University of Southern California Dissertations and Theses

Applications Of Stochastic Processes To Economic Development

(USC Thesis Other)

Applications Of Stochastic Processes To Economic Development

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content This dissertation has been microfilmed exactly as received 67-418 PATEL, Ramaribhai Chhaganbhai, 1931- APPLICATIONS OF STOCHASTIC PROCESSES TO ECONOMIC DEVELOPMENT. University of Southern California, Ph.D., 1966 Economics, general University Microfilms, Inc., Ann Arbor, Michigan APPLICATIONS OP STOCHASTIC PROCESSES TO ECONOMIC DEVELOPMENT A Dissertation Presented to the Faculty of the Graduate School University of Southern California In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy (Economics) by Ramanbhai Chhaganbhai Patel June 1966 UNIVERSITY O F SO U TH ERN CALIFORNIA TH E GRADUATE SCHOOL, U N IV ERSITY PARK LO SA N G EL.ES, CA LIFO RN IA 9 0 0 0 7 This dissertation, written by _ R^aj^h,ai...Cb]tiagiaDtihai..P.at.e3 . .......... under the direction of hi,S....Dissertation Com mittee, and approved by all its members, has been presented to and accepted by the Graduate School, in partial fulfillment of requirements for the degree of D O C T O R O F P H I L O S O P H Y Dean Date ..Ju.ae..l966........... iSERTATION jCO , Chairman FOREWORD Professor Gerhard Tintner of the University of _ Southern California has recently revived the idea of using stochastic theory in economic analysis. One particular i !facet of his efforts in this direction has been to apply j [the theory of stochastic processes to the problem of |economic growth. Such stochastic theory of economic development is, however, based exclusively upon the theory |of discontinuous stochastic processes. This dissertation I extends the theory from its present state to a step further i ! iky applying continuous stochastic processes of diffusion i type to the economic development process. The inspiration for this work arose through the j [author’s association with Prof. Tintner as his research I assistant during the period 1963-65. The author would like :to take this opportunity to express his gratitude to Prof. Tintner for such inspiration and for his assistance and guidance throughout this research work. Due gratitude is also expressed to the other members of the author's dissertation committee, Prof. Richard A. Bilas and Prof. Richard E. Beckwith, for their advice and assistance, iSpecial appreciation is due to Prof. Bilas for his help and i j counsel throughout the author's graduate program. | Finally, the author wishes to thank his colleague John Cownie for his assistance in computer work. TABLE OF CONTENTS FOREWORD............................................... I list OF TABLES..................................... • . * CHAPTER I. INTRODUCTION .................................. Interest and Importance of Economic Development .............................. Need for and Nature of A Theory of Economic Development . . . .. . .. « Deterministic Nature of Traditional | Economic Growth Theory .................. i | Stochastic Phenomena in Economic Development............................ | Need for A Stochastic Theory of Economic G r o w t h .................... The Stochastic Approach and Temporal Dependence ............................. Stochastic Processes: A More Plausible Approach ............. ................. Purpose and Scope of the Present Study . . | II. TOWARDS THE THEORY OF STOCHASTIC PROCESSES I | AND ECONOMIC DEVELOPMENT . . .............. The Notion of Stochastic Process ......... V - ' : Definitions and Fundamental Properties . . X V CHAPTER — PAGE Introduction to the Theory of Markov Processes ...................... 27 Continuous Markov Processes ........... 37 Simple Stochastic Growth Model ......... 41 Economic Evolution .................... 48 III. SOME STOCHASTIC MODELS OF ECONOMIC GROWTH .................................. 54 Introduction ...................... 54 Poisson Processes ...................... 56 Birth and Death Processes.............. 63 4 Further Results and Concluding Remarks ........................... 75 IV. LOGNORMAL DIFFUSION PROCESS. OF ECONOMIC DEVELOPMENT .................... 79 Introduction ........................... 79 Genesis of The Lognormal Distribution . 81 Lognormal Diffusion Process of Economic G r o w t h ............... 86 Estimation of the Parameters of the M o d e l ............................. 91 Empirical Results I .................... 94 Influence of Exogenous Factors ......... 105 Empirical Results II .................... 109 Appendix A ............................. 120 V CHAPTER PAGE V. MULTIVARIATE LOGNORMAL DIFFUSION PROCESS......................... 143 Bivariate Lognormal Diffusion Process............................... . 144 Empirical Results I .................... 153 Influence of Exogenous Variables .... 159 Empirical Results II ....................... 164 Multivariate Lognormal Diffusion Process ........... . . . . . . . . . 168 Empirical Example . . . . . ' . . 174 Appendix B ....................... 181 VI. SUMMARY AND CONCLUSIONS......... 189 Introduction -18 9_ Summary......................... 191 Conclusions and Comments....... 196 Suggestions for Further Research .... 202 Concluding Remarks ................ . . . 206 BIBLIOGRAPHY ......................................... 208 LIST OF TABLES TABLE I. Estimates of (3, yj b, a and their Variances for the Lognormal Diffusion Process Applied to the National Income Data of the U. S. (1946-63), Netherlands (1948-60), Canada (1948-60), India (1948-63), and Ecuador'(1950-62) .... I II. The Trend and the Variance of the National i j Income of the U. S. (1946-63), i Netherlands (.1948-60), Canada (1948-60) , India (1948-63), and Ecuador (1950-62) . III. Estimates and 95% Confidence Limits of the Real National Product of the U* S. for 1965-1970 ........................... IV. Estimates and 95% Confidence Limits of the Real National Product of Canada for 1965-1970 ...................... V. Estimates and 95% Confidence Limits of the Real National Product of Netherlands for 1965-1970 ........................... VI. Estimates and 95% Confidence Limits of the Real National Income of India ■ % for 1965-1970 ........................... VII. Estimates and 95% Confidence Limits of the PAGE I 96 | i 98 99 99 100 100 TABLE Real National Product of Ecuador | for 1965-1970 ............... ' . ! VIII. Estimates of P, y; b, a and Their Variances for the Lognormal Diffusion Process Applied to the National Product Data of the U. S. for 1900-1957, U. K. for 1880-1940, Netherlands for 1920-1938, Germany for 1880-1913, India for 1900- 1958, Canada for 1920-1938, Japan for | 1900-1938 .................................. IX. The Trend and the Variance of the National Product of the U. S. (1900-1957), U. K. j (1880-1940), Germany (1880-1913), Canada (1920-1938), Netherlands (1920-1938), India (1900-1958), Japan (1900-1938). . . X. Estimates of the Parameters of the Modified Lognormal Diffusion Process with Exogenous Influence of Government i j Expenditure Applied to the National Product Data of the U. S. (1946-1963) i and India (1948-1962) .... ............ | XI. Variances of the Estimates of the Parameters of the Modified Lognormal Diffusion Process with Exogenous Influence of Government Expenditure viii TABLE Applied to the National Product Data | of U. S. (1946-63) and India (1948-62). . i j I XII. The Trend and the Variance of the i I j National Product of U. S. (1946-63) i and of the National Income of India (1948-62) XIII. Estimates and 95 Per Cent Confidence Limits for National Income X(t) of India for 1964-65 to 1970-71 XIV. Expected Growth Rates of National Income of India during the Fourth Plan . . j j XV. Estimates and 95 Per Cent Confidence i j Limits for National Product X(t) of U. S. for 1964-1971 XVI. Estimates of the Parameters of the | Bivariate Lognormal Diffusion Process I j Applied to the National Product Data of the U. K. (1880-1940), U. S. (1948-63), and Ecuador (1950-62) XVII. Variances of the Estimates of the Parameters of the Bivariate Diffusion Process Applied to the Data of U. K. (1880-1940), U. S. (1946-63), and Ecuador (1950-62) .... XVIII. The Estimated Trends for Xj^(t) and X2 (t) of the Bivariate Diffusion Process Applied to PAGE 111 111 j i I | i 114 113 117 155 155 ix TABLE PAGE U. K., U. S., and Ecuadorion Data .... .156 XIX. Estimates for the Index of Output and Capital Stock of the U. K. for 1964-1970 ......................... 157 XX. Estimates for National Product and Investment of the U. S. for 1964-1970 . . 158 XXI. Estimates for National Product and Capital I Stock of Ecuador for 1964-1970 158 j XXII. 95 Per Cent Confidence Ellipses for the Bivariate Lognormal Diffusion Process for the Year 1966 .................... . . 159 XXIII. Estimates of the Parameters of the Modified Bivariate Diffusion Process Applied to the National Product Data of U. S. (1946-63) and Ecuador (1950-62) . . 165 XXIV. Variances of the Estimates of the Parameters of the Modified Bivariate Diffusion Process Applied to the National Product Data of the U. S. (1946-63) and Ecuador (1950-62) ......... 165 XXV. Estimated Trends for X^ (t) and X2 (t) of the Modified Bivariate Diffusion Process Applied to the U. S. (1946-63) and Ecuadorion (1950-62) Data ...... 167 XXVI. Estimates of Gross National Product Xi(t), TABLE i Private Consumption X 2 (t), Domestic Investment X^ (t), and Government Expenditure X4 (t) of the U. S. for 1964-1971 based on the Multivariate Lognormal Diffusion Process of Growth . PAGE . 177 CHAPTER I INTRODUCTION I ! j Economic prosperity has remained a central theme in j I 1 i ! human society. Interest in problems of economic development is as old as human history. Investigations into such ! problems have led to many theories describing the nature of j economic growth process. Following an introduction to the I i !need for and nature of an adequate theory of economic 'development, an outline of the stochastic approach in the j ; j ieconomic growth analysis and of the aim and scope of the j ipresent study is provided in this chapter. j 1 ' f ; | I i jInterest and Importance of Economic Development j It is often maintained that a measure of happiness can be expressed as a function of the ratio of possessions !to desires. Human desires are evidently unlimited. From j ! |the stone age to the present nuclear world, man has striven! i l jto satisfy his wants, which have no end. Comparison and 'competition are basic characteristics of man and serve to ! ■ intensify his constant efforts to equate possessions with jdesires. Even if human desires are assumed limited to the j {basic propensities of natural desires, for example, the I I I desire for high consumption and leisure, the urge to { {procreate and care for those alive, the desire to provide j for the future, et cetera, the sum total of these desires { will grow at least in proportion to population growth., if not more, and hence will call for material advance. It is, therefore, clear that if some given standard of living is to be maintained, real income must grow in proportion to desires. If living standards are to rise over time, the jgrowth in real income must exceed that of population. 1 Concern with the problem of economic growth is not new. A central issue of classical economics, starting with Adam Smith, was the growth of capital as the chief mean of raising income and the living standards of the people. I Recently, concern with economic growth has made a reappear- !ance under the impetus of increasing international parity i ;programs to alleviate the striking differences in the I levels of income of various nations. Nearly all the under developed countries have within the past decade set up official agencies to plan and promote economic development. 1 i The need for economic development in underdeveloped areas j stems from the human instinct of keeping up with his peers.j ! I iPerhaps a more important consideration militating in favor j : I of economic development is the fear of ’instability' in I i ;international relations. Some degree of economic equality j i among the various regions is a must to safeguard the j | prosperity of developed nations and the survival of our j \ j ! civilization in general. j ; I Economic growth has thus become an important objec- | ; l tive of public policy in developed and underdeveloped nations alike. It has been put forward as an aim of the world community in United Nations debates, indicating the strong social demand for it. The need for economic develop ment and for international assistance to hasten it, is a ^recurrent topic of discussion and debate in the economic and social organizations of United Nations and other specialized agencies the world over. ■Need for and Nature of A Theory of Economic Development The interest and desire to accelerate economic i ■growth then call for an investigation of how economic growth occurs and an understanding of the mechanism under lying the growth process. The need for a theory of |economic development is then an obvious corollary of the i ! —* j fundamental problem of raising economic prosperity. Even ■ if the issue of economic development is not publicly debated, the problems confronting us^hn this field are ' i 1 ■ ! t i ■perplexing enough to arouse the scientific curiosity of j I ' ' I ! every social scientist. History has evidenced the nations | whose standards of living have remained at strikingly low j levels for centuries. There are some countries where per jcapita income has doubled or trippled during the last j i J J hundred years, while others have witnessed little or no progress. Yet, there are probably no adequate theories to • explain the phenomenon of economic development. | Since economic development became the subject of explicit analysis about two centuries ago, many attempts have been made to characterize the process of economic development by an abstract theory based on a few simple axioms. Such axioms are chosen from the regularities found in a group of observed economic phenomena. The need for a I ■theoretical model capable of providing a logical explana- i tion to the economic dissimilarities observed in the various nations has created in recent years a real revival ;of interest in the longe-range theories of economic growth^! 1 i I The purpose of the theoretical growth models is to isolate j I ! the factors that lead to the prosperity or poverty of jnations. The explanation of past trends of an economy j | given by an econometric model provides an understanding of jthe principal determinants of economic progress. In the j ! “ i :first place such an explanation is useful because of the ; possibility that the past behavior of the economy may |repeat itself in the future. Much more important is the j I knowledge gained from such a study about the operation of | ; i , i • the economy. This knowledge is necessary and useful both j I : :for a full understanding of the development of the economy | :under conditions different from those of the past and for i ' i ; I jthe considerations of means of economic policy. Irrespec- ) | i i tive of the nature of the instruments of public policy , ! ; i ! ^A survey of current work in the theory of economic ; growth is found in E. D. Domar's paper on “Economic Growth:| An Econometric Approach," American Economic Review, Vol. 42 j j May, 1952, pp. 479-495. i adopted in some form of development program, some sort of : analysis of the present and future performance of the economy is needed to make them effective. The main theme of growth analysis is to investigate the trend of economic level and growth factors affecting it- I More specifically, the economic level is characterized in terms of a few important economic variables like national j j income, capital stock, employment, et cetera, each properly; normalized for differences in factors like population, natural endowments, et cetera; the trend exhibited in such j : ' j !standardized variables is investigated. j Deterministic Nature of Traditional Economic Growth Theory I i i i j Many models studied in growth analysis indicate j jvarious trends observed in empirical data. Such analysis is customarily carried out in aggregative or multi-sectoral deterministic models. The classical growth models of Smith,; i i . I !Ricardo, and the more recent ones of Schumpeter, Harrod, j ; j i 2! |and Domar are a few examples of shch deterministic models, j i j |However, nearly all of these theoretical analyses of | economic growth, although very helpful in characterizing I See B. Higgins, Economic Development (New York: j W. W. Norton & Company, Inc., 1959); W. J.Baumol, Economic j j Dynamics (second edition; New York: The Macmillan Company, j 11959); E. D. Domar, Essays in the Theory of Economic Growthi (New York: Oxford University Press, 1957); R. F. Harrod, j Towards a . Dynamic Economics (New York: The Macmillan j iCompany, 1956); J. R. Hicks, A Contribution to the Theory j I of Trade Cycle (Oxford: Oxford University Press, 1950). i j 6 the process of economic development in different aspects, particularly in explaining past history, have been basi cally very restrictive in that they are deterministic wherein the influence of random elements are neglected. The traditional economic theories are erected on the static or equilibrium level based on analogies in physics such as the concept of equilibrium or stationary state and deter ministic laws of motion. However, physics has progressed from the deterministic to stochastic and quantum theories, j :and laws of motion are studied in terms of probability ! i ' I distributions of particles. There seems then an obvious j Ineed for modifying the analytical tools in economics, I especially when the nature of economic phenomena is even I | jmore probabilistic than physics. I ; ( Stochastic Phenomena in Economic Development One reason why an economic process changes in time i ; is that it is acted upon by so called 'stochastic elements',, |that is, 'random factors'; another influence is the workingj I of the internal factors of the economic system. Then, the ! i i :deterministic dynamic system specifying the path of deve- ;lopment with a rigid law seem far from reality. Histo- i ' I !rically, it is a fact that chance factors have played a j j | !significant role in the economic development of many j i i 1 . i |nations. It is evident that , in any realistic situation, j j ‘ | 'the economic variables like population, demand and supply | . . . 7 variations, et cetera, which specify the course of growth I of an economy by their interactions, are probabilistic in nature. The decisions about the capacity variations and the changes in demand are not made in a world of certainty, but in one of imperfect knowledge and uncertainty. Human behavior is fundamental in economic activities of produc tion and exchange, and it changes in an unpredictable manner, although there is certain regularity even in the ;randomness. Economic growth then involves a dynamic i i . !interacting system of linked changes. The interactions I I among the economic activities are many and of a complex i I variety. As a result, the random elements in the growth j ' ' {process are innumerable and of great importance. Events i |such as earthquake or war break-out, revolution or social unrest, innovations or resource discovery, irregular weather conditions or epidemic spread, et cetera, are a ;few examples among many random elements that have isignificant effects on the economic activities and levels i |of a country. i 'Need for A Stochastic Theory of Economic Growth In so far as random factors influence economic ! activities, the deterministic models of economic develop- | ment are inadequate and incomplete descriptions of the | growth process of an economy as they neglect the effects : of random factors in their analyses. This was clearly ! 8 - ! 1 3 ! pointed out explicitly by Haavelmo (63, p. 64) when he j i mentioned that: "Exact models of the type we have been discussing belong, of course, to the world of fiction. Nobody expects such models to depict accurately the facts that we aim to 'explain'." It is, therefore, clear that there is a need for the | introduction of stochastic elements in the growth models. The analysis of probabilistic aspects of economic growth which takes into account uncertainty, acquires a crucial and important role. The stochastic approach in economic i ! : growth analysis is more justified than the deterministic theory, especially when the purpose is not only to explain j ; past performance of an economy but to predict its future i I j course of development. This is because the deterministic j I models are inadequate to take into account future uncertainty, that is, role of chance factors, while the i stochastic models are more appropriate for future j ‘ projections and hence more useful for policy purposes. j I . ! j j The Stochastic Approach and Temporal Dependence i Recognizing the need and importance of stochastic : elements in the theory of economic development, we come up I ■ to the problem of how to incorporate them in the analysis, j i i I i ; _ • _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ j J ■ 3 * The number m the parentheses refers to that of I Bibliography. Page number refers to that of the I reference. Throughout this dissertation, such reference | ! system is adopted wherever possible. 9 i ;Very frequently the stochastic elements are introduced into econometric models of growth using one or the other of the following: (i) errors of measurement in observations, and i(ii) errors in equations because of an incomplete theory jdue to the omission of some factors. In most of the j [dynamic models in the literature, the approach taken is to ineglect errors of observations and to consider only dis- j i turbances in equations assuming that the influence of j factors omitted is of less significance. However, the arbitrary assumption of statistical independence of the j i ;distrubance terms over time in such models seems objection-| iable, particularly when it is found to be incorrect in many !empirical analyses.^ I I Further, Haavelmo (63, p. 64) mentions a third point ! of view that "The idea"that 'the assumption of exact laws is empirically meaningless1. . . . it may be necessary, ifrom the very beginning, to think in terms of more flexible; I'relationship' than those involved in an exact, determini- I ! ! i ;stic model." Temporal dependence and significance of i 3 j ’empirical data in economics appears to have been stressed i [by Schumpeter (134, pp. 8-9) when he writes that: The individual household or firm acts, then, i according to empirically given data and in an equally [ j empirically determined manner. . . But every one will i : cl±ng as tightly as possible to habitual economic [ methods and only submit to the pressure of circumstancej i I----------------- ] 4See Tintner (147, pp. 270 ff.) 10 as it becomes necessary. Thus the economic system will not change capriciously on its own initiative but will be at all times connected with the preceding state of affairs. Many other workers in the field of economic analysis take a similar point of view of temporal dependence of economic activities. Rostow (127), and other stage i ! theorists consider this as the very basis in their growth ! analyses where they attempt to characterize economic ' development as the successive attainment of various j economic levels or stages, treating the development process! ;as a series of interdependent moves between levels of | ieconomic activities with a forward momentum gained from j i I jpast history. The familiar notion that the past governs ! ! I e i i ;the present and present governs the future appears more j ! relevant in the process of economic progress. The temporalj dependence between economic levels, that is, between the values of the economic variables that characterize the economy and specify its growth path, is clearly seen in the! I following hypotheses of Rostow (127, p. 53): j I I In the earlier stages of growth, primary and j supplementary growth sectors derive their momentum j essentially from the introduction and diffusion of changes in the cost-supply environment (in turn, of course, partially influenced by demand changes); while , the derived-growth sectors are linked essentially to j j changes in demand (while subject also to continuing ' | changes in production functions of a less dramatic j i character).... j : I j At any period of time it appears to be true even in i a mature and growing economy that forward momentum is maintained as the result of rapid expansion in a limited number of primary sectors, whose expansion has I ... """...... 1 1 I significant external economy and other secondary effects. Stochastic Processes: A More Plausible Approach Thus, it appears that the stochastic approach, where all the relevant economic variables are treated as random i jvariables and the temporal dependence of their values is explicitly considered, is more appropriate in characteri zing economic growth than the usual approach taken in the economic dynamics of specifying the economy by a set of functional relationships between the economic variables j land superimposing temporal independent random shocks. It | ; 1 i I shows, therefore, that a more plausible approach of j {stochastic process, which involves time in an essential ! way, be taken in characterizing economic development process. Haavelmo {63, p. 65) first suggested that “In the case of dynamic models the natural and more realistic counterpart of an exact model is the notion of stochastic { processes: (italics in the original), " and showed that : economic development can be characterized by means of a | ' i suitable stochastic process. i Stochastic process implies (i) not proceeding according to any immutable law but (ii) atleast partial : I j i I dependence on random or chance factors. It is called a i i ■ I random or stochastic process, usually the later adjective j | . f ;is preferred, because random might convey the idea that [ ! ' " | I every stochastic process is purely haphazard; whereas, in I ; 12 many cases, such as the growth process of population, the growth of economic variables, et cetera, the development of the process appears to follow a certain regularity iunder an influence of a certain basic phenomenon. ; . i | To understand what we mean by stochastic process, j | | Iconsider a system characterized by a number of variables, :Xi, X2 , * , Xk, say. For example, the state of the | economic system specified by the values of a set of dynamic ; ! variables like output level, population size, capital j stock, et cetera, interacting among themselves over time. | ■This set of values at a given instant of time defines a ! point in the phase-space of the system, and it describes I I a path as the state of the system varies over time. The j > | i space } { of all possible values that the variables can i assume is called the 'state space', and the elements x c that is, the different values that can be assumed by the variables are called the 'states'. The dynamic variables : j \ jare considered as statistical variates with a joint i I probability distribution, and are treated as random I functions of time or sometimes of space and time, because it is generally not possible to specify the values of the Ivariables 'exactly' due to the influence of a variety of j I < e I jchance factors as discussed earlier. The change with time 1 i I jof the economic system can be expressed by the changes in j ! t | the values of these variables, which come about through thej working of the internal system and the external forces, and' i 13 interactions between them. Such a transition of the jvariates between different states, can then be described j jby the statistical description of the probability distribu- ! tion of the variables. The probability distribution of the variables depends on the'transition probabilities per unit of time specified in terms of certain elementary events joccuring in a small interval of time, say t to t + At, At >0. The specification of these probabilities for the jtransition of the states from one set to another depends I i I on the assumed mechanism of the growth process. The Variables, when considered with such stochastic dependence between their values at two or more instants of time, form I a stochastic process. Thus, a model which specifies the j jcomplete joint probability distribution of the values of the different variables at each point of time would be a stochastic model, and the whole process, conceived as a continuous development in time would be called a stochastic ! jprocess (or probability process). The variables in a ' ! ] | istochastic process are thus considered as random functions i i ; of the independent variate t (time), and we try. to express j !the phenomena of economic process in the form of stochastic | jdifferential equations or in terms of equations for the j jprobability distribution function of the system. Thus, j i ! ithe theory of stochastic processes in broad terms, concerns j ■with the corresponding wider theory of the statistics of j ! \ change. 14 It may be stated here that Doob (35) has defined a ;stochastic process as 'the mathematical abstraction of an j jempirical process whosedevelopment is governed by probability laws1. Obviously, then the term 'stochastic process' refers to a mathematical abstraction, or a theoretical model representing an empirical process and not I to the empirical process itself. In our case, the empirical process involved is the growth of an economy and |the model is the system of equations for the probability i distribution function of the economic variables characteri zing the economy, which depends on certain assumptions 'concerning the probabilistic manner in which the economy i Jcould develop or is developing. Purpose and Scope of the Present Study In this dissertation, the applications of some of the stochastic processes to the observed set of time series j of economic variables are considered. The purpose of the I analysis is to discover and explain the nature of the igrowth process of economy exhibited by the temporal inter dependent set of observed series of the important aggregate economic variables such as national income, capital stock, !et cetera. ! 1 I I Since the idea of applying the theory of stochastic j :processes to the economic evolution introduced in 1954 by Haavelmo (63), many attempts have been made in specifying } ;the economic growth process by a variety of stochastic processes, and formulating models of economic growth using the methods made available by the mathematical theory of | jstochastic processes. In all of these studies, a special isub-class of stochastic processes, called Markov chains l ! jor processes, have been used. This is because of the concept of chain dependence in such processes reflecting the similar phenomenon in the economic process. We shall i discuss the concept of chain dependence and the Markov ! i 5 1 property of stochastic processes later in detail. In this| I 1 dissertation, attention will be restricted to this class j •of stochastic processes, namely, Markov processes. Also, j f i primary concern in this study is with applications, rather ;than with abstract formulations and fundamental theory of stochastic processes. Hence, without going very deep into the rigorous mathematical theory underlying such processes, |a more heuristic approach, that is often considered in ! I ; 'applied mathematics, will be adopted throughout this I dissertation. For a rigorous theoretical discussion of j Markov processes as well as other classes of stochastic I processes, the treatise of J. L. Doob (35) and references [listed therein may be referred. I : i i ----------------- ~~ ! ^Named after Russian mathematician A. A. Markov who j ;first developed such concept of chain dependence and did I 'basic work on the class of processes with such property of ! chain dependence. Further, the objective of any abstract model or theory is not only to provide a description of empirical phenomena with a reasonably good approximation but to allow the estimation of relatively dependable parameters for prediction and policy proposals or for simple explanation. The necessary empirical verification of a theory is then essential if theory is to be of any use. It is this empirical work which is mainly emphasized in our study and, therefore, throughout the present study, empirical examples ! i ! are given wherever possible. Lack of empirical analysis, I ! I :if any, is mainly due to paucity of suitable data and j : ■ i ; I inadequacy in the available data for the purpose of j | j I analysis. I I ! | j ! The stochastic growth models developed so far using ] I the theory of Markov processes, although useful in characterizing the process of economic development, are I restrictive from the point of view of the assumption I .underlying them about the discrete nature of the variables.! ' I ; ] ■The economic variables describing an economy, such as j national income, capital stock, et cetera, are in fact continuous rather than discontinuous variables. This study: ]therefore proposes the approach of diffusion processes, j ; i |where the variables (economic) are assumed continuous, as j ; i | t :a more appropriate analysis in the study of economic growth jprocess. Thus, our chief objective in the present work is ] to extend the stochastic theory of economic development a step further from its present state so as to provide more realistic stochastic models of economic growth. Further, the present study also tries to verify the ! ifamiliar notion that there are similarities between the i process of economic growth in different countries and periods in so far the basic phenomena are concerned. This is investigated by applying the diffusion process analysis j to the empirical data of both developed and underdeveloped ; countries as well as for different time periods. j I In order to provide a somewhat systematic and brief | iaccount of the stochastic theory of economic growth which j has been developed so far and to subsequently extend it j ;coherently, chapter II offers an introductory attempt to j !formulate a general framework of the theory of stochastic j 1 i processes and to show how the concept of stochastic process; arises in the study of real phenomena of economic growth. Chapter III deals with the historical background of I the theories of stochastic processes used in economic i 'growth models, and a brief review of some of the represents-; tive type of these models is provided along with discussion: of some of the methodological aspects in the analysis, j Chapters IV and V are then devoted to the approach of j |diffusion processes. A single variate model is discussed l I at length in chapter IV followed by empirical results for i ■ ' |many countries. An attempt is made to generalize the univariate model to -a multivariate model to study a number : i 18 ,of economic variables and the interdependence among them; this enables formulation of a system of relationships between the various economic variables. Also, along with studying the past growth paths for some of the countries over a long period of time, the analyses in these chapters extend further to suggest the analytical policies to be adopted for achieving alternative growth paths that would be possible under varying conditions of the instrument I variables, ,such as government expenditure, investment allocations, et cetera. i Finally, chapter VI offers a summary of the I ifindings and major conclusions, and mahes tentative I I suggestions for further research. I j . i i CHAPTER XI TOWARDS THE THEORY OF STOCHASTIC PROCESSES AND ECONOMIC DEVELOPMENT The previous chapter has established our interest i • in the theory of stochastic processes and their applica tions to the economic growth process. It is the purpose iOf this chapter to provide the necessary general framework i 'of the stochastic theory of the economic growth process | and set the stage for subsequent chapters where specific Imodels of economic development followed by empirical results are discussed. The rather vague discussion of the aims and scope of, this dissertation, which was inevitable without a precise system of notation and further descrip- i j tion of the theoretical aspects of stochastic processes, should also be cleared up with such general introduction. The Notion of Stochastic Process The concept of a stochastic process was first met in chapter I. We shall reconsider it more precisely now from our practical point of view. All the problems in which a variate t arises explicitly can be viewed in the |following general framework. t Consider a system S of any kind whatever that | i - I I evolves randomly in time. This, in a broad sense, can be j called a stochastic process. To say that a system evolves j 20 I randomly in time means that it is capable of being in different'states at different times but the exact state at any time can not be predicted precisely due to the influence of random events. Let uu denote a possible state of S, and Cl the set of all possible states uu of the system S. We shall hold fi constant through time, though such an 6 assumption is not restrictive. i Let U ) . j . denote the state in which the system S is at j I time t; being one of the states u u of Q . Let X be the magnitude of any characteristic of the system S, that is, X is a function of the state uu of the system S, at any given time. In other words, there exists a number X{uu) which is a function of the state uu , defined for uu varying I j in , and such that X takes the value X(uu) when S is in i Estate uu. At any time t, X will thus take the value j X(t) =x(cu-t) , an<3 - X(t) is thus a statistical random variable | but one which generates a random function as t varies. We ;say that this random function is connected with the i ] ! ; i I transition process of S, and therefore, a stochastic ! process is difined as a family of random variables |x(t), t^o}, over a space % of all possible values x that |the random variable X(t) can assume. Perhaps a more ; ^In case a particular state can not recur, we may |assign probability zero to such a state, and thus it is ipossible to include the case, where changes over time, Iwith little modification. 21 i generalized specification of a stochastic process can be made as followss Definition: Let T denote a set of real numbers on a I itime axis such that at each point t in T an observation is ;made of a random variable X(t,U)) occupying a point U ) in the state space fi. Then, a stochastic process is specified by ; the family of random variables jx(t,uu), tcT, uucfij. The random evolution of a system S means that the j j changes in the state of S depend on random events; succes- j j sive changes of the states being the results of a sequence J i of random events constituting an over all event E to which ; the entire evolution of S is related through the probabi- : • i . lity law of random events. The evolution of S can be considered to have started at any specific time, which is usually taken as t=0; although it can be t=-°° or any other arbitrary value. IDefinitions and Fundamental Properties j i Having defined a stochastic process, the fundamental,! ‘ j problem is to investigate the evolutionary nature or the j form of evolution of the system under consideration. The form or law of evolution of a system is studied by the |probability law of the random variable of a stochastic process that reflects the casual mechanism of evolution. j I Thus, the expression for the probability that at time t thej !system is in a certain state is investigated. In terms of 22 the random variable X(t) , we want the expression for the probability that the random variable X(t) will assume the lvalue x at time t, that is, we seek P-jx(t)=x]-. i The probability law can assume any form depending on the casual nature of the phenomena of the system. Processes may then be classified according to the law which deter- Imines the evolution of a system S, and also according to the type of variables, namely, continuous or discrete. The |former type of classification is more fundamental than the | later- The following general classification of stochastic 1 ! processes is based on the type of values that the state ( jvariable X and the time variable t assume. i . [ ! If a random variable X of a stochastic process takes i |only discrete values in the state space, the process is i said to-be discrete in space; and similarly if the time variable t takes discrete values, the process is said to be discrete in time. j | If a stochastic process is discrete in space but j ; i I continuous in time, it is called a discontinuous process. If a process is continuous with respect to both the state variable X and the time variable t, it is called a continuous process or a diffusion process. ! More precisely, the difference between these types I of processes lies in the way the variables take different 'values. Consider the sequence in time where changes in the state of a system S occur as a region T, whatever T may be. 2 3 The variate t varies in the region T. One possibility is that the changes of the state may occur only at particular jconsecutive times, say, tQ, t-^, t2, . - - #tn, given in advance; and therefore not random. The region T is then the set of values (tn), where S remains in the same state between two consecutive instants tn and and the process is said to be discrete. In such case, one usually assumes that the instants (tn) at which changes of the state pccur are in an arithmetic progression 0, 1, 2, . . . ,n j which can always be achieved by a time transformation. I | The second possibility, which is more relevant to ! t jour analysis, is that T is an interval (T^, T2), finite or i iinfinite, with S capable of changing at any instant t of T. i The sequence is then said to be permanent. Depending on i ! the type of values that the random variable X assumes, we have two types of such permanent processes? one is called i idiscontinuous, where X takes discrete values, and the other j jis called a continuous or diffusion process, where X is a i icontinuous variable. ; If a change of state consists of a sudden change or jump at a certain time t from a state E to a different ptate E 1, where S has remained in the earlier state for an I j (interval (t-s, t) prior to t, the permanent process where S j !changes state only by such jumps is said to be discontinuous If a state changes continuously such that between the times t and t+dt, dt>0, the system S should pass from a ! ! 24 state E to a state E' differing infinitesimally from E, the i (permanent process where S evolves only by such continuous f jchanges of states is called a continuous or diffusion i (process. It may be noted that in a discontinuous permanent process, the successive times tQ, t-j_, t2 , . . . /tn, * * •/ (where a jump occurs, are random instants of time and not j fixed in advance. This is what distinguishes discontinuous i processes from discrete processes. In the latter case, S I also evolves by jumps from one state to another but here | : the jumps occur at fixed times rather than random instants 1 i ;of time. We shall be concerned with the discontinuous and i 1 ! i • , r |diffusion type processes as these are more relevant to the j ! | | nature of the phenomena of economic growth than the I i ' j discrete processes. In considering the classification of stochastic iprocesses according to the underlying probability law that : | s ;determines the nature of evolution of a system, it is j ■ (possible to conceive an infinite variety of laws of evolu- j j jtion.»The problem here is not make an exhaustive list of these, but rather to extract certain types of laws which jcorrespond sufficiently to practical phenomena, especially j |the phenomena of economic growth in which we are primarily j i iconcerned in this study. Also, we wish to consider those j probability laws that are simple enough to lend themselves | | to an advanced treatment. j 25 Consider a simple random function X(t) of a stochastic process. Between the times t and t+At, At>0, [ ■ i jx is subject to the increment AX=X(t+At) - X(t) . The value I j ' ~ [which X will take at the future instant of time t+At is a I jirandom variable which may depend on the values of t and At |as well as on the past evolution of the process up to and j 'including the time t. The past evolution is summarized by i a sequence of values assumed by X at the instants t defined j j i I It is then clear that the simplest law of evolution which can be imagined and of which there are several practial examples, is that for which AX is a random i i 'variable independent of the past evolution of the process I ;up to and including the time t, whatever the values of t i and At-are. It is this simple type of law of evolution which characterizes the so called additive process, or more ‘ precisely process with independent increments. However, jthe phenomena in which we are interested in this disserta tion is not of the nature where such temporal independence [holds. In fact, as we discussed in chapter I, the plausible situation in economic processes is that of [temporal dependence. In that case, the probability law of |AX depends on the past history of the process as revealed |by the values assumed by the random variable X(t) up to and |including present time t. Consider a simple case of dependence of AX on the 26 past behavior of X. Let the probability law of AX depend not on the entire past evolution of S but only on its final value, namely, the value assumed by X(t) at time t. This case is called a Markov process. Here we deal with a process where the evolution of the system depends only on its immediate past value. In other words, the entire influence of the past with respect to the present instant t is contained in the knowledge of X(t) . This Markov property, namely, the restriction that the future probabi- ! ' ' lity behavior of the process is uniquely determined once the state of the system at the present time is given, is |not a serious restriction as might appear at first. It is | ^equivalent to expressing the probability distribution of i jthe size of an economy tomorrow solely in terms of its size to-day, taking into account random influences in the growth ; process. In reality, of course, the economy tomorrow is ilikely to be related not only to its present state, but | I ' jalso to its past levels in many different time periods. j “ ~ i Nevertheless, the Markov restriction frequently enables us j 'to formulate models that are both mathematically tractable and useful as first approximation to a stochastic picture ;Of reality. Rather, it will be seen later that the Markov | I ! jprocesses show a reasonably good fit to the empirical data. | !To the extent that such processes reveal their ability to jaccount for empirical observations, it is suggestive that j the underlying assumption behind the Markov processes may 27 be true in economics. So far the future evolution of an economy subsequent to t is concerned, the assumption that s I all the necessary information with respect to past behavior of the- economy is contained in its present state appears a j plausible approximation to the reality. jlntroduction to the Theory of Markov Processes We discussed briefly in the preceding section the j general nature and characteristic of a Markov process and its relevance in economic applications, particularly in the I theory of economic development. In this section we attempt ; I • ! :to present some broad aspects of the theory of Markov j j j jprocesses without going deep into rigorous mathematical I j i details. This is because our primary interest is in the |applications of such theory to economic growth models* For a more extensive account of the theory of Markov chains, ,see Bharucha-Reid (14), Barlett (10), Doob (35), Feller I j(41, 43, 44), Parzen (122), and Takacs (143). | j Let tg, t^, . . . , tn be any number of instants of { time? and let X(tQ>, X(t^), . . . , X(tn) be the sequence i i ; [ of values assumed at these instants of time by the random variable X(t) of a stochastic process |x(t), tcT^. We define the following conditional probability to express the j i |probability that the random variable X(t) assumes a certain value at time when the history of the process, that j iis, the values assumed by X(t) at previous instants of time,! t0/ tlf t2/ * * * • are ^ cnown: (2.1) X(t^_) =^n“l ' * * * 1 X(t1)=x1/ X(t0)=xQ} As mentioned earlier, if the structure of the stochastic jprocess {xtt), t C t} is such that the conditional probabi- i lity (2.1) depends only on the value of X(t) at the moment j tn and is independent of all the previous values of X(t) at the moments preceding tn, the process is a Markov process. jMore precisely, we define a Markov process as follows: ; ' | j Definition: A stochastic process lX(t), t ’c Tj is J I ! Icalled a Markov process if for n=l, 2, , and ^arbitrary t c T (m=0, 1, 2, . . . , n)., where tn t. , . I i m u / j . i . . , tn, and for every real x and y, the following equality is satisfied I I I ! ( 2 .2 ) p{x(tn)<y / X(tn_1)=x, X(tn_2)=xn_2, • - • / i j h | ! X(t1)=x1, X(tQ)=x0} = p{x(tn)<y / X(tn„1)=x} | ; i for every real xn_2, . . . , Xq. According to (2.2), the conditional distribution of j !a random variable X(t„), under the condition that the i ! n I For each chapter, mathematical expressions are numbered consecutively following the chapter number, for example,(2.1) indicates expression 1 of chapter 2. 29 j random variable X(tn_;L) takes the value x, does not depend j ,on any information about the values taken by the random variable of the process in the moments preceding tn_-^. For any t-^ c T and t2 C T, where t-^<t2, we define the conditional distribution function F(t^, x; t2 , y) by •the formula (2.3) F(tlf x; t2, y) = p{x(t2)< y / X(ti) = x}. ! ;This distribution function (2.3) is independent of the j I i values of X(t) at t<t^, and it fully determines the proba- j I ! ! I bility distribution of X(t2) conditional on X(t1)=x. This i jis also referred to as the transition distribution function. We shall discuss this conditional distribution later. How ever, we note that it should satisfy the usual conditions I to be a probability distribution, namely (a) F(t^, x? t2, y) is continuous from left in y, ; (b) lim F(t , x; t , y) = 0 and y . . >—co “ ! i • * j . i * lim F(t^, x; tg, y) = 1. j | y-> • +oo | In addition to the conditional probability distribu-i ition (2.3), we may consider the so called absolute or priori probability function p{x(t)=x}', which is the ! probability that X takes a value x at time t. Then, the {probability distribution function is (2.4) ff[x(t)<x] = p{x(t)<x}. | 30 If the evolution of the process is considered to have started at the initial time t=0, the distribution function (2.5) f[x (0)<x ] = p{x (0)<x} i is called the initial probability distribution of X. 'Analytically, a Markov process is obviously determined by ; its transition probability function and initial probability j distribution. Thus the initial probability law of X as x I ivaries over and the conditional probability of the i ;process fully specify the temporal law provided the process i I is a Markovian type. We note that the random variables of |a Markov process are clearly not independent but with the dependence extending over one unit of time. Thus the concept of a Markov process is obtained by abstraction from any empirical process associated with some1 system whose state changes with time according to some probability law specified by the transition probability andt :initial probability distributions, in such a manner that I the probability of the system going from a given state at j ;time r to an another state at time t > t depends on the :state at time T and is independent of the state of the ' ;system at times prior to r. : i | Markov process may be discrete or permanent, conti- I I ; nuous or discontinuous, and there are further distinctions ! depending on the nature of Q and )(. Thus, there exists ! ! a wide variety of Markov processes. However, only certain j ' 31 ! specific types of Markov processes, which appear to be adequate in-economic growth models, will be considered here in some detail. Before we discuss their general properties we give the following few useful definitions: Definition: A Markov process jx(t) , t C t} is cai]_e(^ homogeneous if for any arbitrary t-^c T and t2CT, where t]_< 12, the transition probability function depends only on the difference (t2-t2 )=t (say). Definition: A stochastic process jx(t) , tCT^ is called a process with homogeneous increments if for any t-^c T and t^c T, where t]_<t2, the probability distribution function of £x(t2)-X(t^)J depends only on the difference (t2-t1)=t (say). Definition: A stochastic process jx(t), t C Tj is called stationary if for any arbitrary number h>Q, the j distribution function of the random variable X(t) at times i . iti# t2, • • - , t is independent of the random variables X(t-L” th) , X(t2+, h), . . . , X(tn+h) at times t^+h, t2+h, . . . , tj^+h, whatever be the values of t]_, t2, . . . , tn and the interger n. We note that a stationary process may also be a i I iMarkov process. Though a stationary process is not, in general, subject to time transformation, a Markov process !is independent of time transformation. This means that its 32 probability law remains unchanged by any transformation of the time axis, that is, by any change of the origin. We now introduce the notion of a discontinuous Markov process and derive its functional equations. Let jx(t), t TJ be a Markov process, where % denotes the state space associated with it, that is, ^ is the space of all |the possible values x on a real line which the random variable X(t) can assume. Let us consider T as the set l-^t^oj', that is, real values on the positive axis 0 ^t < ». I ;We recall that for a Markov process the conditional i i 1 distribution function j(2.6) F(r, x* t, y) = p[x(t)<y / X(r)=x] ■is defined for every t and r, where 0 ^ t < t, and this I ! ■distribution function is independent of the values of X(t) for t < t . We assume the usual conditions for F(t , xy t,y); | to be a probability distribution function. Also, we assume j : ■ ithat F (t , x; t, y) is a continuous function with respect toj it and r. Then, the so called Chapman-Kolmogorov equation j ;takes the form j l 00 : (2.7) F(T, x ; t, y) = J F(s, z? t, y) dzF(7f xy s, z) ! — C O i i ! i ‘ } |where r<s<t. If F(r, x ; t, y) has a density function (with1 1 respect to y) f(T, x; t, y) , then (2.7) is C D i (2-7) 1 f(t, x; t, y) = \ F(s, z; t, y)f(r, x; s, z)dz. 33 where r<s<t. The equation (2.7) or (2.7) ' is obtained by- considering the probability of a transition from x to an event jx(t)< yj- through all points s c (r, t) , and hence it can be given the following interpretation. The passage from the state x at the instant r to a state less than y at ! the moment t through a state z at the intermediate moment s <c (t , t) is a product of the two transitions: (i) at the instant s the process is in a state belonging to the interval (z, z+dz) and the probability of this event is i ;f( r , x ; s , z)dz; (ii) from the state z at the moment s the process passes to a state less than y at the instant t and the probability of this event is F(s, z; t, y). i ; We now introduce two new functions: (i) the i I |intensity function q(t, x) and (ii) the relative transition probability function P(t, x, y) defined by | (2.8) q(t, x) = lim P-fx(t+At)-X(t) ^ 0 / X(t)=x\ I At-»OAt * * i ! P(t, x/ y) = lim P-fx(t+At)<y. / X(t)=X/ AX(t)^o}. I At—^O u J | These functions q(t, x) and P(t7 x, y) are non- I negative/ and P(t, x, y) as a function of y is a distribu tion function. The intensity functions have the following |interpretation: q(t,x)At + o(At) is the probability that j X (t) will undergo a random change in the interval (t, t+At) ;when X(t)=x; equivalently/ if at the moment t the process !is in the state x, then the probability that at the moment 34 (t+At) it will be in a state different from x is equal to q(t, x)At + o(At); hence, l-q(t, x)At + o(At) is the probability that no change will take place. Similarly, P(t, x, y) is the conditional probability of X(t) assuming a value less than y at time (t+At), when it is known that X(t)=x and that it has undergone a change in the interval (t, t+At) . If the changes of state occur by jumps as above, the ! process is called a purely discontinuous process. Analyti-j * I j I cally, a purely discontinuous process is defined as follows: i | Definition: A Markov process jx(t), t C t| is j Ipurely discontinuous if for every t c T, j l-q(t,x)At+q(t,x)P(t,x,y)At+o(At) j (2.9) F(t, x;t+At,y) = / for x<y, x)P(t,x,y)At+o(At) for x^y. In other words, if x<y, the process may be in a j I state <y at the moment (t+At) provided that it was earlier i |in a state x at the instant t, due to either there was no s | |change of state during the interval (t, t+At) or a change I of state occurred, but at the moment (t+At) the process was in a state less than y. If y^x, the process may be in a istate less than y at the moment (t+At) only if a suitable |change of state occurred. If we put E(x,y)=l when x< y and E(x,y)=0 when x^y, i ithe condition (2.9) can be restated as follows: 35 (2.10) F (t, x; t+At, y) = £l-q(t, x)At^E(x, y) +q(t/ x)P(t/ x, y)At + o(At) W. Feller (39, 41) in his fundamental paper derived the following asymptotic functional equations that describe discontinuous Markov processes for a small At in the limit as At — > 0: (2>11j d- = q(T,x)[F(T,X;t/y) r+c° ~ ] -j r oF(T/z?t/y)dzP(T/X,z) J, « ^ - o o (2-12) 5F (T/X; t, y) = _f q(t, z)d F(r,x;t, z) St J —0 ° • f “ C O +[ q(t,z)P(t,z,y)dzF(r,x;t,z) | These equations (2.11) and (2.12) describe the j | discontinuous Markov processes and are called the Feller integrodifferential equations;(2.11) is called the backward :equation, since it involves differentiation with respect |to the earlier time T; while (2.12) is called the forward iequation, since it involves differentiation with respect to the later time t. If we turn from the real line | to a case where the i random variable X(t) can only assume a denumerable number j j of values, which we may denote by the non-negative l integers, that is, £ is the set jx = 0, 1, 2, . . . ,}, ' [ the transition probability distribution function F(T,x;t,y) I defining the Markov process TX(t), t ^ Or is reduced to t a point function P(r, i; t, j), which we will write as f 36 I I P^j(t, t) . This expresses the probability that X(t)=j j i conditional upon X(r) assuming the value i at time r<t. j { Similarly, P(t, x, y) reduces to (t) in this case. Here ji and j are the initial and final states respectively. In I ' j Ithis case, the Feller integrodifferential equations (2.11) i I jand (2.12) reduce to the system of differential equations (2.13) dP±j(t,t) = qi{T){Pij(r,t)-^ Pik(r)Pkj(r,t)}, k=0 9r SPij(T,t) « | (2.14) — - = (t)P±j(r,t) qk (t)Pkj(t)Pik(r,t), ! k=0 | : where i,j =0,1,2, . . . . The first system of equations j ! ’ i | (2.13) is called the backward differential equations, and j | j |the second system (2.14) is called the forward system of !differential equations. These two infinite systems of ; differential equations, which describe the Markov process Iwith continuous time taking non-negative values 0, 1, 2, . ! i j , were first derived by Kolmogorov (92) in the paper : i i j which played a fundamental role in laying the foundation in i I the development of the theory of Markov processes. Hence, j :the above equations are called the Kolmogorov differential I equations. : | The equations (2.13) and (2.14) depend explicitly onj iboth r and t, and hence are not time-homogeneous. If we i : | irestrict our analysis to the time-homogeneous case, we havej i ' i (2.15) p.. (r, t) = P, . (t - r) ’ L. . _ - 1 -J I Thus, we assume that the transition probabilities depend |only on the duration of the time interval, and not on the instant of time. We can, in this case, then put r=0 and consider the transition probabilities Pj_j (t) . The exis- tance and uniqueness of solutions of the Kolmogorov differential equations have been studied by several workers iunder different conditions. We shall not go into their theoretical discussions here but consider a few applica tions in the next chapter. There we shall derive the differential-difference equations describing the probabi lity laws of the specific processes, and obtain their solutions and discuss their properties. Continuous Markov Processes In the preceding section, we discussed the theory of discontinuous Markov processes in continuous time defined on non-negative integers. Such processes are characterized by the chance of a transition in a small interval of time being very small, but the size of the transition , if it occurs, being sufficiently large. On the other hand, in | ithe continuous processes that we are now going to study, ; j the probability that some change will occur in any small j I ! ;interval of time At is almost one; but for small At, the i ! I |change will also be small. Such continuous Markov t ■ I i i j processes defined on the real line are called diffusion jprocesses. We consider in this section, a general theory ' ' 38 n of diffusion processes m brief, and will look at its specific applications in chapters IV and V. In contrast to the discrete variables in continuous time, where the conditional probability P^j (t, t) is studied, here we are concerned with a continuous probabi lity distribution F(t, x ; t, y) of the random variable X(t) of the process. We assume that X(t) takes any value x on the real line, that is, % = jx: -°°< x <»j., and let the j region T = -jo ^ t where the variable t assumes any i real value. We recall that (2.16) F(r, x; t, y) = P-jx(t) = y / X(r)=xj-, r < t denotes the transition probability distribution of the process jx(t) , t ^ 0 j", and, for fixed x and r, it is a ;conditional distribution function in y and a continuous function in t. The following assumptions are introduced |to define a continuous Markov^process. These assumptions are taken to be satisfied everywhere in the region T. (a) Probability that |X(t+At) - X(t)J £ 6 > 0 I conditional on X(r)= x is an infinitesimal of order of i ‘magnitude lower than the time interval At, that is, (2.17) lim ~ f dyFft, x; t+At, y) = 0 At-*0 |y-x| s 5 y i I --------------- : 7 j See Blanc-Lapierre and Fortet (16), Doob (35), !Feller (44, 48), and L4vy (98) for detailed studies in the theory of diffusion processes. | 39 (b) The following limits exist i r (2.18) lim At (y-x)d F (t, x; t + At, y) = m(t, x) J V y Atr-K) [y-x| < 6 and 1 r 2 2 1.2.19) lira. _ J (y-x) d^F (t, x; t+At, y) = cr (t, x) >0 At-^0 At fy—x| <8 These limits (2.18) and (2.19) are called the infinitesimal mean (mathematical expectation) and variance of the change in X(t) respectively. However, the existence of these limits does not necessarily imply that F(r, x; t, y) has i !first and second order moments. The two formulas (2.17) I land (2.19) can be alternatively written in the following I i j forms: i . . ;(2.20) p{ |x(t+At)-X(t)| : > 8 / X(t)=xj- = o (At) , 2 2 (2.21) P-j£x(t+At) -X(t) J / X(t) =xj- = cr (t,x) At+o (At) . Thus, according to (2.20) the probability of large i ichange of state in a small time interval is very small (of ;order less than At), and according to (2.21), for any At of i small magnitude, the probability that |jX{t+At) -X(t) J ^ 0 i is positive. We then analytically define the continuous Markov process as follows: Definition: A Markov process -jjx(t) , t c tJ- is called continuous if for every t c T and every 8 > 0, the l |relations (2.17), (2.18) and (2^19) are satisfied. 40 If the density function f(r, x; t, y), that is the derivative of F(r# x? t, y) with respect to y exists, then the formulas (2.18) and (2.19) take the form, respectively (2.18)' lim v t f (y-x)f(t, x; t+At, y)df=m(t, x) Atr*0 $-x| <6 y and I Q) * l i m a 4. r (v —x ) 2 f (t . x : t i + A t . v l d In addition to the above conditions, we assume that (2.19)' lim A-t I (y-x) 2f (t,x; t+At, y)df = cr2 (t,x)>0 Atr-»0 | y - X | <6 for all t, x; t, y, the partial derivatives of the first land second order of f(T, x? t, y) exist and are continuous with respect to every argument. Instead of denoting the ; infinitesimal mean and variance of the change AX in X(t) by m(t, x) and cr (t, x) , we shall use the notation adopted in the literature to denote them by b(t, x) and a(t, x) respectively. Under the various assumptions made above, the density function f(r, x; t, y) is a solution of the ifollowing parabolic partial differential equations: I (2.22) land Bf 1 d2f Bf — = — a(r, x) — 7 + b(r, x) — 2 Bx^ Bx Bf 1 32La(t, y)f] B^b (t, y)fl (2.23) — -------- 2--------------------- Bt 2 By By where f = f ( t / x ; t, y). These equations are called backward and forward Kolmogorov diffusion equations respectively. The forward Kolmogorov diffusion equation (2.23) is also known as the Fokker- Planck equation. As before, we call the process homogeneous if the itransition probabilities depend only on the time difference (t - t) . Hence, for a homogeneous diffusion process, we jwrite the probability density function f(t-T;x,y); or as jf(t, x, y) assuming t=0. An important point in the homo- sgeneous case is that the limits a(t,x) and b{t,x) depend only on x and not on t, that is, these are then a(x) and b(x) as functions of x only. In physical applications the form and properties of the distribution function of a diffusion process depend on the basic partial differential equations which are characterized by the coefficients a(t,x) and b(t,x). Then, j | - | idifferent hypotheses regarding these coefficients give rise j to a set of different probability distributions and hence the choice depends on how well a real physical phenomenon ' i can be characterized by a specific structure of the coeffi-! j j !cients reflecting the mechanism of the physical system | ! i !under investigation. We shall consider the question of j 1 i plausible specification of the coefficients a(t,x) and j ib(t,x) in order to describe the economic growth process as a diffusion process in later chapters. j i I ! ! |Simple Stochastic Growth Model j i The phenomena of growth in general and that of j ieconomic development in particular can be easily treated 42 by the theory of Markov processes within the framework of .stochastic processes discussed in brief in the above sections. To illustrate the application of the theory of stochastic processes in an economic growth model/ we may consider a simple model of birth type process. The word 'simple' implies only that the model we consider is a | simplified representation of the complex economic growth process and not that it is simple or trivial from mathe- imatical point of view. Of course, a simple stochastic I I i jprocess may not be adequate to describe the complex nature ! I I :of the economic growth process. However, we shall restrictj : ourselves to a simple example at this moment, as our j ! j .purpose here is to show how the above theory of stochastic j | | iprocesses is useful in economic growth models. More j adequate economic growth models in terms of Markov processes will be considered in the following chapters. In; 'order to illustrate the difference between a stochastic | Imodel and a deterministic one, we first consider a simple j deterministic model for economic growth and then its i i I stochastic analogue. i Let X denote the variable that describes the state iof economy as it evolves, and x(t) a real-valued and j i . ! i jcontinuous function denoting the value of X at time t. Thej Ivariable X may be taken as a certain measure of economic J t ' i I level of the complex economy, for example, X may denote j ! { real national income over-all or per capita in the 43 aggregative theory of economic development. To describe the growth of the economy, we consider the following | [postulated mechanism for the manner in which the value of X I jean change: Consider that (i) at time t the economy is at the level x, and ! (ii) that the economy advances to a higher level ;such that the increase in the small interval (t, t+At) is proportional to the present level at time t, that is, we have the relation x (t+At)-x (t) = Ax(t) = lx(t)At, where X>0J The constant I is the proportionality constant of birth or ! growth rate. This leads to the differential equation : dx(t) j (2.24) dt (t). i | j - If we assume that the economy at the initial time I !t=0 (say) is at x ( 0 ) = x q >0 , then the solution of the equation (2.24) is i(2.25) x(t) = xoexp(Xt). i ! The distinguishing feature of the deterministic I solution (2.25) is that it shows that, whenever the initial ;value is Xq, the economy willalways be at the same level for a given time t > 0. The important point in such deter ministic is that the growth path of the economy is rigidly [fixed once the initial level xn is known and remains at the! I ! |same level. Serious objection to this can be raised since ;it does not take into consideration the large number of j ! ! ■ random or chance factors that influence the economy. 1 I 44 We therefore consider the stochastic analogue of the above deterministic model. Let X(t) be an integer-valued random variable representing the size of the economy in terms of real national income at time t, and let X{0)=xQ>0 be the initial size at time t=0. In stochastic approach, we attempt to find an expression for the probability that j [ at time t the economy will be at a level x instead of deter-^ mining a functional equation for X(t). Thus, we seek here Px (t) = p{x(t) = x}. In order to formulate the stochastic model, assumes (i) if at time t the economy is in the state x (x = Xq, xq-KL, . . . ), the probability of a transition from x to x+1 in the interval (t, t+At) is equal to XxAt + o (At) , X>0; (ii) the probability of a transition from x to a state different from x+1 during the time interval (t, t+At) iis o(At), where o(At) denotes the higher order terms in At j which can be neglected for small At; and hence ; i ! (iii) the probability of no change during the time I interval (t, t+At) is 1 - XxAt + o(At). Under these assumptions, we have the following :relation : i 1(2.26) Px (t+At) = (l-\xAt)Px (t) + X(x-l)Px_L(t)At + o(At)j In the limit, as At——► O , we obtain the system of differen- j [ j tial difference equations j 45 dP (t) (2.27) x = - XxPx (t) + \(x-l)P ^(t), x=xQ, x +1, dt ■The equation (2.27) is the stochastic analogue of the equation (2.24). Under the assumption X(0) = x^ and further assuming the initial condition Px (0) = 1 for x = xQ, the solution of the equation (2.27) is (2.28) = p{x (t) = xj- = (x_i)c ^x_x ^ exp (-\x0t) £l-exp (-Xt) J32 X0 for x ^ Xq. We observe that, for fixed X and x^, we have for ;every pair (x,t), x i x and t s 0, a number P (t), ; O X |0 £ Px (t)^l which is the probability that the random variable X(t) will assume the value x at time t. It is of interest to note that the deterministic model is a special |case of a stochastic model, in the sense that its results hold with probability one. In the application of stochastic precesses to the economic growth model, it is not, however, the probabilities P„(t) that are usually of primary importance but some characteristics of the probabi lity distribution of X(t) in which we are interested, like its moments or its asymptotic behavior, et cetera. One of the economist's main duties is to attempt to describe it in terms of a certain number of typical values, for example. mean values that are characteristics of what we call general conditions. Consider then the mean or expected economic level. Let m(t) = E'jx(t)}'. denote the mathematical; jexpectation of the random variable X(t). Then, by | i t ; definition i i ■ C D (2.29) m(t) = T xP ft) = x exp(Xt) ! i —' x 0 ' x=0 We note that the mean economic level (2.29) in the j ; t stochastic model is the same as the economic level (2.25) ; ;obtained in the deterministic case. In view of such correspondence, we can state that equation (2.24) describes! i .the mean level of the economy while equation (2.27) takes j iinto consideration the random influences. However, it may j i I be pointed out here that such correspondence between the two models does not hold in general; rather there are many cases where deterministic solution is not the same as the j w l stochastic mean. i i j ! If we wish to avoid the somewhat unrealistic ! assumption made above of taking integer values by economic j variable like national income X(t), the diffusion-equation : :analogue of the above model can be obtained by assuming j I _ f |X(t) as a continuous random variable and considering the j I infinitesimal mean and variance, that is, the coefficients ! ; 'l j :b(t,x) and a(t,x), as proportional to the instantaneous ! I i state of the economy, and independent of time. Under these 47 assumptions, we have : (2.30) a(x) = ctrx and b (x) = gx where of and p are constants, a > 0. The coefficient { 3 can jbe interpreted as representing the 'drift' in the economy i jwhich can be either positive or negative depending on l ! [whether the economy in general moves ahead or drifts backward. With the coefficients as specified above, the forward Kolmogorov equation takes the form j (2 31) af(t,x) S-[xf(t,x)] S[xf(t,x)] ; — j r ~ = 1 e s ’ j where f(t,x) denotes the probability distribution function j of X (t) at time t, 0 < x < Solution of the Kolmogorov j i i :equation is very complicated in this case. However, i ! j , , v 9f(xo*t,x) assuming that x f (xn; t, x) and x ----------- tend to zero dx as x — 0, it can be shown that the expected economic level i ;at time t • C O 1(2.32) m(t) = E'jx(t)}’ = xffx^; t,x)dx yields the differential equation (2.33) = m(t) dt Therefore, solving (2.33), we obtain (2-34) m(t) = xQexp(pt) ; 48 It is interesting to note that the expected level given by ;(2.34) is the same as that given by (2.29) in the case of a simple birth process of discontinuous type. | jEconomic Evolution ! | In the foregoing discussion we dealt with the general introduction of the theory of Markov processes and showed its relevance in the model of growth process. How ever, we have left unsolved the issue of how to choose and measure the relevant descriptive characteristics of i economic progress, and the questions of how the economy ! devolves and which Markov process would adequately describe j ! - J |such an economic evolution. It is difficult to give * i ! jcorrect answers to such problems on a really broad basis. j The problem of a meaningful definition of an economy is closely tied to another general problem of aggregation. What specific aggregate criteria would be relevant as ; • i |indicators of general economic prosperity is rather prob- j ilematic. However, the following things are usually i ' j ^associated with prosperity. High level of per capita pro duction, large amount of physical capital per head, high standards of general education and technical skill, high : i ■average expectation of life, etc. There are also other I ’ descriptive characteristics that characterize the basic structure of an economy, viz, propensities and attitudes j ■ i of people towards changes in the economic and social 4 9 environment. To combine all such descriptive characteris tics and inadequately defined phenomena into a few aggre- i jgate indicators of the general conditions of the economy is I a ever-present problem in economics, However, we often construct and use certain macro-variables, the meaning of which is given only implicitly by the nature of the theoretical framework in which they are used. In view of the analytical methods we wish to use in this study, the following variables appear to be sufficient for our purpose-1 (i) Some index of total productive output, e.g., real national income. <ii) Some index of the total accumulated capital. (iii) Size of the population. (iv) Total investment outlay. (v) Total government expenditure. These variables are assumed to specify the economic Isystem. Of course, we can conceive of the economic system j ; i las composed of various economic activities interacting | I lamong them in time, and these activities generate what we call income. The meaning of 'income' here is in the sense ; of 'value added to a product or services' by an economic activity. An economic system is then the network of inter-j i |change of such components of a society's income that are ! i i i reckoned in terms of a monetary unit at a time of inter- j : j I change. Certain deep-rooted and slow changing cultural j factors taken as important growth determinants than the 50 physical resources then determine a society's rate of population increase, its basic work-leisure pattern, the intensity and composition of final demand, division of skill and labor, etc. With the diffusion of economic activities under such a complex structure, the economic system evolves in time which can be characterized by a suitable stochastic process. Owing to the complex nature of the growth process through the diffusion of various economic activities, it ; is necessary to formulate and integrate models of the type, ! 'called macro-models, which are concerned with the growth ini 'aggregate economic variables that are assumed to reflect I the general conditions of the economy and its growth ! | | ; process. Should an adequate model of this type be formu- j lated, it may be then necessary and useful to formulate models, which deal with the micro-economic processes within [the over all economic system, and which are possibly jresponsible for generating the economic growth we observe :in macro-model^. The equations describing the levels of the macro-variables at time t, say X^ (t), X 2 (t), etc., derived on the basis of a postulated mechanism by which :growth in these variables is assumed to take place may, I [however, not reflect the internal processes (micro-procr- j : ’ I esses) responsible for changes in these macro-variables. j ; _ I With the conceptual framework outlined above and I the analytical theory of stochastic processes described in 51 ! i the preceding sections as a background, we wish to review in the following chapters some of the applications of Markov processes upon which a stochastic theory of economic ■development could be built. We try to formulate macro dynamic models of economic growth based on the fundamental characteristics of economic evolution, such as national income, capital stock available, etc. If we succeed in describing the economic system by means of a general schema of Markov processes embodying certain properties of the system, there is a definitional question to be discussed, viz., the question whether the system as thus depicted, will by its own workings produce the commonly observed trend, and if so, under what conditions. The problem of meaningful definition Of the trend in the stochastic approach is rather arbitrary. Corresponding to the time path of an 'exact1 model, the natural definition of trend of an economic variable X(t) of a stochastic process may ;be taken as the mathematical expectation of the variable. ;However, there are other ways of characterizing the stoch- I astic process X(t) which may correspond to the 'exact' or 'deterministic' part of the model, for example, by re placing the stochastic variables by fixed variables, that I is, assuming the random elements identically equal to zero.; We shall adopt the expectation E"{x(t)j" of the variable X(t) ! of a stochastic process as the more plausible definition of the trend. This is partly out of conviction that a random 52 variable can be adequately described by the two main i ^characteristics: (i) the central value (mathematical lexpectation), and (ii) a measure of dispersion about this I ! I central value (variance), and partly due to statistical i convenience and analytical simplicity. Our chief interest in the following chapters will I then be to develop stochastic models of economic growth so i as to yield the type of trends commonly observed in the I \ i empirical data, and to attempt a statistical verification j of the hypotheses underlying such models. It has already ' i | been stated earlier that, depending upon the assumptions made regarding the casual mechanism of evolution of a stochastic process in terms of transition probabilities, j ! ■ > i the model can tahe on various forms. It may be noted that | : i the same distribution function may arise in various ways or different distributions may produce similar trends. For I example, the negative binomial distribution not only arises: i from the hypothesis of heterogeneity between various |groups, but also from the 'contagion' hypothesis in time? ! or as noted earlier the exponential trend in national income arises under a discontinuous birth process as well ;as under a diffusion process of proportional growth. Thus,j | we realize that the total distribution itself will not j 'discriminate between these two possibilities; and a more ;detailed study of changes in time will be required to investigate the plausibility of one or the other. We shall 53 discuss these questions of the suitable probability ;distribution functions in the economic growth phenomena in I l |the next three chapters while considering various j |applications of the theory of Markov processes. Whether i I we can answer such questions with success or not, the fact i remains that one casual mechanism can generate the same statistical distribution, and that it is a possible warning against considering a unique manner in which an observed distribution may have arised. i CHAPTER III | SOME STOCHASTIC MODELS OF ECONOMIC GROWTH j i ! [Introduction i I i A systematic attack on stochastic theory of economic development has taken place only recently. The flow of literature on the subject of economic growth models based on the theory of stochastic processes began with the work of Haavelmo (63), who in 1954 first thought that the i I evolution of an economy can be described by a stochastic process and considered the well-known poisson process as a ;reasonable first approximation to the process of economic j j j [growth. Since then, for some time, the idea of applying j i i the theory of stochastic processes to the economic evolution was apparently overlooked by the researchers engaged in the study of economic growth models. Tintner (149) in 1961 considered the problem of economic growth again within the framework of the theory of stochastic processes. Starting with the work of Tintner, a good deal of attention has recently been paid to the stochastic approach in economic analysis and a number of stochastic models of economic growth have been formulated and tested : 1 ■ i [with empirical data. Among these, the principal works are ! ’ i of Tintner and Sen Gupta (133, 150, 151) and Narayanan j 1 (115). These models describe, under various hypotheses, j the growth phenomena in the principal variables of an j I economy. In this chapter, we give a brief review of such models which provide a fairly good approximation to the growth of an economic system. We shall consider only a few i ! |representative types of these models, especially those that j are based on the so called discontinuous Markov processes, i namely, the poisson process, the birth and death processes,1 and their variants, etc. One thing common in these models is that they are formulated in terms of discrete variables taking non negative discontinuous values by jumps. Further, the basic| :assumption in all these models is that the changes in I economic variables behave as if their genetic process is a I ! j |random walk type. Secondly, the more important assumption j i in such models is the temporal dependence between the values of the economic variables. Historically, the first attacks on the statistical problems of the analysis of time' series, where dependence of observations in time is more :important, were made independently by the Russian mathe matician and econometrician E. Slutsky (137) and by the English statistician G. U. Yule (164) in the year 1927. Kendall's work (86) on the temporal dependence in price |data is an another example of early recognition of the f i problem of dependence over time in economic analysis. The j ;approach of these people, however, was a general one and j was not explicitly based on the theory of stochastic 56 j processes. ! Although the stochastic models of economic growth jare formulated considering the assumption of temporal i idependence of economic time series data, the nature of the jvariables differs from model to model depending upon the I lunderlying assumptions about the probabilities for the changes occurring in the variables during a small interval ; At of time. We shall briefly review these aspects while discuss;- | , ing the models one by one in this chapter. In each case, I : ■ i we shall see how the different assumptions regarding the nature of transition of the process as specified by transi-j ition probabilities of the changes in the variables lead to j ’ i the various system of the differential-difference equations| describing the probability distribution of the process. We shall also discuss the solutions of these equations and investigate their properties. It must be noted, however, that the following brief review does not attempt to enumerate entirely the literature on the subject. Poisson Processes The poisson process is the simplest among the dis- i I continuous processes that have found useful applications :in the economic growth models. It is generated when the possible values of the variable of the process are non- \ I i | negative integers, and there is a constant probability X ' 57 per unit of time for a unit increase in the variable. In I j .other words, when a process is of the type where the chance jof an increment in the variable during the time interval At l jis XAt+o(At), X > 0 and the chance of more than one change i in the interval At is o(At), then the variable X(t) of the ■process follows the poisson probability law with the parameter It. \ In terms of an aggregative theory of economic development, consider the simplest assumption that economic : development is measured by the real national income denoted | by a random variable X(t). With the assumption that the economic growth process is specified by the variable X(t), ithe development of the process can be described by the \ process "[x(t), t ^ o}. Assume that the variable X(t) is a | discontinuous variate taking discrete values in the state space % = {x: 0,1,2, . . .J, and that it is a Markovian type variable, that is, the state of the economy at time t 'can be described by the values of the random variable X(t) with the property (3.1) p{x(t) < y / X(t) = x} s p{x(t) < y / X(t') = x'} for all x' and t' £ t, where r< t. Let Px (t) =P^X(t) =xj,xc)(; The following assumptions about the Markov process ■with regard to the possible changes in the variable X(t) I occurring during a small interval (t, t+At) of time specify! the manner in which the economy is assumed to proceed: 58 (i) The probability of a change in the interval (t, t+At) is XAt + o(At), where X is a positive constant. (ii) The probability of more than one change in the interval (t, t+At) is o(At). (iii) The probability of no change in the interval j(t, t+At) is 1 - XAt + o(At). We note that the assumption of constant X implies | i that the probability law of the change in X(t) depends only on At and not on either t or x. Also, it is assumed here ! : ' I that the random variable AX = X(t+At) - X(t) is independent i of the values of X(f) for r< t. \ The above hypotheses can be restated in terms of the j transition probabilities of X(t) during a small interval j I 1 (t, t+At) of time as follows: J (3.2) p{x(t+At) = x+1 / X(t) = x} = XAt + o(At) (3.3) p{x(t+At) = x+k, k > 1 / X(t) = x} = o (At) 1(3.4) p{x(t+At) = x / X (t) = x} = 1 - XAt + o(At) ! These assumptions lead to the following relation for Px (t+At), (3.5) Px (t+At) = (1-XAt) Px (t) + XPx-1(t) + o (At) | i ! 1 from which the system of differential-difference equations j is obtained in the limit as Atr—*0 as j 5-9 dPx (t) r ( 3 - 6 ) ________= X Px-1 (t) - Px (t) , x=0,1,2, . = . dt This characterizes the poisson process. Assuming the initial condition j (3•7) Px (0) = p{x (0) = x} = 1 for x — 0 = 0 for x ^ 0 we can easily see that the solution of (3.6) is given by (3.8) = exp (-It) (Xt)X/xl Thus (3.8) is the poisson distribution giving the probability that at time t ^ 0 the economic system is in the state xc^, and the mean of the distribution is . : f (3.9) m (t) = E-jx (t) j- = Xt i The above results can be interpreted as follows. If we regard the unit of income defined in the sense of 'value added' as an income particle, X(t) denotes the total number; of income particles generated by the economic activities at; |time t. Then the assumptions characterizing the poisson process imply that the income emissions occur at various distinct instants of time with probability equal to almost one; and the number of income units generated is finite for; : iany finite interval of time. Further, it is implict that ! r i the income generations by the economic activities are I random and independent of each other. The solution (3.8) j can then be interpreted as the probability that there are x number of income units generated during a time interval of length t, and the mean m(t)=Xt given by (3.9) shows that assumptions can hardly be adequate to explain the income growth process of a complex economy. However, with a j suitable origin and scale transformation, a generalized version of such poisson process can be and has been employed in the analysis of economic growth. Tintner ^nd others (114) and Narayanan (115) considered the following transformation We note that the mean (3.12) and the variance (3.13) are jthe mean growth of the economy is linear. Of course, such a simple process under very simple (3.10) Y(t) = at + utX(t) j :where at and.ut are some definite functions of time t, and \ i j ;X(t) is the random variable of the simple poisson process. : Then Y(t) follows a generalized poisson probability distribution taking the values a-j-, at+ut, at+2ut, . . . , with associated probabilities given by (3.11) y = at, at+ut, at+2ut, . The mean and the variance of Y(t) are given by (3.13) v(t) linear functions of t. Thus, the trend (mean growth) of national income Y(t) over time describes a straight line if we assume at and ut as constants. In most of the aggregate economic variables, linear trend is a fairly good approximation, and hence the generalized poisson process explains the economic growth phenomena reasonably well. Narayanan (115) applied the theory of generalized poisson process to the empirical data of national income of India for the years 1948-49 to 1961- 62, and obtained the estimates of the parameters a^=a/ ut~u and \, using the two methods of estimation, the simple and weighted regressions, and observed the following estimated trends for the two methods respectively: (3.14) m(t) = 78.0385 + 4.270t (simple regression) (3.15) = 82.0598 + 3.734t (weighted regression) These results show a fairly good approximation to the observed data. Also, he considered at as a function of an instrument variable like government expenditure, say G(t), at time t, that is, (3.16) at sag + a1G(t) a simple linear function of G(t) suggested by the Keynesian multiplier analysis. Assuming ut^u, some policy proposals 'for achieving alternative growth paths are investigated in j the empirical study by Narayanan. ■ i We thus have the tentative support to the theory of j poisson process as a first approximation to the economic 62 growth process. However, the two major drawbacks of the poisson process are thats (i) the probability law for the change in the variable X(t) is independent of time t, and (ii) the changes are occuring independently of each other, :by assuming A as a constant. The real situation is rather contrary in economics. To consider more realistic situa tion, the above poisson model can be generalized in two directions. One approach is to consider the parameter X a function of time, that is, X=X(t), and second one is to treat Xa function of state space, that is, X=X(x). The first type of generalization, called time-dependent process, has been first given by Kendall (81, 82} while discussing the stochastic models of population growth. Narayanan (115) uses the same empirical data of Indian national I income in the model of poisson process modified on the basis of Kendall's work. He uses a liner function ;(3.17) X(t) = X0+X1G(t) of government expenditure G(t) at time t and obtains a Imodified linear trend I t ; (3.18) m(t) = E{Y(t} - atu[x0t+Xx^ G(j) j-1 However, one can take any suitable functional form for X(t) 'appropriate to the situation under investigation. In the ; i general case, the expression for the trend is ■ (3.19) m(t) = at + uf c X(r)dT The important case in the economic growth appears to I - ■ I be that of 'contagion' hypothesis. As we pointed out in the preceding chapter that the forward momentum of the jeconomy depends on its present state. In that case, the jchance of an income increase in a small time interval At is j inot constant, but it depends on the income level that has I already occurred. Then the growth process needs more adequate characterization than the poisson process. Let the chance of an unit increase in income depend on its present level, that is, (3.20) P-|x(t+At) = x+1 / X(t)=x} = [l0+ ^XftjjAt + o (At) ! the contagion hypothesis can be shown to lead to the so called 'negative binomial distribution'. Such result has been first established by A. G. McKendrick (102), long ago in 1914 while working with the medical applications. We shall discuss this and some other similar cases in brief ■ i in the next section. Birth and Death Processes We mentioned in the above section that more appro priate case in the economic growth process is to consider the transition probabilities to depend upon the present j i state of the system. We consider this case in this j section. There is also another important aspect that needs| l | to be considered. This is to remove the unrealistic 64 assumption made in the poisson process that national income always increases by jumps unit by unit. It is rather more Hrealistic to assume that in the economic growth process, a i jchange may be either an increase or a decrease. Such | Jprocesses are called birth and death type processes, which j ; j ;appear more appropriate and plausible for the analysis of economic development where we have the economic variables, for example, national income, sometimes decreasing in value rather than always increasing. The first application of the birth and death type of processes in the analysis of economic growth models has j been considered by Tintner and Sen Gupta in 1962 (131, 133, ■ ■152). In these studies, many variations of the birth and i 1 I > • I death processes have been treated with empirical examples. ! Later, Narayanan (115), in his empirical study of Indian economic data, used some of these and other similar type of processes with suitable modifications. — i In the general birth and death process, the follow ing assumptions are made about the possible changes in the value of the random variable X(t) of the stochastic process during a small time interval (t, t+At): (i) assumption about stationary independent incre ments which postulates the following conditions: if at time : t the system jx(t) , t^oj- takes the value x c. % = |x: 0, 1, j 2 \ i 65 (a) the probability of a transition from x to x+1 in the small interval (t, t+At) is A.(x)At + o(At), i j (b) the probability of a transition from x to (x-1) |in the small interval (t, t+At) is p,(x)At + o (At) , j | (e) the probability of a transition between any two specific states of the system, x and (x+s) is independent | of the initial position. (ii) The probability that the present state remains unchanged during the small interval At of time is 1 - £l(x) > + ^{xjjAt + o(At). (iii) The probability of a transition during the ! small interval At of time from a state to a state other j I • i i • i | than the neighboring state is o(At). j : 1 The \ (x) and m-(x) are any functions of the state x of the system g.t time t. More formally, the transition probabi lities in the birth and death process can be written as (3.21) P"{x{t+At) =x+l / XftJ^x} = X(x)At + o(At) (3.22) p{x(t+At) =x-l / X(t)=x} - f i (x) At t o(At) (3.23) p{x(t+At)=x / X(t)=x} = l-[\(x)+n(x)]At + o(At) and all other transition probabilities are of order of magnitude smaller than At. These assumptions lead to the j relation between the various probabilities as follows: 6 6 | (3.24) Px (t+At) = X (x-1) Px_i (t) At + jl- X(x)+p,(x) Atjl^ft) • + 1J. (x+1) (t) At + o(At) j Talcing the limit as At --> 0, the relation (3.24) gives the j [following system of differential-difference equations j 1(3-25) dPx(t) = X(x-l)P3r_1(t) - ["X (x) + | J - (x) lp (t) dt X 1 L J X + ( j , (x+l)Px+1 (t); for x c The system (3.25) is to be solved with the initial conditions (3.26) Px (0) = 1 for x = Xq = 0 otherwise. ■ Considering various alternative assumptions about I j the arbitrary functions X (x) and |J.{x)7 Sengupta and Tintner ! (133) have studied a number of probabilistic models of economic growth as follows: (I) Linear birth process type growth model: where i ;the assumptions are made as X (x) = Xx, X > 0 and (x) — 0, i ; that is, the probability of a transition in the interval (t, t+At) is proportional to x, the constant of propor tionality being X. In this case, (3.25) becomes (3.27) dPx (t) ----- = -XxP (t) + X(x-1)Px-1(t) for x s Xq i dt i j : where Xq denotes the initial value of X(t) at time t = 0. j | t The solution of (3.27) is then i 67 j X-xQ (3.28) ^x (t) = x-lcx-xo C1 ~ exp(-Xt)] jwhere Xq is the initial value of X(t) at t = 0. If we i |approprlately choose the units so as to make x^ = 1, (3.28) ! jean be written more simply as r nX-1 (3.29) Px (t) = exp(-Xt)[l - exp(-Xt) for x=l, 2, 3, .. This is obviously a negative-binomial type distribution, called the Yule-Furry distribution, which has the mean and the variance as given by (3.30) m(t) = E{x(t)} = exp(Xt) 1(3-31) v(t) = V^X(t) j" = exp (Xt) ~[exp(Xt) - lj" ! For the general case, where initial value is x , the mean 0 and the variance are (3.32) m(t) = E-jx(t) j- = x^exp(Xt) (3.33) v(t) = vjx(t)j- = Xgexp(Xt) |exp(Xt) - lj- This birth process was first studied by Yule (163). in 1924 in connection with the mathematical theory of evolution, and later a similar model was used by Furry (50) in 1937 |while dealing with cosmic ray phenomena. It is to be noted that the mean growth of the ' ■ \ i iprocess, given by (3.30) or (3.32), follows the exponential! ; f law in contrast to the linear trend in the poisson process.: 68 i J The exponential trend in economic time series has been advocated by many economists. The exponential solution in j | lHarrod-Domar type deterministic growth model is well known, j jThe economic meaning of the proportional growth rate (or j i jbirth rate) in terms of economic models can be interpreted ! by treating 1 as the product of the two structural coe fficients of the Harrod-Domar type growth model, that is, X = orcr where a is the saving coefficient and a denotes the marginal output capital ratio. Writing X=a'<7, the Harrod- Domar type growth model dx (3.34) = Xx j |has the solution j : ! : i (3.35) x(t) = xQexp(Xt) where Xq is the initial value of x(t) at t = 0. This is same as (3.32). (II) Linear birth and death type growth model: here the random variable X(t) of the process is assumed to j : experience both positive as well as negative changes by jumps in time randomly such that the changes are propor tional to the present value of the random variable, that i^, | X (x) and j j . (x) are assumed to be linear functions of x. ; . i Thus, in the growth model of the linear birth and death ' type process, X(x) = Xx and p,(x) = [ i , x , X > 0, > 0, chara- | \ cterize the stochastic distribution of the national income . 69 X(t) over time. The difference between this and the Furry process is the addition in the former of a growth retarding; |effect proportional to the present state of the economy as j i | | specified by y (x) = yx. The economic interpretation of the j !growth rate X is that it reflects the average increase in [ i i ! . j jreal income resulting from an additional dose of real j i capital formation, and similarly the death rate y repre sents an average decrease in real national product resul ting from any decrease in investment caused by the increased size of the total capital stock, or any misdirec tion of investment. Any such retarding effect may be subsumed under the rate y. In this case, the probability distribution of X(t) I I turns out to be the negative-binomial type j (3.36) Px (t) = [l-of(t)][l-p(t) ] [p(t) ]X 1, x=0, 1, 2, .., where (3.37) a (t) = y£exp{(X-y) t]--l j / fxexp{ (X-y) t}-y J j ;(3.38) P(t) - xj^exp^(X-y) tj— lj/ £ Xexp|(X-y) tj~y J , i The mean and the variance of X(t) are easily obtained from (3.36) as (3.39) m(t) = E-|x(t) j " = exp"[(X-y) t| : (3.40) v (t) = vjx(t)]-= exp{(X-y) t]£exp-[(X-y) tj. -lj r ” ~ i i ! We observe that the formulas (3.39) and (3.40) are similar 1 i to those (3-30) and (3-31) except that, here we have net birth rate (X - j j , ) in place of X in the pure linear birth iprocess. If the initial value of X(t) at t=0 is Xq , the I expressions corresponding to (3.32) and (3.33) for the i i jmean and the variance are obtained by multiplying (3.39) i i and (3.40) respectively by Xq . We note that if X = p, the expected rate of growth is zero and the mean national income would be stationary. “ \ A more general model of linear birth and death process has been considered by Tintner and others (133, 151, 152), and by Narayanan (115). In such general model, the birth and death rate functions are specified as j 1(3.41) X (x) = Xq + X-^x , Xq , X-j ^ > 0 i (3.42) p(x) = p0 + p^x , | j. q , m- x > 0 where pQ is assumed as zero. The constants X^ and are interpreted as birth and death rates respectively, and Xq ; is interpreted as a constant immigration rate. i j The probability distribution in such case is j 'obtained as | (3 43) p (t) = rxo+ximtt^ //Xlrxo^o+xi) • • j (Xq+^ix^ : x L * 5 J L X -I I r m(t) 1* i LXq+X.m (tX ' i I I where m(t) = e( X (t>] is the mean of the distribution. i 71 The expression for 'the trend m(t) for this distribution (3.43), however, comes very complicated in the sense that the parameters involved are non-linear functions jWhen suitable change of origin and scale is considered as IY{t) = a + u.X(t) in studying the empirical examples of ! t 'growth in the national output of different countries, Narayanan (115) faced with the difficulty of estimating the parameters being non-linearly involved in the trend (3.44) m(t) = E-|y (t) j- = A + Bexp(Ct) where A = a-uX0/ (X ^ ^ ) , B = uXQ/ (X-^p^) , C = (X^i^) , a^. = a, and u^ = u. The method suggested by Tintner (148) in such cases to estimate the parameters, however, did not yield satisfactory results in his empirical study. (Ill) Non-homogeneous birth and death processes: Kendall (81,82) first studied the time-dependent birth and death processes of a more general nature, and used them in the population growth models. The distinct feature of such processes is that the birth and death rates are assumed as functions of time and not functions of the state variable as discussed above. Thus, X(t) and |Ji(t) are considered in place of X(x) sand p . (x) ; this shows that the secular jvariations take place in birth and death rates over time. This case seems more suitable than the earlier one of i ! i |homogeneous process in the theory of economic evolution j where the structural change takes place in time due to the | changes in productivity and technological innovations/ etc. Kendall (81) has obtained complete solution for the general case/ and showed that the distribution of the random variable X(t) of such process still has the modified ‘ geometric from (3.36) but (3.37) and (3.38) take the form The mean and the variance of X(t) for this non- homogeneous process are given by rnean and the variance of X(t) can be obtained by multi plying the expressions (3.49) and (3.50) by Xq . It is easy to see that in case of homogeneous process/ that is, X(t)=\ and ( j , (t) =y>, the expressions (3.49) and (3.50) simplify to those (3.39) and (3.40) derived above. the transformation Y(t) = at + utX(t) , the mean and the (3.46) p(t) = 1 - 1 / w(t) where (3.49) m (t) (3.50) v (t) = v{x(t)} = exp{-2Y(t) }JQ X(r)f^(T) ]exp-jv (t)}; jWe note that in case X(0) = Xq > 1, the expressions for the! Introducing the two functions at and ut as before by: 73 variance of X(t) are easily obtained as (3.51) m(t) = E{y(t)} = at + utexp{-y(t)} (3.52) v(t) = v{y(t)} = u^J0|[x(t)+ M < (r) ]exp{y (t) | no empirical investigation of economic growth models based I j on such non-homogeneous processes seem to have been made \ because of the difficulties of obtaining efficient estimate due to lack of an adequate method of estimation in such processes. (IV) Nonlinear birth and death type growth models: Sengupta and Tintner (133) as well as Narayanan (115) have iexamined such type of models of economic growth. A genera lized form of a stochastic version of the growth model, j which can give rise to the logistic type trend of economic i growth, can be obtained by introducing non-linearities in the birth and the death rate functions as (3.53) Mx) = a(x2-x) | (3.54) | J . (x) = p (x-x^) | i i ! j ■where a, P are positive constants, and x, x^, x2 are ; i positive integers, x^< such that X(t) at t = 0 lies in’ the interval (x^, x2). These assumptions, however, lead to- the system of second order partial differential equations icorresponding to (3.25) and for which there is no explicit j . i !solution available so far. Therefore, P„(t) is unknown in \ X \ ' - | such case. However, it is possible to obtain a logistic ! 74 analogue by considering the specific type of time functions a - t, ut, X(t) and i x (t) on the lines suggested by Kendall 1(81) and Tintner and Sengupta (151). It has been shown i ! 'first by Feller (40) and later by Kendall (81) that the ! Jmean value m(t) in this case satisfies the following j j i | jdifferential equation j dm(t) (3.55) “ = (ax2 + pxj^Jmtt) - (a+£)M(t) where M(t) is an unknown function representing the second order moment of the process "|x(t) t ^ 0j". However, this does not correspond to the corresponding result in the j ■ deterministic model of logistic growth, showing that the I I ! 'development of a deterministic process and the mean develop- j jment of its stochastic anologue are not always the same. j ! Narayanan (115) has made an attempt to obtain a first approximation to the logistic trend (3.56) b(t) = k/1 + a-exp(-bt) by considering appropriate time functions. He obtains | j (3.57) m(t) = E Y(t) = at + ut[xl + (xQ-x^m^t) + (x2-x0)m2 (t) ] where Y(t) = at + utX(t) , and j (3.58) m1 (t) = [<* + pexp{_(a+p)t}]/(crf-p), ! | i | : (3.59) m2 (t) = a£l - exp|- (a+p) tj-J/(a+0). j i i The trend m(t) is also expressed'as' (3.60) m(t) = at+ ut£ (a'x2+Px1) (< xx2-Xq ) - (Px-^Xq) }e t j ( a r + P ) i jwhich can be easily seen as a first approximation to the \logistic trend L(t) given by (3.56) if we expand L(t) as an j infinite series (3.61) L(t) = K^l-a-exp (-btj +a2exp (~2bt)-a^exp (-3bt) + . . . Further Results and Concluding Remarks Xt will have become apparent from the above brief discussion of various discontinuous stochastic processes and the economic growth models based on them that the j 'method of formulating the general equations for the Markov ■processes is very simple provided the possible 1 transition^' i I 'occurring in a small interval of time (t, t+At) are j suitably listed. Even when the process involves more than one variable, the problem is simple and the results of the univariate processes can be easily extended further by replacing the scalar random variable X(t) by the vector random variable £x^ (t), X2 (t), . . . . , X^(t)J, where the ■ scalar components Xj_ (t) # i = 1, 2, . . • , k represent the various economic variables that jointly characterize the economic conditions or the state of the economy at time t. j j t I In this case the state space ) ( becomes k-dimensional, the j ’ • j states being represented by the points x = (x^, X2 » •• - .,j ■ j x^.). Sengupta and Tintner (133) considered the specific j 76 case of the multiple stochastic processes where the variables are the principal components of the variables |x^(t) and X2 (t) denoting the net outputs of two mutually 1 1 [exclusive sectors of an economy, and formulated a linear |interdependent economic growth model. A similar approach I 1 [has been taken by Narayanan (115) using the multivariate variant of the poisson process and obtained some empirical results for the U. S. economy. A detailed account of such investigations1 would seem somewhat out of place here. However, we shall consider a multivariate stochastic process of diffusion type that would be more appropriate to describe the economic growth process in the chapter V. One: I ! or two points that arise in the discussion of the stochas- !tic models need to be mentioned before we conclude this j chapter. First, all the models discussed so far deal with the' ■discrete variables Which seem to be unrealistic in the i :economic applications. It can be hardly expected that economic variables, like national income, capital stock, et cetera only assume discrete values and the changes in i their values occur only by jumps. A second important point [is that of non-linearity in the stochastic models, namely, !the mean development path of the stochastic process and thej 8 ! For a general theory of multidimensional stochastic i processes, see Arley (5). development path of the deterministic model are not the same. This leads to an inverse problem: given a determi nistic model, what are the conditions under which a corres ponding stochastic model (or its analogue) may yield the | i jmean value that is same as the solution of the deterministic model. Here the correspondence is meant with respect to the first order. There may be several stochastic models with the same mean value, for example, the exponential law results in a number of different cases of stochastic models but it is the solution for a single deterministic model; : while the higher order moments are different for the ; [ : various stochastic models. Thus, in the stochastic models of economic growth, we have two main problems.. One is to I | j jchoose the type of variables appropriate to reflect the : = I nature of economic changes, and the second is to formulate a suitable stochastic model which can explain the growth phenomena adequately in terms of the empirical tests. We :shall investigate these aspects in the next chapter and ; propose a plausible model of economic development in terms j i of continuous stochastic processes. Further, we shall attempt to apply the theoretical models of economic growth based on the theory of diffusion j |processes to a few empirical examples. Also, the models j | ‘ ; based on the theory of discontinuous processes, although j j ! j i j provide the commonly observed trends in the economic j variables and thus can be useful in describing the economic j 78 I i ] growth phenomena, the problem of estimation needs attention.; The difficulties experienced in obtaining efficient I ^estimates of the parameters of the probability distribution ; i of the discontinuous type stochastic process from the i _ jobserved data are considerable and very important. However, i ■investigation into these problems of estimation is out of j i \ place here in view of the purpose of the present chapter of ; providing a brief review of the theory of the economic growth process based on the discontinuous stochastic processes. We shall, however, attempt to over come some of; i the difficulties of estimation in our proposed models in j :the later chapters by using the efficient method of j j estimation, namely, the method of maximum likelihood. Also,! | j :the more important question from the analytical point of view of characterizing the process of economic growth in terms of more than one variable will be investigated in the subsequent analysis. ^See Kendall (83) and Bartlett (9, 10) for details. CHAPTER IV LOGNORMAL DIFFUSION PROCESS OF ECONOMIC DEVELOPMENT j I ! i | Introduction j We discussed in the previous chapter some applica- : i tions of stochastic processes of the discontinuous type to ; economic growth models. The common feature of such models is that the economic variables assume discrete values. This seems somewhat unrealistic especially in an aggregate i (analysis of the economy. Economic development may be j : j discontinuous for an individual firm or an industry but for ! Ithe aggregate population of a large number of economic j i I ! I tunits, firms and consumers, the assumption of continuity j in the economic development seems more reasonable in view of the transition of the economy from one level to another in a small interval of time being certain, although the transition may be relatively small. We therefore propose iin this chapter a continuous stochastic process of the ! i ' j diffusion type so as to reflect the above characteristics of the economic development process. The likelihood of such a characterization of economic development process can be seen to be suggested in the following passage of Rostow i ; I(127, p. 46)s Perhaps the most important thing to be said about the behavior of these variables in historicalT"cases of take-off is that they have assumed many different forms. . . . The rate and productivity of investment can rise, and the consequences of this rise can be j diffused into a self-reinforcing general growth process ; by many-different technical and economic routes, under | the aegis of many different political, social and ! cultural settings, driven along by a wide variety of | human motivations. j I In the simplest aggregative theory, consider the [ i . ! ! assumption that economic development is measured by a j ■ ! single continuous variable X(t) which may denote real j national income at time t. Our first concern is to discuss the way in which economic development proceeds, that is, to characterize the stochastic process jx(t), t s o]-. The choice of a particular process, that is, the form of the transition probability distribution function of X(t), depends on the extent to which the underlying statistical I distribution of X(t) can be derived from realistic ele- j mentary assumptions regarding economic changes and on the degree to which empirical observations agree with the theoretical model. So far as the latter point is concerned, it is not difficult to note that in economic data, posi tively skewed distributions fit as a rule. The use of the j lognormal or Pareto distribution, Mandelbrot (104, 105), Pareto (121), Davis (29), rather than the normal distribu tion has been suggested by a number of investigations which seem to point out its usefulness as a convenient apprOxiamation in a number of fields of economics, j Aitchison and Brown (1), Bowley (19), Fama (38), Granger | ! and Morgenstern (57), Gibrat (55), Houthakker (74), Kalecki (77), Lydall (100), Orcutt, et al. (118), Prais and Houthaker (124). We will therefore attempt here to investigate the possibility of X(t) following lognormal probability law. ! Although the success with which the lognormal I probability law graduates empirical data in economics is a sufficient criterion of its worth, it is important to have a plausible probability theory to yield such a frequency distribution for X(t). It is the more fundamental basis ! | j in the theory of probability that may help us to obtain a clear insight into the underlying mechanism of the economic growth process, and in turn often suggest a wider applica- i i I : t i tion of such a theory. Furthermore, the knowledge of the i ! _ | elementary assumptions on which such a probabilistic theory; can be based would enable us more easily to modify the probability law to meet new conditions. We therefore spend isome time in discussing the way in which the continuous !stochastic process {x(t), t £ o}, characterizing the state : ; i 'of the economy at time t, can lead to a lognormal distribu-; tion of the variate X(t). Genesis of The Lognormal Distribution Since the first introduction of the lognormal dis tribution by Galton(52) and McAlister (101) as a distribu- tion arising from a theory of elementary errors combined by a multiplicative process, many theories have been advanced based on the fundamental proposition of Galton that in many situations in nature the process of underlying; change or growth is multiplicative rather than additive- j The method of translation treated by Kapteyn (79), Wicksellj i j (161), Edgeworth (18), and Galton-McAlister (52/ 101), and i imore recently by Johnson (76) and Draper (36); the theory of proportionate effect discussed by Gibrat (55) and its variants considered by Kalecki (77) and Champernowne (22); and the theory of breakage put forward by Kolmogorov all in essence aim to establish the conditions under which a i i variate defined as a product of a number of elementary variates tends to be lognormally distributed. j I i The argument advanced by Kapteyn (79) and Wicksell j i I (161) runs as follows. Consider a positive random variable! representing the size of some organ and assume that the variate is the outcome of a joint effect of a large number of independent causes acting in ordered sequence during the; time of growth of the organ. Let the variate X initially have the value X^ and that after the jth step in the i process it is Xj/ reaching its final value Xn after n steps. At the jth step the change in the variate is then assumed as a random proportion of a function 4>(Xj_-^) of the value xj_i already attained; thus ( 4 . 1 ) X j - X . ^ = e ;. . c i ) ( X . _ 1 ) j where the set "[ejj" is mutually independent. The change at 8 3 any step is therefore not independent of the value attained previously except for the case 4>(X) =1- Thus the process is conceived to be a discrete Markov type process of one- step dependence. In the special case <£{X) = X, that is, where the change in the variate is a random proportion of the pre viously attained value of the variate rather some more complicated function of it, the law of proportional effect reduces (4.1) to (4.2) X. - X. i = e. .X. , 3 j j-l which can be also written as ! (4.3) (X - X )/ X. =6. i 3 3 1 D - l 3 so that (4.4) V |(X - X )/ X. 1 = y e Z. L 3 3-1 J-U / 3 j=l 3=1 Suppose the magnitude e^ of the effect at each step very ;small, ! n Xn (4.5) y Xjrl] « ^ « - logXn - logxo whence log Xn= log XQ +e^+e^+. . . + ^ is asymptoti cally normally distributed by the central limit theorem; ;thus X^ is lognormally distributed for large n.^ Thus, by ■ l°The natural or Naperian logarithm will be used j throughout the present study. Hence no specific notation is^ here used to indicate that the "log" is to the base e. 84 ; I assuming the random increments proportional to the | variables to which they apply, the distribution of the variate is obtained as a lognormal. Although the working of the above model (4.2) of the j i i i I proportionate law assumes an ordered sequence of events in j ! i ■time, it is possible to interpret the model without such an assumption, as pointed out by Aitchison and Brown (1, p. 25).! One may suppose that at any point of time, the distribution of the variate X(t) arises from a large number of effects j acting simultaneously. The outcome of such a large number of different effects acting in accordance with the model (4.2) is then to produce a lognormal distribution for X(t). j At other points of time the distribution of X(r) may be { i • ! regarded as similarly produced, again leading to a j lognormal law, but with possibly different parameters. It is this approach that appears more likely in our analysis, :as we shall see below. 1 A similar approach on the lines of Kapteyn, but i ■ ; ■ 1 ;assuming continuity instead of discreteness, has been taken I by Kolmogorov (93) who advanced the so called breakage theory to explain the occurrence of lognormal distribution. The breakage theory is equivalent to the formal theory of iproportionate effect except it is in terms of distribution j functions rather than variate values. Here a set of ! elements are considered where each element has an associa- j ted positive measure, the dimension or magnitude of the element., and the distribution of elements with dimension j i^ x, denoted by F(x), is studied under the growth of the elements by a proportionate effect at each breakage opera- Ition. Then the distribution is shown to follow the ! i |lognormal law for a large number of breakage operations on i the elements. j If we then regard the economy as composed of a large; number of income generating economic activities, and con sider the 'income' or 'value added to a product or a service' as the associated measure of the size of the : I economic activity, the growth process of the economy can be| i characterized to proceed in a manner similar to the j : j breakage theory of Kolmogorov or, equivalently, it can be j | . j 'explained by the interpretation of the law of proportionatej change under a joint impact of a large number of effects working simultaneously. The assumption of the hypothesis of proportionate effect viewed from a joint effect in the I economy of a multitude of repercursions due to an economic : i activity, say of investment, analogus to the breakage operation seems more reasonable in economic process. It is; not uncommon to hold that a change in the level of economic activities as measured in the sense of 'value added* would,; I in all probability, be proportional to its existing level, j : i The law of proportionate change has more to recommend in i ;economics, especially in the process of income generation, j if it is not incorrect to believe that it is as easy for a : 8 6 rich person to increase income by 100 dollars from his present income of 10,000 dollars as for a poor person to i I jraise income by 10 dollars from his present income of 1,000 ! jdollars. In other words, an income change of ten dollars Ifor a 1,000 dollar income has equal probability to an i i jincome change of 100 dollars for a 10,000 dollar income, ivining (159, 160) suggested the lognormal curve for the distribution of the changes of income payments by the individual States in America. Another common hypothesis considered in economics which supports the law of propor tionate effect is that the price change of one dollar for a jten dollar stock has probability equal to a price change of Iten dollars when that stock is selling at 100 dollars. j i • i ! ! Such an assumption for a price change is found frequently j in the literature, Davies (28), Fama (38), Kendall (86), Laurent (96, 97), Mandelbrot (106), Moqre (107). i iLognormal Diffusion Process of Economic Growth Having discussed the plausible nature of the develop-! : l ment of national income, we now characterize the stochastic: process {x(t) , t ^ 0 j" by the law of proportional change and derive the transition probability function of the process as follows. i * i | Let X(t), the national income at time t, be the | random variable of a continuous stochastic process of a \ \ i 'Markovian type defined on the interval (0, Qf the real line, that is# X(t) is a Markovian random variable depending on a continuous time parameter t, and it assumes i f | values in the state space 3 f = 0 < x < “. j | Let the transition probability density for X(t) | (4.6) f (T,x;t,y) = p|jX(t)=y / X(r)=xJ, 0 < x,y < <» j exist for every r and t, 0 ^ r < t, satisfying the backward and forward Kolmogorov equations. Assume that the random variable is continuous with probability one. J Consider now the coefficients b(t/x) and a(t,x), the! \ I ■ infinitesimal mean and variance of the change AX(t) in the j I ;variable X(t) during a small interval At of time t, which j |characterize a particular process of the diffusion type as i j 1 i ' (4.7) e[~AX(t)j = b(t#x)=btx, and vfAX(t)] = a(t,x)=atx2> 0! where bt and at > 0 are functions of time. Consider a simple case of bt=b and at=a/ where a, b are constants, a > 0. We thus assume the expected change and its standard ; j ;deviation in national income as proportional to the instantaneous national income. Assuming a(t,x) > 0 implies1 that some change in the national income takes place in any interval At of time, and that such a change is small for small At. This is the basic assumption about the mechanisml i of income growth, which seems to describe fairly well the j plausible characteristics of the development of the economy] 88 With the coefficients as specified above, the backward and forward Kolmogorov equations for the diffusion process {x(t) , t S : o]' are (4.8) - |f = + bx |f (4.9) ~ = + (2a-b)y + (a-b)f dt dy^ Sy The probability density function satisfying these diffusion equations (4.8) and (4.9) is then given by the lognormal density function — 1 (4.10) f ( t , x ; t,y) = y-1j^2JTY (t-T)J expj-^2v (t-r) J Tlogy-logx-8(t-r) } | ! where y = a and 3 = (b-a/2). i The characteristics of this distribution are easily derived from its moments given by (4.11) E[x(t) ]k= [x(T)]k exp[k(3+ky/2) (t-T) ], k=l,2, . . .j j From (4.11), we have the mean and the variance of X(t) as j (4.12) E^X(t)] - X(r) exp[(3+y/2) (t—r)] = a(t), say, (4.13) v[x(t)] =£x (t)] expj^2 (P+y/2) (t-T)]{exp^v{t-T)] - l} 2 2 (t)j [\(t)] = v(t), say I where £_7] (t)] = exp^a(t-r)] - 1. We note that T | (t) is the j i coefficient of variation of the distribution. The two ! 89 measures of departure from normality, namely, the ^coefficient of skewness Yi(t) and the coefficient of jkurtosis Y2 (t) are given by (4.14) Yl(t) = [71(t)]3 + 371 (t) ; (4.15) y2(t) = [Tl(t)]8 + 6[Tl(t)]6 + 1s[t] (t)]4 + lejj (t)f It is easily seen from (4.14) and (4.15) that the distri bution is positively skewed and it has positive kurtosis. The value of T ) (t) would depend on y an< 3 . if the latter is ; small, which would often be the case, the distribution would not be very different from the normal in shape. I The median and mode of the distribution, X(t) and |X(t) respectively, are given by (4.16) X(t) = X(r) exp{p (t-r) ] • I (4.17) X(t) = X(T)exp{(3-Y) (t-T)} I | showing their relative positions with the mean as 5 i | (4.18) E[x(t)| > X(t) > X(t) which again emphasizes the positive skewness of the ; distribution. If we consider the initial time as zero, that is, j IT = 0, which can be done by a scale transformation of time, |and assume that p[x (o) = XqJ = 1, then we have the mean aft) and the variance v(t) of X{t) as j (4.19) o? (t) = E^X(t)!1 = XQexp(bt) j ; (4.20) v(t) = v£x(t)j = x^exp (2bt) ["exp (at) - lj j j i jwhich are both expenential functions of time t. The trend j i : j , I !given by (4.19) shows the exponential growth in national j income, which appears to be a good first approximation to i the empirical data. The dynamic path of the economy as icharacterized by the national income variable X(t) can be |evaluated by (4.19) using the initial condition, namely, ! i the value of X(t) at t = 0, and .the value of the parameter | ;b. If the value of b is unknown, its estimate obtained j j | j from empirical data can be used for evaluating the future j I course of development of X(t). j ! . t An undesirable feature, however, observed in characterizing the growth of national income as a time ;process involving the transition probabilities as con- j ! :sidered above, is that the variance of the income distribu-i i i | tion increases as the process continues as shown by ( 4 . 2 0 ) . j This seems, apparently, contradicting the empirical i |evidence. Of course, this weakness of considering the .economic growth process as a lognormal diffusion process |may not be real as it appears. Probably it may be a I I jgenuine underlying tendency but frustated by counteracting j i i ipolicies of governments and of the parties or economic ! \ units involved in income determination so that the variance I 91 is constant. Under constant corrective measures of exogen ous agents to stabilize fluctuations in income and secular ! / jchanges in the behavior of the endogenous units, the |economy may not reflect the true nature of the increasing i variance in the empirical data. Some evidence in support of the above argument is I !observed in the analysis of income distribution of professionals made by Aitchison and Brown (1, pp. 108 f.) (and in the analysis of worker's earnings made by ROy (128, i 129). Further, the variances in the national income data Ifor different time periods in case of the U. S., Canada, ;India, and a few other countries showed significant results j ;supporting the thesis of increasing variance in the income |as time passes. It appears, therefore, that our model of i j ! income growth based on the lognormal diffusion process is a plausible one. |Estimation of the Parameters of the Model ! j We now derive the estimates of the parameters of the \ i ;lognormal diffusion process. We mentioned in the earlier ; | chapters the unsatisfactory approach of least squares method of estimation which is usually adopted in the literature. We try, therefore, the best available method I ! |of estimation, namely, the method of maximum Likelihood. j I We may note that we apply the method to the transformed j I data in logarithms, and then obtain the estimates of the j 92 parameters of the original variate. This is perfectly j legitimate in view of the property of maximum likelihood j estimates of being invariant under monotonic transformation.! Let Xq# x^_, . . . , xn be the observed states of | i X (t) at times tQ, t^, . . . , t respectively/ and let j |p[x(0) =xgj = 1. The joint probability of the observations, j that is, the likelihood of the observed data under the assumed model is (4 -n/2 n. -1 ~h r 1 71 21) L = (2ttY) ^ (t^-t^i) exp[-~(fe— t ^ ) j {logx^-logx^-Ud^-t )} ] | i and the logarithm of the likelihood is I ; i i n n r - l j : (4.22) log L = C-T logx^-Ss 7 log ~ V [(t -t„_i) ! 0?-l . Qf=l - ^ " O'-! „ i |The maximum likelihood estimates are then obtained by |solving the following equations: ! <4*23) = ^ [ lo9V - lo9xa-l- I V V l 11 = 0 j < 4 ' 2 4 > ^ 7 = [ < t c > ' - t « - l > 1 { l o 3 x a - 1 ° 9 x g - l - ? ( V t g - l ) } ] j I I dy ■ , Qf=l 1 ! i = 0 t ; i ]The estimates obtained on solving (4.23) and (4.24) are as j follows: i 93 n n (4.25) % = £ (logx^-logx^) / £ (t^-t^) 0 1= 1 Q?=l (4 '26) ^ " I ' | l o 5 ;xQ '-l o 5 x Q f - i “ P J ] / n Qf=l Hence the maximum likelihood estimates for a and b are (4.27) a = y and £ = (p+y/2) !The dispersion matrix of the estimates (P, y) i-s given by the inverse of the information matrix I: (t-r) 0 ! (4.28) I = n 0 1/2 y jHence, |(4.29) V(P) = y/n(t-T) ! . i(4.30) V(y) = 2y2/n i i i r | (4.31) C<£, y) = 0 |We assume that the observations are taken at equal inter- i jvals of time, say one year, that is, ttQ'-to'-l^ = 0 1 = 12, . . , , n, and let tQ= 0, then the estimates and their I sampling variances are n (4.32) p = £ (lo9xa- lo^xa_i) / n oi=l n ^ V I ^ o (4.33) y = ) (logxQ,- logx^-p) / n cx=l - | (4.34) V ((3) = y/nt — > 0 for large n and t From these, the estimates, of b and a and their sampling variances are easily obtained as (4.36) a = y and b = (§ + y/2) I(4.37) V(a) = 2y2/n and V(b) = (y2/2n) + y/nt We note that the variances (4.34), (4.35) and (4.37) of the estimates tend to zero for large n and t, and hence the estimates (§, y) and (b, a) are consistent estimates. Using the property of asymptotic normality, an approximate confidence interval for X(t) can be obtained using (4.19) and (4.20) as (4.38) p{|x(t) - E[x(t)]| £ tc / yvj^X(t) ] } s (1 - a) where (1-a) is the confidence coefficient and t^ is the lOOcv per cent point of Student's t distribution with (n-2) degrees of freedom. Thus the confidence limits are (4.39) X^(t) = E[x(t)] - ta J v[x(t) ] | " - | (4.40) X^j (t) = E^X(t)J + ta.-V Vj'X(t) J | I where ^(t) and Xy(t) denote the lower and upper limits respectively. I i Empirical Results I ! i We now present empirical results of fitting the j j lognormal diffusion process to the national income data of j 95 i some -underdeveloped as well as developed countries. The behavior of the national income X(t) is examined using the simple diffusion process of proportional growth for the U. S., U. K., Germany, Canada, Netherlands, Japan, India, and Ecuador for which reasonably good data are available for sufficiently long period of time.'*'-*' To have a rough I idea about the time behavior of the estimates of the i structure of the economic growth process, we have used time I | I i series corresponding to different time periods wherever ;possible. i I Consider first the results obtained for the recent data for the U. S. (1946-1963), Canada (1948-1960), Netherlands (1948-1960), India (1948-1963), and Ecuador (1950-1962). Here X(t) represents the real national ■product (gross) of the U. S. in billions of dollars, and J of Ecuador in millions of sucres, in the year tj while X(t) denotes the real national income of India in billions of rupees in the year t. For Canada and Netherlands, X(t) ! f ! denotes the index of real national product (gross) with j ! 1913 = 100 base. We assume that X(t) takes the observed j ; I value in the initial year in each case with probability onei The maximum likelihood estimates and their vari- | ! | I ances are observed as presented in the Table I. i | Based on the estimates given in the Table I, the j . ! ■ . i ; ^See the Appendices A and B for the data and their sources. j TABLE I ESTIMATES OP p, y; b, a AM) THEIR VARIANCES FOR THE LOGNORMAL DIFFUSION PROCESS APPLIED TO THE NATIONAL INCOME DATA OF THE U. S. (1946-63), NETHERLANDS (1948-60), CANADA (1948-60), INDIA (1948-63), AND ECUADOR (1950-62) Country ? v(B) y = a V(y)=V(a) V(b) u. S. .0327 l54(.544)/t lO3(•9255) 156(.101) .0332 107{.252)+104 (.544)/t Netherlands .0493 104(.56l)/t 103(.6728) 10?(.754) ,0496 107(.189)+104 (.561)/t ! Canada .0397 103(.l23)/t 102 (.1482) 106(.365) ,0405 107(.913)+103(.123)/t India .0317 104 (.333)/t 103 (.4995) 10?(.332) ,0319 108(.831)+104 (.333)/t Ecuador .0517 104 (.466)/t 103(.559l) 10?(.520) ,0519 107(.130)+104 (.466)/t expressions for the trend and variance are as shown in Table II. The fitted trend values are given in Tables I I | through V of Appendix A. The positive values of the estimates 0.0332, 0.0405, 0.0496, 0.0319, 0.0519 of b for the five economies during the last two decades show that the real national product is igrowing in each case at the rate of 3.32, 4.05, 4.96, 3.19, I and 5.19 per cent respectively. Based on these estimated j igrowth rates, we tried to predict the values of the ! |national income for the period 1965-1970 and also tried to | ■obtain the 95 per cent confidence limits. The results of i i jthe prediction and the confidence limits based on our j ! ! I -theory are given in Tables III through VII. | Looking to the results obtained, it appears that the ; | lognormal diffusion process is a fairly good first approx imation to the process of economic growth. To investigate I |further the plausibility of the lognormal diffusion process| [ t jas the inherent characteristic or mechanism of the process i !of economic development, we applied our model to the long- i j i !term economic development of the following countries: i I ; U. S., U. K., Germany, Canada, Netherlands, India, Japan, ;for which reasonably good data on national product are I {available for long periods of time.-^ As before, X(t) i i | |denotes the real gross national product (domestic product j 12 See Appendix A for the data and their sources. TABLE II THE TREND AND THE VARIANCE OF THE NATIONAL INCOME OF U. S. (1946-63)/ NETHERLANDS (1948-60), CANADA (1948-60), INDIA (1948-63), AND ECUADOR (1950-62) Country Trend: E^X(t)] Variance: v[x(t)] U. S. (282.5)exp' (.0332) t} 2 (282.5) exp (.0664)t} exp' {lO3(.9255)t} - 1 ] Netherlands (194.0)exp- (.0496) t j : 2 (194.0) exp (.0992)t} exp-{l03(.6728)t} - 1] Canada (246.7)exp- (. 0405) tj- 2 (246.7) exp (.0810)t} exp[l02(.1481)t} - x] : India (86.5) exp- (.0319) t} 2 (86.5) exp (.0638)t} exp-{l03(.4995)t} -i] : Ecuador (8089) exp- (.0519) tj- (8089)2exp (. 1034) tj- exp[l03(.5591)t]-. . - 1 ] VC 00 TABLE XII 99 ESTIMATES AND 95 % CONFIDENCE LIMITS OF THE REAL NATIONAL PRODUCT OF U. S. FOR 1965-1970 i < i Year t Trend 95 % confidence limits e[_x (t) J Lower Xj^t) Upper Xq (t) 1965 19 530.90 381.70 680.10 ;1966 20 548.83 390.55 707.11 1967 21 567.36 399.65 735.07 1968 22 586.52 409*03 764.01 1969 23 606.02 418.67 793.97 1970 24 626.79 428.58 824.99 TABLE IV ESTIMATES AND NATIONAL 95 % CONFIDENCE LIMITS OF PRODUCT OF CANADA FOR 1965- THE REAL 1970 Year t Trend 95 % confidence limits s[x(t)] Lower XL (t) Upper X^j(t) 1965 17 491.05 320.21 661.88 1966 18 511.34 328.22 694.46 1967 19 532.47 336.48 728.46 1968 20 554.47 345.01 763.93 1969 21 577.38 353.79 800.97 1970 22 601.24 362.84 839.64 100 TABLE V ESTIMATES AND 95 % CONFIDENCE LIMITS OF THE REAL NATIONAL PRODUCT OF NETHERLANDS FOR 1965-1970 Trend 95% confidence limits Year t E[x(t)] Lower X-^ft) Upper Xu (t) j 1965 17 451.21 345.76 556.67 1966 18 474.18 360.13 588.23 1967 19 498.32 375.16 621.48 1968 20 523.69 380.87 656.51 1969 21 550.35 407.30 693.40 1970 22 578.36 424.46 732.25 TABLE VI ESTIMATES AND NATIONAL 95 % CONFIDENCE LIMITS OF THE REAL INCOME OF INDIA FOR 1965-1970 Year t Trend s[x(t)] 95% confidence limits j Lower XL^ft) Upper X^j(t) 1965-66 17 148.83 119.54 178.12 i i 1966-67 18 153.65 122.53 184.77 1967-68 19 158.64 125.63 1 191.68 1968-69 20 163.78 128.81 i 198.75 1969-70 21 169.09 132.08 206.09 1970-71 22 174.58 135.47 213.69 101 TABLE VII ESTIMATES AND 95 % CONFIDENCE LIMITS OF THE REAL NATIONAL PRODUCT OF ECUADOR FOR 1965-1970 Trend 95% confidence limits Year t E[x(t)] Lower X^(t) Upper Xyft) 1965 15 17559.57 14048.12 21071.02 1966 16 18490.78 14671.31 22310.25 1967 17 19471.37 15324.98 23617.76 1968 18 20503.96 16010.47 24997.45 1969 3.9 21591.31 16729.18 26453.44 1970 20 22736.32 17482.60 27990.03 in case of Japan) in the year t for each of the countries considered except for the Germany, U. K., and India where it denotes net national product and national income. The imaximum likelihood estimates and their variances in these j : I leases are obtained as shown in the Table VIII. i | The trend and the variance of X(t) based on the i . i I estimates of Table VIII are as shown in Table IX. The i l ' ■ 1 fitted trend values are given in Tables VI through XII of I Appendix A. i i We observe that except, for the U. S. and Japan, i j the growth rate of national product during the early i jtwentieth century is around two per cent. Both the U. S. J i :and Japan show the comparatively higher growth rates of i TABLE VIII ESTIMATES OF ,g, yj b, a AND THEIR VARIANCES FOR THE LOGNORMAL DIFFUSION PROCESS APPLIED TO THE NATIONAL PRODUCT DATA OF U. S. FOR 1900-1957, U. K. FOR 1880-1940, NETHERLANDS FOR 1920-1938, GERMANY FOR 1880-1913, INDIA FOR 1900-1958, CANADA FOR 1920-1938, JAPAN 1900-1938 Country A P v(p) A A y - a V(y) A b v(b) U. S. .0319 -4 10 (.698)/t -2 10 (.3977) -6 10 (.554) .0339 -6 -4 10 (.139)+10 (.698)/t :U. K. .0207 I04 (.291)/t -2 10 (.1732) -7 10 (.999) .0215 -7 -4 10 (.499)+10 (. 291)/t Netherlands .0207 -4 10 (.682)/t -2 10 (.1247) -6 10 (.172) .0214 -7 -4 10 (.432)+10 (,682)/t Germany .0241 104 (.787)/t 102 (.2596) -6 10 (.408) .0254 —6 —4 10 (.102)+10 (.787)/t India .0144 104 (.559)/t 102 (.3244) g 10 (.362) .0160 -7 -4 10 (.891)+10 (.559)/t Canada .0173 -3 10 (.304)/t -2 10 (.5470) -5 10 (.332) .0200 -6 -3 10 (.831)+10 (.304)/t J apan .0361 -4 10 (.132)/t -2 10 (.7656) -5 10 (.202) .0399 -6 -3 10 (.496)+10 (.132)/t TABLE IX THE TREND AND THE VARIANCE OP THE NATIONAL PRODUCT OF U. S. (1900-1957), U. K. (1880-1940), GERMANY (1880-1913), NETHERLANDS (1920-1938), CANADA (1920-1938), INDIA (1900-1958), JAPAN (1900-1938) Country Trend: E^X(t)] Variance: v[x(t)] U. S. (38.2)exp (.0339)t} (38.2)2exp{(.0678)t} exp{l02(.3977)t} - l] U. K. (102.8)exp (. 0215) t j - (102.8)2exp{(.0430)t} exp{l02(.1732)t} - l] Germany (219.4)exp (.0254)t} (219.4)2exp{(.0508)t} exp{l52(.2596)tj - l] Netherlands (117.2) exp (.0214)t} (117.2)2exp{(.0428)t} f -2 1 1 expjlO (.1247)tj - lj Canada (103.7)exp (.0200)tj (103.7)2exp{(.0400)t} exp{lO2(.5470)t} - l] India (51.1) exp (.0160) t} 2 r i (51.1) exp|(.0320)tj r -2 1 1 exp\10 (.3244)tj - lj Japan (26.3) exp (.0399)t} (26.3)2exp{(.0798)t} exp^lO2(.7656)tj - lj 103 104 three and four percent respectively. The reasonably good fit of the lognormal diffusion to the long-run economic development of such a wide variety of economies suggests and supports our theory of economic growth as a Markov process following a law of proportionate effect. i Further, it is interesting to note that in all the | : I A | cases, except the U. S., the estimate 0 increases in time A while the estimate y appears to decrease. This suggests the stabilizing tendency of economic growth. Such a result| i iis a further verification of the inference drawn by Solow ! ; i ; i | (138) that the rate of growth in the economy adjusts itself j j |to any given rate of population growth and eventually j ] ! ' ! reaches a steady proportional expansion. Such a behavior is the characteristic of the well known logistic or 'Gompertz curve which has been advanced as the true trend of i economic development in the long-run in many investigations, Klein (91), Tinbergen (146), Tintner (147). If we consider! ’ . 1 I the logistic trend as the valid theory of long-run economic ■ development, the relevance of our results is very clear in ;so as far they show the so called phases or stages of j :economic development, namely the low growth rate in the j initial stage of development, then a faster rate in the ; j _ ! |second stage and then a steady rate. It might be that the t I :basic mechanism of the economic growth process is exponen- i ; tial but it is influenced by some exogenous factors in time i i 'so as to produce varying growth phases. We will attempt ' 1 0 5 below to investigate possible influences of government |expenditure taken as an exogenous economic variable. Such an analysis may even help in formulating some policy pro posals for acceleration or stabilization of economic growth. Influence of Exogenous Factors sider the parameters of the process as functions of such i variable. ! Let us assume a linear effect of government expendi ture G(t) on Pt as j One way the influence of exogenous factor on economic growth can be considered in our theory is to con- j j (4.41) 0t = pQ + PxG(t) j where Pq , p^ are constants and G(t) denotes the government j expenditure at time t. We then have: 1 t t (4.42) r T ,t where t t say. With the infinitesimal mean and the variance of |change in the national income variate X(t) defined as | (4.43) b(t/ x) = btx and a(t, x) = af cx2 where at = aQ > 0 and b^ = bQ+ b^G(t), the transition | 106 1 probability density function of X{t) satisfying the jKolmogorov diffusion equations is i l (4.44) f(T,x?t,y) = y {^iry (t-r)} exp{_-^2y (t-r) ] £logy-logx-P0 (t-rJ-p^Gtt)] J } i 'where y=a0, p0= ( bO~ao/2) ' and ^l=:bl* The maximum likelihood estimates of y, Pq, and P]_ obtained by maximizing the likelihood of the sample values I of X(t) at (n+1) instants of time ta, a=0t 1, 2, . . n jare as follows: i (4.45) (4.46) i (4.47) i j j ;Let the observations be assumed to be taken at equal n t n P o= V {{logX a-logx0,_1)-?1[G(t)] “ }/ T f V V i 01=1 Qf-J- Q-=l n Y' {(1°gxa-1°gxa_i)[G(t)]t" (ta-t^x)1} q/=1 a-i. -^{(logxa,-1°gXQ,_1)}j’ {[G (t) ]^“_i}/ £ < W : Pl= oi=l o!— L a=l ar-1 a=l Q f- 1 .. . . f f - l (W i v] x I ‘V W 2 n 107 intervals of time, say unity in suitable units, that is, (t —tQ(_^) = 1 for a = 1, 2, 3, . . ., n. If it is further assumed that t = 0 and G(t) a step function 0 (4.48) G(t) = G^ for tff_i < t ^ t^, a = 1, 2, 3, so that ,n 'a (4.49) r G (t) ] “ = G(t) L J t a - 1 J t c v - 1 . dt = G.(t -t _) = G„ t^ or a-1 ® In that case, our estimates are n n |(4.50) PQ={ ^(logxa-logxa_1) - £(Ga)}/n j a=l of=l (4.51) Y {(Ga) (logx^-logx^)}- ^(logx^logx^) Y (G^/n Ck'-l- - Q pb 0^=1 }2/ n ct=l or= 1 n a « 2 (4.52) y = y(logxQ,-logxa_1 - P0 - P ^ ) / n a=l ' ! j I |The dispersion matrix of the estimates (p0, j^, y) is givenj i ' jby the inverse of the information matrix I, ! (4.53) I _ n § Ga «si “ n 2 (S Ga) V t 0 a=l 1/2V Therefore, (4.54) V((L) = y/nt 108 (4.55) V(Pq) - yt/n( g G )2 _ ar-l “ (4.56) V(y) = 2Y2/n Hence, the estimates of a^, bg, and b^ and their variances are obtained as (4.57) bQ = (Pq+y/2) / V(£0) =(Y/nt) + y2/2n (4.58) , V(bL) - Yt/n(JLGa)2 | ( 4 . 5 9 ) aQ = y / V(aQ) = 2Y2/n I ~ A A A |We observe that the variances of the estimates (P()#Pi#y) A x A and (bQ,b^,aQ) tend to zero for large n and t when Gq, is |constant or increasing. Thus the estimates are consistent estimates. The probability density function of X(t) with tQ=0 and P-fx(0)=xrtT = 1 with (t -t n) = 1 for all a is i UJ a of—J L -h l ( 4 . 6 0 ) f(xQ?tfx) = x1|27TY.j- exp-j- (2ytJ | 2 | [logx-logXo-Pot-Pj^^Sj^G,,,] } I The moments of the distribution (4.60) are given by ! i I ( 4 . 6 1 ) E^X(t)] = [ x ( 0 ) Jexpjk |(P0+Y/2)t + i “ 1/ 21 * # » • Hence, the mean and the variance are ( 4 . 6 1 ) s[x(t)] = X(0)exp(bQt + h± j l ^ ) ( 4 . 6 2 ) v[x(t)J = [ x ( 0 ) ] e x p ^ ^ Q t + b ^ S ^ ) j.|exp(aQt) - lj- 109 Using the expressions (4.62) and (4.63) for the mean and the variance of X(t), we can obtain the confidence limits for a confidence coefficient (1-a)100 percent as (4.64) XL (t)=E[x(t) ]-ta7v[x(t) ], Xu (t)=E[x(t) ]+tc^/v[x(t7] where t^ is the (100) o r per\ cent point of the t distribution I i 'with (n-2) degrees of freedom. j If we take G(t)=0 for all t, that is, assume no influence of government expenditure, we note that all of the above, results reduce to those of the preceding case, as jit should be, if we consider (3^=0 in the above results. j Empirical Results II I ' We now apply the above variant of the lognormal diffusion process model of economic development to the empirical data of national product and government expendi ture of the U. S. for 1946-1963 and India for 1 9 4 8 -1 9 6 2 . I Here X(t) represents real national product (or income) in I I i the year t measured in billions of dollars at constant jprices of 1954 for the U. S. and in billions of rupees at :constant prices of 1948-49 for India; and G(t) denotes the government expenditure during the year t measured in the : same units as X(t) but at current prices for the two i j economies. Further, we assume in each case that X(t) ; takes the observed value in the initial year with 13 See Appendix A for the data and their sources. 110 probability one. The results of our empirical analysis are given below. The maximum likelihood estimates of the parameters of the process and their variances are found as shown in the following Tables X and XI. TABLE X ESTIMATES OF THE PARAMETERS OF THE MODIFIED LOGNORMAL DIFFUSION.PROCESS WITH EXOGENOUS INFLUENCE OF GOVERNMENT EXPENDITURE APPLIED TO THE NATIONAL PRODUCT.DATA OF THE U. S. (1946-1963) AND INDIA (1948-1962) Country a A A Pi = b-L A A y = a0 A *>0 U. S. 0.02714 104(0.7929) -3 IS3 (0.9241) -3" 0.0276 India 0.01963 10 (0.8763) 10 (0.5338) 0.0199 The expressions for the mean and variance of X(t) j ;are shown in Table XII. The fitted trend values are shown ■ !in Table XIII and XIV of Appendix A. j i -4 ! i We conclude from the positive values (0.793)10 and i -3 * j ;(0.876)10 of b-^ that there is an influence of government ' 'expenditure G(t) on the growth of economy in each of the ;two cases. However, we note that the influence of j !government expenditure is negligible for U. S. case, as mayj \ I !be expected for such a private free-enterprise orianted j j economy. The effect of government expenditure is sub- i 1 . istantial in the case of India, thus providing further TABLE XI VARIANCES OF THE ESTIMATES PARAMETERS OF THE MODIFIED LOGNORMAL DIFFUSION PROCESS WITH EXOGENOUS INFLUENCE OF GOVERNMENT.EXPENDITURE APPLIED TO THE NATIONAL PRODUCT DATA OF U. S. (1946-63) AND INDIA (1948-62) ' Country V(Po) V(Bi) = V(bx) V(y) = V(ao) V(b0) -4 -4 t 2 U. S. 10 (,5435)/t 10 (.5435)t/( S.G ) Q, =l a India 104 (.4106)/t IQ4 (.4106) t/( 1 ^ ) 2 -6 -4 -7 10 (.1004) 10 (,5435)/t + 10 (.2511) 10? (.4384) 104 (.4106)/t + 10?(.1096) TABLE XII THE TREND AND THE VARIANCE OF THE NATIONAL PRODUCT OF U. S. (1946-63) AND. OF THE NATIONAL INCOME OF.INDIA (1948-62) Country Trend: E[x(t)] Variance : v[x(t)3 U. S. 282.5exp[(.0276) t+104 (.793)^1^] 282. 5lxp[(. 0552) t+103 (. 1586)2 G^j(expp)? .924) t]-l) p —3 2 -2 f c ” ] —3 India 86.5expj(.0199) t+10 (.876)2^J 86.5expj(.0398) t+10 (. 1752)E Gj(exp|0(.534) tj-l) 111 | ~ ’ ”” 112 evidence of the important role of government in maintaining ieconomic growth in an underdeveloped economy. Further, such a result of an exogenous influence provides us with useful policy proposals and helps in planning for a desirable growth rate. Consider, for example, some specific hypotheses regarding government expenditure G(t) in the period of planning. Based on such possible hypotheses about G(t), we can then evaluate the jpath of national output during the planning period and also i I jthe corresponding confidence limits using the expressions j jfor the trend and the variance. Then the optimum hypothe- jsis can be chosen easily by comparing the various alterna tive time paths of national output or comparing their I igrowth rates. We give below some results for India showing i : 1 national income estimates under some selected levels of I G(t) during her planning period. We also present similar jresults for the U. S., although such results are not of | 1 I imuch significance for the U. S. economy. However, it I i jprovides us with a basis of comparing earlier results where ino exogenous effect of government expenditure is considered.) ) i Consider first the following different hypotheses of; igovernment expenditure outlay during the third and forth plan periods (five-year plans, 1961-66 and 1966-71 j |respectively) of India: i (I) Yearly 1.5 billions increase in government i j expenditure during the third and fourth plan periods. ’ ’ ” ' ’ ' 113 (IX) Yearly 2.0 billions increase in government |expenditure during the third and fourth plan periods. i | (III) Yearly 2.5 billions increase in government !expenditure during the third and fourth plan periods. (IV) Yearly 2.0 and 3.0 billions increase in government expenditure during the third and fourth plan jperiods respectively. The results obtained under these alternatives are shown in Table XIII. Based on the estimates of national ; income given in Table XIII, the growth rates under the i different hypotheses of government spending are calculated ! |for various sub-periods of the fourth plan and are shown i I ;in Table XIV below. TABLE XIV EXPECTED GROWTH RATES OF NATIONAL INCOME OF INDIA. DURING THE FOURTH PLAN Period Expected Growth (I) Rates under (II) the Hypotheses of G(t) (III) (IV) 1966-1968 5.01 5.27 5.54 5.61 1968-1970 5.30 5.66 6.03 6.20 1970-1971 5.40 5.79 6.20 6.44 Thus, depending upon what growth rate is desired, I the level of government expenditure may be undertaken. i : Similar results for the U. S. are derived under the TABLE XIII ESTIMATES AND 95 PER X{t) OF CENT CONFIDENCE LIMITS FOR NATIONAL INDIA FOR 1964-65 TO .1970-71 INCOME Hypotheses about G(t) (I) (II) Year t E[x(t)] . xL (t) X[J(t) E[x(t)_ XL(t) Xgft) 1964-65 16 149.1 119.0 179.2 149.5 119.4 179.7 1965-66 17 156.1 123.6 188.6 156.8 124.2 189.4 1966-67 18 163.6 128.6 198.7 164.7 129.5 200.0 1967-68 19 171.8 134.0 209.6 173.3 135.2 211.5 1968-69 20 180.5 139.8 221.3 182.7 141.5 224.0 1969-70 21 190.0 146.0 233.9 193.0 148.3 237.6 1970-71 22 200.2 152.8 247.6 204.1 155.8 252.5 H TABLE XIII (continued) . Hypotheses about G(t) Year t fi[x(t) ] (III) xL(t) Xu(t) E[x(t)] (IV.) xL (t) 1 Xu(t) 1964-65 16 149.9 119.7 180.2 . 149.9 119.7 180.2 1965-66 17 157.5 124.7 190.3 157.5 124.7 190.3 1966-67 18 165.8 130.3 201.3 165.9 130.4 201.4 1967-68 19 175.0 136.5 213.4 175.2 136.6 213.7 1968-69 20 185.0 143.2 226.8 185.5 143.6 227.4 1969-70 21 196.0 150.7 241.4 197.0 151.4 242.5 1970-71 22 208.2 158.9 257.5 209.6 159.9 259.2 1 7 1 116 j following hypotheses about government expenditure G(t) during 1964-1971: (I) Yearly increase of 4.0 billions in G(t) during the period 1964-1971. t (II) Yearly increase of 4.0 billions in G(t) during 1964-1967, and a yearly increase of 5.0 billions in G(t) i I during 1968-1971. (Ill) Yearly increase of 5.0 billions in G(t) during jthe period 1964-1971. (IV) Yearly increase of 5.0 billions in G(t) during !1964-1967, and a yearly increase of 6.0 billions in G(t) i I during 1968-1971. i t j The estimates and the corresponding 95 per cent i i j confidence limits for the national product based on the | I above alternative government outlays during the period 1964-1971 are shown in Table XV. : i ! We may compare the results of Table XV with those j ! I jof Table III. We find that the dynamic path of national j j product is altered by the high government spending but it | i is less significant in each year. This shows that the j U. S. economy is less likely to be influenced significantlyI I by government expenditure. This again reflects the nature I iand characteristics of a free private enterprise orionted ! ! i >economy. j i On the lines shown above, other possible important i Iexogenous variables in the growth process of an economy can i 1 TABLE XV j ! ESTIMATES AND 95 PER CENT CONFIDENCE LIMITS FOR NATIONAL PRODUCT X(t) OF U. S.VoR 1964-1971 Year t E^X(t) (I) xL(t) Hypotheses Xu(t) about G (t) E[x(t) ] (II) xL(t) Xu(t) 1964 18 514.7 374.1 655.3 514.7 374.1 655.3 1965 19 533.4 383.6 683.1 533.4 383.6 683.1 1966 20 552.9 393.6 712.2 552.9 393.6 712.2 1967 21 573.4 404.1 742.7 573.4 404.1 742.7 1968 22 594.8 415.0 774.6 594.8 415.0 774.7 1969 23 617.2 426.3 808.0 617.3 426.4 808.2 1970 24 640.6 438.2 843.0 640. 9 438.4 843.4 1971 25 665.2 450.6 879.7 665.7 451.0 880.4 117 TABLE XV (continued) Year t E[x(t)] (XII) xL(t) Hypotheses x0 (t) about G(t) E[x(t>] (IV) xL(t) Xu(t) 1964 18 514.7 374.1 655.3 514.7 374.1 655.3 1965 19 533.5 383.7 683.3 533.5 383.7 683.3 1966 20 553.2 393.8 712.6 553.2 393.8 712.6 1967 21 573.8 404.4 743.3 573.8 404.3 743.3 1968 22 595.5 415.2 775.8 595.5 415.5 775.6 1969 23 618.2 427.1 809.3 618.4 427.2 809.5 1970 24 642.0 439.2 844.9 642.3 439.4 845.3 1971 25 667.1 451.9 882.2 667.6 452.3 882.9 ! 119 I ] be introduced in the model of diffusion process of economic jgrowth by characterizing the parameters and other values I Jas some functions of such variables. Furthermore, such i modification can also be considered with regard to some endogenous variables such as investment or foreign trade. However, a more realistic approach regarding important endogenous variables in the economic growth process is to characterize a multivariate stochastic process wherein all the endogenous variables are interdependent and they proceed simultaneously in time as a time dependent process. We shall consider this latter approach in the next chapter, wherein we try to generalize our single variate diffusion process to a multivariate process. We shall also present some further empirical results obtained for such generalized model using postwar data of the U. S. and a few other countries. APPENDIX A TABLE I GROSS NATIONAL PRODUCT, X(t), OF UNITED STATES FOR 1946-1963 AT CONSTANT PRICES OF 1954 {in Billions of Dollars) Year t X(t) B [ x ( t ) ] 1 9 4 6 0 2 8 2 . 5 2 8 2 . 5 0 1 9 4 7 1 2 8 2 . 3 2 9 2 . 0 4 1 9 4 8 2 2 9 3 . 1 3 0 1 . 9 0 1 9 4 9 3 2 9 2 . 7 3 1 2 . 0 9 1 9 5 0 4 3 1 8 . 1 3 2 2 . 6 3 1 9 5 1 5 3 4 1 . 8 3 3 3 . 5 2 1 9 5 2 6 3 5 3 . 5 3 4 4 . 7 8 1 9 5 3 7 3 6 9 . 0 3 5 6 . 4 2 1 9 5 4 8 3 6 3 . 1 3 6 8 . 4 6 1 9 5 5 9 3 9 2 . 7 3 8 0 . 9 0 1 9 5 6 1 0 4 0 2 . 2 3 9 6 . 7 6 1 9 5 7 ■ — 1 1 4 0 7 . 0 4 0 7 . 0 5 1 9 5 8 1 2 4 0 1 . 3 4 2 0 . 8 0 1 9 5 9 1 3 4 2 8 . 6 4 3 5 . 0 0 1 9 6 0 1 4 4 3 9 . 9 4 4 9 . 6 9 1 9 6 1 1 5 4 4 7 . 7 4 6 4 . 8 7 1 9 6 2 1 6 4 7 4 . 8 4 8 0 . 5 7 1 9 6 3 1 7 4 9 2 . 9 4 9 6 . 8 0 Source: Data given in the references (155) and (156) 122 TABLE II INDEX OF GROSS NATIONAL PRODUCT, X(t), OF CANADA . FOR 1948-1960.AT CONSTANT PRICES.OF 1913 {Base 1913=100). Year t x(t) E^X(t) ] 1948 0 246.7 246.70 1949 1 253.9 256.89 1950 2 271.7 267.51 1951 3 288.0 278.56 1952 4 315.1 290.07 1953 5 326.3 302.06 1954 6 314.6 314.54 1955 7 340. 3 327.54 1956 8 373.1 341.08 1957 9 376.3 355.17 1958 10 380.0 369.85 1959 11 399.3 385.13 1960 12 397.5 401.05 Source: Indices built by Maddison (103). TABLE III i INDEX OF GROSS NATIONAL PRODUCT, X(t), OF NETHERLANDS FOR 1948-1960 AT.CONSTANT PRICES OF 1913 {Base 1913=100). Year t x(t) E [ x ( t ) ] 1 9 4 8 0 1 9 4 . 0 1 9 4 . 0 0 1 9 4 9 1 2 0 9 . 2 2 0 3 . 8 7 1 9 5 0 2 2 1 6 . 8 2 1 4 . 2 5 1 9 5 1 3 2 2 2 . 9 2 2 5 . 1 6 1 9 5 2 4 2 2 7 . 5 2 3 6 . 6 2 1 9 5 3 5 2 4 7 . 5 2 4 8 . 6 7 1 9 5 4 6 2 6 4 . 9 2 6 1 . 3 2 1 9 5 5 7 2 8 5 . 8 2 7 4 . 6 3 1 9 5 6 8 2 9 6 . 9 2 8 8 . 6 1 1 9 5 7 9 3 0 4 . 3 3 0 3 . 3 0 1 9 5 8 1 0 3 0 8 . 8 3 1 8 . 7 4 1 9 5 9 1 1 3 2 4 . 2 3 3 4 . 9 7 1 9 6 0 1 2 3 5 0 . 6 3 5 2 . 0 2 Source: Indices built by Maddison (103). T A B L E I V R E A L N A T I O N A L I N C O M E , X ( t ) , O F . . F O R . 1948-49 T O 1963-64 A T C O N S T A N T P R I C E S O F 1948. (in Billions of Rupees) 124 I N D I A Year t x ( t ) E ^ X ( t ) ] 1948-49 0 86.5 86.5 1949-50 1 88.2 89.3 1950-51 2 88.5 92.2 1951-52 3 91.0 95.2 J 1952-53 4 94.6 98. 3 1953-54 5 100.3 101.5 1954-55 6 102.8 104.7 1955-56 7 104.8 108.1 1956-57 8 110.0 111.7 1 1957-58 9 108.9 115. 3 1958-59 10 116.9 119.0 1 1959-60 11 118.8 i 122.9 1960-61 12 127.3 | 126.9 | 1961-62 13 130.6 131.0 1962-63 14 133.1 135.2 ; 1963-64 15 139.1 139.6 ! i I Source: Estimates of National Income, (New Delhi: \Central Statistical Organization/ Government of India, lApril 1963, April 1965). TABLE V INDEX OP GROSS NATIONAL PRODUCT, X(t), OP ECUADOR FOR 1950-1962.AT CONSTANT PRICES.OF 1960 (in Millions.of Sucres) Year t x ( t ) f i [ x ( t ) ] 1 9 5 0 0 8 0 8 9 8 0 8 9 . 0 0 1 9 5 1 1 8 7 1 2 8 5 1 7 . 9 7 1 9 5 2 2 9 4 7 2 8 9 6 9 . 6 9 1 9 5 3 3 9 8 0 5 9 4 4 5 . 3 6 1 9 5 4 4 9 9 4 5 9 9 4 6 . 2 6 1 9 5 5 5 1 0 8 6 1 1 0 4 7 3 . 7 3 1 9 5 6 6 1 1 3 7 9 1 1 0 2 9 . 1 6 1 9 5 7 7 1 2 0 1 2 1 1 6 1 4 . 0 5 1 9 5 8 8 1 2 3 2 2 1 2 2 2 9 . 9 6 1 9 5 9 9 1 3 0 8 8 1 2 8 7 8 . 5 3 1 9 6 0 1 0 1 4 0 9 4 1 3 5 6 1 . 5 0 1 9 6 1 1 1 1 4 6 7 1 1 4 2 8 0 . 6 8 1 9 6 2 1 2 1 5 0 3 8 1 5 0 3 8 . 0 0 Source: Memorial del Gerente del Banco Central (Ecuador) , 1 9 6 2 . (Report of the Central Bank, Ecuador). 126 TABLE VI GROSS NATIONAL PRODUCT, X(t), OF UNITED STATES FOR 1900-1957 AT CONSTANT PRICES OF 1929 (in Billions of Dollars) Year t X(t) fi[x(t) ] i 1900 0 38.2 38.20 1901 1 42.6 39.52 1902 2 43.0 40.58 1903 3 45.1 42.29 11904 4 44.6 43.75 1905 5 47.9 45.26 11906 6 53.4 46.83 11907 7 54.3 48.44 1908 8 49.8 50.11 1909 9 55.9 51.84 1910 10 56.5 53.63 11911 i 11 58.3 55.48 11912 12 61.1 57.40 1913 I 13 63.5 59.38 ■ 1914 14 58.6 61.43 : 1915 15 60.4 63.55 11916 16 68.9 65.75 ! 1917 1 17 67.3 68.01 : 1918 18 73.4 70.36 i 1919 19 74.2 72.79 127 TABLE VI (continued) j i Year t X(t) E[x(t)] 1 9 2 0 2 0 7 3 . 3 7 5 . 3 1 1 9 2 1 2 1 7 1 . 6 7 7 . 9 0 1 9 2 2 2 2 7 5 . 8 8 0 . 5 9 1 9 2 3 2 3 8 5 . 8 8 3 . 3 7 1 9 2 4 2 4 8 8 . 4 8 6 . 2 5 1 9 2 5 2 5 9 0 . 5 8 9 . 2 3 1 9 2 6 2 6 9 6 . 4 9 2 . 3 1 1 9 2 7 2 7 9 7 . 3 9 5 . 5 0 1 9 2 8 2 8 9 8 . 5 9 8 . 7 9 1 9 2 9 2 9 1 0 4 . 4 1 0 2 . 2 0 1 9 3 0 3 0 9 5 . 1 1 0 5 . 7 3 1 9 3 1 3 1 8 9 . 4 1 0 9 . 3 8 1 9 3 2 3 2 7 6 . 4 1 1 3 . 1 6 1 9 3 3 3 3 7 4 . 2 1 1 7 . 0 6 1 9 3 4 3 4 8 0 . 8 1 2 1 . 1 0 1 9 3 5 3 5 9 1 . 4 1 2 5 . 2 8 1 9 3 6 3 6 1 0 0 . 9 1 2 9 . 6 1 1 9 3 7 3 7 1 0 9 . 1 1 3 4 . 8 0 1 9 3 8 3 8 1 0 3 . 2 1 3 8 . 7 1 1 9 3 9 3 9 1 1 1 . 0 1 4 3 . 5 0 1 9 4 0 4 0 1 2 1 . 0 1 4 8 . 4 5 128 Table VI (continued) Year t x ( t ) f i [ x ( t ) _ 1 9 4 1 4 1 1 3 8 . 7 1 5 3 . 5 8 1 9 4 2 4 2 1 5 4 . 6 1 5 8 . 8 8 1 9 4 3 4 3 1 7 0 . 2 1 6 4 . 3 6 1 9 4 4 4 4 1 8 3 . 6 1 7 0 . 0 3 1 9 4 5 4 5 1 8 0 . 9 1 7 5 . 9 0 1 9 4 6 4 6 1 6 5 . 6 1 8 1 . 9 8 1 9 4 7 4 7 1 6 4 . 1 1 8 8 . 2 5 1 9 4 8 4 8 1 7 3 . 0 1 9 4 . 7 6 1 9 4 9 4 9 1 7 0 . 6 2 0 1 . 4 8 1 9 5 0 5 0 1 8 7 . 4 2 0 8 . 4 3 1 9 5 1 5 1 1 9 9 . 4 2 1 5 . 6 3 1 9 5 2 5 2 2 0 5 . 8 2 2 3 . 0 7 1 9 5 3 5 3 2 1 4 . 0 2 3 0 . 7 7 1 9 5 4 5 4 2 1 9 . 3 2 3 8 . 7 4 1 9 5 5 5 5 2 2 4 . 9 2 4 6 . 9 8 1 9 5 6 5 6 2 2 8 . 9 2 5 5 . 5 0 1 9 5 7 5 7 2 3 6 . 0 2 6 4 . 3 2 Source: Estimates given by Kendrick (88) i ! i 129 TABLE VII NET NATIONAL, PRODUCT, X(t), OF UNITED KINGDOM FOR 1880-1940 AT CONSTANT PRICES OF 1913 (in 10 millions of sterling pounds) |Year t X(t) E^X(t) ] 1880 0 102.8 102.80 1881 1 109.6 105.04 1882 2 113.8 107.33 1883 3 117.8 109.66 j 1884 4 120.2 112.05 j 1885 j 5 123.5 114.49 Il886 1 6 128.1 116.98 11887 7 131.3 119.53 1888 8 139.0 122.14 i 1889 9 145.0 124.80 1890 10 154.4 127.51 1891 11 151.3 130.29 1892 12 153.6 133.13 1893 13 150.1 136.03 1894 14 164.6 138.99 1895 15 174.6 142.02 1896 16 178.3 145.11 1897 17 184.1 148.27 1898 18 189.4 151.50 1899 19 191.2 154.80 130 TABLE VII (continued) 1 l ■ — ■ ■ - Y e a r t x(t) E[x{t) ] 1 9 0 0 2 0 1 8 8 . 0 1 5 8 . 1 7 1 9 0 1 2 1 1 9 4 . 9 1 6 1 . 6 2 1 l 1 9 0 2 2 2 1 9 6 . 6 1 1 6 5 . 1 4 1 9 0 3 2 3 1 9 2 . 9 1 6 8 . 7 3 1 9 0 4 2 4 1 9 4 . 9 1 7 2 . 4 1 j 1 9 0 5 2 5 2 0 1 . 3 1 7 6 . 1 6 1 9 0 6 2 6 2 0 7 . 6 1 8 0 . 0 0 1 9 0 7 2 7 2 1 1 . 4 1 8 3 . 9 2 j 1 9 0 8 2 8 2 1 0 . 8 1 8 7 . 9 2 1 9 0 9 2 9 2 1 7 . 6 1 9 2 . 0 2 1 9 1 0 3 0 2 1 6 . 5 1 1 9 6 . 2 0 1 9 1 1 3 1 2 2 2 . 1 2 0 0 . 4 7 1 9 1 2 3 2 2 1 6 . 6 2 0 4 . 8 4 j I 1 9 1 3 3 3 2 3 7 . 1 2 0 9 . 3 0 1 9 1 4 3 4 2 3 1 . 7 2 1 3 . 8 6 i j 1 9 1 5 3 5 2 2 5 . 4 2 1 8 . 5 1 ! 1 9 1 6 3 6 2 1 9 . 4 2 2 3 . 2 7 1 9 1 7 3 7 2 1 5 . 6 2 2 8 . 1 4 1 9 1 8 3 8 2 0 8 . 1 2 3 3 . 1 0 j 1 9 1 9 3 9 2 0 2 . 6 2 3 8 . 1 8 | 1 9 2 0 4 0 1 9 7 . 3 j 2 4 3 . 3 7 ! 131 TABLE VII (continued) Year t x(t) E[x(t) ] 1921 41 192.1 248.67 1922 42 195.2 254.08 1923 43 193.4 259.62 1924 44 212.1 265.27 1925 45 215.3 271.05 1926 46 212.1 276.95 1927 47 232.0 282.98 1928 48 232.4 289.15 1929 49 235.0 295.44 1930 50 229.0 301.88 1931 51 223.6 308.45 1932 52 221.8 315.17 1933 53 237.4 322.04 1934 54 247.2 329.05 1935 55 261.5 336.22 1936 56 275.9 343.54 1937 57 281.6 351.02 1938 58 280.0 358.66 1939 59 291.1 366.47 1940 60 355.5 374.46 Sources Data compiled by Narasimham (116, Appendix) from the estimates given by Deane and Cole (31) . 132 | TABLE VIII REAL NATIONAL INCOME, X(t), OF GERMANY FOR 1880-1913 AT CONSTANT PRICES OF 1913 (in .00 millions of marks) i [Year t x(t) 1 i 11880 0 219.4 219.40 1881 1 232.2 225.06 1882 2 246.1 230.86 : 1883 3 267.3 1 236.81 i | 1884 i 4 268.1 242.91 1885 5 270.5 249.18 i i 1 1886 6 261.5 255.60 ! i | 1887 7 253.7 262.19 j ! 1888 8 258.0 268.95 1889 9 276.3 275.88 1890 10 300.1 283.00 i | 1891 11 318.5 290.29 | i 1892 f 12 333.5 297.78 j 1893 13 347.1 305.45 | 1894 j 14 347.7 313.33 ; 1895 15 345.3 321.41 i | 1896 1 16 350.4 329.69 1 i 1897 ] 17 338.4 338.19 1 i ; 1898 I 18 361.5 346.91 ' 1899 19 376.8 355.86 133 t TABLE VIII (continued) Year t x(t) a[x(t)] 1900 20 390.3 365.03 1901 21 401.6 374.44 1902 22 397.7 384.10 1903 23 405.0 394.00 I 1904 24 405.8 404.16 1905 25 445.2 414.58 1906 26 453.6 425.26 1907 27 470.5 436.23 1908 28 475.2 447.47 1909 29 456.6 459.01 1910 30 495.0 470.85 1911 31 419.1 482.98 1912 32 469.0 495.44 1913 33 486.9 508.21 from Source: Data compiled by Narasimham (116 the estimates given by Hoffmann and Muller , Appendix) (73) . TABLE IX INDEX:OF GROSS NATIONAL PRODUCT, X(t), OP CANADA FOR 1920-1938,AT CONSTANT PRICES OP 1913 (Base 1913=100). Year t x{t) s[x(t) ] 1920 0 103.7 103.70 1921 1 94.3 105.79 1922 2 101. 9 107.93 11923 3 108.3 110.11 1924 4 108.1 112.34 1925 5 112.8 114.61 |1926 6 122.7 116.92 i 192 7 i [ 7 132.8 119.28 1928 8 143.7 121.69 1929 9 143.9 124.15 ,1930 10 138.3 126.66 1931 11 120.7 129.22 1932 12 110.5 131.83 1933 13 101.7 134.49 1934 14 113.5 137.21 1935 15 122.2 139.98 1936 16 127.8 142.81 1937 17 140.8 145.70 1938 18 141.5 148.64 Source: Indices constructed by Maddison (103). TABLE X INDEX OP GROSS NATIONAL PRODUCT, X(t) , OF NETHERLANDS FOR 1920-1938 AT ..CONSTANT PRICES OF 1913 (Ba.se 1913=100) . Year t x ( t ) E [ x ( t ) ] 1920 0 117.2 117.20 1921 1 123.4 119.73 1922 2 128.1 122.32 1923 3 132.8 124.96 1924 4 137.5 127.66 1925 5 143.8 130.42 1926 6 151.6 133.24 1927 7 156.3 136.12 1928 8 164.1 139.07 1929 9 168.8 142.07 1930 10 168.8 145.14 1931 11 157.8 148.28 1932 12 153.1 151.48 1933 13 150.0 154.76 | 1934 14 148.4 158.10 1 1935 15 151.6 161.52 | 1936 16 159.4 1 165.01 ! j 1937 17 171.9 168.58 j j 1938 18 .70.3 172.22 | I Source: Indices constructed by Maddison ! (103). ! . . . . . . . . . . . . . . . . . . . . . . . . . J 136 TABLE XI REAL NATIONAL INCOME, X(t), OF INDIA FOR 1900-1958 AT CONSTANT PRICES OF 1948 (in Billions of Rupees) Year t X(t) |1900 0 51.1 51.10 1901 1 49.3 51.92 1902 2 56.0 52.76 1903 3 55.5 53.62 1904 4 51.5 54.48 1905 5 55.0 55.36 1906 i 6 57.8 56.26 1 11907 7 48.5 57.16 a908 8 55.4 58.09 1909 9 61.7 59.03 1910 10 62.4 59.98 j 1911 11 61.1 60.95 1 1912 12 62.0 61.93 1913 13 59.8 62.93 1914 14 61.3 63.95 1915 15 66.6 64.98 1916 16 70.6 66.03 1917 17 70.6 67.10 1918 18 59.1 68.18 1919 19 68.3 69.28 TABLE XI (continued) 137 i .,'■■■■ s j Year t X(t) E[x(t) ] 1920 ] 20 64.7 70.40 1921 21 66.2 71.54 11922 i 22 70.3 72.69 1923 23 65.9 73.87 1924 24 70.2 75.06 11925 25 70.6 76.27 j 1926 26 71.1 77.51 1927 27 72.9 78.76 11928 28 74.9 80.03 11929 29 77.0 81.32 1930 30 76.3 82.64 1931 31 75.3 83.97 1932 32 75.5 85.33 1933 33 77.2 86.70 1934 34 78.7 88.10 1935 35 80.2 89.53 1936 36 81.7 90.97 1937 37 84.2 92.44 1938 38 80.4 93.94 1939 39 85.7 95.45 1940 40 86.5 96.99 1941 41 87.4 98.56 138 I TABLE XI (continued) Year t x ( t ) EjjjX(t) ] 1 9 4 2 4 2 8 8 . 1 1 0 0 . 1 5 1 9 4 3 4 3 8 6 . 8 1 0 1 . 7 7 1 9 4 4 4 4 8 9 . 3 1 0 3 . 4 1 1 9 4 5 4 5 8 9 . 0 1 0 5 . 0 8 1 9 4 6 4 6 8 8 . 8 1 0 6 . 7 8 1 9 4 7 4 7 8 6 . 7 1 0 8 . 5 1 1 9 4 8 4 8 8 9 . 4 1 1 0 . 2 6 1 9 4 9 4 9 8 8 . 7 1 1 2 . 0 4 1 9 5 0 5 0 9 1 . 2 1 1 3 . 8 5 1 9 5 1 5 1 9 8 . 7 1 1 5 . 6 9 1 9 5 2 5 2 1 0 0 , 3 1 1 7 . 5 6 1 9 5 3 5 3 1 0 2 . 8 1 1 9 . 4 6 1 9 5 4 5 4 1 0 4 . 8 1 2 1 . 3 8 1 9 5 5 5 5 1 0 9 . 9 1 2 3 . 3 5 1 9 5 6 5 6 1 0 9 . 0 1 2 5 . 3 4 1 9 5 7 5 7 1 1 6 . 7 1 2 7 . 3 6 1 9 5 8 5 8 1 1 7 . 8 1 2 9 . 4 2 j Source: Data given by Mukerjee (112) and extended ! j further by Narasimham (116, Appendix) are used. | TABLE XII GROSS NATIONAL- (DOMESTIC) PRODUCT* X(t), OF JAPAN FOR 1900-1938 AT CONSTANT PRICES OF 1934-36 (in Billions of Yens) Year t X(t) s[x(t) ] 1900 0 47.3 47.30 1901 1 50.2 49.38 1902 2 44.3 51.55 1903 3 50.2 53.81 1904 4 50.5 56.17 1905 5 46.2 58.64 1906 6 58.8 61.22 1907 7 57.0 63.91 I 1908 8 60.2 66.71 1909 9 59.8 69.64 1910 10 58.0 72.70 1911 11 66. 2 75.89 1912 12 71.6 79.23 : 1913 13 71.0 82.71 1914 14 72.0 86.34 1915 15 73.3 90.13 1916 16 80.9 94.09 1917 17 83.4 98.23 1918 18 95.2 102.54 1919 19 113.1 107.05 TABLE XII (continued) 140 Year t x(t) E[x(t) ] 1920 20 89.4 111.75 I 1921 21 90.6 116.65 1922 22 91.9 121.78 1923 23 101.7 127.13 j 1924 24 112.4 132.71 1925 25 121.5 138.54 1 1926 26 122.6 144.63 i 1927 27 123.5 1 150.98 | I 1928 28 128.9 1 157.61 1929 29 131.5 164.54 1930 30 127.8 171.76 1931 31 129.3 179.31 | 1932 32 143.4 187.18 i 1933 33 158.2 195.40 j 1934 34 162.2 203.99 i 1935 35 175.8 212.96 1936 36 189.3 222.30 i 1937 37 200.0 232.07 | i 1938 38 213.4 242.26 from Source: Data compiled by Narasimham the estimates given by Okawa and others (116# Appendix) (117). i TABLE XIII GROSS NATIONAL PRODUCT, X(t) , AND GOVERNMENT EXPENDITURE, G(t) , OP UNITED STATES FOR 1946-63 AT CONSTANT PRICES OF.1954 (in Billions of Dollars) . . 1 Year i t G(t) x(t) s[x(t)/ G (t) ] 1946 0 43.9 282.5 282.50 1947 1 37.2 282.3 291.26 ■1948 2 42.1 293.1 300.42 1949 3 47.2 292.7 309.98 1950 4 45.1 318.1 319.80 1951 5 63.3 341.8 330.40 1952 6 77.7 353.5 341.75 1953 7 84.3 369.0 353.67 1954 8 75.3 363.1 365.75 1955 9 73.2 392.7 378.17 1956 10 72.9 402.2 391.01 1957 11 75.0 407.0 404.35 1958 12 79.3 401.3 418.29 1959 i f 80.1 428.6 432.74 1960 14 79.9 439.9 447.68 1961 15 84.3 447.7 463.30 1962 16 90.2 474.8 479.68 1963 17 93.7 492.9 496.78 Source: Data given in the references (155) and (156). 142 i TABLE XIV j REAL NATIONAL INCOME, X(t) , AT CONSTANT PRICES OF 1948-49 i AND GOVERNMENT EXPENDITURE, G(t), AT CURRENT PRICES • OF INDIA FOR.1948 .(in Billions -49 TO 1961- of Rupees) 62 Year t G(t) X(t) E[x(t) / G (t)] 1948-49 0 8.5 86.5 86.5 1949-50 ^ 1 8.1 88.2 88.9 1950-51 2 8. 3 88.5 91.3 1951-52 3 8.8 91.0 93.9 1952-53 4 9.0 94.6 96.5 1953-54 5 9.8 100.3 99.3 1954-55 6 11.0 102.8 102.3 1955-56 7 12.9 104.8 105.5 1956-57 8 14.8 110.0 109.0 1957-58 9 17.5 108.9 112.9 1958-59 10 18.5 116.9 117.1 1959-60 11 17.4 118.8 121.3 1960-61 12 19.3 127.3 125.8 1961-62 13 23.5 130.6 131.0 Source: Estimates of National Income, (New Delhi: iCentral Statistical Organization, Government of India, iApril 1963, April 1965). CHAPTER V MULTIVARIATE LOGNORMAL DIFFUSION PROCESS In the previous chapter, a lognormal diffusion process was proposed for the development of national income i ;and its empirical verification was attempted using national l income data of many different countries. Both on a priori theoretical ground and empirical results, the theory of thej lognormal diffusion process as characterized by the variatej ;X(t) representing the national income appears to be a valid| ; theory of economic development. It can be, however, hardly| | i ! said that economic development is specified by a single I : j I variable like national income. It is more plausible and j f j | more adequate to consider the evolution of an economy in terms of a number of interdependent, important economic variables, like national income, capital stock, employment, ; i . . . . ! i et cetera, which jointly determine the state of the economy j | I ! at any given time. I t I : I > It is then the purpose of the present chapter to ; i ; extend the univariate lognormal diffusion process to a i multivariate process Which takes into account the inter- ' dependence among the economic magnitudes that specify the j i I | state of economic development. In what follows, we ! ! i j | consider first a bivariate case and then extend it to a j ! I | general case of p variables. Each case will be followed ; 144 by empirical examples. Regarding empirical illustrations, we were forced to confine the analysis to only a few i jvariables and also to a few countries due to the lack of I adequate data. Our theory, however, is quite general and it should not be taken as restricted to only a few vari- I i 'ables because of the paucity of data. Introduction of exogenous variables is also con sidered in case of two variables. Such a modification is ; straightforward in the general case of the multivariate lognormal diffusion process and is therefore not repeated | further. Bivariate Lognormal Diffusion Process Let national income and capital stock represent the i |two random variables (t) and X2 (t) , respectively, which may be considered as the variables characterizing the process of economic development at time t. Assume, as before, the variables of the process |x^(t), X2 (t), t s o} continuous and of the Markovian type defined on the 'positive quadrant of a real plane, that is, {xx(t), x2(t)} is a Bivariate Markovian random variable depending on a continuous time parameter t, and assuming values in the i state space ^ : 0 < x-^, x2 < “. Let the transition probability density function be (5.1) f(r, x-^ x2? t, y-L, y2) = pj^X1(t)=y1, X2(t)=y2/ X1(t)=x1, X2(t)=x2] where 0 < x^, y^ < 00, 1=1, 2, for every r and t, 0 ^ t < tf j satisfying the backward and forward Kolmogorov diffusion equations. We assume the random variable {x^ (t), X2(t)} continuous with probability one. I Consider the coefficients bi(t; xi, X2), ;a^(t; x^, x^), and c (t; x^, x2^ ' infinitesimal means, variances, and covariance, respectively, of the changes in the variables X1(t), X(t) during a small interval At of time as follows, i (5.2) bi(t, xx, x2) = b ±(t)xi , i=l, 2 i ! ! (5.3) a± (t, xx, x2) = a± (t)x? , i=l, 2 | | (5-4) c(t, xx, x2) = c(t)x1x2 ! i i ! . [ I I jwhere b^(t), a^(t), and c(t) are functions of time t. As a; simple case, we may consider these functions linear or even: constant. Let us specify them as constants, namely, I j (5.5) bi(t) = l>i , a± (t) = a± , c(t) = c , i=l, 2 j ; ! jwhere a^ > 0, b^, and c are constant, i=l, 2. We thus | assume that the expected changes and their standard j I deviations are proportional to the instantaneous size of the random variables. With a^ > 0, i=l, 2, it implies that some change in the values of the variables takes i place in any small interval At with probability one. It | thus assumes the hypothesis of proportional effect as ; considered before in Chapter IV, which appears to describe 146 fairly well the growth phenomena of the aggregative economy. With the ahove simple specifications of (5.5), the backward and forward diffusion equations are (5.6) - 2£. = + CX1X2- ^ - + ar 2 1 1 ax| x 2 ax1 ax2 2 2 2 ax| + b-iX-i^- + b~x0- ~ J- J-Sx - l 2 2 9x 2 ,c ,, S£ _ 1 2a2f , a2f , 1 2a2f (5‘7> aT - “2 ^ ayf + cyiy2 5 ^ 2 + T a y a^r + (2a1-b1+o)y1||- + (2a2-b2+c) y2-|£ 2 | j + ( a 1 + a 2 - b 1 - b 2 + c ) f The probability density function satisfying these diffusion! equations (5.6) and (5.7) is obtained as the bivariate j I • ’ I lognormal density function j I - __i j, i (5.8) f (T,x1,x2;t,y1,y2) = (y1y|1{2i7{t-T)j ‘ {y- ^ (l“p2) } 1 - exp£|-2 (t-T) (1-/Q2) }• Qj/1 2 !where Q [{logy1-logxi-p1(t-T)]/Y1 + {logy2~logx2 i 2 ■ I -P2(t-r)}/Y2 - 2p{logy1-logx1-p1(t-r)} i {logy -logx -p (t-T).}/ Vy1Y2] where Y-=a-/ P . = (b.-a./2) , i=l, 2 and p=c//a a . 1 l 1 j . 3b 2 147 The characteristics of the distribution given by i i i (5-8) are easily obtained from its moments j (5.9) E{[xx(t)] [x2( t ) ] } = [x1(T)]jexp { j Y j / 2 ) ( f “T)]} P2+k' Y2//r2^ t"'T^}^ e x p ^ k p f Y i ^ ^ j t - r ^ | 3/ i/ 2 , 3 , . • • , From (5.9), we get the means and variances of the variables; as ! (5.10) E[^(t)] = X^r) exp{(3i+Yi/2) (t-T)} = a± (t) , say, i=l, 2 i 2 j (5.11) V X (t) = [ x ± ( t )] exp{2(Pi+yi/2) (t-T)} j {exp[Yi(t-r)] -1 } 2 2 = {«i(t)} {lli (t)}/ say, i=l, 2 | where 71j_(t)= {exp^Yj[ (t-T)j - l} ■ a* Also, the covariance and correlation coefficient between the two variables are given by 2 n i=l ■ expj^p(Y1Y2) 2 (t-T)j - l} (5.12) c[xi(t), x2 (t)] =. n X^ (T) exp|(P±+Yi/2) (t-T)} = of (t) a2 (t) 6 (t) , say 148 where 8 (t) = exp£p an<^ hence, the ;correlation | (5.13) R^ft), X 2 (t)] = 6 (t)/{'Tl1 (t)Tl2 (t) } 2 I I j If we reduce the initial time T to zero by a | j suitable time scale transformation, and assume that ■ ( 0 ) = Xj 0 , X2 (0 ) = x2 q! = 1 , the expressions for the means, variances, covariance, and correlation can be I written as under. (5.14) E^ft)] - xi^0 exp(bit) , i=l, 2 , = ai(t), say. I j 1(5.15) vj^Xi(t)] = (xi Q )2 exp(2bit) |exp(ait) - l} | 2 2 = ■ja^ft)^; {\(t)} , say, i=l, 2 , where ^(t) = ^exp(a^t) - lj , i=l, 2 . (5.16) cfx-^tt), X2 (t) J = (t) o?2 (t) 6 (t) , 8 (t)=exp(ct) - 1. (5.17) R[xx(t), X 2 (t)]= 6 (t)/{Tl1 (t)Tl2 (t)} I Thus, the trends for X-^ (t) and X2 (t) are exponential ;functions of t, given by (5.14). The dynamic path of the j I ; j economy as characterized by the exponential trends of | |national income X-^(t) and capital stock X2(t) can be j : i i 1 ! evaluated using their initial values x^ n, i=l,2, at time I j x i u I ; i t=0 and the values of the parameters a^, b^, i=l, 2, and c : of the diffusion process. The values of the parameters can i be estimated from the empirical data using the maximum ilikelihood method of estimation as follows: i i | Let x.. n, x. ,, . . ., x, be n observed pairs of jl / u i / x x i n values of (t) at tg, t-^, . . ., tn moments of time, i=l,2, and let (0) q, X2(°)~x2,o} = Then the likelihood of the observations is ! (5.18) L = ( 2 i r T n { V l Y 2 ( l - p 2 ) } j exp{[-2(l-p2) | where < V V l > ^ 1 + {1o9X2,„ j -logX2,a- r P2 (W l )i/'<2 - 2P{lo9xl,f f i i J -1o9*1, of-l-Pi <ta-to-l) }{lo9x2, crlo9x2, ff-1 i Maximizing logL with respect to the parameters, the estimates are obtained as follows: (5.19) B-j= §. (logx. -logx. 1 / S. (t -t ,), i=l, 2 1 a=l i ><x 1 ,0?-! ' ct=l of of—1 150 (5.21) p = j j f l w J {lo9x2, a-lo9x2, a-l-h ^ J i If the observations are assumed to be made at equal intervals of time of unity/ that is, (t -t ,) = 1/ a=l, 2, a cv-1 !3, . . o, n and if tQ= 0, then the maximum likelihood estimates are (5.22) ^ (logxi#Qf-logXi^ Q?_1) / n / i = 1, 2 (5.23) ya = J 1(logXii(r-logx.ja_1-Bi)2/ n , i = 1, 2 j 2 {(logx -logx -jL ) (logx„ -logx„ o?=lL I/O' 3 1, Q?-l 1 * 2,a y 2/CV-l 2j | (5.24) p =--------------- /A A n v ’ V1Y2 iHence, the maximum likelihood estimates of the parameters ■ai7 kf7 i=l/ 2, and c of the bivariate lognormal diffusion process are ! ( 5 . 2 5 ) f i ± = ( B i + Y i / 2 ) , i=l, 2 j ( 5 . 2 6 ) a±= y± , i-1, 2 I ( 5 . 2 7 ) c = P'Jy-L y 2 The information matrix for the estimates is ’ (5.28) 1= -1 V1 0 -1 V , -1 -1 ; where and V2 are the following information matrices | for the two sets of estimates ($i/p2) anc^ (Yl'Y2'p) 151 respectively: (5.29) V71- nltzrl (l-p2) (5.30) v21= n 2(1-p ) i / y i - p / ( y 1 y 2 > Vy2 2 (2-p2)/2Yi ~P^2^ly2 "P/yi (2~P2)/2y2 “P/Y2 2(i+p2)/d-p2: Hence the variance-covariance matrices of the two sets of ft ft* A estimates (f^, P2) and (Y]_* Y2/ p) are (5.31) V, = n(t-r) Yl p (yxy2) Y 2 (5.32) V2= - n 2y'_ 2P2Y1Y2 2Y' Y ^ d - P ) Y2P(1_P J n 2, 2 (1-p ) The variance matrices of the estimates (b., b ) and 2 A A A (a^, a2# c), and the covariance matrix between them are obtained respectively as (5.33) Sl= n (a“/2)+a1/(t-r) (c2/2)+c/(t-T) (a /2)+a„/(t-T) (5.34) S9= - ^ n (5.35) Sl2= - n 2a. 2c 2a, 2a^c 2a2G (aia2 + ° > an c a2° 152 iTherefore, for the r = 0 case, the sampling variances of j the estimates are (5.36) V(Pi) = Yi/nt , i = 1, 2, (5.37) V(y±) = 2y?/n , i = 1, 2 1(5.38) and j (5.39) |(5.40) f I !(5.41) V(p) = (1-p2)2 A 2 V(b^) = (a^/2 + a^/t)/n = a2/2n for large t,i=l,2 V(a±) = 2a^/n , i = 1, 2 V(c) = (aia 2 + c )/n ; We note that the variances of the estimates given I jby (5.36) through (5.41) tend to zero for large n and t, t land hence the estimates (p-^, P2/ Y]_* V2» ^ and ^1' A A A . :al» a2' c) are consistent estimates. j I Further, using the property of asymptotic normality | 153 ;an approximate confidence ellipse for X2 (t)| can be obtained as follows. The quadratic form 1(5.42) Q^Ct), X2 (t)J = [x^t) - ^(t), X2(t) - a2(t)J C Xxft) -o'! (t) x 2 ( t ) - < * 2 ( t ) where C is the variance covariance matrix of -Jxq (t), X2(t)j" as given by (5.15) and (5.16), follows asymptotically a Chi-square distribution with two degrees of freedom. Under | |the asymptotic theory, an approximate confidence ellipse I of confidence coefficient lOOtl-cv) for the values of X. (t) ! 1 is given by the inequality (5.43) p{Q[xx(t), X2(t)]^x^}^ (1-aO 2 where is the upper (100) o' per cent point of the Chi- i I square distribution with two degrees of freedom. Empirical Results I We now present the empirical results of the above j |model of a bivariate diffusion process applied to the I development of the aggregate economic variables of the U.K. jwe also apply the model to the economy of Ecuador as an |illustrative case of an underdeveloped economy. The more i |recent data of the U. S. are also considered. Here we consider the two variables as national product (t) and i 154 i I (Capital stock X£(t) at time t. In the case of the U. S., | investment is considered instead of capital stock. This appears more appropriate in a short-term analysis. Consider first the results for the long term data of the U. K. for 1880-1940.* Here X-^(t) represents an index of total output with base year 1913, and X^(t) is capital ! Sstock measured in billions of sterling pounds at constant prices of 1913. Similar results for Ecuador (1950-62) for |national product Xi(t) and capital stock X0(t), both | ‘ L * ■ measured in billions of sucres at constant prices of 1960, and for the U. S. (1946-63) for national product X^(t) and investment X2(t), both measured in billions of dollars at constant prices of 1954 are obtained as follows. Assuming that the probability of taking the observed lvalues by the two random variables for the initial time as [ i one, the maximum likelihood estimates and their sampling (variances for the parameters of the diffusion process |'[x^(t), X2 (t) , t ^ o} are obtained as shown in Tables XVI land XVII. 5 I The estimated trends for the economic variables are i J (given by the expressions as mentioned in Table XVIII. The jtrend values evaluated for the periods of analysis are J jobserved as given in Tables I, II, III of Appendix B. j j We observe that the results are similar for the *See Appendix B for the data and their sources. 155 TABLE XVI : ESTIMATES OF THE PARAMETERS OF THE BIVARIATE LOGNORMAL DIFFUSION PROCESS APPLIED TO THE NATIONAL PRODUCT DATA OF U. K. (1880-1940), U. S. (1948-63), AND ECUADOR (1950-62) i jCountry h ^2 A al A a2 A c U. K. .0266 .0141 102(.8304) 105 (.7084) 104 (.5290) U. S. .0332 .0397 103 (.9255) 101{.2441) 102 (.3802) Ecuador .0392 .0519 103{.5591) 103(.1899) 104 (.9896) TABLE XVII VARIANCES OF THE ESTIMATES OF THE PARAMETERS OF THE BIVARIATE DIFFUSION PROCESS APPLIED TO THE DATA OF U. K. (1880-1940), U. S. (1946-63), AND ECUADOR (1950-62) Country V(bx) v (£2) v(ax) V(a2) V(c) U. K. -3 10(.277)/t +10^.115) -6 10(.236)/t -32 +10 (.836) -5 10(.460) -11 10(.334) -8 10(.205) U. s. lot.544)/t -7 +10(.252) 10^.144)/t +lot•175) lot.101) lot-701) 10;|. 218) Ecuador -4 10(.466)/t +lo].130) -4 10 (. 158)/t +10^.150) -7 10 (.521) -8 10 (.601) lif.966) 156 TABLE XVIII THE ESTIMATED TRENDS FOR X;l (t) AND X2 (t) OF THE BIVARIATE DIFFUSION PROCESS APPLIED TO U. K., U. S., AND ECUADORION DATA Country E[Xl(t)] U. K. (110.2)exp(0.0266)t (72.78)exp(0.0141)t U. S. (282.5)exp(0.0332)t (42.40)exp(0.0397)t Ecuador (8089.0)exp(0.0519)t (22611.00)exp(0.0392)t developed economy of the U. K. and for the developing economy of Ecuador in so far as the relatiort between the development of national product and capital stock is concerned. In each case a small covariance is observed i j between the changes in the two variables, showing nearly a : constant change in each country although the capital stock increase is at a higher constant level in Ecuador than in the U. K. This latter reflects the tendency of high j i j investment in developing economies. The high correlation i between the changes in national income and investment in case of the U. S. shows that the change in investment |increases as changes in output increase. This is the ! famous theory of warranted growth, which is revealed by j | the approximately equal growth rates of output and investment. The high value of a.2 (relatively) in the U. S. case shows the fluctuating behavior of changes in s investment in the period considered for the analysis. 157 Thus, our results show a constant relationship between the changes in capital stock and national output; and an increasing relationship between the changes in output and investment. Based on our above model, we estimate the values of the variables by evaluating the trends given in Table XVIII. The estimated values obtained for 1964-1970 are shown in the following Tables XIX, XX, and XXI, respectively, for the U. K., the U. S., and Ecuador. Also, | we give the 95 per cent confidence ellipses for the two ; variables for the year 1966 in Table XXII. | TABLE XIX ESTIMATES FOR THE INDEX OF OUTPUT AND CAPITAL STOCK OF THE U. K. FOR 1964-1970 ! Year t e[x2 (t)] j - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1964 44 337.24 131.45 1966 46 346.34 133.31 1968 48 355.69 135.20 1970 50 365.29 137.12 TABLE XX 158 ESTIMATES FOR NATIONAL PRODUCT OF THE U. S. FOR 1964- AND INVESTMENT -1970 Year ..... . . t e[x x (t)j e[x2 (t)j 1964 18 513.57 86.68 1965 19 530.90 90.20 1966 20 548.83 93.85 1 1967 21 567.36 97.66 1968 22 586.52 101.61 | 1969 23 606.32 105.73 1970 24 626.79 110.02 iYear TABLE XXI ESTIMATES FOR NATIONAL PRODUCT AND CAPITAL STOCK OF ECUADOR FOR 1964-1970 ElXj. (t) e[x2 (t j j 11964 14 16740.65 39142.65 ! 11965 15 17633.36 40707.45 1966 16 18573.67 42334.80 1967 i 17 19564.13 44027.20 i 1968 I 18 20607.40 45787.26 11969 1 t 19 21706.30 47617.69 I 1970 20 22863.81 49521.28 159 TABLE XXII 95 PER CENT CONFIDENCE ELLIPSES FOR THE BIVARIATE LOGNORMAL DIFFUSION PROCESS FOR THE YEAR 1966 Country 95 Per Cent Confidence Ellipses U. K. 104 (.2023)X^(43)-103(.7858)X1(43)X2 (43) 2 +0.1923X2(43)+0.09074X1 (43)-51.0042X2 (43) +3383.9711 £ 5.99 u. S. ~3 2 -3 10 (,3788)X1 (20)-10 (.5562)X±(20)X2 (20) -3 2 +10 (.3846)X2(20)-0.3636X1(20)+0.2331X2 (20) +88.84 £ 5.99 Ecuador -6 2 -2 10 (.3552) Xx (16) +10 (. 15071X3^ (16) X2 (16) /■ O +10 (.2019)X2 (16)-10 (.6323)XX (16) -IQ1(.1408)X2(16)+356.81 ^ 5.99 Influence of Exogenous Variables As in the univariate analysis considered in the ^previous chapter, the introduction of exogenous variables ;like government expenditure in the bivariate model is simple and can be treated similarly. We consider below first the theory using government expenditure G(t) at time t as the exogenous variable and later present empirical results. 160 As government expenditure is one of the components of the income flow and, therefore, either competes or compliments investment, it seems more appropriate to jconsider its influence on investment from either point of j Iview rather than on capital stock. On the other hand, investment can he taken as an exogenous variable and, if it is, it would be more realistic and appropriate to consider national product and capital stock as the variables of the J i ibivariates diffusion process. Accordingly, therefore, we i i ! |shall present two illustrative applications of the ; exogenous variable: (i) government expenditure G(t) as exogenous and national product X^(t) and investment X^(t) i las the endogenous variables for the U. S. (1946-63), and j j (ii) investment as exogenous and national product X-^(t) and capital stock X2 (t) as endogenous variables for Ecuador |(1950-62). i I j Following the univariate approach, consider the j jparameters of the bivariate growth process as functions of 1 an exogenous variable, say government expenditure G(t), as (5.44) bi(t) = ki + b^ft) i = 1, 2 iwhere k^, b^, i = 1, 2 are constants. Taking corresponding linear effects 1(5.45) p±(t) = K± + p±G(t) iwe have then i With such specifications, the transition probability density function for jx-^t), X2 (t) , t ^ o} satisfying the Kolmogorov diffusion -equations is (5.48) f (r,x1,x2; t/yL,y2) = (y1y2) ’1{2ir (t-r) J {yiY2(1“4 iwhere exp{[-2 (t-T) (1-p)] Q'j, I Q'= {logy1-logx1-K1(t-r)-PL[G(t]T)/Y1 - 2p{logyi | -logx1-K1(t-r)-P1j^G(t)]_=}-[logy2-logx2-K2 (t-t) -P 2[G ^' t]T}/^YxY2 + Jlogy2-logx2-K2(t-T) -P2[G (t)]T} / Y2 jwhere Yi=ai/ K^= {k^-aj/2) # P^b^, i=l,2 and p=c/vaia2 Assuming that the observations are taken at equal intervals of time of unity, that is, (ta“' t : - Qf_^) =1, a = 1,2 . . .,n and taking initial time t^= 0, and further assume p|xi(0)=x1 q, X2 (0)=x2 0j- = 1; consider G(t) a step function as j (5.49) G(t) = G for t . < t ^ t , a=l,2, . . .,n j cy ^ ^ iso that | (5.5°) J a G(t)dt = Gt (tQ,-t^1) = G , <*=1,2, . . •,n tff-l- - O f 162 Then the maximum likelihood estimates are obtained as (5 51) V {0,li(lo® Ki,a-lo3* i 1oi-l)_® i J i (0«)}/n ' 1=1,2 (5.52) ?±- [ ^ { ( G q. ) (1°9xilQ.-l°9xi,a-l)} ' J! Uogxi; ff-logx±, „-i) /n ]/{J? ) -^S^Ga)j/n j . 2 (5.53) yL= j 1{(logxi_a-logxi;C(_1)-Ki-P1 (Ga)] /n , i=l,2 (5.54) p = J 1[{(logxlia-logx1(a.1)-K1-g1(Gj{(fogx2j0, _logx2 ,«-l)-*a- ® 2 (<5«>}}'n^VlY2 Hence, the maximum likelihood estimates of the parameters of the bivariate process are ! 1(5.55) ki=(fci+Yi/2) , i=l, 2 (5.56) £jL=$i , i=l, 2 I 1(5.57) a.=y, , i=l, 2 I x ____ (5.58) c =Pa/y^Y2 The sampling variances of the estimates are easily derived as (5.59) V(lcjL) = a^/nt + a^/2n , i=l, 2 (5.60) V(b±) = ait/nf^G^)2 , i=l, 2 j (5-61) V(a-) = 2s£/n , i = 1, 2 I X X i 1(5.62) V(c) = (a^a2 + c^)/n |(5.63) V(jo) = (l-p2)2 163 We observe that the variances of the,estimates tend to zero for large n and t where G(t) is constant or increasing, and therefore the estimates are consistent. | From the moments of the distribution of £x]_ (t), X2 (t)] jgiven by ■ ■ 3 r -P (5.64) E{[x1 (t)] [x2(t)] }=[xi(0 ) ] expjjfcKj+jY^Jt: j I exp{j ' [ (K2+j ' Y2/2) t+P2aSiGQr]} exp{jj'pv/YLY2 t}, j, j'= 1, 2, 3, . . . jwe obtain the means, the variances and the covariance of jthe variables easily as follows: (5.65) Means: E^X^ (t)] = X^(0) exp{kjLt+bi^S1 GQ-}, i = 1, 2 (5.66) Variances: V^X^ (t)] = (0)] exp { 2 (kit+b^f^GQ,) } ■|exp(ait) - l} | (5.67) Covariance: C^X-^(t), X 2 (t)j (0)j exp(k^t i - - ! t . | tb^I^G^) |exp(ct)-lj i Thus, the trends for X-^ (t) and X 2 (t.) as given by (5.65) can jbe evaluated using the initial values X^(0 ) and X2 (0 ) of jthe variables and the estimates of the parameters of the r jdiffusion process for given values of the exogenous jvariable G(t). Also, using the property of asymptotic normality, an approximate confidence ellipse of confidence 1 6 4 coefficient 100 (1-a?) for {x^ (t), X^ (t) J" can easily be obtained as | | (5.68) P { Q [ X 1 (t) , X 2 (t)] £ Xq/} ^ (1-Qf) J where ofx^t), X 2 (t)] = ^ (t) - E ^ (t) }, X2 (t)-e{x2 (t) }] v{xl(t)} c{x1(t), x2(t)} -1 X1 (t) -E- c{xx(t), x2(t)} v{x2(t)} X2 (t)-E-[x2 (t)} : follows asymptotically a- Chi-square distribution with two 2 idegrees of freedom, and where x is the upper (1 0 0 )a per O ' i icent point of the Chi-square distribution. i | If any other exogenous variable than government ! iexpenditure is considered, say investment I(t), then the |only change in the above analysis to be made is to replace :G(t) by I(t). [Empirical Results II i • • We now fit the modified model with the exogenous variable as government expenditure G(t) to the U. S. data (1946-63) of national product and investment, and as ■investment expenditure I (t) to the Ecuadorian data .(1950-65) iof national product and capital stock. The maximum likelihood estimates of the parameters and their sampling variances are obtained as shown in the following Tables XXIII and XXIV. f TABLE XXIII | i ESTIMATES OF THE PARAMETERS OF THE MODIFIED BIVARIATE DIFFUSION PROCESS APPLIED j TO THE NATIONAL PRODUCT DATA OF U. S. (1946-63) AND ECUADOR (1950-62) . Country £l £.2 A ai A a 2 A A c p U. S. Ecuador .0276 .1241 .0936 .0162 lot-7930) - -lot.2298) - 2 10 (.1197) lot-1269) -3 10 (.9237) -3 10(.4836) - 1 1 0 (.2400) -3 10 (.1669) - 2 10 (.3739) .80 -3 10 (.1406) .50 TABLE XXIV VARIANCES OF THE ESTIMATES OF THE PARAMETERS OF THE MODIFIED BIVARIATE DIFFUSION PROCESS APPLIED TO THE NATIONAL. PRODUCT.DATA. .OF U. S. (1946-r63) AND ECUADOR (1950-62) Country v(£x) v(*2) v(6x) v(fi2) Y(ax) V(a2) V(c) U. S. Ecuador -7 10 (.251) -4 +l0(.543)/t -8 ' 10(.974) -4 +l0(.403)/t -4 10 (.169) - 2 +10 (. 141)/t - 8 10(.116) , 4 +10(.139)/t -4 10(.543)t * K g*)2 Qf=l “ lot.403)t ( ^Gq ,)2 0F=1 “ - 2 10(.141)t <Ji G « )2 lot.139)t t 2 cv=l a - 6 1 0 (.1 0 1 ) -7 10(.390) -4 -5 10 (.678) 10(.213) _ 8 — 8 10 (.464) 10(.837) M Cl Oi 166 The negative value of £^ and relatively high value A ,of b-^ for the U. S. economy indicate that government i ! | expenditure competes with investment while influencing jincome positively, the latter being consistent with the jmultiplier concept. Of course, the negative value of lean be interpreted alternatively as a stabilizing effect of government spending, reducing investment incentives by increased demand for public goods and services. On the !other hand, the negative value of in the case of Ecuador | shows an adverse effect of investment on the growth of |national output which seems difficult to interpret. May be the result has occurred due to high investment associated with small changes in national income, and small investment I i with large changes in income during the period of analysis. This is not entirely improbable if we consider the time lag in the effect of investment to work itself out. With the above interpretation of our results, the model seems plausible to describe the economic growth process as a stochastic process. Based on the estimates obtained above, the trends for the economic variables are ]shown in Table XXV. Evaluating these trends for the 'periods of analysis, the estimated values corresponding to the observed values of the variables are obtained as given in Tables IV and V of Appendix B. Based on our theory, we also tried to predict the 'values of the variables Xjft) and X2 (t) for the year 1966 167 TABLE XXV ESTIMATED TRENDS FOR X1 (t) AND X, (t) OF THE MODIFIED .BIVARIATE DIFFUSION. PROCESS APPLIED TO U.: B. . (1946-63) AND. ECUADORION (1950-62) DATA Country 1 E[xx (t)] e[x 2 (t)] : !u. s. (282.5) exp{o. 0276t (42. 5)exp{o. I241t + ‘l54 (0.7930) -102 (0.1197) J lGJ Ecuador (8089.0)exp{o.0936t (998.0)exp{o.0162t - 104 (0.2298)a|1Ia} +104 (0.1267)^1^) and also tried to obtain their joint confidence interval of (95 per cent confidence , considering government expenditure I ' I of 106 billions of dollars, that is, G(20) = 106, and investment of 2800 millions of sucres, that is, 1(16) =2800 respectively for the U. S. and Ecuador during 1966. The results are as follows. The estimates for 1966 for the U. S. are X-^ (20) =552.9 and X2(20)=83.5, and those for Ecuador are X±(16)=17190.4 and X2(16) =44157.6. The corresponding 95 per cent confidence ellipses for 1966 for the U. S. and Ecuador are, respectively, as follows: -1 5700.7 19321.2 X1 (20)-552.9 X2(20)- 83.5 Xx(16)-17190 X2 (16)-44157 4296.9 -T-l 2295619 1829004 5214603 Xx (20)-552.9 X2(20)- 83.5 Xx (16)-17190 X2 (16) -44157^ £ 5.99 £ 5.99 where 5.99 is the upper 5 per cent point of the Chi-square .distribution with two degrees of freedom. i j jMultivariate Lognormal Diffusion Process i | We now generalize the bivariate lognormal diffusion 'process to a multivariate process where more than two variables/ say p ^2, are considered. The extension of the i diffusion process to a p-variate case is analogus to the bivariate case. Consider a vector X(t) of p economic ; variables characterizing the state of the economy at time t |The evolution of the economy can be considered by the joint development of these p variables as follows. Let X(t) be a random vector of p continuous variables of a stochastic process { x ( t ) , t ^ oj" of the Markovian type such that each component Xj_ (t) , i=lr 2, ..,p :assumes values in the interval (0, °°) of the real line, and jit represents the economic variable. Assume that the ! transition probability density function f for X(t), ( 5 . 6 9 ) f[r,x(T) ; t , £ ( t ) ] = p [ x ( t ) = y / X(T)=x 0 ^ r < t < 0 0 satisfies the backward and forward Kolmogorov equations: 169 where the functions b£^t,x{t)j and j£t,x(t)], i,j=l,2,. .,p jare the infinitesimal means and variance-covariances of the J i jchanges in X(t) during a small interval of time dt, and the | jspecific process is characterized by a particular set of jthese functions. Consider the following specification of these coefficients: (5.72) b±£tfx (t)j = £>1^ / 1=1/ 2, . . p (5.73) a^jj~t,x(t)J = ajLjXj^Xj , i,j=l, 2, . . ., p where b^, a^j are constants such that the matrix A of the 'coefficients a•• is positive semidefinite. We thus assume jthat the changes in X(t) during a small interval of time dt have the expectation and the standard deviation and the covariances as proportional to the present values of X(t)* Assuming A positive semidefinite implies that the economic jvariables X(t.) do undergo changes in the small interval dt of time though the magnitudes of the changes would be small for the small dt. In light of the constantly changing nature of the economy through the interactions of a complex ;of economic activities and the diffusion amongst them, the i l above specification for the nature of development of X(t) jseems to describe fairly well the growth process of the economy. With the coefficients defined by (5.72) and (5.73) jthe backward and forward Kolmogorov diffusion equations become as (5.75) -2£ = -L / r a1aijyiyjf>. - .g 9£fia« i dt 2 i,j=l dy^ dyj i=l dy^ i I i ;The transition probability density function f satisfying the diffusion equations (5.74) and (5.75) turns out to be the multivariate lognormal density function (5.76) f (T/Xyt,^) = J^y^V^ir) (t-Texp j-J^2 (t-T)J q| where Q =£logy-logx-J,(t-T)] f^flogy-logx-J^t-r)] , where :S=((°‘ ij)) = ((a^)) = A, i,j=l, 2/ . . ., p, and J3= (b-^a) , ;U~ ^iij- / * ^ " 1/2/. ../P« I The characteristics of this distribution given by i ] (5.76) are easily derived from those of the well-known I multivariate normal distribution. Denoting logX(t) by Z(t), jthe density function of Z(t) is the multivariate normal ! ■ - I with mean vector and covariance matrix as (5.77) E[z(t)J = [z(T)+£(t-T)] (5.78) v£z(t)j = 2 (t-T) = V, say 'respectively. Prom the moment generating function of JZ(t) » given by (5.79) E^exp-^6’jZ(t) = expj^Q' “ £z(t) +J3(t-T) } + hj$’ vej jthe moments of X(t) are easily obtained, from which the theoretical means and covariances for X(t) can be derived. If we reduce the initial time T as zero and take i I time interval of unity between successive states by a proper time scale transformation, and further assume that |pjx(O)=Xq| = 1, then the means and covariances of X(t) are j (5.80) E^Xi(t)J = Xi (0) expO^t) , i=l, 2, . . p (5.81) v[xi(t)j = Xi (0) exp(2bit) |exp(aiit) - l{, i=l,2,..,p j (5.82) cj^Ct), Xj (t^XitOjexp^tJXj (Ojexptbjt) j {exp(a±jt) - l}, i^j=l,2, . ..,p j | (5.83) R^X^t), Xj(t)] = {exp (a± jt) - l} / [{exp (a±it)-l} | " {exp(ajjt) - l}]^ i,j=l,2, . . ,p i | We observe that the dynamic path of the economy as 'specified in terms of the development of the variables X{t) jis given by the exponential trends (5.80). Also, the i variances (5.81) are exponential functions of time. Given the initial conditions, namely, the values of X(0) at time t = 0 and the values of the parameters b and A, the future | course of development of the economy can be evaluated from !the above expressions for trends. The values of the parameters can be taken as the estimates obtained from the empirical data of past periods. Such estimates are easily derived by the maximum likelihood method. We obtain the estimates using the maximum likelihood method to the - 172 transformed variables logX(t) and using the invariance property under monotonic transformation of such estimates. Let X.., x . . , . . ., x be the observed states of the —U “l —n : random vector X(t) at times tg, t^, . . tn respectively, iand let pjx(O) = xQ| = 1. The joint probability of the f [observations, known as the likelihood function, is (5.84, I - exp[-J5(tQ,-ta_1) Qa]}, ; where Qa=[logx^-logx^-L-j} (tff-tQ_ 1 )]s ogx^-logx^ 'Maximizing logL with respect to the parameters j3 and £, we i 'obtain the estimates as l I | | (5.85) $i~(^-O9xiof— logx^ q,_ i) 1^ ' 2,..,p | (5.86) ^ij= ~ a=i{(‘ ^a'-taf-l) ^logxioj-logxi, a-i-Bi ( ^"af— l^J i ) | [l09xjQ.-1°9xj,a-l-Sj(V ta-l, ]}'i'3=:1- 2' - "P I Then the maximum likelihood estimates for b and A are 'obtained from those of the parameters Ji and £ as !(5.87) bi = Bi + ^ii/2 , i=l, 2, . . ., p | (5.88) a±j = c^j , i, j=l, 2, . . ., p I i | j Under the assumption that the observations are taken ! at equal intervals of time of unity in suitable units, that 173 is, (‘ kcT't'a-l) = 1 ^or a = 1' 2, . . n, and t^=0, the imaximum likelihood estimates are obtained as ! 1(5-89) Pi=jE;|^lo9xia-logxi,a-l)/n ' 1=1' 2, . . p i i ; (5.90) 5'ij=a|1j(logxiC l ,-logxif a_i-Pi) (logxja-logxj # o-i-Sj)^- i/3 1» 2, - • . » p and the estimates for b and A in this case are obtained from (5.87) and (5.88) using the estimates given by (5.89) j j and (5.90). j ; j The asymptotic sampling variance-covariances of the i * [estimates jJ and 2 are easily obtained as (5.91) v(p±) ~ aii/nt' i = 1, 2, . . ., p (5.92) c(p±/ pj) — j / , i 7^ j = 1, 2, . • .,p (5.93) V(ff±j) (®"ijb ^ii^ j j )/n f - * ■ / 3 2, * * • / P (5.94) C (^i j ' 5i-j') = <aii-aj3’ + ^i-j^ij')/^ . i/ i . / j / j # ^ * * • * f P j We note that the variance-covariances of the i I estimates tend to zero for large n and t, and hence the * ■ * n [estimates J3 and S are consistent estimates. The sampling variances of the estimates b and A can be easily obtained from those of the estimates and 2 as (5.95) V(b^) = a^/nt + a?^/2n , i = 1, 2, . . ., p 174 ■N 2 (5.96) V(a±j) = (a.^ + ai;Lajj)/n , i = 1, 2, . . p j A jand these also tend to zero for large n and t, and hence b I A jand A are also consistent estimates. | Further, using the property of asymptotic normality, I f jwe can derive a joint confidence ellipsoid for the distribution of X(fc) as follows: The quadratic form (5.97) o[x(t)] = {x(t) - EfxttjJj-'c^lxtt) - E[x(j:)]} "i i Where C is the covariance matrix of X(t)/ follows |approximately a Chi-square distribution with p degrees of freedom. Hence an approximate confidence ellipsoid of jconfidence coefficient 100(1-a) for the values of X(t) is i jgiven by the inequality j i(5.98) p{o[x(t)] £ Xa} * (l-o;) 2 where %a is the upper (100) or per cent point of the Chi- t square distribution with p degrees of freedom. Empirical Example We now present an empirical illustration of a four j yariate lognormal diffusion process applied to the postwar ! data of the U. S. for the period 1946-63. Due to limited data available for the analysis, we restrict our analysis to only the following four aggregate variables: Gross national product X-^(t), Private consumption expenditure |X2 (t), Gross domestic investment Xgft), and Government 175 expenditure X4 (t), at time t measured in billions of dollars at constant prices of 1954. The maximum likelihood estimates of the parameters and their variances are observed as follows: (5.99) § = Hence, ' V ’ 0 . 0 3 2 7 4 3 " P 2 0 . 0 3 1 5 7 0 A . 0 3 0 . 0 2 7 5 2 6 A _ 0 . 0 4 4 5 9 9 _ I (5.100) £ 0.001103 0.000344 0.003851 0.000876 0.000447 0.001963 -0.000454 0.024637 -0.003793 0.013111 (5.101) b i > t * H i 0 . 0 3 3 2 9 4 b 2 0 . 0 3 1 7 9 4 f i 3 0 . 0 3 9 8 4 4 S . 0 . 0 5 1 1 5 5 (5.102) A = (a..) =103 xd 1.103 0.344 0.447 3.851 1.963 0.876 ■0.454 24.637 -3.793 13.111 The sampling variances of the estimates are found as - 6 (5.104) VfCT^) = 10 0.1431 0.0365 0.0235 176 1.5980 0.8957 0.8745 0.3568 71.4200 19.8500 20.2200 iHence, the variances of £ and A are (5.105) V(£) = 10 — -4 0.678_ " 0.03578 0.263 -6 0.00588 + 10 14.500 17.85240 7.710 5.05583 t=l,2, (5.106) V(a±j) = V(&±j) The empirical trend values for X(t) are given by (5.107) E^X-j^t)] = (282. 5) exp(0.033294t) , t =1, 2, ; (5.108) E[x2 (t)] = (192. 3) exp (0. 03l794t) , t =1, 2, j(5.109) E[x3(t)J= (42.4) exp (0.039844t) , t =1, 2, 1 (5.110) Efx4 (t)] = (43. 9) exp (0.051155t) , t =1, 2, We observe that the dynamic paths of the variables X(t-) given by the trend expressions (5.107) through (5.110) | depend on the initial state of the economy and the growth i |rates during the period of our analysis which are estimated as 3.3 , 3.2 , 4.0 , and 5.1 per cent respectively for the four variables. The estimated trend values on the basis of !these growth rates and the initial values of the variables. ...... ... ......... ’ .......... ’ 177.... that is, during the year 1946, are calculated and shown in Table VI of Appendix B. Further, based on our theory, we tried to predict the values of the variables X(t.) for the years 1964 through 1971. These predicted values are shown ; in the following Table XXVI. TABLE XXVI ESTIMATES OF GROSS NATIONAL PRODUCT X1(t), PRIVATE j CONSUMPTION X2 (t) , DOMESTIC INVESTMENT X3 (t) , j AND GOVERNMENT EXPENDITURE X4 {t) OF U. S. ' FOR 1964-1971 BASED ON THE MULTIVARIATE LOGNORMAL DIFFUSION PROCESS OF GROWTH . Year t EfXct)] e[x 2 <t>] E[x3(t)] e[x4 (t)] 1964 18 514.39 340.82 86.86 110.24 1965 19 531.80 351.83 90.39 116.03 1966 20 549.80 363.19 94.07 122.12 1967 21 568.42 374.93 97.89 128.53 1968 22 587.66 387.04 101.87 135.28 1969 23 607.56 399.56 106.01 142.38 1970 24 628.13 412.45 110.32 149.85 1971 25 649.40 425.77 114.81 157.71 The expressions for the variances and correlations ! i t j are (5.111) vfx-^t)] = (282.5)2 exp (0.066588t) {exp (0.001103t)-l} (5.112) v£x2(tJ] = (192*3)2 exp(0.063588t) {exp(0.000447t)-l} 178 (5.113) v£x3(tjj = (42.4)2exp(0.0796881) {exp(0.024637t) -l} (5.114) vfx4 (t)| = (43.9)2exp(0.1023l0t){exp(0.013111t)-l} where t = 1, 2, 3, . . . . Also, for t=l, the correlations between the variables are | i,j = 1, 2/ 3, 4. i I We note that the observed correlation coefficients reflect the nature of the relationship between the incre- ! jments per units of time in the variables analyzed. The high correlations 0.74 and 0.60 between the changes in national output and consumption with investment indicate I jthe familiar concepts of multiplier and acceleration I principles. Also, the negative correlation -0.21 between the increments in investment and government expenditure show the tendency of maintaining the aggregate expenditure Iby accelerating the public expenditure whenever investment !falls or increases slowly. I j Further, using the asymptotic normality theory, we |tried to obtain the joint confidence ellipsoid for the values of X(t) for the year 1966, that is, for t=20. The approximate confidence ellipsoid of confidence coefficient (5.115) R[x±(t), Xj (t)] = 0.50 0.74 0.23 1.00 0.60 -0.20 1.00 -0.21 *0 * " 1.00 179 of 95 per cent for X(20) is obtained as under. j (5.116) jx1(20)-549.8,X2(20)-363.2/X3(20)-94.1,X4 (20)-122.l] ! -1 6742.112 1378.333 4141.114 1186. 234~ X^O)-549.8" 1184.568 1367.944 -400.626 X^O)-363.2 5634.912 -839.187 X (20) - 94.1 3 4471.385 X(20)-122.1 _ 4 _ £ 9.49 'where 9.49 is the upper 5 per cent point of the Chi-square ;distribution with four degrees of freedom, j The above empirical results show some interesting |characteristics of the U. S. economy. The empirical trends | show that the economy is progressing at an average growth rate of about 3.3 per cent and the growth benefits nearly all go to higher standards of living as evidenced by nearly j the same average growth rate (3.2 per cent) in private consumption. Further, maintenance of such a constant growth in national product calls for a high investment rate. The growth rate in domestic investment has been |around 4 per cent during this period. More important, jhowever, is to note the high rate of growth in government jexpenditure, namely 5 per cent approximately. This is jprobably because of the instable characteristic of I jinvestment on one hand, and deliberate attempts of i i government to stabilize economic activity as evidenced by 180 the large variances in investment and government expenditure, and by the negative correlation between the j |changes in their levels- The negative relationship between |the changes in investment and government expenditure jpossibly reflects the substitutability and competitive | jbehavior of these two important components of the matured ■economy of the U. S- This nature of public expenditure is also evidenced with regard to private consumption by the negative correlation between the changes in the two. jWherever the growth in consumption is low, public consump- i tion presumably makes up the deficiency by accelerating its increase. l | The correlation coefficients may be interpreted as j __ |some sort of propensities, for example, the correlation of 0.50 between the changes in consumption and national income |can be regarded as a form of induced change in consumption increase for a unit change in output increase. Similarly, the correlation 0.60 may represent the induced change in investment increase for a unit change in consumption increase, and so on. These results thus provide the lempirical support to the theory of multiplier-accelerator. | We considered in this chapter the extension of the univariate lognormal diffusion process to two and more than two variables, and obtained some empirical results which indicate that our theory appears reasonable in describing ithe economic growth process. APPENDIX B 182 TABLE I INDEX OF OUTPUT (1913=100 Base), X-, (t) , AND CAPITAL STOCK (in 100 millions of sterling pounds), X2 (t), OF UNITED KINGDOM FOR 1880-1940 AT CONSTANT PRICES OF 1913 Year t x2(t) E[xX(t)] x2(t) E[x2 (t)] 1880 0 110.2 110.20 72.78 72.78 : 1882 1 114.4 113.17 73.61 73.81 1884 2 121.8 116.23 74.53 74.86 1886 3 114.4 119.36 75.40 75.92 1888 4 120.7 122.59 76.24 76.99 1890 5 134.1 125.89 77.24 78.09 1892 6 130.7 129.29 78.25 79.19 1894 7 127.4 132.78 79.19 80.31 1896 8 138.6 136.37 80.20 81.45 1898 9 148.6 140.05 81.32 82.61 1900 10 155.1 143.83 82.51 83.78 1902 11 153.5 147.71 83.72 84.97 1904 12 154.2 151.69 84.97 86.17 1906 13 166.5 155.99 86.27 87.39 1908 14 173.4 164.31 87.68 88.63 1910 15 169.2 168.74 89.00 89.89 1912 16 185.6 173.30 90.44 91.16 1914 17 196.7 177.98 92.00 92.46 TABLE I (continued) Year t xx(t) E[x-L(t) ] x2 (t) E[x2 (t)~] 1916 18 178.9 182.78 93.47 93.77 1918 19 164.0 187.71 94.92 95.10 1920 20 176.8 192.78 96.34 96.44 1922 21 136.5 192.78 97.38 97.81 1924 22 175.6 197.98 98.90 99.20 1926 23 180.5 203.32 100.16 100.60 1928 24 202.5 208.81 101.85 102.03 1930 25 208.2 214.45 ' 103.61 103.48 1932 26 185.9 220.24 105.00 104.94 1934 27 202.1 226.18 106.68 106.43 1936 28 201.2 232.28 108.61 107.94 1938 29 209.8 238.55 110.51 109.47 1940 30 216.3 244.99 111.01 111.02 Source: Data constructed by Hoffmann (72)- TABLE II GROSS NATIONAL PRODUCT, Xi(t), AND DOMESTIC INVESTMENT, X2(t), OF UNITED STATES FOR 1946-63 AT CONSTANT PRICES OF 1954 (in Billions of Dollars) Year t x x ( t ) e[x i (t)] x 2 ( t ) e[x ( t ) ] 1946 0 282.5 282.5 42.4 42.4 1947 1 282.3 292.0 41.5 44.1 1948 2 293.1 301.9 49.8 45.9 1949 3 292.7 312.1 38.5 47.8 1950 4 318.1 322.6 55.9 49.7 1951 5 341.8 333.5 57.7 51.7 1952 6 353.5 344.8 50.4 53.8 1953 7 369.0 356.4 50.6 56.0 1954 8 363.1 368.5 48.9 58.3 1955 9 392.7 380.9 62.5 60.6 1956 10 402.2 393.8 63.1 63.1 1957 11 407.0 407.0 57.8 65.6 1958 12 401.3 420.8 49.0 68.3 1959 13 428.6 435.0 61.7 71.0 1960 14 439.9 449.7 60.2 73.9 1961 15 447.7 464.9 57.5 76.9 1962 16 474.8 480.6 65.2 80.0 1963 17 492.9 496.8 67.7 83.3 Source: Data given in the references (155) and (156). TABLE III GROSS NATIONAL PRODUCT, X1(t)/ AND CAPITAL STOCK, X2(t)# OF ECUADOR FOR 1950-1962 AT CONSTANT PRICES OF 1960 (in Millions of Sucres) 1 1 jYear t xx(t) e[x x (t)] X2(t) (t) ] 1950 0 8089 8089.00 22611 22611.00 1951 1 8712 8520.35 23364 23514.91 1952 2 9472 8974.70 24107 24454.96 1953 3 9805 9453.29 24674 25432.59 1954 4 9945 9957.39 25124 26449.30 1955 5 10861 10488.38 26613 27506.66 1956 6 11379 11047.68 28490 28606.28 1957 7 12012 11636.80 29772 29749.87 1958 8 12322 12257.34 30986 30939.17 1959 9 13088 12910.97 32297 32176.01 1960 10 14094 13599.46 33639 33462.30 1961 11 14671 . 14324.66 34487 34800.02 1962 12 15038 15088.54 36150 36191.21 Source: (Ecuador), 1962 Memorial (Report del Gerente del Banco of the Central Bank, Central Ecuador). TABLE IV GROSS NATIONAL PRODUCT, X1 (t) , DOMESTIC INVESTMENT, X2 (t) , AND GOVERNMENT EXPENDITURE, G(t), OF UNITED STATES FOR 1946-1963 AT CONSTANT PRICES OF 1954 (in Billions of Dollars) Year i t G (t) xx(t) Efx^tJ/Gtt)] X2 (t)E[x2 (t)/G(t)] '1946 0 43.9 282.5 282.5 42.4 42.4 1947 1 37-2 282.3 291.3 41.5 45.9 1948 2 42.1 293.1 300.4 49.8 49.4 1949 3 47.2 292.7 310.0 38.5 52.9 1950 4 45.1 318.1 319.8 55.9 56.7 1951 5 63.3 341.8 330.4 57.7 59.5 1952 6 77.7 353.5 341.7 50.4 61.4 1953 7 84.3 369.0 353.7 50.6 62.8 1954 8 75.3 363.1 365.7 48.9 65.0 1955 9 73.2 392.7 378.2 62.5 67.4 1956 10 72.9 402.2 391.0 63.1 69.9 1957 11 75.0 407.0 404.3 57.8 72.4 1958 12 79.3 401.3 418.3 49.0 74.5 1959 13 80.1 428.6 432.7 61.7 76.7 1960 14 79.9 439.9 447.7 60.2 78.9 1961 15 84.3 447.7 463.3 57.5 80.7 1962 16 90.2 474.8 479.7 65.2 82.0 1963 17 93.7 492.9 496.8 67.7 83.0 Source: Data given in the references (155) and (156).; 187 TABLE V GROSS NATIONAL PRODUCT, X-, (t) , CAPITAL STOCK, X, (t) , AND INVESTMENT, I (t) , OF ECUADOR FOR 1950-1962 (in Millions of Sucres) i Year t I ( t ) x x ( t ) E [ x 1 ( t ) / I ( t ) ] x 2 ( t ) E [ x 2 ( t ) / I ( t 1950 0 998 8089 8089.0 22611 22611.0 1951 1 1218 8712 8637.7 23364 23337.5 1952 2 1055 9472 9258.2 24107 24037.5 1953 3 1430 9805 9838.1 24674 24876.8 1954 4 1837 9945 10357.0 25124 25878.3 1955 5 1971 10861 10870.0 26613 26965.9 1956 6 1917 11379 11422.0 28490 28080.0 1957 7 1890 12012 12009.9 29772 29230.1 1958 8 1803 12322 12653.2 30986 30393.8 1959 9 1971 13088 13279.7 32297 31671.1 1960 10 2081 14094 13901.9 33639 33048.2 1961 11 2193 14671 14516.0 34487 34534.1 1962 12 2410 15038 15081.7 36150 36186.2 j Source: Manorial del Gerente del Banco Central :(Ecuador), 1962. (Report of the Central Bank, Ecuador). TABLE VI GROSS NATIONAL PRODUCT, X, (t) , PRIVATE CONSUMPTION, X2(t), DOMESTIC INVESTMENT, X3 (t), AND GOVERNMENT EXPENDITURE, X4 (t) , OF UNITED STATES FOR 1946-1963 AT CONSTANT PRICES OF 1954 (in Billions of Dollars) Year t x1(t) *jxi(t)] x2(t) % 2 It)] x3(t) Ejx3(t)] X4(t) 1946 0 282.8 282.5 192.3 192.3 42.4 42.4 43.9 43.9 1947 1 282.3 292.1 195.6 198.5 41.5 44.1 37.2 46.2 1948 2 293.1 301.9 199.3 204.9 49.8 45.9 42.1 48 . 6 1949 3 292.7 312.2 204.3 211.5 38.5 47.8 47.2 51.2 1950 4 318.1 322.7 216.8 218.4 55.9 49.7 45.1 53.8 1951 5 341.8 333.7 218.5 225,4 57.7 51.7 63.3 56.7 1952 6 353.5 345.0 224.2 232.7 50.4 53.8 77.7 59.7 1953 7 369.0 356.6 235.1 240.2 50.6 56.0 84.3 62.8 1954 8 363.1 368. 7 238.0 248.0 48.9 ■58.3 75.3 66.1 1955 9 392.7 381.3 256.0 256.0 62.5 60.7 73.2 69.6 1956 10 402.2 394.1 263.7 264.3 63.1 63.1 72.9 73.2 1957 11 407.0 407.4 270.3 272.8 57.8 65.7 75.0 77.1 1958 12 401.3 421.2 273.2 281.6 49.0 68.4 79.3 81.1 1959 13 428.6 435.5 288.9 290.7 61.7 71.2 80.1 85.4 1960 14 439.9 450.2 298.1 300.1 60.2 74.1 79.9 89.8 1961 15 447.7 465.5 303.6 309.8 57.5 77.1 84.3 94.5 1962 16 474.8 481.2 317.6 319.8 65.2 80.2 90.2 99.5 1963 17 492.9 497.5 328.9 330.1 67.7 83.5 93.7 104.7 i Source: Data given in the references (155) and (156).; CHAPTER VI SUMMARY AND CONCLUSIONS j The purpose of this final chapter is to restate the Imain idea and objective of the present study and to briefly review and summarize results. At the end, some of the possible directions in which further research can extend the analysis to a more complete theory are explored. | i Introduction ! In the evolution of any scientific discipline there i is a period in which mathematical theories are developed jwhich supplement the qualitative or verbal theories; such I is the case in economics, especially in the theory of economic growth. Many attempts have been made in recent j !times to characterize the process of economic growth by an I abstract theory based on a few simple axioms derived from empirical phenomena relevant to economic development. Such ' [theories express, in the form of some set of equations, a postulated mechanism or model that generates a set of :theoretical observations which on comparison with empirical idata suggest the plausibility of the model. Nearly all the economic growth models, however, are deterministic in nature, that is, where the course of economic development I Jis determined by a rigid path specified in terms of few important economic variables which characterize the state 190 of economic conditions. Although these models are useful as a convenient first approximation of the reality, they are far from adequate in view of the stochastic nature of economic activities; also these models neglect the role of chance elements in the behavior of the economy. Economic activities such as investment and production are influenced by the so called random factors or stochastic elements. j j The state of uncertainty and imperfect knowledge is a rule ; rather than an exception in economic decisions. In so far I ;as the random elements and chance influences are inherent | fin the economic growth process, the need for a proba bilistic model in order to adequately describe the economic growth process is evident. It is this basic idea of developing a stochastic theory of economic growth process which has been the centre of discussion in this i i dissertation. More specifically, this study has attempted | |to develop a stochastic theory of economic growth using the | ;theory of stochastic processes of the diffusion type so as to overcome the unrealistic assumption of considering | ieconomic variables as discontinuous as is usually made in i I such a theory. To this extent the present study has ;departed significantly from the literature on stochastic ! models of economic growth, although it lies in the general realm of the theory of stochastic processes. Apart from extending the stochastic theory of economic development t i from its present state to a step further in terms of 191 continuous variables and thus provide a more realistic model of economic growth, our main aim in the present work has been with the applications rather than the more abstract formulation of a growth model. It has been the J empirical work emphasized in the present study which has ! i helped to provide useful conclusions regarding the structural similarities or dissimilarities of various economies. :Summary Having emphasized the need for a stochastic theory of economic development in the introductory chapter as j restated above, we began in Chapter II with the exposition |of a general theory of stochastic processes and its relevance in the economic growth process. Based on a simple assumption of temporal dependence of a Markovian ;characteristic about the evolutionary nature of the economy and considering the development of national income as a discrete birth process with a birth rate, a simple model of economic growth showed an interesting result; the mean development path of such a stochastic model is given I by the same exponential function x(t) = xQexp{Xt) as the j |development path of the corresponding continuous I j !deterministic growth model with a proportional growth rate I X, where x(t) denotes the national income at time t and Xq I is its initial value at time t = 0. Of course, this 192 similarity does not hold in all cases. There are more .cases in which the trend values of the stochastic model [differ from the dynamic path of the corresponding ; [deterministic model. On the otherhand, there are cases i |where many different stochastic models give rise to the ■same trend as shown by the negative binomial distribution [under the two distinct hypotheses of 'heterogeneity' and 1 contagion1. j i We then reviewed in Chapter III some of the j ; | applications of Markov processes of the discontinuous type [proposed in the literature on economic growth models. The [growth models based on discontinuous processes/ such as the jwell-known poisson process, birth and death processes and |their generalizations showed useful results in so far they ! [yield the type of short term trends commonly observed in the empirical data. However, such models have a drawback I [in that they are based on the unrealistic assumption of considering the economic variables as discontinuous and in that,the economy proceeds by discrete jumps. A more serious weakness of these models is that they are 1 inadequate to be applied to empirical test due to the difficulties involved in the estimation of the parameters of the models from the observed data. Thus the usefulness of such models are seriously limited on both grounds; lack (of empirical results to support the theory due to ! estimation difficulties and the unrealistic assumption of [ 193 .discontinuity in the economic variables. | Chapters IV and V proposed continuous stochastic [processes of the diffusion type in an effort to overcome | both the above drawbacks. We considered in these chapters i ja simple stochastic theory of the economic growth process, 'specified by a single economic variable X(t), say national income, or two or more economic magnitudes X^) , as a j | continuous stochastic process using the theory of Markov j I processes. The simplest economic growth process ' ! I constructed in Chapter IV to generate a skew distribution lis based on the 'law of proportionate effect1, which yields |a lognormal probability distribution law for the variable jx{t) of the process. In light of the joint impact of a jlarge number of random effects working simultaneously under the influence of certain interlinkages of economic activities (for example, investment or production activity),! i 1 the assumption of the hypothesis of proportionate effect j appears reasonable in the economic growth process. j i Our theory then states that changes in the economic i variable over non-overlapping intervals of time can be 'viewed as a product of a large number of random effects. ; Besides the above assumption regarding the growth mechanism of the economic variable which seems to describe reasonably well the plausible characteristic of the development of the economy, the Markov process describing the generation of lognormal distribution for X(t) further 194 .assumes that the transition probability depends on the I time spread between the initial state of the economy and i I the final state. This gives rise to the exponential trend e| jx(t)j = X(0)exp(bt) for the development of X(t) , which can be interpreted, in the first place as a linear development in logarithms of the variable and, in the second place as the result of a simple law of growth whereby the annual s increase in the variable is proportional to the magnitude j of the variable already reached. The economic meaning of j 'the proportional growth rate b in terms of the model of |income growth can be interpreted by treating b as the |product of the two structural coefficients of the Harrod- Domar type growth model, that is, b = c x ( T where a is the ! . |saving coefficient and O’ denotes the marginal output- capital ratio. Writing b = o-cr, the Harrod-Domar type j ! growth model =^-= bx has the solution x(t) = x(0)exp(bt), dt Iwhere x(0) is the initial value of the national income x(t) i jat time t = 0. | | | The weakness of the time-dependent lognormal jtransition probability law is that the variance of the Idistribution depends on time t and hence increases as the iprocess develops in time. This weakness in considering ithe economic growth process as a lognormal diffusion | jprocess is difficult to interpret in light of the contrary j evidence in empirical data, except that it may be regarded as the true underlying nature but is stabilized by 1 9 5 counteracting exogenous, factors such as deliberate jgovernment policies or actions by parties or economic |institutions involved in income determination. Some i I evidence does support this argument. Brown (1) and Roy j ( 1 2 8 , 1 2 9 ) . i The theory of lognormal diffusion process taken as a suitable first approximation to the economic growth process is thus considered as a stochastic model of economic ;growth. The parameters of such process are then derived j i I by the most efficient method of maximum likelihood, and the | I empirical verification of the model is considered by ^applying it to a number of economies. The results obtained i i |appear to be satisfactory. Observing satisfactory [ 'empirical results, we endeavored to modify our theory to include exogenous variables and obtained tentatively useful! results for policy proposals. Assuming that public ; I ;expenditure has a linear effect on economic growth as a I j I first approximation, various hypothetical policies regarding government expenditure were considered which would be used for accelerating or stabilizing the growth |rate at a desired level. Some important conclusions drawn ;from the empirical results are discussed later. Recognizing the fact that economic development can hardly be characterized by a single variable like national income, we tried to extend univariate model to a multivariate model in Chapter V so as to include a complex 196 of economic variables X(t) that jointly specify the state I of economic conditions at time t. The characterization of ! |such a generalized model is similar to the single variate |case both in regard to the simple case of endogenous | ivariables as well as its modified version considering the j influence of exogenous variables. Empirical verification is also made in each case which yield satisfactory results. | Some of the main results and conclusions are given below. i i :Conclusions and Comments | The main results and conclusions of the present I study can be briefly stated as follows: E 1. The process of economic growth as specified in (terms of the development of an important economic variable (or variables can be explained as a diffusion process of a Markovian nature following a lognormal probability density i law. Three implicit considerations are to be emphasized here. First, the assumption that the growth of an economy i follows the development of a stochastic process of a Markovian type means that the state of the economy at time t is described by the value of the random variable X(t) : with the Markov property: Probability distribution of X(t) given the value of X(r) is identically equal to the probability distribution of X(t) given the value of X(t') for all t‘£ t whenever r < t. Such an assumption implies : that future development is independent of the past except 197 that it depends on the present level. The hypothesis, that the level of the economy is subject to random shocks and has a certain inertia so that in any given period t it is j determined partly by the random effects experienced in that period and partly by the level of the economy in the j preceding period, appears plausible in light of the observed statistics and the available empirical results. Such theory has a close relation with the model advanced i [by Slutsky (137) who showed that the level of economy is | |determined by random shocks; however, this is not t |determined by the shocks in the present period alone, but by a weighted average of the shocks experienced in the current period and in the past periods. The difference between this and the Markov characteristic is that, according to the former, the current income level is | affected by past random shocks as such, while the latter j assumes the effects of past random shocks only in so far as | i ; | I these have affected immediate past income. There is, of j 1 i j |course, a genuine distinction here. i | Secondly, the lognormal probability law for the i I transition probability is the result of the basic assump- jtion underlying the mechanism of growth process, namely i jthe 'law of proportionate effect'. This can be possibly [explained by the theory of proportional linkages or the I jmultiplier-accelerator interactions. There is a third point implicit in the model that needs to be mentioned, 198 ■namely/ the evolutionary nature of the growth process. The variance of the random variable increases with time; this is contrary to the assumption of a constant variance in a stationary process. 2. As a result of the lognormal distribution of the growth process, the mean growth path of the variable comes out as an exponential trend. Such a trend is very common in economic data and has been used with success in graduation of empirical data. To this end the empirical data support our above model of characterizing the economic growth process as a diffusion process based on the [hypothesis of proportional change, and further subject to the Markov property. 3. A third important result of this study is that the theory of diffusion process of economic growth is applicable to a wide variety of economies; this indicates jthat the structural framework of the nature of economic I growth process is basically same in form although the values of the parameters are different in different situations and at different times or phases of growth. This is a very significant result observed from the ! empirical fit of the model to the different national | economies. i | 4. The empirical results are even more interesting. jThe simple growth model applied to real national product j [data for a recent period was found appropriate for the 199 developed economies of the U. S., Canada, and the Netherlands as well as for the developing economies of India and Ecuador. The estimated growth rates for these I economies during the last two decades are estimated to be i J3.3, 4.0, 5.0, 3.2 and 5.2 per cent respectively. Also, i the results obtained for these and many other countries for ;long-term data on real national output indicated reasonably good fit. These empirical results clearly show that the 1 theory of economic growth as a Markov process following | |a law of proportionate effect is a reasonably valid theory i ;of economic development. i 5. An interesting result from the empirical I |investigation is observed regarding the time behavior of f !the estimates of the growth model. In all cases except the U. S., the estimates of the parameter [ 3 of the lognormal process was found increasing in-time while that i of the parameter y, representing the variance of the changes in the variable, showed a decreasing tendency. This tends to show the stabilizing tendency of economic growth as time proceeds. Such a conclusion can be ;interpreted either as a genuine characteristic of economic |growth or as a result of deliberate policies of stabilize- I ! tion in recent time. If we regard the well-known logistic trend as the valid theory of long-run economic development l I !and consider our exponential trend as a first approximation to the true logistic trend, then the relevance of the above 200 results is very clear in so far as they show the different phases or stages of the logistic curve. Alternatively, it may be that the true mechanism of economic growth process !yields the exponential trend as our model explains, but it i j I is influenced by some exogenous factors in time so as to | |produce varying growth phases. i i j 6. The low value of the parameter p-^ representing the structural coefficient of the influence of the j |exogenous variable, government expenditure, on national ! i j jincome in the case of the U. S. relative to India indicates ;the private free-orionted enterprise nature of the U. S. :economy while the high value of in the case of India {provides the evidence of the important role of government in maintaining the economic growth in an underdeveloped country. ! | 7. The results obtained for the multivariate model ! I j applied to recent data (1946-63) of the U. S. for gross national product, private consumption, domestic investment, and government expenditure, are found equally significant, and lead to supporting conclusions to the above. The |empirical trends for the variables analyzed show that the i economy is progressing at an average growth rate of about three per cent and the benefits of this growth in real income nearly all go to higher consumption as indicated by i (the nearly same growth rate of 3.2 per cent in private iconsumption. Further, maintenance of such a growth in 201 national product is evidenced by the high investment growth :rate of around 4 per cent. A more important result is the I |high rate of public expenditure. Such results can be i I i interpreted as the instable characteristic of investment on one hand, and the deliberate stabilization policy of j government on the other hand. This is shown by the large variances in the changes in these variables coupled with j j the negative correlation between the two. The results ; obtained for the correlations between the changes per unit i : of time in the variables analyzed imply even more ! [interesting conclusions. The negative relationship between the changes in investment and government expenditure possibly reflects the substitutability and competitive behavior of these two important components of the matured i economy of the U. S. Whenever the growth in investment falls, public consumption presumably makes up the I i j | deficiency by accelerating its growth and thus tries to ! stabilize income growth. Further, the high correlation coefficients of 0.74 !and 0.60 between the increase in national output and consumption with investment increases indicate the working [of the familiar multiplier-accelerator principles. Then, i the correlation coefficients between the changes per unit | of time in the various variables can be interpreted as some form of measure of induced change in one for a unit change i iin the other. 202 Suggestions for Further Research Although statistical tests are lacking to determine to what extent the stochastic growth models are better than the conventional deterministic ones, it is obvious that the stochastic approach is preferable in so far it takes into account the role of chance effects and thus coincides with reality more closely. It is clear from the present study ! i that both from a priori reasoning and from the supporting j empirical results, our theory gives reasonably good results! i i by describing the economic growth process as a lognormal | 'diffusion process. However, it is difficult to claim that |this theory is complete and the best stochastic theory of j |the economic development process in the absence of any I formal statistical tests. Obviously, there are many questions that call for further research, and it seems j appropriate to mention a few suggestions here. i j It is evident from our present study that a more i appropriate analysis taking into account the role of chance factors in the development of an economy can be considered I Iin terms of the development of a stochastic process of the diffusion type. This study only considered a special sub-class of stochastic processes, namely Markov processes of a simple type. Here then is a possibility to depart i |from the present study and investigate the possibility of Jcharacterizing the economic growth process as a stochastic process of the type following the properties other than 203 Markov property. Even within the framework of Markov processes, it appears more appropriate to consider | i different assumptions regarding temporal dependence and j J alternative hypotheses about the nature of the transition probability. An alternative hypothesis to the one-step time dependent Markov characteristic, as considered in the present study, may be to consider a k-step time dependent I i | Markov process which would be more suitable if the present j state of economy is not only dependent on the immediate i i past but also on many past periods. There is more reason ;to hold this hypothesis true in the development of an economy in view of the fact that investment works with lags l * |and a gestation time. j : Secondly, more important is the question of the proper characterization of the transition probability i function by suitable hypotheses regarding the nature of the (underlying mechanism of the growth process of economic j (variables. It will have become apparent from the present I jstudy that the method of forming the general equations for [a diffusion process is very simple provided that the possible transition occuring in a small interval of time |At is suitably listed. The question of the proper form of i jthe distribution of the variable (or variables) needs more I (attention than the consideration of a realistic elementary (hypothesis about the nature of development of the growth process. It would be of use if it yields a trend that can j 204 be fitted to empirical data with success. To this end, it is suggestive to explore the possibility of characterizing a stochastic process which yields the well-known logistic law of growth that has been used with much success in graduation of empirical data describing the growth process i in many situations, both in economics and outside of economics. Assuming the hypothesis that the growth rate falls off linearly with an increase in the variable X(t), the differential equation in the deterministic model of where the growth rate b is taken as a linear function bgtl-x/k) of x, bg and k being constants. This (6.1) then as the well-known 'logistic' law. A corresponding diffusion equation analogue of the logistic growth process can be obtained as ; growth takes the form | (6 .1 ) = bQx(l-xA) yields the solution (6.2) x (t) = kx(0) exp(bQt)/{k + x (0) [exp (bQt) - l]} dt 9x2 9x 0 < x < " where a, P and y are constants, and f is Iunder the assumption that the infinitesimal mean is equal 205 to instantaneous change in the value of X(t), and the infinitesimal variance of the changes in X(t) during a small interval of time dt is proportional to the instanta- jneous value of X(t), namely i I I I ;(6 - 5) b (x) = ax - 0x^ and a(x) = yx I The solution of (6.3) appears to have not been available ! i in the literature. j i Finally, the analysis of exogenous factors attempted in this study can be extended to a more extensive level jboth in coverage and in respect of analytical methods. The | analysis can be enlarged in terms of many exogenous variables in a variety of ways which may yield quite interesting results. Another approach that seems more ■useful would be to consider a different type of influence of exogenous factors than the simple linear effect as 1 ■assumed in the present study. Possibly, exogenous relation- I I ships of a variety nature, a priori determined, may be !introduced as side restrictions subject to which the jdiffusion equations may be solved. j One can even expand the present study further by I considering the study of the conditional mean growth paths i i Jof the various endogenous variables given the other variables, both endogenous and exogenous. It is interest ing to note here that the system of relationships given by such conditional trends can be regarded as an equilibrium j 206 model for the economic system. To this end/ the theory of stochastic processes is a more general theory, as the } imodels usually considered in the literature are implicitly i ; included in it. The advantage, of course, is that the |parameters of such a system come out as functions of more j ibasic structural parameters in the general stochastic case. ! Furthermore, the usual difficulties experienced in dynamizing economic model in the deterministic case no longer remain in the stochastic model. Of foremost importance is the absence of identification and estimation i |problems in the stochastic models. Without going much |deeper into these aspects here, it seems appropriate to conclude with the hope that in future research some of these and any other considerations would be investigated. ;Concluding Remarks | In conclusion, it is clear from the present study i that one should take into consideration the role of random l effects in the analysis of economic development, and one | way this can be done is to consider the economic growth I i jprocess as a realization of a stochastic process. More j specifically, it is easily seen from the detailed discussion of this dissertation that a fair approximation to the real situation can be obtained by considering a lognormal diffusion process following the Markov property i as the process of economic development. It is our hope, 207 then, that this dissertation may prove helpful in raising an interest in the study of economic variables which can j jbe considered as realizations of a stochastic process, and ! |in clearing the ground for further advances in the t |analytical evolution of formulating economic growth models by a uniform and self-contained treatment of the more general problems of the stochastic theory of economic development. The attempt made in this dissertation to develop a stochastic theory of economic growth in a way ithat will interest both the economist and the mathematician would be worthwhile if it accelerates further research in ! this field. BIBLIOGRAPHY BIBLIOGRAPHY ! Aitchison, J., and Brown, J. A. C. The Lognormal Distribution. Cambridge: Cambridge University Press, 1957. Alexander, S. S. "Price Movements in Speculative Markets: Trends or Random Walks," Industrial Management Review of M. I. T., Vol. 2, pp. 7-26, 1961. { _. "Mr. Harrod's Dynamic *Model," Economic Journal, Vol. 60, No. 240, pp. 725-739, 1950. Anderson, T. W. An Introduction to Multivariate Statistical Analysis. New York: John Wiley and ■ Sons, Inc., 1958. j j i Arley, N. On The Theory of Stochastic Processes and ; Their Applications to The Theory of Cosmic Radiation. New York: John Wiley & Sons, Inc., j 1948. I | Armitage, P. "A Note on the Time-homogeneous Birth ! Process," Jour.Roy.Stat.Soc., B, Vol. 15, pp. 90-! 91, 1953. j Bachelier, L. J. "Theorie de la speculation," Ann. Sci. Norm. Sup., Vol. 3, pp. 21-86, 1900. Bailey, N. T. J. The Elements of Stochastic Processes ■ with Applications to the Natural Sciences. New York: John Wiley & Sons, Inc., 1964. ! j Bartlett, M. S. "Some Evolutionary Stochastic j Processes," Jour. Roy. Stat. Soc., B, Vol. 11, ! pp. 211-229, 1949. j ________. An Introduction to Stochastic Processes. London: Cambridge University Press, 1955. ________. Stochastic Population Models in Ecology and | Epidemiology. London: Methuen, 1960. j ) ________. Essays on Probability Theory and Statistics, j London: Methuen, 1962. I i i Baumol, W. J. Economic Dynamics. New York: The j Macmillan Company, 2nd ed., 1959. 210 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. Bharucha-Reid, A. T. Elements of The Thery of Markov Processes and Their Applications. New York: McGraw-Hill Book Company, Inc., 1960. Billingsley, P. "Statistical Methods in Markov Chains," Ann. Math. Stat., Vol. 32, pp. 12-40,1961. Blanc-Lapierre, A. and Fortel, R. Theorie des fonctions ale'atoires. Paris: Masson et Cie, 1953. ________. The Theory of Random Functions, Vol. 1. New York: Gordan and Breach Science Publishers, 1965. Bowley, A. L. F. Y. Edgeworth's Contributions to Mathematical Statistics. London: Royal Statis tical Society, 1928. Bowley, 0. "The Action of Economic Forces in Produ cing Frequency Distributions of Income, Prices, and other Phenomena: A Suggestion for Study," Econometrica, Vol. 1, pp. 358-372, 1933. Central Bank, Ecuador. Memoria del Gerente del Banco Central, Ecuador, 1962. Central Statistical Organization, Government of India.j Estimates of National Income, April 1963, April 1965. Champernowne, D. G. "A Model of Income Distribution," Economic Journal, Vol. 63, ~p>p. 818-351, 1953. ; Cohen, A. C. "Estimating Parameters of Logarithmic- | normal Distributions by Maximum Likelihood, " Jouri Amer. Stat- Asso., Vol. 46, pp. 206 ff., 1951. 1 ' i Cootner, P. H. "Stock Prices: Random versus Syste- ; matic Changes," Industrial Management Review of i M. I. T., Vol. 3, pp. 24-45, 1962. _______ .(ed.) The Random Character of Stock Market Prices. Cambridge, Mass.: The M. I. T. Press, j 1964. Cowles, A. "A Revision of Previous Conclusions Regarding Stock Price Behavior," Econometrica, Vol. 28, pp. 909-915, 1960. — Davies, G. R. "The Logarithmic Curve of Distribution" Jour. Amer. Stat. Asso., Vol. 20, pp. 29 f., 1925.; ' ! 28. 29. 30. 31. 32. 33. 34. , 35. 36. 1 37. t I [ 38. 39. ! 40. 211 ________. "Pricing and Price Levels," Econometrica, Vol. 14, pp. 219-226, 1946. Davis, H. T. The Analysis of Economic Time Series. Bloomington, Indiana: The Principia Press, pp. 394 f., 1941. Theory of Econometrics. Bloomington, Indiana: The Principia Press, pp. 208 f., 1941. Deane, Phyllis and Cole, W. A. British Economic Growth 1688-1959: Trends and Structure. London: Cambridge University Press, Domar, E. D. "Capital Expansion, Rate of Growth and Employment," Econometrica, Vol. 14, pp. 137-147, 1946. "Economic Growth: An Econometric Approach, 1 1 Amer. Econ. Rev., Vol. 42, pp. 479-495, 1952. Essays in The Theory of Economic Growth. New York: Oxford University Press, 1957. Doob, J. L. Stochastic Processes. New York: John Wiley & Sons, Inc., 1953. Draper, J. “Properties of Distributions Resulting from certain Simple Transformations of the Normal Distribution," Biometrika, Vol. 39, pp. 290 ff., 1952. Fama, E. F. "Mandelbrot and the Stable Paretian ! Hypothesis," Journal of Business, Vol. 36, pp. 420-429, 1963. j ________. "The Distribution of the Daily First Differ-1 ences of Stock Prices: A Test of Mandelbrot's j Stable Paretian Hypothesis," Journal of Business, ! Feller, W. "Zur Theorie der Stochastischen Prozesse (Existenz und Eindeutigkeitssatze),“ Math. Ann., Vol. 113, pp. 113-160, 1936. "Die Grundlagen der Valterraschen Theorie des Kampfesums Dasein in wahrscheinlichkeitstheor-j etischer Behandlung," Acta Biotheoretica, Vol. 5,! pp. 11-40, 1939. j 212 41. 42. 43. 44. 45... 46. 47. 48. 49. 50. 51. 52. 53. 54. _________. "On the Integrodifferential Equations of Completely Discontinuous Markov Processes," Trans. Amer. Math. Soc., Vol. 48, pp. 488-515, 1940. ________. "On the Logistic Law of Growth and Its Empirical Verification in Biology, " Actabiotheor, Leiden, Vol. 5, pp. 51 ff., 1940. ; ■ i ________. "On the Theory of Stochastic Processes with Particular Reference to Applications," Proc. j First Berkeley Symposium on Mathematical Statis- j tics and Probability, pp. 403-432, 1949. j 1 ________. "Some Recent Trends in the Mathematical Theory of Diffusion," Proc. International Cong ress of Mathematicians (Cambridge, Mass.), Vol. 2, pp. 322-339, 1950. ________- "Diffusion Processes in Genetics," Proc. Second Berkeley Symposium on Mathematical Statis tics and Probability, pp. 227-246, 1951. ________. "Two Singular Diffusion Problems, " Ann. Math.! Vol. 54, pp. 173-182, 1951. _______. "Diffusion Processes in One Dimension," ! Trans. Amer. Math. Soc., Vol. 77, pp. 1-31, 1954. j I ' _____• An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: John Wiley & Sons, Inc., 1957. Fisz, M. Probability Theory and Mathematical Statis- ; tics. New York: John Wiley & Sons, Inc., 1963. Furry, W. "On Fluctuation Phenomena in the Passage of| Higher Energy Electrons through Lead," Phys. Rev. , I Vol. 52, pp. 569 ff., 1937. j I Gaddum, J. H. "Lognormal Distributions," Nature, London, Vol. 156, pp. 463 f., 747 f., 1945. Galton, F. "The Geometric Mean in Vital and Social ; Statistics, " Proc. Roy. Soc. . , Vol. 29, pp. 365 ff. , ' 1879. j t i ________. "Co-relations and Their Measurements, chief-' ly from Anthropometric Data," Proc. Roy. Soc., | Vol. Ill, pp. 135 ff., 1888. I j ________. Natural Inheritance (Loindon)1889. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 213 i i Gibrat, R. "Une loi des repartitions ^conomiques: l'effect proportionnel," Bull. Stat. gen. Vol. 19, pp. 469 ff., 1930. Godfray, M. D., Granger, C. W. J., and Morgenstern, O. "The Random Walk Hypothesis of Stock Market j Behavior," Kyklos, Vol. 17,_pp. 1-30, 1964. j Granger, C. W. J. and Morgenstern, O. "Spectral J Analyses of New York Stock Market Prices," Kyklos, ! Vol. 16, pp. 1-27, 1963. Grenander, U. "Stochastic Processes and Statistical | Inference,” Ark. Mat., Vol. 1, pp. 195 fff, 1950. ; _______ . "On Empirical Spectral Analysis of Stochastic! Processes," Ark. Mat., Vol. 1, pp. 503 ff., 1951. ! _______ •i and Rosenblatt, M. "On Spectral Analysis of; Stationary Time Series," Proc. Nat. Acad. Sci., Washington, Vol. 38, pp. 519 ff., 1952. i _______ . "Statistical Spectral Analysis of Time Series arising from Stationary Stochastic Pro- ! cesses," Ann. Math. Stat., Vol. 24, pp. 537 ff. ' 1954. | _______ . Statistical Analysis of Stationary Time j Series. New York: John Wiley & Sons, Inc., 1957. ! Haavelmo, T. A Study in the Theory of Economic Evolution. Amsterdam: North-Holland Publishing Company, 1954. Hamberg, D. Economic Growth and Instability. New York:; W. W. Norton and Company, 1956. j I t Harrod, R. P. Towards a Dynamic Economics. London: Macmillan & Co., Ltd., 1948. | Hart, P. E. and Prais, S. J. "The Analysis of Bus iness Concentration: A Statistical Approach," Jour. Roy. Stat. Soc., A, Vol. 119, part 2, pp. j .150-191, 1956. j i Hicks, J. R. "Mr. Harrod's Dynamic Theory," Economica, Vol. 16, No. 62, pp. 106-121, 1949. _______ . A Contribution to the Theory of Trade Cycle. Oxford: Oxford University Press, 1950. \ 214 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. ________. Capital and Growth. Oxford: Oxford University Press, 1965. Higgins, B. Economic Development. New York: W. W. Norton & Company, Inc., 1959. Hill, T. P. "An Analysis of the Distribution of Wages and Salaries in Great Britain,” Econometrica, Vol. 27, pp. 355-381, 1959. Hoffmann, W. G. British Industry, 1700-1950. Trans lated by Henderson, W. O. and Chaloner, W. H. (Oxford: Basil Blackwell), p. 333 f., 1955. ________, and Muller, J. H. Das Deutsche Volksein- kommen 1851-1957. Unter Mitarbeit von Heinz Konig and others. J. C. B. Mohr, (Paul Siebeck) Tubin gen, 1959. Houthakker, H. S. "The Pareto Distribution and the Cobb-Douglas Production Function in Activity Analysis,” The Review of Economic Studies, Vol. 23, pp. 27-31, 1955. - "The Econometrics of Family Budgets," Jour. Roy. Stat. Soc., A, Vol. 115, pp. 1-28, 1952. Johnson, N. L. “Systems of Frequency Curves gener ated by Methods of Translation," Biometrika, Vol. 36, pp. 149 ff., 1949. ~ Kalecki, M. "On the Gibrat Distribution," ™ Econometrica, Vol. 13, pp. 161-170, 1945. . "A Macrodynamic Theory of Business Cycles," Econometrica, Vol. 3, pp. 327-344, 1935. Kapteyn, J. C. Skew Frequency Curves in Biology and Statistics. Astronomical Laboratory, Groningen: Noordhoff, 1903. ________, and Van Uven, M. J. Skew Frequency Curves in Biology and Statistics. Groningen; Hoitsema Bros., 1916. Kendall, D. G. "On the Generalized Birth and Death Process," Ann. Math. Stat., Vol. 19, pp. 1-15, 1948. 82. . "On Some Models of Population Growth Leading to R. A. Fisher's Logarithmic Series Distribution," 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 215 Biometrika, Vol. 35, pp. 6-15, 1948. ________. "Stochastic Processes and Population Growth," Jour. Roy. Stat. Soc., B, Vol. 11, pp. 230-264, 1949. ________- "Some Analytical Properties of Continuous Markov Transition Functions," Trans. Amer. Math. Soc., Vol. 78, pp. 529-540, 1955. ________. "Deterministic and Stochastic Epidemics in Closed Population," Proc. Third Berkeley Sympo sium on Mathematical Statistics and Probability, Vol. 4, pp. 149-165, 1956. Kendall, M. G. "The Analysis of Economic Time Series," Jour. Roy. Stat. Soc., A, Vol. 116, pp. 11-34,1953. ________, and Stuart, A. The Advanced Theory of Statistics, Vol. 2. New York: Hafner Publishing Company, 1961. Kendrick, J". W. Productivity Trends in the United States. National Bureau of Economic Research, New York. Princeton: Princeton University Press, pp. 293-95, 1961. Keynes, J. M. "The Principal Averages and the Laws of Error which lead to them, " Jour. Roy. Stat. Soc., Vol. 74, pp. 322 ff., 1911. Klein, L. R. Textbook of Econometrics. Evanston, Illinois: Row, Peterson and Co., 1953. ________. An Introduction to Econometrics. Englewood Cliffs, N. J.: Prentice-Hall, Inc., 1962. Kolmogorov, A. N. "Uber die Analytischen Methoden der Wahrscheinlichkeitsrechnung, " Math. Annalen, Vol. 104, pp. 415-458, 1931. ________. "Uber das Logarithmisch Normale Verteilungs- gesetz der Dimensionen der Teilchen bei Zerstuc- kelung, " C. R. Acad. Sci., U. R. S. S., Vol. 31, pp. 99 ff., 1941. ________. "On Some Problems Concerning the Differen tiability of the Transition Probabilities in a Temporally Homogeneous Markov Process having a Denumerable Set of States" Ucenve Zapiski Mat. Moskov Go s. Univ., Vol. 148, pp. 53-59, 1951. 216 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. Kuznets, S. S. National Product Since 1869. National Bureau of Economic Research, New York, pp. 52-60, 1946. Laurent, A. G. A Stochastic Model for the Behavior of a Set of Price Index-numbers and Its Applica tions. East Lansing, Michigan: Michigan State University Research Memo. 37, 1957. ________. "Comments on Brownian Motion in the Stock Market," Operations Research, Vol. 7, pp. 806-7, 1959. Levy, P. Processus Stochastiques et mouvement Brownien. Paris: Gauthier-Villars, 1937. Lorenz, M. 0. "Methods of Measuring the Concentra tion of Wealth," Publications of. the Amer. Stat. Asso., New Series, Vol. 70, pp. 2D9 ff., 1905. Lydall, H. P. "The Distribution of Employment Incomes," Econometrica, Vol. 27, pp. 110-15,1959. McAlister, D. "The law of the Geometric Mean," Proc. Roy. Soc., Vol. 29, pp. 367 ff., 1879. McKendrick, A. G. "Studies on the Theory of Continu ous Probabilities with Special Reference to Its Bearing on Natural Phenomena of a Progressive Nature,” Proc. Lond. Math. Soc. (2), Vol. 13, pp. 401 ff., 1914. Maddison, A. Fluctuations in Trade and the Trans mission of Cycles. Banca Nazionale Pel Lavoro, Vol. 56, pp. 127-195, 1962. Mandelbrot, B. "The Pareto-Levy Law and the Distri bution of Income," International Economic Review, Vol. 1, pp. 79-106, 1960. ________. "Stable Paretian Random Functions and the Multiplicative Variation of Income," Econometrica. Vol. 29, pp. 517-543, 1961. ________. "The Variation of Certain Speculative Prices," Journal of Business, Vol. 36, pp. 394- 419, 1963. Moore, A. "Some Characteristics of Changes in Common: Stock Prices," The Random Character of Stock Market Prices. P. H. Cootner (ed.) Cambridge, 2X7 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. Mass.: The M. I. T. Press, pp. 412-474, 1964. Moran, P. A. P. The Statistical Processes of Evolutionary Theory. Oxford: Clarendon Press, 1962. Morishima, M. Ecruilibrium. Stability and Growth. Oxford: Clarendon Press, 1964. Moyal, J. E. "Discontinuous Markov Processes," Acta Math., Vol. 98, pp. 221-264, 1957. _______ . "Stochastic Processes and Statistical Physics," Jour. Roy. Stat. Soc., B, Vol. 11, pp. 150-210, 1949'i Mukerjee, K. "A Note on the Long-term Growth of National Income in India, 1900-01 to 1952-53,” Paper Presented at the 2nd Indian Conference of Research in National Income, 1960. _______ . Levels of Economic Activity and Public Expenditure in India. Bombay, India: Asia Publishing House, pp. 57-58, 1965. Mukerjee, V., Tintner, G., and Narayanan, R. "A Generalized Poisson Process for-the Explanation of Economic Development with Applications to Indian Data," Arthaniti, Vol. 7, 1965. Narayanan, R. Some Stochastic Models of Economic Development. Unpublished M. S. thesis. Univer sity of Southern California, 1965. Narasimham, G. V. L. Economic Models of Growth and Planning. Unpublished Ph. D. dissertation, University of Pittsburgh, 1964. Okawa, K., Shanohara, M., and others. The Growth Rate of Japanese Economy Since 1878. Tokyo: Kinokuniya Bookstore Co., pp. 230 ff., 1957. Orcutt, G. H., Greenberger, M., and others. Micro- analysis of Socio-economic Systems. New York: Harper and Row, Inc., 1961. Osborne, M. P. M. "Brownian Motion in the Stock Market," Operations Research, Vol. 8, pp.145-73, 1959. 218 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. ________- "Reply to Comments on Brownian Motion in the Stock Market," Operations Research, Vol. 7, pp. 807-811, 1959. Pareto, V. Manuel d'Economie Politique, 2nd ed. Parzen, E. Stochastic Processes. San Franciscos Holden-Day, Inc., 1965. Pearson, K. "Contributions to the Mathematical Theory of Evolution: Skew Variation in Homogene ous Material," Phil. Trans., Vol. 186, pp. 343 1895. Prais, S. J. and Houthakker, H- S. The Analysis of Family Budgets. Cambridge University Press, 1955. Rao, C. R. Advanced Statistical Methods in Biometric Research. New York: John Wiley & Sons, Inc.,1952. Rosenblatt, M. Random Processes. Oxford: Oxford University Press, 1962. Rostow, W. W. The Stages of Economic Growth. Cambridge: Cambridge University Press, 1962. Roy, A. D. "The Distribution of Earnings and of Individual Output," Economic Journal, Vol. 60, pp. 489 ff., 1950. ________. "A further Statistical Note on the Distri bution of Individual Output," Economic Journal, Vol. 60, pp. 831 ff., 1950. ________. "Some Thoughts on the Distribution of Earnings," Oxford Economic Papers, New Series, No. 3, pp. 135 ff., 1951. Sengupta, J. K. and Tintner, G. "A Stochastic Programming Interpretation of the Domar-type Growth Model," Arthaniti, Vol. 5, 1963. ________. "On Some Aspects of Trend in the Aggregative Models of Economic Growth,” Kyklos, Vol. 16, pp. 47-61, 1963. ________. "An Approach to A Stochastic Theory of Economic Development with Applications," in the volume of Essays in honour of Michal Kalecki, Problems of Economic Dynamics and Planning. 219 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. PWN-Polish Scientific Publishers/ Warszawa, 1964. Schumpeter, J. A. The Theory of Economic Development. Cambridge: Harvard University Press, 1934. Business Cycles, 2 vols. New York: McGraw- Hill Book Company, 1939. Simon, H. A. "On a Class of Skew Distribution Functions," Biometrika, Vol. 42, pp. 425-40,1955. Slutsky, E. "The Summation of Random Causes as the Source of Cyclic Processes," Econometrica, Vol. 5, pp. 105 ff., 1937. Solow, R. "A Contribution to the Theory of Economic Growth," Quarterly Journal of Economics, Vol. 70, pp. 65-94, 1956. Stone, J. R. N. "Prediction from Autoregressive Schemes and Linear Stochastic Difference Systems," Proc. of the International Statistical Conferences, Vol. 5, 1947. Sprenkel, C. M. "Warrant Prices as Indicators of Expectations and Preferences," Yale Economic Essays, Vol. 1, pp. 178-231, 1961. Sweezy, P. M. The Theory of Capitalist Development. New York: Oxford University Press, 1942. Symposium on Stochastic Processes. Jour. Roy. Stat. Soc., B, Vol. 11, pp. 150 ff., 1949. Takacs, L. Stochastic Processes. New York: John Wiley & Sons, Inc., 1960. Takashima, M. "A Note on Evolutionary Processes," Bull. of Math. Stat., Vol. 7, pp. 18-24, 1956. Theil, H. Economic Forecasts and Policy. Amsterdam: North-Holland Publishing Company, 1958. Tinbergen, J. On The Theory of Trend Movements. L. H. Klaasen, L. M. Koyck, and H. J. Witteveen (eds.). Amsterdam: North-Holland Publishing Company, 1959. Tintner, G. Econometrics. New York: John Wiley & Sons, Inc., pp. 208-9, pp. 270 ff., 1952. 220 ; 148. 149. 150. 151- 152. 153. 154. 155. 156. 157. 158. ________. "Eine neue Methode fur die Schatzung der Logistishen Funktion," Metrika, Vol. 1, pp. 154- 157, 1958. ________. "A Stochastic Theory of Economic Develop ment, " in H. Hegeland (ed.) Money, Growth and Methodology. Lund, Sweden: Gleernp Publishers, pp. 59-61, 1961. _______, et. al. "A Simple Stochastic Process for the Explanation of the Trend of Regional Develop ment, " Los Angeles: Department of Economics, University of Southern California (mimeographed), 1963. ______ , and Sengupta, J. K. "Ein Verallgemeinerter Geburten-und Todesprosess zur Erklaerung des Deutschen Volkseinkommens, 1851-1939," Metrika, Vol. 6, pp. 143-47, 1963. ________, and Thomas, E. J. "Un Modele Stochastique de Development Economique avec Application a 1'Industrie Anglaise," Revue D 'Economie Politique, Vol. 73, pp. 278-80, 1963. ________, and Patel, R. C. "A Lognormal Diffusion Process Applied to Economic Development of India," Los Angeles: Department of Economics, University of Southern California (mimeographed), 1965. ________, and Patel, R. C. "Multivariate Lognormal Diffusion Process of Economic Development,” Los Angeles: Department of Economics, University of Southern California (mimeographed), 1965. U. S. Department of Commerce, Bureau of the Census. Historical Statistics of the United States, Colonial Times to 1957, A Statistical Supplement. U. S. Government Printing Office, 1960. ________. Statistical Abstract of the United States, 1964. U. S. Govern. Printing Office, 1965- Van Der Wijk, J. "Incomens-en Vermogens Verdeling, " Netherlands Economic Institute Publication, No. 26, Haarlem: De Erven F. Bohn, 1939. Van Uven, M. J. "Logarithmic Frequency Distribu tions, " Proc. Acad. Sci. Amst., Vol. 19, pp. 533 ff., 1917. 159. 160. 161. 162. 163. 164. 221 Vining, R. "Regional Variation in Cyclical Fluctuation viewed as a Frequency Distribution," Econometrica, Vol. 13, pp. 183 ff., 1945. ________. "The Region as a Concept in Business-cycle Analysis, 1 1 Econometrica, Vol. 14, pp. 37 ff., 1946. Wicksell, S. D. "On Logarithmic Correlation with An Application to the Distribution of Ages at First Marriage," Medd. Lunds Astr. Obs., Vol.84, 1917.! i ________. "On the Genetic Theory of Frequency," Ark. Mat. Astr. Fys., Vol. 12, pp.l ff., 1917. Yule, G. U. "A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, ; F. R. S.," Phil. Trans. Royal Soc., London, B, Vol. 213, pp. 21-87, 1924. ! ________. "On a Method of Investigating Periodicities| in Disturbed Series, with special Reference to j Wolfer's Sun-spot Numbers," Phil. Trans., A, Vol. 226, pp. 267 ff., 1927. I I I

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Applications Of Stochastic Processes To Indian Agriculture

PDF

Approaches To Economic Policy For Stabilization And Growth

PDF

A Theory Of Regional Economic Growth: Growth Poles And Development Axes

PDF

Some Effects Of Monetary And Fiscal Policy On The Distribution Of Wealth

PDF

On The Dynamics Of Planned Economy: General Theory With Empirical Analysis Of The Hungarian Experience

PDF

The Economics Of A Non Profit Enterprise In The Dental Health Care Field

PDF

Some stochastic models of economic development

PDF

Production Functions In Korean Manufacturing: An Analysis Of Trade Policyand Its Effect On Technological Change

PDF

Optimal And Adaptive Control Of A Stochastic Service System With Applications To Hospitals

PDF

Some External Diseconomies Of Urban Growth And Crowding: Los Angeles

PDF

Export Instability And Economic Development: A Statistical Verification

PDF

Appraisal Of Developmental Planning And Industrialization In Turkey

PDF

The Economics Of Sugar Quotas

PDF

European Economic Integration And African States

PDF

Application of benefit-cost analysis to evaluate the impact of the Greater Chao Phys Project to the economy of Thailand

PDF

A Generalized Economic Derivation Of The ''Gravity Law'' Of Spatial Interaction

PDF

Pollution, Optimal Growth Paths, And Technical Change

PDF

A Constrained Price Discriminator: An Application To The U.S. Post Office

PDF

External Trade And Economic Growth In Developing Countries With Special Reference To Iraq

PDF

Valuation Adjustment In Input-Output Tables, Economic Structure, And Economic Systems: A Comparative Empirical Analysis

Asset Metadata

Creator Patel, Ramanbhai Chhaganbhai (author)

Core Title Applications Of Stochastic Processes To Economic Development

Degree Doctor of Philosophy

Degree Program Economics

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag economics, general,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Tintner, Gerhard (committee chair), Beckwith, Richard E. (committee member), Bilas, Richard A. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-109380

Unique identifier UC11360050

Identifier 6700418.pdf (filename),usctheses-c18-109380 (legacy record id)

Legacy Identifier 6700418.pdf

Dmrecord 109380

Document Type Dissertation

Rights Patel, Ramanbhai Chhaganbhai

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA