h igher costs in the per iod in which investment o c c u r s , and in all later p e r i o d s , due to maintenance c o s t s . T h e r e f o r e , the level of in f ras t ruc ture in any per iod should be increased until the present value of the cost of an extra unit is just equal to the reduct ion in d iscounted costs due to the delay in construct ion which that unit makes poss ib le . Th is t rade-of f , then , is the fundamental determinant of the optimal investment strategy in this model. A statement of the model and its solution follow in Section 3. 11 3. S O L U T I O N T O T H E B A S I C P R O B L E M In this sect ion , we f ind an optimal control for our basic 3 problem and we show that, among the class of piecewise cont inuous (PWC) cont ro ls , it is un ique . For re ference , the problem as modified in Section 2 is stated below: minimize = I { Y K + l} d t + F ( K ( T ) ) . . . . (3 .1) subject to C = K K = I Kt) € [o, I j C ( o ) = CQ C (T) = C K(o) T 0 i \ i l (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) where 6, y, 1 ,^ T , and C-p are f ixed posit ive constants and K Q , C Q are non-negat ive constants . T h e r e are several points about the s t ruc tu re of the problem which should be noted at this s tage . F i r s t l y , the minimization is taken over all controls I which are PWC on [0 , T ] . S e c o n d l y , we will not A funct ion u(t) is PWC on [ 0 , T ] if it is cont inuous at all but a finite number of points in [ 0 , T ] , and it admits finite limits from the left and r ight at each of these points of d iscont inu i ty . 12 initially speci fy the terminal function F ( K ( T ) ) as Equation (2.14); we need only assume that F is lower semi-cont inuous on [ 0 , ° ° ] to ensure the existence of an optimal c o n t r o l . T h i r d l y , we also need to assume that 0 < C. - C - K T « L-T 2" • • • - ( 3 - 8 ) T O O -!-! z u to guarantee that the set of feasible controls is non-empty . If the second inequali ty in (3.8) is v io la ted, then any admissible control I, with state trajectories C and K, satisfies T T C(T) = C0 + J K ( 0 d t < CQ t j ( | H t + K0) dt 0 0 Z < C T . T h u s there are no feasible controls because the time horizon is too short to complete construct ion on time, even if investment and construct ion proceed at the maximum possible rate. S imi lar ly , if the f i rs t inequality in (3.8) is v io la ted, then every control I with values in [0/ljyj] results in C ( T ) > C ^ , which means that the terminal condit ion in Equation (3.6) cannot be sa t is f ied . T h e f i rs t inequality thus eliminates the tr ivial case in which the level of in f ras t ruc ture in place pr ior to the per iod of optimization is already greater than that 4 T h e dist inct ion between feasible and admissible cont ro ls , as def ined in C la rk [2, p. 89],. will be followed in this thes is . 13 requi red to ensure completion of the project by time T . T h e solution to the problem stated above depends pr imari ly on an application of the Pontryagin Maximum Principle (see [1] ) , which descr ibes necessary condit ions for an optimal con t ro l . S t r ic t ly s p e a k i n g , in attempting to .solve any control problem, one should use the Maximum Principle in conjunction with other theorems which establ ish the existence of an optimal control among the class of admissible cont ro ls . T o ensure that the optimal investment strategy for this problem is one which can be phys ica l ly implemented, we have rest r ic ted the class of admissible controls to the class of PWC funct ions which satisfy the control constra int in (3 .4) . B y re fe r r ing to well -known existence theorems,^ one can easily ver i fy that there is an optimal control among the class of measurable cont ro ls . With this information, the Maximum Principle may be appl ied to identify candidates for an optimal control which is measurable, but not necessar i ly PWC. We shall see , however , that these measurable controls are equiva lent , in the sense that they di f fer only on a set of measure zero , and there fore , they must all result in the same value for the objective funct ional . From this class of optimal measurable cont ro ls , one which is PWC shall be uniquely spec i f i ed . T h i s control must be optimal among the class of piecewise cont inuous c o n t r o l s , and therefore it will p rov ide the solution to our problem. See , for example, Berkov i tz [1, p . 61] or Lee and Markus [7, p. 233]. 14 We proceed with the Maximum Pr inc ip le . T h e Hamiltonian for this problem is -St (3.9) If I and ( C , K ) are an optimal control and its response , respec t ive ly , then there exist a constant , A Q, equal to 0 or 1, and funct ions y and A, absolutely cont inuous on [ 0 , T ] , such that i) c W ^ K W j a ^ ^ a U C O ) - m a x - M ^ K U J . Z ^ o . / l f t X A C t ^ a . e ; z f c £ o . ; l M ] . . . ( 3 .10 ) ii) the vector ( A 0 / AM, A a ) ) is never 0 on [ 0 ,T ] (3.11) iii) JUL AK = 0 6t f a . e . . . . . (3.12) J iv) - (ju(T),A(T)) ~ A0(o, F'(K(T))) J _ U T } X (o,oo) / or A ( T ) = - A D F ' ( K ( T ) ) . . . . . (3.13) Since y is absolutely cont inuous , (3.12) implies that = constant = j± 0 ( t i T . . (3.14) The case in which X Q equals 0 is re fer red to as the abnormal form of the problem. It can easily be shown that if X N = 0, then one 15 of the inequalities in (3.8) must be an equal i ty . T h i s makes the problem tr ivial to solve because there is only one feasible con t ro l . (See Append ix A . ) C l e a r l y , the more interest ing form of the problem is the normal one, when A N = 1, and (3.8) holds with str ic t inequal i ty . In order to make the form of the solution more t ransparent , we consider f i rs t the case in which the initial levels of social capital and construct ion are both zero ( i . e . , K n = CQ = 0), and there is no terminal payoff ( i . e . , F = 0). T h e n the t ransversa l i ty condit ion (3.13) is just A(T) = 0. ... .(3.15) From Equation (3.9), with AQ = 1, the Hamiltonian may be written as •Hv, .-./.. 16) so by (3.10) a control I which is optimal must satisfy f | M when <rU) > 0 \ 0 when cf (t) < 0 > l(t) a . e . . .(3.17) where cT(t) = A(t) - e . . . .(3.18) is the switching f u n c t i o n , and A ( t ) , the adjoint var iable for K ( t ) , is determined by 16 x = r e A(T) - o 0 i t 4 T ( 3 . 19 ) f o r some c o n s t a n t y . E x a m i n a t i o n o f (3 .18 ) a n d (3 .19 ) s h o w s t h a t t h e r e is n o s i n g u l a r s o l u t i o n to the p r o b l e m , b e c a u s e a is s t r i c t l y c o n c a v e . T h e r e f o r e , f o r a g i v e n v a l u e o f y . E q u a t i o n (3 .17 ) d e t e r m i n e s a c l a s s o f m e a s u r a b l e c o n t r o l s w h i c h d i f f e r o n l y o n a s e t o f m e a s u r e z e r o , a n d c o n s e q u e n t l y , e a c h h a s t h e same c o s t . F r o m t h i s c l a s s o f m e a s u r a b l e c o n t r o l s , we s h a l l w o r k w i th t h e o n e w h i c h is PWC a n d t a k e s the v a l u e 1^ a t r o o t s o f o, t h a t i s . ( t ) | M w h e n tfU) ) o [ o w h e n <f (t) < 0 / 0 v< t { T (3 .20 ) What r e m a i n s to b e s h o w n , t h e n , is how to d e t e r m i n e t h e opt ima l v a l u e ( s ) , o f y . In f a c t , t h e r e is o n l y o n e v a l u e o f y f o r w h i c h t h e c o r r e s p o n d i n g PWC c o n t r o l s p e c i f i e d b y E q u a t i o n (3 .20) s a t i s f i e s the t e r m i n a l c o n d i t i o n C ( T ) = C ^ . . T h i s b e c o m e s c l e a r a f t e r s t u d y i n g t h e d e p e n d e n c e o f t h e a d j o i n t v a r i a b l e X(t) = A ( t ; y ) o n t h e p a r a m e t e r y . F r o m ( 3 . 1 9 ) , X ( t ; y ) is c o n c a v e a n d h a s a r o o t a t t = T . F i g u r e 1 d i v i d e s t h e fa mi ly o f c u r v e s { X ( t ; y ) : y e R } in to t h r e e g r o u p s , a c c o r d i n g to the s i g n o f the e x p r e s s i o n A ( T j y u ) = r e -jm . Figure 1 T h e Adjoint Funct ion X(t;y) for Al ternate S igns of X (T;y) 18 We can dispense with those values of y for which X ( T;y) >_ 0; for such values of y, X ( t;y) is negative on [ 0 , T ) , and therefore , the — (St cor respond ing switching func t ion , a(t) = X ( t ; y ) - e specif ies a control which is not feasible. Integration of (3.19) yields A(t) - A(o) - x ( 1 ~ e " 5 t ) ' M i 6 and since ACT) - o , A(o) = MT - x (1 - c " S T ) , so 5 A C t j A ) = Ait) ^ I ( c ' * T - e 5 t ) + y u ( T - t ) . . • . ( 3 .2 i ) T h e c u r v e X ( t;y) reaches its maximum at t = t ( y ) , where t(jUL) = ~± t^Aj . . . . (3.22) T h e funct ion t is decreasing and for f ixed t < T , X(t;y) is an — 6 T increasing funct ion of y. T h e r e f o r e , as y increases from y = ye the c u r v e X ( t;y) pivots clockwise about the point (T ,0 ) and "stra ightens out" , as shown in F igure 2. Compar ing these c u r v e s with the g raph of e shows how the zeros of the switching function a(t) depend on y. Let y = y * be the value of y for which X ( t;y) is just tangent — (5 t to the c u r v e e . Let the point of tangency occur at t = t*. (See "~ 6 t c u r v e C in F igure 3.) For y > y*, the c u r v e s X ( t;y) and e intersect at two points t Q = t Q ( y ) and t 1 = t ^ y ) , with t Q < t* < t 1 < T . A s y increases , t Q decreases and t 1 increases . T h e points t Q ( y ) and t . j(y) are the zeros of the switching funct ion a ( t ) , so the control I speci f ied by Equation (3.20) is Figure 2 The Adjoint Funct ion A ( t ; y ) for Increasing Values of} 20 where l(t) = o 1 to W,o; (3.23) (3.24) K o : = C 0 From the state Equations (3.2) and (3.3) with initial condit ions = 0, the response trajectories for this control a re : K(t) -o In ( t - U IM (1,-0 = K o x< t < 10 t 0 i t N< t, (3.25) I 0 i CU) » { j IM CS -10) ds io C(t) t K ds and evaluation of the integrals g ives us C(t) -o ( t < i t, < t «T ' 0 1 / 2 • l M - ( t - - t c ) 2 _ |„ (t-0-(T - k l i i ) 0 4 t < to , t0< t U ; 1 t, < t * T . (3.26) Since t Q and t 1 a re . funct ions of y, so is C ( T ) , the level of c o n s t r u c -tion at time T . That i s , C(T) 2 CtT^ U U W ) s l M (1,-0 (T - iaii ) . • .(3-27) 21 T a k i n g partial der ivat ives of Equation (3.27) with respect to t Q and t^. we have <k(T) = a to (T-O < 0. . . . . (3.28) C U C T ) = l r t ( T ~ {,) > 0 . a t , which is exactly what common sense would p red ic t . It is clear from F igure 3 that ^ < 0 6jd so we conclude that and Ai, > o dM. deer) = <k(T) . aic + dp. c)C(T) ) o (3. 29) T h e r e f o r e , C ( T ) is monotone in y , so there can be only one value of y for which C ( T ) = C y . That i s , for only one choice of y is the control descr ibed by Equation (3.23) an admissible con t ro l . Choos ing y determines the switching funct ion a(t) (Equation (3.18)) and the switch-ing times t Q and t^. T h e s e , in t u r n , speci fy the response trajectories for cap i ta l , K, and c o n s t r u c t i o n , C , which satisfy the terminal c o n -straint C ( T ) = C j . , for only one value of y . T h i s establ ishes the uniqueness of the optimal c o n t r o l . T h e economic interpretat ion of the trajectory for social capital which cor responds to this control is clear from Equation (3 .21) , 22 which may be rear ranged as A(t) = yu(T-t) - x ( e - e ) . s T h e adjoint variables X and u can be interpreted as the present value shadow pr ices of social capi ta l , K, and c o n s t r u c t i o n , C , along the optimal t ra jectory. T h e f i rs t term, y ( T - t ) , represents the present value of the reduct ion in total costs due to the lower levels of in f ra -s t ructure requ i red in later per iods if an extra unit of capital is invested at time t. The second term. is the addit ional cost in maintenance and wages which would accrue over [ t , T ] with that extra unit of i n f r a s t r u c t u r e . T h e d i f ference of these two terms, A( t ) , is therefore , the marginal benefit of investment in social capital at time t. Since the marginal d iscounted cost of — 6 t investment at time t is e , the decision of whether or not to invest in more capital at time t depends on whether or not Aft) is greater — 61 than e . T h i s is precisely what is stated in Equations (3.17) and (3.18) , which descr ibe the optimal investment s t ra tegy . It is obvious that the form of the solution does not change if we now consider the more general initial condit ions T t K ( o ) = K0 > 0 C(o) = C 0 >/ 0 23 T h e response trajectories change from K and C , def ined in (3.25) and (3.26) to K and C , where K ( t ) 2 K ( t ) + K 0 . . . . (3.30) C ( t ) = C C t ) + K 0 t + C 0 . . . . . (3.31) T h u s , the level of construct ion at time T is still a monotonically increasing funct ion of u, and the optimal value of u is the one which satisfies C l T . i . U U U ) ) - C T - ( C c + K 0 T ) . So f a r , the d iscussion of the solution to the basic problem has assumed that the terminal funct ion is identical ly ze ro . T h e simplest form of terminal funct ion is one which is l inear in K ( T ) , b e -cause this makes the t ransversa l i ty condit ion for A (T) independent of K ( T ) and allows the same method of solution as when F = 0. For example , we might consider the terminal funct ion d i s -c u s s e d in Section 2, Equation (2 .14) , F ( K ( T ) ) = _DI e r t K(T) • • • - ( 3 - 3 2 ) s and investigate the qualitat ive change that this terminal funct ion induces in the optimal solut ion. It is easy to see that the optimal investment strategy is still to invest at the maximum rate , 1^, over a sub- in te rva l [ t Q m , t ^ ] of [ 0 , T ] . Let [ t Q , t 1 ] be the optimal investment per iod when F = 0. We shall show that if t f l > 0, then 24 C < L < tm < t . . . . (3.33) and m , m t, " t 0 < t« " t 0 . . • • . (3.34) That i s , investment in in f ras t ruc ture starts ear l ie r , and does not last as long when the model includes the maintenance costs of in f ras t ruc ture over the operational phase of the fac i l i ty . If we denote the adjoint var iable for K by A m ( t;y) = A m ( t ) , then the t ransversa l i ty condit ion (3.13) becomes A " ( T ) = -m. e 8 T s - d < 0 • • • - ( 3 -35) S Since the adjoint Equations (3.12) do not c h a n g e , A m ( t;y) and A(t;y) (for the case F = 0) d i f fer by the constant d , for any g iven value of y . That i s , A W ( t ; * ) 5 A (t; M ) - m e " S T = A ft; JU) - d . • • . - . (3 .36 ) 5 For a g iven y, let t Q m ( y ) < t ^ t y ) < T be the solutions to the equation m -St A ( t ; M-) = C if these two c u r v e s in tersect , and let t g(y) < t ^ y ) < T be the solutions to the equation (51 B y Equation (3 .36) , if the c u r v e s A ( t;y) and e c r o s s , they must 25 26 cross below the c u r v e A ( t;y), as in F igure 4, so that t 0U) < tJV) < CijJi) < t, U) . . . . . (3.37) Since the funct ion C (T , tg , t . j ) is decreasing in t n and increasing in t^, (3.37) implies that C ( T , t. mU), t"U)) < C ^ U M J ^ U ) ) . • • • -(3-38) Let y m and y^ be the optimal values of y for the problem with and without the non-zero terminal f u n c t i o n , respect ive ly . Because the solutions to both problems must satisfy C ( T ) = C-j., • • • • (3» 39) Recall ing that C ( T , t n m ( y ) , t ^ f y ) ) is increasing in y, (3.38) and (3.39) imply that y m > y^. We shall therefore consider the c u r v e s A m ( t;y) for y > y°. Let y 1 > y° be the value of y such that A m ( t ; y 1 ) c rosses the c u r v e s e ^ * and A(t;y°) at t Q(y°), as shown in F igure 5. Then = A m ( U M ° ) ; y U ° ) t d o r , Am(t0(xx°);V) - Am (^  U'); / ) = d 28 Since A m(t ;Ai') " A (t;/) - U ' V M T - t ) which is decreasing on [ 0 , T ] , and t Q ( y 0 ) < t ^ y 0 ) , it must be true that \(t<(u°);rf) ~ A™ (t, ; M°) < d . T h e r e f o r e , A (t, ; V) < Am (t , (xi°);M°) + d = A (t, U°); M°) so the c u r v e X ( t ;y ) c rosses the c u r v e e at times to t Q (y ) and t ^ t y 1 ) , where CU') = t0U°) < t,m(M1) < UM°) . . . . . (3.40) T h u s , C ( T , t 0 m ( y , ) # t 1 m ( y ' ) ) < C T , and we conclude that m i PL > /J. . . . . . (3.41) 2 1 m 2 Simi lar ly , if y > y is the value of y such that X ( t ;y ) crosses e 6 t and X ( t ; y ° ) at t ^ f y 2 ) = t ^ y 0 ) , then the other in ter -section of X m ( t ; y 2 ) with e ^ * occurs at a point . tJV) < t 0U°) / so c C T / i 0 m U l ) / t r U 1 ) ) > C T , 29 which implies that 2 m JUL < /A (3.42) From (3.41) and (3.42), and by monotonicity of tQm{\i) and t ^ f y ) , it follows that and C V ) i (t,B(M'),t,(*')) . (3.43) as F igure 5 indicates. Simpl i fy ing notation to t j m ( y m ) = t.^ and 0 m t.(y ) = t., where the t. and t. (i = 0,1) are the optimal switching times, the condit ions in (3.43) imply that t Q m < t Q < t ^ < t 1 # which is one of the results (Equation (3.33)) which we wanted to show. T h e second result follows eas i ly . From Equation (3.39) and (3.27), ( C - C ) ( r - C i C ) - ( t ) - 0 ( T - k l i j .(3.44) and by Equation (3.33), to + t t } t 0 -h t , 2 2 m . . .(3.45) Equations (3.44) and (3.45) imply that ( C - C ) = (t,-t0) ( T - k ± i ) T to +t, < U - U 30 which is the inequality we claimed in (3 .34) . T h e r e f o r e , the investment intervals for the two problems o v e r -lap, but with the non-zero terminal funct ion (3 .30) , the investment starts ear l ier , ends ear l ier , and lasts for a shor ter time, than with the terminal funct ion identically zero . T h i s will be d iscussed fu r ther in Section 4. T h e condit ions in (3.33) and (3.34) have been shown to be true, when tg > 0, i . e . , when construct ion (and investment) doesn't start r ight at time t = 0. We should mention what happens when t^ = 0. In this c a s e , the optimal control (for F = 0) is ' l M 0 s< t ^ t , l(t) = 0 , t , < t v< T because the smaller root , t Q ( y 0 ) , of the switching funct ion is negat ive . A similar argument to the one above shows that CV) < tau°) < o < CV) = t,u°) . In this case , therefore , the optimal control does not change when the terminal funct ion (3.32) is included in.':the objective funct iona l . Th is concludes our descr ipt ion of the solution to the basic model. T h e following section extends this d iscussion by ana lyz ing some qualitative aspects of the solut ion. 31 4. Q U A L I T A T I V E B E H A V I O U R O F T H E S O L U T I O N A s we have already s e e n , the unique optimal solution to the basic problem is a b a n g - b a n g cont ro l , with only one in te rva l , [ t 0 , t . j ] , d u r i n g which investment o c c u r s . T h e investment may begin at time t = 0, and it always ends before time T . T h e most obvious question which can be raised at this point i s : how does this investment pol icy depend on the parameters of the model? In this sec t ion , we d iscuss the quali tat ive behaviour of the solution which results from changes in the parameter va lues . O u r analyt ical resul ts are numerical ly exemplif ied in the accompanying tables, which descr ibe the optimal investment interval c o r r e s p o n d i n g to part icular parameter va lues . T h e numerical method of solution and the computer program for it are summarized in A p p e n d i x C . Some attempt has been made to determine an initial set of parameter values which are real ist ic , based on data for boomtowns in the western United States . (See Mehr and Cummings [8]). Calculat ions to estimate reasonable ranges of parameter values are contained in Append ix B . T h e initial set of parameter values comprises the f i rs t line in Table I, and other combinations have been chosen to prov ide numerical examples of the qua l i -tative changes in the solut ion. In all c a s e s , we have assumed that the initial levels of social capi ta l , K Q , and construct ion C Q , are both zero. T h i s is not an unreal ist ic assumpt ion, and as we point ^For example, the proposed location of a new mine is often too distant from any town or c i ty to make use of ex ist ing social i n f r a s t r u c -tu re , so that a while new town must be buil t to serv ice the labour force associated with the construct ion and operation of the mine. 32 out below, the qualitative change in the solution c o r r e s p o n d i n g to a posit ive value for K Q or C Q is easy to predict analyt ica l ly . T h e parameters in the model can be separated into three g r o u p s , namely: (i) those parameters which affect the response trajectories C and K, but do not appear in the adjoint funct ion A ( t;y), (ii) those which affect the adjoint f u n c t i o n , but not the response trajector ies, and (iii) one parameter which appears in the response trajec-tories and the adjoint func t ion . T h e following d iscuss ion concentrates on each of these g roups separate ly . The parameters K g , CQ, 1^, and Cj do not affect the adjoint funct ion A ( t;y). T h e y determine the initial and terminal condit ions and control cons t ra in ts , and thus enter the solution only through the response trajectories. T h e r e f o r e , when one of these parameters is c h a n g e d , the qualitat ive change in the solution is obvious from Equation (4.1) c r ~ Uo + K 0 T ) = c(T , u>d, t, (p.)) =(Vt0)(T- * 4 ^ ) . ' M IM ... .(4.D if we recall that dc ) o , it? < o , a n d ik ) 0 . cU dtt 33 C T - ( C 0 + K 0 T ) If the ratio — i is decreased (by a change in any M parameter except T ) , then ( t 1 - t Q ) [T ^—] must also decrease , caus ing y ° , the optimal value of y, to decrease , and t Q and t 1 to increase and decrease , respect ive ly . Th is result is intuit ively c lear . If the level of completed c o n s t r u c t i o n , C-p, is r e d u c e d , or the rate of investment in i n f r a s t r u c t u r e , Ijyj, is increased , costs can be reduced by star t ing the investment later , and by invest ing for a shor ter per iod of time. Th is effect can be observed in Table I. Lines 1 and 4 di f fer only in the value of Cj. For the smaller value of Cj, the investment per iod [tp,t^] is reduced to [6 .1 , 6.4] from [4.8, 7 .3] . S imi lar ly , lines 6 and 1 show 6 6 that increasing I,, from 2.0 x 10 to 5.0 x io s h r ink s the investment interval from [2 .7 , 8.2] to [4 .8 , 7 .3 ] . T h i s leads us to cons ider a "l imiting" form of the problem, in which 1 ^ = +°°. Success ive ly l a rger , but f inite values for 1 ^ shr ink the investment interval (as shown in Table II) so that when 1 ^ = + 0 ° , we expect that the investment " interval" should just be a point , tg = t.j. T h e cor respond ing control would have to be an impluse c o n t r o l , i . e . , investment which occurs only at t = t Q , at which time the level of in f ras t ruc ture jumps from K(0) to K ( t Q ) = K ( T ) , in o rder to just complete const ruct ion by time T . T h e control problem with I,, = + o ° is solved in Section 5. A second group of parameters, which includes 6, y, and m, appear in the solution only through the adjoint funct ion A ( t;y). T A B L E I Behav iour of Solution with Changes in Alternate Parameters Set 6 Y - C T T m y t 0 ( y ) t ^ y ) V ' o 1 0. 10 1.0 5 . 0 x 1 0 7 5 . 0 x 1 0 6 10 0.0 0.60163 4.7917 7.3304 2.5387 2 0.10 2.0 5. 0 x 1 0 7 5 . 0 X 1 0 6 10 o.o 1.06640 5.2461 8.3876 3.1415 3 0.10 0. 5 5 . 0 X 1 0 7 5 . O x i O 6 10 0.0 0.36363 4.0198 6.0298 2.0100 4 0.10 1.0 5. O x i O 6 5 . O x i O 6 10 0.0 0.58923 6.1098 6.3759 0.2661 5 0. 10 1.0 5. O x i O 7 5 . O x i O 7 10 0.0 0.58923 6.1098 6.3759 0.2661 6 0.10 1.0 5. O x i O 7 2 . O x i O 6 10 0.0 0.64636 2.6996 8.1845 5.4849 7 0.02 1.0 5. O x i O 7 5. O x i O 6 10 o. b 1.00620 0.1471 1.2205 1.0734 8 0.10 1.0 5. O x i O 7 5. 0 X 1 0 6 20 0.0 0.22133 1 4 . 7 9 1 7 17.3304 2.5387 9 0.10 1.0 5. 0 X 1 0 7 5. 0 x i 0 6 6 0.0 0.89753 0.7917 3.3304 2.5387 10 0.10 1.0 5. O x i O 7 5 . 0 X 1 0 6 10 0. 1 0.67839 3.8785 5.8199 1.9414 4= T A B L E II Behaviour of Solution as I,, Increases Set 6 Y C T •M T m y t Q ( y ) y y ) y t o 1 0.10 2.0 I.OxlO7 5.0x:105 10 0 1.140977 3.573500 8.859900 5.286400 2 0.10 2.0 I.OxlO7 5.0x10 6 10 0 1.025205 6.818855 7.526201 0.707345 3 0.10 2.0 1.0X10 7 I.OxlO7 10 0 1.023506 7.009777 7.365340 0.355563 4 0.10 2.0 1.0x10 7 1.5x10 7 10 0 1.023187 7.071745 7.309025 0.237280 5 0. 10 2.0 I.OxlO7 2.0x10 7 10 0 1.023075 7.102366 7.280389 0.178023 6 0.10 2.0 1. 0x10 7 3. 0x10 7 10 0 1.022995 7.132732 7.251444 0.118712 7 0. 10 2.0 I.OxlO7 5.0x10 7 10 0 I. 022954 7.156834 7.228070 0.071236 8 0. 10 2.0 I.OxlO7 I.OxlO8 10 0 1.022936 7.174795 7.210415 0.035620 9 0.10 2.0 1. 0 xl 0 7 I.OxlO 2 0 10 0 1.022931" 7.192656 + 7.192657 - o . o o o o o f U l 36 A change in one of these parameters will shift the interval over which investment occurs either towards T , or away from T , with an accompanying increase or decrease , respect ive ly , in ( ^ - t g ) , the length of the in te rva l , in o rder to meet the condit ion C ( T ) = Cj. T h e analysis in Section 3 on the terminal funct ion F ( K ( T ) ) = m e ~ S T K ( T ) , 8 where m represents unit maintenance costs for social cap i ta l , shows that increasing m from zero to some posit ive level moves the interval [tg,t.|] c loser to t = 0, and decreases (t^-t^) (see (3.33) and (3.34)) . T h i s effect appears in Table I, lines .1 and 10 , where sett ing m = 0.1 shifts [t 0 , t . j] from [4 .8 , 7.3] to [3 .9 , 5 .8] . Of c o u r s e , if the root tg(y^) is negative for m = 0, then the f i rs t switching time is t^ = 0, and increasing m to a posit ive level does not change the solut ion. Th is predict ion is i l lustrated in lines 4 and 6 of Table III. It is clear from the analysis in Section 3 that the same qualitat ive change in the solution occurs if m is increased from some posit ive level m ] to a h igher level m 2 , and this result makes economic sense. When m is increased , the marginal cost of social capital is h i g h e r . Since K ( T ) , the optimal level of social capital at time T , is that level for which marginal costs are equal to marginal benef i ts , an upward shift in the marginal cost c u r v e results in a lower optimal level for social capi ta l . That i s , K (T) < K ( T ) , so from Equation (3.25) , ( t . . m - t n m ) < t n - t n . T h e r e f o r e , in order to meet the terminal TABLE 111 Sensi t iv i ty of Solut ion to Changes in T and m, with Negative t . (y) Set 6 Y C T 'M T m y t Q ( y ) t,(y) 1 0.02 1.0 5 . 0 x 1 0 7 5. O x i O 6 10 0.0 1.00620 0.14710 1.2205 1.0734 2 0.02 1.0 5 . O x i O 7 5 . 0 x 1 0 6 20 0.0 0.82378 1 0 . 1 4 7 1 0 11.2205 1.0734 3 0.02 1.0 5 . O x i O 7 5 . 0 x 1 0 6 10 0.1 1.46370 - 3 5 . 0 1 2 0 0 1.0557 1.0557 4 0. 02 1.0 5 . O x i O 7 2 . O x i O 6 10 0.0 1.01280 - 2 . 1 7 9 7 0 2.9289 2.9289 5 0.02 1.0 5 . O x i O 7 2 . O x i O 6 20 0.0 0.82509 9.27820 1 1 . 9 4 0 0 2.6618 6 0. 02 1.0 5 . 0 x 1 0 7 2 . O x i O 6 10 0. 1 1.5918 0 - 4 3 . 8 1 7 0 0 2.9289 2.9289 38 condit ion C ( T ) = C-j., the investment interval for the model which i n -cludes the terminal function (or has a larger value of m ) must start ear l ier . I .e. , t Q < t Q . A n increase in either of the other two parameters, 6 , and y , appears to have the opposite effect on the solution from an increase in m. As 6 or y increases , [tp,t^] shif ts towards T . T h i s observat ion re l ies, however , on the numerical resul ts presented in Table I. Lines 1 and 8 reflect the change in the solution cor respond ing to a change in 6, and lines 1 , 2, and 3 show the optimal investment interval for di f ferent values of y . Tha t these qualitative effects are exhibi ted for all posit ive values of 6 and y ' s more di f f icul t to prove than the cor respond ing resul t for the parameter m. T h e simple graphica l analysis which was used to prove results for m relies on the fact that 3A/3m is not a funct ion of t, which is not the case for either 3 A/9 6 or 3A/3y. ; T h u s , the same method of proof cannot be appl ied in this case . T h e behaviour of the investment in te rva l , co r respond ing to changes in <5 and y , can be intuit ively motivated by appeal ing to the effect of d i s c o u n t i n g . A s the discount rate 6 increases , costs imposed in the fu ture are weighted less , so that it pays to postpone investment unti l a time closer to T , in spite of the fact that this delay must be accompanied by a longer per iod of inves t -ment, and therefore , a larger final level of social cap i ta l . S imi lar ly , an increase in y , which we can think of as an increase in the wage rate for labour , means that the total wage bill is increased for a g iven investment—construct ion po l icy . Because costs are d iscounted at a 39 posit ive rate , it is therefore better to wait longer to start investment and c o n s t r u c t i o n , even though more social capital will ultimately be r e q u i r e d . T h e one parameter which h a s , so f a r , not been accounted fo r , is the time horizon T . Changes in this parameter affect both the terminal condit ion C ( T ) = C-j- and the adjoint funct ion A ( t;y), so one might expect that the c o r r e s p o n d i n g changes in the optimal solution would be complicated. However , numerical resul ts indicate a simple relat ionship between T and the;optimal investment in te rva l , [ tg , t^] . These results show that if t^ > 0, then chang ing T to T + A will change the investment interval to [tg+A, t^+A], p rov ided that t^+A > 0. For example, by comparing lines 8 and 9 of Table I, we see that decreasing T by 14 u n i t s , from 20 to 6, simply shifts [t Q ,t.|] 14 units towards the o r i g i n , i . e . , from [14.79, 17.33] to [0. 79, 3.33] . If tg+A < 0, however , investment over [0,t ; 1+A] is not a feasible c o n t r o l , since it results in C ( T ) < C ^ , so the optimal investment interval extends from t = 0 past t = t ^ A . (Compare the value of t ^ t g in lines 4 and 5, Table III). Examination of the switching function allows us to ver i fy these results analyt ica l ly . Suppose that y Q is the optimal value of y for the problem with time horizon T , and let t 0 = to^1Jo^ a n c l t 1 = t ^ y g ) , with t Q > 0, be the optimal switching times. If the time horizon is changed to T+A, it suff ices to show that no (i) t Q + A and + A are roots of a ( t ;T+A,u) for some value of y = y . I .e. , a(t_+A ; T+A , y I = a(t,+A ;T+A ,u..) = 0, A 0 A i A (ii) the switching times t Q + A, t 1 + A satisfy the terminal c o n -dition for construct ion with time horizon T+A. I .e. , C(T + A/ VA, VA) = C T If, in addit ion to (i) and ( i i ) , t Q+A S 0, then the investment interval [t Q+A ,t, j+A] describes a feasible c o n t r o l , and since the value of y which satisfies (i) and (ii) is un ique , this control is optimal. We claim that the value of y which satisfies (i) is def ined by ~ S A (II 91 A A = Mo C • • • - ( 4 ' 2 ) T o see th is , we evaluate the switching funct ion a ( t ;T+A,y^) at t = t Q+A and t = t ^ A . From Equation (3.18) and (3 .21) , for i = 0 ,1 , d - ( t , , A ; T + A , A ) » | e s ( T , A ) - £ , . ) e S ( t ' , i ) + ^[T * A - (t,t A ) ] [ r c - ( p i e ) + A(T-t,) . b y (1.2) = o , I .e. , t Q+A and t^+A are zeros of a ( t ;T+A,y^) 41 Claim (ii) is obvious from Equation (4.1) (with K 0 = C Q = 0). i f t Q and t 1 are the optimal switching lines for time horizon T , then C ( T ; t 0 / t , ) = l M ft-0(T ~ t ^ j ) = C T . T h i s equation is also satisfied for switching times t Q+A , . t^+A, with time horizom T + A. T h i s shif t in the investment interval co r respond ing to a change in T agrees with the predict ions of a simple economic ana lys is . For a g iven size of fac i l i ty , Cj, investment rate , 1^, "wage" rate , y, and d iscount rate , <5, there is an optimal length of time, A 2, to complete construct ion of the fac i l i ty . T h e total construct ion time determines the total amount of social capital r e q u i r e d , o r , the length of the investment in terva l , A 1 # since investment always occurs at a constant rate. Since costs are d i s c o u n t e d , the construct ion should not be star ted unti l A 2 units before the end of the time hor i zon , T . T h i s * * ( A 1 , A 2 ) investment-construct ion policy remains optimal for any time * * horizon T > A 2 . However, if T < A 2# the time horizon is too short to allow the optimal construct ion time, so decreases to just cover the time hor i zon , and consequent ly , the length of the investment in terva l , b, , must increase. T h e exact relat ionship between A j and A 2 can be de r ived from the terminal condit ion on C ( T ) , in Equation (3 .27) , which relates T , t Q , and t 1 . T h i s equat ion, stated in terms of A 1 = t j - tg and A 2 = T - t Q g ives us 42 A, ( A, - 4 ) = - (4.3) T h u s , the optimal investment-construct ion policy as a function of T can be summarized as , A 2 = A2(T) = T , r 4 T < A , ' 2 / A : , T >/ A \ 5 A , ( A 2 ) - A 2 - ( A 2 2 - T : 2 ) # t 3 2_Cj (4.4) T h e equation for A j follows from (4 .3 ) , and the constant l i s the smallest time horizon for which there exists a feasible c o n t r o l , g iven C T and l M (with C Q = K Q = 0). T o sum up the thesis to this point , t h e n , we have completely solved the basic model which we der ived from Cummings' and Schulze 's model, and have car r ied out a sensi t iv i ty analysis of that solut ion. T h e remainder of the thesis cons iders two extensions which ar ise natural ly from this basic model. H3 5. E X T E N S I O N T O U N B O U N D E D C O N T R O L S E T T h e f i rs t extension to the basic model which we shall d iscuss is one which has been suggested in Section 4 , as a resul t of the sensi t iv i ty analysis on the parameter 1^. It was clear that success ive ly larger values for 1^ caused the optimal investment interval [ t^,^] to cont inuously s h r i n k , and we hypothesized that in. the limit, as 1^ approaches +°° , the optimal investment policy should be an impulse control which occurs at a certain time t in the interval [ 0 , T ] . F i r s t , we shall define the extended control problem more c lea r ly . T h e basic problem in Section 3 assumes that there is a maximum rate of investment al lowed. I .e. , 0 < |(t) < l M < + °° , 0 ( t 4 T . If the problem is changed so that there is no upper bound on the rate of investment, the c o r r e s p o n d i n g control constra int is 0 < |(t) 4 + 0 0 0 x< t x< T . We shall refer to this modified control problem as P ^ . 7 T h e condit ion "I (it) = + °°" signif ies that I is an impulse control which causes an instantaneous (but finite) jump in K (the level of social capital) at time t. T h e class of admissible controls for P includes all controls which oo 7 T o avoid unnecessary algebraic c lu t te r , it will be assumed throughout the section that C = K = 0 , and F ( K ( T ) ) = 0 . 44 are PWC on [ 0 , T ] , p lus the set of impulse controls which are PWC on [ 0 , T ] , except at a finite number of points in [ 0 , T ] , at which finite jumps in K o c c u r . A s a f i rs t step in f ind ing a solution for P we observe that the Maximum Principle a n d , in fac t , the existence theorems re fer red to in Section 3 are not val id for this problem, because the control set has changed from [0 , 1 ^ ] to [ 0 , + ° ° ) , and is therefore not compact. For tunate ly , an al ternat ive method of solution is avai lable. We begin by examining more r igorously what happens to the optimal solution for the control problem with f in i te -va lued 1 ^ , as 1^ + + ° ° . ' Proposit ion 1. Let P n be the basic control problem with 1^ n, for n\ = 1,2,-". Let I be the optimal control for P^, with response trajectory (C , K ), switching times t^1"1^ and t ^ n ^ , and cost T -St J"n - J ( l n } = e I rKn • |nj dt o T h e n 3 a unique t A e [0 ,T ) such that (i) t Q ( n r ) f t* and t ^ " 1 + t* as n -* +» . . . . (5.2) S ( T - t 4 ) r -, (ii) t* solves the equation e |_S U - j - 1 j + V - Q . (5. 3) Note that piecewise cont inuity of controls rules out funct ions I (•) for which l iml ( t ) =+°° or lim. I(t) = +°°, for some t e [ 0 , T ] . t+T t->T+ 45 (iii) K n (T) » K (iv) J N I K+ e I * Y ( \ --o(l-U) (5.4) (5.5) Proof (i) T h e existence of t* can be deduced by examining several equations from the solution to the basic control problem with 1^ < 0 0 . From the analysis in Section 3, the construct ion response at time T , scaled by l M is g iven by c f i o U U t / t ) ) 5 ^ j j = ( t r O f T - W i ] • • • - ( 5 - 6 ) 1 V 2 IM Recall ing that C is a monotone increasing function of y, the optimal value of y is the value for which C (t„U), t,U)) = ... .(5.7) T h e r e f o r e , if 1^ increases , the optimal value of y must decrease . S ince 3t at * T T — < 0 and •?— > 0, the decrease in y implies that the switching 8 U d y r r 3 times t Q and must increase and decrease , respect ive ly . For any finite value of I,,, t A < t , . T h e r e f o r e , M 0 1 H6 lim t0 = ^ l(v) -> + 00 both ex is t , and and lim = t u IM->+ oo 0 * t 2 ^ t u < T (5.8) From Equations (5.6) and (5.7) = c IN] (5.9) T a k i n g limits of both sides of Equation (5.9) as 1^ ->• 0 0 lira ( t , - t , ) ( T • 0 IM + oo ^ or (t u - tL) (T ~ h±U 2 0 . (5.10) The second factor in Equation (5.10) is pos i t ive , by (5 .8 ) , so the f i rs t factor must be zero. I .e. , t u = = t , 6 [ O , T ) oo T h u s , in terms of the sequence of problems {P^ J" _-| '• w e have lim i (n) l - - + l i m t , - X. which proves (i) . 47 (ii) A s we noted p r e v i o u s l y , increasing values of 1^ = n resul t in decreas ing optimal values of y = y ^ . S ince t ^ " ' and t ^ n ^ converge monotonically to a single point t* , the sequence { y ^ } must converge monotonically to a value y * such that the c u r v e s X(t;y*) and - f i t 9 e are just tangent at t*, as shown in F igure 6. T h e r e f o r e , ( y * , t*) is the unique solution of the equations A l t * ) M-*) = e with t* < T . Subst i tu t ing Equation (3.21) and (3.19) for X and X, we have X ( e - e j + / L ( T - t j - e . . . . . ( 5 . i n r c s U - A = - s e S U . ( 5 - 1 2 ) Solv ing Equation (5.12) for y* and subst i tu t ing in Equation (5.11) leaves an equation in t*: y \c - e J + (Y+ s ) £ ( T - t J = e 8 which algebra will reduce to Equation (5 .3 ) , thus ver i f y ing part (ii) of the Proposi t ion . T h e c u r v e X(t;y*) also appears in a prev ious sect ion; see F igure 3. Figure 6 Adjoint c u r v e A ( t;u^ for increasing n 49 (iii) By Equation (3.25) the final optimal level of social capital for K „ ( T ) - n(tr-C) Equation (5.9) rewritten for P n says that (L W , (n) \ L In) i (n) \ n (t, " t0 )/T - t0 t l , J = C T so subst i tu t ing (5.13) into (5.14) and solv ing for K n ( T ) , we have K n (T) = C_l {X - i j n \ t,(n) and taking limits on each s ide , Equation (5.2) g ives us lim K n(T) = C j _ n-> oo T _ t i which is what we wanted to show. (iv) T h e last condit ion which we must check is the behaviour of the sequence of costs -{J n ) . Because the set of feasible controls for P n is proper ly contained in the set of feasible controls for P n + - | , J { l n + 1 } < J { l n } . In fac t , uniqueness of the solution to P n + 1 makes the inequality s t r i c t , so {J n } E | j{ I n }| is a monotonically decreas ing sequence of posit ive numbers , which must , therefore , have a limit. 50 The limit in Equation (5.5) can be ver i f ied by evaluat ing the cost integral for the optimal control I . T h u s T J n = j e ( r K n +l n) dt t, o tn) -bl Yn (t-t0'n)H n| dt + I e 4 t V K n ( T ) d t • • • . (5.15) , trO ,tn> by Equation (3.23) to (3.25) and definit ion of K n and l n . Integrating the f i rs t integral by p a r t s , we have An) - s t n e I 01) r(t-CViUt st n r 1 e ° l ° I s 2 i - e 4 -S^Vta) .<")} t n e i - e T t i n c St 0 tn) r 1 - e - I K n(T) C 6 . . (5.16) by Equation (5 .13) . T h e second integral in (5.15) is simply T Y K n ( T ) C dt = l K n ( T ) -ST-1 £ J (5.17) 51 So from Equations (5.15) and (5 .16) , r w oo l i m i [ H n e r w oo \S h - S l 0 l In) 1 - £ - 8 ( + r - * r ) i l.m r K n (T) e r w co h 8T I +1) lim n e S / n -> oo S (n) 1 - e ST (5. by Equation (5 .4 ) . T o evaluate the limit on the r ight side of Equation (5 .18) , we expand e — 5 ( t 1 (n ) — (n)) j n g | ^ a c | a u r j n s e r i e s . T h u s J l e h 1 ri(rl) = n e 0 [i.- ( i - o U w > - C ) + OCr-lT)2), Si (n) e " n(l!"'-0[l - 0(C-C e S ' , W K n ( T ) [ l - Ofr-C1)] C a s n -» oo Subst i tu t ing this result into Equation (5.18) gives us the requ i red expression for lim J , which concludes the proof of the proposi t ion. n+oo 52 A glance at the results in the preced ing proposit ion suggests a candidate as a likely solution to P w > As n -*• 0 0 , the sequence of optimal controls f l n } approaches an impulse control 1^  , determined by (5.3) and (5 .1 ) . It is only reasonable to expect that if has a solut ion, it should be the control I. . 1 0 Theorem 1 below will ver i fy t * that this guess is c o r r e c t . The proof of the theorem, however , requires a prel iminary result which is stated in the following lemma. Lemma 1. Given e > 0 and a control I which is feasible for P , 3 a 0 0 control I and an inteqer N such that V 2 N , I is feasible for P , and 3 n n J { f } < J{ | . } + e. T h e proof of the lemma is contained in A p p e n d i x D. Bas ica l ly , the lemma shows that the cost of any impulse control can be approximated (with arb i t rar i ly small er ror ) by the cost of a control which is PWC and of finite value everywhere . We proceed now to our main resul t . Theorem 1. T h e impulse c o n t r o l , I. , def ined below, is optimal for 1* P . OO 0 otherwise .(5.19) K^ (t) = i . K 4 where t A is def ined in Equation (5 .2 ) . t / w — L - 0 1 * . . . . (5.20) ^ T h i s d iscussion does not address the question of uniqueness of the solution for P 53 Proof: It is clear that the control I. is feasible for P . In order t * 0 0 to conclude optimality of I it suff ices to show that 1 * J Til] feasible for P T h e ifimum does exist because the set ( j {I } I I feasible for P is bounded below by zero . St ra ight forward calculation shows that T J - a * f -si K , e t J e Y K , d t K, e s t l + I K e * % * lim Jn by Equation (5 .5 ) , so Equation (5.21) is equivalent to the two inequalit ies j i i ) feasible for P Since l „ is feasible for P , for every n n 0 0 ' • ' 00 C {Jllf I feasible for P T h e r e f o r e , 00 ( J i l l feasible for P or from ( 5. 5), 54 J * >/ feasible for P T h e second inequali ty in (5.22) follows from the Lemma. Given e > 0 and any control I feasible for P ^ , 3 a control I and an integer N such that f is feasible for P , V > N , and J{T} s J {I } + e. S ince I is n n n optimal for P , n J n = JJUl x< J T ( | ) N< t I # V n > N , and by taking the limit of the sequence J , as n °° T< = lim Jn 4 + £ . • • • -(5.23) n - > o o T h e inequality above is true for any e > 0, from which we may conclude that ^ J { l } , for any control I feasible for P . T h u s , J * is a lower bound for feasible for P > , so ( { J f l ) | Ifeasible for P^ ] T h i s ver i f ies the second inequality in (5 .22) , which implies the optimality of I. for P . r 7 t* 0 ° With the results proved in this sect ion , we have solved the extended vers ion of our basic control problem in which there is no upper bound on the rate of investment , and instantaneous jumps in the level of social in f ras t ruc ture are permit ted. Under these more 55 general cond i t ions , we have shown that the optimal investment strategy is a "one-shot" instantaneous investment in social capital at a time t* , which occurs before the end of the time horizon ( t=T), and is uniquely determined by the d iscount rate , the wage rate, and the time hor i zon . 56 6. OPT IMIZAT ION O F T H E TIME HORIZON The second extension to the basic model which we d iscuss in this thesis is the optimization of the time horizon T . It is not unreasonable to assume that the decis ion-maker faced with the job of p lanning the construct ion of the facil i ty and investment in in f ras t ruc ture may f i rs t have to determine when the facil i ty should go into operat ion, in which case treat ing the time horizon as a var iable to be opt imized, rather than as a specif ied parameter, makes the model more real ist ic . A n examination of the objective funct iona l , however , shows that when T is opt imized, the cost can be made arb i t rar i ly small by choosing a large enough value of T . T o be spec i f i c , let J ( T ) be the cost of the optimal control for the basic model with f ixed time horizon T . Th is cost has been evaluated in Equation (5.15) to (5 .17) , and may be simplif ied to the form J ( T ) - In * i J i f l * i ) e"Mi-c 4 N -YA where A j = t j - tg and A 2 = T - t Q , def ined by Equation (4 .4 ) , are the optimal lengths of time for investment and construct ion for time horizon T . If (A^,A2) is the optimal policy for some time horizon T , then investment and construct ion are both started A2 per iods before T. Th is po l icy - is obviously feasible (although perhaps not optimal) for all time horizons T > T , with cost (6.1) 57 j (T ) = e • J(T) which approaches 0 as T + » . T h e r e f o r e , the objective functional for our basic model does not admit an optimal value of T which is f in i te . T h e reason that the infinite time horizon is optimal is simply that the objective functional for the basic model includes only c o s t s , and no benef i ts . Presumably , the reasons for cons t ruc t ing any facil i ty include some form of benefi ts which accrue d u r i n g the operational phase of the fac i l i ty , after the completion of its cons t ruc t ion . In this c a s e , choosing an optimal value of T is equivalent to balancing h igh c o n -struct ion costs (from small T ) with a loss in benefi ts (from large T ) . For the purposes of this d i s c u s s i o n , we assume that the benefi ts accrue at a f ixed rate 3 over time, as that the present value of the stream of benefits over [ T , ° ° ) is oo B(T) = f p e ^ d t - P e" S T . . . . . (6-2) T o T h e problem of optimizing the time hor i zon , then , is one of maximizing the d i f ference between benefi ts and cos ts . G iven a part icular time horizon T , the decis ion-maker will choose an investment-construct ion strategy which minimizes his cos t , so the appropr ia te cost function for this problem is J ( T ) , descr ibed in (6.1). O u r problem, therefore is to determine a value (or values) T = T * which maximize B ( T ) - J ( T ) over the domain [ T , ° ° ) . 58 T h e benefit funct ion B ( T ) has a v e r y simple form which is easy to handle ana ly t ica l ly , but the cost func t ion , J ( T ) , is a complicated function of several parameters. From Equat ion(4.4) , J ( T ) can be expl ic i t ly expressed as a function of T on each of the intervals * * [ T , A 2 ) and [ A 2 , ° ° ) wh ich , together , consti tute its domain: J(T)» f (T) = IM 8 CJ (T) 2 U -ST T t [l, A * ) ... where T is the smallest feasible time hor i zon , given other parameter * va lues , A2 is the largest time horizon for which the optimal inves t -ment policy starts at t f l = 0, and A^ = A..j(A 2). With J ( T ) written as two di f ferent func t ions , f and g , a v e r y simple argument establ ishes the main result of this sect ion , which is stated below in Proposit ion 2. T h e proof of the proposit ion d e p e n d s , in par t , upon the fact that J * * is di f ferentiable at A 2 . Cont inu i ty of J at A 2 is o b v i o u s , s ince * * f and g are cont inuous , and f ( A 2 ) = g ( A 2 ) . T h e r e f o r e , from the * * following lemma, which shows that f ' ( A 2 ) = g ' ( A 2 ) , we conclude that J'( A_ ) ex is ts . Lemma 2. Let A 2 be the largest time horizon for which the optimal investment policy starts at t Q = 0, and let f (T ) and g ( T ) be def ined * * by (6.3), with domains [x, 0 0) and (- 0 0 , 0 0 ) . Then f'(A_) = g'(A_).' 59 Proof: Differentiat ion of f and g y ie lds £i0 = ( y G ) e V < T ) + r e ( 6 A f -A T'(T)) IM /6 =• A, (T)( ( Y+ O) e - r e } + Y<SA,e . . . . (6.4) (6.5) CJ (T) = (Y + s)(e -Ij e + Y6 A, e lM/6 A t T - t n e s e c o n c " term in both equations if the same, so that f (O - g'M - A>>')((m)eSir -re'4**} = "A, A * - A ; . . . . (6 .6) after di f ferent iat ing A ^ T ) (Equation (4 .4 ) ) . T h e r e f o r e , our claim * * that f'(A 2) = g'(A 2) will follow if we show that the r ight side of Equation (6.6) is zero . For th is , we require an equation which implicitly defines A 2 in terms of the parameters of the model. Because A 2 is the largest value of T for which t Q = 0, the f i rs t root , t Q ( y ) , of the — 61 switching funct ion a ( t;y) = A ( t;y) - e must be ze ro , and the * second root, t^(y) must be A^, as shown in F igure 7. T h e two relevant equations are 60 A ( t ; / i ) -6t Figure 7 Switching times for T = A-which are equivalent to 61 l(e"»* - e 8 A ' ) • ^ K - A , ' ) - e " " (s.7) 1 ( e 1 - 0 t U A * = 1 . . . ,(6.8) s Subtracting (6.8) from (6. 7) results in the equation I (1 - e S A*) - / * A * = e 8 * * - i • (6.9) Solving Equations (6.7) and (6.9) for y, equating them, and multiplying * by 6 A i , we are left with 6A*yu s A* j ( m ) e S A ' - r e S A L J = ( Y * S ) 0 - e ^ * 1 ) A * - A * . . . .(6.10) * Since these two expressions for 6A^y correspond to the two terms in Equation (6.6), the right side of Equation (6.6) is zero, which proves our claim. We proceed now to our main result. Proposition 2. Let B(T) and J(T) be defined by Equations (6.2) * and (6.3), respectively, and let T maximize B(T) -J (T) on t x , 0 0 ) . * * Then either (i) T e (x, A_ ), or 62 (ii) E v e r y T e [ A ~ , ° ° ) maximizes B ( T ) - J ( T ) , and max{B ( T ) - J (T) } = 0. Proof: We shall compare the two exponential funct ions B ( T ) and g ( T ) , * which defines J ( T ) on [ A _ , ° ° ) . From Equation (6 .3 ) , we note that g(T)= g ( ° ) ^ T ; g(o) - i M J ^ + i j e ^ d - e ^ * ) and from Equation (6 .2 ) , B(T) = 6(0)6 ST 8(o) = A h T h e r e f o r e , depending on the sign of B ( 0 ) - g ( 0 ) , the c u r v e B ( T ) lies below g ( T ) for all T , above g ( T ) for all T , or B is coincident with g . We show that the f i rst two possibi l i t ies cor respond to (i) in the prop+i osi t ion, and that (ii) may arise when B and g are ident ical . If B(0) < g ( 0 ) , then B'CT)- g'(T) = -6 [ B ( O ) - g(o)J &'ST > o V T so B ( T ) - g ( T ) has no maximum. Since J ( T ) = g ( T ) on [ A 2 , 0 0 ) , this means that B ( T ) - J ( T ) has no maximum on [A2,°°). T h u s , if * * * there \s_ a value T which maximizes B ( T ) - J ( T ) on [ x , ° ° ) , T e [ T , A 2 ) On the other h a n d , if B(0) > g(0) then B ' ( T ) - g ' ( T ) < 0 for all T . T h e r e f o r e , by Equation (6 .3 ) , 63 BCD- J'(T) 2 B'(T) - gVr) < 0 ; T e [A* (<*>)•• • .(e.nj * where equali ty at T = A 2 follows from Lemma 2. It is clear that B _ * and J = f have cont inuous der ivat ives on ( T,A 2], so Equation (6.11) * implies that B ( T ) - J ( T ) is decreas ing on [A 2-£,°°), for some e > 0. T h u s , over its ent ire domain [ x , 0 0 ) , B ( T ) - J ( T ) is maximized for some ~k it T = T e [ x , A 2 ) . For both of the above c a s e s , we can eliminate the possibi l i ty * that T = T . From Equation (4 .4 ) , A,'(T) = 1 " T > - oo as T - > Z + so Equation (6.4) implies that f ' (T ) -* as T •+ x +. T h e r e f o r e , lim B ' C O - j ' d ) = - x 8 ( o ) c - lim f ' ( T ) = + oo ; regardless of how large or small B(0) i s . T h i s means that B ( T ) - J ( T ) is always increasing in [x, x+e], for some e > 0, so if T is optimal, * T > x. T h e preceding remarks show that if B(0) ^ g ( 0 ) , then (i) is t rue ; that i s , if any optimal values of T ex is t , they must be in the * interval (x, A 2 ) . T h e remaining possibi l i ty is that B(0) - g ( 0 ) , which means that 8(T) - J ( T ) = 0 , T 6 [ A * , oo) . . . .(6.12) 64 so the optimal time horizon really depends on the funct ion f ( T ) = J ( T ) , for T e ( x , A 2 ) . If B (T ) > f (T ) for some T e ( x , A 2 ) , then (i) is t rue . * Otherwise , B (T ) - F ( T ) < 0 on ( T , A 2 ) , so Equation (6.12) implies * case ( i i ) . I .e. , every T E [ A 2 , ° ° ) maximizes B ( T ) - J ( T ) , and max ( B ( T ) - J ( T ) } = 0. Q . E . D . From a pract ical point of view, this proposit ion makes sense . Case (ii) may be viewed as a somewhat pathological s i tuat ion; f i rst of a l l , only a relatively small set of parameter values will sat isfy B(0) - g ( 0 ) , and second ly , since the maximum net d iscounted return B ( T * ) - J ( T * ) is always zero in this case , it is unl ikely that any planner would even undertake the project . Case ( i ) , however , tells us that if there is_ an optimal time horizon T , over which the facil i ty should be c o n s t r u c t e d , T will always be smaller than A 2 < T h e r e f o r e , by * Equation (4 .4 ) , the optimal pol icy c o r r e s p o n d i n g to time horizon T will be to start the project at time tg =. 0. T h i s is really not s u r p r i s -ing at a l l , for the following reason: if the decis ion-maker can achieve a posit ive net discounted re turn R us ing an investment interval [ t0 , t j ] / with t Q > 0 and time horizon T , he can clear ly increase his net re turn to e 0 • R by star t ing investment at t = 0, invest ing for the same length of time as be fore , and completing the construct ion of the facil i ty at t = T - tg . Fur thermore , if his or iginal investment * policy is optimal for the time horizon T , then T - tg = A 2 , and * e 6 t ° R = [ B ( 0 ) - g ( 0 ) ] e " 6 A 2 > o, so B(0) > g ( 0 ) , and 65 B'(/£)-g'(A*) = - 6 T BCo) - g(o)] edaa < 0 . T h u s , he can increase his re turn by choosing a time horizon shorter * than A 2 , even though this will force him to increase his final level of investment in social i n f ras t ruc tu re . In order to obtain s t ronger results than those stated in Proposit ion 2, we need to know something about the cost funct ion J ( T ) = f ( T ) on [T,A*) . It is easy to deduce that J ( T ) is d e c r e a s i n g , us ing the fact that it is the cost of the optimal policy for time horizon T . Unfor tunate ly , more useful results regard ing the shape of this funct ion are di f f icul t to prove for all possible parameter va lues . However, the problem of optimizing T has been solved numerical ly for several sets of parameter va lues , and for these part icular c a s e s , we found that the funct ion f is convex and dominates g , except at T = A 2 , where the two c u r v e s are tangent . F igure 8 i l lustrates f and g for one set of parameters. We see that for this cost f u n c t i o n , the optimal time horizon T has a very simple relat ionship with the benefit funct ion B ( T ) . If B(0) < g(0), then B ( T ) < g ( T ) s f ( T ) , V T e [T,°°), so B ( T ) - J ( T ) is negative and increases to zero as T +°°; * thus there is no optimal time horizon T < ° ° . In the unl ikely case that B(0) = g(0), B ( T ) E J ( T ) on [ A * , ~ ) and B ( T ) < J ( T ) on * * [ T , A 2 ) , so the set of optimal time horizons is the interval [ A . , , 0 0 ) . Of c o u r s e , the only in terest ing case from the point of view of the decis ion-maker is the one for which B(0) > g(0), because this provides 66 E E :EEE •.tip: IEEE itn .:±r: EEE :EEf EE; EE ri:'.: E :: : • iii; E E ii: t EEE :rr: EEE*: :HE E'E E EE: \ LrE" ™n M ::Hi 3a'; ~an 161 i-— • e r \ ' I CJ C " ; '•IE EEz SI; -EE I—r—!—1 >"i—1 E E E:: : E \ \ E E E E'E.E <E :• SE = ( IE1 oi: in! p :::rr: 3;_: K C n:ri H E ; \ : R;.|.. - i i i 5-' w I 0 E EE! jSl'i- OM 10 Z:rE M; !E!-11'V E E itiO EE: E* r HE: E-E -ir-: :r: :T < 3(' )) )65 X K )' H E f: E E 3, ( 0 ) Hi H7 5C 10 1 0 : '. 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In fact , there is a unique * * * optimal time horizon T , and as we predicted T e ( x , A 2 ) . (Check the graphs of B 2 and B 3 in F igure 8.) Fur thermore , as B(0) increases , — 6 T the slope B ' (T ) = - 6 B ( 0 ) e decreases for every T , so that the * point where B ' (T ) = J ' ( T ) must shift to the left; i . e . , T decreases . T h i s effect is i l lustrated b y c u r v e s B 2 and B^ in F igure 8, with * B 2 ( 0 ) = 8 . 2 and B 3 ( 0 ) - 9.0. T h e cor respond ing values for T are approximately 5.2 and 5.1. A l though the analysis is restr ic ted to values of the parameter which y ie ld a cost c u r v e like the one in F igure 8, it does cor respond to what intuition would dictate a decis ion-maker to do: the greater the per per iod benefits from the fac i l i ty , the sooner the faci l i ty should be in operat ion. These results cor respond to the basic model, in which inves t -ment occurs at a maximum rate 1^ < ° ° . However , the problem of optimizing the time horizon can also be considered for the extended model in Section 5, in which impulse controls are al lowed, and as one might expect , the analysis is simpler in this case . T h e optimal control with time horizon T is an impulse investment which occurs at t = t*, and t* is uniquely determined by Equation (5 .3 ) , 6 + Y It is clear from this equation that as T c h a n g e s , the di f ference T -remains constant . T h u s , the cost of this impulse c o n t r o l , given by 68 Equations (5.4) and (5.5), can be written as a simple funct ion of T , J(T) = C i T-L ( , t r ) . e s ( H l ) - r -ST 1 -6T = a e . . ( 6 . 1 3 ) for any time horizon T > 0. S ince our benefit funct ion B ( T ) = B(0)e is of the same form, the d i f ference B(0) - a determines the optimal values of T . If B(0) < a, the B ( T ) - J ( T ) increases to 0 as T approaches +°°, so there is no optimal value of T which is f ini te. If B(0) = a , then any t is optimal, and max ( B ( T ) - J ( T ) | T > 0} = 0. F ina l ly , if B(0) > a , then [B(0) - a] e is posit ive and decreas ing on ( - 0 0 , 0 ?) so B ( T ) - J ( T ) has no maximum on ( 0 , ° ° ) . Of c o u r s e , the pract ical interpretat ion of this result is simply that if benefits are high enough to offset the cost of construct ion of the fac i l i ty , the project should be completed as soon as poss ib le . Th is completes our d iscuss ion of the optimization of the time hor izon . T h e treatment of the problem presented here is by no means comprehens ive , but it does provide some general analytical results which make sense and can be used as a basis for numerical analysis of the problem. 69 7. C O N C L U S I O N T h e results presented in this thesis indicate the benefi ts which can be der ived from keeping analytical models relat ively simple in s t r u c t u r e . In contrast with Cummings 1 and Schulze 's model in [5] , our linear model of in f ras t ruc ture investment can be completely solved analyt ica l ly . We were also able to c a r r y out a sensi t iv i ty analysis of the solution and to extend the scope of the problem to take account of a broader class of c i rcumstances . In par t icu la r , the l inearity of the model facilitates analysis of the general project p lanning problem in which economic benef i ts , as well as c o s t s , must be c o n s i d e r e d . T h e s t ruc ture of the model was also shown to be amenable to numerical solution using parameter estimates der ived from actual data . In conc lus ion , t h e n , when compared to more complex models, such a model would seem to better sat isfy the requirements of pract ical pol icy problems. 70 B I B L I O G R A P H Y [1] B e r k o v i t z , L . D . , Optimal Control T h e o r y , S p r i n g e r - V e r l a g New York Inc. (New Y o r k , 1974). [2] C l a r k , C . W . , Mathematical Bioeconomics, John Wiley and S o n s , Inc. (New Y o r k , 1976). [3] C l a r k , C . W . , F . H . C l a r k e , and G . R . M u n r o , "The Optimal Exploitation of Renewable Resource S t o c k s : Problems of I r revers ib le Investment," Econometrica , 47 (1979), 25-47. [4] Cummings, R . C . and A . F . Mehr , "Investments for Urban Infra-s t ruc ture in Boomtowns," Natural Resources J o u r n a l , 17 (1977), 223-240. [5] Cummings, R . C . and W . D . S c h u l z e , "Optimal Investment Strategy for Boomtowns: A Theoret ical A n a l y s i s , " American Economic Review, 68 (1978), 374-385. [6] Cummings, R . C , W . D . S c h u l z e , and A . F . Mehr , "Optimal Municipal Investment in Boomtowns: A n Empirical A n a l y s i s , " Journal of Environmental Economics and Management, 5 (1978), 252-267. [7] Lee , E . B . and L. M a r k u s , Foundations of Optimal Control  T h e o r y , John Wiley and S o n s , Inc. (New Y o r k , 1967). [8] Mehr , A . F . and R . G . Cummings, "Time Ser ies Profile of Urban In f rast ructure Stocks in Selected Boomtowns in Rocky Mountain S ta tes , " Los Alamos Scient i f ic Labora tory , Informal  Report LA-6687-MS (1977). [9] Northeast Coal S t u d y : Report of the B . C . Manpower Sub-Committee on N . E . Coal Development, V i c t o r i a : Min ist ry of Economic Development (November, 1976). 71 A P P E N D I X A T H E A B N O R M A L FORM O F T H E C O N T R O L P R O B L E M We shall refer to the problem descr ibed in Equations (3.1) to (3 .8 ) . T h e condit ions in Equations (3.9) ;to (3.14) are consequences of the Maximum Pr inc ip le . Assume that A Q = 0. T h e n from Equation (3.9) the Hamiltonian is simply fclO^KJ^A) = + Al . . . . ( A . i ) and the adjoint Equations (3.12) and t ransversa l i ty condit ion (3.13) imply that A(t) = y a ( T - t ) J 0 ( t N< T ( A . 2) Subst i tu t ing (2) into (1) , we have X(t ; K,l,/U) = yUK + yll(T't)- ( A . 3) By condit ion (3.11), y £ 0, and from (3 .10) , if I is an optimal c o n t r o l . l ( t ) = 0 , i f jutT-t) < 0 ] I M , «f / U . ( T - t ) > 0 a . e . (A.4) 72 T h e r e f o r e , if y < 0, I (t) = 0 a . e . T h i s control is feasible only if the initial and terminal condit ions are such that C T " C 0 " K 0 T • • ' ' - ( A - 5 ) Simi lar ly , if y > 0, the optimal control must be l(t) = 1^  a . e . , which is feasible only if Cy - CQ = \JAJ_ 2 + K0T ... .(A.6) Z Equat ions (5) and (6) c o r r e s p o n d to equality in (3.8). In either c a s e , there is only one feasible con t ro l ; the constra ints on the problem are too restr ic t ive to allow any optimization. 73 A P P E N D I X B S E L E C T I N G AN IN IT IAL S E T O F P A R A M E T E R V A L U E S T h e initial set of parameter va lues , which appear in the f i rs t line of Table I, were chosen with the following points in mind. For the basic model speci f ied in Equations (3.1) to (3 .7 ) , with C Q = K Q = 0, the parameters which had to be speci f ied were: 6, y, 1 ,^ Cj, and T . The values had to sat isfy inequality (3.8) which can be simplif ied to z (when C Q = K Q = 0.) Most of the data which have been used come from reports and art icles concerned with boomtowns in the Rocky Mountain states of the United States ( i . e . , Co lorado, U t a h , New Mexico, Wyoming; see [2] , [4 ] , [6 ] ) . S ince these data series are expressed for the most part in terms of 1975 do l la rs , this monetary unit was used for the parameter va lues . T h e following notes summarize how the initial set of parameter values was obta ined. j: [0 .5 ,2 .2] "wages" for labour plus unit maintenance costs for i n f r a s t r u c t u r e . T o "estimate" a value for y, we had to refer back to the or iginal model in which L (labour) was a var iab le . 74 y K = wages for labour + maintenance costs for ex ist ing social capital = oo • L + m • K + m • K where to = annual wage for construct ion labourers k = j- = f ixed social capital / labour ratio m = annual maintenance cost per unit of social capi ta l . These three variables were "estimated" as follows: u : [5,000, 15,000] $ / year . 5 From Table I in [4] , a weekly wage of $105.30 was c i t e d . T h i s is equivalent to approximately $5,500 per y e a r , so $5,000 - $15,000 was set as a range for oo in 1975 do l la rs . 75 k : [7,000-10,000] $ i n f r a s t r u c t u r e / l a b o u r e r . In [8] , a range of 2,800 to 3,900 $ in f ras t ruc ture per person has been suggested as a "norm" for per capita in f ras t ruc ture levels (p) in non-boom communities, i . e . , communities in which per capita in f ras t ruc ture is fa ir ly K stable. To obtain a range for k = -j- for our model, . this range for p was multiplied by the factor 2.5 persons per labourer (an estimate for average family size suggested in [6]) . T h i s gave a range of 7,000 to 9,750 for k, which was rounded to [7,000, 10,000]. m: [0.01, 0.10] unit maintenance costs for infra-s t ruc ture in [6] , where m has been def ined as main-tenance and depreciat ion costs per unit of i n f r a -s t r u c t u r e , this parameter has been ass igned a value of 1/30 = .0333, based on the assumption of a th i r ty year lifetime for social capi ta l . T h e interval [0 .01, 0.10] contains values with the same order of magnitude. 76 With these ranges for a i , k , and m, y e [0.5,2.2.] . For the initial set of parameter va lues , we took y = 1.0. C y : [3,000,8,000] man-years or [21,80] million in f ras t ruc ture d o l l a r - y e a r s . In the original model, the state equation and terminal condit ion for construct ion were: C = L C ( T ) - C T so the size of the facil i ty was measured in man-years . For example, [9] states that the manpower content for the construct ion of coal -mining facil it ies may run in the order of 3,000 to 5,000 man-years . K In our model, j - = k = constant , and the state equation and terminal condit ion are C = K C(T) " C T .-. c = k c c ^ k c T . S o , for numerical solutions to our model, C T is measured in units of " in f ras t ructure d o l l a r - y e a r s . " If k e [7,000, 10,000], then the widest range for C y = k C y is 21,000,000 to 80,000,000 in f ras t ruc ture dol lar -y e a r s . A n initial value of C _ = 50,000,000 was c h o s e n . 77 l M : [150,000, 10,000,000] $ / year - maximum allowable rate of investment in social i n f r a s t r u c t u r e . T h i s range was based on data in [6] , on annual investments in in f ras t ruc ture for each of twenty-s ix towns in the Rocky Mountain states. A n "average" value of $5,000,000/year was used for the initial set of parameter va lues . T : With l M = 5,000,000, Cj = 50,000,000, T had to be chosen so that T > A M = 20 « 4 .5 ; the initial value for T was T = 10 y e a r s . _6_: A single "real ist ic" value for the social rate of d iscount would have been dif f icult to Ghoose. One purpose of computing numerical solutions to the basic problem was, in fac t , to test the sensi t iv i ty of the solution for di f ferent values of 6 in a wide range. T h e initial value was a rb i t ra r i l y set at 6 =0.10. 78 A P P E N D I X C M E T H O D O F N U M E R I C A L S O L U T I O N FOR T H E B A S I C M O D E L A method for numerically so lv ing the basic problem, for g iven parameter va lues , follows natural ly from the analysis in Section 3 which shows that the optimal solution is un ique . Al l of the solutions were obtained us ing a computer program which f inds roots t Q ( y ) and t ^ y ) for the switching funct ion <f(t; yu) = A(t) Jl) - £St in a search for the optimal value of y. In the p rogram, the f i rs t trial value of y is chosen so that — 6 t the c u r v e A(t;y) intersects the c u r v e e at t = 0, i . e . , so that MO) = 1 . Since A ( i ; / i ) *J_{thT- 0 $t) t ^ ( T - t ) , b 1 A(O;/JO = 1 —^ r (e6T-i) = 1 1 + 1 d - cbl) b J For each trial value of y, the roots t Q ( y ) and t ^ y ) are calculated us ing the subrout ine R Z F U N which is based on Muel ler 's 79 method. (The routine is available in the publ ic file *NUMLIB at the U B C computing cent re ; documentation for the subrout ine is given in U B C N L E Zeros of Nonlinear Equat ions , F e b r u a r y , 1977). Convergence to the optimal value of y is tested by comparing construct ion at time T , C ( T ; t Q ( y ) , t . j ( y ) ) , with C y . (In the p rogram, these two values are actually "normalized" by 1^). In the subsequent t r i a l , y is decreased or i n c r e a s e d , a c c o r d -ing to the s ign of C ( T ; t g ( y ) , t^(y)) - C T . T h e step size by which y changes is decreased when a success ive value of y g ives a response e r ror for construct ion wh ich , in magnitude, is greater than or equal to the prev ious response e r r o r . T h i s eliminates the possibi l i ty of "bounc ing" back and for th on ei ther side of the optimal y, in a loop which doesn't c o n v e r g e . A l ist ing of the computer program follows. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 30 20 10 IMPLICIT R E A L * 8 ( A - H , 0 - Z ) E X T E R N A L SW R E A L * 8 X ( 2 ) , I M A X COMMON D E L , U , A , B NR=2 M A X I T = 50 R E A D ( 5 , 1 0 ) E 1 , E 2 , E 3 , E 4 , E 5 , E R R , H FORMAT(10D8 .1 ) WRITE(6,30) E 1 , E 2 , E 3 , E 4 , E 5 , E R R , H ITER=0 R E A D ( 5 , 2 0 ) D E L , G A M , C T , I M A X , T FORMAT(6D10 .3 ) T E S T = C T / I M A X WRITE(6 , 3 0 ) D E L , C A M , C T , I M A X , T E S T , T F O R M A T ( / / 7 D 1 8 . 3 ) E D T = D E X P ( - D E L * T ) R = G A M / D E L B = 1 .DO+R C = R * E D T 80 20 ULOW=GAM*EDT 21 W R I T E ( 6 , 3 0 ) E D T , R , B , C , U L O W 22 U = (1 D 0 / T ) * ( 1 . D 0 + R * ( 1 . D 0 - E D T ) ) 23 40 ITER=ITER+1 24 IF ( ITER . G T . 2 0 0 ) G O T O 1 5 0 25 A=C+U*T 26 W R I T E ( 6 , 5 0 ) I T E R , H , U 27 50 F 0 R M A T ( / I 5 , D 1 6 7,D25 16) 28 C S E E IF U IS G T LOWER B O U N D 29 I F ( U . G T . U L O W ) G O T O 53 30 U=(U+H+ULOW)/2.D0 31 H = (U+ULOW)/2 .D0 32 C S E E IF R O O T S EXIT 33 53 T S = ( - 1 . D 0 / D E L ) * D L O G ( U / ( G A M + D E L ) ) 34 Y=SW(TS) 35 I F ( Y . G E . 0 . D 0 ) G O T O 55 36 H=H/2.D0 37 GO T O 120 38 55 X(1)=T 39 X(2)=-0.5D0 40 C FIND Z E R O S O F SWITCHING F U N C T I O N 41 C A L L D R Z F U N ( S W , N R , M A X I T , X , I N D , E 1 , E 2 , E 3 , E 4 ) 42 IF(IND . E Q . 1)GO T O 120 43 T0=X(1) 44 T1=DMAX1(0 .D0 ,X (2 ) ) 45 C C A L C U L A T E C ( T ) / I M A X FOR THIS C O N T R O L 46 C I= (T0 -T1 ) * (T - (T1+T0 ) 2 .D0 ) 47 C T E S T FOR O P T I M A L I T Y 48 ERL=ERR 49 E R R = C I - T E S T 50 A B E R = D A B S ( E R R ) 51 I F ( A B E R . G E . D A B S ( E R L ) ) H=H/2.D0 52 I F ( U / H . G T . 1.D16JGO T O 150 53 WRITE(6 ,60 )X (2 ) ,T0 , C I , E R R 54 60 F O R M A T ( 4 6 X , 201 9. 7, 2D24.16/) 55 IF ( E R R . L E . E 5 ) GO T O 120 56 IF ( E R R . G E . E 2 ) GO T O 100 57 S T O P 58 C D E C R E A S E U 59 100 U=U-H 60 GO T O 40 61 C I N C R E A S E U 62 120 U=U+H 63 GO TO 40 64 150 S T O P 65 END 66 D O U B L E PRECISION F U N C T I O N SW(X) 67 IMPLICIT R E A L * 8 ( A - H , 0 - Z ) 68 COMMON D E L , U , A , B 69 S W = A - B * D E X P ( - D E L * X ) - U * X 70 R E T U R N 71 END 81 A P P E N D I X D P R O O F O F LEMMA 1 Lemma 1: Given e > 0 and a control I which is feasible for P , , 0 0 3 a control I and an integer N such that V n 2 N , I is feasible for Pn , and J {? } s J { l } + e . Proof: We f i rs t consider the case in which I is an impulse control at only one instant , t = 0, and is PWC on ( 0 , T ] . Let ( C , K) be the response trajectory for I, and Let A K be the increase in K at t = 0. Since K E I > 0, K is n o n - d e c r e a s i n g . We want to define a PWC control I which approximates I in c o s t , so K, the response trajectory for I , must be "close" to the trajectory K. In add i t ion , I must be feasible , so K must sat isfy T f K(t)dt = C • • • - ( D . D o o r , equ iva lent ly , T I (K-K) dt = 0 . . . . (D .2 ) 0 We approach the feasibil i ty requirement as follows. Cons ider any number a e (0 ,T ) which is small enough so that the stra ight line through the or ig in and (a , K(a)) c rosses the trajectory K(t) only at t = a and 82 K(a) - T = OL 'or ' T > C (D.3) as shown in F igure 9. For each e [ a , T ] , define a 2 e [ a 1 # T ] as OL m m It | K(t) = U-oi,) , if l.-rt, < K(T) I T , if le.-a, > K(T) }. . . C D . U ) Let the funct ions K (t) and I (t) be def ined by a, a. a, a 1 K.- (t) l « ' t # t 6 CO,* ,) K ( t ) , t e (<X2;TJ U ( 0 s K (t) , t t [O,T] (D.5) (D.6) T h e n K is piecewise smooth, so I is PWC. When a, = a. a, a. 1 a, a. 1 T K , - ( 0 d t c\ T K w o ((t)dt = | KloO-tdt + j K(t)dt 0 <*- o< T < f K(t) dt = C T ' and if a 1 = T , Figure 9 Graph of K a, a, 84 J Krfol(i)dt = j K M tdt - KCoiJ -T >C o ' 1 o a by ( 3 ) . For f ixed a, the integral K (t)dt changes cont inuously 0 0L,a^ with o^, so there must be exactly one value of e ( a , T ) for which T K*<t)dt - CT • — ( D - 7 ) I .e. , for each a, there is a unique = g(a) e (a,T) with a =h(a) e[a,,T] so that K , v satisf ies (7). T h i s means that 2 1' a , g(a) I , , is feasible for P , for large enough n . Hencefor th , for a g iven a, we shall refer only to the feasible control I = I . * and 3 ' a a , g(a) its response K = K , . . We claim that ^ a a, g (a) lim = . . . . . (D .8 ) a->0+ From the preced ing def in i t ions, it is clear that g and h are cont inuous and that j(a) 1 0 as o(l 0 . . . . (D .9 ) L t oo as d l 0 . . . (D. l0) ; L-gC"-) - ^ ( g C o O ) I A K as o ( | fl . . . (D.11) 85 T h e behaviour of h ( a ) as a -> 0 depends on the control l ( t ) . If l(t) > 0 on ( 0 , p ) , for some p > 0, then K(t) is s t r ic t ly increasing on ( 0 , p ) , so by Equation (11) and (4) , h ( a H 0 as a i 0, as shown in F igure 10(a). T h e other possibi l i ty is that l(t) = 0 on the interval (0 , p ) for some p s 0, and in this c a s e , h ( a ) I p as a H (shown in F igure 10(b) ) . T h e limit in Equation (8) can be ver i f ied separately for each of these two possibi l i t ies . Because the controls I and I are identical on the interval a [ h ( a ) , T ] , •"ll-,] " J( l ) hfa) I e 6t { t U }dt - j e " { y K + i } d t h(«0 si o q(«0 9W e 5 t ( r K a t i j d t - j e S t [ r K +1} dt ht-) 9 W AK AK (D If the f i rs t term on the r ight side of Equation (12) is integrated by p a r t s , we obtain qfrO e (n.-t + i j d t Figure 10 Tra jectory K(t ) determines p : (a) p = 0; (b) p > 0 87 = r I (1 7* * •696*) 0 / b 1 »*Q(«) 6 J (D.13) In o rder to take the limit of Equation (13 ) , as a -> 0, we expand e 5 9 ( a ) in a Maclaurin se r i es , so the f i rs t term on the r ight side may be written \ I, -gW j 1 - 0(g(«)) i r 1 (D.14) Subst i tu t ing this express ion into Equation (13), we have g(«0 lim j . C 6 t { Y + IccJdt <*-»0 0 Um i d <x-*0 g c « 0 • - 0(g («))'] - r = AK • 1 (D.15) by (9) and (11). T h e second integral on the r ight side of Equation (12) is easily d isposed of: 9W lim j e { YK + !} d-t g(«) = lim I e 5 t 1 Y K +1 j di = = 0 .. . . . (D . 16) 88 because the integrand is cont inuous (for small enough a ) , and bounded on ( 0 , g ( a ) ] . S imi lar ly , the th i rd integral in Equation (12) also approaches 0 as a -> 0, a l though this should be ver i f ied for the case in which lim h ( a ) = p > 0. A s we noted ear l ier , p > 0 iff l (t) = 0 for a - *0 t e ( 0 , p ) , in which c a s e , K(t ) E A K on ( 0 , p ) . T h e r e f o r e , j t [r IVcjCoO - Ka)] - l ( t ) ) dt 9W KW K(«*) KG*) f -St ( -st ( -${ r i w . - q W i e dt - Y ) e - K ( i ) d t - j e i a ) d t J ab*) g(cO gC«0 P A P P YAK ) e dt - Y J e • K(t) dt - j & • l(t) dt 0 0 0 9 Y A K J -St e dt - Y e 5 tAK dt - 0 0 as a —* 0 . . ( D . 17) Combining Equation (12) to (17) ver i f ies our claim in Equation (8) , ie. T h u s , g iven e > 0, we may choose a > 0 such that 89 Let N = smallest integer >. sup x { l (t) } (< °° because I is PWC) . T h e n the control ? = I is feasible for P , V 2 N , and J{f} 5 J {I } + e . a n n ' T h i s completes the proof of the lemma for a control I which has only one impulse, at t = 0. T h i s proof can easily be extended to cover all other feasible controls for P by const ruc t ing a PWC control I , which approximates I at each of its k ( < ° ° ) i m p u l s e s , ^ as indicated in F igure 1 1 . T h e proof can also be modified for the case of an impulse at T = T , by def in ing an approximat ing control with h(a) ^ g(a) <a<T. However , any control with an impulse at x = T cannot be optimal, for the same control without the jump in capital at T is feasible and costs less . For our application of the lemma in Section 5, this case can be expl ic i t ly exc luded from the lemma. Figure 11 Const ruct ion of K when K has more than one jump 

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