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Pollution, Optimal Growth Paths, And Technical Change
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Pollution, Optimal Growth Paths, And Technical Change
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I I 72-17,525 WONG, Robert Eing, 1938- POLLUTION, OPTIMAL GROWTH PATHS, AND TECHNICAL CHANGE. University of Southern California, Ph.D., 1972 Economics, theory University Microfilms, A X ERO X Com pany, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED POLLUTION, OPTIMAL GROWTH PATHS, AND TECHNICAL CHANGE by Robert Eing Wong 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) February 1972 UNIVERSITY O F SO UTHERN CALIFORNIA TH E GRADUATE SC H O O L UNIV ERSITY PARK LOS A N G ELE S, CALI FO R N IA 9 0 0 0 7 This dissertation, written by ........ Robert Eing._ Wong,.......... under the direction of Z t A . S . . . Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Gradu ate School, in partial fulfillment of require ments of the degree of D O C T O R O F P H I L O S O P H Y Dean Date Feb_ruary 1972 DISSERTATION COMMITTEE Chairman PLEASE NOTE: Some pages may have indistinct print. Filmed as received. University Microfilms, A Xerox Education Company ACKNOWLEDGEMENTS The completion of this dissertation has been supported generously by the members of my dissertation committee consisting of Professors Gerhard Tintner and John Niedercorn of the Economics Department and Profes sor Lucien Neustadt of the Electrical Engineering Depart ment . As chairman of this committee, Professor Tintner has provided valuable guidance toward a meaningful develop ment of this dissertation. His publications involving the application of optimal control to Economics have been excellent references. Professor Niedercorn's original work and ideas in the area of growth and pollution have been essential to the initiation of this dissertation. His aid in the shaping of the Economic model of pollution was fundamental to my research. Professor Neustadt's support in the way of mathematical analysis has been very substantial. His theorem in optimal control is the basic tool for my analysis. Professor Neustadt's gener ous contribution of his own time has been very much appreciated. - ii - ill A special thanks must go to my wife, Ellen, and my sons, Michael and Mark, for their patience and under standing during my efforts to attain the doctoral degree. ABSTRACT This dissertation represents a contribution to the Optimal Economic Growth literature. This extension of the current mathematical economic literature centers about the effects of pollution on economic growth and the requirements that must be placed upon technological progress in order that future generations may have a satisfactory standard of living. The basic aggregate model is of the one sector Cobb-Douglas type with three factors of production: labor, capital and energy. The planning of the future of the model economy is based upon the allocation of produc tive output to investment and energy in time to maximize the economy’s discounted net consumption per capita. Three models of technology were considered: a constant state of technology, an exogenous neutral technical change model and a pollution abatement technical change. New proper ties of optimal growth paths were derived using optimal control theory. These are presented in mathematical terms as well as graphically by comparison with the - iv - properties of the current literature on Optimal Economic Growth. Of particular interest is the discovery of a technical change multiplier. A use of this multiplier was made to arrive at a policy implication regarding the direction of efforts toward improvement of technical effi ciency and pollution abatement technology. The results indicate that the future of the model economy depends upon the relationships between technical advance, both in efficiency and pollution abatement, and population growth. Specific relationships were derived which determine the requirements on the rates of techno logical progress in order to ensure the future generation will live above the subsistence level. TABLE OP CONTENTS Chapter Page ACKNOWLEDGEMENTS .................................. ii ABSTRACT.......................................... iv LIST OP ILLUSTRATIONS............................. viii I. INTRODUCTION.............................. 1 II. POLLUTION AND ENERGY...................... 11 III. ECONOMIC MODEL............................ 20 IV. THE ’ ’POLLUTED" TURNPIKE AND CONSTANT TECHNOLOGY.............................. 31 V. ADDING NEUTRAL TECHNOLOGICAL CHANGE ........ 74 Case of No Critical Pollution Level........ 8l Case Where Economy Reaches Critical Pollu tion .................................... 89 VI. THE POLLUTION TURNPIKE AND POLLUTION ABATEMENT TECHNOLOGY .................... 97 Case of No Critical Pollution................ 102 Case Where Critical Pollution Level is Reached................................ 109 - vi - vii Chapter Page VII. POLLUTED OPTIMAL GROWTH PATH, NEUTRAL EFFICIENCY AND POLLUTION ABATEMENT PROGRESS.................................. 119 Case Where No Critical Pollution Level Exists.............................. 124 Case With a Critical Pollution Level .... 130 VIII. SUMMARY AND FINAL REMARKS.................... 133 APPENDICES.......................................... 144 LIST OF SYMBOLS................................ 145 A: GOLDEN RULE OF ACCUMULATION................ 147 B: A THEOREM ON THE NECESSARY CONDITIONS FOR A MAXIMUM IN PROBLEMS WITH BOTH BOUNDED CONTROLS AND BOUNDED STATE VARIABLES ... 151 C: COMPLETE SOLUTION TO THE PROBLEM WITH A CONSTANT STATE OF TECHNOLOGY - NO CRITICAL POLLUTION LEVEL . . .................... 156 BIBLIOGRAPHY ....................................... l6l LIST OP ILLUSTRATIONS Figure Page 1. Graphical Representation of the Samuelson and Cass Optimal Growth Paths .............. 6 2. Flow of Economic Goods, Pollution and Factors of Production ...................... 21 3. Investment Share History with a Constant State of Technology and No Critical Pollution Level ............................ 53 4. Energy Share History — Constant State of Technology, No Critical Pollution Level ... 5^ 5. Output Allocation Path — Constant State of Technology, No Critical Pollution Level ... 56 6. Total Pollution Time History — Constant State of Technology, Critical Pollution Level Reached.............................. 70 7. Output Allocation Path — Constant State of Technology, Critical Pollution Level Reached.................................... 71 8. Net Consumption Time History — Constant State of Technology, Critical Pollution Level Reached.............................. 72 9. Capital History — Neutral Technical Change, No Pollution.............................. 85 10. Investment Time History...................... 87 11. Output Allocation Paths ...................... 88 12. Total Pollution Time Path.................... 93 13. Output Allocation Paths ...................... 9^ - viii - ix Figure Page 14. Net Consumption Time Path.................... 96 15. Total Pollution Time Path.......................105 16. Energy Share Time P a t h .........................106 17. Output Allocation Paths ...................... 108 18. Total Pollution History ...................... 115 19. Energy Per Capita History.......................116 20. Output Allocation P a t h .........................117 21. Investment Share History .................... 126 22. Energy Share History ........................ 127 23. Output Allocation Paths ...................... 128 CHAPTER I INTRODUCTION Economic growth and pollution abatement are desir able goals of our society that are in conflict. The economic growth witnessed in the twentieth century has been phenomenal with technological advance accelerating the growth with amazing continuity. However, along with this growth, this society has also witnessed the accumula tion of pollution generated essentially as a by-product of economic growth. Our natural environment has been deteriorating at a rate consistant with our economic growth. The rate of pollution generated may over any short time period be small but the accumulated effects are mounting and definitely represent an economic and social cost to our society. Thus, limiting the accumulation of this pollution is required. But the question to be exam ined is how adversely does pollution affect economic growth and what alternatives do we have to achieve some desirable level of satisfaction between the two goals. This is especially important for future generations. - 1 - 2 What activities can the economist engage in to study and influence policy makers? In terms of current activity the major catagories of activities are: I. Examining the economic costs of pollution. II. Evaluating the alternatives for implementing government pollution reduction policies and determining the effects of each alternative. III. Analyzing the interrelationship between economic growth and pollution reduction. Most of existing economic literature on economics and the problem of pollution could probably be placed in the above three catagories. Category I consists of both theoretical and empiri cal endeavors. On the theoretical side of these studies, there has been a number of approaches to this analysis. Most of these treat the problem from a narrative or gra phical point of view using the classical tools of analysis. Empirical studies are ones in which meaningful results will "^For example, G. W. Sorenson, "The Effects of Changes in Air Quality on Tourist-Related Industries in a Region: A Theoretical Analysis," Paper presented at the Western Economic Association Conference, Davis, California, August 28, 1970; J. Buchanan, "A Behavioral Theory of Pollution," Western Economic Journal, Vol. 6, No. 5 (December. 1968), pp. 3^7-358; H. Wolozin, The Economics of Air Pollution (New York: W. W. Norton Co., 19&6). 3 not be forthcoming for some time since determining measure ment standards and obtaining good quality data will take much more study and time. However, some work has been developed in this area and studies have been initiated 2 and proposed. Until the empirical work on the economic costs of pollution has made some headway, the work being done in categories II and III will remain theoretical. Category II is policy oriented and relates to the govern ment instruments to control pollution. The possible tools available to government would include: 1) Direct Regulation including licenses, permits, compulsory standards, zoning, registration and equity litigation. 2) Payments including direct payments or subsidies and reductions in collections that otherwise would be made. Examples are subsidization of particular control devices, reduction of local property taxes on pollution control equipment, payments for decreases in the discharge of pollutants and tax credits for investment in 2 For example, R. Ridker, Economic Costs of Air Pollu tion (New York: Frederick A. Praeger, 1967); G. Tintner, R. E. Bellman, and J. H. Niedercorn, A Proposal for Inter disciplinary Research Relevant to Problems of Our Society (Los Angeles: University of Southern California, 1970). control equipment. 3) Taxation including schedules of charges or fees for the discharge of different amounts of specified pollutants and excise or other taxes on specific sources of pollution. This dissertation falls into Category III. We will be concerned with the interaction of economic growth and pollution. In particular we shall be concerned with an extension of the current optimal growth literature to include the effects of pollution. Optimal growth theory is a sector of mathematical economics which originated with the work of Ramsey^ in 1928. The Ramsey paper derived the aggregate savings required to maximize the integral of utility function of consumption over time. The classi cal calculus of variations was the mathematical tool which he used to obtain his "bliss" solution. Over 30 years elapsed before extensions of the Ramsey paper were made. Only in the last decade have Samuelson, Koopmans and Cass1 1 P. P. Ramsey, "A Mathematical Theory of Saving," Economic Journal, Vol. 38 (1928), pp. 5^3-59. ^P. A. Samuelson, "A Catenary Turnpike Theorem Involving Consumption and the Golden Rule," American Economic Review, Vol. 55, No. 3 (June 1965), pp. 466—96; T." C.' XOOpmahs, "Objectives, Constraints and Outcomes in Optimal Growth Models," Econometrics, Vol. 35, No. 1 (Janu ary, 1967), pp. 1-15; D. Cass, "Optimal Growth in an Aggre gate Model of Capital Accumulation: A Turnpike Theorem," Econometrics, Vol. 3^, No. 4 (October, 1966), pp. 833-^9; 5 revived optimal growth theory as a formal area of study for economists. Their analysis is slanted toward a neo classical theory of growth. Others such as Tintner, Stoleru, and Chakravarty^ have examined optimal growth theory with a slant toward planning and development in underdeveloped countries. The scope of this dissertation will revolve around the consumption turnpike theorems of Samuelson and Cass. Samuelson showed that: ...in the problem of planning an economy's program of production and allocation of out put, as the planning period becomes indefinitely large, its optimal program must spend an indefinitely large fraction of time arbitra rily near a given level of capital dubbed the "turnpike" level. The Cass version of the turnpike theorem replaces the "arbitrarily near" by "on." These two turnpikes are shown in Figure 1. The difference in the theorems is based upon a difference in the one sector models assumed. Capital accumulation is the control variable of the A. D. Nazarea, "Generalized Theory of Extremal Consumption Growth," Metroeconomica, Vol. 22 (1970), pp. 116-132. ^G. Tintner, "What Does Control Theory Have to Offer," American Journal of Agricultural Economics, Vol. 51, No. 2 (May, 1969), pp. 383-93; L. G. Stoleru, "An Optimal Policy for Economic Growth," Econometrica, Vol. 33, No. 2 (April, 1965), pp. 321-48; S. ChaFravarty, "Optimum Savings with Finite Planning Horizon," Inter national Economic Review, Vol. 3> No. 1 (September, 1962), pp. 338-55. 6 capital, k kpp •. kf k -- o Cass Path ii 1 1 *1 - Samuelson Path -----1 ------------------------- 1 -----*- time t initial t final Figure 1. Graphical Representation of the Samuelson and Cass Optimal Growth Paths kQ = initial capital level k = turnpike capital or golden rule level of capital kf = specified final level of capital 7 theorems and a utility function of consumption over time is to be maximized. Both have shown that the turnpike is a modification of the golden rule or golden age of capital accumulation derived by Phelps and Robinson.6 The turnpike theorems have been extended by Shell and 7 others to include technical change and multisectors models. The contribution of this dissertation is to extend the Cass and Shell versions of optimal growth to include pollution. This extension is an original contribution. As far as this author is aware, the presentation of a portion of the dissertation at the 1970 Western Economic O Association Conference at the University of California, Davis was the first to appear at a conference or in the literature. The contribution derives the properties and effects along a "polluted turnpike." Since that time, ^E. S. Phelps, "The Golden Rule of Accumulation: A Fable for Growthmen," American Economic Review, Vol. 51, No. 4 (September, 1961), pp. 636-43; J. Robinson, "A Neoclassical Theorem," Review of Economic Studies, Vol. 29, No. 3 (June, 1962), pp. 219-26". 7 K. Shell, Essays on the Theory of Optimal Economic Growth (Cambridge, Mass.: MIT Press, 1967). 8r. e . Wong, "Optimal Growth with Production In hibited by Pollution Generation," Western Economic Asso ciation Conference (Davis, California: August, 1971). 8 other treatments of optimal growth and pollution have appeared. Brock has derived a "polluted golden age" and Zeckhauser et al. have derived a "murky age."9 Dorfman10 recently remarked that "capital theory has become so profoundly transformed that it has been rechristened growth theory." The basis for this statement stems from optimal growth theory that Ramsey, Samuelson, Koopmans and others have developed. This "rechristening" is the result of using capital accumulation as the central controlling force in the growth of an economy wishing to maximize a utility function of consumption over time. The more recent literature has been facilitated by the development of optimal control theory by the Russian mathematician L. S. Pontryagin.^ This dissertation also ^W. A. Brock, "A Polluted Golden Age," Unpublished paper, University of Rochester, 1970; R. C. D'Arge and K. C. Kogiku, "Economic Growth and the Natural Environment," Paper presented at the Econometric Society Meetings (Detroit: December, 1970); R. Zeckhauser, Michael Spence and Emmett Keeler, "The Optimal Control of Pollution," Paper presented at the Conference on Economic Growth and the Natural Environment (Riverside, California: University of California Riverside, April, 1971); R» E. Wong, "Optimal Growth, Technical Change, and Pollution," Paper presented at the Conference on Economic Growth and the Natural En vironment (Riverside, California: University of California Riverside, April, 1971). 10R. Dorfman, "An Economic Interpretation of Optimal Control Theory," American Economic Review, Vol. 59, No. 5 (December, 1969) ,“pp^“5T7-3rr:------------ •^L. S. Pontryagin, V. G. Boltyanskii, R. V. Gam- krelidze, and E. P. Mishchenko (ed. by L. W. Neustadt), 9 determines the optimal growth path of capital. However, another factor of production, energy, will share impor tance with capital as a controlling force for a model economy. This use of energy as a factor in optimal growth theory is also an extension of the existing literature. As will be discussed in the next chapter, energy will be the factor responsible for the pollution in the model assumed for this dissertation. A discussion of this assumption and the role of energy in economic growth will be discussed in the next chapter. Chapter III will specify the assumptions of the dissertation in the form of mathematical expressions. The basic model is a one sector Cobb-Douglas economy. Chapter IV will concisely state the optimal growth problem, derive the properties along the "polluted" golden rule and exam ine the implications of pollution on future generations. The results imply Malthusian conclusions in the case where a constant technology exists. Chapter V considers the addition of an exogenous neutral technical progress function to the model and derives the requirements for assuring a satisfactory level of consumption for future generations. Chapter VI introduces a pollution abatement The Mathematical Theory of Optimal Processes (New York: Interscience Publishers, Inc., 1962). 10 technology which has the characteristics of a constant rate of progress toward pollution reduction. The require ments on this rate of advance in order to assure a consump tion happy future are derived. Chapter VII analyzes the problem when both neutral technological progress and pollution abatement advance exist in the model economy. The basic analyses of interest here are the different effects of various combinations of rates of progress between the two states of technology for productive efficiency and for pollution abatement. The last chapter summarizes the basic conclusions of the dissertation and discusses the areas of further research. CHAPTER II POLLUTION AND ENERGY In the past, we have treated many environmental resources as free goods that could be used as lavishly as desired. In the distant past, such resources as air and water could essentially be considered free since the rate at which man polluted such resources was low enough that the environment could assimulate the pollution withoii any problem. However, with the great technological ad vances accelerating productive activity, the rate of pollu tion generated has exceeded the ability of our environ ment to absorb the pollution. One can no longer consider air and water free. Industrial products are presently underpriced because they do not Include the cost of using "free” environmental resources. As a consequence, the well being of our society has improved in the direction of Inexpensive, mass-produced products while deteriorating In another direction — an increasingly polluted and un pleasant environment. - 11 - 12 Since our society desires both material goods and environmental quality, some adjustment must be made to compromise these two desires. The adjustment must come about by the correct pricing of the pollution of our environmental resources. There are two extremes to the pollution problem. One extreme is the continuation of the high rate of productive activity with no regard for the quality of our environment by ignoring any cost of environ ment resources. The other extreme is the ecological alarmists1 view in which all productive activity generating pollution should be stopped. This view essentially con siders environmental resources as having an infinite price. The rational view would be to assign the appropriate price to our environmental resources which in turn will yield the appropriate cost of pollution. This dissertation takes such a view. Any theoretical endeavor on an economic problem of our society requires an abstraction of reality since it would be impossible to consider every factor affecting the problem simultaneously. The most important factors should be included and examined. By isolating the most significant variables, general relationships can be derived which provide implications toward the real world. Even though, in economic theory, there exists a large body of literature not necessarily directly applicable to public 13 policy the former has represented an Important point of departure toward further research. Theoretical abstrac tions may take the form of mathematical models or narrative expositions. For this dissertation an important abstraction relates to the use of labor, capital and energy as the factors of production. In both the classical and modern theoretical literature, labor and capital have become standard as the factors of aggregate production. The addition of energy represents an extension of the usual assumptions. The dependence of modern industrial nations upon the use of energy is undebatable. Their standard of living and security of existence bears a direct relationship to their use and consumption of energy. With less than 1/17 of the world’s population, the United States consumes more than 1/3 of the world's energy. In other words, the aver age consumption of energy per capita in the United States is six times the world average. The standard of living is five times the world's average. There is no doubt that the United States' affluence is related to its energy consumption. However, the affluence is being threatened by the generation of pollution of which energy production is a heavy contributor. At the present time there is a growing concern over the available sources of energy. This is aggrevated by the fact that energy sources are pollution generating. Also, the dependence upon fossil fuels has led us to ravaging of our lands to get at the coal, oil and gas. Air pollution regulations in some places forbid the use of high-sulfur fuel. Most available coal, especially east of the Mississippi and nearest the large markets, is in the medium-to-hlgh sulfur category. With most of our low-sulfur coal in the West, transportation to eastern markets at a reasonable price presents difficulties. The point to be made here is the fact that energy is a major pollution producer and restrictions to reduce this pollution is threatening the United States' economic growth. In fact, crises are developing relative to the problem of energy and pollution. The above discussion presents one crisis, that is, restriction of energy sources due to pollution is creating a relative shortage in sources of energy needed for future economic growth. Another crisis which our society faces is that related to the actual conversion of energy sources to energy. The current crises exist in our metropolitan areas, especially in New York and Los Angeles. There has been a tremendous influx of population to the greater metropolitan areas in the last fifth years. This alone would increase the energy requirements for these highly populated areas. However, technological advances have accelerated the requirements for energy and the residents of the metropolitan areas have at tained an affluent level of living which they have become accustomed to. This affluent living and the continued growth of the greater metropoli tan areas have placed heavy requirements for in creased amounts of energy to satisfy the needs of the populace. The attempts to provide more energy are being thwarted by pollution regulations and public outcrys. In Los Angeles, the Department of Water and Power (DWP) has the responsibility for pro viding the energy for the city residents1 needs. In 1970, it attempted to continue construction of an additional steam-powered generator within the City of Los Angeles. However, the county of Los Angeles' Air Pollution Control District (APCD) would not permit the construction. The DWP lost a lawsuit against the APCD for permit to build the additional generator. The decision of this lawsuit was based upon the fact that there existed no evidence that the cit" residents' basic needs could not be met with the available generators. It was felt that the available generators may not be sufficient for all "peri pheral" needs of the community but that the basic need for energy would be satisfied. DWP has fore casted a grim future without the additional plant. A similar dilemma exists in New York City. Consolidated Edison Company is responsible for New York's electrical energy production. Here the problem is much broader. The city has had a number of electrical power "brownouts" and the famous "blackout" in 1965. In order to avoid such problems in the future, the Consolidated Edison Company (Con Ed) has been attempting to expand its energy production through several methods, but each is meeting opposition. The at tempt to increase the output of present conven tional generating plants has been opposed due to the air pollution caused by fossil fuel burning required by these types of generators. Justly or not, Con Ed has won the reputation of being one of the city's worst offenders with its highly 17 visible plants in Manhattan and Queens. The Con Ed expansion in Queens is meeting opposition by the Environmental Protection Administration which must approve the expansion. Con Ed has a nuclear facility on the Hudson River. This plant was to have supported the area's electrical energy requirements and reduced the pollution generally produced by the conventional power plants. On the contrary, the plant has had problems in service and its thermal and chemical pollution has been termed excessive. State offi cials have been threatening to close down the plant unless the steps are taken to reduce the excessive amounts of pollution. Con Ed has also proposed hydroelectric generation through an ingeneous scheme that would use water from the Hudson River. In off-peak periods, power would be used to pump river water into a reservois high on a mountain, 50 miles north of New York; then when power use rises, the stored water would be used to spin generators as it flows back into the river. But this plan, too, has been stalled for years by conservationist 18 forces, who are angry over what they fear will be desecration of an unspoiled wilderness area. These specific cases have been cited due to their direct relation of the pollution and energy. Undoubtedly, there are many sources of pollution such as certain industrial and chemical wastes that are not directly related to energy production. However, we would be justified, in developing an abstraction of reality, to consider only pollution generated by energy production. This is based upon the fact that the pollution generally of most concern is that produced as a by-product of energy generation. Most air pollu tion and certain forms of water pollution fall into this category. Not only is air pollution due to energy conversion by automobiles and power plants very undesirable and costly but attempts to reduce the pollution will have strong implications for the level of productivity and affluence of future generations. Economic growth is definitely being retarded by attempts to limit pollution caused by energy production. This chapter has attempted to justify the model to be described in more detail in the next chapter. The model will assuifie that In our abstraction of reality pollution Is generated as a by-product of using energy in the productive process. CHAPTER III THE ECONOMIC MODEL The economic model can be summarized by the dia gram of Figure 2. The flow of economic goods is seen to consist of three factors of production, capital K, energy E0, and labor L producing consumer goods, energy and capital goods ( AK ) for further production as well as the by product, pollution, ( AP ). Capital is a stock which is accumulated while the energy produced is completely con sumed in the productive activity. Pollution is also stock which accumulates with further production. In the following discussion each of the system variables will be further specified as to their characteristics for the assumed model. In terms of the system, the model can be described as an aggregate one sector model of the Solow^ ^ • R. M. Solow, "A Contribution to the Theory of Economic Growth," Quarterly Journal of Economics, Vol. 70, No. 1 (February, 195b), pp. b*J-94. - 20 - 21 4 K Produc- tion (K,L,EJ Population Consumer Figure 2. Flow of Economic Goods, Pollu tion and Factors of Production 22 type. Labor L, is provided by a population, N which is assumed to be growing at some constant rate p. Although over the long run such an assumption is not particularly realistic, it is a standard assumption in the economic growth literature and shall be retained here. If the labor force is a constant proportion of the total popu lation, then L = PN That is, the growth rate in the labor force is equal to that of total population. Capital stock, K, is assumed to grow at a rate dependent upon the investment, I, and the rate of capital depreciation, y. This can be written as K = I - yK The investment variable is one of the two controls which this model economy has at its disposal to steer its course toward attainment of its specified objectives. The model essentially assumes that there is a planning board which 23 will determine the proper investment at the proper time to reach this economy’s goals. The planning board will make the necessary plans over a given planning time interval. It should be mentioned that investment and the resultant accumulated capital is generally the prime control of interest in the current literature on optimal growth theory.^ in fact, capital has been the center of economic growth theory in general. Although this dis sertation will still regard capital as a very important control, it will be given the same importance as another factor, energy. Energy is the final factor of production to be considered in this model. To this author's know ledge, the use of energy has never been considered in Economic Growth Theory literature. This is a serious omission, since it is quite obvious, as previously men tioned, that energy has been a basic force in the economic growth of the United States and all other countries for that matter. The study of energy by urban and resource economists has been extensive and in-depth. Yet, economic growth economists have ignored its implications in their theoretical analyses. 2For example, see the papers by Dorfman, Samuelson, Koopmans, Cass, Shell, op. cit. 24 Energy is an output of production which is imme diately consumed in order to continue the production pro cess. Energy is thus a flow variable rather than a stock variable such as labor and capital. All of the three factors in this economy will be fully employed in the production process. All energy produced is consumed. All available labor is employed and all capital stock is utilized. The three factors of production are combined together for production according to a Cobb-Douglas func tion. Symbolically, the output Y of the production ac tivity is given by Y = AL%aE6. where t p = partial elasticity of production of labor at= partial elasticity of production of capital g = partial elasticity of production of energy A = production constant which indicates the state of technology Initially, there will be assumed a constant state of tech nology. This will be extended in later chapters to in clude a time dependent neutral technological change factor. The following neoclassical features of the production 25 function are also assumed: i) constant returns to scale, i.e., i|i+a+6 = 1 ii) positive a,6 iii) diminishing marginal rate of substitution between factors iv) all available factors are employed. As far as I am aware of, this use of energy in the Cobb- Douglas production function was first suggested by Niedercorn.3 The desired aggregate output, Y, can be separated into three distinct categories of use: consumption, C, capital investment, I, and energy required for current production, Eq. Thus, Y = C + I + EQ Although this equation is a simple one which in some sense represents an accounting of the economy's uses of output, it is an important fundamental equation. This equation provides a statement of the allocation of our output to three difference alternatives. In other words, it states the basic economic problem of where to allocate the output of our production in order to attain our economic objec- ^John H. Niedercorn, unpublished paper. 26 tlves. As stated in the prior paragraphs, investment and energy are the control variables of the model economy. Since the above equation holds, consumption is determined once output, investment and energy have been determined. If the planning board decides upon more production by allocating more of output to investment and energy, then there will be less consumption for the economy’s popula tion. If the planning board decides on less production, then more will be allocated to the population’s consump tion. It should be noted that this model assumes that energy is never used as a final product in itself, but as a means to assist in final product consumption or in capital usage. For example, energy is used to aid house hold items functioning or it may be used to aid manufac turing machinery perform its function. However, there is an undesirable by-product generated, pollution (P). Pollution is an accumulated stock variable. In this analysis, pollution generation is assumed to be linear function of the amount of energy produced. Also, there exists a dissipation of pollution such that the net production rate of pollution is given by • P * 0EQ - aP where 0 and a are constants. The assumption that energy 27 is a major source of pollution is a reasonable one as discussed in Chapter II. Pollution as a function of energy was also suggested by Niedercorn. Since pollution is a stock that has very definite adverse affects upon the quality of life, it cannot be allowed to increase indefinitely. Therefore, it can be assumed there is a critical or maximum level of pollution which the planning board will not allow to be exceeded. We shall examine the effects upon a growth when the cri tical level is reached. The stock of pollution at the critical level will be designated as Pq. Finally, the planning board must decide upon an objective for the economy. It will be assumed that the desirable direction of growth for the economy is toward maximizing a discounted net consumption over time. The term net is used here to indicate that gross consumption designated by C above only encompasses that obtained directly from the output of production Y. The harmful effects of pollution will be a cost that degrades con sumption so that net consumption will be designated by C - zP where Z is a constant. But, this net consumption will be ^Ibid. 28 discounted in time so that the objective to be maximized by the planning board is T f (C-£P)e~6tdt 0 In other words, the total amount of discounted net con sumption over the planning period T is to be maximized. The final value of T will also be referred to as the plan ning horizon. 6 is the discount rate. If the discount rate is positive, then the planning board prefers current and near future consumption to consumption in the distant future. In order to reduce the number of variables to which we must address ourselves, it is convenient to eli minate the labor variable by defining all other variables in terms of labor. Thus, ct 6 0 6 3 y = (Y/L) = A(K/L) (Eq/L) = Ak E (1) k = (K/L) = (K/L) - (K/L2)L = (I/L) - (yK/L) - (K/L)(L/L) = i - (y+p)k (2) p = (P/L) = (P/L) - (P/L2)L = (Eq/L) - (aP/L) - (P/L)(L/L) = E - (a+p)p (3) y = c + i + E 29 where as indicated y = (Y/L) k = (K/L) P = (P/L) i = (I/L) E = (E /L) o A further specification of our model is that capital invest ment can be expressed as i = sy where s is the capital output coefficient. This, in effect, has transformed our control variable from invest ment i to the capital output coefficient s. The value of s can be between the limits 0 and 1. So, y = c + sy + E where c = (C/L) and c = (l-s)y - E The pollution constraint is given by pt Pc 1 P - PLoe (4) (5) 30 where L is the number of laborers at the beginning of o the planning period. Thus, equations (1) - (5) repre sent the economic model under consideration. The objective can be redefined in per capita terms as maximizing the per capita discounted net con sumption over a specified time period. Specifically, in mathematical terms, this is written T -fit Max / (c-Ep)e dt (6) s,E 0 -fit where e represents the discount factor. s and E are the economy's controls to attain the maximum. For long term growth analysis, T is large and may approach infinity. The objective function is shown to maximize the consumption per unit labor and not consumption per capita. However, the amount of labor is a given percen tage of the population. Thus, maximizing the consumption per unit labor is equivalent to maximizing the consumption per capita. Similarly, since only a constant separates the per labor and per capita variables, the per capita terms may be used in place of the per labor unit. CHAPTER IV THE "POLLUTED" TURNPIKE AND CONSTANT TECHNOLOGY This chapter will treat the growth problem using the model specified in the previous chapter assuming a constant technology. The problem can be concisely stated as determining the path of the control variables s(t) and E(t) and the phase variables K(t), p(t) which maximizes discounted net consumption per capita T -61 J = / (c(t)-Io(t))e dt 0 subject t o ' * ' c(t) = (l-s(t))y(t) - E(t) C f t ^ y(t) = Ak (t)E (t) k(t) = s(t)y(t) - (y+p)k(t) p(t) = 0E(t) - (a+p)p(t) - 31 - 32 p(t)L ept < Pp o — o s(t) + E(t)/y(t) j< 1 E (t) > 0 s(t) >_ 0 where E, 6, a, 6, y, p, a, 0, P , LQ, A are constants defined in the previous chapter. This is a problem in optimal control theory with a mixed set of inequality constraints on the control and phase variables. That is, we have a combination of one inequality constraint which involves the control and phase variables and another inequality constraint involving a variable and time. A theorem covering the necessary condition for an optimal solution for this case has been derived by Neustadt.^ The theorem is stated in Appendix B. The main stream of existing optimal economic growth literature^ represents an extension of the Ramsey1 * ^We will use y(t) to denote Aka(t)E^(t) throughout the rest of the dissertation except where it may be more meaningful to use the full expression. In some instances (t) may even be omitted from y(t) as well as other phase variables. 2L. W. Neustadt, An unpublished theorem obtained through private communications. To appear in a forth coming book. 3 Dorfman, Samuelson, Koopmans, Cass, Shell, op. cit. problem In which the integral of a utility function of consumption is to be maximized. The control variable is generally savings, investment or capital. The original 5 one sector model has been extended to n-sectors. The treatment of the optimal growth problem in this disser tation is another extension of the existing literature but in several facets. First, it considers growth in relation to the problem of pollution.^ This added dimension to growth theory in general and optimal growth theory more specifically had not previously been treated. Secondly, this dissertation treats a specific model, using, as an abstraction of reality, energy as the source of pollution. As has been previously mentioned, energy is not generally treated as a major factor of production in the usual treatment of growth although its importance can hardly be debated to the contrary. A third extension to the existing optimal growth literature is related to the application of a mathematical theorem. This speci fically is in the use of a mixed set of inequality i | Ibid. ^Shell, op. clt., pp. 17-29. ^Professor W. Brock of the University of Rochester has informed me of his independent approach to this pro blem. His paper, op. clt., is a discrete and general approach to the pollution problem. 3^ constraints. The model has an inequality constraint on the control variables and a phase variable as well as an inequality constraint on a phase variable and time. The usual optimal economic growth literature considers only inequality constraints on the control variables.? We now proceed to the necessary conditions for a solution to the problem. Model with No Critical Pollution Level Let us first consider the case in which there is no pollution constraint, that is, no critical pollution level exists, and the condition p(t)L ept < P~ is deleted. o — o The analysis of this problem will procede by applying the theorem in Appendix B. In order to use the theorem directly, a transformation must be made. This is that T J = x°(T) = f (c(t)-Ip(t))e~6tdt 0 o where x satisfies the equations ^Dobell and Ho have considered an optimal growth problem of an inequality constraint on the control varia bles and an inequality on one state variable. A. R. Dobell and Y. C. Ho, "Optimal Investment Policy," IEEE Transactions on Automatic Control, Vol. AC-12, No. 1 (February, 1967), pp. 4-14. -51 x (t) = (c(t) - Ep(t) )e x°(0) = 0 In the notation of the appendix, the phase and control vectors are composed as follows: x = x u = s E The control process is expressed by x°(t) x(t) = k(t) = ' * • ct 1 -6 t ((1-s (t)) Ak0 1 (t)E? (t)-E(t)-Ep(t))e s (t) Aka (t)E^(t) - (p4p)k(t) 0E(t) - ( c J 4p ) p (t ) The inequality constraint in the appendix B notation is X (x(t) ,u(t) ,t) = s(t ) + (E(t)/Aka (t (t) )-l < 0 The condition at the end points is expressed as 36 where k and p are specified values of capital and pollu- o o tion given at the initial time point. Suppose there are values of the control and phase variables which are optimal in the sense that the objec tive function is maximized. Then by the theorem in Appen dix B, there exist differentiable auxiliary functions ip (t), \p (t), and \ p (t) and a piecewise continuous function k P o £(t) such that (t) = -ip (t)(aS(t)y(t)/k(t) - (y+p)) (7) K k + £(t)(E(t)a/k(t)y(t) _ _ _ - i j > 0(t) (l-s(t)) (ay(t)/k(t))e -6t i p (t) = (t)(a+p) + ip (t)Ee (8) p P u < J j Q(t) = 0 and such that the conditions of Appendix B hold, where the bar above the control and phase variables designates values along the optimal path, i p (t) and i j > (t) are the k p auxiliary variables and can be interpreted as the shadow prices of capital and pollution respectively in terms of the consumption objective function. Mt) is a Lagrange multiplier function associated with the control-phase variable inequality constraint. 37 S(t) is equal to zero when ?(t) + (E(t)/y(t)) < 1 and equal to a non-positive value when the constraint holds in equality form, i.e., when s(t) + (E(t)/y(t)) = 1. It is assumed that the final pollution level at time T in this first case, where pollution is not limited, will not be specified. Thus the transversality conditions state that ^k(T) = 0 ^p(T) = 0 = v0 - 0 Thus, since ^Q(t) is constant, ^0(t) = VQ- Assuming that a reasonable solution to the problem exists, is a positive constant and may be set equal to 1 since the vector ^ can be scaled by a positive constant without changing the necessary conditions. Since (8) is a . linear non-homogeneous equation in i j / p and time, it can be solved explicitly. The solved expression is: -<$t (cr+p+6) (t-T) i j ; (t) = (£e /(cr+p+6))(e -1) (9) P The shadow price of pollution \p is by equation (9), an P explicit function of time. The term within the square brackets is negative but increasing exponentially to zero at the planning horizon. The terms in the right bracket at time t=0 approaches -1 as T approaches infinity. Thus, the shadow price of pollution at the initial planning time approaches -(E /a+p+6) as T approaches infinity and at the termination of the planning period, it is zero. This negative shadow price for pollution is expected. Note that if there is no deterioration of consumption due to pollution, i.e., if E = 0, then if> is identically zero as expected. The optimal control functions E(t) and s(t) are determined by application of the theorem in Appendix B. Define 8 H(K> ,k,p ,s ,E,t )= ((1-s ) AkaE -Ep-E)e P + i | ; k (s AkaE ^- (vi+p ) k) +ip (0E-(a+p)p) P 39 Using (ii) of Appendix B, we have at each instant of time H ( t | > (t) , i | > (t) ,k(t) ,p(t) ,s(t) ,E(t) ,t) K P = max H ( i | > . (t), i p (t),k(t),p(t),s,E,t) s>0 K P E>0 s + (E/AIca(t)Ee)<l It is of interest to study the maximization of H indicated above and to examine the bahavior of s"(t) and E(t) along the optimal path. Let us initially consider the characteristics of s"(t) while assuming E(t) is given. Later in the chapter E(t) will be examined with the F(t) obtained for specific situations in the following analysis We may write H as: H ( ( t) ,if>p(t) ,k(t) , p (t) , s , E (t) , t) (10) - 61 _a _B = s(^ (t)-e )Ak (t)E (t) + k _ a _B _ _ -6t (Ak (t)E (t) - Ep(t) - E(t))e -ip (t) (u+p)k(t) + i p (t) (0E(t)-(a+p)p(t)) P Since s appears linearly in one term, the maximization of H with respect to s depends on the value of the coeffi cient of s in (10). The constant state of technology A is positive in order to make sense. Also, ka(t) is positive and E^(t) is non-negative. If E^(t) is zero, then a 40 singular situation exists since the coefficient of s is zero and therefore H is invariant to s. For this par ticular singular case, s may be arbitrary since output y(t), will be equal to zero and any value of s will give the same result with respect to the optimal path. The results by the very definition of s are given; s is the share of output that goes to investment. But since _3 _ output is zero when E (t) is zero, then any s(t) will still yield a zero share to investment. Essentially, over this portion of the optimal path, s is an ineffec tive control. Since this type of singular situation in which the output is zero is not an economically plausible one, let us assume that E(t) is positive. This assumption, of course, cannot be consistent with later results re garding E(t) when it is examined with respect to H. Referring back to the coefficient of s in (10), if ^(t) > e then s(t) = 1 - (E(t )/Alc“ (t)E6 (t)) = 1 - (E(t )/y (t) ) since this will make the Hamiltonian take on its largest value with respect to s. Similarly, if . — 61 ^(t) < e , then ?(t) = 0 41 since only by making s zero can one prevent the first term from being negative. The first case represents the growth path In which there will be no consumption allo cated from the output. However, if pollution is generated then there will actually be a negative net consumption as there will be a cost for the polluting effects. The second case where investment will be zero, represents a case where net capital will be decreasing and consumption will be allocated a large share of the output. Thus, the first situation indicates an economy with a relatively low level of capital while the second case is one with a relatively high level of capital. Consider the situation in which i f * k(t) = e"6t In this case, the H function is invariant to variations in the control s and at each instant of time, 3H(ip1 .(t), ijr(t), k(t), p(t), s, E(t)) - E___________________________= o 3s ^The arguments of H are those of (10). In the sub sequent discussion, the arguments of H, the auxiliary, phase and the control variables will be omitted unless there may be some source of confusion. The argument of the auxiliary, phase and control variables is, of course, time. For example, will replace Yk(t). *12 This is the singular arc condition of optimal control theory if it exists over a finite period of time. Only recently have necessary conditions for the singular case Q been derived. In the engineering and applied mathematics literature, the singular arc is generally considered as an unusual case which occurs as an oddity. As will be discussed, this case is of basic importance in optimal growth theory. Since this is true, we shall in the rest of this study consider only this case. We shall assume that the given initial and final conditions are such that the entire planning period will be on a singular arc. It is assumed that such a solution exists. Three conditions existing at each instant of time of the singular arc are: 3 H ( i | / k(t) , t f / p(t) ,k(t) ,p(t) ,s,E(t)) = 0 / 3 H(if>k(t) , i | > p(t) ,k(t) ,p(t) ,s ,E(t)) dt \ 3 s 2 / 3H(i|> (t),i|» (t),k(t),p(t),s,E(t)) d / k P __ dt2 ' 3 s = 0 (11) = 0 Q ^Summaries of these conditions are found in: A.E. Bryson, Jr. and Y. C. Ho, Applied Optimal Control (Waltham, Mass.: Blaisdell Publishing Company, 1969); D. H. Jacobson, "A New Necessary Condition of Optimality for Singular Con trol Problems," SIAM Journal of Control, Vol. 7, No. * 1 (November, 1969), pp. 578-95. These general conditions yield the following characteris tics of the singular solution for our model: - 6t \(t) = e (12) y (t) = y + 6 + p (13) k (y(t)/y(t)) = (k(t)/k(t)) (14) where y.(t) is the partial derivative of output, y, with J n . respect to k. (Note that we have used the notational convenience stated in footnote 1, Chapter IV.) Equation (12) states that the shadow price of capital along this optimal path is equal to the discount factor. Thus, its value decreases exponentially with time. However, this value Is that assigned by the planning board at time t = 0. If one considers the shadow price assigned at any given time, t, (that is, no discount considered) then the shadow price is 1. Equation (13) is derived by taking the time deri vative of (12) and substituting (7). The equation shows that the marginal productivity of capital will equal the sum of the capital depreciation, the discount rate and the rate of population growth. If y = < 5 = 0, then one has yk(t) = p which is the condition for the golden rule of accumulation 1° developed by Phelps, Robinson, and others. Thus, we have a modified rule of capital accumulation. Since yk(t) = (ay(t)/k(t)) equation (13) provides (k(t)/y(t)) = (a/y+6+p) (15) So, the capital-output ratio is a constant as is shown by equation (14) also. I1 Cass, Samuelson, and Shell have obtained turn pike theorems regarding solutions for single sector models. These turnpike theorems indicate that the optimal growth path will be along the golden rule or consumption turnpike path except possibly near the initial and terminal points. Equation (13) is the same condition which they derived along the "modified" golden rule growth path. We also have a condition, equation (14), that the per capita rate of capital growth is equivalent to that of per capita output growth. For the case in which only labor and capital are factors of production, these growth rates are equal to zero along the modified golden rule path, that is, equilibrium conditions exist. This has been referred to 1(^Phelps, op. cit. ; Robinson, op. cit. ; also, see Appendix A. 11Cass, op. cit. ; Samuelson, op. cit. ; Shell, op. 45 in the growth literature as "balanced growth." As shall be shown later, the existence of pollution will prevent this condition from equalling zero in our model. Thus, the singular solution of the model in this paper is a "polluted" turnpike or golden rule of accumulation. Equation (14) can be used to derive another inter esting property of the polluted turnpike. If one differ entiates the Cobb-Douglas function with respect to time, the following results: (y(t)/y(t)) = a(k(t)/k(t)) + B(E(t)/E(t)) Substitution of equation (14) yields (y(t)/y(t)) = (g/l-a)(E(t)/E(t)) (16) This equation states that the total output growth is a constant proportion of the growth of energy. But since the rate of capital growth equals the rate of total output growth, it is also a constant proportion of energy growth. Since g < 1-a the capital and total output growth is less than the energy growth. Although, we have not obtained an explicit expres sion for the energy control over time, we have derived a property for energy growth. We will expect energy growth cit. 46 to be greater than that for the total output and capital. Symbolically, (y(t)/y(t)) = (k(t)/k(t)) < (E(t)/E(t)) This, of course, occurs when energy growth Is positive. If energy growth is negative, then the output and capital growth will not be decreasing at a rate less than that of energy. These growth rates are all equal when they are zero. An explicit function for the investment share of output, s, is required. The Cobb-Douglas function and equation (14) will provide that (a-l)(Ic(t)/k(t)) = -g(i(t)/E(t)) Substituting the equation for k, . (ft-1). (sy(t )-(y+p)k(t)) = -3 (E(t)/E(t)) k(t) or solving for the optimal investment share function, s(t) = (a/p+6+y)(y+p+(3/l-a)(E(t)/E(t)) (17) with the aid of the golden rule condition. As shown before, the term outside the parenthesis, (a/p+6+u), is the capital output ratio. This indicates that the amount of capital formation is a function of the rate of energy per labor growth. If the growth rate of energy is zero, that is, 47 energy is a constant, then l5(t) = (a (y 4p )/p 46 4y ) (18) which is the Cass solution for the case in which energy is not considered as a factor of production. Note also that if 6=0, then the share of output to capital will equal a when the energy growth equals zero. The investment share equals the marginal elasticity of production for capital. Let us refer to this as the reference investment path. Since a <1, by assumption, and a, 8, p, 6, y are all non-negative, then the optimum program for s will vary in the same direction about the reference investment path as the growth rate of energy. That is, s will be greater than, equal to, or less than, the reference value depending upon whether the energy growth is positive, zero or nega tive . Now that we have examined the characteristics of an optimal investment program, s"(t), attention can be made with respect to the characteristics of the optimal energy program, E(t). For this portion of the study, we shall limit ourselves to the "polluted" golden rule path or the singular arc case analyzed in the previous discus sion. In order for us to be able to do this, it is required that _ _ _ _ _ _a _3 (s(t),E(t),k(t),t) = s(t)+(E(t)/Ak (t)E (t)) -1 < 0. But, if this expression (with s(t) replacing s"(t)) is used in equation (10) to determine s"(t) through (ii) of Appen dix B, we find that the singular case or the "polluted" golden rule will not appear. This occurs since equation (10) will not be a linear function of the investment share s. Thus, we shall assume that For if _ _ _a _3 s(t) + (E(t)/Ak (t)E (t)) = 1 then 01 ^ E(t) = (l-s(t))Ak (t)E (t) or x(s"(t) ,E(t) ,k(t) ,t) < 0 throughout the rest of the analysis. Along the "polluted" golden rule of accumulation the H function can be written as -St H(e , i j > (t) ,k(t) ,p(t) ,s(t) ,E(t) ,t) P a 3 _ = (Ak (t)E (t)-Ep(t)-E(t))e -St -St e (p+p)k(t) +1^p(t) (0E(t)-(a+p)p(t)) 5^ where e has been substituted for ^ (t). To determine E(t), we must apply (ii) of Appendix B. We have at each instant of time H(e“6 t (t),p(t),k(t),F(t),E(t), t) = max H(e-<St ,1 b (t) ,p(t) ,k(t) ,F(t) ,E,t) E>0 p This requires that we satisfy the equation 9H (e“6t, i j / (t),p(t),k(t),s(t) ,E,t) = 0 T e p at each instant of time along the optimal path. This equation results in the following condition: (F(t )/y(t)) = (p/l-if> (t)9efit) (19) This expression can be further determined in terms of time by use of the expression for in equation (9). Thus, P (20) E(t) = ________________ 0______________ (a+p+6) (t-T) 1 - 9E (e -1) a+p+6 ^(1/1-0) _ _a ' E(t) = )___________0Ak (t)_________ (a+p+6)(t-T) 1 - 9 E (e -1) a+p+5 It must be emphasized that this solution only 50 holds as long as the critical pollution level has not been reached. This separate case will be treated later in this section. Note that if pollution was a neutral or cost less by-product, that is, E=0, or if energy was not a generator of pollution, 9=0, then the share of production going to energy would be equal to 3, that is (E(t)/y(t)) = 3 This states that the share of output to energy is a con stant and equal to the marginal elasticity of production for energy. But, we have shown above that when the dis count rate is assumed zero, then the investment share equals the partial elasticity of capital, a. This was the reference investment path. Thus, for this restricted case, the output shares are a to investment, 3 to energy and l-a-3=^ for gross consumption. This is an equilibrium or balanced growth case. The growth rate of each factor in per capita terms is zero. Let us call this the a3 reference growth path. For 9,£ not equal to zero, then as t+T, the production share to energy will approach 3 from below. This is due to the negative value of the term within the parenthesis of the denominator. For T large, most of the initial growth path of the economy will be near i(t) = 3 ST(t) " i + 0Z a+p+6 51 The deviation from a 3 share of the output for energy depends upon the rate of pollution generation 0 and the pollution cost factor £. If these are large relative to the pollution decay rate (o), the population growth rate (p) and the discount rate (6), then the energy share may be significantly reduced from 3. If these pollution factor rates are on the order of the values of the other rates then the deviation from 0 will be relatively small since the pollution factor rates appear multiplicatively where as the other rates appear in an additive manner. The equations for the energy share and the capital share (s) indicate that production has been curtailed due to the pollution that is being created. As the above analysis indicates, the presence of pollution will shift the allocation of output between capital, energy and con sumption from the pollution-free or aB reference growth case. The shift will, in turn, reduce the level of produc tion output from that which a pollution-free world would yield. Thus, pollution has two basic effects. It has a direct effect on net consumption via a cost function and it has an indirect effect in terms of reducing total production through a shift in the allocation of output among the factors. The shift being a reduction in the use of the pollution generating factor. These shifts from the aB reference path may be 52 shown with the aid of control-time and control plane diagrams. These diagrams will only consider the turnpike portion of the growth path. Figure 3 illustrates the fact that the share of output to investment for the pollu ted turnpike is increasing with time. During the initial portion of the path the s value for the polluted turn pike may lie above the reference (a& trajectory) path. However, it may not necessarily lie above since it depends upon the value of the discount rate. If the discount is zero, then the polluted turnpike s definitely lies above that for the reference value, a. If the discount rate is large enough and the rate of energy growth small enough, then it is possible for a polluted s to lie below the reference value. Figure 4 presents the time history of the energy share E/y. This situation will have the polluted energy share always below that of the reference or pollution free path as shown. The value of the share will increase with time until the planning horizon T is reached. When this occurs then the energy share of the polluted path will match that of the reference case and will equal the mar ginal elasticity of production for energy. How far below the reference path will depend upon the costs of pollution and the pollution generating capability of energy. Figure 5 provides a control share plane diagram 53 Investment share, s polluted turnpike a reference time T 0 Figure 3. Investment Share History with a Constant State of Technology and No Critical Pollution Level Energy Share, E/y reference path 8 polluted turnpike Time 0 T Figure 4. Energy Share History — Constant State of Technology, No Critical Pollution Level 55 showing the path of s and E/y in reference to the aB reference path. This latter path is represented by a point (a,B). The figure illustrates a triangle bounded by the lines s=0, E/y=0 and s+E/y=l. Any point inside this triangle represents an admissible control to the growth problem. Since the maximum value of any share is 1 by definition, this is the maximum value of either s, E/y or their sum. The inadmissible control shares are shown by the shaded area. The movement of the growth path is depicted by the curve between t=tQ to t=T. The path is from some value of s above a and E/y below 3 to a point where E/y equals 3 and s is still above a. This is essentially a translation of the time paths in figures 3 and 4 to the control share plane. Obviously, if this is an admissible solution, it must lie within the triangle. Any solution lying within the triangle must be a singular arc and ^ = e . If the solution is on the boundary of the triangle where s=0, i.e., on the E/y axis, then the shadow price of capital is less than the discount factor. If the solution is on the hypotenuse of the triangle, then the shadow price is greater than the discount factor. The share to consumption is the difference between the total output and the sum of the shares to investment and energy. In figure 5, the path is shown to initially 56 Investment Share, s s=l s + E/y t=T |t=t Figure 5. Output Allocation Path — Constant State of Technology, No Critical Pollution Level 57 begin at a value of a and g such that consumption is greater than if the <x6 path was followed. This is seen geometrically by examining the path in relation to the broken line q^q2. a + g = Oq-L = 0q2 If the path is inside the triangle Oq^qj then the con sumption share of output is more than that in a pollution free economy. If the optimal growth path exists outisde the 0q^q2 triangle, then the consumption share is less than in the a6 growth economy. Thus, the path shown will move from a consumption share greater than the reference share to a state less than the reference share. However, this may not necessarily always hold true. The starting point could possibly begin outside 0q1q2 and end at the point shown as t=T. For example, the total path might be just the end portion of the trajectory shown outside of Oq^qg* In this case, the consumption is always less than the ag reference case. The actual path depends upon the value of the parameters in the model. Before going to the next subsection, let us review the various characteristics obtained in the present sub section. I. Major Assumption: There exists a solution in which the shadow price of capital equals the J> i , rate of discount, ^k(t) “ e™ , will hold throughout the growth path. II. General Characteristics: i) Constant capital output ratio. (k(t)/y(t)) = a p+6+y ii) Growth of output and capital are equal. (y(t)/y(t)) =(k(t)/k(t)) iii) The absolute magnitude of energy growth is greater than that of output. (y (t )/y(t)) = _ L _ (f(t)/E(t)) 1-a iv) The marginal productivity of capital is equal to the sum of the population growth rate, the discount rate and the capital depreciation rate. This is the modified golden rule. y(t)=p+ 6 +y v) The investment share of output is given by vi) The energy share of output is given by k P+S+M (y+P+ (E(t)/E(t)) 59 E(t) = 6 y(t) (a+p-hS ) (t-T) 1 - __®2__(e -1) a -tp +5 III. Case where pollution does not exist, 0=0. i) Balanced growth condition (y(t)/y(t)) = (k(t)/k(t)) =(E(t)/E(t)) = 0 ii) Constant share of output to investment s(t) = ---§---- (U+P) P+fi+M iii) Share of output to energy is equal to the partial elasticity of production for energy (E(t)/y(t)) = p I V . Case where pollution exists, 2 ,0 ^ o. i) There is a shift of output shares relative to the pollution free case. There is an increase in the share to investment and a decrease in the share to energy, the pollu tion generating factor, ii) The share to investment is increasing with time. iii) The share to energy is increasing with time, 60 iv) The net consumption for future generations may or may not be satisfactory depending upon the value of the parameters involved. The smaller the values of E and 0, the unit cost of pollution and the rate of pollution generation by energy, the better off will be our future generations. In this portion of the analysis, we have assumed tacitly that there is no critical level of pollution or equivalently the critical level is an infinite level. The pollution level at the horizon is important in esti mating the net consumption level. The horizon level of energy is designated Ej». As an approximation to the corresponding pollution level assume p(t) - 9Ef - (a+p)p(t) This can be solved to obtain P(t) = 0Ef + (p _ 0lfl)e"(a+p)t 0+p o a+p For large t, p will approach (eE^/o+p). Through using the final or maximum level of energy, this is maximum level of pollution per laborer. PQ is the pollution level at t=0. The net consumption per laborer at the horizon 61 Is approximated by Cf * (l-s)yf - (l+(£0/a+p))Ef where y^ designates the production per capita at the hori zon. This expression indicates that a finite level of net consumption can exist for the future generations in this model. How satisfactory this level of consumption will be dependent upon the specific values of constant parameters, E and 0 relative to a and p. However, an assumption for the above expression was that the critical pollution level is infinite. The analysis of the next subsection will examine the case in which a finite critical pollution level exists. Before going on to the next subsection, it should be emphasized that the above treatment has been with the case where no critical pollution level is defined. In the following, this case with an inequality constraint holding will be analyzed. Our purpose in this chapter and those to follow have not been to solve the complete mathematical problem but to examine the characteristics of the necessary conditions of the solution in economic terms. We have especially focused our attention on the effects of pollution on the turnpike or golden rule por tion of the optimal growth path. 62 Optimal Paths Where the Critical Pollution Level is Reached Let us reconsider the problem with the constraint pt p(t)LQe - Pc < 0 where pt pLQe - Pc = x(P>t) and xtPjt) is the constraint function in Appendix B. Sup pose that the total pollution level is initially below the critical P . If the planning period T is long enough, and there is net pollution generation, then at some point in the growth path the critical level P will L / be reached. This will be assumed to be the situation for the analysis of this subsection. The H function in this case will be defined as a 8 -6t \p , k,p,s,E,X,t) = ((l-s)Ak E -Ip-E)e (21) K P + t | > (sAkaE^-(y + p)k) k + \p ( 0E- (a+p )p) P pt - A(0E-(a+p)p)LQe Applying (i) of Appendix B, ^k(t) is still defined by equation (7), but now i p (t) is characterized by -fit pt = i p (t)(a+p) + le - A(t)aL e (22) p p o 63 This differs from equation (8) by an added term. To obtain the optimal investment and energy share programs, we must again apply (ii) of Appendix B. As before we shall first examine the maximization of the H function with respect to the investment share s while assuming the optimal program for energy, E(t). The H function (analogous to equation (10)) can be written in the form H(iJ>k(t) ,^p(t) ,k(t) ,p(t) ,s,E(t) ,A(t) ,t) = s(^k(t)-e”6t)Ak0l(t)E6(t) + < J > (t)(0E(t)-(a+P)p(t)} P + \(t)(V+P)k(t) _a _3 _ _£t + (Ak (t)E (t)-E(t)-Zp(t))e -X(t)(Loept(6E(t)-(o+p)p(t)) Note that the control s appears linearly as it did in equation (10). Thus, maximizing H as required by (ii) in Appendix B will again provide the three possibilities obtained before. That is, if then s(t) ■ 1 - (E(t)/y(t)) if (t) < e”^, then F(t) ■ 0 k if = e > then we have the singular case. A(t) has the properties stated in (v) of Appendix B. Pro perties (v,a) and (v,b) require that A(t) be non-negative. For the singular case, which is of most interest, it can be shown that the optimum investment program is given by equation (17). Thus, the expression for the in vestment program in terms of energy has not changed despite the addition of a pollution related constraint. The examination of the optimal energy program will procede using the results of the investment program analysis above. The H function is rewritten as H(e"6t, i | > ,k,p,s,E,A,t) = ^p(6E-(a+p)p) (23) -5t -St -e (y+p)k+(y-E-Ep)e Pt -A(L e (6E-(a+p)p)) -St ° where e has replaced i i > ^ . Applying (ii) of Appendix B to obtain E(t) as we did in the previous section, the following equation results for the optimal share of output to energy. E(t) _________3__________ (24) _ pt St y(t) l -(^p0 -*L0e )e If the constant is never met then the multiplier A is constant. 65 But condition (v) in Appendix B requires that X(T) = 0 at the planning horizon. Then X(t) must be zero all along the growth path and the expression for the energy share is exactly as was obtained in the previous section, i.e., equation (20). If the pollution level P is met during the turn- pike growth path then this expression for the energy share will have several characteristics. \b is the shadow P price of pollution and if the solution is to make sense then it will be a negative value except possibly at t=T. x(t) is a positive value except at the planning horizon where it has a value of zero. Thus, the net value of the terms within the parenthesis in the denominator is nega tive in value except at t=T and the energy share is less than 3 except at t=T. Again, as in the case where the critical pollution level was never reached, we see that the share to the pollution generating factor is less than for the pollution free case. If the turnpike conditions hold to the end of the planning period, t=T, then since \jjp=0 and x(t)=0, the energy share at the planning hori zon will be equal to the marginal elasticity of production for energy. However, the energy share expression (2*0 above does not yield an explicit expression for the value of energy. This is determined by use of the constraint, pt 66 t)L e o Therefore, p(t)Loe = Pc. -Pt p(t) = (Pr/L )e (25) 0 o If the constraint is to be maintained in time, pt pt p(t)L e + pp(t)L e =0. ° o • Substituting the expression (0E(t)-ap(t)) for p and re arranging terms, pt (0E(t)-op(t))LQe = 0. In determining the optimal energy function, this leads to _ _ -Pt E (t) = (a/0)p(t) = (a./0 ) (Pc/LQ)e (26) (f(t)/E(t)) = -p Energy is seen to decrease with time exponentially as long as the constraint x(P»t) continues to hold. Let us now examine the expression for the invest ment share of output. From equation (17), s(t) = (a/(y+p+6))(y+p+(3/(a-l))p) Since all the components of this expression are constant parameters, the share to investment is a constant. By taking the time derivative of the Cobb-Douglas function, 67 using equation (14) and rearranging, (k(t)/k(t)) = (y(t)/y(t)) = (3/l-a)P That is, along the constraint boundary of the turnpike path, we see that the rate of growth of per capita out put and capital are constant and negative. However, the rate of negative growth is less than that for energy. The share to energy is seen as _ -(l-a-3/l-a) pt (E(t)/y(t)) = (Ec/yc)e (27) where and yc are the energy and output levels at the beginning of the period along which the pollution level is critical. The share diminishes with time. We now have expressions for the investment and energy shares in the case where the critical pollution level has been reached. However, we have to consider if it is at all possible for this model economy to get off the maximum pollution level after such has been reached. This possibility would seem doubtful as continued popu lation growth would require more output in order to provide a given level of consumption. But since the total pollu tion level must stay constant or be reduced, such output increase is not possible. Thus, one would expect that the economy must continue to exist at the pollution level P . 0 68 Probably the most important property that we need to examine is that of the per capita consumption level over time. Since the maximum level of pollution has been reached, our economy has adjusted its output downward due to the reduction of use in one of the factors, energy. This, in turn, will affect consumption. By substituting our expressions for energy, output, and pollution (equa tions (26), (27), (25), respectively) into the equation for net consumption, -(ep/l-a)(t-t_) _ _ -p(t-tc) C (t) = (l-s)y e -(Ec+p )e N c c where subscript c designates the value of the variable at the time when the critical level of pollution was reached, s has been shown to be a constant. As time becomes very large, CN becomes arbitrarily small. Thus, this model exhibits the Malthusian tendency. That is, due to the growing population, consumption per capita will become smaller and smaller. As an alternative to future starvation, one might expect that such an economy would increase its value of the critical pollution, Pq. This, however, would only buy some time and eventually, any arbitrarily small level of consumption could be reached. Thus, In an economy with a constant state of tech nology and a critical pollution, the consump tion level in time will become arbitrarily small as the time horizon becomes very large. As in the Malthusian thesis, one would expect that as the consumption level approaches subsistence, population would become stabilized and life would continue at this level. Our model does not extend beyond such a subsistence point. Figures 6 through 8 present illustrations of the important variables involved in this model. Figure 6 shows the total pollution time history. The total pollu tion increases with time until the maximum level is reached. Figure 7 provides a diagram of the investment share versus the energy share. At some point tQ, the path is on the turnpike and moves to point t at which time the critical pollution level is reached. The value of s then shifts lower and the value of E/y may become higher at time tc. From this point, s is constant and E/y moves toward zero. Figure 8 presents the consumption history. The consumption will decrease exponentially after the critical pollution level is reached. It should be mentioned that depending upon the target value of capital K at the planning horizon, the 70 Total Pollution, P P c Time, t 0 t Figure 6. Total Pollution Time History — Constant State of Technology, Critical Pollution Level Reached 71 Investment Share, s s+E/y t=t t=0 t=t Energy Share Figure 7. Output Allocation Path - Constant State of Technology, Critical Pollution Level Reached 72 Net Consumption, Time, t c Figure 8. Net Consumption Time History - Constant State of Technology, Critical Pollution Level Reached 73 growth path will move off the turnpike with either a no consumption or no capital investment policy near the end of the planning period. The results are Malthusian and obviously the model suffers from the same basic shortcoming as a constant state of technology was assumed. In the following chapters, two other models of technology will be con sidered. These are: a) A neutral technical progress function with a constant rate of technological advance; b) A pollution abatement technology factor which represents progress toward reducing the amount of pollution generated by energy. This tech nology function also assumes a constant rate of progress. Our main concern will be to examine the requirements of technical progress in order that future generations may be assured a satisfactory level of consumption. CHAPTER V ADDING NEUTRAL TECHNOLOGICAL CHANGE A significant omission of the Malthusian theory of population was his neglect of technological progress. His conclusion that man would always approach a subsis tence level of consumption was based upon a model with no consideration for technical progress. Technical progress has been an accelerating factor in our society’s phenomenal economic growth and excluding it would leave a significant void in any analysis. Malthus’ forecast for future genera tions was incorrect just as that of the previous chapters’ conclusion would be incorrect if not extended to include technological advance. Let us refer to the model of Chapter IV as the "Alarmist’s’ 1 model. This seems to be an appropriate choice of terms since there has appeared in much of today's literature and news many "scientific" accounts of how this earth may not survive the twentieth century due to the pollution problem. They, of course, do not allow for the effects of innovation and technical change. The model of technological change that shall be considered here is an exogenous neutral technical progress function. The equation for production is now given by Yt a 8 y(t) = Ae k (t)E (t) (29) where Y is the constant rate of technical advance. The type of technical change is exogenous to the model and such change has been referred to as essentially "manna from heaven." This technical change is neutral since it neither saves nor uses capital, that is, it is one which produces a variation in the production relation, itself, but does not affect the marginal rate of substitution of capital for energy. This definition of neutral change was made by J. H. Hicks.'1 ' This type of model specifies that output will increase in time even though the same amounts of labor, capital and energy are used. Thus, the technical progress is essentially an efficiency factor which grows with time. The procedure to analyze the effect of adding progress will be the same as in the previous chapter. First, let us state the optimal growth problem. We wish to "^J. R. Hicks, Theory of Wages (Oxford, England: Oxford University Press, 193*1) • 76 T -fit maximize J = / (c(t)-Zp(t))e dt 0 subject to c (t y(t k( t p(t p(t s(t E(t s (t = (l-s(t))y(t) - E(t) = AeYtka(t)E6(t) = s(t)y(t) - (y+p)k(t) = 0E(t) - (a+p)p(t) Lnept < P o — c + E(t)/y(t) < 1 > 0 > 0 The Neustadt Theorem in Appendix B can be directly applied so that given an optimal path, there will exist shadow prices defined by i f » k(t) = -ipk(t) (as(t)y(t)/k(t)-(y+p)) +£(t)(E(t)a/k(t)y(t)) -ip0(t) (l-s(t)) (ay(t)/k(t) )e -fit (30) 77 it (t) = t y (t) (cr+p) + rfi (t)Ze’6t-A(t)cL ePt P p o o (31) T | ) Q(t) = 0 These equations are identical in form to (7) and (21), therefore, the addition of technical progress has not changed the functional relationships for the shadow price of capital or pollution. As before, the final pollution per capita level, p(T), and the final capital per capita level, k(T), will not be specified. This leads to the transversality conditions V » p (T) = 0 Also, i|»k ( T ) = 0 < j J n(T) = V = 1 o as discussed before in Chapter IV. Since ^ (t) is a con- o stant, its value will then be 1 along the entire path. The H function is defined in a form analogous to equation (21), that is, H(^k,^p,k,p,s,E,X ,t) = s(Tj>k-e <St)AeYtkaEe + i| ) p ( 0 E - ( a + p ) p ) - i { ; k(u+p)k + (continued) 78 yt a 3 -St + (Ae k E -E-Ep)e pt -X(L e (0E-(a+p)p)) (32) o Condition (ii) of Appendix B requires that at each instant of time along the optimal path, H(if> (t) , i | » (t) ,k(t) ,p(t) ,s(t) ,E(t) ,t) K p = max H(^, (t) f \p (t),k(t),p(t),s,E,t) s>0 k P E>0 o c ^ s+(E/Ak (t)E ) < 1 We may examine this maximization by assuming E(t) is known and study the characteristics of the above condition with respect to the s"(t) program. The H function for this study is written as H(^k(t),i|» (t),k(t),p(t),s,E(t),t) -St yt_a _3 _ =s(^(t)-e )Ae k (t)E (t)+^p(t) (6E(t) -(a+p)p(t)) - i ^ k(t) (y+p)k(t) yt_a 3 _ -St +(Ae k (t)E (t)-E(t)-Ep(t))e P t _ -X(t)(L e (0E(t)-(a+p)p(t)) o The polluted golden rule portion of the optimal growth path is of most interest. That is, we wish to 79 examine the case of the singular arc. By examination of (32) and (ii) of Appendix B, this singular case in the optimal path occurs when the shadow price of capital equals the discount factor, or symbolically, -fit i p (t) = e K As an assumption, we may assume that such a condition can exist and will continue over the entire growth path. For this assumption to be valid, the capital requirement k(T) at the planning horizon must be at such a level that the transversality condition yields -6T i | > k(T) = e This required capital level at t = T will be determined in the analysis to follow. The singular arc conditions (11) must again be satisfied. These are: -fit ¥fc(t) = e yk(t) = p+5+y • • (y(t)/y(t)) = (k(t)/k(t)) which are similar to equations (12), (13), and (1*0. The capital output ratio can be determined from the second of these equations so that 80 (k(t )/y (t)) = (a/(p +<5+y)) This is the condition relative to capital which will keep the growth path along the polluted golden rule. Although these equations seem to be identical to those in Chapter IV, there exist different relationships which were used in the derivation of these conditions. In particular, by differentiating equation (29), • « i (y(t)/y(t)) = Y+a(k(t)/k(t))+$(E(t)/E(t)) This leads to • • (y(t)/y(t)) = (y/(1-a)) + (B/(l-a))(E(t)/E(t)) (33) Note that the growth rate of output may be larger than energy growth depending upon the rate of technical progress. If the energy growth is negative, the output growth could still be positive. This is a new result which can be compared with equation (16). These conditions along the polluted golden rule can be solved for the optimal investment share of output 8, s"(t)= a (u+p+ y + B E(t) ) (3*0 p+6+y ( 1-a 1-a J This equation differs from equation (17) by an added term. This investment policy, iT, along the capital accumulation 81 turnpike can occur only if 0 . < s < (y-E)/y. But one notes that there is no guarantee that (y/d-a)) will not be so large as to make ? larger than (y-E), i.e., 7 y will take on very large values. Let us assume for the present time that y will be small enough that this turn pike situation will exist. By maximizing the H function with respect to the control energy, and letting s = sT, the general expression E(t) = 3__________ x _ (^ 9_ L0ePt)e6t is obtained. This is similar to the equation (24) derived in Chapter IV. CASE OF NO CRITICAL POLLUTION LEVEL Suppose we first consider the case where the critical pollution level is never reached. Then, the investment share of output is given by (34) and the energy share is given by E(t) _____________B " “ (a+p+6)(t-T) 1 - 91 Ve -1) \a+p+6 J where we have made use of equation (9) in the equation for the energy share above. Let us now consider some properties of the resulting growth path, for the purpose 82 of comparison with the results in Chapter IV. First, one notices that the share of output to energy has not changed, This means that as the amount of output increases due to Hicksian efficiency, the amount of energy will also grow along with it. This can be shown as follows. Let g equal the denominator for the energy share equation, , N (a+p+6)(t-T) g(t) = 1 - I 91 Ue -1) (a+p+6 J The optimal energy equation becomes E(t) = (S/g(t))y(t) and (E(t)/E(t)) = (y(t)/y(t)) -(g(t)/g(t)) Substituting equation (33) in this equation, the following results: = 1 _ ( - 1= S L \ g - CU l-a-3 \l-a-8 J (35) E (t) E(t) x"a"p U-a-py g(t) where (a+p+6)(t-T) g(t) _ -8Ze sr(t) r ^ (a+p+6) (t-T) 1 - f 91 V e \q+p+6 j -1) If the rate of progress y is positive, the energy rate of growth is greater than if no progress existed. 83 But this energy rate of growth is even greater than that of the rate of technical advance, y . Thus, there is a multiplier effect where the multiplier is (l/(l-a-3)). The first property can be concisely summarized: If there is a neutral technical advance and if there exists no critical pollution level, then the energy rate of growth along the polluted golden rule path will be greater than the rate of technical advance y by a multiplicative factor, This property will in turn, affect the output growth rate since output is a function of energy. This is over and above the increase just due to the efficiency factor. We then expect multiplier action to also exist in rela tion to output. Using (33) and the above relationships, the growth rate of output can be shown to be Thus, the multiplier for output growth is also (l/(l-a-B)). We may state the following result: (1/(l-a-3))• y (t) y (36) Suppose that neutral technical progress exists along the polluted golden rule path and further suppose that the pollution level is below critical, then the rate of growth of output per capita will increase by more than the rate of technical progress. The appropriate factor of multiplicative increase is (1/(l-a-3))• A third multiplier property which relates to capital does not need to be detailed since the growth rate of capital is equal to the growth rate of output. Therefore, the same multiplier property will hold. Figures 7, 10, and 11 illustrate the comparison between the constant technology case and the Hicksian progress case. Figure 7 depicts the time history of capital in the case when there is no pollution. When the economy is pollution free and no technical progress occurs, we have the steady state, balanced growth situation where output, capital and energy per capita are constant. Figure 9 shows this by a solid line labeled y = 0 . Where there exists a positive rate of technical progress then the rates of growth for output, capital and energy must all be equal, constant and non-zero. The relationship is that • • • (y(t)/y(t)) = (K(t)/Ic(t)) = (E(t)/E(t)) = (y/d-a-3)) illustrating the multiplier effect. Thus, the capital 85 Capital, k Y>0 y=0 Time, t 0 Figure 9. Capital History - Neutral Technical Change, No Pollution 86 growth in figure 9 is shown as an exponentially increasing function of time. Both cases are shown to begin at the same point. Figure 10 considers the case where pollution exists and depicts the investment allocation with time. Suppose that no technical progress occurs, Y=0, then the invest ment share may begin above a and increase exponentially with time. However, given that Y>0, then the share s will be greater by an amount (Y/(l-°0). The no-progress case is shown as a solid curve and the positive progress case is shown as a dashed line. Figure 11 illustrates the growth path on the allocation plane of s and E/y. The share to energy will not change due to the Hicksian progress but the share to investment will increase as was shown in figure 10. The comparison of share between the two cases, Y>0 and Y=0, are shown with the former in a broken curve and the latter in a solid curve. The difference between the two curves in terms of s are equal for all values of E/y. Before considering the case in which the critical pollution level is reached, we may note the level of con sumption along this polluted turnpike with technical advance. The net consumption per capita in time may be determined explicitly by solving for the growth of output, energy, investment share and pollution. It was determined 87 Investment Share, s Y>0 a Time, t 0 Figure 10. Investment Time History Investment Share, s 88 Y>0 > E/y Energy Share Figure 11. Output Allocation Paths 89 in Chapter IV that the net consumption per capita in a constant technological world would be satisfactory depend ing upon the parameters involved. Since output growth with technical change is greater relative to energy growth than without technical change (equation 33), net consumption will be higher than that found in Chapter IV. How much higher will depend upon the rate of techni cal change. CASE WHERE ECONOMY REACHES CRITICAL POLLUTION which a critical pollution level is reached in our world of technological change. Assume that at some time tc, the critical level is reached. Using the analysis of the previous chapter, the rate of energy growth was determined to be -p (equation (26)). Also using equations (33) and (3*0, the following set of equations resuls: It can be shown by the same argument used in Chapter IV following equation (26) that the optimum growth path will never move away from the maximum pollution level We are now in a position to examine the case in (E(t)/E(t)) = -p (37) (y(t)/y(t)) 90 once it has been attained. Applying equations (37) and the explicit expression for pollution (25), the following equation for net per capita consumption is obtained. (£#*- v -p(t-te) Qn = (l-s(t))yce -(Ec+Zpc)e where t_ = the time at which the critical pollution level was first reached yc = production level when the critical pollution level was first obtained P c = pollution level per capita at the time when critical pollution level was first reached. The equation contains two terms which are of interest in examining the fate of the future generations in our model. The second term consists of a constant (E_+ p ) times V C a factor which decreases exponentially with time. Thus, the second term becomes small as the time becomes large. This indicates that the first term is of major interest to us. Whether this first term increases, remains constant or decreases with time depends upon the value of the exponent of the exponential factor. In particular, the sign of Y “ BP is the key to the destiny of future generations. If t y < ep 91 then the net consumption will decrease with time exponen tially. As long as the rate of technical change is posi tive, then the rate at which net consumption decreases is less than if technical progress did not exist. Thus, the technical progress in this case has bought some time but has not eliminated the eventual subsistence level found in Chapter IV. Suppose that the rate of technical progress is equal to the product of the partial elasticity for energy and the population growth, y = Bp then the net consumption will increase to a maximum given by c (t) = (l-s(t))y n c Thus, we may expect future generations to be able to attain some level of net consumption, but how satisfactory this level is depends upon the output at the time when the critical pollution level is reached. This level of output is dependent upon the values of the parameters of our problem. The case when Y > Bp will provide an ever increasing net consumption with time, that technical progress has rescued our economy from a 92 Malthusian result. Thus, In an economy of neutral technical pro gress where a critical pollution level has been reached, future generations can be guaranteed a satisfactory level of con sumption only if Y > 3p Equality between these two terms will pro vide some maximum level of net consumption which may be satisfactory depending upon the magnitude of the constant parameters in the model. This completes the analysis considering the neutral tech nical progress in the model economy. In Chapter VII we will consider it in conjunction with pollution abate ment technology. Before proceeding to the chapter on the pollution abatement technology, it would be illuminating to illus trate the results of this last subsection with some diagrams. Figure 12 Illustrates the fact that total pollution will reach the critical level PQ and remain there. Figure 13 shows the share to investment and to energy increasing until the critical pollution level is reached. Then investment share will immediately drop Total Pollution, P h 93 P c Time Figure 12. Total Pollution Time Path 94 Investment Share, s s+E/y t=t E/y Energy Share Figure 13. Output Allocation Paths 95 below a and the share to energy may ’ ’ jump" to a higher value still lower than $. From this point, the growth will move along a constant value of s and a decreasing value of E/y. The time rate of decrease is p, the population growth rate. Figure 14 presents the net consumption history diagram. The consumption share of output is initially decreasing with time, however, because of the technologi cal advance and its multiplier, the consumption itself may be increasing. This is the case we have shown in figure 14. At time t_, when the critical pollution level is reached, the total pollution may drop by a sig nificant amount. The rest of the growth path depends upon the magnitude of the progress rate in relation to the population growth rate. When y<3p, the consumption will decrease exponentially. When y=$p then the net consumption will approach a maximum level. This level may or may not be a satisfactory level. If y>3p then consumption increases exponentially. We are now prepared to analyze technical progress directed at pollution abatement itself. 96 Net Consumption, cn Y>Bp Y<Bp Figure 14. Net Consumption Time Path CHAPTER VI THE POLLUTED TURNPIKE AND POLLUTION ABATEMENT TECHNOLOGY The specific case that will be analyzed is one in which the production technology is assumed constant, but there exists research and progress toward pollution abate ment. The problem can be concisely stated in mathematical terms as determining the growth path of s(t), E(t), k(t), p(t) in order to maximize T / (c(t)-Ep(t) )e ^dt 0 subject to equations c(t) = (1-s(t))y(t) - E(t) a 3 y(t) = Ak (t)E (t) (38) k(t) = s(t)y(t) - (y+p)k(t) p(t) = e_irt0E(t) - (a+p)p(t) and inequalities - 97 - 98 pt p(t)L e < P o — c s(t) + E(t)/y(t) < 1 s(t) >_ 0 E(t) > 0 -TTt where e is the pollution abatement technology which explicitly appears only in equation (38). i t is the rate at which progress in pollution abatement advances. This is a model in which abatement progress advances at a constant rate similar to the neutral technology advance. Given that a solution exists for this problem, then there exists auxiliary variables i J j (t), * | > k(t) which are defined by -6t pt i | » (t) = (t) (a+p)+Ee -X(t)L e (39) p P o ^k(t) = ~^k(t)(a?(t)y(t)/k(t)-(y+p)) +£(t)(E(t)a/k(t)y(t) (^0) -5t +ip (t) (s(t)-l) (ay(t)/k(t) )e o i (t) = 0. o (t), (t), and \b (t) are continuous. The multiplier ° p k function £(t) is zero when x(k(t),s(t),E(t),t) < 0 and 99 £(t) <_ 0 when x(k(t) ,s(t) ,E(t) ,t) = 0. As before, the pollution and capital level at t = T will not be specified so that the transversality conditions ((iii) in Appendix B) yield that V ( T ) = 0 i | » k(T) = 0 *o(T) = vo Since < J > 0(t) is a constant, and assumed not zero, we may set \pQ equal to one without altering any necessary condi tions of the theorem in Appendix B. The H function is defined as H(i|>k, i | > p,k,p,s,E,t) = -6t a 3 -ift s(^k-e )Ak E +ip (e 0E-(a+p)p) P a 3 -<5t - i | ; k((y+p)k) + (Ak E -E-Zp)e pt -7ft -X(LQe (e 6E-(a+p)p)) (41) Condition (ii) of Appendix B requires that at each instant of time, H ( ( t) , i p (t) ,k(t) ,p(t) ,s(t) ,E(t) ,t) k P max H(ip, (t),t p (t) ,k(t) ,p(t) ,s,E,t) s>0 * P E70 _a _3 s+(E/Ak (t)E )<1 100 To perform the maximization of H, consider first examining the optimal program for s with the energy program E(t) given. H may be written as: (t) , i | » (t) ,k(t) ,p(t) ,s,E(t) ,t) ■ K P 0 6 ^ s(^k(t)-e"6t)Ak (t)E (t) + (Aka(t)E3(t)-Ep(t)-E(t))e“6t -ipk(t) (y+p)k(t) + ip (t )(6E(t )-(a+p)p(t D ir +X(t)(LoePt(e_7rt0E(t)-(a+p)p(t))) Performing the maximization of H with respect to s we again find that since the s control appears linearly, the H-function will take on a maximum depending upon the value of the coefficient ( i | ; k(t)-e-5t). If ^k(t) > e_<5t then s"(t) = 1 - (E(t)/y(t)) (42) ^k(t) < e-<^ then s(t)=0 (43) The first situation (42) is a no-consumption path while the second is a no investment path. The optimal growth path of most interest to us is the one which corresponds to the singular arc relative to the control s. The singular arc occurs when i | > k(t) = e_(^t. Along the singular arc, the shadow price of capital will equal the discount factor. The necessary conditions along this singular arc, 101 equation (11), provides that (3 y(t)/3 k(t)) = yk(t) = p+p+6 (44) • • (y(t)/y(t)) = (k(t)/k(t)) (45) These are equivalent to the expressions derived previously. Suppose initially, that the critical pollution level has not been reached, but eventually with time it is attained. The general relationship for the energy share of output is obtained by satisfying the maximum condition (11). This is E(t) _ 8 /- It IT N ^ 7 --------- =rt------- rp^iryt st— (46) ytt) l-(*pe e-XL0e )e This expression is used to determine the energy level as long as the critical pollution level has not been attained. This equation differs from that in Chapter IV by the expo nential term involving pollution abatement. With time, the term in the denominator will approach one faster than without technological progress in pollution abatement. The share of output for investment is derived again to be s(t) ® ( a 3\fE(t)\~> (47) ( a 'NCu+p-/' B \ fE(t)\A \ p+5+y )C Va-l) ) Since a<l, the term in the parenthesis is positive when 102 energy growth is positive. CASE OP NO CRITICAL POLLUTION The first case to consider is when the pollution level is never to reach the critical level, then equation (39) can be solved so This expression indicates the same general property as that in Chapter IV. That is, the share of output going to energy initially begins at a level less than 3. This level is the same as in the case in Chapter IV, Prom this level, the energy share will increase with time to the value of 3, the partial elasticity of production for energy. This is also the property of the similar case in Chapter IV. However, the path along which the energy share variable moves is different due to the pollution -fit (a+p+5)(t-T) (t) = 2e (e P a+p+6 -1) Substituting this into expression (46) E(t) = y(t) 6 (a+p+6)(t-T) -1) (48) E (t) y(t) 3 (49) -(a+p+6)T 1 + Z9(1-e a+p+6 103 abatement technology. This energy share path Is closer to the value of along the path. This will be shown diagrammatically later. The equation for the energy share (48) can be written as E(t) = (g/g(t))y(t) where - (6 +tt ) t (a+p+6 ) (t-T) g(t)=l - Z Q e_______ (e -1) a+p+6 Taking a time derivative, • • • (E(t)/E(t)) = (y(t)/y(t)) - (g(t)/g(t)) using • • (y(t)/y(t)) = (g/(l-a))(E(t)/E(t)) • • (E(t)/E(t)) = -((l-a)/(l-a-8))(g(t)/g(t)) where (a+p+6) (t-T) -(6+ir)t g(t) -((a+p*iT)e +(6+tt) )6Ze (50) gttl (a+p+6) (t-T) -(6+u)t — (6+tt)t a+p+6-E0e e +S9e The substitution of this expression into equation (47) gives us an explicit function of time for the investment share of output. Using the above expressions, we may extract some 104 qualitative information which may be displayed diagramma- tically. Let us do so but, in particular, present a comparison of the case with and without the pollution abatement technology. Figure 15 depicts the reduction in pollution per capita with time as pollution abatement. This figure, of course, only presents one possible compari son in that the initial level of pollution will play a major role in how these paths will move. However, the figure does show the divergence between the time paths with and without pollution abatement which is the major point to be brought out by the figure. The figure shown assumes that the values of E and initial pollution level are such that there is initially a positive growth in the pollution level. With a long enough time period, the case with pollution abatement will eventually cause the pollution per capita level to decrease in time and approach zero. Figure 16 presents the comparison of the path of the energy share with time. Again the dashed line illus trates the case where there exists no pollution abatement advance. This path begins at time zero with a value equal to equation (49). With time, the path will move toward 3 and will be equal to 3 at the time horizon. For the case where there are advances made toward the reduction of pollution generated by energy, the path is 105 Total Pollution No Pollution Abatement with Pollution Abatement Tech nology 0 Time Figure 15. Total Pollution Time Path 'CiM 106 Energy Share 3 E/y 3 (0 Time Figure 16. Energy Share Time Path 107 shown by the solid curve. The initial and final points for this curve are exactly the same as in the dashed line. However, the economy with the pollution abatement advance will experience a higher share of output to energy between the initial and final points. That is, the energy share path arches closer to 3 in the case where pollution abate ment exists. Figure 17 presents a control plane diagram which depicts the growth path of the two control variables s and E/y in time. For the case where pollution advancement exists, s(0)= a fu+P+/ B ^ y+p+6 (i-a-81 -(a+p+6)T (a+p-TrUee +Z9(6+tt) -(a+p+6)T a+p+6-£ee +E0 „ (51) which is greater or less than 8 depending upon the value of the constant parameters. Suppose it is positive. The energy share is given at t=0 by equation (50). At the final time T, the energy share is 0 and the investment share is / \ f -(6+ir)T s(T)=/ a \ )p+p+8E9e_______ V (52) \]i+p+6 / ^ l-q-8 J For the case where pollution abatement technology is a constant, ir=0, - zee-(q+P+5)T__________JU 53) -l+(eE/(a+p+6))(l-e"(a+p+6)T) ) r (o) = ( q NC+p+T e ^ ly+p+6 It U-q-8' 108 Investment Share, s s=l s + (E/y) t=T t=0 — o E/y, Energy Share Figure 17. Output Allocation Paths 109 and s(T) = f o \ Hp+p+(B0E/(l-o-0))> (5*0 [m + p + s I C J Thus, the starting points differ for the two cases. The values of the energy share In both cases are the same but the Investment share of output in the pollution abatement progress model is higher than without pollution abatement. This is seen by examining equations (51) and (53). How ever, the reverse is true at the planning horizon. Equa tions (52) and (5*0 show that at the time T, the output share to investment is greater for the case where there is no progress in the abatement of pollution through tech nical change. These properties of the growth paths are shown in figure 15, the two paths representing the con stant pollution technology case. Thus, the change in investment share of output is less when there is pollution abatement advance. CASE WHERE CRITICAL POLLUTION LEVEL IS REACHED Suppose the critical pollution level P is reached c along the optimal growth path, then pc - and p(t) = (p0/Lo)e pt 110 or (p(t)/p(t)) = -p Using the fact that _i_. -irt _ p(t)= e 9E(t) - (a+p)p(t) we can derive _ _ (ir-p)(t-t ) E(t) = EQe c (55) where E = gpe eLo t = time at which the critical pollution level is reached. The energy per capita is exponentially changing function of time where the rate of change is (f(t)/E(t)) = tt - p The investment share of output is given then by s(t) = ( o t ^ (yi+p+ B (tt-p)) (56) \y+p+6/C 1-a J This is a constant which depends upon the parameters of the problem. The output growth is given by (y(t)/y(t)) = ^B(tt-p) ^ and ^B(ir-p) (t-tn ) ^ j 111 y(t) = yc e (57) where yc = output level at point of critical pollution (1/a) aA (ci/(l-a)) _ 3 E c p+6+y The expression for capital is given by k(t) = fay(t)\ ^p+6+y j The net consumption per capita is then given by (g (tt-p \ \ 1-0 t c ) _ (T T “p) (t —t ) _ -p(t-t ) Cn(t) = (l-s)yce -Ece -z;Pce where s is constant and the subscript c denotes the value of the variable when the critical pollution level is reached. We note dire results if there is no pollution abatement progress, i.e., tt=0. If this happens then net consumption will decrease exponentially with time and future generations of our model economy can expect a life of low consumption. Suppose our economy does progress at a constant rate of pollution reduction through technological advance. But suppose the rate of progress is not as great as the rate of population growth, that is i t < p 112 As long as this rate of progress in reducing pollution is less than the growth of population, the net consumption per capita will decrease exponentially with time. The only thing that such a rate of progress buys is time. The Malthusian result will still eventually result only not as soon. Using the same method as was done in Chapter IV, it can be shown that with it < p , the optimal growth path will never move away from the critical pollution level. If our economy was able to provide progress in abatement of pollution at a rate equal to the rate of popu lation growth then we can expect a level of net consumption which is growing exponentially with time. The gross con sumption will at least be as good as existed at the time when the critical pollution level was reached. The first and second terms of equation (58) are constant and the pollution per capita term is decreasing. Since the pollu tion level is held constant, the cost of this pollution will be spread over more people with time due to the constant growth of the population. Whether or not this consumption level is satisfactory depends upon the para meters in equations (55) and (57). The key is (a/0), the ratio of the pollution dissipation rate and the pollu tion generation rate. Of course, things will become even better when 113 the rate of technological advance in pollution mitigation is greater than the rate of population growth. This is seen in equation (55). For ir>p , the equation indicates that the energy will increase exponentially with time. If the time difference between T and t is small, then we would expect that there might only be a short period in which the economy would expect to experience a deprived level of consumption. The energy level would eventually rise and accordingly so would the total output level, leading to more net consumption. This would occur only over the short period between t and T. On the other hand, if the time difference between tc and T is large, that is, the critical pollution level was reached much ahead of the planning horizon, then energy according to equation (55) could take on arbitrarily large values. If this is the case, then we could expect the energy level to reach that which corresponds to not being at the critical pollution level. This would then mean the optimal path would move off the pollution con straint and could ignore it the rest of the way. This is possible in this case, in contrast to that treated in Chapter IV, due to the fact that pollution abatement technology will allow the total energy to increase without a corresponding increase in total pollution. This latter case is of particular interest to our 114 economy and some of the possible characteristics of this case can be illustrated by our phase and control plane diagrams. Figure 18 shows the time path of total pollu tion. The path begins below the critical level, moves to and along the critical level between times t and t . c o During this time, the pollution technology is improving the pollution problem until t is reached at which time the economy can move off the critical path. Figure 19 presents the energy per capita path with time. As long as the critical pollution level is not yet reached, the energy per capita will increase as shown until t is reached. At this point, the critical pollution level was reached and there had to be an adjustment downward in energy per capita produced. This is shown by the dashed line at tc< But due to the pollution abatement advance being larger than the population growth, the energy level will increase with time even though growth is along the critical pollution level. When time tQ is reached, the energy path will continue to increase in an environment in which pollution will be decreasing. Figure 20 illustrates the control path. Between times t=0 and t , the path is moving away from the origin as both s and E/y are increasing. At tc, the values of s and E/y will be discontinuous and the path moves in the direction shown between the two points t . During the time c 115 Total Pollution, P P c 0 t, Time t t c o Figure 18. Total Pollution History 116 Energy per Capita, E it- p T T <p t, Time Figure 19* Energy Per Capita History Investment Share, s s+E/y t=0 — G Energy Share Figure 20. Output Allocation Path at which the growth path is along the critical pollution, the controls will move from t to t0. Along this path, s is constant and E/y is increasing. When t is reached, the energy share will move toward with the path shown between t and t=T. o The results of this chapter can be summarized. Given pollution abatement technological advance, the model economy can expect only relatively short periods of deprived net consumption if the rate of pollution abatement progress i r is greater than the pollution growth rate p. If n=p, then the net consumption level near the end of the planning horizon can expect to reach some maximum level of consumption set by the constants of the problem. This may or may not be a satisfactory level. If tt. < p, then the Malthusian results will develop. Chapter VII will examine the case where neutral technologi cal advance may augment the deficiencies of the pollution abatement technology. CHAPTER VII THE POLLUTED OPTIMAL GROWTH PATH, NEUTRAL EFFICIENCY AND POLLUTION ABATEMENT PROGRESS We have seen in Chapters V and VI that the basic problem is whether progress in the form of either pro ductive efficiency or pollution abatement can overcome the burden brought about by population growth. We have seen that with population growth comes more pollution if a golden rule level of consumption is desired. But given that some critical level of pollution exists then even the golden age consumption will degenerate toward subsis tence levels without the aid of technological progress. Refutation of the Malthusian thesis is based upon the presence of technical progress. Up to the present time modern man has been able to accelerate economic growth at a pace which outstripped population growth and affluence has resulted, not the despair which Malthus predicted. In the previous two chapters, we have examined the requirements of technical progress to sustain a satisfactory - 119 - 120 level of consumption in the golden age. The technical progress took the form of productive efficiency and pollu tion abatement. Basically, the requirements were that that rate of progress In these two forms of progress had to be larger than some function of population. That is, Y > $P and tt > p If the inequality was reversed then our model society could expect despair for future generations and gloom would be inevitable. However, these requirements on the two types of progress were considered separately. The economy under present consideration is expected not only to have progress in efficiency of production but also to work toward re ducing the costs incurred by pollution. Thus, the tech nical requirements obtained in previous chapters would appear to be too stringent and some relaxation is available if both types of progress exist. This is the substance of the analysis in this chapter. The optimal growth path is that in which we have T -6t maximized / (c(t)- p(t))e dt 0 subject to 121 c(t) = (l-s(t)y(t)) - E(t) yt a 6 y(t) = Ae k (t)E (t) k(t) = s(t)y(t) - (p+p)k(t) -irt p(t) = e 9E(t) - (a+p)p(t) (59) pt p(t)L e - P < 0 o c ~ s(t) + E(t)/y(t) - 1 0 E(t) > 0 s(t) >_ 0 This Is the model to which we shall address ourselves in this chapter. The shadow price equations are • ^k(t) = -^k (t) (as(t )y(t)/ic(t) - (p+p)) + 5(t)(E(t)a/k(t)y(t)) -6t + ^Q(t)(s(t)-l)(ay(t)/k(t))e -61 pt ^p(t) = ^p(t) (cf+p)+^0(t)Ze -X(t)LQe (60) ^Q(t) = 0 As shown these equations have not changed in form from the basic model in Chapter IV. The H function is defined by 122 H(1»k,¥p,k,p,s,E,t) = -St yt a 3 -irt s( -e )Ae k E + i | ^ (e 0E-(a+p)p) k p yt a 3 -St (y+p)k) + r { < 0(Ae k E -E-£p)e pt -irt -X(LQe (e 0E-(a+p)p)) The transversallty conditions require that ip (T) = v Yo o i p (T) = 0 P since we have not constrained the final pollution level nor the discounted consumption over time. As before, setting i p (t) = 1, will not affect the necessary condi- o tions for the optimal growth path. From (ii) of Appendix B, we have at each instant of time H ( i | » (t) , i | > (t) ,k(t) ,p(t) ,s(t) ,E(t) ,t) k p = max H ( i | » (t),i|< (t) ,k(t) ,p(t) ,s ,E,t) s>0 k P E>0 _ a _3 s+(E/Ak (t)E )<1 Let us consider this maximization by assuming the optimal E(t) and examining the i"(t) that results by maximizing H with respect to s. H is rewritten as 123 H ( i | » k(t) , i | i p(t) ,k(t) ,p(t) ,s,E(t) ,t) -fit yt_ = s ( i | j k(t)-e )Ae k(t)E(t) -irt _ + i f ; (t) (e ©ETt)-(a+p)p(t)) - i ( ; k(t) ((y+p)k(t)) yt_ a _3 _ _ -6t +(Ae k (t)E (t)-E(t)- p(t))e P t “T T t ___________ _ -X(t)(LQe (e 0E(t) - (a+p)p(t))) By maximizing the H function with respect to the control variable s, the following results: If +k(t) -fit > e then s(t) = 1-E(t)/y(t) If -fit < e then s(t) = 0 If *kct) = e-S* then a(t) = ( a \(y+p+ y + 6 ly+p+fi/v. 1-a 1-a (61) E(t) Equation (61) was derived by a method similar to that in Chapter V. As we have done before, we shall be concerned r j l . strictly with the case where ijj (t) = e“ which is that K corresponding to the polluted golden rule path. The energy share of output is obtained along this polluted turnpike path by maximizing the H function with respect to energy. The result is 124 E(t) = S (p—u)t st )e (62) y(t) -irt l-(i^p(t)e 0 -X (t)LQe The above equations provide us with the general equations of the optimal path we are concerned with. CASE WHERE NO CRITICAL POLLUTION LEVEL EXISTS As has been done in the previous chapters, let us first examine the case where no critical pollution level exists. In this situation and the equation for the shadow price of pollution will be given by through solving equation (60). Substituting this in equation (62), E(t) _ B y(t) - i -/e-*tei\(el0+p+s,u-TJ. U+P+6 / Using the fact that (E(t)/E(t)) = (y(t)/y(t)) - (g(t)/g(t)) (66) X(t) = 0 (cx+p+6) (t-T) ( \ /ke___V -1) (63) (65) and 125 where and then -irt \ (a+p+6) (t-T) g(t) = 1 -f e___01:) (e -1) V p+a+6 At t = 0 g(t) g(t ) " E(t) E(t) ) j(0) E (0) -irt (a+p+6)(t-T) e 9E( (Tr-a-p-5 )e ^1. -irt (p+a+6 ) (t-T) a+p+6-e 6£(e -1) 1-a \ f y ^ g(t) I [ t= z\ - I t f y -(a+p+6)T 1-a] ) y _ 9^( ( tt- p - p - 6 )e -1)^ l-a-3/;1-a -(a+p+6)(T) a+p+6—0£(e —1) At t = T, j(T) = E(T) l-a-8 We see then that this case has the same initial and ter minal point in the s-E/y plane as the model in Chapter (Compare equations (64), (67) and (68) with (6l)). However, the path between these two points is different The path will move toward the final point faster than found in Chapter V and spend more time in that vicinity This is seen in figures 21, 22, and 23. (67) (68) Investment Share, s 126 With Pollution Abatement No Pollution Abatement a t, Tifffe T Figure 21. Investment Share History 127 Energy Share, E/y With Pollution Abatement No Pollution Abatement 0 Figure 22. Energy Share History Investment Share, s 128 With Pollution Abatement No Pollution Abatement + E/y t=T t=0 — e E/y Energy Share Figure 23. Output Allocation Paths 129 As was shown in Chapter V, the model economy exhibits a multiplier property: Given the model economy works under the system described by the equations in (59) and that the growth path of the economy is along the polluted golden rule, then growth rate augmentation due to the rate of technical progress, y , is given by a multiplier, (l/(l~a-3)), times the rate of tech nical progress. Thus, along the optimal path dubbed the ’ ’ polluted” turn pike or golden rule, technical progress is used to even greater advantage for output growth than would be expected from initial knowledge of the rate of technological advance. That is, at first glance, one would expect the augmentation to output growth by technical advance to be no more than Y • The above analysis has derived some important properties of the optimal growth path when a critical pollution level does not exist. A final remark should be given to the net consumption level for the particular case. Since there is technical advance which is reducing pollution generation through energy production, there will be less and less pollution being produced as time passes 130 by because of the more efficient production activity. Therefore the level of pollution will be higher than with out the productivity advance. The main point is that even though pollution as well as output is being increased, there will be a larger and larger difference between the gross consumption and the cost of pollution. The techni cal advance is an exponential function of time and the net consumption is a linear relationship. Thus, the economy under consideration could expect a gluttonous situation in the distant future. CASE WITH A CRITICAL POLLUTION LEVEL advances in production and pollution abatement. What then are some of its properties and what can future generations expect in terms of net consumption? Assume that the planning horizon is of a long period. At some finite time, the economy reaches the critical pollution level, P . Then the following relationships for growth 0 are derived from equations (59), (6l) and (65). Suppose now the economy which has both technical ly+p+S J (U+P+ v+3 (tt-p ) 1-a ) (E(t)/E(t)) = v - p 131 The expression for net consumption is now ^(y+B(tt-p ) ) (fr-tp) cn(t) = (l-s(t))yce 1_a _ (ir-p) (t-t ) _ -p(t-t ) (69) - Ece c - Epce c The exponential rate of change for output is more com plicated than before but still simple enough to analyze. We have shown before that if the rate of progress in pollution abatement advance is equal to or greater than the population growth rate then future generations can be assured of at least as much consumption as existed at the time the critical pollution level set in. We wish to see how technical progress can aid when pollution abate ment technology cannot keep up with the population growth. Thus, assume it < p. Then the second and third terms of (25) will diminish exponentially with time. The impor tant term is the first one. In particular, the rest of our analysis is with Y +$ (tt-p ). Suppose Y < 13 ( tt—p) | . In this situation there will be an exponentially decreasing consumption which means possible despair for future generations. If, on the other hand, Y = | B(ir-p) | 132 then net consumption will increase with time and future generations can expect consumption to approach a maximum of d-s)yc For the final case, Y > | 3 ( it— p ) | An obviously gluttonous species will evolve as consumption will increase indefinitely. In this model with Hicksian neutral technical progress, we have four variables involved with the control of consumption in the future. The pollution abatement progress rate tv was shown in the previous chapter to be the potential hero of our society. However, if this parameter is not up to the task then we must depend upon technical efficiency in production to take up the slack. We have seen that the rate of technical advance desired is not required to make up the entire difference between the population and pollution abatement advance. The amount of compensation desired of the technical advance depends upon the partial elasticity of production of energy. This might be expected since energy is the polluting factor and is only one of three factors in the production process. CHAPTER VIII SUMMARY AND FINAL REMARKS The previous chapters have provided a mathemati cal investigation of the interrelations of economic growth, pollution and technological change. The scope of the investigation was to delve into the effects of adding pollution to the existing body of optimal economic growth literature. The analysis was theoretical and assumed a one-sector Cobb-Douglas model. Three types of technical progress were considered: 1) Constant State of Technology 2) Neutral Technological Advance 3) Pollution Abatement Technological Advance. The source of pollution is a factor of production, energy. The one-sector economy was assumed to have as its objec tive, the maximization of net discounted consumption per capita over time. The problem is one in optimal control theory. Although the more general solution was examined initially, the analysis mainly focused on the polluted - 133 - 134 golden rule growth path which corresponds to the singular arc of optimal control theory. The other two alternative paths, which correspond to no-consumption or no-investment paths, were briefly described but later omitted since these represent extreme situations and could justifiably be omitted to concentrate on the polluted golden rule path. A major assumption then was that the golden rule paths existed and that the initial and final conditions of the problem were such that the golden rule condition occurred along the entire optimal growth path. Along the polluted golden rule growth paths, regardless of the technological advance model considered, the following properties of growth were derived: i) The shadow price of capital is equal to the discount factor, ii) There is a constant capital-output ratio, iii) The magnitude energy growth is greater than the magnitude of output or capital growth, iv) There is a shift in the allocation of out put away from the pollution generating factor, energy, to the other endogenous factor, capital, v) The marginal productivity of capital is equal to the sum of the population growth rate, the discount rate and the capital 135 depreciation rate. This property is dubbed the polluted golden rule. These five properties have been compared to that of the existing literature on optimal economic growth. Proper ties (iii) and (iv) represent new contributions to the literature. In each examination with the various technical change models, a case was considered in which there was a critical level of total pollution which represents a maximum limit which the economy would permit. When this limit was reached, the economy would have to reduce its energy production. In situations where such a limit was reached, it was determined whether there could possibly exist a desirable consumption level for future generations. In the model with a constant state of technology, the net consumption level was found to decrease with time. This was due to the fact that a positive population growth requires more output in order to keep the same standard of living. However, given that one has already reached the limit in total pollution, a constant level of total energy must be maintained. Therefore output is restricted and the overall consumption will decline. Since there is no technological progress available to increase output or reduce pollution, this leads to a dim future for inhabitants of this economy. The main reason for this 136 dismal outlook is the continued population growth and the output restriction placed by the critical pollution level. The results are Malthusian in that consumption will continue to decrease toward the subsistence level. At subsistence, the population growth rate would have to be zero, since consumption could not decrease any lower or else no population could exist at all. Thus, the population level at this point would stay constant and any "excess" members to this economy would die from starvation. The living would survive at subsistence. The addition of an exogenous technical change factor will at least buy some time as to when a subsistence standard of living would be reached. The analysis deter mined the rate of neutral productive progress required in order that future generations could be assured a desirable level of net consumption. The requirement is that Y > 3p that is, the rate of technical progress must be greater than the product of the partial elasticity of production for energy and the population growth rate. Since $ is less than one, we see that the rate of technical advance does not have to be as large as the population growth rate. The factor 3 in this condition indicates that since energy is the polluting factor and technology change is 137 neutral, the Increase in productive efficiency will only have to compensate any decrease in output which is caused by reduced levels of energy. As discussed above the lower energy level is caused by the pollution restriction. If the above condition for technical progress is satis fied, the economy’s net consumption can become arbitrarily large with time. In examining this problem, a very interesting property of the optimal growth path was derived. This was the existence of a technical progress multiplier. By taking a simple time derivative of the Cobb-Douglas function, one expects that the addition to output growth due to technical change would be y. However, due to the feedback caused by the allocation of the higher output, the following multiplier was found: 1 1-a-B where a and 6 are the partial elasticity of production for capital and energy. The product of this multiplier and the rate of technical advance will provide the increase in the output growth caused by the Hicksian progress factor. The third technical change factor analyzed is one directed at pollution abatement. This type of progress assumes that with time, advances are made toward the 138 reduction of pollution generated. As with the neutral efficiency factor this type of progress will at least buy our economy some time with respect to the time when subsistence may set in. To insure a happy consumption for the future in this model * the following condition must exist: TT>p . i r is the rate of pollution abatement progress. The condition requires the pollution abatement progress rate to be greater than the population growth. If this require ment is met, then the economy can expect to move off the critical pollution level at some time in the future. The economy will eventually move toward pollution free growth. The final analysis in this dissertation examined the combination of both the neutral and the pollution abatement technical factors. Here, the two rates of tech nical change complement each other so that requirements on each, in order to maintain a satisfactory level of consumption, will be eased. The situation of most con cern would be where the advances in pollution abatement were not enough to set off the population growth, that is, ir<p. In this case the neutral technical advance would have to compensate for this deficiency. This requires that y > | 3(ir-p) | . 139 Note that the rate of neutral technical advance does not have to compensate for the entire difference between the pollution abatement advance and population growth rate since g is less than one. The compensation is less than the difference by a factor equal to the partial elasticity of production for capital. A second possible situation would be if the neutral technical advance was not sufficient to sustain a desired level of consumption, i.e., Y < 3p then the pollution abatement advance must take up the slack. The requirement on this advance is that t t > I (y~Bp) I 3 Since g is less than unity, the pollution abatement ad vance must do more than just make up for the difficiency of the neutral progress. The general requirement for a guaranteed future of satisfactory consumption in the combined technology case is y + g(ir-p) >0 Because of the technical progress multiplier, small changes in the rate of neutral technical progress will provide much larger changes in the output growth. This can be to the economy's advantage if the neutral 140 technical advance rate Is as easy to Improve as the pollu tion abatement advance rate. However, the multiplier will be to the economy’s detriment if the neutral progress rate is more difficult to improve. There are two distinct policy implications that result from the analysis performed in this dissertation. The first is that: An economy which takes into account the cost of pollution due to production should appropriately shift its use of the factors in production away from the polluting factor to the non-polluting factors. In this dissertation, the polluting factor is energy and the non polluting factor is capital. This would be an expected result. For the particular economy of interest, the production process will become more capital intensive than it would be without considering pollution. An example of this policy in practice might be the emphasis of mass rapid transit rather than the automobile for commuter transportation. The mass rapid transit system would be more highly capital oriented than the automobile. Huge capital development is needed for the rapid transit system but its overall energy use and associated output is much lower than that of the 141 automobile. A second policy implication is that regarding priorities for research and development. This dissertation implies that: Research and development priorities should emphasize progress in increased productivity through efficiency in production rather than progress in direct development of methods to reduce pollution. This policy is implied by the fact that a multiplier effect exists with productivity advances. Such a multiplier does not exist with advances in the pollution reduction. The policy implies that the economy will reap more bene fits by increasing its ability to produce more goods with a given quantity of factors for production, while holding pollution at a given level, than by using up its research resources to reduce pollution directly. That is, since productivity is increasing, an economy can afford to reduce its use of a given polluting factor without a corresponding reduction in its desired consumption level. This fact is emphasized by the technical change multiplier effect. On the other hand, if advances are made only with respect to reduction of pollution, the economy’s output per capita will always be limited. 142 This dissertation represents a point of departure for further theoretical research in the area of optimal economic growth, pollution and technical progress. The importance of the interrelations of economic growth, pollution and technical progress are obvious. This inves tigation has obtained some specific results for a specific model of an economy. The main advantage to the use of a specific model is that the study may provide more depth in the results and specific results will lead to policy implications which are generally desirable. However, there are a number of specific models that may be con sidered in further research. These would include the following: * Different objective functions * Other production functions such as the Constant Elasticity of Substitution (CES) function * Other types of technical advance such as embodied technical progress * Different types of Pollution Abatement Advance functions * Other functions for defining capital and pollution generation. More general extensions would include the following: * Multi-sector economy * Cost of research for neutral and pollution abatement technical progress * Adding more sources of pollution. Obviously, the many changes and extensions would alter some of the properties of the optimal growth path derived in this dissertation. Whether or not it would change the policy implications obtained remains to be seen. APPENDICES LIST OP SYMBOLS A = constant of technology state c = gross consumption per capita C = gross total consumption cn= net consumption per capita Eq= total energy E = energy per capita H = Hamiltonian I = total investment i = investment per capita k = capital per capita K = total capital L0= total labor at t = 0 L = total labor N = total population p = pollution per capita P = total pollution s = investment/output ratio t = time T = planning horizon or final time u = control vector - 1^5 - 146 v = control vector y = output per capita Y = total output a = partial elasticity of production for capital 0 = partial elasticity of production for energy ijj = partial elasticity of production of production for labor y = neutral technical advance rate 0 = pollution/energy ratio 6 = discount rate p = population growth rate X = La Grange Multiplier for the pollution con straint 5 = La Grange Multiplier for control constraint a = pollution dissipation rate y = capital depreciation rate shadow price of capital i^p= shadow price of pollution Z = cost of pollution unit in consumption units ipQ = shadow price of the objective function c f > = proportion of population in labor force f t = rate of progress in pollution abatement APPENDIX A GOLDEN RULE OP ACCUMULATION Since the golden rule of capital accumulation is of significant importance to this dissertation, it is derived here for reference. The traditional theory of growth suggests that the well-being of the future is limited only by the willing ness of the present generation to save. By foregoing present consumption, society can raise the growth rate and thereby enjoy a higher level of consumption in the future. Neoclassical theory, however, suggests that this proposition is false. The growth rate cannot be permanently raised by an increase in the fraction of income that is saved and invested because of the presence of diminishing returns to capital. Although the equilibrium rate of growth cannot be affected by a change in the fraction of income saved, the saving ratio is an important determination of the level of per capita consumption that society may enjoy. This can be illustrated by considering the effect of some extreme - 147 - 148 assumptions about saving behavior. A society that saves and invests nothing at all will have zero capital stock and therefore no output or consumption. At the opposite extreme, a society that is so frugal that it saves most of its output will have a large level of output, but it will have very little consumption. These extremes suggest that somewhere in the middle there must be some optimum savings ratio that maximizes per capita consumption. As will be shown: The optimum savings ratio is such that the following capital to output ratio will exist k _ a y P where p is the population growth rate and a is the partial elasticity of capital. Equivalently, the marginal productivity of capital is equal to the population growth rate yk - P These are the original conditions for the golden rule of capital accumulation. The text of this dissertation has derived modified versions of these due to the more general model used. For this appendix, the traditional model for derivation will be used. The problem is to derive the capital output ratio at which the maximum sustainable consumption per capita will exist. The model is given by the following equations 149 with the symbols having the meanings defined in the text of the dissertation. C = (l-s)Y K = I = sY N = PN We note that the capital model takes on its simplest form, and when this is the case there is no consideration for a depreciation rate. Let these equations be transformed into per capita terms so that c = (c/N) = (1-s)(Y/N) = (l-s)y (A-l) • • • • 2 k = (K/N) = (K/N) - (KN/N ) = s(Y/N) - (K/L)(N/N) = sy - pk (A-2) The fact that the interest is with a sustainable economy indicates that the level of consumption and capi tal per capita must be constant. Thus, k = sy - pk = 0 Using (A-l) y - c = sy and k = y - c — pk = 0 or c = y - pk Maximizing with respect to k, 150 I s = i z _ 0 = o and 3 k 3 k p Z J L = p (A-3) 3 k M which is the desired expression. If the production function is of a Cobb-Douglas form, and where 1-a a Y = AL K a y = A'k A ’ = A/(f> then ( ( ) = constant proportion of population which is in the labor force, 3y _ ay 3k “ k and using (A-3) k _ a y P This provides us with the necessary background on the golden rule of capital accumulation. APPENDIX B A THEOREM ON THE NECESSARY CONDITIONS FOR A MAXIMUM IN PROBLEMS WITH BOTH BOUNDED CONTROLS AND BOUNDED STATE VARIABLES The mathematical analysis of this dissertation is based upon a theorem derived by Professor L. W. Neustadt. This theorem was obtained from Neustadt through private communications. The general problem posed by this disser tation involves a maximization problem where two inequality constraints exist. One inequality constraint is a func tion of a phase variable and time, the other is a function of the two control variables and one phase variable. Necessary conditions for a problem involving both of these types of constraints has not yet appeared in the litera ture. The theorem is stated in general terms. Suppose the control process is described by x(t) = f(x(t), u(t), t) (B-l) where x is a differentiable (n+l)-dimensional real vector function of time t and u is an r-dimensional - 151 - 152 real piecewise continuous vector function of time, and 0. _< t . < _ T. The function f is continuously differentiable in x and continuous in u, x and t. The vector x 0 * ] has the components (x , x , x ) and f has the com ponents (f°, f1, ... fn). The function to be maximized is J = x°(T) (B-2) There are two scalar constraints which must be satisfied at each instant of time. These are X(x(t), t). < 0 x(x(t), u(t), t) , <_ 0 where x and X are given scalar valued functions. Also the following must be satisfied at the initial and final times: X(x(0), x(T)) = 0 where x Is a given m-dimensional vector-valued function. X must be once differentiable, x twice differentiable and x once differentiable. THEOREM Suppose x(t), u(t) maximizes J = x°(T) 153 subject to x(t) = f(x(t), u(t), t) 0 < t. _ < T x(x(t), t)j<0 x(x(t), u(t), t) _< 0 0<t<T X(x(0), x(T)) = 0 u^(t) > 0 for all t, 1 Then there exists an m-dlmensional row vector - ( 1 a number v° > 0, scalar, piecewise continuous multiplier functions A(t), £(t) and an auxiliary (n+1)-dimensional row vector function <J>(t) which is differentiable such that (i) \p(t) = -ip(t)f (x(t), u(t), t) A -£(t)x(x(t),u(t),t) A +A(t)Q (x(t),t) x where Q(x,t) = j ? (x,t)f (x,u(t) ,t) + x.(x,t) A Ti f is the Jacobian matrix of partial derivatives of x components of f with respect to the components of x. X , x a n d Q are row vectors obtained by taking the X X X partial derivatives of Xj X and Q with respect to the components of x. x*. ds the partial derivative of x with respect to t. (ii) ' ty(t)-X(t)x (x(t),t)} f(x(t),u(t),t) X = sup _ ' fy(t)-A(t)$ (x(t) ,t) }f (x(t) ,v,t) V£0)(x(t),t) x where ai(x,t) = ’ {v: v >_ 0 for all i, x(XjV,t) < 0} (iii) The following transversality conditions are satisfied: if»(0) - A(0)x (x(0) ,0) = ~vX (x(0) ,x(T)) x xx \p(T) = v x v (x(0),x(T)) + (v°, 0, 0) 2 where xY and X are the Jacobian matrices obtained by 1 2 taking the partial derivatives of X components with respect to the components of the first and second arguments of X which we denote as x^ and x^ respectively. (iv) The following inequality holds: 155 for all v such that v* >_ 0 for all 1 and where x u has the obvious meaning. (v) The multiplier function A(t) is a. non-increasing b. constant on every interval on which x(x(t), t) <0 c. equal to zero at T d. continuous from the right on the open inter val (0, T). (vi) The multiplier function £(t) is a. non-positive b. zero for all t in which x(x(t),u(t),t), <0 This completes the theorem. APPENDIX C COMPLETE SOLUTION TO THE PROBLEM WITH A CONSTANT STATE OP TECHNOLOGY - NO CRITICAL POLLUTION LEVEL This appendix provides a complete solution to the situation in Chapter IV in which there is no critical pollution level. In Chapter IV, the following equations expressed the system along the optimal growth path. y(t) = Aka(t)E3(t) (C-l) k(t) = s(t)y(t) - (y+p)k(t) (C-2) p(t) = 0E(t) - (a+p)p(t) (C-3) s(t) = (a/p+y+6)(fi+p+(e/l-a)(E(t)/E(t))) (C-4) E(t) =______________gy(t) (C-5) (a+p+6)(t-T) 1 - 8£ (e -1) a+p+6 A solution to this system of equations would provide expressions to the paths of the phase and control variables. - 156 - First, let us examine equation (C-5). This can be re-expressed as E(t) = (3y(t)/g£t)) where ( J.Z.) \a+p+6 / (a+p+$)(t-T) g(t) = 1 (e - 1) Since then where and (a+p+6 / a time derivative, we can obtain (E(t)/E(t)) = (y(t)/y(t)) - (g(t)/g(t)) (y(t)/y(t)) = (3/1-a)(E(t)/E(t)) (E(t)/E(t)) = -((l-a)/(l-a-3))(g(t)/g(t)) (a+p+6)(t-T) g(t) _ - 6 Z e___________________ gttT (a+p+6)(t-T) f -IL..\ \a+p+6 1 (a+p+6)t e~(a+p+6)T e ia+p+5) (t-T ) 1-/ 0Z \(e -1) 158 We can now solve the equation for energy growth for E by use of the separable variables technique, so E, E / (dE’/E') = t -(a+p+6)t' Q / ' {dt ’ t (e 0 -Z_6L-\ (a+p+6) +/9J_\ (a+p+61 -(a+p+6 )T -(a+p+6)t1 e )} where Eq I s a constant and E1 and tf are variables over which the integrals are to be taken. These are essentially changes in notation from E and t for convenience. This can be solved to obtain the following: E (t) = = E, (1+ 9S )-( QZ \ C a+p+6) (a+p+6) (a+p+6)(t-T) ((l-a)/(l-a-6)) (C-6) Then s(t) a (li+P+ y+p+6 B6Z ( 1-a \ 1-a (l-a-B / (a+p+5)t -(a+p+6)T \ 1- (a+p+6)(t-T5 9Z (e -1) a+p+6 (C-7) Now, the condition set by assuming the singular solution will hole along the entire growth path is that (k(t)/y(t)) = (a/y+p+6) (C—8) But k(0) is a known quantity so that y(0) = ((y+p+6)/a)k(0) Knowing that, a 3 y(0) = Ak (0)E (0) So a ^ E(0) = (y(0)/Ak (0)) 1/3 = ((y+p+6)/Aa) Using (C-6), 1/3 EQ = ((y+p+6)/Aa) ((l-a)/(l-a-3)) , x -(a+p+6)T 1+ 6Z - ( QZ \ e a+p+6 |a+p+6l (C—8) Using the relationship between y and E, v (a+p+6)(t-T) y(t)=E(t) 1 - ( ez\(e -1) (C-9) 160 We now have explicit expressions for time of y, k, E and s. There remains only a time path for the pollution. Unfortunately, an analytic expression for this variable is not available. 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Wong, Robert Eing
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Pollution, Optimal Growth Paths, And Technical Change
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