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Essays in comparative dynamics Davidson, Russell 1977

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ESSAYS IN COMPARATIVE DYNAMICS by RUSSELL DAVIDSON B.Sc, University of Glasgow, 1963, Ph.D., University of Glasgow, 1966. A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY THE FACULTY OF GRADUATE STUDIES (Department of Economics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1977 in Russell Davidson, 1977 In presenting th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l ica t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ttk A U S O S S T (l77 i Research Supervisor: Professor G.C. Archibald. ABSTRACT. Problems in the theory of economic dynamics are tackled both by theoretical arguments and by use of specific examples. The work is divided into three essays. The f i r s t treats optimal control theory from an economic point of view, giving an exposition of the mathematical theory in terms of economic concepts. The idea of the marginal worth of time i s introduced and found to be useful in a variety of problems. An interpretation i s given of the phase planes of optimal control problems as demand-and-supply diagrams. The second essay makes use of the techniques developed in the f i r s t to solve the problem of when and how a firm faced with adverse economic circumstances w i l l choose to go out of business i f i t s operations depend on a stock of some fixed asset that depreciates over time. A straightforward catalogue i s presented of different possible outcomes. The third essay deals with a model of urban housing. It contains two main sections. In the f i r s t , an equi-librium state i s described i n which demand by tenants for housing is met by supply from landlords who act as p r o f i t maximisers over the whole period of time that their property exists. The rent paid for any particular dwelling i s assumed to depend on i t s state of upkeep, which in turn depends on how much i s spent by a landlord on maintenance. The equilibrium i s found by a procedure analogous to that regularly used in general equilibrium theory, namely by finding a fixed point of a mapping in a (here infinite-dimensional) i i vector space. In the second section of the essay, i t i s assumed that some externality arises which adversely affects urban l i f e and which provokes people to move out to suburbs. The consequences of this are studied and two different kinds of dynamical evolution can be distinguished. One, in which house construction i n the suburbs i s slow enough to make i t necessary for new construction to continue in the cit y , tends not to be disastrous for the city; the other, in which a l l urban construction stops when the externality arises, usually leads to complete decay of the city. Throughout the thesis there i s an emphasis on the differences i n approach between static or quasistatic problems and dynamic ones. i i i CONTENTS. Abstract i Contents i i i Li s t of Tables v L i s t of Figures v i Acknowledgements v i i INTRODUCTION 1 CHAPTER I: 7 An Economic Interpretation of Optimal Control Theory. 1. Dynamic Programming and Optimal Control 9 2. Discussion of* the Algorithm 19 3. Transversality 23 4. The Marginal Worth of Time 30 5. The Phase Plane 41 CHAPTER II: 45 A Stock Adjustment Model and the Problem of Optimal Exit. 1. Specification of the Model 46 2. Comparative Dynamics 55 CHAPTER IIIA: 64 A Model of Urban Housing. 1. The Landlord's Profit 66 2. The City Dweller's Choice of a Home 71 3. Equilibrium between Landlords and Tenants 72 CHAPTER IIIB: 84 A Model of Urban Decay. 1. The Flight to the Suburbs 86 2. A Particular Case of the Model of the City 92 i v 3. The Case of Rapid Suburban Construction 102 4. The Case of Slow Suburban Construction 111 5. Summary and Conclusions 119 POSTSCRIPT 123 BIBLIOGRAPHY 127 LIST OF TABLES. Table I: Stock and Shadow-Price Equations. v i LIST OF FIGURES. Figure 1 28 Figure 2 29 Figure 3 ' 42 Figure 4 50 Figure 5 51 Figure 6 54 Figure 7 56 Figure 8 57 Figure 9 60 Figure 10 61 Figure 11 69 Figure 12 69 Figure 13 94 Figure 14 94 Figure 15 98 Figure 16 98 Figure 17 101 Figure 18 114 v i i ACKNOWLEDGEMENTS. It i s with much pleasure that I acknowledge the great deal of help and support that I have received from many members of the UBC Economics Department. F i r s t , my heartfelt thanks go to Don Paterson, who rescued me from Ec. 100 and persuaded me to study economics more seriously. When I joined the department, I was encouraged and stim-ulated by Chuck Blackorby, David Donaldson, Curt Eaton, Keizo Nagatani, Phil Neher and many others. The debt I owe them is immense. Special thanks must go to Chris Archibald, chairman of my thesis committee, who, no less an intellectual companion than anyone in the; department, managed also to make me comply with deadlines and the requirements of procedure, things I am not very good at. It i s the friendship offered me by a l l the people I have named that I rejoice in most, and what I say here must be quite inadequate. Grateful acknowledgement i s made of financial support from the Canada Council, in the form of a Doctoral Fellowship. INTRODUCTION. The common theme of the three essays of this thesis i s , as the t i t l e would suggest, economic dynamics. A desire to see into the future, with especial regard for one's own fortunes and well-being, has been implanted in mankind from the beginning. Necromancy and augury are among the time-honoured techniques for accomplishing this desire - they have a much longer history than s c i e n t i f i c , or even pseudo-scientific, analysis, and numerology and astrology are their spiritual children. The study of economics i s often grouped with the above practices, not necessarily to the distress of i t s professional advocates, but very much to the denigration of i t s avowed means and ends. Imputations of wizardry can be hard to deny although they can be ignored by practitioners of mature sciences. Ignoring i s perhaps the wrong response, for i t provides no corrective to super-st i t i o n when needed. Thus the natural philosophers following Galileo and Newton were content that they had escaped from the mental chains of medieval thought and alchemy, but did not always understand that their achievementsVwere taken by many as just somewhat better magic than what was fashionable i n earlier times. So i t was that their spells were seized upon by lesser men as panaceas for a l l earthly disorders, and even by some great men struggling with matters less tractable than the inanimate physical universe. For Marx had his incantation quite correct. The "laws of motion of capitalism" i s a phrase that has called up a s p i r i t by no means exorcised 2 i n our times. The great d i f f i c u l t y with such sp i r i t s i s that no one knows i f they can ultimately be taught the catechism and made to s i t with a l l proper appurtenances and apparel i n the halls of true science and knowledge, or i f they are of the other kind of s p i r i t , demons, which do nothing but torment our minds with vain longings that can never be satisfied. I have sported with this s p i r i t , then, i n these essays. My leaning i s towards the hall of science, and my effort i s directed towards developing a l i t t l e piece of economic theory that w i l l help the understanding of what sorts of questions about economic dynamics can reasonably be treated by use of the wits and intuition that the human race has succeeded in getting for i t s e l f at the present time. Discovering the most sensible questions to investigate i s one of the d i f f i c u l t things about the study of comparative dynamics. It i s for this reason that I do not start off, in decent fashion, by defining what I mean or what i s generally meant by comparative dynamics. Comparative statics i s nowadays an established discipline, i t s rules laid out as formulae to be applied mechanically. In a sense, i t would be good i f comparative dynamics were in the same case, so that there i s in this thesis a good deal of laying down of rules and procedures. But to define i s to lim i t , and since no one knows where i t might be best to set the limits of comparative dynamics, i t i s foolish to define with any precision. An alternative way to go forward i s to look at specific problems that involve dynamical considerations, and that i s the way chosen here. It points up the conclusion that comparative dynamics has a quite different flavour from comparative statics. It i s not just that the well-known techniques of total differentiation followed 3 by attempts at signing bordered Hessians no longer seem so immediately applicable, but also that the economic matters that are to be taken into account have no analogue in static problems - they are not just extensions of static concepts with time thrown i n as an extra variable. The tone of the three essays that comprise the thesis varies somewhat from one to another. This i s due to the variety of tasks undertaken. In the f i r s t essay, the aim i s to consolidate as a standard box-of-tricks for economists' use the mathematical theory of optimal control. There are numerous economic studies that make use of this theory, often very expertly. Consequently my emphasis has been on expounding, on demystifying as i t were, and the resulting tone i s allusive and sometimes chatty, my hope being that i t i s also evocative and illuminating. The second essay arose from attempts to solve two seemingly unconnected problems. The f i r s t one, which in my mind has attached to i t the name of "the bankrupt railroad", was suggested to me by Professor Archibald and deals with the response of a firm which possesses substantial fixed assets when i t runs into d i f f i c u l t i e s . The second problem was posed to the Economics Department at large by Professor Nagatani and i s , essentially, the one ex p l i c i t l y treated here. The two problems are the same on a suitable level of abstraction, both contain a basic and typical question i n economic dynamics, and both are solved by the same device, the explication of which forms the content of the essay. The use of mathematics i n the last essay of the thesis i s much heavier than i n the other two, and the effect of this i s - to my regret - to make the tone much heavier too. But i t i s unavoidable at present - instances are very rare of a new sort of investigation being presented for the f i r s t time i n i t s most perspicuous form. I hope that the gain in understanding i s worth the cost. 4 Optimal control theory is what makes comparative dynamics possible, at least in a practical sense. F.P. Ramsey's pioneering article (Ramsey (1928)) makes no use of i t because i t wasn't invented then, but i t i s quite fascinating for someone who knows the modern theory to see how much of i t appears there i n only a slightly different form. His formal tool i s of course the calculus of variations, and i t can now be seen that, i f one works hard enough at i t , (Hadley and Kemp (1971)) optimal control theory can be devvoed from the calculus of variations. But the insights gained by physical scientists as well as economists from the twin notions of the principle of optimality (Bellman (1961)) and the maximum principle (Pontryagin et al (1962)) are much more numerous than those that the classical approach can provide. It i s striking that Ramsey supplements his calculus of variations argument with an economic line o'f reasoning that he ascribes to Keynes and that has much of the s p i r i t of the two principles which underlie optimal control theory. The f i r s t essay, then, is i n the tradition of Ramsey-cum-Keynes and Bellman, and away from the work of Hadley and Kemp. Growth models have been at the centre of economic dynamics at least since Ricardo, and certainly with Marx, although i t was not exactly growth in the modern sense that Marx was concerned with. Modern interest i n growth theory began with Harrod (1939) and Domar (1946), but no one at that time conceived of economic dynamics as other than a descriptive discipline that could, to be sure, warn against some kinds of danger. (Ramsey's arti c l e stands outside this discussion - i t was really much ahead of i t s time.) Rates of capital accumulation were derived from purely ad hoc 5 descriptions of saving behaviour, and the social benefits arising from different specifications of behaviour compared. It would be wrong to deny the name of comparative dynamics to these exercises, but there was precious l i t t l e economising going on i n the models used. That i s , there were no economic agents exercising rational choice over their behaviour. This was true also of the celebrated von Neumann growth model (von Neumann (1945)) , and even of the subtle model proposed by Solow (1956) with i t s exogenous savings ratio. Phelps (1961), i n his critique of Solow1s model, returned in s p i r i t to Ramsey, and the word "optimal" reappears i n the growth literature, just at the time when by a fortunate coincidence U.S. and U.S.S.R. military needs caused Bellman and Pontryagin respectively to produce what i s now called optimal control theory. By the time of writing of articles by Kurz (1968) and Hahn (1968), this theory was taken for granted by growthmen, and a l l sorts of problems to do with s t a b i l i t y and so forth were being encountered. These problems have led sophisticated mathematical economists l i k e Brock (1976), Brock and Burmeister (1976), Brock and Scheinkman (1975) (see also other articles l i s t e d by these authors in the cited works and the "Symposium on Hamiltonian Dynamics in Economics" published as the Feb. 1976 issue of the Journal of Economic Theory) to study intently the sta b i l i t y of dynamical systems and to propose the resurrection of Samuelson's suggestion that the assumption of st a b i l i t y can yield meaningful theorems in economics. (I cannot resist citing Hatta (1977) as a beautiful example of an article that uses this suggestion with great effect i n comparative statics.) 6 Of more immediate relevance, i t seems to me, than this very technical work i s the systematic use of optimal control theory i n microeconcmic studies as opposed to the macroeconomics of growth. The principal example of this that I have i n mind i s Mortensen's article on wage and employment dynamics (Mortensen (1970)), which has been followed by various articles on job search by employees and labour hiring by employers written from an ex p l i c i t l y dynamical point of view. In this literature the optimising behaviour of economic agents comes to the fore. It i s certainly premature to claim that economic theory contains a worthy set of "micro-foundations" for macroeconomic phenomena, but i t i s hard to doubt that such foundations w i l l be necessary for a decent understanding of them. It i s my hope, then, that applications of optimal control theory w i l l ultimately provide enough insight into comparative dynamics in microeconomics that one w i l l be able to return to growth theory and actually succeed in predicting a l i t t l e b i t of the future of our fortunes. CHAPTER I AN ECONOMIC INTERPRETATION OF OPTIMAL CONTROL THEORY In this essay, optimal control theory, the cornerstone of comparative dynamics in i t s present state of development, i s discussed from an economic point of view. Much of the subject matter w i l l seem routine to those who regularly use optimal control methods, but although there are many economists among these people, I have the feeling that most of them share my view that the theory has not yet been f u l l y " c i v i l i s e d " , that i s , translated out of the language of engineering physics, the discipline responsible for i t s creation in i t s modern form, into a language of prices, margins, demands and so on easily comprehensible to economists and such as to make the optimal con-t r o l equations satisfying to economic intuition. A stylised problem is treated at f u l l length in the remainder of this essay, and i t is my hope that the treatment, which is throughout presented in economic language, w i l l act as a sort of translation of the theory. In section 1, the well-known optimal control algorithm is derived from f i r s t principles. A form of argument much used in duality theory is found useful i n deriving the equation of motion for the co-state variable. Then, in section 2, comes the main effort of "translation". The algorithm i s picked apart and a l l of i t s variables, functions, and equations discussed as economic entities. After this work, i t becomes possible in section 3 to discuss the matter of transversality conditions as a series of stories in economics, by use of no more than the everyday arguments of economic reasoning. In section 4, I have discussed the notion of time as a factor of production 8 and the r e l a t e d concept of i t s marginal worth. I t turns out to be possible to define t h i s concept quite p r e c i s e l y , and to use. i t e f f e c t i v e l y i n the treatment of autonomous problems and problems with discounting — these problems are worked out as examples. F i n a l l y , i n section 5, i t i s shown that, i n many cases, the phase-plane, with i t s saddlepoint equilibrium ^ and catenary-like motion i n the v i c i n i t y , can be seen as a s t r a i g h t -forward supply-and-demand diagram with adjustment mechanisms around equilibrium included: j u s t what i s required by Samuelson's correspondence p r i n c i p l e i n f a c t (Samuelson [1947], p. 253 et seq., p. 350). I have tried to keep a l l the discussion of t h i s chapter uncomplicated as possible as regards the mathematical equipment used. Much greater ge n e r a l i t y i s e a s i l y a ccessible with l i t t l e extra e f f o r t , but i t was not at a l l my aim to i n d i c a t e the richness of optimal control theory or i t s applications — that i s done i n (probably) hundreds of e a s i l y a vailable references. (For example, Bryson and Ho [1969] , Bellman [1967], Lee and Markus [1967]). My aim, on the contrary, was to demonstrate that the theory could j u s t as well have been developed by economists, using t h e i r own t o o l s , techniques and language, as by engineers, i f only t h e i r r e -search needs i n the l a t e f i f t i e s and e a r l y s i x t i e s had been so clamant and t h e i r work so heavily funded. Such a demonstration could well be quite p o i n t l e s s , but i t does seem to me that the concepts developed here are both u s e f u l , and i n some cases, novel. So f a r as I know, no one, not even Arrow [1968], has yet pointed out i n the economics l i t e r a t u r e that the marginal worth of time i s constant along optimal paths; and the i n t e r p r e t a t i o n of the phase-plane as a supply-and-demand diagram i s c e r t a i n l y new. 9 1. Dynamic Programming and Optimal Control The problem to be considered in this section is that of maximising the functional J[c] = f F(k,Ct) dt (1) where c, the argument of J, i s a function of time t, defined on the inter-val t Q t <_ t^, and the function k, defined on the same interval, is given as the solution of the ordinary differential equation k(= = f(k,c,t) (2) dt with boundary condition k(t Q) = k Q (3a) We r e s t r i c t the domain of functions c over which maximisation takes place by imposing another boundary condition k(t f) = k f (3b) The problem is controllable i f there exist functions c which allow equation (3b) to be satisfied. Thus k i s completely specified once c is given. In the expressions F(k,:c,t) and f (k,c, t) , what is meant'is that both F and f are functions of three variables, and that each is evaluated at the point (k(t), c(t) , t) . But for simplicity the arguments of k and c w i l l be omitted unless confusion is l i k e l y . The variable names, k and c, are meant to be suggestive. One may think of k as a capital stock and of c as a rate of consumption. Then the differential equation (2) w i l l express the rate of capital accu-mulation (investment) as a function of the existing capital stock, k (let the labour supply be fixed, for example), and the rate at which output is 10 consumed. A common form that eguation (2) might take is k = g(k) - Sk - c where g is a production function and 6 a depreciation rate. The equation t e l l s us that consumption plus net investment, c + k, equals total output g(k) less the amount of output, 6k, needed to offset depreciation. The function F(k,c,t) may perhaps measure the u t i l i t y of consumption, or the p r o f i t a b i l i t y of producing consumption goods, or in general the worth of some benefits. The fact that J[c] is given as an integral means that these benefits accrue addi-tively over time. If F is interpreted as a u t i l i t y function, then this form of J[c] implies additive u t i l i t y in the intertemporal sense; that i s , no intertemporal complementarity. Since this notion is a r t i f i c i a l , i t is probably better to construe F as a measure of p r o f i t . I shall try to refer to i t consistently as the "bene-f i t " . The equations (1) and (2) define a standard optimal problem, and i t i s as well to introduce the standard terminology now. The function c, which determines, both directly via i t s appearance in the benefit func-tion F, and indirectly via k, the value of "total benefit" J, i s called the control Variable. We can assume that c can be chosen quite arbitrarily, or, i f i t is preferable, we can restr i c t i t in some way. The two cases are equally easy to handle, conceptually at least. It is important to understand that the control variable c, and only c, is at our disposal. Within the confines of whatever restrictions are imposed, i t is to be chosen so as to maximise benefits, that i s , J. The other variable, k, is 11 not at our disposal, except indirectly. Once c i s chosen, k i s determined by equation (2) and one of the boundary conditions (3) . Consequently, k i s called the state variable. i t affects benefits (by appearing as an argument in F) and so has to do with the state of. affairs. i In order to find the controller (as i t is often called), c*, which gives a maximum of J[c], we shall use the approach of dynamic programming. This approach rests on a very general principle, the Principle of Optimality (see Bellman [1961]). According to this, whenever there exists an optimal way of achieving some end or of carrying out some activity which proceeds by a sequence of steps, each step contributing additively to a "performance criterion" (for our purposes, -the criterion i s J[c]), then i f one breaks in on the sequence part of the way through, the s t e e p s from that point of break-in u n t i l the end of the sequence must be optimal for the problem defined over those stages alone. For our problem/ this means that i f c* is the optimal controller for the problem of equations (1) and (2) with boundary conditions (3), then the same function c*, is the optimal con-t r o l l e r for the problem: t max J[c] = I f F(k,c,t) dt 1 with k = f(k,c,t) and boundary conditions (4) k(t 1) = k*(.t ) k(t f) = k f where by k*(t^) we mean the value of k arrived at by time t^ i f the path defined by c* has been followed from time t with k(t ) = k . The Principle of Optimality is both general and t r i v i a l . Its truth follows from a short veduotio ad absurdum argument: If another function, c**, say, defined for «t <_ t f gave a larger value of J[c] than c*, then the function defined by c*(t) t < t < t, rc*(t) t Q < t < t l 3 (t) = < (c**(t) t <_ t<<_ t f would yield a larger value of J[c] over tn <_ t <_ t than would c*. But this contradicts the definition of c* as the optimal controller. This kind of argument is precisely the one used in a l l stepwise optimisation procedures. Back to the problem at hand. Let us define another functional closely related to J[c]: t J(k , t, C ( T ) ) E / f F(k (T) , C ( T ) , T ) dx t<x<t * where k is defined by k = f(k,c,t) k(t) = k ; k(t f) = k . This then i s a functional of the function c, defined on t <_ x <_ t^, with extra dependence on the numbers k and t. This new J is just the perform-ance criterion to be maximised for'a problem like the old one in a l l aspects except i t s time horizon and i t s i n i t i a l value of the state va r i -able. Now l e t us imagine that the maximisation has been performed. Then we w i l l write: J*(k ,t) = max J(k , t , t c ( x ) ) T • C ( T ) ^ f t f 13 The star indicates that.the maximisation has been done. There w i l l be a certain function, c*, say, which causes J to take on i t s optimum value J*. That i s : J*(k ,t) = J(k , t, C*(T)) t<T<t Now fcf J(k. , t, C * ( T ) ) = / F(k * ( T ) , C * ( T ) . t )dT t<T<t f t [k* is the solution of the defining differential equation with c = c*]. The integral can be s p l i t up: J(k , t c * ( x ) ) = / ^ + 6 t FdT + f * . FdT t ' t<T<t f t t + 6 t t Now for the Principle of Optimality: the second term here F must be the optimised value for the problem beginning at t+St with boundary condition k(t+ 6 t ) = k*(t+St). This means in fact: J(k , t, C*(T)) = J*(k ,t) t<x<t = / ^ + < S t F ( k * ( T ) , C*(T) , x )dx + J*(k*(t+ 6 t ), t + 6 t ) . (5) It should be noticed here that J* i s simply an ordinary function of the two variables k^_ and t, once the maximisation has been done. If we may cheerfully assume a l l the continuity and diffe r e n t i a b i l i t y that we need, 14 then J * can be expanded i n a Taylor series about ( k t , t ) . (It i s not the i n t e n t i o n here to worry about t e c h n i c a l d e t a i l s , and such "nice" properties w i l l always be taken f o r granted. There are e n t i r e l y reasonable s u f f i c i e n t conditions for these properties to hold. For statement and proof of Taylor's theorem, see Hardy. (1952), p. 286.) Thus we may write: J * ( k * ( t + 6 t ) , t + 6 t ) = j*(k*(t) , t) + J*(k*(t) , t ) 6 k + J*(k*(t) , t ) 6 t + o ( 6 t ) . * * The notation i n t h i s eguation i s as follows: and are the two par-t i a l d e r i vatives of J * ; 6k = k*(t+<5t) - k*(t) = k*(t + 6 t ) - k by the boundary condition; the expression o ( 6 t ) denotes any quantity, X, say, such that lim. — = 0 . 6t->0 In the equation above, o ( 6 t ) s i g n i f i e s terms proportional .to higher powers of 6 t than the f i r s t (for more d e t a i l s , see Hardy [1952], p. ,183) . I t follows at once from the d i f f e r e n t i a l equation (2), which the function k* must s a t i s f y that: * dk 5k = — — fit + o(St) dt = f(k » c * ( t ) , t ) S t + o ( 6 t ) . Therefore equation (5) becomes: t + 6 t •J*(k » t) = / F ( k * ( x ) , c * ( x ) , T)dT + j*(k , t) + [ j * ( k . , t ) f ( k . f c * ( t ) , t) ';; t 1 t t + J*2(k , t ) ] 6 t - ' + 0 ( 6 t ) . 15 The integral of course can be written: / £ + 6 T F ( K * ( T ) , c * ( x ) , T)dx = F(k , c*(t) t)6t + o'(St); so that J*(k t, t) = [F(k t,c*(t), t) + J*(k t,t) f ( k t , c*(t), t) + j*(k ,t)]6t + J*(k t,t) + o(5t) There are now only two terms which involve the control variable ex p l i c i t l y , and in these only i t s value at time t appears. The optimality for other times is b u i l t into the definition of J*. For time t, i t can be made quite explicit: J*(k t,t) = J*(k f c,t) + J * ( k t , t ) 6 t + max [F(k t,c(t),t) + J*(k f c,t) f ( k t , c ( t ) , t ) ] 6 t c(t) + o(6t) . This i s true because c* (t) is the optimal controller. If we let <5t-*0, we obtain, recalling the definition of o(6t): J*(k ,t) + max [F(k ,c(t),t) 1 c(t) + J*(k t,t) f (k f c,c(t) ,t)] =0 (.6) where the value of c(t) which yields the maximum i s c*(t). This last equation i s called the Hamilton-J'acobi equation, by analogy with an 16 equation known for some time in classical mechanics. Because of i t s newly-discovered use in dynamic programming, i t i s also called the Hamilton-Jacobi-Bellman equation (see Bellman [1967], chapter 5). Equation (6) already contains the whole of the optimal control algorithm for our problem. A l l that remains for us to do i s to dig out of i t the well-known form of the algorithm. The content of the equation can be expressed particularly simply i f we make the following definition: H(k,c,X%t) = F(k,c,t) + Xf (k,c,t) (7) This function of the four variables k, c, X, t, is called the Hamiltonian of the problem. In classical mechanics, the Hamiltonian i s identified with the energy of a system: i t is the function which determines the evolution of a mechanical system through time. In economics, the mean-ing of the Hamiltonian i s somewhat similar, but i t i s susceptible, a l l the same, to an intuitively appealing and purely economic interpretation. By use of equation (7), equation (6) can be written: * it J 2(k f c,t) + max H(k ,c(t), J (k ,t),t) = 0 c (t) * with the Hamiltonian function evaluated at X = J^( k t , t ) . From the defini-tion of c*(t), we have: max -H(k ,c(t) , J^k^t) ,t) = H(k t,c*(t), J (k jt^t) c (t) This i s the f i r s t part of the optimal control algorithm: the optimal con-t r o l l e r c* maximises, at each point in time, the value of the Hamiltonian. * The Hamiltonian as we are using i t , however, depends on J1 (k^,t). But let 17 us simply treat this quantity as an unknown (as yet) function of time X(t), and then pin down i t s behaviour by a differential equation. To do this, i t i s necessary to recall that k = k* (t) . The function X(t) i s then de-fined: X(t) = J*(k*(t) ,t) , (8) so that * dk* * * * J l l IT + J12 = J l l + J 1 2 ( 9 ) This last quantity can be obtained directly by differentiating equation (6) with respect to k^ _. But perhaps the most illuminating way to proceed, because of the similarity of the technique to one used constantly (see Gorman[1968]) in duality theory, i s to note that equation (6) says: * max {J (k*(t),t) + F(k*(t),c,t) c + J*(k*(t),t) f (k*(t) ,c,tj}= 0. Therefore, for any k ? k* (t) , J 2(k,t) + F(k,c*(t),t) + j * ( j ^ t l f (k,c* (t) , t) <0 because c*(t) i s the optimal controller i f k = k*(t), but not i f k ^  k*(t). Equality occurs in this expression only i f k = k*(t), which i s thus a max-imum point of the expression. So then the derivative with respect to k is zero for k = k*(t): J^ 2(k*(t),t) +F (k*(t), c*(t),t) + J* 1(k*(t) ,t) f(k*(t), c*(t),t) + J*(k*(t),t) f x ( k * ( t ) , c*(t),t) =0. 18 But from equations (8) and (9), * * A = J f(k*,c*,t) + j = -F 1(k*,c*,t) - * f (k*,c*,t) 5H(k*,c*,A,t) -9 k ( ' And this i s indeed the second equation of the optimal control algorithm. * We have therefore ju s t i f i e d the use of J^(k*(t),t) as A (t) , the co-state variable. Now we-have a l l that we need. The equations (2), (10), and (3) give k = f(k,c,t) = | Y (k,c,X,t) (11) A = - — (k,c,A,t) (12) k(t Q) = k Q; k(t f) = k f (13) to be satisfied by the optimal paths of the variables k, c, A, and further we know (p.*17) that H(k*,c*,A*,t) = max H(k*,c,A*,t) (14) c This i s a l l of the optimal control algorithm. It is enough to determine the optimal path completely, since (14) gives c* as a function of k* and A* which can be substituted into equations (11) and (12), two first-order differential equations, with two boundary conditions, eguation (13), and thus quite determinate. 19 Sometimes the algorithm i s quoted with a Hamiltonian, H*(k,A,t) which i s the result of the maximisation of equation (14). The differential equations are given simply as k = ||1 (k,A,t) 3H* A = - — — (k,A,t) (boundary conditions as before) ok This i s no different from the algorithm as we have quoted i t , since, for instance, (k,A,t) = |S- (k,c*,A,t) + |S- (k,c*,A,t) 1^  3H but -— = 0 for c = c* by the maximality relation (14) . [When the maximum 3c of equation (14) is not an interior one, but i s on the boundary of the admissible set of c's, then the argument must be modified in a way familiar enough i f one knows the Kuhn-Tucker equations. The result is unchanged]. 2. Discussion of the Algorithm The derivation of the optimal control algorithm of the preceding section has been rather mathematical, and i t was based on the Principle of Optimality, which i s a very general principle. It i s possible to give much more economic insight into the workings, now that we have seen a l l the relevant mathematical relations. The whole problem can be thought of as being the finding of an expression for derived demand. The economic notion behind derived demand for anything, be i t labour, capital or any-thing else which yields benefit not only (or not at all) directly but also 20 by being used in some way, is that, i f efficiency of use is guaranteed somehow (as, for instance, by competition for resources in a state of perfect information) then the worth of a stock of the thing i s given by the value of the maximum benefit the stock can yield (that i s , i f i t i s used optimally) and the derived demand is the worth at the margin. When the problem of eguations (1), (2), (3) is posed then, what the answer, J * ( k 0 , t Q ) t e l l s us is the worth of a stock k Q which can be used between the times t and t^ with the constraint, equation (2). If an economic agent is given the choice of having the stock k^ under those conditions or not, i t follows at once that J*(k ,t ) is the demand price (assuming that the agent is rational) that he is willing to pay 3J* to have the stock. If k is divisible.on the other hand, then is 0 ^ 8k Q the demand price of another unit of stock — the marginal worth of the stock. These remarks apply just as well at any time t ( t <t<t^) to J*(k(t),t). A l l of the above is just an elaborate statement of the definition of the maximum benefit a stock can yield, and consequently the derived demand for i t . But i t enables us to interpret the various parts of the optimal control algorithm. From equation (6), i t is seen that c*(t) is the value of the control variable at time t which maximises F(k(t) ,c(t)t) + J* (k(t) ,t) f (k(t) ,c(t) ,t) (15) Now F(k,c,t) i s the integrand of the objective functional J, and so i t measures the rate at which (at time t) benefits are accruing. We may * c a l l i t the current rate of satisfaction. J (k(t),t) has just been 21 identified as the marginal worth of stock — the derived demand for another unit of i t . But f(k,c,t)-is just k, the rate of accumulation of stock. So the second term in the expression (15) measures the rate of accumulation of benefits to be reaped in the future. Thus the whole expression (15) measures the total rate of acquisition of benefits, both current and expected and i t i s not hard to see why c*(t) i s chosen to maximise i t . The optimal c*(t) gives that particular trade-off be-tween current satisfaction, F(.k,c,t), and future satisfaction which maxi-* mises total benefit. Since, once X(t) is substituted for in expression (15), we have just the Hamiltonian, i t follows that the Hamiltonian is the rate of acquisition of benefits — or, i n other words, the benefits that derive from the duration of one unit of time. This latter interpre-tation w i l l be much expanded later. In the language of economic dynamics, the requirement that c*(t) should maximise expression (15) i s the requirement of equilibrium at each moment, l e t us say of "current equilibrium". That i s , i t expresses an equilibrium between current needs as expressed through direct demand for current satisfaction and future needs as expressed through the derived demand for stock, X(t), as usual for Lagrangian multipliers in economics, is the equilibrating price, i f , as has been implicit throughout this * discussion, "benefit" i s taken as numeraire. (For X = = marginal worth (i.e., benefit) of stock). With this in mind, i t is not surpris-ing that equations (11) and (12) provide the rest of the economic dyna-mics, that i s the link between successive current equilibria. Equation (11) i s our physical restraint, given to us exogenously, and can be 22 thought of, as in the example given in the last section, as a production function. So much is very easy. But now i t is clear that eguation (12) i s a price adjustment mechanism reflecting the changes over time in the marginal valuation of stock. What is the sense of this price equation? One remark here as a b i t of a digression. Now that A i s identified as a (shadow) price, or mar-ginal evaluation of something, i t is plain why equation (12) emerged (see p. 17) through an argument borrowed from duality theory. Duality theory reflects various symmetries between quantities and prices, and i t s tech-niques l e t us go from conclusions about one set of these variables [eq. (6)] to conclusions about the other [eq. (12)]. Let us write out equation (12) more explicitly: A' = - — = - F k(k,c,t) - Af k(k,c,t) , whence T + f, (k,c,t) + r F, (k,c,t) = 0 (16) A k A k The f i r s t term is the rate of accrual of capital gains to holders of stock (A is i t s price). f (k,c,t) is the own rate of return on stock, since i t i s JC the increment to the rate of accumulation k arising from one unit more of stock. F^(k,c,t) is the increment to the current rate of satisfaction from one unit more of stock, and since -^ is the price of satisfaction in terms of stock, the third term in equation (16) is the revenue to a stock holder from providing this increment of satisfaction. Equation (16), then, i s a zero net-profit condition, and i t consequently confirms the identification of A with a shadow price. It says that the total rate of return on the 23 marginal unit of stock is zero: in one unit of time, and for this marginal unit, the increase (in stock units) of i t s value ( A / A ) plus the produced increase in stock (f^) plus the value (in stock units) of the increased current satisfaction (rr F, ) add up to zero. A k A point of possible confusion: What about "normal" profits? Surely the rate of return should equal the rate of interest, not zero? The answer is easy: A is a price quoted at time tg, since J* is measured always in the same way. That i s , J*(k(t),t) i s defined as fcf max / F(k(x), C ( T ) , T) dr which is the value, as seen from t^, of the part of the programme from t to t^. Consequently, i f there is a rate of interest, A / A is the rate of capital gains (in the usual sense) minus the rate of interest: for example, what one usually calls "no capital gains" corresponds to present-value prices de-clining into the future at the interest rate, i.e., ^ - = - r . A f u l l e r treat-ment of this matter w i l l be given later. 3. Transversality The problem considered so far, as defined by eguations (1), (2), (3) has been that of maximising the functional J[c] for a fixed range of time t to t f , and with a requirement that, at time t f , the stock or state variable k should take on the value k^. It is frequently the case in problems in economics that neither t^ nor k^ is specified, but that both can be chosen optimally. There is conceptually no d i f f i c u l t y whatever to this. The function J*(k ,t ) is worked out for a range of accessible values of t 24 and k^, a l l that i s needed is to find out the point where the appropriate f i r s t derivatives vanish. It is convenient now to include as arguments of J* the terminal quantities t^ and k^ (dropping the bar for clarity) and to suppress kg and t^, i t being understood that these are given and fixed. Thus we define: t J ( t f , k f ) = / F ( k * , c * , T ) dx where k*(x,t^,k^) and c * ( r , t f , k ^ ) satisfy the optimal control equations for the problem with the boundary conditions: k(t Q) = k , k(t f) = k f . The name of s e n s i t i v i t y analysis (see Hadley & Kemp [1971]) is given to the problem of computing the first-order partial derivatives of J. We proceed directly: -rr— (t ,k ) = / dx{F — — + F — — } •3kf f f t Q k o k f c 3k f Now we may use the optimal control equations to note that: and Then F + Xf =0 (maximum principle: H is maximized by c*) c c X = - M = - F - Xf . Sk k k =-!X^{w-t^%i (i7» 25 where ' by T T - i s meant t h e t o t a l d e r i v a t i v e o f f ( k * ( T , t f , k f ) , c * ( x , t f , k f ) ,t) •F t f 9k* 7ith r e s p e c t t o k f . The term .dx X -^r— can be i n t e g r a t e d by p a r t s t o W3 — T" I „ , . 0 f y i e l d : 9k*. t fcf ^ r f c f , . 9k* - X ( t J + A dx X (18) f t Q d k f The l a s t s t e p f o l l o w s f o r t h e s e r e a s o n s : The v a l u e o f k * ( t f , t f , k f ) i s k f by 9k* d e f i n i t i o n , and t h e v a l u e o f k * ( t , t ,k ) i s k . Hence — (t=t ) = 1; '0"-f f' 0" 3k f (t=t ) = 0. Then the o p t i m a l p a t h k* s a t i s f i e s k*(x) = f ( k * , c * , x ) , so 9k* 3 k f ' ~ "0 3k* t h a t — — i s j u s t t h e d e r i v a t i v e w i t h r e s p e c t t o k o f f ( k * ( x , t ,k ) , d iC _ i x. c * ( x , t _ , k _ ) T T ) , t h a t i s , — — as d e f i n e d above. T h u s , . f i n a l l y : f f d k f . | £ - ( t . k j = - X ( t . ) (19) 3k,. f , f f T h i s r e s u l t i s h a r d l y s u r p r i s i n g . S i n c e X(t) was d e f i n e d t o be 3J* — — ( k ,t) f o r any time t between t n and t _ , i t was i n t e r p r e t e d as the ok t u r t m a r g i n a l worth o f s t o c k a t time t . What e q u a t i o n (19) says i s t h a t i f t h e t e r m i n a l c o n s t r a i n t k^ i s i n c r e a s e d by one u n i t , t h e n the v a l u e , J , o f t h e o p t i m a l programme i s d e c r e a s e d by th e m a r g i n a l worth o f s t o c k a t t h e t e r m i n a l t i m e , t f . I t c o u l d h a r d l y have been o t h e r w i s e : i f one r e l a x e s t h e t e r m i n a l c o n s t r a i n t k^, the b e n e f i t from t h e r e l a x a t i o n (at t h e margin) must be j u s t t h e m a r g i n a l b e n e f i t o f s t o c k a t time t ^ . 26 Now the economic i n t e r p r e t a t i o n o f t h e o t h e r p a r t i a l d e r i v a t i v e 3 J 9 t , o f J , Viz. -rr—, i s c l e a r . I t w i l l t u r n o u t t o be t h e m a r g i n a l worth o f "f time a t t h e e n d p o i n t o f t h e programme. The transversality conditions, which a r e d e f i n e d t o be the r e q u i r e m e n t s which must be f u l f i l l e d by the o p t i m a l c h o i c e s o f k^ and t ^ w i l l thus be n o t h i n g o t h e r than t h e r e q u i r e -ments t h a t t h e m a r g i n a l worth o f s t o c k a t t h e end o f the programme and the m a r g i n a l worth o f time a t t h e end o f the programme s h o u l d b o t h be z e r o : a v e r y i n t u i t i v e l y s a t i s f y i n g economic r e q u i r e m e n t . What the n i s t h e m a r g i n a l worth o f time a t t h e end o f the 8 J programme? I t i s ( t ' , k ) which from the d e f i n i t i o n o f J i s j u s t : ot^ t f 9 J BtT^f'V = F(k*(Wkf) f c * ( t f , t f ,k f) , t f ) »^f , r 3k* 8c*-, + ••I.I d T t F . - r — + F -—} . t k 8 t f c 8 t f By an e x a c t l y s i m i l a r argument t o t h a t which l e d t o e q u a t i o n (18), one o b t a i n s : g - ( t f , k f ) = F ( t . ) - [X ^-]tf 8 t f f f f 8 t f t Q . f n d f , d f . where F ( t ^ ) i s i n t e r p r e t e d o b v i o u s l y , as the v a l u e o f F a t time t ^ a l o n g 8k* the o p t i m a l p a t h . C l e a r l y ( t = t Q ) = 0. S i n c e k * ( t f , t f , k f ) = k f by d e f i n i t i o n , i t f o l l o w s t h a t k 1 ( t f , t f , k f ) + k 2 ( t f , t f , k f ) = 0 27 where k. denotes the partial derivative of k* with respect to i t s 1st, it * it 3 k* 2nd argument. But k = k = f (k*,c*,t) , so that T — ( t = t ^ ) , which i s just * r k 2 ( t f , t f , k f ) , equals - f ( k * ( t f ) , c * ( t f ) , t f ) (where some arguments of k* and c* have been omitted). Thus: |^-(t f,k f) = F(t f) + X ( t f ) f ( t f ) (in obvious notation) = H(t f) (20) In other words, the marginal worth of time at time t^ i s just the Hamiltonian at time t^. The two transversality conditions to be satisfied by an optimal choice of t^ and k f are X(t f) = 0; H(t f) = 0 (21) two equations for two unknowns. It can happen frequently that k^ and t^ are neither completely specified nor completely free to be chosen. It may be that some relation must be satisfied by them, say S(t^,k^) = 0. This modification i s easily handled. The problem has become: maximise J(t^,k^) subject to S(t^,k^) = 0. The Lagrangian i s : L = J ( t f , k f ) - p s(t ,k ) and the first-order conditions for a maximum are therefore 3J 3S , , 3S 28 3kf 9k £ S ( t f , k f) - 0. That's a l l ! But i t i s perhaps worthwhile to clear up a common miscon-ception of what i s meant by the transversality conditions. Figure 1 depicts an optimal path, k*(t), ending at the optimal point (tf, kf) of the curve whose equation is S(tf, kf) = 0. /> k* ^ S ( t f , k f) - 0 : J t Fig 1. As drawn,, the path k*(t) does not intersect the curve S at right angles, and i n general there i s no reason for i t to do so (in physics sometimes there i s - hence the confusion). But at each point on the path, a l i t t l e arrow i s drawn, and this points in.the direction of the vector (H(t),„-\(t)). The directions are as drawn i f both H and X are positive, as w i l l often be the case in applications. This vector i s indeed at right angles (trans-verse) to the curve S. Why so? This i s just the content of eq (22), which t e l l s us that the vectors (H,-X) and (S t,S k) are par a l l e l at ( t f r k f ) . (Partial derivative notation). But (St,S. ) i s the gradient vector of the 29 function S (VS as i t i s frequently written), and the gradient vector i s the normal to the line of constant S, i.e., the curve S(tf, kf) = 0. This kind of reasoning i s perfectly familiar to engineers and physicists, but perhaps less so to economists. They would more lik e l y reason as follows: the function J ( t f , kf) i s just the maximum worth Cderived u t i l i t y perhaps) from a programme ending at ( t f , k f ) . We are interested i n maximising i t subject to a constraint, Vis. S(tf, k^) = 0. Thus usual procedure ca l l s for us to draw indifference curves, that i s , l o c i of points yielding the same u t i l i t y J, and then to find where an indifference curve i s tangent to the line of constraint. This i s shown in Figure 2. indifference curves > Fig 2. 30 At the tangency point, the gradients of J(t^,k^) and S(t^,k^) coincide, i.e., VJ = VS, or more expl i c i t l y : 3J 8S 9J 8S dtl = P 9tT ? 9kT = P Bk7 S = ° f f f f These are just the transversality conditions (22). Maybe i f economists had created optimal control theory, they would be called the tangency conditions. A few more remarks. The usual notions of quasiconcavity (convexity) can obviously be brought into play here so as to obtain sufficient conditions for a maximum. The requirement i s easy to impose on S, but a b i t trick i e r on J. Often the constraint S takes the form of fixing t^ or k^ while leaving the other variable open to choice. Then S w i l l be something like k^ - k^ = 0, say. The transversality conditions becomes simply (for this example) H(t f) = 0, k f = k f. (The other equation, -A(t^) = p has no content, since p, too, is unknown). There is s t i l l enough to determine the problem f u l l y : one equation (H=0) for one unknown (t^). 4. The Marginal Worth of Time The nature of time i s a considerable mystery even to physical scien-t i s t s . The shift from statics to dynamics in economic analysis involves d i f f i c u l t i e s whose source is precisely the nature of time. This essay cer-tainly does not claim to solve a l l these d i f f i c u l t i e s . But I think i t is f a i r to say that certain economic insights can be had from the considerations of optimal control theory, which are helpful precisely in the search for better understanding of intertemporal economic problems. 31 It was pointed out as unsurprising that the co-state variable X should, as a Lagrange multiplier, be interpreted as a price and a marginal worth — of stock, that i s , the state variable. But sensitivity analysis revealed that the worth of time at the end of an optimal programme was the Hamiltonian, and that i s not so unsurprising. Again (p. 21) <• i t was seen that the Hamiltonian was also the rate of acquisition of benefits — the benefits accruing during one unit of time. This interpretation i s not exactly the same as calling the Hamiltonian the marginal worth of time, but i t is close. Moreover, i t holds for a l l times between t and t^, not just at t^. Yet again, the Hamilton-Jacobi-Bellman equation [eq. (16)] can be interpreted in this vein. Because of equation (14), 'equation (6) can be written as J*(k ,t) = -H(k ,c ,X ,t) (23) in obvious notation. Now J* is not quite the same as J, being a function of different arguments. To clear this matter up, let us now include a l l the arguments of the maximised functional, and write t J*(t,k. , t _ , k j = max / dxF (k (x) ,c (x) ,x) t r t t c subject to k = f (k,c,t) k(t) = k ; ' k(t f) = k f Then equation (23) can be written 32 and i t i s clear that this result, too, is saying something about the Hamil-tonian and the marginal worth of time. But what exactly? It says that i f , at time t in the course of an optimal programme, one skips a unit of time, without anything else changing — k stays the same, as well as t^ and k^ — and then proceeds optimally after this skip, then the toss of total benefit i s H, evaluated at time t. In this precise sense, then, H is the marginal worth of time, at time t. If an intertemporal maximisation is being carried out, and i t may be a much more general one than that specified by equations (1), (2), (3), time i s an input, or factor of benefit, more or less like any other. I must say "more or less", because i t i s only in some regards that i t is like any other. It has a marginal worth, or price, certainly, and H, the Hamiltonian, comes close to measuring i t . But in what circumstance does i t measure what an economic agent would be willing to pay for a unit of time? Each instant of time is unique, with i t s own p r o p e r t i e s — i t is a heterogeneous input, and plainly at some times one would pay much for a few golden moments like the ones just experienced (excuse the language, but time i s a mystery and there-fore liable to provoke mystical talk). The d i f f i c u l t y , a l l too well known, is that moments of time, in their f u l l individuality, cannot be either skipped or replicated, and so in no conceivable market could anyone receive, in exchange for whatever payment, an extra unit of time, at time t, valued to be sure at just H(t). The physical world is not like this. The laws of motion are im-mutable, and what can happen at time t can happen at time t 1 , for a l l t and t \ (This principle, the homogeneity of time, was stated in a clear, and false, form by Newton, and once i t was corrected and cast into a better 33 form by Einstein became a major ingredient of the Principle of Relativity). The physical world deals with inanimate objects, drab, dull and neither (pace Samuelson) pro f i t maximising nor a l t r u i s t i c . Thus, for a l l these reasons the Hamiltonian, as defined by physicists for physical purposes, is quite exactly the marginal worth of time — or as they would say, the infinitesimal generator of time translations, when these are taken as ele-ments of the Poincare group. In economics, i t is sometimes reasonable to maximise one's bene-f i t s as i f time were homogeneous, and the distant future as v i t a l as the present. Ramsay [1928], after a l l , told us i t was immovat to discount. When this i s so, the Hamiltonian is the marginal worth of time, as w i l l be seen in a moment. What an economic agent can perfectly well buy in an appropriate mar-ket i s an extra unit of time in which to complete his programme. A student may bribe an invigilator for an extra few minutes to finish writing an exam; a big firm w i l l bribe the government for more time to comply with anti-pollution laws; options can quite legally be written into contracts to allow a contracting party more time, at a price, to carry out his obligations; most familiar of a l l , payment of interest w i l l buy time to repay a debt. Where does a l l this sound economic sense appear in optimal control theory? The 8J* , answer xs easy: as — — (t,k ,t .k-). This is•the worth of an extra unit of dtf t f r time tacked on at the end of the programme. That unit i s what i t i s , and may be quite different from any preceding or subsequent unit, but i t is a definite, unmystical unit, and i t s worth i s easily determined, as 9J*/8t^. At this point, economic science gives us a very large bonus. We a l l have the feeling that when a number of inputs contribute to output or 34 u t i l i t y or whatever — benefit in general — then the optimal result is achieved when the quantities of the inputs are chosen so that the extra worth deriving from a unit of further expenditure on any one of the inputs i s the same as that from any other. Time is only one input, but a hetero-geneous one, and so each instant is like a separate input. The result which corresponds to the above familiar one i s this: i t does not matter at what time t an economic agent purchases an extra unit of time to be tacked on at the end; the extra, or marginal, worth i t provides w i l l always be the same, along an optimal path. This result i s not exactly the same as the usual one about inputs, but i t i s just as useful, and i t is proved in the same way. For, because J* i s defined as an integral, we can write, J * ( t Q , k 0 , t f , k f ) = J*(t 0,k Q,t,k*) + J*(t,k*,t f,k f) * where k^ i s the value, k* (t) of k along the optimal path from t Q to t f , at time t. Now consider a slightly longer programme, lasting t i l l t^ + dt. If * one were constrained to pass through the point (t,k^) on the new trajectory, the (constrained) maximum worth to be had would be J*(t 0,k Q,t rk*) + J*(t,k*,t f + dt,k f) 9 J * * = J * ( t 0 , k Q , t f , k f ) + — ( t , k t , t f , k f ) d t + o(dt) But this must be less than the unconstrained maximum, which is 9 J * J * ( t 0 , k Q , t f + dt,k f) = J ^ W W + 3 ^ - ( t o ' k o ' t f ' k f ) d t + o ( d t ) 9 J * * 9 J * Therefore - r — (t,k , t _ , k j < T — (t„,k .t_,k ) . If one considers a shorter ot,. t f f — dt-. 0 0 f f f ' f 35 programme, of length t^-dt, i t i s plain that: 8J* * J * ( t 0 , k 0 , t f , k f ) - ^ - ( t , k t , t f , k f ) d t < J * ( V k o , t f , k f ) " I r 7 ( t0' k 0' tf kf ) d t whence | ^ ( t , k ; , t f , k f ) > | ^ . ( t o , k o , t f , k f ) But the two results show that f ^ ( t ' V V V = | ( t o ' V V V ( 2 4 ) for a l l t. And this i s the desired result. The extra worth from purchasing an extra unit of time, at time t [left-hand side of equation (24)] i s the same for a l l times t between t Q and t^. The argument i s just as easy in words. The gain in worth from being given an extra unit of time at t cannot be more than that from being given i t * at t^, or else the path from t to t^ through k^_ could not be optimal. De-layed information (i.e., greater length of time available) cannot be more valuable than the same information provided earlier. On the other hand, the loss from being deprived of a unit of time at time t must be at least as great as that of being deprived of i t at t^ for exactly the same reason: delay of bad news cannot make things any better. But at the margin, the gain from a unit of time i s the same as the loss from being deprived of i t , and so both are equal to the gain from the extra unit purchased at t^, and therefore con-stant along the optimal path. This result, like the familiar one about i n -puts, characterises optimal paths, and i s proved by exploiting optimality. I said above that this result was a large bonus. I shall now give two examples which w i l l , I trust, vindicate that statement. The f i r s t 36 is the example of autonomous systems, that i s , those drab, dull ones like physical systems or Ramsay's moral ones, where one instant of time is exactly like any other. What this means i s that the functions F and f of equations (1) and (2) cannot depend expli c i t l y on t. Consequently, neither does H, which i s nothing but F + Af. In fact, since J*(t,k,t f,k f) = max/tf F(k(x) ,c(x) )dx [subject to k = f(k,c) and k(t) = k,k(t f) = k f] t +t' = m a x / ^ F(k(i) , c ( T ) ) d t (25) [subject to k = f(k,c) and k(t+t') = k,k(t +t') = k ], we have J*(t+f,k,t^+t',k^) = J*(t,k,t^,k^) or else, more simply, J*(t,k,t^,k^) is a function of k^, k^ and t ^ - t alone. (That i s , time is homogeneous, and only time differences matter: equation (25) says that i f both t and t^ are translated an equal amount, then J* is not changed). But then, the result 3J* we have obtained, namely that is constant along the optimal path, means at 3J* also that - , i.e., H(t), is constant along the optimal path, since, i f o t 3 J * J*(t,t_) = P(t - t ) , say, suppressing k - dependences, i t is immediate that 9 t 3J* £ f 3 t x We have shown in this case that H(t) is indeed the marginal worth of 3t time. Now, on p. (19) , i t was pointed out that once the maximum principle, equation (14), was used, equations (11) and (12) were two first-order differential equations for k and X. Knowledge that H = constant is the same as knowledge of a f i r s t integral of these equations. In other words, i f one substitutes c* = c*(k,A) from equation (14) into H(k,c*,A) = constant, one 37 obtains the equation of the trajectories generated in the phase plane [k,A)-space] by the equations (11) and (12). These are usually called the optimal t r a j e c t o r i e s . If one decides to look at one of them, that is i f one fixes the value of H, then one may solve H(k,c*(k,A),A) = H for either k or A, substitute the solution into equation (11) or equation (12) and get a single differential equation with just one unknown function of time. For example, one may obtain k = h(k), say. This equation can always be solved by inte-grating T-T-. , as follows: (The solution has been reduced to quadrature, as the old books on applied mathematics say). This very pleasing result i s quite general i f there i s only one state variable, k. It i s not so useful in the case of several state variables (a vector k of state variables) each with a corresponding co-state variable, but i t is not entirely worthless either. The second example i s the case where the heterogeneity of time enters only because of Ramsay immorality, that i s , discounting of future benefits. Here, 9J*/9t^ turns out to be an interesting quantity although i t i s no longer H. The Hamiltonian can be written as: H(k,c,A,t) = e F(k,c) + Af(k,c), so that the optimal control equations are as follows: 9H 9c e_ < 5 tF (k,c) + Af (k,c) = 0 A k 9H 9A = f(k,c). (26) . 38 — fit If now one puts A = pe , the resulting system of equations in k and p (and c) i s autonomous: F (k,c) + p f (k,c) = 0 c c P - Sp = - F k - p f k k = f(k,c) (27) As in the ful l y autonomous case, i f a f i r s t integral of this system can be found, the problem i s essentially solved. But H i s not a f i r s t integral. Before deriving what the f i r s t integral i s , i t i s worthwhile to spend just a few moments more on H. It i s not constant, and i t s total time derivative can be calculated: • d-H „ • 3H 3H 3H ' 3H h^'c'x't] = 3t: + 3 k k + 3c" c + 3H = -— + Xk - kX, by equations (26) dt 3H/3t. All the time dependence of H is the explicit part, everything else cancels out along an optimal path. This calculation provides an alternative proof of the constancy of H in autonomous systems. The real constant i s 3J*/3t^. We may observe that J*(t+dt,k ,t f,k f) = / f_.e~ 6 TF(k*(T) , c * ( T ) ) d T (28) t+dt where k * ( x ) and c * ( x ) define the optimal path and where in particular k * ( t + d t ) = k and k*(t f) = k f. 39 The integral of equation (28) can also be written, after changing the dummy variable of integration x to x' + dt, as t — c i t e ~ 6 ( d t ) / t f e" 6 TF(k*(x'+dt),c*(x'+dt))dx' Similarly, t -dt , J*(t,k t,t f-dt,k f) = / e" TF(k(x),c(x))dx where k and c define the optimal path with boundary conditions k(t) = k , k(t f-dt) = k But k(x) must equal k*(x+dt) and similarly c(x) must equal c*(x+dt), since the hatted and starred pairs of functions are defined by the same autonomous system, equation (27), with the same boundary conditions once the translation by dt has been attended to for k* and c*. Hence J*(t+dt,k t,t f ,kf) = e - < S ( d t ) j * ( t , k t , t f - d t , k f ) . This yields J*(t,k t,t f,k f) + dt -|^= e " 6 ( d t ) [J*(t,k t,t f,k f) - dt + o(dt) 8 J * = J*(t,k t,t f,k f) - d t [ — + &J*] + o(dt) and thus 8 J * 8 J * = H - SJ*. [Equation (23)] 40 So that the constant along an optimal trajectory, and also the marginal worth of time, is H - 6 j * . This result i s useful in a quite different way from the autonomous system result. There H = constant gave the equation of the optimal trajectories, but here, since J* is what is to be found by solving the optimal control problem rather than being a known function, H - 6J* = constant is of no help in finding the equation of the trajectories. But these after a l l are given by the autonomous system (27), which may yield a f i r s t integral by direct methods. Then, i f that is so, the trajectories are known, and consequently H can be calculated along them directly. At the end of any trajectory, J* = 0 by definition, and so the constant appropriate to that trajectory i s known. In particular, i f the end of the trajectory i s charac-terised by the transversality condition H = 0, the constant is zero. This means that J* i t s e l f , the worth of the optimal programme, can be obtained at any point on a trajectory, without f u l l y solving the problem and carry-ing out the integration of equation (1). This information can be very valuable. The case of the H = 0 transversality condition is especially interesting. For then we get the very agreeable economic result that, for an optimal programme for which time is not a constraint — and so is chosen optimally — the remaining value of the programme, at any point in i t s course, i s obtained simply by solving H - 6J* = 0, i.e., J* = 7 H ; 0 in other words simply by taking the capitalised value of the Hamiltonian at the given rate of discount, 6. The remaining worth, J*, is equivalent to a perpetual benefit stream of size H, at interest rate 6. These last results w i l l be of great use in the other two essays of this thesis. 41 5. The Phase Plane In this f i n a l section, l e t us re s t r i c t our attention to problems which are either autonomous or time-dependent only through an exponential —61 discounting factor e . Equation (27) gives the autonomous optimal control equations for k and the "current-value" shadow price p: F (k,c) + pf (k,c) = 0 c c | = " Fk " P fk + 6 P k = f(k,c) (27) Let the solution of the f i r s t of these equations (the maximum principle) be written as (28) c = c(k, p) This may now be substituted into the remaining two equations to yield a closed system. The phase plane of the problem i s constructed by taking k as abscissa,P as ordinate, plotting the two lines p=0, k=0, and drawing the trajectories of the solutions k(t), p(t) of equations (27) for varying i n i t i a l conditions. Frequently the result w i l l resemble Figure 3. 42 P r fc - 0 p rt 0 k Pig 3. The lines k = 0 and p • 0 can be best interpreted by imagining a static world. F i r s t of a l l , i f an economic agent could perceive only a spot price., p, of stock, and further presumed that this price would last for ever, he would wish to purchase an amount, k, of the stock such that i t s marginal worth equalled p *- that i s , such that the marginal unit y i e l d -ed zero net p r o f i t . How much benefit does a stock k, costing pk, yield? In a short space of time, At say, i f our agent chooses a value c of the control variable, the direct benefit i s F(k, c)At. The stock has changed by an amount kAt, that is f(k, c)At, which, since the price stays fixed at p ^ i s worth pf(k, c)At. The original stock i s s t i l l worth i t s cost pk, but interest has been forgone on this sum to the value of 6pkAt. The net benefit i s thus (F(k f c) + pf (k, c) - 6pk)At. The first-order conditions for maximising this are just F c + P f c " 0 ; *k + p f k " 5 p = °< 43 and i t is immediate from equations (27) that this yields the p=0 line in the phase plane. Thus this line can be interpreted as the static (derived) demand curve for stock. Next, one can imagine a holder of stock, rather than a purchaser. If his holding i s k, what would i t cost for him to produce an extra unit of stock? The cost w i l l be the difference in benefit received over a short time At in the following two sets of circumstances; (a) the stock k is maintained unchanged during At, (b) the stock k i s increased by one unit during At. For (a), the control variable c must be chosen so that k =0, that is f(k,c) = 0. Let the solution to this equation be c = c(k). Benefit received is thus F(k,c(k))At. For (b) the control variable must be chosen as c(k) + Ac, say, where kAt = 1, that is f(k,c(k) + Ac)At = 1. Since f(k,c(k)) = 0, this means that AcAt = l/f c(k,c(k)) (29) The benefit received is then F(k,c(k) + Ac)At = F(k,c(k))At + F (k,c(k))AcAt. c The benefit forgone in producing one unit of stock i s , therefore, from equa-tion (29) : F (k,c(k)) -Fc.(k,c(k))AcAt = - f C ( k f C ( k ) ) c It i s reasonable to c a l l this quantity the supply price of stock, p, say . s Then the variables p ,k and c(k) satisfy the pair of equations s F (k,c) + p f (k,c) = 0 c c c f(k,c) = 0 , and from eqs (27). this gives pre-cisely the k = 0 line i n the phase plane. This line can then be 44 interpreted as the static supply curve for stock. These arguments show that, whert the phase plane looks like Fig. 3, i t can indeed be understood as an ordinary demand-and-supply diagram, with adjustment mechanisms provided. These mechanisms are the key to comparative dynamics, as w i l l become clear in the examples of the remaining two essays of the thesis. To conclude this essay, some comments are in order on the limitations of the analysis presented. Throughout, only one state variable, k, has been considered. Problems can of course easily arise in which two or more are needed — natural resource models are the most obvious example; see, for example, the bibliography in Clark [1974]. Such problems,where naturally a two-dimensional phase plane is no longer sufficient, are harder to treat than the problem of this essay in the same measure as multi-product general-equilibrium models are harder than one-product partial equilibrium ones. On the other hand, where there i s only one state variable, more than one control variable can be handled as easily as can one — the model of the following essay is an example of such a case. The development and analysis of the opti-mal control algorithm, as well as of the transversality conditions, i s appli-. cable to a problem with a vector of state variables with virtually no modifi-cations. It i s the subsequent analysis, both mathematical and economic, that is d i f f i c u l t . The technical limitations of the presentation of this chapter are manifest. It i s enough to work through the book of Hadley and Kemp [1971] to become aware of the great variety of subtle mathematical points which can arise in optimal control questions. But Hadley andKemp state their aim as being to write a mathematical textbook with examples drawn from economics: mine has been the obverse — to simplify the mathematics so as to c l a r i f y the economics. 45 CHAPTER II A STOCK ADJUSTMENT MODEL AND THE PROBLEM OF OPTIMAL EXIT This essay builds on the work of the preceding one. A specific model i s treated which allows the economic interpretation of optimal control theory to be used for working out comparative dynamics. The results are not particularly d i f f i c u l t . This fact may seem surprising to anyone who has studied the work of Oniki (1973) and i t may be worthwhile to ask why. Oniki has successfully (I believe) followed Samuelson's instructions for comparative statics i n the case of comparative dynamics. That i s , he has computed expressions for the (infinitesimal) changes induced at each instant i n a l l the endogenous variables in an optimal control problem by changes in any of the exogenous variables. It i s not surprising that these expressions are compli-cated in their general forms and, besides, usually impossible to sign. But often the interesting economic consequences of a change in exogenous variables are restricted to a small number of those calculated by Oniki. Again, as w i l l become clear in this essay and the next, i t can be the effects of a finite change in an economic environment which are of real interest rather than a tendency, expressed by some derivatives, produced by infinitesimal change. These remarks are i n no way meant as a slight on Oniki's extremely valuable work (for he has a real compendium of results i n very general form), but instead are intended to point out the substantial difference between his approach and mine. There are questions - of detail rather than of essence -that can quite legitimately be called questions in comparative dynamics which I shall not consider at a l l in what follows, because I think i t i s more f r u i t f u l at the moment to concentrate on matters which are distinc-tively related to dynamics and time. Perhaps fortunately, perhaps be-cause of my choice of models, these matters lend themselves to a reason-ably simple treatment. The detailed questions l e f t unanswered can, after a l l , be approached by Oniki's methods, which are chiefly an extension of those of comparative statics. The main point that I want to emphasize is that I shall be concerned with non-infinitesimat changes in exogenous variables. In Section 1, the model of stock adjustment to be considered i s specified. It can be thought of as providing an elementary paradigm for comparative dynamics, as I wish to look at i t in this thesis. The optimal control equations are written down, and the phase plane drawn. Then in Section 2, a (finite) change is made in the cost structure for production of the stock. The dynamic adjustments made as a result are analysed, with special emphasis being given to the possibility of exit from business. The methods of Chapter 1 are used extensively in this section. 1. Specification of the Model The question considered in this section is the following: i f a firm keeps an inventory of finished goods out of which to meet demand for i t s product, what w i l l be the effect, short-run and long-run,' on the size of this inventory of an increase in production costs? We may observe at once that i f only long-run effects are to be considered, the traditional comparative statics methods for long-run equilibria are sufficient to 47 answer the q u e s t i o n . The model which i s s e t up a l l o w s a d e f i n i t e q u a l i t a -t i v e answer t o t h a t s t a t i c q u e s t i o n and the n t h e p a t h t h e f i r m f o l l o w s i n g o i n g from i t s o r i g i n a l s i t u a t i o n t o i t s f i n a l one i s examined. F i r s t , a l t h o u g h o p t i m a l c o n t r o l t h e o r y a l l o w s t r e a t m e n t o f s i t u a t i o n s where t h e f i r m i s not i n e q u i l i b r i u m b e f o r e i t s c o s t s rise, there a r e t o o many k i n d s o f d i s e q u i l i b r i u m f o r an i n v e s t i g a t i o n o f a l l o f them t o be u s e f u l . C o n s e q u e n t l y we s h a l l assume t h a t the f i r m s t a r t s i n a p o s i t i o n o f l o n g - r u n e q u i l i b r u m . (Comparative statics c e r t a i n l y r e q u i r e s t h i s ) . The model: t h e f i r m ' s o b j e c t i v e i s t a k e n t o be the m a x i m i s a t i o n o f t h e d i s c o u n t e d sum o f i t s p r o f i t s i n t o the i n d e f i n i t e f u t u r e . L e t the f l o w r a t e o f s a l e s be s, and l e t t h e f l o w o f revenue from t h i s s a l e s volume be R ( s ) . L e t t h e f l o w r a t e o f p r o d u c t i o n be y, and t h e f l o w o f c o s t from t h i s be C ( y ) . The f u n c t i o n s R and C w i l l be assumed t o have t h e u s u a l c o n v e n i e n t p r o p e r t i e s o f d i f f e r e n t i a b i l i t y and d i m i n i s h i n g r e t u r n s . L e t the i n v e n t o r y s t o c k be denoted by k, and l e t c o s t s o f h o l d i n g t h i s s t o c k be d i r e c t l y p r o p o r t i o n a l t o k, 0k, say. Then t h e d i s c o u n t e d sum o f p r o -f i t s i s ( d i s c o u n t r a t e = r) : OO — y t" J = / e 1 [R(s) - 0k - C ( y ) ] d t . Output goes i n t o i n v e n t o r y u n t i l s o l d , so t h a t k = y - s. I t i s n e c e s s a r y t h a t k >_ 0, y>_0, s >_ 0. In o r d e r t h a t t h e r e s h o u l d be a purpose f o r h o l d i n g i n v e n t o r y , t h e assumption i s made t h a t the s a l e s f l o w cannot exceed some f r a c t i o n o f i n v e n t o r y : 48 s •••<_ qk. (1) This last constraint can be thought of as a crude model of the technology of distribution. If i t takes a certain time for goods to be shipped from a factory to points of sale, then only a limited quantity determined by the size of inventory, can be used to restock shelves when their previous contents have been sold. The precise form of inequality (1) is in any case not c r i t i c a l to the results to follow. A l l that matters is that there should be some constraint on sales volume related to inventory size. Similarly i t w i l l become apparent in the analysis below that the direct cost, 0k, of holding inventory need not have that particular form, and in fact may be zero without affecting qualitative results, so long as the discount rate, r, i s positive, since a carrying cost (forgone interest) is thereby introduced. The. Hamiltonian can now be formed. Following the rules of Chapter I, we obtain: H(k,p,s,y,t) = e" r t[R(s) - 0k - C(y) + p(y-s)] where k is the state variable, p the (current value) co-state variable, s and y control variables. The maximum principle requires that we maximise H with respect to s and y for any admissible k and p over the feasible set of s and y: 0 <_ s <_ qk y >_ 0. 49 The optimal values s* and y* are as follows: s*= y* = 0 i f P >_ R' (<>.} s'jpl i f R' (01 ^p >_ R1 (qk) qk i f p £ R' (qk) 0 i f P <_ C'(0) y(p) i f p >. C (0) where the functions s and y are the inverse functions of R' and C' and there-fore s a t i s f y the following i d e n t i t i e s : R' (s (p) ) = p C (y(p)) = P The d e r i v a t i v e s , R' and C , are presumed to be monotonic functions: R', the marginal revenue, decreasing, and C , the marginal cost, increasing. The maximised Hamiltonian i s now: M(k,p,t) = e~ r t(R(s*) - Ok - c(y*) + P ( y ^ - s * ) ) , (2) and from t h i s one may obtain the shadow-price equation: p = rp - e r tM k [eq(I-27)] Five d i f f e r e n t regions of the phase plane can be distinguished and the forms of the shadow-price equation as well as of the stock equation k = y-s are l i s t e d below i n the table, while the regions of the phase plane are depicted i n F i g . 4. 50 TABLE 1 STOCK AND SHADOW-PRICE EQUATIONS k = y - s p = rp - M^ e r t I. p>,R'(0) y(p) y(p) rp + 9 P .< R* (0) II. p >_ R* (qk) p> C (0) y(p) s(p) y(p) - s(p) rp + © III. p < C'(0) p >" R' (qk) s(p) -s(p) rp + 0 IV. p <_ C (0) p < R« (qk) qk -qk rp + 0 + q(p-R'(qk)) V. p > C*(0) p <_ R' (qk) y(p) qk y(p) ~ qk rp + 0 + q(p-R'(qk)) Fig 4. 51 (Unless R' (0) > C*(0), production w i l l never be profitable). With this information, i t i s now possible to draw the p =» 0 (static demand) line and the k = 0 (static supply) lin e . The result i s i n Fig 5, along with the senses of the optimal paths in the several parts of the phase plane. p f Fig 5. The equation of the p = 0 line is , = qR' (qk) 9_ p r+o . ~ r+q which plainly must l i e beneath the line p = R'(qk), as drawn. The k = 0 line i s just the p-axis in region IV: in region V, i t has equation p = C(qk) in Region II, i t is the line p = p, where p satisfies y(p) = s(p). It i s evident that p i s the value of p where the curves p = R* (qk) and p = C (qk) intersect. It is possible to verify directly for this model that the p = 0 and k = 0 lines are respectively the static demand and supply curves. To do so i t is convenient to imagine the firm divided into three departments, production, inventory and sales. Then the demand curve<gives the prices (in static situations) that the sales manager i s willing to pay the inventory manager for the latter's maintaining an inventory of a given size. The supply curve gives the prices that the production manager requires to be paid to supply goods sufficient to maintain inventories of given sizes. The analysis of Chapter 1, Section 5, makes i t clear that the above statements are true but i t may be illuminating to check them explicitly. F i r s t , the supply curve (attention w i l l be restricted to Region V : " nothing of extra interest appears in the other regions.) In a static s i t -uation inventory is constant, and so;in Region V, s = y = qk, a l l constants. We wish to know the marginal cost of a unit of stock in an inventory of size k. Such an inventory calls for a production rate, y, equal to qk. If one extra unit of stock i s to be produced in one unit of time, then this calls for a production rate of y+1 for this unit of time and a marginal cost of C (y) = C (qk). It should be noted that this i s the cost of the last unit of stock taken from the flow of production. Next, the demand curve. This gives the net revenue achieved by the inventory manager from the sale of the last unit of stock to the sales manager. Let this revenue be denoted by p. Since sales volume i s s = qk, the revenue obtained by the sales manager from one more unit sold i s R'(qk), and this then i s what w i l l be paid to the inventory manager for i t ^gainst this, there are some charges to be borne by the inventory depart-ment. To provide one more unit (over unit time, say, although this condition does not affect the result) the inventory level must be raised by — units because of the turnover constraint s 4 qk. The cost of this increase for one unit of time is rp/q + 0/q: interest cost plus holding cost. Thus net revenue = p = R'(qk) - rp/q - 0/q, whence p = qR'(qk)/r+q - 0/r+q. This agrees with the equation of the p = 0 line. The point (k,p) in Fig. 5 is the saddlepoint of the optimal paths in the phase plane. It should now be clear that i t i s indeed just the long-run equilibrium maintained by the inventory manager in our f i c t i t i o u s decen-tralised scheme of the firm. It i s then the point that we assume the firm occupies when an exogenous cost increase takes place. The neighbourhood of this point i s shown in Fig. 6, along with the stable and unstable arms leading to and from i t , and the sense of the adjustment paths. 54 2. Comparative Dynamics Now that the optimal control problem has been worked out, we shall see that some comparative dynamics questions are not very d i f f i c u l t . There is assumed to take place a change in the cost function C(y). It is not the function C i t s e l f but i t s derivative C , the marginal cost func-tion, which defines the k = 0 line i n the phase plane (see Fig 6 ) and an increase in C(y) for a l l y.does not necessarilyCmean anT.increase (shift up^ > wards) in C (y) — an increase in fixed costs alone, for example, leaves c\y) unchanged. But C (y) w i l l increase i f the cost change i s , for example, a specific tax on output produced or inputs used in production. Besides this i s the usual state of affairs meant when one speaks of a supply curve being shifted up because of increased costs. Let us begin with this case. The long-run effect i s no more d i f f i c u l t than the most elementary demand-and-supply analysis. The new long-run equilibrium (saddlepoint) li e s on the p = 0 line further up than the old (k,p). The next question i s : Does the firm wish to adjust towards the new saddlepoint or go out of business? Let us.for the moment assume that posi-tive profits were being made at (k,p) (i.e., our firm was intramarginal in i t s industry), so that i f the cost change is small enough, i t is s t i l l worthwhile to stay in business. Now the stock k is not instantaneously adjustable, but i t s shadow price may well change discontinuously with the cost increase (since we are considering a finite change in costs). Since the firm's optimum policy is to move to the new saddlepoint, we can t e l l in fact that i t must move along a stable arm to get there. Thus immediately after the change, i t goes to the — g point (k,p ) as shown in Fig. 7. 56 P k=0 before stable arm (new costs) p=0 k k Fig 7. The "impact effect" of the change can thus be read off at once: shadow price increases from p to p s; stock decreases moriotonically from k to new equilibrium value; sales ( =» qk) behave like k; production ( = y(p)) drops suddenly, but increases as new equilibrium i s reached. In this case, then, the long-run comparative statics result that stock w i l l decline i s of the same sign as the impact effect, and indeed the effect at a l l intermediate times. What i f profits are completely eroded and the firm wishes to leave the industry? It may s t i l l do so optimally. The transversality conditions (see eq (1-21)) for an optimal exit path are that at the endpoint M(k f ? p£, t f ) = 0 and Pfkf =0. It i s plain from Fig 5 that such endpoints, i f one begins from an i n i t i a l stock holding of k, can be found only i n ""•271 Region IV, where M =^R(qk) 0k C(0) - pqkl e . c(0) of course i s just fixed cost. Further, since i n Region V the equation for the state variable k i s k = *-qk, i t follows that k w i l l never f a l l to zero i n a f i n i t e time, 57 so that the endpoint must l i e on the k-axis. The endpoint, or point of exit,, i s then given by p f = 0^  R(qkf) = Gkf + C(0). It i s s t i l l not clear that there is any trajectory starting from a point on the line k « k that ends at (k-, 0). case 2: no optimal exit path P»R'(qk) Fig 8. 58 I t can be seen from F i g . 8 that whether or not there i s one depends on where the unstable arm reaching i n t o Region IV h i t s the k-axis. The c r i -t i c a l state of a f f a i r s i s when the unstable arm ar r i v e s at the k-axis j u s t at the point k f. I f i t i s to the l e f t [Case 1] we see that an exit'.- path does e x i s t , i f to the r i g h t [case 2] then not. Case 1 i s quite easy to understand. There does e x i s t a path at the end of which, i . e . , at the point (k^,0),the t r a n s v e r s a l i t y conditions f o r optimal p r o f i t are s a t i s f i e d . For a fir m which does not intend to stay i n business, there i s no doubt that t h i s i s the path that i t i s best to follow. But i n Case 2, although there are paths that end up on the k-axis, fo r a l l of them the Hamiltonian there i s p o s i t i v e , which implies, according to the discussion of Chapter I, that a path l a s t i n g a longer time would be more p r o f i t a b l e . I t i s a f a m i l i a r r e s u l t of "turnpike" theory (see, f o r example, Dorfman, Samuelson and Solow [1958] and Radner [1961]) that the clo s e r an optimal path passes to the saddlepoint, or "turnpike", the longer i t l a s t s . Thus the most p r o f i t a b l e of a l l the motions s t a r t i n g on the k = k l i n e i s , i n Case 2, the stable arm. Now i t may be permissible f o r the firm to shut down at once with-out cost, or at some f i x e d cost. I t w i l l of course prefer t h i s course i f the best p r o f i t a v a i l a b l e by staying i n business e i t h e r f o r a f i n i t e or for an i n f i n i t e time i s s u f f i c i e n t l y negative. In Case 1, then, the firm has three options: to stay i n business along the stable arm, to e x i t optimally, and to shut down i n s t a n t l y , at cost S, say. In Case 2, there are only two options: the stable arm or instant shutdown. We s h a l l f i r s t of a l l see that i n Case 1, optimal e x i t i s always preferred to the stable arm. 59 The Hamiltonian for this problem involves the time only through - r t an exponential discounting factor, e , and so we know that the marginal worth of time, constant along a l l optimal paths, is M - r J * , where J* is the worth of pursuing the path. The stable arm ends, at time i n f i n i t y , at - r t the saddlepoint. Because M is multiplied by the factor e , i t is plain that M - r J * = 0 at the saddlepoint at time i n f i n i t y . Therefore, for the — s stable-arm path, starting at (k,p ) at time 0, the worth is J* = i M(k,pS,0) = ^ (R(qk) - Ok - C(y(p S)) + p S(y(p S) - qk)) [see equation (2)]. At the end of the exit path, J* = 0 by definition and M = 0 by the transversality condition. Along this path, too, then, the mar-ginal worth of time is zero. The worth of the path is J* = - M(ic,pe,0) e r — Q where (k,p ) is the beginning of the exit path. But now we observe that J > J . This follows because p S < p S, and in Regions IV and V anywhere e s below the k. = 0 line — = ke ? t < °- Thus we confirm that i f an optimal exit ' y p • - -path is available, i t is preferred to the stable arm. Possible shapes for optimal exit paths are depicted in Fig. 9. A l l four shapes can be realised in appropriate circumstances. In a l l cases, stock and sales decline monotOnically, but the shadow price may.first increased and i t may give rise to continued production for a time, with or without an interruption immediately following the cost change. I possible k _'s k Fig 9. Case 1 i s now f u l l y analysed. Optimal exit by a path shown i n Fig 6 i s chosen i f j * a i M(ic, p e, 0) > -S e r and instant shutdown i s preferred otherwise. Case 2 i s just as easy: the stable arm i s followed i f J* = - M(k, p s, 0) > -S, s r and otherwise there is instant shutdown. The "perverse'1 case of Cty) becoming greater for a l l y but C'(y) becoming less i s no doubt more l i k e l y to lead to exit than the usual case discussed above. Most of the possible outcomes are shown in Fig 10 with no further comment. 61 The intermediate case i s where the cost increase i s confined to fixed costs, without change in marginal cost. Then the phase plane does not change, and the firm may remain at the saddlepoint with reduced p r o f i t i f the unstable arm ends to the right of the point (k f, 0) defined by H(k f, 0) = 0, or exit along the path leading to (kf, 0) i n the other case, or f i n a l l y shut down at once i f that i s cheapest. This model, i n i t s "usual" rather than "perverse" form, i s one where 62 q u a l i t a t i v e comparative s t a t i c s gives an unambiguous answer f o r the d i r e c t i o n of change of k because of a conjugate p a i r (Samuelson [1947a] i n the equations that define the l o c a t i o n of the saddlepoint. The impact e f f e c t i n compara-t i v e dynamics i s of the same sign as the long-run e f f e c t , as are e f f e c t s at intermediate times (even i f e x i t i t a k e s p l a c e ) . One i s tempted to believe that the presence of conjugate p a i r s i n equations de f i n i n g saddlepoints may have stronger ( i . e . , dynamic) consequences than j u s t the well-known s t a t i c ones. Against t h i s i s the warning conveyed by F i g . 10. This whole matter of impact and long-run e f f e c t s i s discussed at some length by Nagatani [1976], who also draws att e n t i o n to the extreme d i f f i c u l t i e s of signing impact e f f e c t s i n problems with more than one state v a r i a b l e . The subject i s f a s c i n a t i n g , and much remains to be done to elucidate i t . The work of Epstein [1977]., which treats the Le C h a t e l i e r p r i n c i p l e i n a dynamic context, seems to me to i n d i c a t e how progress can be made. In expounding the model used i n t h i s essay, I have been precise i n s p e c i f y i n g the economic meaning of a l l the v a r i a b l e s . I hope that i t i s c l e a r even so that the same mathematics w i l l describe other problems i n other branches of the "theory of the firm." Of p a r t i c u l a r note i s the micro-theory of investment. In t h i s context Lucas [1967] and Gould [1968] have constructed models of intertemporal p r o f i t maximisation where adjustment costs a r i s e when a change i s made i n c a p i t a l stock, or number of workers em-ployed, or even i n rate of investment. Lucas, i n p a r t i c u l a r , has pointed out the need to consider e x p l i c i t l y the matter of entry and e x i t of firms i n an industry when one seeks to explain aggregate investment. For t h i s purpose, as well as f o r others r e l a t i n g to inventory cycles and the l i k e , the model of 63 this chapter should be exp l i c i t ly relevant. But there is a capital-theoretic element involved in almost a l l of a firm's decisions. The formal s imilar i ty of maximising models with stocks of productive capital , inventory, and labour should mean that a l l of them can be somewhat better understood by means of the techniques used in this essay. In the sense of these remarks, then, the model presented in the next essay is an example which shows how, in one case at least, matters of aggregation and general equilibrium can be handled when entry and exit :are taken as endogenous. 64 CHAPTER IIIA A MODEL OF URBAN HOUSING. This chapter and the next make up the last essay of the thesis. Another model of intertemporal maximisation i s set up, but this time in a general equilibrium context, with both sides of the market ex p l i c i t l y modelled. I t was the claim of Chapter II that interesting comparative dynamics results are li k e l y to be obtainable only i f the i n i t i a l state, on which a perturbing influence i s supposed to act, i s one of equilibrium. Consequently the present chapter w i l l be devoted to determining the long-run equilibrium of the model, and then in Chapter IIIB a disturbance w i l l be made and i t s results analysed. Although the discussion of this chapter cannot s t r i c t l y be called comparative dynamics, i t i s , I trust, s t i l l of substantial interest in i t s own right. In order to determine the equilibrium i n the model between the forces of demand and supply, even i n a steady state, i t i s necessary to take into account the details of an intertemporal pro f i t maximisation. It w i l l turn out f i n a l l y that solving the equilibrium equations means locating a fixed point of a mapping, just as i n standard general equilibrium theory (see for example Arrow and Hahn (1971)), but here the mapping acts on a function space: i t i s a highly non-linear integro-differential operator. Extensions of Brouwer's fixed-point theorem apply to function spaces of i n f i n i t e dimensionality just as well as to the finite-dimensional spaces most commonly used i n economic theory however, and so no great new technical d i f f i c u l t y i s encountered. The proofs of Brouwer's theorem and i t s extensions proceed by contradiction and are not constructive, and so these theorems give no help i n finding e x p l i c i t solutions of equilibrium equations. Another general principle, the contraction mapping principle, i s constructive on the other hand, and the possibility of using i t i s explained at the end of this chapter. In fact, i n Chapter IIIB an explicit solution w i l l indeed be found for a rather simplified version of the model of this chapter - the simplification being necessary to make the discussion of dynamics at a l l tractable. Here, then, the aim i s to characterise the equilibrium state by (complicated) equations and to show how existence and uniqueness may be demonstrated in some circumstances. The model i s one of a city inhabited by utility-maximising tenants who live in dwelling-places provided for them by absentee landlords who maximise profits. There are no owner-occupiers. Uncertainty, i s abstracted from completely, and perhaps a word of justification for this i s called for. The overall aim of this thesis i s elucidation of some topics i n comparative dynamics. Although the effects of uncertainty are, very properly, the object of much study at present, even the comparative statics of a stochastic equilibrium i s not yet a solidly-based technique. Consequently, to make any progress in comparative dynamics, I found i t impossible to give any attention to uncertainty. I lament this drawback, and hope that economic theory w i l l soon be able to do better. This hope i s not motivated only by intellectual curiosity, for central to any assumption that makes an economic agent into an intertemporal maximiser i s that he should have well-defined (even i f stochastic) expectations about the future over which he i s maximising. But the future i s always uncertain, even in the presence of futures markets. Throughout this essay, a presumption i s made of rational expectations a la Muth (1961). This means simply that economic agents are endowed with perfect foresight of what the model predicts.on the assumption that they do have 66 perfect foresight. If i t i s accepted that uncertainty i s to be ignored, then I feel that a rational expectations hypothesis i s the next most honest thing. Besides, i t leads to a much cleaner and more self-contained theory than would an alternative hypothesis involving an expectations-generating mechanism leading to frustration and constant planning revision. (See however Goldman (1968) on this subject: such a hypothesis could be made manageable by his kind of scheme.) Section 1 contains the model of the landlords' behaviour. They have an optimal control problem to solve which i s not very different from that of Chapter II. Then i n section 2 comes the model of the tenants. Their behaviour i s taken to be governed by instantaneous u t i l i t y maximisation, and the assumption that they can move from one dwelling-place to another costlessly. This scheme, however unrealistic, i s usual enough i n demand analysis - a consumer's intertemporal considerations are only beginning to be noticed, and would certainly be an unwanted complication here (see Diewert (1974)). i n section 3 the long-run equilibrium between landlords and tenants i s worked out i n the sense discussed above. (1) The Landlord's Profit A city of any age contains buildings of widely differing dates of construction. Most often, the older buildings, however solid their structure, are not kept up very well, and provide the not very comfortable, run-down housing of the poor. New buildings on the other hand are regularly f i t t e d out with furnishings of great luxury, and are expected to be inhabited by high-income people. In the model to be discussed i n this section, each dwelling w i l l be assumed to be characterised by two properties only: the age of the structure, v, and the level of upkeep, or "comfort", k. This variable, k, i s a stock of upkeep, not a flow. Dwellings are owned by absentee landlords, who invest in upkeep so as to maximise the discounted stream of expected rents they receive. Gity dwellers are distributed over a spectrum of incomes, according to which, as well as to their tastes, they choose, by maximising their u t i l i t y , a level of upkeep from the selection offered by the landlords. The term "city dweller" should be understood to mean an entire household rather than an individual, although only one u t i l i t y function w i l l be allowed to each household. The income of any household w i l l be assumed to be constant over time. City dwellers are presumed to be quite lindifferent to the age of the buildings they inhabit - only.upkeep i s a characteristic entering their u t i l i t y functions. The municipal authority, on the other hand, cares only about the age of a building. After i t has existed for some time, T, i t must be torn down to make room for new construction, i f new construction i s i n fact profitable. If landlords are in a state of perfect competition, each one w i l l per-ceive a profile of rents available or expected to be available at any time t in return for a level of upkeep k: l e t this be denoted by the function R(k,t). Any level of upkeep w i l l depreciate unless maintained by investment, and so, i f I i s the level of investment, one may imagine that upkeep changes over time according to the equation k = I - 6k, where 6 i s the depreciation rate. Let the cost per unit time of level I of investment be C(I). Then each landlord w i l l wish to maximise J 5 / e ~ P t ( R(k,t) - C(I) ) dt o subject to k • I - 6k, I 5t 0, k i 0. p i s the discount rate, v the age of the building. Following as usual the rules of Chapter I, we may form the 68 Hamiltonian with a current-valued shadow price p: H = e-Pfc( R(k,t) - C(I) + p(I - 6k)) . The control variable i s I, and H i s to be maximised with respect to i t . The optimal value, I*, i s given by: I(p) i f p>C'(0) I* - { 0 i f p < <r<0) (1) where i(p) satisfies the identity C'(l(p)) - p. The marginal cost function, C, i s assumed, as usual, to be positive and increasing. Further, the function C i s taken to be independent of time, which means that technological progress i s ignored. In fact this r e s t r i c -tion i s not very important to the analysis that follows, and could be relaxed at the cost only of complication. The maximised Hamiltonian i s M(k,p,t) - e" p t(R(k,t) - C(I*) + p(I* - 6k)) and so the price equation i s p = pp - R'(k,t) + 6p (2) where the dash denotes a derivative with respect to k. The stock equation i s of course just k = I* - 6k (3) and so the phase plane i s as shown in Fig i i . The equation of the p = 0 l i n e , the demand curve, i s p - _ J L _ R'(k,t) p + 6 and that of the K = 0 l i n e , the supply curve, i s p = C'(6k). Above the line p = C(0), the stock equation i s k = I(p) - 6k, and below, k "" Fig 1 2 . i t i s just k = -6k. Because R i s an explicit function of t i n general, the phase plane i s not static. The k = 0 line does not move, and so the saddlepoint must always be on i t , but the p => 0 line w i l l s h i f t about. This fact introduces no real conceptual d i f f i c u l t i e s , but i t makes the analysis harder. The present chapter, though, i s concerned only with a state of long-run equilibrium, i n which the time-dependence of R disappears. Long-run equilibrium w i l l mean zero economic p r o f i t for landlords. That i s , the worth of optimally exploiting a newly constructed dwelling 11' w i l l be equal to the construction costs. These w i l l be composed of two parts: the building cost, assumed fixed and constant, and the cost of the > i _ i n i t i a l state of comfort, k, say. Since the line p •= C'(6k) i s the supply curve (marginal cost curve) for comfort levels i n an already constructed building, i t i s reasonable to suppose that costs are lower on the construction site than when tenants are around to object to inconvenience. The marginal cost curve for k w i l l thus be supposed to l i e beneath, the k = 0 l i n e , as shown i n Fig 1 2 . The i n i t i a l point, (Jc, p), on the optimal trajectory must, by the usual transversality argument, l i e on this curve. Similarly, the f i n a l point must have zero shadow price for optimality: the optimal trajectory must end on the k-axis. The actual trajectory i s then uniquely defined by the total lifetime of the build-ing, T. Paths which go closer to the saddlepoint take longer: the standard turnpike result. The optimal path also defines the f i n a l state of upkeep, k, say. Because the time-dependence i s now restricted to the discount factor, the worth of the building can be computed by noting that M - pj* i s constant along the path: the result i s : 1) To avoid any d i f f i c u l t i e s to do with rent of the land that dwellings are on, i t i s easiest to assume that land i s so abundant that i t i s a free good. 71 J* = — (M(k, p, 0) - M(k, 0, T)) . (4) P 2. The City Dweller's Choice of a Home. Each cit y dweller i s endowed with a u t i l i t y function, D, the arguments of which for our purposes are k, the comfort level of his dwell-ing, and X, a Hicksian composite of everything else he buys. The price index for the composite i s denoted by P. He i s assumed to have an un-changing income y: later y w i l l be used to index the inhabitants of the c i t y , and so i t w i l l be possible to l e t different people have different tastes by allowing U to depend on y. We may write then: U - U(k, X} y). The problem of maximising U i s slightly different from the usual one, as k does not measure so much a quantity as a quaivty of dwelling. Each inhabitant i s presumed to have a completely inelastic demand for exactly one dwelling. Then his budget constraint i s : R(k) + PX » y. (5) The first-order conditions for a u t i l i t y maximum are: U, - XR-(k) k u x = XP along with equation (5). (X i s a Lagrange multiplier = marginal u t i l i t y of income.) If P i s constant - as i t must be i n long-run equilibrium -there i s no loss of generality in setting i t equal to unity. Then the first-order condition may be written thus: R'(k) - fi(k, R(k) , y) (6) where fi i s a marginal rate of substitution: fl(k, R(k), y) = ux(k, y - R 0«>' y) For further analysis a l l we shall need i s this function ft, at least for this chapter. It should be noted here that, although the u t i l i t y function depends e x p l i c i t l y on y and thus can in principle accommodate any sort of preferences for each c i t y dweller, i t w i l l i n the next section be assumed that, for any reasonable rents prof i l e R(k), a higher income y w i l l lead to a choice of a higher k according to eq (6). This assumption, probably quite reasonable, i s needed to avoid technical complications. 3. Equilibrium between Landlords and Tenants. Let us assume that there are N inhabitants of the c i t y , and so i n equilibrium N dwellings. Let the income distribution of these tenants be described by a cumulative distribution function F, so that there are NF(y) people with income less than or equal to y. Let y_ fie the lowest income, y the greatest, so that F(y_) =0 and F(y) =1. Now i n the short run} the supply of dwellings i n a given state of upkeep i s completely inelastic. In fact, l e t the distribution of states of upkeep be represent-ed by a function G, such that the number of dwellings with a comfort level less than or equal to k i s NG(k). It i s convenient to abstract from a l l market imperfections and assume that at each moment demand and short-run supply are i n instantaneous equilibrium. (In long-run equilibrium, G does not change over time, and this instantaneous equilibrium i s identical to the f u l l one.) Once G i s given, then R(k), the rents p r o f i l e , should be completely determined by demand, that i s by eq (6). Each tenant, faced with the p r o f i l e R(k) and given his income y, can solve eq (6) to determine the comfort l e v e l , k, that he w i l l purchase. The result, withithe assumption stated at the end of the preceding section, w i l l be a one-to-one increasing relation between y and k - l e t us write i t as y = y(k) K H with y a function of k rather than Viae versa. If the function R i s specified, then y(k) i s determined by eq (6). Contrariwise, i f the function y i s specified, the rents profile R(k) can be recovered from eq (6), which i s now simply an ordinary di f f e r e n t i a l equa-tion for R, i f a boundary condition i s available. The desired boundary condition i s obtained from the zero-profit condition of long-run equilibrium, which equates the expression in eq (4) with total construction costs. Although i t i s a long-run condition, i t i s a legitimate determinant of instantaneous equilibrium because, i f new construction i s to take place at each moment (and i t must since demolition goes on continuously), then the expected rents prof i l e at a l l times during the l i f e of a dwelling must guarantee zero pr o f i t at each instant. The short-run instantaneous equilibrium, determined purely by demand in the presence of an inelastic supply G(k), can now be written down. Since the number of tenants occupying dwellings of comfort level less than k i s NG(k)1, and this number, by the assumption of the monotonicity of y(k), i s just the number of tenants with incomes less than y(k), that i s , NF(y(k)), i t follows that y(k) = F _ 1(G(k)) (8) The inverse function F" 1 i s always well-defined because F i s a cumulative probability distribution. Now i n the short run, the functions F and G are both exogenously given, and so therefore i s y by eq (8). The rents profi l e R(k) can now be obtained as discussed above. Eq (8), with our assumption of instantaneous tatonnement, holds good also i n dynamic situations. In the long run, I t Is of course the supply of comfort which must adjust f u l l y to obtainable rents. In fact, since the discussion of the landlords' profits shows exactly what time-path upkeep levels w i l l take given a rents p r o f i l e R(k), we can now compute the supply spectrum G(k), given R(k) i n a long-run situation. The result of this computation, along with the short-run one giving R from G, w i l l simultaneously determine both functions and complete our general equilibrium analysis. We are dealing with a steady state i n which a l l buildings last for a time T and i n which the distribution of building ages i s rectangular. Thus the number of these which have an age, v, greater than some age t < T, i s N(l - t/T). For each age t , there i s a unique optimal upkeep, k(t), say, given by the optimal path shown i n Fig 12. (Unique because k declines monotonically along the path.) The function k(t) satisfies the optimal equations (2) and (3). The number of dwellings of upkeep level less than k(t) i s therefore also N(l - t/T), that i s : To proceed, then, we must solve eqs £'(2) and (3) for the function k(t). The solution w i l l be i n terms of the rents profile R(k). Each of the optimal equations for the variables k(t) and p(t) can be solved for one variable i n terms of the other. From eq (2) we obtain G(k(t)) - 1 - t/T. (9) d -(p+6)t, -(p+6)t dt (pe e (P - (p + 6)p) e-(p+«)t R'(k) whence T -(p+fi)(t'-t) p(t) - / dt' e R'(k(t')), (10) t since p(T) =0 by the terminal transversality condition. Similarly from eq (3) we obtain : T A k(t) = k e 6 ( T " t ) - / dt' e 6 ( t _ t> I(p(t')), (11) t where k (see Fig 121 i s the upkeep level at the moment of demolition, and where the function I(p) must be taken as having the value zero for p <_ C'(0) (eq (1)) . As can be seen from Fig 12, there i s an age of build-ing, t Q say, past which no further maintenance i s done, since p <^  C(0) . Therefore eq (11) can be rewritten as: k(t) = k e 6 ( T _ t ) - / ° d f e 4 ( t ' " t ) K p ( f ) ) for t < t Q (12) t k(t) = ke**1"*) for t >_ t 0 (13) Now eqs (10) and (11) can be combined to yie l d : kct) = - / t o dt' .«(t'-t) i ( \ d t - .-(P+«(t--t*) t t' xR'(k(t"))) (14) for t < t — o F i r s t of a l l we observe that eqs (14) and (13) i n effect provide the answer to the long-run supply problem i n the face of an (exogenous) rents profile R(k). This i s so, since i f we invert the function k(t), given by eqs (14) and (13) i n terms of R(k) and other exogenous functions, we obtain the supply spectrum G(k) from eq (9) : G(k) - 1 - lLk_L T where by definition t(k) and k(t) are inverse functions one of the other: t(k(t)) • t and k(t(k)) = k. The time t Q , which has so far been defined only in words, i s given by the equation C'(0) - / T dt' e - ( p + 6 ) ( t ' " l o ) R-(ta«<*-t*>|J (15) to (see eqs (10) and (13)). The boundary value k i s perhaps best kept as an exogenous parameter at 76 this point, since i t s value depends on the construction-site marginal cost schedule for k (Fig 12). For our present purposes, i t contains a l l we need to know about this schedule, since the point (k, p), which l i e s on i t , i s just the point (k(0), p(0)) given by eqs (10) and (12) i n terms only of k and other known quantities. The next step, then, i s to combine eqs (6), (8), (9), (13) and (14), which express the short-run equilibrium and the long-run equilibrium, so as to determine simultaneously both R(k) and G(k). This step i s the "general equilibrium" step, and i t may be worthwhile to point out the similarities between our equations and the usual general equilibrium ones. It has already been remarked that the short-run equilibrium i s character-ised by the forces of demand. The distribution function G(k), describing as i t were the "quantities" of upkeep available, corresponds to a vector of quantities in a conventional general equilibrium economy. The R(k) which, for a given G(k), i s derived from eqs (6) and (8), then can be seen as a function corresponding to a vector of market-clearing prices. The long-run equations determine supply; that i s , the G(k) derived from eqs (9), (13) and (14) for a given R(k) gives the upkeep levels called forth by the rents R(k). Thus this part of the calculation corresponds to writing down a set of supply functions - a vector of quantities supplied i n response to a vector of prices. It can thus be seen that our model i s indeed formally analogous to a general equilibrium one, with vectors (finite-dimensional usually) of quantities and dual price vectors replaced by functions i n a space of cumulative probability distributions and dual functions in some suitable dual space. The calculations involved i n this general equilibrium step proceed as follows. From eq 1(14), which i s the solution of the optimal control equations, some changes of variable lead us to an equivalent equation, (17), 77 for the inverse function t(k). Then the demand equation, (6), i s solved formally for the rents function R(k) i n terms of t(k). This allows the equation for t(k), eq (17), to be written i n terms of t(k) alone; this w i l l be eq (20). The question of the existence and uniqueness of a solution for this equation i s then taken up. F i r s t , a series of technical manipulations gives eq (26), which i s just eq (20) much simplified. Then a fixed-point theorem i s invoked to conclude the analysis. It Is convenient to begin with some manipulations of eq (14). The function t(k) inverse to k(t) i s defined, via eq (14), by: 6(T-t(k)) f T 6(t'-t(k)) k =» ke - J dt e t(k) , , T -(p+6) ( t " - f ) . xi( / dt"-*e R'(k(t"))J (16) t * where, to save writing two equations at each step, we make the convention I(p) =0 for p G"(0). Eq (13) i s now therefore subsumed in eq (14). A change of variable can be performed i n each of the integrals i n eq (16), from a time-variable to a state-variable. Thus, l e t f = t(k'); t" = t(k"). Tnen i t i s easy to see that /« « . . - « » « > «t'"t*» R'<kCt»» - / * ' « . ( . - £ « • > } . - ' » • « " « * • > - ' * ' » since t(k) = T, t(k') - t ' and k(t(k")) •» k" by definition. Similarly the integral / dt' e 6 ( t ~ t ( k ) ) f ( t ' ) , for any function ¥ of t', can t(k) be expressed as: k 6(t(k')-t(k)) / dk'(-3£(k')) e Y(t(k'))-k a k 6t (k) Thus eq (16) becomes, on multiplication by e : 78 ke 6t(k) = ke tdk' (-^(k')) k dk 6t(k') i f / dk" (-££.(k")) e (P+6) (t(k")-t(k')) R'(k"J.) . (17) The above eguation holds for a l l k between k and R. The demand eguation (6) can be written as follows: R'(k) = ft(k, R(k), y(k)) where the function y(k), as in eq (7), gives the income of the tenant who chooses comfort level k. But from eqs (8) and (9) we know tliat y(k) can be expressed as: y(k) = F" 1(G(k)) T ' with the same function t(k) as i n eq (17). Therefore we obtain the following ordinary di f f e r e n t i a l equation for R(k): For a unique solution, a boundary condition i s needed. We shall simply take R(k) as R, and f i n a l l y pin down R by the zero-profit condition. Meanwhile R w i l l be treated as another parameter of the model. In fact, just as a knowledge of k was enough information about the construction-site marginal cost schedule for present purposes, so i s R enough information about fixed construction costs. If R i s given, then we may infer, by reasoning backwards, what the cost of constructing a new dwelling must be. It i s as legitimate to regard R as truly exogenous as i t would be to introduce a constant C, say, as the cost of a new dwelling with b u i l t - i n comfort level k, and then solving for R - and i t i s very much simpler. (18) A standard Lipschitz condition (see, as a convenient i f not very com-plete reference, Bryson and Ho (1969a)) i s then enough to guarantee the existence and uniqueness of the solution to eq (18). The assumption that such a condition i s satisfied i s not at a l l stringent, and i s hereby made. The solution can then be formally written as: R(k) - A(t(k) ; R) (19) where A(...; R) i s some, i n general non-linear^operator acting on the space of continuous functions t from the interval (k, k) into the non-neg-ative real l i n e . (Or, i f we wish to be more precise, although i t would seem to be of l i t t l e benefit, functions such that t(k) =» T, t(k) « 0.) Let A'(t(k); R) denote the function of k which i s the derivative with respect to k of A(t(k); R). Then eq (17) becomes: k ef i t ( k ) = j^ST _ j k d k . ( _ d t ( ] 0 ) e6t(k') k dk x !( ^ dk» (^t(k»)) e - ( P + 6 H t ( k " ) - t ( k ' , ) A ' ( t ( k " ) ; R)) k dk (20) This i s now a non-linear integro-differential equation for the function t(k) written i n terms of exogenously given quantities only. Once we are satisfied that i t possesses a unique sensible solution, our problem i s solved, since G (k) i s just 1 - t ( k> , and R(R)s i s just A (t (k); R) . A l l that T ' ~ remains to be shown, then, i s this matter of existence and uniqueness. Let us make the definition: T(k) S e 6 t ( k > . (21) Then x ** (k) - the dash denotes the derivative - i s equal to sf^-(k) e 6 t ^ ) , dk and eq (20) can be written i n terms of T(k) as follows: k kxlk) - k e 6 T - U dk' (^x'(k')) p+6 k' p+26 x l({tlk')} « - / dk" (-x'(k")) {T(k")}-^S r ( T ( k " ) ; R ) ) (22) where r(x(k"); R) = A'( — log x(k"); R) and i s just another nonlinear 6 operator. k d Now kr(k) - ke 6* = / — ( k ' x ( k ' ) ) dk' (since x(k) - e 6 T) k dk k = / (x(k ) + k'x'(k')) dk'. k It follows that eq (22) can be written k / dk' fr(k') + x'(k') fk' - ~ T(x(k') ; R ) ) } - 0 (23) k 6 with the non-linear operator T defined by: T(x(k') ; Rl -p+6 k' _p+26 l({t(k')}®"-/ dk" (-x'(k")) {x(k"))}"^~ r(x(k"}; R ) ) . 6 * (24) The integral sign i n eq (23) i s plainly unnecessary: _ x'(k) = i  X(k) k - i T(x(k) ; R) 6 This equation w i l l be re-integrated i n a moment. Meanwhile l e t us take note of some restrictions that we wish to place on admissible functions 6T x(k). In addition to the boundary condition x(k) » e , we require that x(k) i s always greater than unity and decreasing, because of the physical interpretation of t(k). Again, since l(p) = 0 for p £C'(0), it.*is necessary that, for k less than some k Q, we have simply: T(k) - (k/k> e O A , i.e., t(k) = T - - log ( k A ) . 6 Whatever the behaviour of x(k) for k > k Q, i t follows from eq (18) and i t s solution eq (19) that, for k <^  k Q , r(x(k); R) = A'(t(k); R) - A'(T - i - log (k/k); R) . If this last result i s substituted into eq (24), KQ i s determined as the greatest k for which T(x(k); R) i s zero - that i s , k Q i s determined com-pletely by the exogenous quantities, as are also x(k 0) and t ( k Q ) . Bearing in mind the definition of x(k), eq (21), we may now inte-grate eq (25) from k Q to k. The result i s : k ! t(k) - t(k ) - / dk' (26) k Q 6k' - T(t(k'); R) with the operator T now redefined in an obvious manner to act on t(k) instead of x(k). Eq (26) i s a rather straightforward looking non-linear equation, and we may ask directly about the existence and uniqueness of i t s solution. A solution i s plainly a fixed point of the operator k A(t(k)) = t(k Q) - / dk' ~ k Q 6k' - Tit(k'); R) It should be pointed out here that k, the starting-value of the state variable on the optimal trajectory that we are looking for, i s s t i l l endogenous. I t is determined by the equation t(k) = 0. But we shall never be interested i n values of t(k) for k > k, and so eq (26) could be rewritten as: t(k) = max (A(t(k)), 0). Let us t r i v i a l l y redefine the operator A to be the right-hand side of this equation. Then i t follows that the denominator 6k' - T(t(k'); R) , which i s positive ( => 6k Q ) at k' = k Q , must always be positive for k' < k, for otherwise i t would vanish and make t(k) equal to minus i n f i n i t y . We conclude then that A maps positive decreasing functions t(k) (k Q <_ k < ») into positive decreasing functions. The boundary value t(k Q) i s preserved, and the operator i s clearly bounded. The contraction mapping principle (see Krasnosel'skii (1964) and Kleider et al. (1968)) can be invoked i n cases such as the present to prove existence and uniqueness of a fixed point. Bounded continuous functions defined on 5(k0, ») form a Banach space with norm defined by: | t 1 - sup | t(k) | (27) k 0<k«» The contraction mapping principle says that i f A maps a bounded region of this space (such as positive decreasing functions with || t || £ t d ^ ) ) into i t s e l f , and i f , for any two functions tjjk) and t£(k) belonging to this region, the Lipschitz condition 11 At! - At 2|| < o|| t ! - t 2 || (28) holds with a < 1, then there exists a unique fixed point of the operator A i n the region, which can be calculated from any starting function t Q(k) by iterations: t n(k) = At n_!(k). We need only ask, then, under what conditions the inequality (28) i s sat-i s f i e d . The iterations can of course be thought of as steps i n some tatonnement process. In fact, many cases i n which Brouwer's theorem i s used can be so interpreted, and often the contraction mapping principle would be applicable to these cases. If so, because of i t s constructive nature, i t i s greatly to be preferred. One instance of the use of the contraction mapping principle i n economic theory can be found i n Brock (1972), where i t i s used i n connection with equilibrium forecasting. It i s clear from eq (26) that A i s a continuous operator when the norm of eq (27) i s used. I have not succeeded in demonstrating that the inequality (28) i s always satisfied for any exogenous functions 0, P and I, but this i s not, I fee l , a very urgent matter. The contraction mapping principle i s a very stringent sufficient condition for existence and uniqueness, and when i t s requirements are not satisfied, there i s often no d i f f i c u l t y i n proving at least existence by other fixed-point theorems, many of which are presented i n Krasnosel•skii*s book. One may quite easily s i t down and find out i f inequality (28) i s or i s not satisfied i f Si, P and I are given. When, i n the next chapter, specific choices of these functions are made, this w i l l be done, and the solution w i l l i n fact be e x p l i c i t l y derived. The specification of the long-run equilibrium i s now finished. Because i t i s necessarily a steady atate, the solution functions G(k) and R(k) do not depend on time. This means that the equilibrium i s rather insensitive to the expectations-generating mechanism postulated for the landlords. As well as the rational expectations assumed at the beginning of this chapter, expectations of rents remaining as they are at any mom-ent would give the same result. In the dynamic analysis of Chapter IIIB however, expectations become more important, and whatever results are found depend on how they are generated. The conditions for "entry and exit", too, are simple because of the steady-state nature of the solution. Again, these conditions assume greater interest i n a dynamic context. 84 CHAPTER IIIB. A MODEL OF URBAN DECAY. The long-run equilibrium treated i n Chapter IIIA i s the starting point for this chapter. Into the steady state of that equilibrium w i l l come a disturbance. Specifically, i t w i l l be imagined that a consumption externality arises, adversely affecting housing, but without other effects. This i s readily modelled by replacing the u t i l i t y function of the tenants by U(u(k, a), X; y) (29) where U i s the same u t i l i t y function as before, but where the worth to a tenant of a level of comfort k i s no longer simply k but i s measured by the function u(k, a). The parameter a measures the externality: we assume that u(k, 1) = k, and that uffl > 0. The separability of k and a from the other variables in the argument of U expresses the assumption that the externality affects only housing. The specification that u(k, 1) = k means that the value of unity for a corresponds to the absence of any externality: i f , as we suppose, the externality that arises i s adverse, i t must, since u a > 0, correspond to a value of a less than unity. For l example, i f pollution of some kind, from a smoky factory chimney for instance, i s the source of the externality, then clean air means that a - l , and a can be thought of as the inverse of some measure of concen-tration i n the a i r of noxious fumes. The s p i r i t of comparative statics would have us now consider the derivatives with respect to a , evaluated at a = 1, of a l l the endogenous variables of the model. We shall not do so, but rather look at the results of a sudden discrete change from a - 1 to a value of a less than 1 . Again, comparative statics would focus attention on the functions R(k) and G(k) that measure rents and supply of comfort, and would take account only implicitly of the po s s i b i l i t i e s of entry and exit from the landlord business (by taking a zero-profit condition on new construction as one of the equilibrium equations). But here, entry and exit are considered quite ex p l i c i t l y , and windfall gains or losses become a feature of our model. These disequilibrium phenomena i n fact give the distinctive colour to the economic tale to be told. For, once the externality has arisen, the bright idea w i l l no doubt occur to someone that the nuisance of building a house i n the suburbs and commuting daily i s outweighed by the advantage of escaping the externality, and so he moves out of the city. It w i l l presumably be the richest per-son who moves, since his desired standard of comfort, k, has become effect-ively unobtainable. But his departure lowers the demand for housing, and therefore also the revenue p r o f i l e , R(k), perceived by landlords. If, as i t i s reasonable to suppose, the marginal revenue p r o f i l e , R' (k), i s also lowered, i t can be seen from Fig 12 that the landlords w i l l respond by spending less on upkeep. This further lowers-the available standards of comfort, and so, in a cumulative process, more and more city-dwellers are pushed to the margin where i t i s more advantageous to move to the suburbs. Suburban housing w i l l no doubt take time to build, and w i l l be subject to increasing costs. If so, ultimately a cut-off point may be reached where poor people are l e f t i n the city, trapped there in decaying slums ho longer kept up by the landlords, but without enough income to afford the now expensive suburban housing. The above discussion provides a possible scenario of events. If i t i s to be modelled, then i t i s clear at once that the conditions for entry of a firm into the suburban housing business and for exit from (and poss-ible re-entry into) the urban construction and renting business must be ex p l i c i t l y l a i d down. In section 1, then, a model of the suburbs i s appended to our exist-ing model of the ci t y , and entry and exit are discussed. Then i n section 2, the assumption i s made of a horizontal marginal cost schedule for up-keep in the city. This permits a great simplification of the urban model and the exp l i c i t form of the long-run equilibrium i s worked out. Other specific functional forms for exogenous quantities are chosen here and in later sections so as to make i t possible to write down explicit express ions for some of the endogenous functions. In section 3, i t w i l l be seen that two distinct states of affairs can arise i n the coupled city/suburb model, according to whether or not the rate of new suburban construction i s fast enough to leave unoccupied dwellings standing in the city. The case in which i t i s i s analysed i n section 3, and the case in which i t i s not in section 4. After this, i t i s possible i n section 5 to catalogue the various modes of urban decay that can be generated by the present model. 1. The Flight to the Suburbs. The model proposed here for suburban affairs i s simpler than the ci t y model. In particular, intertemporal considerations for suburban landlords, tenants and/or owner-occupiers w i l l be abstracted from. We shall i n fact lose, interest i n a person more or less at the moment he or she moves to the suburbs. The only quantities of interest w i l l be the 87 rental price (for one unit of time) that must be paid for a suburban -:r-.:-'-. dwelling of comfort K at a time when M dwellings exist i n the suburbs, and the cost (expressed as a rental price) of constructing such a dwelling. This information i s enough to allow calculation of the margin between c i t y and suburb, and, provided only that suburbanites stay i n the suburbs once they get there, knowledge of affairs at the margin i s a l l that we shall need. A very elementary sort of geography i s implied i n a l l t h i s . Both the c i t y and the suburb are treated as points, or as completely undifferent-iated areas. People may make only a binary choice of where to l i v e -notions of better or worse neighbourhoods are ignored. Our programme for this section i s as follows. F i r s t we shall specify the response by the construction firms that build suburban hous-ing to any given state of demand. In particular entry and exit of these firms w i l l be discussed. Then demand considerations w i l l be taken up, and i t w i l l be seen that demand for suburban dwellings i s determined by the margin of indifference between city and suburban l i v i n g , this margin being determined by conditions i n the city. Lastly the two sides of the market w i l l be brought together to give the dynamics of suburban construct-ion. Let us then write S ( K , M) for the rent of a suburban dwelling of comfort level < at the time when exactly M suburban dwellings exist, and H ( K , M) for the cost of constructing i t , expressed as a rental charge. The p r o f i t obtained by a suburban construction firm for putting up this dwelling i s thus S ( K , M) - 5(K, M) per unit time, or, more sensibly, the capitalised sum :L(S(K, M) - S ( K , M)), where p i s the discount rate. P 88 Next, we must enquire how many firms are engaged at any time in suburban construction, how fast they can put up a dwelling, and what state of competition prevails among them. For simplicity, we may assume that each firm can put up exactly one dwelling per unit time, regardless of i t s comfort level, K . The construction rate at any moment i s then just m(t), where the function m(t) gives the number of firms i n the business at time t. These "firms" are to be thought of as entrepreneurial units in comp-etiti o n , and so i t seems reasonable to require that the profit that each makes per unit time should be the same across firms at each moment. Why, in a state of competition, should there be any pr o f i t at a l l ? After a l l , i n the city, a zero-profit assumption was used to characterise equilibrium. Various answers can be given: adjustment costs of one kind or another, rezoning costs, or, i n general, costs of entry. I f , for instance, there are many potential entrants into the suburban construction business, and each perceives a barrier to entry (in dollar terms) of a different size, then, the more firms are i n existence, the higher profits must be - here "profit" means earnings over and above the prime cost of a l l used-up factors of production. At a l l events, p r o f i t may certainly exist i n a competitive industry where entry and exit are not instantaneous adjust-ment processes, even i f no dollar cost i s involved. But once firms do exist and are in competition, i t i s a reasonable assumption that the firms w i l l so bid for business that every dwelling under construction at a given moment, noismatter what i t s comfort level K , w i l l yield the same prof i t . Otherwise, low-profit firms would constantly have an incentive to underbid high-profit firms. 89 If the above reasoning i s accepted, then we may assume that, i f exit of a firm i s costless, or rather, i f , whatever exit costs may be, each firm can be imagined to exit after each dwelling that i t i s building i s completed and subsequently to re-enter i f the pro f i t rate i s satisfactory, then, the number of firms, m(t), i s an increasing function of the p r o f i t per dwelling alone, that i s : m(t) - f(n(t)), ( f > 0) (30) where Tr(t) = S (K , M) - E ( K , M) (31) i s the pr o f i t . Our assumptions mean that S (K , M) - E ( K , M) must be V independent of < over the existing spectrum of comfort levels K , so that ir(t) i s well defined. The cost function E ( t c , M) i s of course exogenously given, and we shall assume that E K > 0, E M > 0, the second of these conditions expressing increasing costs in suburban construction as more dwellings are put up. A justification of this i s no more than an invoking of the law of diminish-ing returns with some factor (available drained land or some such) fixed. The state of affairs i n the suburbs i s determined by M(t), the number of finished dwellings, and we have by definition that: M(t) - m(t) - f(ir(t)) (32) Since M(0) = Q (the externality begins at time zero), eq (32) gives the dynamics of the suburb once ir(t) has been f u l l y expressed i n terms of the exogenous variables. This means that we must now pin down S(K, M), presumably by the forces of demand. Potential suburbanites have the alter-native of l i v i n g i n the c i t y , and so we may begin our demand analysis there. A city-dweller of income y has, by eq (29) , u t i l i t y : U - U(u(k, a), y - R(k); y). (R(k) i s , as usual, the rents prof i l e i n the city at some moment - tenants have no intertemporal considerations;) The first-order condition for maximising this u t i l i t y i s vu(k, o)U (u(k, a), y-R(k); y) R'OO = _ * t|x(u(k, a), y-R(k); y) - u^k, a)fl(u(k, a), R(k) , y) ( 3 3 ) where Q i s the marginal rate of substitution introduced i n eq ( 6 ) . Let the solution, k, of this first-order condition be written k = k(y). In general there i s some disadvantage associated with l i v i n g i n the suburbs rather than i n the city (else why was there no suburb before the externality?) and i t i s convenient to model this fact by assuming that a standard of comfort i n the suburbs K enters u t i l i t y functions as U ( K , o_) for some <x_ satisfying 1 > a_ > a. This number a_ does not of s s s s course correspond to an externality, but rather an in t r i n s i c disadvantage of suburban l i f e . Then, analogously to eq ( 3 3 ) , we obtain for the suburbs S ' ( K ) = u K ( K , a s ) f i ( u ( K , a g) , S ( K ) , y) ( 3 4 ) Let the solution of this equation be K = K(y). (Time-dependence, as expressed v i a either t or M, i s suppressed for the moment for the sake of clearer notation.) Then, for a person of income y to be at the margin of indifference between city and suburb, we must have that: U(u(k(y), a), y~R(k(y)); y) - U(u(ic(y), a ), y-S(ic(y)); y). ( 3 5 ) s From this equation and eq ( 3 4 ) we can now determine ir (t). From eq ( 3 1 ) i t follows that S ' ( K , M) - E ' ( K , M) (dashes here denote different-iation with respect to K ) , so that eq ( 3 4 ) becomes: E ' ( K , M) - u]c(<, O g j f i t u d c , ct g), ir(t) + S ( K , M) , y). (36) Now, i f at time t, there are M dwellings in the suburbs, then there are also M people in the suburbs, out of a total of N people altogether, and so the income of the person at the margin, y(M) say, must satisfy the equation N(l-F(y(M))) - M, i.e. y(M) - F" 1 f*t*I) . (37) v N ' S t r i c t l y speaking, for this to be true we must make an assumption like that of section 2 of Chapter IIIA that people leave the city i n descending order of income. For any specific choices of the exogenous functions, this i s a matter to be verified, not assumed. But for the moment, the assumption i s a l l that i s necessary. If, then, F *(^jffi i s substituted for y into eq (36), the value of K which satisfies the equation, K(M, ir(t)), say, (time-dependence i s explic i t again) i s the level of suburban comfort chosen by the person at the margin. Consequently we may use eq (35) to obtain: U(u(k(y(M)), a), y(M) - R ( k ( y ( M ) ) , t) ,- y ( M ) ) - U ( U ( K ( M , IT(t)) , a ), y(M) - ir(t) - S ( K ( M , i r(t)), M) ; y ( M ) ) (38) s The left-hand side of this equation involves, i n addition to M and t , only the exogenous functions and quantities, U , u, a, y(M) (via F), and the functions k(y) and R(k, t) which depend only on the state of affairs i n the ci t y . For the purposes of the model of the suburbs, these last are taken as given, and so the left-hand side can be regarded as a known function of M and t. But, on the right-hand side, U, u, y ( M ) , a g, E and the function K (M, ir(t)) are a l l exogenous or derived directly from exogenous quantities. The result i s eq(38) i s an equation which can be solved for ir(t) as a function of M and t alone. When this solution i s substituted into eq (32), the dynamics of the suburban model have been f u l l y specified. 2. A Particular Case of the Model of the City. In this section, the marginal cost function C ( I ) for invest-ment i n upkeep in ci t y dwellings w i l l be assumed to be a constant, c, for values of I between zero and 6k, and for higher values of I i n f i n i t e . The aim of this assumption i s to make especially simple the optimal upkeep path that city landlords w i l l follow. That i t does so can be seen by observing that the function I(p) of eq (1) becomes equal to the constant 6k i f p > c, 0 i f p < c, and indeterminate between these values for p = c. This i s in fact an instance of a "bang-bang" control (see Bryson and Ho (1969b)). The result i s that i f the state variable k has the value k, i t w i l l remain unchanged at that value for as long as p > c. If the further assumption i s made that the construction-site marginal cost schedule i s given by the scene function C'(6k) as gives the k = 0 line, then, so long as a dwelling lasts long enough to receive some positive amount of main-tenance, i t w i l l be put up originally with comfort level k and maintained there u n t i l , at the end of i t s l i f e , i t decays according to the equation k = -5k. In Fig 13, the phase plane, analogous to that of Fig 12, i s drawn for this case. The exact location of the p = 0 li n e , with equation p = _ _ i — R'(k, t) p + 6 (dash denotes differentiation with respect to k), affects only the details of the "exit path", so long as (l/(p + 6))R-*(k, t) ^  c for a l l t (39) As usual, i f specific choices of the exogenous functions and quantities of the model are made, i t i s necessary to verify that this inequality i s satisfied. To proceed, then, we shall f i r s t compute, for the steady state as described in Chapter IIIA, the supply function G(k) and then the rents function R(k). On the way, some specific choices w i l l be made for some of the exogenous functions. Then various checks willbe made to ensure the consistency of the solution with the various assumptions that have been made. In the course of these checks, i t w i l l turn out that the model can be understood rather more generally than has been stated so far, and this w i l l be explained. Finally we shall see that eq (26) of Chapter IIIA gives the same solution as the one obtained here. We begin by observing that, i n a steady state with a rectangular distribution of building ages, the comfort supply function G(k), i s given by: G(k(t)) = 1 - (t/T) (eq (9)) while _ k (t <^  t*) k ( t ) = { - • *rt t .y k e-6(t-t«), (t>_t*) (40) where t* i s the (still-to-be-determined) building age at which maintenance stops. Thus: G(k) - 1 - - (t* + k log ) for k < ic T * Jk } (41) and G(k) = 1. We may notice further that k = ke-«(T-t*) (42) (k, as usual, denotes the upkeep level at the time of demolition.) With the help of eq (41) we may deduce from eq (8) the function y(k) to be used i n eq (6) in order to determine R(k). We have: y(k) - F~x(G(k)) - F-Ml - Mt* + i- log (kA))> (k < k) (43) T 6 ^ ( G t _ 1 ( i ( t * and y(k) is now the set of a l l incomes above F _ 1 ( l - (t*/T)). Fig 14. In what way can this transformation of the function y(k) into a corr-espondence be justified? Since i t i s impossible, because of the i n f i n i t e marginal cost, for a comfort level exceeding k to exist, we may write symbolically R(k) = 0 0 for k > k. The budget set of a tenant of income y, given by points (k, X) satisfying the inequality R(k) + X <^y, i s then truncated by a vertical line at k = k (see Fig 14, in which for c l a r i t y the budget lines of different households have been rescaled so as to coincide.) Tenants whose income i s less than F _ 1 ( l - (t*/T)) w i l l be on indifference curves l i k e 1^ tangent i n the usual way to the boundary of the budget set (cross-hatched). Tenants whose income exceeds F~l(1 - (t*/T)) w i l l be on indifference curves like I3, with no tangency at the corner of the budget set. The person whose income i s exactly F ~ * ( l - (t*/T)) w i l l be on indifference curve I 2 / tangent to the non-vertical part of the boundary of the budget set just at the corner. A l l this means that eq (6) gives R(k) just as before for k £ k ^ k~ i f y (k) i s interpreted simply as F _ 1 ( l - (t*/T)). This i s an appropriate time to introduce some more specific choices of exogenous functions. We shall be able, after doing so, to perform a l l the verifications necessary to ensure that the model makes sense. Accordingly, l e t us set: F _ 1(x) - y_+bx (0<yx<_l) (44) U(k, X; y) = k aX. <a,b > 0) (45) (Later, k w i l l be replaced by u(k, a) i n eq (45) .) Eq (44) gives us a rectangular distribution of incomes as i n Fig 15 , and eq (45) i s just a Cobb-Douglas u t i l i t y indicator. From eq (45) we obtain the marginal rate of substitution fi, as follows: 96 U k(k, y - R(k); y) ft(k, R(k), y) = U x(k, y - R(k); y) a(y - R(k)) (46) k With these simplifications we get from eq (43): y(k) - y + b f l - - ( t * + i . log (k/k))) for all k < k < k, — T S _ and eq (6) becomes: -W*(k) = y + b f l - i ( t * + ^ l o g (k/k ) ) ) i - R(k) a T 6 This i s a linear differential equation for R, and i t i s solved as follows: f r (kaR(k)) = ak3""1 ( ~R'(k) + R(k)) dk v a ' = a k a _ 1 ( y + b ( l - ~ ( t * + ^ log (k/k))) by the equation. T O Both sides of this last equation can now readily be integrated between k and k: k _ kaR(k) - kaR(k) = a/ dk' ( k ' ) a _ 1 ( y + b ( l - i.(t* + =- log ( k A ) ) ) ) — — k v — T O % (47) a-l Here we must notice that an indefinite integral of k log k is the funct-ion (l/a^)k a(a log k - 1 ) . (This can be checked by differentiation.) Making use of this result, and recalling eq (42), we obtain: k*R(k) - k^tk) - (k a - k a) (y - — (log k + h) — — — 6T — a + —• (k a log k - k a log k) , OT — — whence: R(k) = ( k A ) * ^ ) + (y - JL ) (1 - ( k A ) a ) + ~ - log (kA) a«T — 6T — This, then, i s the f i r s t of our unknown functions (for long-run equilibrium) and we may now directly check i t s properties. Different-iation gives: R'(k) = i - {a(kA) a(y_ " R(k) - ~ - ) + — } k - aST ST = ~ {a(kA) a(Y ~ RW) + -~ (1 - ( k A ) a ) ) (48) k — " ~ ST — It i s clear at once that R'(k) i s always positive and monotonically decreasing, as one would wish. ( y - R(k) i s positive because i t i s the X chosen by the lowest-income person - the part of his income not spent on housing in fact.) Next we can examine condition (39). We have that R'UO « j _ _b j e-a6(T-t*) + } k v *~ ~ oT * oT This i s expressed i n terms of t * , and for condition (39), either*: t* must be put i n terms of c or Vice Versa. From eq (15) the link i s found: c = e - ( 0 + 6 ) t ' R - ( 4 9 ) . 0 It w i l l be i n order then, to use. t* as the exogenous parameter that must satisfy conditions which w i l l allow condition (39) to hold. By use of eq (38), eq (49) can be evaluated, and the result i s : , ^ * i e V if , v r ^ w b) -a6(T-t*) (p + 6)c = Ha(y_ - R(k)) - e x l±l ( e < a 6 -P> W-t*) _ 1 } + J l { 1 + 6, ( 1 _ e - p ( T - t * ) ) } a6 -p 6T p 98 Plainly a sufficient condition for inequality (39) to hold i s that p + 6 a6 - p (a6-p)(T-t*) -1) < 1 and (1 + *)(1 - e - P < 1 P Sufficiently small T - t * (which i s equivalent to sufficiently small c) allows both of these inequalities to be satisfied. A l i t t l e manipula-tion shows that a condition which approximates both inequalities i s (p + 6) (T - t*) < 1 It i s of some interest to investigate the consequences of a choice of exogenous quantities such that condition (39) i s not satisfied. If the marginal revenue function R' i s used as calculated i n eq (48) and the phase plane drawn, the result w i l l look like Fig 16. But there i s of course no reason for this not to be quite correct. Furthermore, none of the above analysis needs to be changed, except to replace k by k , the saddlepoint s value of k. This follows because the i n i t i a l point of the optimal upkeep path must s t i l l l i e on the k = 0 line, and the only feasible path ending at k i s one which consists of staying at the saddlepoint un t i l age t* (possible since 1(c) i s indeterminate and may be set equal to <5kg.) and then exiting along the unstable arm, reaching the k-axis at age T. Another point emerges clearly from this discussion. It was stated in Chapter IIIA that treating k as exogenous was tantamount to writing down an exogenous con-struction-site marginal cost schedule. But here, once c and k (the point at which the schedule goes to infinity) are given, our assumption has been that the schedule was known. Consequently, k should no longer be a para-meter free to be chosen, and we can see i n fact how i t i s to be determined. 100 Fig 16 shows that k must be at the end of the unstable arm, and i t i s this fact that pins k down. Eq (49) gives, for exogenous c, the quantity t* as a function of k, since the function R'(k), given by eq (48), involves only k and other exogenous parameters. (R(k) i s exogenous.) But for k to l i e on the unstable arm, that i s , the path leading out from the saddlepoint, we require that k e ^ T - t * ' , which i s the value of the variable k at time t* on the path ending at k at time T, should equal k s, the saddlepoint k given by the equation c(p + 6) ** R'(kg) . We require i n fact that c(p + 6) = R'(ke6<T-**(k>)). (50) This i s an equation for k in terms only of exogenous parameters. We may now classify the cases that can arise by the use of eq (50). Let the solution of eq (50) be written as k = Mc) , and l e t k g = k g(c) be just ke 6 (T-t*(k(c))) ^  i f k g ( c ) >^  k f w e g e t t h e c a s e i n i t i a l l y proposed, shown schematically in Fig 17(a) . The intermediate case k g(c) = k i s shown i n Fig 17(b) , and the case k g(c) <^ k i s shown i n Fig 16 . One case remains, and i t i s distinguished from the others not so much by the value of c as that of T. I t could i n principle happen (we shall not be much interested i n this possibility) that T i s so short that calculation gives a negative value for t*(k(c)). In this case, the optimal path starts at p = c as usual, but at a lower value of k than k s -as shown in Fig 17 (c) . We notice that i n a l l cases no dwellings exist with a k greater than max(ks, k);, either because.of technological imposs-i b i l i t y or because of insufficient demand. After this demonstration that our i n i t i a l analysis w i l l work i n a l l cases with k redefined as max(k3, k), we shall nonetheless stick with our 101 Fig 17. interpretation of k as technologically imposed - for reasons of dynamics. Once the steady state i s l e f t , and demand becomes time-dependent, k (c) s w i l l also be variable. It i s much easier to deal with a technologically fixed k. The last matter to be attended to i n this section i s to see that direct use of eq (26) gives the same result as the one we have obtained. This i s now quite t r i v i a l . The operator T that appears i n eq (26) i s now equal simply to 6k for k = k, and zero for k < k. The (unique) solution such that t(k) = T i s immediate: k t(k) = t* + J dk' — - = t* + - log (k/k) for k < k. k «*' 6 This i s in accord with eq (40). For k = k the right-hand side of eq (26) i s not defined, as we require to make sense of the result that k(t) - k for a l l t such that 0 <. t <_ t*. 3. The Case of Rapid Suburban Construction. This section presents the f i r s t part of the discussion of the dynamics of the coupled model of city and suburb. The suburb i s model-led as i n section 1 and the ci t y as i n section 2. Specifications of some more exogenous functions are made, and the meaning of the t i t l e of this section, "rapid suburban construction", i s given. Then, the rents function R(k, t ) , now a function of time as wellr.as of comfort, i s obtained i n terms of the supply function G(k, t) v i a the demand d i f f e r -ential equation. The supply function G(k, t) is next determined by consideration of the ci t y landlords* optimal control problem. A partic-ular case of the dynamical evolution, that i n which a l l urban maintenance stops immediately after the externality appears, i s treated f i r s t . The function G i s then easily written down, and consequently also the rents function, R. A series of checks has then to be undertaken to determine when the particular case applies - these checks are concerned with the dynamics both of the city and of the city-suburban margin. Other possible regimes of dynamical evolution are discussed following the checks. Lastly, i t i s verified that the evolution of the model i s stable, in the sense that i t i s indeed the richer city-dwellers who f i r s t become dissat-i s f i e d and move out to the suburbs. F i r s t then, l e t us specify in a particularly simple form the function f which appears in eq (30) and which links the rate of construction i n the suburbs, m(t) , to the pro f i t per dwelling, ir(t). Let m i f ir > 0 f(ir) - { 0 i f ir < 0 (51) and f(0) i s indeterminate between zero and m. This means (eq (32)) that the number of dwellings in the suburbs at time t i s M(t) = mt for so long as positive profits exist over an unbroken time interval. What eq (51) says i s just that there i s a perfectly inelastic response of entry by exactly m firms for any positive p r o f i t whatever, and instant exit of a l l of them i n the face of loss. At the margin of exactly zero p r o f i t , there may be any number between zero and m. This choice of the function f makes our calculations much simpler than would any other choice, and does not obscure the dynamical questions in which we are principally interested. The rate at which demolition goes on i n the city i s , at least for time T after the imposition of the externality, given by N/T. (Long-run equilibrium, with a rectangular distribution of building ages, prevails befpre this time.) For this section, we consider only the case m > N/T, that i s , the case i n which positive profits i n suburban construction cause more dwellings to be b u i l t there than are simultaneously being demolished in the c i t y . The result i s , of course, more dwellings than people, and so there are unoccupied dwellings in the city. We may now specify the form of the function u(k, a), which expresses how a comfort level enters a tenant's u t i l i t y function i n the presence of the externality a. A particularly easy form i s u(k, a) = ak. (52) For this choice of u, then, and with the marginal rate of substitution given by eq (46), the demand equation i n the city, eq (33) , becomes: R'(k, t) = ( 5 3 ) k just as before. The demand equation i s not unchanged in general - that i s just a felicitous result of eq (52) . Time dependence has been made exp l i c i t , and dashes denote differentiation with respect to the upkeep variable. The fact of unoccupied dwellings means that the rent charged on the least comfortable inhabited dwelling i s zero - dwellings any less comfort-able have become free goods. This observation provides the boundary condition to accompany eq (53). The zero-profit condition no longer applies of course, since we assume that the externality arrives unexpect-edly. Let the lowest upkeep level of any inhabited dwelling at time t be k Q U t ( t ) . Then R ( k o u t ( t ) , t) = 0 and so eq (53) gives: k R(k, t) - ak" a / dk' ( k ' ) a _ 1 y(k', t) W W where y(k, t) i s , as usual, the income of the person inhabiting a dwelling of upkeep level k at time t. Next, l e t us define the upkeep supply function G(k, t) as follows: NG(k, t) = number of dwellings in existence (not necessarily inhabited) at time t of upkeep level ^ k . We may express k t ( t ) in terms of G. 105 Since the number of uninhabited dwellings at time t i s just (m - (N/T)), we have NG(k Q U t(t), t) = (m - (N/T))t. (54) Similarly, equating .numbers of dwellings to numbers of people, one obtains for y(k, t) the equation: NF(y(k, t)) + (m - (N/T) )t = NG(k, t) , so that y(k, t) = F - 1{G(k, t) - ((m/N) - dA))t}, k whence R(k, t) = ak" a / dk' ( k ' ) a _ 1 {y + b(c(k', t) '•out^ " (* - b t ) } (55) N T ' by use of eq (44). We have now reached the stage where, i f we can find G(k, t ) , the problem i s done. To find G(k, t ) , we must, as always, consider our optimal control problem for landlords. F i r s t , since at time zero (the moment of imposition of the externality) G(k, 0) i s just the long-run equilibrium function given by eq (41), we may see from eq (55) that R(k, 0) i s less than the equilibrium R(k) by just k_aR(k)/kar a quantity which i s always positive, (see eq (47).) On the other hand, a decrease i n R(k, 0) means, because of eq (53), an increase in R'(k, 0), so that the impact effect of the externality i s "perverse", i n the sense of Chapter II, in that the total revenue schedule f a l l s , but the marginal revenue schedule rises. There i s nothing perverse economically of course: the lower rents leave more money for other things, and the marginal rate of substit-ution shifts in favour of housing. The impact effect i s not, of course, the whole story. If i t were, then landlords would tend to maintain dwellings to a greater age than i n equilibrium - eq (49) shows that larger R" means a shorter time interval T - t*. There are in fact circumstances in which the landlords optimal 106 response i s to cease a l l maintenance, and we shall now consider this case. Any dwelling of age v less than T - t * (t* w i l l throughout denote the equilibrium t* given by eq (49)) i s at time t = 0 i n state k. It w i l l r'inish i t s l i f e at time T - v, at which time the shadow price of upkeep, p(t), w i l l be zero. At time t = 0, then, we get from eq (10) that T-v p(0) = / dt e _ ' P + 6 ) t R'(k(t), t) 0 where k(t) i s the upkeep at time t. If no maintenance i s done for t > 0, then k(t) = ke" 6 t, and so p(0) = / T " V d t e"< p + 5 ) t R'(kV 5 t, t) (56) 0 But no dwelling can reach age T and s t i l l be receiving a positive rent. Further, there i s always a positive time interval during which no rent i s received and for which therefore R" — 0, since R(k, t) = 0 for a l l k < k Q u t ( t ) . The upper limit on the integral i n eq (56) can thus be extended to », and i t i s clear at once that p(0) i s the same for a l l dwell-ings of age less than T - t * . If, then, p(0) as given by eq (56) i s less than c, there w i l l indeed be no maintenance after t*=P0. Let us now complete the analysis of this case. It i s immediate that NG(k, t) = NG(ke 5 t, 0) - Nt/T for k < ke~5t and NGlke~ 6 t, 0) = NGlk, 0) - Nt/T-From eq (41) we obtain: G(k, t) = 1 - ~ ( t * + - log (k/ke 6 t)) - -T o T = 1 - i ( t * + i . log (k/k)) for k < ke~ 5 t, just as before T 6" - -St ( 5 7 ) and G(ke ,0) = 1 - t A . The function k Q u t ( t ) comes from eq (54) and i s : *aut<« - ke-^ T- t*- t< ( m T/ N ,- 1J>. (58) , ,. _ fit Of course this makes sense only i f k Q u t ( t ) < ke" , since the right-hand side here i s , at time t, the greatest existing upkeep level. This means that T - t* - mTt/N > 0, i . e . that t < _E (T - t * ) . Once t = — (t - t * ) , raT mT the free-good upkeep level coincides with the greatest upkeep level, and a l l c i t y housing i s free. A simple, but tedious, calculation of eq (55) gives R(k, t ) . The result i s : R(k, t) = ( y - J P - J f l - ( k o u f c ( t ) A ) a ) + ^ | l o 3 (kAout(t)) (59) From this one can calculate eq (56). The upper limit of the integral i s the time t for which ke - 1 ^ = k Q U t ( t ) , which means that t = (N/mT) (T - t * ) . (60) The answer i s : x ( e ((a«mT/N) -p) N (T-t*) /mT _ ^ + _ b _ ( l - e-pN(T-t*)/mT) ( 6 1 ) p6Tk Thus, f i n a l l y , i f this quantity i s less than c, there w i l l indeed?be no maintenance after t== 0, and eqs (57):, (58) and (59) provide the solution to the problem. After time t = N(T - t*)/mT, no more rent can be collect-ed i n the city, and property continues to deteriorate exponentially. For this fearsome tale to proceed to the end, that i s , complete desertion of the city, i t i s necessary that profits remain non-negative i n the suburbs. The continued lack of maintenance in the ci t y means that 108 the maximum u t i l i t y to be had there steadily declines. Therefore, for as long as anyone l e f t in the city can afford i t , there w i l l be a steadily growing incentive to move to the suburbs. Even i f , after some time, profits there f a l l to zero, a rate of construction less than m can be maintained. Let us now choose a specific form for the suburban construction cost function H ( K , M) : 5(K, M) = C(l + hM) * IK. (h,U, C > 0) This comprises a fixed building cost C(l + hM) which grows with M, the number of already existing suburban buildings, and a linear variable cost IK, dependent on the b u i l t - i n standard of comfort. Eq (36) can be solved with this choice of the function H and the other choices made previously. Eq (36) becomes: I = i-faaty - ir - C(l + hM) - l<)) K S with solution K = a (y - ir - C(l + hM)) . Ma + 1) From this we may calculate the u t i l i t y obtainable i n the suburbs by a person of income y, when the p r o f i t there i s it and there are M dwellings in existence. We obtain (eq (45)): Usub = TifhF < y - * - c < 1 + h M » 2 <62> where i t i s necessary that y > TT + C(l + hM) i n order that the rent paid does not exceed total income. For a city-dweller of income y who at time t avails himself of the highest obtainable standard of comfort, u t i l i t y attained i s : U c i t y = o k e " 6 t t y " Rtke" 5*, t ) ) . (63) From eqs (58) and (59) one has: R(ke- 5 t, t) - (y - J L . ) ( l - B-*6W-*-W*») a6T + k(T - t* - *£?_) (64) T N Now, while the pr o f i t ir i s positive, we know that M = mt. The income of the person at the margin of indifference between cit y and suburban liv i n g is then (eq (37) and eq (44)): F ^ d - (mt/N)) = y + b ( l - (mt/N)) • Therefore the marginal condition, eq (38), which gives ir, can be written down by equating the right-hand sides of eqs (62) and (63). The result i s , at time t : JSs {y + b ( l - (mt/N)) - ir - C(l + hmt) }2 (65) JUa + l ) 2 = a k e - f i t {(y - J U e-a6(T-t*-(mtT/N)) + + _± } — aST T as , whence ir (t) = y_ + b (1 - (mt/N)) - C (1 + mht) -St (a + 1) ( &«ke { _ _b_ , e~a5 (T-t*-(ntT/M)) a<xs aST + kit* + _ i )})** (66) T aS This i s clearly a decreasing function of t. In those cases, then, where the c i t y does not simply empty directly, a time, t 0 , say, w i l l be reached when ir becomes zero. This time t Q i s calculated by setting the right-hand side of eq (66) equal to zero. After this, i t i s the fact that ir = 0 i f any more suburban construction occurs that determines the dynamics. Let us, for the sake of simplicity, assume that t Q > ¥ (t i s the time when rents in the city go to zero - 1 1 0 eq (60)). If not, the following analysis w i l l be more complicated, but s t i l l feasible. With t > t, the marginal condition, eq (38), when ir = 0, reads: a c ts •fy + b ( l - M/N) - C(l + hM) }2 I T a + 1) ^  - c£e^ t {y. + b { 1 ~ ( 6 7> and this can be solved directly for M as a function of t. We must notice that since .the right hand side of this equation i s always positive, M can never exceed the value, M, say, which makes the left-hand side zero: _ N(y + b - C) M =u—== b + hNC If M < N, there i s a fraction of the city population which can never afford suburban housing, and i s l e f t trapped i n the decaying city, albeit with free housing. Eventually, demolition w i l l catch up with these people, and housing w i l l no longer be free. Our present model does not say what w i l l happen then, but i t must certainly be something breaking the pattern of preceding events. (Government subsidies, changed municipal rules, r i o t s , i l l e g a l squatting are a l l possibilities.) There i s one more matter to be checked before this section i s con-cluded. It has been assumed a l l along that the margin between city and suburban l i v i n g was such that people with incomes greater than the marginal one preferred the suburb, people with lower incomes the city. This assumption must be verified. At time t , a person with income e less than the marginal one w i l l attain a u t i l i t y less than that of the marginal _ -St person by ^ ^ ^ ^ ake e (compare eq (63)). or by 2aa sc sub " ~ ~ (y - ir - C(l + hM)). (compare eq (62)) A(a + l ) 2 I l l We require then that (y — IT — C(l + hM)) > otke~fit. Ka + l ) 2 If in this inequality, ake - 6^ i s replaced by the expression for i t obtained from the marginal condition i t s e l f , eq (65), the requirement becomes: 2(y - R(ke~ 5 t, t)) > y - ir - C(l +hM) , i.e. y + it + C(l +hM) > 2R(ke - < 5 t, t ) . For times t > t, this is t r i v i a l l y satisfied. Inspection of eq (64) shows that 2R(ke _ , 5 t, t) decreases faster with t than does ChM, and so the cond-i t i o n w i l l certainly be satisfied for a l l t i f i t i s at t = 0. A s u f f i c -ient condition i s then y + C > 2R(k, 0), which w i l l always be satisfied i n normal circumstances. The richest person would otherwise be spending well over half his income on rent after the externality - a most unlikely state of a f f a i r s . We may conclude then that the course of events is as described. 4. The Case of Slow Suburban Construction. In the last section, we considered some of the possible outcomes in the event that there were unoccupied dwellings i n the city. The state of affairs i s quite different i f there are not. The reason for this i s simple: i f one excludes the knife-edge case i n which the rate of suburban construction exactly equals the rate of urban demolition, the former must be less than the latt e r i f no unoccupied dwellings exist, and so, i f every-one i s to be housed somewhere, urban construction must s t i l l be going on and must be profitable. We may assume that the p r o f i t for urban construe-112 ion i s exactly zero - this implies competition and no barriers to entry. F i r s t , the effects of continued urban construction can be taken into account so as to provide the distribution of dwelling ages (as opposed to that of upkeep levels) at any time. The distribution of upkeep levels can then be deduced by introducing a function s ( t ) , which gives the time at which maintenance on a dwelling constructed at time t comes to an end. As usual the rents function R(k, t) follows from the demand differential equation. The next step in the analysis i s to obtain an equation for s(t) from the landlords' optimal control problem. This equation i s unfortunately rather involved, and i t i s not possible to provide an expli-c i t solution. However an iterative scheme i s described by which i t may be computed. Lastly, i t i s pointed out that a much simpler result can be obtained i f the s t r i c t rational expectations hypothesis i s relaxed. We assume in this section, that m < N/T. Now the urban demand equation, (53), i s the same as before, but i t s boundary condition i s no longer that R ( k Q u t ( t ) , t) = 0, but rather the zero-profit condition. It should be recalled that we have assumed that urban landlords have perfect foresight, and that their p r o f i t i s then to be calculated with the rent function R(k, t) predicted by the model. At time t = 0 we shall have suburban p r o f i t tr > 0, i f any move to the suburbs i s to take place at a l l . Presumably after some time ir w i l l f a l l to zero, and tnen, since urban construction i s continuing and housing of upkeep level k i s s t i l l available, suburban construction w i l l permanently cease. The only thing that would make i t start up again would (other than another exogenous adverse externality) be an increase i n city rents. Since there are never unoccupied dwellings, the zero-profit condition applies to every building put up after t = 0, and thus R(k, t) w i l l be bounded above and below - only buildings i n existence at t = 0 can incur windfall gains or losses. With our assumption that people who move to the suburbs never move back to the city, the state of affairs where the richest person l e f t i n the city i s just indifferent to moving to the suburbs when R(k, t) i s at i t s highest point i s stable: no more new suburban construction w i l l ^ ever take place (except for replacement of course - we have abstracted from such considerations). It may seem that there i s no reason for R(k, t) to change at a l l from i t s t < 0 value. Indeed there i s no reason for i t to change by very much, but there w i l l be small fluctuations, as we shall now see. While ir > 0, the rate of urban construction f a l l s from N/T to (N/T) - m. This means that the distribution of building ages i s no longer rectangular, in fact, for time t, the distribution w i l l be as i n Fig 18(a). If, as we may for simplicity assume?, once ir reaches zero suburban construction stops for good, urban construction rises again to a rate N/T, and afterwards the age distribution w i l l be as in Fig 18(b) . Let us denote the function graphed in Fig 18 by w(v, t ) , and then w(v, t) dv i s the number of dwellings which at time t have an age between v and v + dv. After sub-urban construction ceases, at time t'f- say, we have w(v, t) = w(v + t 1" - t + nT, t + ) (68) where n i s an integer chosen so that O ^ v + t ^ - t + nT T. 4. It i s as though the function w(v, t ) were/reproduced in each interval of length T as a periodic function, and then propagated i t s e l f forwards like a wave. If there ar f i n a l l y M buildings in the suburbs, construction proceeds so as to keep N - M buildings i n the city: construction rate after t = t^ always equals demolition rate. Whether the function w(v, t) takes on only the values N/T and w(v) N . m T (b) Fig 18. age (N/T) - m as shown in Fig 18 or has some intermediate values attained while R(k, t) rises to i t s highest point, l e t us now make the definition: V(v, t) = number of buildings with age > v at time t. The function V i s of course calculated directly from the function w, but i t i s more convenient to work with V i n the analysis to follow. It can now be seen why the maximum city rent, R(k, t ) , fluctuates. There w i l l be periods when older housing i s scarcer than at others. Consequently the upkeep supply function, G(k, t ) , w i l l at times increase more slowly with k than at others. This means that the whole rents p r o f i l e , R(k, t ) , w i l l assume different shapes at different times, and since the value of a building put up at time t depends on rents from time t to time t + T i n such a way that, with unchanging construction costs, exactly zero profit i s realised, i t follows that R(k, t) cannot always have the same value. It i s reasonable to suppose that i t s fluctuations w i l l be minor - they are certainly bounded - and, clearly, i f a long-run equilibrium with a rectangular distribution i s ever to be achieved, fr i c t i o n a l forces of some kind must operate so as to damp out both the fluctuations of R(k, t) and those of V(v, t ) . We may now finish the present analysis. Let the time at which a building constructed at time t ceases to be maintained at upkeep level k be denoted by s ( t ) . This time i s then determined by the time-depend-ent analogue of eq (15): t+T c - / dt' e - ( e + 6 ) ( t ' - s ( t ) ) R'(&-« t') (69! s(t) ( c i s the marginal cost of upkeep - this equation the the condition that p(s(t)) = c, p(t + T) =0, where p i s the shadow price of upkeep.) The differential equation for the rent function R(k, t) i s eq (53), and i t s solution may be written as follows: k R(k, t) - (k/k) a R(k, t) - ^ j dk' ( k ' ) 3 " 1 y(k', t) (70) k a k where y(k, t) i s as usual the income of the person choosing comfort level k at time t. The function y(k, t) can be expressed i n terms of V(v, t) and s ( t ) . A building constructed at time t', i s , at time t > s ( t ' ) , i n a state of upkeep ke . The number of buildings i n a less good state of upkeep i s V(t - t', t) - that i s , the number of buildings of age greater than t - t", at time t. Equating numbers of people and dwellings as usual, we obtain: -6(t-s(t')) . . _ -,-1 f l y ( k - e - 6 ( t - s  ) f t ) = p - l ^ y ( t _ t f fc)) = y + k V ( t - t', t) - N by eq (44). Therefore: y(k, t) = y + b-vft - s _ 1 ( t - k log (k/k)), t) (71) N 5 where s - ^ i s the function inverse to s. When this i s substituted into eq (70), the function R(k, t) is expressed i n terms of s(t) and V(v, t ) . But V(v, t) can be regarded as known, since i t can be calculated directly from the marginal condition of indifference between city and suburb. Finally then, to complete the set of dynamical equations, there i s the zero-profit condition. If B denotes the unchanging cost of a new build-ing in state k, then this condition i s : s(t) _ e" p t B = / dt e " p t (R(k, t') - cSk)) + J ( t ) , (72) t where J(t) i s the value to be obtained from the building once maintenance stops, that i s : t+T , , J(t) = / dt' e ~ p t R(ke- 6 ( t " s ( t ) ) , t') . (73) s(t) Since R(k, t) i s already expressed i n terms of s ( t ) , so then i s J(t) by eq (73) and so therefore i s the whole of eq (72). Eq (72) i s then an equation of a very involved kind for the function s ( t ) . It would be exceedingly d i f f i c u l t and probably pointless to try to obtain an explicit expression for s ( t ) , even with the many simplifying assumptions so far made. But, as was pointed out before, the fluctuations in R(k, t) are small i n any normal circumstances. An iterative scheme for solving eqs (72) and (69) may therefore be suggested, suitable of course only for numerical computation. A reasonable f i r s t guess for the function s(t) i s s(t) = t + (T - t * ) , where t * i s the long-run equilibrium parameter given by eq (49). The c i t y -suburb marginal condition i s just a5s + b ( l - (mt/N)) - ir - C(l + hmt)) £(a + l ) 2 = ak (y_+ b ( l - (mt/N)) - R(ic, t)) (74) byanalogy with eq (65) , and i f one sets R(k, t) equal to R(k), the pre-externality equilibrium value, and sets ir = 0, an estimate i s obtained for the time t at which suburban construction stops. From this the function V(v, t) i s calculated using eq (68). Then the function y(k, t) follows from eq (71), and hence R(k, t) from eq (70). The quantity J(t) can then be obtained from eq (73) and then eq (72) i s an integral equation for R(k, t ) . It can be solved by differentiating with respect to t: 118 -pe~ p tB » e " p s ( t ) (R(k, s(t)) - cok ) - e~ p t (R(k, t) - cfik ) + J'(t) . This can be solved for R(k, t) in terms of R(k, s ( t ) ) . One may then iterate and write R(k, s(t)) i n terms of R(k, s(s(t))) and so forth; the exponential factors ensure convergence. With the solution R(k, t) in hand, a new expression for R(k, t) comes from eq (70). This can be substituted into eq (69) so as to obtain a second-round estimate of s ( t ) . The whole procedure may be started again, and w i l l in a l l likelihood converge very rapidly. The extreme complication of this result i s due to our rational expectations hypothesis. If landlords are endowed with a l i t t l e less foresight, then things became much simpler. Let us imagine, for instance, that landlords cannot be persuaded to invest in new construction unless the construction cost, B, i s covered by the return t+T _ _ / dt' e ~ p ( t _ t ) (R(k, t') - cfik ) t which would be obtained i f f u l l maintenance continued to the time of demolition. This return of course must be less than that really obtain-able, since i t i s feasible but not optimal. Then eq (72) becomes just t+T _ _ B « / dt' e~P ( t ~ t ) (R(k, tD -.cfik}) , t and this equation i s satisfied by a constant value of R(k, t ) , R(k, t) = R, say. The city-suburb marginal condition , eq (74), i s then satisfied by a unique t = t^ at which TT = 0, and no further suburban construction at a l l occurs for t > t^. The age distribution, V(v, t ) , i s then known completely. The function s ( t ) , which gives the time at which a building constructed at time t i s in fact l e f t to decay has s t i l l to be calculated with some d i f f i c u l t y from eqs (69), (70) and (71), but of course i t i s no longer of much interest. The behaviour of s(t) i n a general way may be seen by inspection. When older buildings are scarce, that i s , when buildings put up i n times of urban construction rate (N/T) - m are nearing the end of their lives, then for any v, V(v, t) i s smaller than usual. From eq (71) i t follows that y(k, t) i s smaller than usual for any k, and therefore by eq (70) that R(k, t) i s greater. From eq (53) , R'(k, t) i s smaller, and so from eq (69) s(t) i s nearer to t and further from t + T than usual. In other words, the equations operate so as to mitigate the scarcity of older buildings by causing maintenance to come to an end sooner. This concludes our formal analysis of the city-suburb dynamics. It remains in the next section to summarise the numerous conclusions of this section and the preceding one, and to gather a few loose ends. 5. Summary and Conclusions. It was not the aim in sections 3 and 4 to provide an exhaustive catalogue of everything that might happen i n the two cases m > N/T and m < N/T. Rather, certain sequences of events were shown to be possible, and some details analysed. There were two reasons for this: the different cases that may arise for various choices of the model parameters are exceedingly numerous, so that an exhaustive catalogue would also be exhausting; and some of these cases, while very similar to each other, can be distinguished by large differences i n complication or ease of analysis. It i s probably worthwhile, then, to supply here a verbal, rather than mathematical, discussion of the information that can be obtained from our exercise i n comparative dynamics. In the rapid suburban construction case, m > N/T, i t i s not necess-ary that a l l c i t y maintenance come to an end at t = 0. There i s a condition for th i s , namely that the quantity p(0) given i n eq (61) should be less than c. What happens i f this condition i s not met? It was pointed out that eq (61) gives p(0) for every building in state k at t = 0 and so i t remains true that there i s only one optimal maintenance path for a l l of them. Some time must pass, then, before general decay starts, and continuity requires that some buildings which had at t = 0 begun their decay should again be maintained for a time. The time t , given by eq (60) , at which city housing becomes free, i s not affected by maintenance after t = 0.. It i s determined only by the difference in the suburban construction rate and the urban demolition rate: i t i s the time when a l l the dwellings of age more than t* at t = 0 have become unoccupied. Consequently i t makes l i t t l e difference to events after t = t whether or not p(0) as given by eq (61) i s less than c If there were no limit to the rate of suburban construction, the number of people in the suburbs, M(t), would be given simply by the marginal condition with zero p r o f i t , eq (67). There would be an instant departure of a positive fraction of the population - M(0) i s not zero i n general. (If a formal solution of eq (67) yields a negative value of M(0), this means that suburban housing i s too dear, and the urban exter-nality, a, not severe enough relative to the suburban one, a g, for any-one to wish to move out.) In any event, M(t) as given by eq (67) specifies the greatest possible number of people in the suburbs at time t so that i n general this number w i l l be min (mt, M(t)). The time t Q i s 121 defined by mt Q => M(t Q), i.e. by T r(t 0) = 0. It may happen that M(t) f a l l s below mt very quickly, so quickly in fact that after a time t, say, there are no more unoccupied dwellings in the cit y . The time t would be defined by M(t) = Nt/T. This equation says that the number of suburban dwellings at time t equals the number of city dwellings demolished between t = 0 and t = t. But i n this case urban construction would have to start again. The dire consequences mentioned i n section 3, riots and so forth, need not take place i f t i s small enough. For once urban construction starts, we are in the situation described in section 4, and the analysis presented there w i l l go through in some cases. At this point, we may see that a solution to the system of equations (72) and (69) may not exist. This w i l l be so i f the incomes of the people l e f t in the city are not large enough for the rents obtainable from new buildings over their lifetime to cover costs. An extreme instance of this would be a state of affairs in which the highest income l e f t i n the city once suburban construction has ceased, y_ + bM/N, i s less than the rent R(k) which would i n a steady state be necessary for eq?.(72) to be satisfied. Now c i t y construction w i l l of course f a i l to be profitable in much wider circumstances than this. It i s then a state of affairs where no new city construction can be profitable that leads to the "dire consequences" of section 3. Even in the case m < N/T i t can turn out that eqs (72) and (67) have no solution. If suburban housing i s cheap enough, and the exter-nality a severe enough, the marginal condition given by eq (74) can, when TT i s set equal to zero, lead to a value of mt sufficiently great that too few people are l e f t in the ci t y for new construction there to be profitable. 122 Again there w i l l be dire consequences. The model presented in this essay i s not of course intended to provide a contribution to r e a l i s t i c urban economics. Por this reason, I shall not attempt to make policy prescriptions for city managers or governments faced with population loss due to migration to suburbs, although, were the model r e a l i s t i c , many such prescriptions would be implicit in the analysis I have presented. On the other hand, I do hope that economic theory i s i n some small measure advanced by this essay. It has been shown how a general equilibrium model can be solved in an intertemporal context; the problems of entry into and exit from business have been e x p l i c i t l y incorporated into the analysis; some indication has been given of the richness of detail that comparative dynamics can provide when f i n i t e rather than infinitesimal changes in exogenous quantities are considered. In the postscript that follows this collection of essays, there w i l l be discussion of some of the many outstanding problems i n comparative dynamics, and what one might do to try to solve them. 123 POSTSCRIPT. Seeing into the future i s an occupation fraught with hazards. It i s ironic^that a world more laden with s t a t i s t i c s and projections, facts and figures than ever i t was in the past i s the one i n which economic theorists are at last trying to come to grips with the effects of uncertainty and lack of foresight. It i s a world, though, where much s c i e n t i f i c research has s t i l l not told us enough about the life-cycles of f i s h or even of trees for poor economists to be able to take into account the facts of physical and biological evolution that are needed for anything that could be called optimal exploitation of renewable resources. It i s a world where the estimates of what remains for us to use of non-renewable resources like o i l and coal are as volatile as the stock market quotations. . It would be wrong of me to i n s i s t on how great a defect i t i s that this thesis does not worry about uncertainty. It i s a defect, but one which can be cured with a b i t of effort. Since von Neumann and Morgenstern (1944) and more especially Arrow's (1971) essays on risk-bearing, economists have made some progress i n the matter. It may even be f a i r to say that the business of making any given economic model into a stochastic one i s just technical. At a l l events, one knows how to tackle the problem. There are some other perhaps less obvious defects i n the theory used i n the thesis, and i t i s time to discuss them. One such defect i s that optimal control theory allows us to maximise only those objective functionals that are integrals. This i s not really a 124 matter of too great concern when one is considering the theory of the firm, in which a discounted stream of profits provides economic incentive, for this i s indeed an integral. But as soon as one looks at the other side of the market, one i s aware that consumers are not i n general l i k e l y to be maximising discounted streams of u t i l i t y . Rather, i t i s usual to suppose that enjoyment today and enjoyment tomorrow are strongly complementary for most people. Theories of job choice, attempts to explain age-earnings profiles and the like must take f u l l account of this fact. No doubt the mathematics of maximising general functionals w i l l be shortly worked out well enough for economists to use i t , but i n the meantime the best way out seems to be to give up a description of events i n continuous time and concentrate instead on objective functions that depend on a large, but f i n i t e , number of discrete variables. An inspiring example of this i s to be found i n a paper by Iwai (1972) i n which a programme of optimal capital accumulation i s worked out for a quite general benefit function i n a model with a discrete time variable. In this thesis, continuous mathematics has been used throughout, not only for time, but even for households and buildings, which must i n the nature of things come i n integral amounts. Approximations are involved in such a procedure as this, and i t can be d i f f i c u l t to know when the graininess or lumpiness of people and things can make a crucial difference i n an economic story. Non-convexities, like uncertainty, provoke a good deal of new economic theory these days. I am inclined to think that the models treated i n the present work are not too much impaired by continuous approximations, but i t i s certainly reasonable to hope that intertemporal 125 models w i l l be developed with discrete households, firms, and so forth, even i f continuous time i s (very properly!) retained. There i s no reason to imagine that aggregation over firms and households, in the sense of bringing them together in a general-equilibrium economy, i s any more d i f f i c u l t a task with discrete variables than with continuous ones. In fact, i t may well be that the individuality of people and managements can be captured better i n a discrete model. Assumptions of perfect competition among identical agents are probably grating on the consciences of most economists nowadays, and i t i s certain that, i f entry and exit of firms are to be objects of inquiry, one must have i n mind a hierarchy of them in order of efficiency or some such attribute. An idea like this one i s behind the modelling of suburban construction activity i n Chapter IIIA. It i s interesting to wonder i f one might construct a general-equilibrium model, with many goods and services, i n which a l l the economic actors had intertemporally defined objectives. The correspondence discovered i n Chapter I between lines of constant state or co-state variables and static supply or demand curves may well be subject to considerable generalisation. It i s tantalising to imagine that the notions already i n use for demand analysis i n a static framework - Hicks aggregation, gross subs t i tut a b i l i t y , and so forth r- might have dynamic counterparts that could help to cut a way through the d i f f i c u l t i e s of many-stock optimal control problems. I should lik e to end this thesis by lamenting the fact that our understanding of intertemporal economic processes;;has not at a l l -126 contributed to a credible theory of speculation. Well-behaved, responsible firms perhaps, but crass speculators no. They might just as well be necromancers or augurers for a l l we understand them - and some are indeed very successful. The glamorous mathematical theory of catastrophes (see, for a serious account of the theory, Brocker (1975)) may provide what i s needed here - i t does discuss motions described by systems of differential equations and the quirks or singularities associated with such motions. 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